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Kerala State Board New Syllabus Plus Two Computer Application Chapter Wise Previous Questions and Answers Chapter 2 Arrays.

Kerala Plus Two Computer Application Chapter Wise Previous Questions Chapter 2 Arrays

Plus Two Computer Application Arrays 1 Mark Important Questions

Question 1.
Write C++ initialization statement to initialize an integer array name ‘MARK’ with the values 70,80,85,90. (MARCH-2017)
Answer:
MARK[4] = {70,80,85,90};
MARK[ ] = {70,80,85,90};
MARK[0] = 70;
MARK[1] = 80;
MARK[2] = 85;
MARK[3] = 90;

Plus Two Computer Application Arrays 2 Mark Important Questions

Question 1.
How memory is allocated for a float array? (MARCH-2016)
Answer:
Memory allocated for the float data type is 4.
Total byte = size of array *4
eg. float n[10];
Here total byte = 10*4
That means 40 bytes.

Question 2.
How can we initialize an integer array? Give an example. (MARCH-2016)
Answer:
Array elements can be initialised at the time of declaration and values are included in braces,
eg: ing n[5]={10,20, 30,40, 50};

Question 3.
Answer any of the following questions 4(a) or 4(b). (MARCH-2016)
a) Define an array. Give an example of an integer array declaration.
b) Consider the following C++ code char text [20]; cin>>text;
If the input string is “Computer Programming”, what will be the output? Justify your answer.
Answer:
a) An array is a collection of elements with the same data type.
eg:- int n[100]; This is an array namely n. We can store 100 elements. The index of the first element is 0 and the last is 99.
b) The output is “Computer”. This is because of c in reads the characters up to space. That means space is the delimiter, The character after space is truncated.

Plus Two Computer Application Arrays 3 Mark Important Questions

Question 1.
Write a C++ program to input 10 numbers into an integer array and find the sum of numbers which are an exact multiple of 5. (MARCH-2017)
Answer:
#include <iostream>
using namespace std;
int main()
{
int n[10],i,sum=0;
for(i=0;i<10;i++)
{
cout<<“Enter value for number”<<i+1
cin>>n[i];
if(n[i]%5==0)
sum+=n[i];
}
cout<<“The sum of numbers which are exact multiple of 5 is“<<sum;
V;
}

Plus Two Computer Application Arrays 5 Mark Important Questions

Question 1.
Write a C++ program to accept a string and count the number of words and vowels in that string. (MARCH-2016)
Answer:
(a) #include(iostream>
# include <cstdio>
# include<cctype>
using namespace std;
void main ()
{
int i, vowel =0, words=1; ,
charstr[80];
cout<< “Enter a string \n”;
gets (str);
for (i=0, str[i]! = ‘\0’; i++)
{
if (str [i] ==32) words ++;
’ switch (to lower(str[i]))
{
case ‘a’: case ‘e’: case ‘i’: case ‘O’: case ‘u’: vowel ++;
}
count<<“The number of words is” <<words; count<<“\n the number of vowels is”<<vowel;
}

Question 2.
Write a C++ program to accept N integer numbers and find the sum and average of even numbers. (MARCH-2016)
Answer:
# include <iostream>
using namespace std;
void main ()
{
float avg, sum = 0.0;
int N, i, no;
cout <<“Enter how many numbers”;
cin>>N;
for(i=0; i<N; i++)
{
cout<< “Enter number”<<i+1; cin >> no; if (no % 2 == 0) sum+=no;
}
avg=sum/N;
cout<< “The sum of even numbers is”<<sum; cout<<“\n the average of even numbers is” <<avg;
}

Question 3.
Answer any of the following questions 8(a) or 8(b). (MAY-2016)
a) Write a C++ program to accept a string and find the length of the string without using built in function. Use a character array to store the string.
b) Write a program to input ‘N’ numbers into an integer array and find the largest number in the array.
Answer:
a) # include <iostream>
# include <cstdio>
using namespace std;
int main ()
{
charstr [80]; int i;
cout <<“Enter a string:”;
gets(str);
for(i=o;str[i]!= ‘\0’;i++)
cout <<“The length of the string is “<<i;
}
OR

b) # include <iostream>
using namespace std;
int main ()
{
int N, no[50], i, largest = 0;
cout<< “Enter how many numbers”;
cin>>N;
for (i=0; i<N; i++)
{
cout <<“Enter number”<<i+1;
cin>>no[i];
if (no[i]> largest)
largest = no[i];
}
cout <<“The largest number is” <<largest;
}

Question 4.
Write a C++ program to enter 10 numbers into an array and find the second largest element. (MAY-2016)
Answer:
#include<iostream>
using namespace std;
int main()
{
int i,j,n[10],temp;
for(i=0;i<10;i++)
{
cout<<“Enter a value for number”<<i+1 cin>>n[i];
}
for(i=0;i<9;i++)
{
for(j=i+1;j<10;j++)
if(n[i]<n[j])
{
temp=n[i];
n[i]=n[j];
n[j]=temp;
}
}
cout<<“The second largest number is “<<n[1];
}

Question 5.
Write a C++ program to convert all lowercase alphabets stored in a string to uppercase. (MAY-2016)
Answer:
#include<cstdio>
using namespace std;
int main()
{
charline[80];
int i;
puts(“Enter the string to convert”);
gets(line);
for(i=0;line[i]!=’\0′;i++)
if (line[i]>=97 && line[i]<=122)
line[i]=line[i] – 32;
puts(line);
}

Kerala State Board New Syllabus Plus Two Computer Application Chapter Wise Previous Questions and Answers Chapter 1 Review of C++ Programming.

Kerala Plus Two Computer Application Chapter Wise Previous Questions Chapter 1 Review of C++ Programming

Plus Two Computer Application Review of C++ Programming 1 Mark Important Questions

Question 1.
_______ is an exit control loop. (MAY-2016)
a) for loop
b) while loop
c) do-while loop
d) break
Answer:
do-while loop

Question 2.
Which among the following is an Insertion Operator? (MARCH-2016)
a) <<
b) >>
c) <
d) >
Answer:
a) <<

Question 3.
Which among the following is equivalent to the statement series b=a, a = a +1? (MARCH-2017)
a) b + = a
b) b = a++
c) b = ++a
d) b + = a + b
Answer:
b) b = a++

Question 4.
A ______ statement in a loop force the termination of that loop. (MARCH-2017)
Answer:
break

Question 5.
_______ operator is the arithmetic assignment operator. (MAY-2017)
a) >>
b) ==
c) +=
d) =
Answer:
c) +=

Plus Two Computer Application Review of C++ Programming 2 Marks Important Questions

Question 1.
How does a ‘goto’ statement work? (MAY-2016)
Answer:
The execution of a program is sequential but we can change this sequential manner by using jump statements. The jump statements are
1) goto statement By using goto we can transfer the control anywhere in the program without any condition. The syntax is goto label;
Eg.
# include<iostream>
using namespace std;
int main()
{
float a,b;
cout<<“Enter 2 numbers”;
cin>>a>>b;
if(b==0)
goto end;
cout<<“The quotient is “<< a/b;
return 0;
end:
cout<<“Division by zero error”;
}

Question 2.
What are the main components of a looping statement? (MARCH-2016)
Answer:
The main components are initialization expression, test expression, update expression, and looping body
eg: for (i=1; i<=10; i++)
cout << i;

Question 3.
Identify the following C++ tokens. (MAY-2017)
a) “welcome”
b) int
c) >=
d) ++
Answer:
a) “welcome” String Literal
b) int-Keyword
c) >= Operator
d) ++-: Operator

Plus Two Computer Application Review of C++ Programming 3 Marks Important Questions

Question 1.
Explain switch statement with an example (MAY-2016)
Answer:
cin >> pcode;
switch (pcode)
{
case ‘C’:
cout <<“Computer”;
break;
case ‘M’:
cout << “Mobile Phone”;
break;
case ’L’:
cout << “Laptop”;
break;
default:
cout << “lnvalid code”;

Question 2.
Rewrite the following C++ code using ‘switch’ statement (MARCH-2017)
cin >> pcode;
if (pcode == ‘C’)
cout << “Computer”;
else if (pcode == ‘M’)
cout<<“Mobile Phone”;
else if(pcode == ‘L’)
cout<<“Laptop
else
cout<<“lnvalid code”;
Answer:
cin >> pcode;
switch(pcode)
{
case ‘C’:
cout << “Computer”;
break;
case ‘M’:
cout << “Mobile Phone”;
break;
case ’L’:
cout << “Laptop”;
break;
default:
cout << “lnvalid code”;

Question 3.
How do continue and break statements differ in a loop? Explain with an example. (MARCH-2016)
Answer:
Break is used to terminate a loop. But continue is used for skipping (bypassing) a part of the code, eg: for (i=1, i<10; i++)
{
if (i%2==0) continue; cout<<i << “, “;
}
Here the output is 1,3, 5, 7, 9
eg: for (i=1; i<10; i++)
{
if (i% 2==0) break;
cout<<i<<“,”;
}
Here the output is 1, that is the loop is quit when i=2.

Question 4.
Explain break and continue statements with examples. (MAY-2017)
Answer:
break statement:- It is used to skip over a part of the code i.e. we can premature exit from a loop such as while, do-while, for or switch, continue statement:- It bypasses one iteration of the loop.
break statement:- It is used to skip over a part of the code i.e. we can premature exit from a loop such as while, do-while, for, or switch.
Syntax:
while (expression)
if (condition) break;
}
Eg.
#include<iostream>
using namespace std;
int main()
{
int i=1;
while(i<10)
{
cout<<i<<endl;
if(i==5)
break;
i++;
}
}
The output is
1
2
3
4
5
continue statement: It bypasses one iteration of the loop.
Syntax:
while (expression)
{
if (condition) break;
}
Eg.
#include<iostream>
using namespace std;
int main()
{
int i=0;
while(i<10)
{
i++;
if(i==5) continue;
cout<<i<<endl;
}
}
The output is
1
2
3
4
5
6
7
8
9
10

Kerala State Board New Syllabus Plus Two Maths Chapter Wise Previous Questions and Answers Chapter 12 Linear Programming.

Kerala Plus Two Maths Chapter Wise Previous Questions Chapter 12 Linear Programming

Plus Two Maths Linear Programming 4 Marks Important Questions

Question 1.
Consider the linear programming problem;
Maximise; Z = x +y , 2x + y – 3< 0, x – 2y + 1 < 0, y < 3, x < 0, y < 0
(i) Draw its feasible region.
(ii) Find the corner points of the feasible region.
(iii) Find the corner at which Z attains its maximum. (March – 2012)
Answer:

(ii) In the figure the shaded region ABC is the feasible region. Here the region is bounded. The corner points are A(1, 1), B(5, 3), C(O, 3).
(iii) Given; Z = x + y

Since maximum value of Z occurs at B, the soluion is Z = (5) + (3) = 8.

Question 2.
Consider the LPP Minimise; Z = 200 k + 500y, x + 2y > 10, 3x + 4y < 24, x > 0, y > 0
(i) Draw the feasible region.
(ii) Find the co-ordinates of the comer points of the feasible region.
(iii) Solve the LPP. (May – 2012)
Answer:

(ii) In the figure the shaded region ABC is the feasible region. Here the region is bounded. The corner points are 4(4, 3), 5(0, 6), C(0, 5)
(iii) Given; Z = 200x + 500y

Since minimum value of Z occurs at A, the soluion is Z = 200(4) + 500(3) = 2300.

Question 3.
Consider the LPP
Maximise; Z = 5x + 3y
Subject to; 3x + 5y < 15, 5x + 2y < 10x, y > 0
(i) Draw the feasible region.
(ii) Find the corner points of the feasible region.
(iii) Find the corner at which Z attains its maximum. (March – 2013)
Answer:
In the figure, the shaded region OABC is the feasible region. Here the region is bounded.
The corner points are

Given; Z = 5x + 3y

Since maximum value of Z occurs at B, the soluion is Z =

Question 4.
Consider the linear programming problem: Minimize Z = 3x + 9y Subject to the constraints: x + 3y < 60 x + y > 10, x < y, x > 0, y > 0
(i) Draw its feasible region.
(ii) Find the vertices of the feasible region
(iii) Find the minimum value of Z subject to the given constraints. (March-2014, SAY-2016)
Answer:

(ii) The feasible region is ABCD.
Solving x + y = 10, x = y we get B(5, 5)
Solving x + 3y = 60, x = y we get C(15, 15)
Hence the comer points are A(0, 10) , B(5, 5), C(15, 15), D(0, 20)
(iii) Given; Z = 3x + 9y

Form the table, minumum value of Z is 6 O at B(5, 5).

Question 5.
Consider the linear inequalities 2x + 3y < 6; 2x + y < 4; x, y < 0
(a) Mark the feasible region.
(b) Maximise the function z = 4x + 5y subject to the given constraints. (March – 2014)
Answer:

(b) 15.2x + 3y = 6

2x + y = 4

Maximum at x = 1.5, y = 1
Maximum value is Z = 11

Question 6.
Consider the linear programming problem: Minimise Z = 4x + 4y Subject to x + 2y < 8; 3x + 2y < 12x, y < 0
(a) Mark its feasible region.
(b) Find the comer points of the feasible region.
(c) Find the corner at which Z attains its minimum. (May – 2014)
Answer:
(a) x + 2y = 8,

3x + 12y = 12

The comer points are 0(0, 0), A(4, 0), B(2, 3), C(0, 4)

(c)

Z attains minimum at (4, 0).

Question 7.
Consider the linear programming problem: Maximum z = 4x + y
Subject to constraints: x + y < 50, 3x + y < 9x, y < 0
(a) Draw the feasible region
(b) Find the corner points of the feasible region
(c) Find the corner at which ‘z’ attains its maximum value. (May – 2015)
Answer:
(a) x + y = 50,

3x + y = 90

(b) Solving the equations we get the points as
O(0, 0) A(30, 0); B(20, 30); C(0, 50)

(c)

Z attains maximum at A(30, 0)

Question 8.
Consider the LPP
Maximise z = 3x + 2y
Subject to the constraints: x + 2y < 10, 3x + y < 15; x, y < 0
(a) Draw its feasible region
(b) Find the corner points of the feasible region
(c) Find the maximum value of Z. (March – 2016)
Answer:

(b) The corner points 0(0,0), A(5,0), B(4,3), C(0, 5)
Z is maximum at B(4, 3), z = 18.

(c)

Question 9.
Consider the linear programming problem: Maximum z = 50x + 40y
Subject to constraints:
x + 2y < 10; 3x + 4y < 24; x, y < 0
(i) Draw the feasible region
(ii) Find the comer points of the feasible region
(iii) Find the maximum value of z. (March – 2017)
Answer:

(ii) In the figure the shaded region ABC is the feasible region. Here the region is bounded. The corner points are A(4, 3), B(0, 6), C(0, 5)
(iii) Given; Z = 50x + 40y

Since minimum value of Z occurs at A, the soluion is Z = 50(4) + 4(3) = 320.

Plus Two Maths Linear Programming 6 Marks Important Questions

Question 1.
A furniture dealer sells only tables and chairs. He has Rs. 12,000 to invest and a space to store 90 pieces. A table costs him Rs. 400 and a chair Rs. 100. He can sell a table at a profit of Rs. 75 and a chair at a profit of Rs. 25. Assume that he can sell all the items. The dealer wants to get maximum profit.
(i) By defining suitable variables, write the objective function.
(ii) Write the constraints.
(iii) Maximise the objective function graphically. (March – 2010)
Answer:
(i) Let x be the number of Tables and y be the number of Chairs. Then; Maximise; z = 75x + 25y
(ii) Furniture constraints x + y < 90
Investment constraint 400x + 100y < 12000
Therefore;Maximise; Z = 75x + 25y, x + y < 90, 4x + y < 120, x<0, y<0
(iii) In the figure the shaded region OABC is the feasible region. Here the region is bounded. The corner points are O(0, 0), A(30, 0) B(10, 80), C(0, 90).

Given; Z = 75x + 25y

Since minimum value of Z occurs at B, the soluion is Z = 2750.

Question 2.
A company produces two types of cricket balls A and B. The production time of one ball of type B is double type A (time in units). The company has the time to produce a maximum of 2000 balls per day. The supply of raw materials is sufficient for the production of 1500 balls (both A and B) per day. The company wants to make maximum profit by making a profit of Rs. 3 from a ball of type A and Rs. 5 from type B.

Then,
(i) By defining suitable variables write the objective function.
(ii) Write the constraints.
(iii) How many balls should be produced in each type per day in order to get maximum profit? (May – 2010)
Answer:
(i) Let x be the number of balls of type A and y be the number of balls of type B. Then Maximise profit is Z = 3x + 5y
(ii) Balls constraints 2x + y < 2000 investment constraint x + y < 1500
Therefore; Maximise; Z = 3x + 5y, 2x + y < 2000, x + y < 1500, x < 0, y < 0

(iii) In the figure the shaded region OABC is the fesible region. Here the region ¡s bounded. The corner points are O(0, 0), A(1000, 0) B(500, 1000), C(0, 1500). Given; Z = 3x + 5y

Since maximum value of Z occurs at C, the soluion is Z = 3(0) + 5(1500) = 7500.

Question 3.
The graph of a linear programming problem is given below. The shaded region is the feasible region. The objective function is Maximise; Z = px + qy

(i) What are the co-ordinates of the corners of the feasible region?
(ii) Write the constraints.
(iii) If the Max. Z occurs at A and B, what ¡s the relation between p and q?
(iv) If q = 1, write the objective function.
(v) Find Max. Z. (March – 2011)
Answer:
(i) From the figure the feasible region is OABC.
Then the comer points are;
A is (5, 0), B is (3, 4), C is (0, 5) and O (0, 0)
(ii) The constraints are 2x + y < 10, x + 3y < 15, x < 0, y < 0
(iii) Given; Z = px + qy

Since maximum at A and B we have;
⇒ 3p + 4q = 5p ⇒ 2p = 4q ⇒ p = 2q
(iv) When q = 1, then p ⇒ 2q ⇒ p = 2
Objective function is; Z = 2x + y
(v) We have; Z px + qy at B Z has maximum ⇒ Z = 2(3) + 4 = 10

Question 4.
A manufacturer produces nuts and bolts. It takes 1 hour of work on machine A and 3 hours on machine B to produce a package of nuts. It takes 3 hours on machine A and I hour on machine B to produce a package of bolts. He earns a profit of Rs. 17.50 per package on nuts and Rs. 7.00 per package on bolts. How many packages of each should be produced each day so as to maximise his profit, if he produced each day so as to maximise the profit if he operates his machines for at the most 12 hours a day?
(i) By suitable defining the variables write the objective function of the problem.
(ii) Formulate the problem as a linear programming problem(LPP)
(iii) Solve the LPP graphically and find the number of packages of nuts and bolts to be manufactured. (May -2011)
Answer:
(i) Let x be the number of packages of nuts produced and y be the number of packages of bolts produced. Then;
Maximise profit is; Z = 17. 5x + 7y
(ii) Time constraint for Machine A; x + 3y < 12
Time constraint for Machine B; 3x + y < 12
Therefore; Maximise; Z = 17.5x + 7y, x + 3y < 12, 3x + y < 12, x < 0, y < 0
(iii) In the figure the shaded region OABC is the visible region. Here the region is bounded. The corner points are 0(0,0), A (4, 0) B(3, 3), C(0, 4).

Given; Z = 17.5x + 7y

Since maximum value of Z occurs at B, the soluion is Z = 17.5(3) + 7(3) = 73.5.

Question 5.
A bakery owner makes two types of cakes A and B. Three machines are needed for this purpose. The time (in minutes) required for making each type of cakes in each machine is given below;

Each machine is available for almost 6 hours per day. Assume that all cakes will be sold out every day. The bakery owner wants to make a maximum profit per day by making Rs. 7.5 from type A and Rs. 5 from type B.
(i) Write the objective function by defining suitable variables.
(ii) Write the constraints.
(iii) Find the maximum profit graphically. (May- 2013, EDUMATE – 2017)
Answer:
(i) Number of cake of type A: x
Number of cake of type B: y
Then profit function is Maximise: Z = 75x + 5y

(ii) 12x + 6y < 360; 18x + 0y < 360
6x + 9y < 360; x > 0, y > 0
Simplifying we get;
2x – i – y < 60………..(1)
x < 20………..(2)
2x + 3y < 120………..(3)
x > 0, y > 0

The feasible region is OABCDO
Solving (1) and (2) we get the point B- (2020)
Solving (1) and (3) we get the point C- (15,30)
A-(20,0), O-(0,0), D-(0,40)
Given; Z = x + y

Since maximum value of Z occurs at B, the soluion is Z = 250 (20, 20).

Question 6.
In a factory, there are two machines A and B producing toys. They respectively produce 60 and 80 units in one hour. A can run a maximum of 10 hours and B a maximum of 7 hours a day. The cost of their running per hour respectively amounts to 2,000 and 2,500 rupees. The total duration of working these machines cannot exceed 12 hours a day. If the total cost cannot exceed Rs. 25,000 per day and the total daily production is at least 800 units, then formulate the problem mathematically. (March – 2014)
Answer:
Let x be the running time for machine A and y be the running time for machine B.
Since machines cannot work more than 12 hours x + y < 12
Since maximum production of two machines is 800 units.
60x + 80y < 800
Maximum cost of production is 25000, 2000x + 2500y < 25000
0 < x < 10, 0 < y < 7

Kerala State Board New Syllabus Plus Two Maths Chapter Wise Previous Questions and Answers Chapter 11 Three Dimensional Geometry.

Kerala Plus Two Maths Chapter Wise Previous Questions Chapter 11 Three Dimensional Geometry

Plus Two Maths Three Dimensional Geometry 3 Marks Important Questions

Question 1.
(i)Find the equation of the Plane with intercepts 2, 3, 4 on X, Y, Z axes respectively.)
(ii) Find the distance of the point (- 1, – 2,3) from the Plane \(\bar{r}\)(2i – 3 j + 4k) = 4 (March – 2011)
Answer:
(i) The equation of the plane is \(\frac{x}{2}+\frac{y}{3}+\frac{z}{4}=1\)
(ii) The equation of the plane is 2x – 3y + 4z = 4
Hence the distance \(=\frac{2(-1)-3(-2)+4(3)-4}{\sqrt{4+9+16}}\)
\(=\frac{-2+6+12-4}{\sqrt{4+9+16}}=\frac{12}{\sqrt{29}}\)

Question 2.
Consider the points A(2,2, – 1), B(3,4,2), C(7,0,6)
(i) FindAB.
(ii) Find the Cartesian and vector equation the plane passing through these points. (March – 2011)
Answer:
(i) \(A B=\sqrt{1+4+9}=\sqrt{14}\)
(ii) Cartesian Equation of the plane is

Vector equation is \(\bar{r}\)(5i + 2j – 3k) = 17

Question 3.
Consider the points A(3, – 4, – 5) and 5(2, – 3,1)
(i) Find the vector and Cartesian equation of the Line passing through the points A and B.
(ii) Find the point where the line crosses the XY Plane. (March – 2012)
Answer:
Let \(\bar{a}\) = 3i – 4j – 5k, b = 2i – 3j + k
(i) Vector Equation is \(\bar{r}\) = \(\bar{a}\) + λ(\(\bar{b}\) – \(\bar{a}\)) \(\bar{r}\) = 3i – 4j – 5k + λ( – i + j + 6k)
Cartesian Equation is \(\frac{x-3}{-1}=\frac{y+4}{1}=\frac{z+5}{6}\)

(ii) Let the point be (x, y, 0)
\(\begin{array}{l}
\Rightarrow \frac{x-3}{-1}=\frac{y+4}{1}=\frac{5}{6} \\
\Rightarrow x=\frac{13}{6}, y=\frac{-19}{6}
\end{array}\)
Then the point on the XY Plane is \(\left(\frac{13}{6}, \frac{-19}{6}, 0\right)\)

Question 4.
(i) Find the Cartesian equation of the plane passing through the point (1, 2, -3) perpendicular to the vector 2i – j + 2k.
(ii) Find the angle between the above \(\frac{x-1}{2}=\frac{y-3}{3}=\frac{z}{6}\) (March – 2012)
Answer:
(i) The equation of the Plane is
2(x – 1) – 1(y, -2) + 2(z+3) = 0
⇒ 2x – 2 – y + 2 + 2z + 6 = 0
⇒ 2x – y + 2z + 6 – 0

(ii) Angle between the line and the Plane is

Question 5.
(i) Find the angle between the lines having direction ratios 1, 1, 2 and \(\sqrt{3}-1,-\sqrt{3}-1,4\)
(ii) If the lines \(\frac{x-1}{3}=\frac{y-1}{2 \lambda}=\frac{z-3}{2}\) and \(\frac{x-1}{3 \lambda}=\frac{y-1}{1}=\frac{z-6}{-5}\) are perpendicular, find the value of λ. (May – 2012)
Answer:

(ii) Since they are perpendicular
a₁a₂ + b₁b₂ + c₁c₂ = 0
⇒ 3 x 3λ + 2λ x 1 + 2x – 5 = 0
⇒ 9λ + 2λ – 10 = 0
⇒ 11λ + 10 ⇒ 2 = \(\frac{-10}{11}\)

Question 6.
(i) Find the equation of the Plane passing through the points (3, -1, 2), (5, 2, 4), ( -1, -1, 6)
(ii) Find the perpendicular distance from the point (6, 5, 9) to this plane. (March – 2013)
Answer:
(i) Cartesian Equation of the plane is

(ii) Perpendicular distance

Question 7.
Consider the vector equation of the planes r.(2i + j+k) – 3 and r.(i – j – k) = 4
(i) Find the vector equation of any of the plane at the intersection of the above two planes.
(ii) Find the vector equation of the planes through the intersection of the above two planes and the point (1, 2, -1). (May – 2013)
Answer:
(i) 2x + y + z – 3 + λ(x – y – z – 4) = 0
⇒ (2 + λ)x + (1 – λ)y + (1 – λ)z – 3 – 4λ = 0
Vector equation is
\(\bar{r}\).(2 + λ)i + (1 – λ)y + (1 – λ)k – (3 + 4)λ = 0

(ii) Since passing through (1,2, – 1) we have;
⇒ (2 + λ)1 + (1 – λ)2 + (1 – λ)( -1) – 3 – 4λ = 0
⇒ 2 + λ + 2 – 2λ – 1 + 1 – 3 – 4λ = 0
⇒ 0 – 4λ – 0
⇒ λ = 0

\(\bar{r}\).(2i + j + k) = 3 is required plane. Since the point (1, 2, -1) is a point on the first plane.

Question 8.
Consider the planes2x + y – 2z = 5 and 3x – 6y – 2z = 7
(a) Find their normal vectors.
(b) Find the angle between these two planes. (May – 2014)
Answer:
(a) Normal vectors are 2i + i – 2k; 3i – 6j – 2k
(b) Angle =

Question 9.
(a) If a₁, b₁, c₁ and a₂, b₂, c₂ are the direction ratios of two lines, then write the condition of its perpendicularity.
(b) Find the angle between the lines \(\frac{x+3}{3}=\frac{y-1}{5}=\frac{z+3}{4}\) and \(\frac{x+1}{1}=\frac{y-4}{1}=\frac{z-5}{2}\)
Answer:
(a) a₁a₂ + b₁b₂ + c₁c₂ = 0

Question 10.
Find the shortest distance between the lines \(\bar{r}\) = i + j + λ(2i – j + k) and \(\bar{r}\) = 2i + j – k + µ(3i – 5 j + 2k) (March – 2016)
Answer:
From the given lines we have;

Question 11.
(i) The equation of the line which passes through the point(1,2,3) and parallels to the vector 3i + 2j – 2k is
(a) \(\bar{r}\) = 3i + 2j – 2k + λ(/ + 2j + 3k)
(b) \(\bar{r}\) = 2i – 5k + λ(3/ + 2j – 2k)
(c) \(\bar{r}\) = i + 2j + 3k + λ(-2i + 4j – 2k)
(d) \(\bar{r}\) = i + 2j + 3k + λ(3i + 2j – 2k)
(ii) Find the angle between the pair lines
\(\bar{r}\) = 2i – 5j + k + λ(3i + 2j + 6k) and \(\bar{r}\) = li – 6k + µ(i + 2j + 2k) (May – 2016)
Answer:
(i) (d) \(\bar{r}\) = i + 2j + 3k + λ(3i + 2j – 2k)
(ii) \(\bar{a}\) = 3i + 2j + 6k; \(\bar{b}\) = i + 2j + 2k
\(\cos \theta=\frac{\bar{a} \cdot \bar{b}}{|\bar{a} \| \bar{b}|}=\frac{19}{7 \times 3}=\frac{19}{21}\)

Plus Two Maths Three Dimensional Geometry 4 Marks Important Questions

Question 1.
Consider the lines \(\frac{x-3}{2}=\frac{y-1}{5}=\frac{z+3}{4}\) and \(\frac{x+5}{1}=\frac{y+2}{1}=\frac{z-3}{2}\)
(i) Find the angle between them.
(ii) Find the shortest distance between them. (March – 2011)
Answer:

Question 2.
(i) Find the vector equation of the Plane Passing through the intersection of the Planes \(\bar{r}\).(i + j + k) = 6 and \(\bar{r}\)(2i + 3j + 4k) = – 5 and through the point (1, 1, 1).
(ii) Express the vector equation \(\bar{r}\).(5i + 3j + 4k) = 0 of a Plane in Cartesian form and hence find its perpendicular distance from the origin. (May – 2011)
Answer:
(i) The Cartesian equation of the given planes are x + y + z – 6 = 0 and 2x + 3 – y + 4z + 5 = 0
The family of such planes is x + y + z – 6 + λ(2x + 3y+ 4z + 5) = 0  …..(1)
Since it passes through (1, 1, 1)

Vector Equation is \(\bar{r}\).(20i + 23j + 26k) = 69

(ii) Cartesian Equation is ⇒ 5x + 3y + 4z = 0 Perpendicular distance from origin is
\(=\left|\frac{5 \times 0+3 \times 0+4 \times 0}{\sqrt{25+9+16}}\right|=0\)

Question 3.
Given the Plane 5x – 2y + 4z – 9 = 0
(i) Find the foot of the perpendicular drawn from the origin to the Plane.
(ii) Write the vector and Cartesian equation of this perpendicular. (May – 2011)
Answer:
(i) The equation of the perpendicular line to the given plane 5x – 2y + 4z – 9 = 0 and passing through the origin is
\(\begin{array}{l}
\frac{x-0}{5}=\frac{y-0}{-2}=\frac{z-0}{4}=\lambda \\
\Rightarrow \frac{x}{5}=\frac{y}{-2}=\frac{z}{4}=\lambda
\end{array}\)
Hence any point on this line is (5λ, – 2λ, 4λ). Let this point be on the given plane then
⇒ 5(5λ) – 2( – 2λ) + 4(4λ) – 9 = 0
⇒ λ = 1/5
Then the foot of the perpendicular is
\(\left(5 \times \frac{1}{5},-2 \times \frac{1}{5}, 4 \times \frac{1}{5}\right) \Rightarrow\left(1,-\frac{2}{5}, \frac{4}{5}\right)\)
Since the line is perpendicular to the Plane and passes through the point \(\left(1,-\frac{2}{5}, \frac{4}{5}\right)\)
The Cartesian equation is \(\frac{x}{5}=\frac{y}{-2}=\frac{z}{4}\)
The Vector equation is \(\bar{r}\) = \(\bar{0}\) + λ(5i – 2j + 4k)

Question 4.
(i) The foot of the perpendicular from the origin to a Plane is P(4, – 2,5). Write \(\overline{O P}\)
(ii) Find the equation of the Plane in vector and Cartesian form. (May – 2012)
Answer:
(i) \(\overline{O P}\) = 4i – 2j + 5k
(ii) Then is perpendicular unit vector to the required plane is \(\frac{\overline{O P}}{\overline{O P} \mid}=\frac{4 i-2 j+5 k}{\sqrt{16+4+25}}=\frac{4 i-2 j+5 k}{\sqrt{45}}\)

The perpendicular distance from the origin is \(\sqrt{16+4+25}=\sqrt{45}\)
Vector equation of the Plane can be written as
\(\begin{array}{l}
\vec{r} \cdot \bar{m}=d \Rightarrow \bar{r} \cdot \frac{4 i-2 j+5 k}{\sqrt{45}}=\sqrt{45} \\
\Rightarrow \bar{r} .4 i-2 j+5 k=45
\end{array}\)
Cartesian from is 4x – 2y + 5z = 45

Question 5.
Consider the lines \(\frac{x-3}{3}=\frac{y-8}{-1}=\frac{z-3}{1}\) and \(\frac{x+3}{3}=\frac{y+7}{2}=\frac{z-6}{4}\)
(i) Express the equation to the lines into vector form.
(ii) Find the shortest distance between the lines. (March – 2013)
Answer:
(i) The vector equation;

Question 6.
Consider the Cartesian equation of a line
\(\frac{x-3}{2}=\frac{y+1}{3}=\frac{z-5}{-2}\)
(i) Find its vector equation.
(ii) Find the intersecting point with the plane 5x + 2y – 6z – 7 = 0 (May – 2013)
Answer:
(i) Vector equation of the line is
\(\bar{r}\) = (3i – 1j + 5k) + λ(2i + 3j – 2k)

(ii) Any point on the line is of the form

Question 7.
The foot of the perpendicular drawn from origin to a Plane is (4, -2, 5).
(i) How far is the plane from the origin?
(ii) Find a unit vector perpendicular to that Plane.
(iii) Obtain the equation of the Plane in a general form. (March – 2014)
Answer:
(i) \(\overline{O P}\) = 4i – 2j + 5k
(ii) Then is perpendicular unit vector to the required plane is \(\frac{\overline{O P}}{\overline{O P} \mid}=\frac{4 i-2 j+5 k}{\sqrt{16+4+25}}=\frac{4 i-2 j+5 k}{\sqrt{45}}\)

The perpendicular distance from the origin is \(\sqrt{16+4+25}=\sqrt{45}\)
Vector equation of the Plane can be written as
\(\begin{array}{l}
\vec{r} \cdot \bar{m}=d \Rightarrow \bar{r} \cdot \frac{4 i-2 j+5 k}{\sqrt{45}}=\sqrt{45} \\
\Rightarrow \bar{r} .4 i-2 j+5 k=45
\end{array}\)
Cartesian from is 4x – 2y + 5z = 45

Question 8.
(a) Equation of the plane with intercepts 2, 3, 4 on the x, y, z-axis respectively is
(i) 2x + 3y + 4z = 1
(ii) 2x + 3y + 4z = 12
(iii) 6x + 4y + 3z = 1
(iv) 6x + 4y + 3z = 12

(b) Find the Cartesian equation of the plane passing through the points A(2, 5, -3), B(-2, -3, 5), and C(5, 3, -3). (March – 2016)
Answer:
(a) (iv) 6x + 4y + 3z = 12
(b) Equation of the plane is

Question 9.
(i) The distance of the plane from the point(1, 1, 1)is
(a) 4 units
(b) \(\frac{1}{\sqrt{3}}\) units
(c) \(\frac{4}{\sqrt{3}}\) units
(d) \(\frac{1}{4 \sqrt{3}}\) units

(ii) Find the equation of the plane passing through (1, 0. -2) and perpendicular to each of the planes 2x + y – z = 2 and x – y – z = 3 (May – 2016)
Answer:
(i) (c) \(\frac{4}{\sqrt{3}}\) units
(ii) Equation of the plane passing through (1, 0, -2) is a(x – 1) + b(y – 0) + c(z + 2) = 0……………..(1)
Plane (1) is perpendicular to the given planes
2a + b – c = 0 ……………..(2)
a – b – c = 0 ……………..(3)
Solving (2) and (3) we get;

Question 10.
(i) The linesx – l = y = z is perpendicular to the line
\(\begin{array}{l}
\text { (a) } \frac{x-2}{1}=\frac{y-1}{2}=\frac{z}{-3} \\
\text { (b) } x-2=y-2=z \\
\text { (c) } \frac{x-2}{1}=\frac{y-1}{2}=\frac{z}{3} \\
\text { (d) } x=y=\frac{z}{2}
\end{array}\)

(ii) Find the shortest distance between the lines \(\bar{r}\) = i + 2j + 3k + λ(i + j + k) and \(\bar{r}\) – i + j + k + µ(i + j + k) (March – 2017)
Answer:

Plus Two Maths Three Dimensional Geometry 6 Marks Important Questions

Question 1.
(i) Find the shortest distance between the line \(\frac{x+1}{7}=\frac{y+1}{-6}=\frac{z+1}{1}\) and \(\frac{x-3}{1}=\frac{y-5}{-2}=\frac{z-7}{1}\)
(ii) Find the equation of the Plane passing through one point(-1, 3, 2) and ± r to the planes x + 2y + 3z = 5 and 3x + 3y + z = 0 (March – 2010)
Answer:

(ii) Let the equation of the plane be ax + by + cz + d = 0….(1)
Since (1) is perpendicular to 3x + 2y + 3z = 5, a + 2b + 3c = 0
Since (1) is perpendicular to 3x + 3y + z = 0
3a + 3b + c = 0
Using (2) and (3) we have;
\(\frac{a}{2-9}=\frac{-b}{1-9}=\frac{c}{3-6} \Rightarrow \frac{a}{-7}=\frac{b}{8}=\frac{c}{-3}\)
(1) = -7x + 8y – 3z + d = 0
Since (1) passes through (-1, 3, 2) we have;
⇒ -7 (-1) + 8(3) – 3(2) + d = 0
⇒ 7 + 24 – 6 + d = 0 = d = -25
Therefore the equation of the plane is
(1) ⇒ -7x + 8y – 3z – 25 = 0

Question 2.
(i) (a) A line makes equal angles with the coordinate axis. Find the direction cosines.
(b) Find the equation of the Plane Passing through (1, 1, -1),(2, 3, 5) an (1, 4, -5)
(ii) Find p and q, if the plane x + py + qz = 0 is perpendicular to the plane 3x + 2y + z = 0 and the line \(\frac{x-3}{2}=\frac{y-2}{3}=\frac{z-1}{1}\) (March – 2010)
Answer:

(ii) Since the Plane x + py + qz = 0 is perpendicular to the Line \(\frac{x-3}{2}=\frac{y-2}{3}=\frac{z-1}{1}\) we have their direction ratios proportional. Plane dr’s is 1, p, q and dr’s of Line is 2, 3, 1.
\(\Rightarrow \frac{1}{2}=\frac{p}{3}=\frac{q}{1} \Rightarrow p=\frac{3}{2}, q=\frac{1}{2}\)

Question 3.
Given the straight lines
\(\bar{r}[latex] = 3i + 2j – 4k + 2(i + 2j + 2k) and [latex]\bar{r}[latex] = 5i – 2k + µ(3i + 2j + 6k)
(i) Find the angle between the lines.
(ii) Obtain the unit vector perpendicular to both the lines.
(iii) From the equation of the line perpendicular to the given lines and passing through the point (1, 1, 1). (March – 2014)
Answer:

Question 4.
(a) Write the Cartesian equation of the straight line through the points (1, 2, 3) and along the vector 3i + j +2k
(b) Write a general point on this straight line.
(c) Find the point of intersection of this straight line with the plane 2x + 3y – z + 2 = 0
(d) Find the distance from (1, 2, 3) to the plane 2x + 3y – z + 2 = 0 (March – 2015)
Answer:
(a) Cartesian equation is [latex]\frac{x-x_{1}}{a}=\frac{y-y_{1}}{b}=\frac{z-z_{1}}{c}\), \(\frac{x-1}{3}=\frac{y-2}{1}=\frac{z-3}{2}\)

(b) Let \(\frac{x-1}{3}=\frac{y-2}{1}=\frac{z-3}{2}=\lambda\)
x = 3λ + 1, y = λ + 2, z = 2λ + 3 are the general point of the line.

(c) Equation of the plane is 2x + 3y – z + 20….(1)
putting the values of x, y, z in (1) is
2(3λ + 1) – 3(λ + 2) – (2λ + 3) + 2 = 0
7λ + 7 = 0 ⇒ λ = -1
∴ Point of intersection is x = -3 + 1 = -2; y= -1 + 2 = 1, z= -2 + 3 = 1
ie; (-2, 1, 1)

(d) Distance = \(\frac{2(1)+3(2)-3+2}{\sqrt{4+9+1}}=\frac{7}{\sqrt{14}}\)

Question 5.
Consider a plane passing through the point (5, 2, -4) and perpendicular to the line
\(\bar{r}\) = (i + j) + λ(2i + 3j – k)
(a) Write the equation in cartesian form.
(b) Find its distance from the point (1, 2, -1).
(c) Find the angle made by it with line \(\frac{x-1}{2}=\frac{y-2}{1}=\frac{z-3}{-2}\) (May – 2015)
Answer:
(a) The equation of a plane passing through the point (5, 2, 4) and perpendicular direction ratios 2, 3, -1 is

Kerala State Board New Syllabus Plus Two Computer Science Previous Year Question Papers and Answers.

Kerala Plus Two Computer Science Previous Year Question Paper Say 2018 with Answers

Time: 2 Hours
Cool off time: 15 Minutes

General Instructions to candidates

• There is a ‘cool off-time’ of 15 minutes in addition to the writing time of 2 hrs.
• You are not allowed to write your answers nor to discuss anything with others during the ‘cool off time’.
• Use the ‘cool off time’ to get familiar with the questions and to plan your answers.
• Read questions carefully before you answering.
• All questions are compulsory and the only internal choice is allowed.
• When you select a question, all the sub-questions must be answered from the same question itself.
• Calculations, figures, and graphs should be shown in the answer sheet itself.
• Malayalam version of the questions is also provided.
• Give equations wherever necessary.
• Electronic devices except non-programmable calculators are not allowed in the Examination Hall.

Part – A

Answer all questions from 1 to 5. Each carries 1 score. (5 × 1 = 5)

Question 1.
The ability of data to be processed in more than one form is called ________
Answer:
polymorphism

Question 2.
Each node in a linked list has a ________ to the next node.
Answer:
pointer

Question 3.
NORESIZE is an attribute of ________ tag.
Answer:
<frame>

Question 4.
In client-side scripting, processing is done by ________
Answer:
browser

Question 5.
VPS stands for ________
Answer:
Virtual Private Server

Answer any 9 questions from 6 to 16. Each carries 2 scores. (9 × 2 = 18)

Question 6.
What is the self-referential structure?
Answer:
Self Referential Structures: A structure contains an element that is a pointer to the same structure.
Eg:
struct date
{
int dd, mm, yyyy;
date *ptr;
};

Question 7.
What is the difference between the two declaration statements given below:
(a) int*ptr= new int(10);
(b) int*ptr = new int[10];
Answer:
(a) Dynamic initialization with value 10.
(b) Dynamic array of size 10.

Question 8.
Differentiate static and dynamic data structures. Give an example for each.
Answer:
The main memory can be allocated in two methods.
i) Static memory allocation
ii) Dynamic memory allocation
When the amount of memory to be allocated is known in advance and memory is allocated during compilation itself, it is referred to as static memory allocation.
int x=10;

When the amount of memory to be allocated is not known in advance and it is required to allocate memory as and when required during run time, it is known as dynamic memory allocation.
‘new’ operator is used for dynamic allocation of memory syntax.
datatype *pointer variable = new datatype;
eg: int *ptr=new int;

Question 9.
Categorize the following tags in HTM L appropriately.
<br>, <hl>, <img>, <table>
Answer:

Question 10.
Explain any two data types in JavaScript.
Answer:
Data types in JavaScript
Unlike C++ it uses only three basic data types
1. Number: Any number (whole or fractional) with or without a sign.
Eg: +1977, -38.0003, -100, 3.14157, etc.
2. String: It is a combination of characters enclosed within double-quotes.
Eg: “BVM”, “[email protected]”, etc.
3. Boolean: We can store either true or false. It is case sensitive. That means can’t use TRUE OR FALSE.

Question 11.
Write the JavaScript code to display “Welcome to Kerala” inside the <hl>, tag as shown in the HTML page.
<HTML>
<Body>
<Script lang= “javascript”>
<hl> …………. </hl>
</Body>
</Script>
</HTML>
Answer:
document.write(“Welcome to Kerala”);

Question 12.
Explain any two components of DBMS.
Answer:
Components of DBMS

• Databases – It is the main component.
• Data Definition Language (DDL) – It is used to define the structure of a table.
• Data Manipulation Language (DML) – It is used to add, retrieve, modify and delete records in a database.
• Users – With the help of programs users interact with the DBMS.

Question 13.
Consider the following table:

(a) Identify the degree and cardinality of the given table.
(b) Identify a suitable primary key for the given table.
Answer:
(a) Degree 4
Cardinality 5
(b) SI. No is the primary key

Question 14.
Write the output of the following PHP code fragment.
function justA fun ($num)
{
$num = $num * 5 + ($num + 6);
return $num;
}
echo justA fun(100);
Answer:
5*100 + (100 + 6)
= 500 + 106
= 106
Hence the output is 606

Question 15.
Compare serial and parallel computing.
Answer:

Question 16.
How does ICT help students in learning?
Answer:
Advantages of E-Learning:

• It can offer various courses to a large number of students from distant locations.
• It saves journey time and money, instructor fees, etc.
• People can enjoy e-Learning at a lower cost.
• It enables people to do courses conducted by national or international institutions.

Answer any 9 questions from 17 to 27. Each carries 3 scores. (9 × 3 = 27)

Question 17.
What is a Pointer in C++? Declare a pointer and initialize with the name of your country.
Answer:
Pointer is a variable which points to memory location of some other variable
Syntax:
data_type ‘variable;
#include<iostream>
using namespace std;
intmain()
{
char*ptr=newchar[10]; .
ptr=”India”;
cout<<ptr;
delete ptr;
}

Question 18.
What is Procedural Oriented Programming? What are the disadvantages of Procedural Oriented Programming?
Answer:
A program in procedural language consists of a list of instructions.
Following are the disadvantages of procedural language:
a) Data is undervalued: Here the importance is on doing things rather than the data. The data has the least importance. That is, data is exposed to all and there is a chance to access or destroy this data accidentally or intentionally by many functions on a program.

b) Adding new data requires modifications to all/many functions: A program may contain many functions and these functions may access different data used in different locations. If we add new data items, we will need to modify all the functions that access the data. This is a laborious task.

c) Creating new data types is difficult: In procedural languages, some built-in data types such as int, float, double, and character are available. Extensibility is the ability to create new data types without major rewriting of codes in its basic architecture. Some programming languages are extensible but procedural languages are not extensible.

d) Provides poor real-world modeling: Procedural programming paradigm treats data and functions are different not a single unit. In this Real-life simulations are not possible.

Question 19.
Write an algorithm to add a new item into a queue.
Answer:
An algorithm is given below
Step 1: If front = 1 and rear=N or front=rear + 1
Then print “OVERFLOW” and return
Step 2: If front = Null then
Set front = 1 and rear =1
Else if rear = N then set rear = 1
Else
Set rear = rear +1
End if
Step 3: Set Queue[rear]=item
Step 4: stop

Question 20.
List and explain any three attributes of the INPUT tag in HTML.
Answer:
<Input> It is used to create input controls. Its type of attribute determines the control type.
The main values of type attributes are given below.

1. Text-To creates a text box.
2. Password – To create a password text box.
3. Checkbox – To create a check box.
4. Radio – To create a radio button.
5. Reset – To create a Reset button.
6. Submit – To create a submit button.
7. Button – To create a button

Question 21.
Consider a string “Gandhiji” Write JavaScript code fragment to do the following tasks:
(a) Convert the string to upper case.
(b) Find the length of the string.
(c) Display the 3rd letter in the string.
Answer:
(a) var x=”Gandhiji”;
alert(x.toUpperCase());

(b) var x=”Gandhiji”;
alert(“The number of characters is “+ x.length);

(c) var x=”Gandhiji”;
alert(x.charAt(2));

Question 22.
What is an FTP client software? Differentiate FTP and SFTP.
Answer:
FTP (File Transfer Protocol) client software:
When a client requests a website by entering the website address. Then FTP client software helps to establish a connection between the client computer and the remote server computer. Unauthorised access is denied by using username and password hence secure our website files for that SSH (Secure Shell) FTP simply SFTP is used. Instead of http.//, it uses ftp.//:

By using FTP client s/w we can transfer(upload)the files from our computer to the web server by using the ‘drag and drop’ method. The popular FTP client software are FileZilla, CuteFTP, SmartFTP, etc.

Question 23.
List and explain different database users in DBMS.
Answer:
Users of Database

• Database Administrator – It is a person who has central control over the DBMS.
• Application Programmer – These are computer professionals who interact with the DBMS through programs.
• Naive users – He is an end-user. He does not know the details of DBMS.

Question 24.
Differentiate DELETE and DROP in SQL. Write the syntax of DELETE and DROP.
Answer:

Syntax:
Delete from <Table Name> where condition;
DropTable <Table Name>;

Question 25.
Name the different types of arrays in PHP. Explain with an example.
Answer:
Arrays in PHP: In PHP array is a collection of key and value pairs. Unlike C++, in PHP we can use either nonnegative integers or strings as keys.
Different types of arrays in PHP
Indexed arrays
Associate arrays
Multidimensional arrays.

Indexed arrays: The elements in an array can be distinguished as first or second or third etc. by the indices and the index of the first element is zero. In PHP the function array() is used to creating an array.
Syntax: $array_name=array(value1, value2, …….);
OR
$array_variable[key]=value;
Eg:$mark=array(60, 70, 80);
$course=array(“Science”, “Commerce”, “Humanities”);

Associative arrays: Arrays with named keys and string indices are called associative arrays.
Syntax:
$varibale_name=array(key1=>value1, key2=>value2,etc);
Eg:
$course=array (“Computer Science”=>”05”, “Commerce”=>”39”, “Science”=>”01”);

Question 26.
Discuss any three information security laws for protecting information shared over cyberspace.
Answer:
Guidelines for using computers over the internet

• Emails may contain Viruses so do not open any unwanted emails.
• Download files from reputed sources (sites).
• Avoid clicking on pop-up Advt.
• Most of the Viruses spread due to the use of USB drives so use cautiously.
• Use a firewall on your computer
• Use anti-virus and update regularly
• Use spam blocking software
• Take backups in regular time intervals
• Use strong passwords, i.e a mixture of characters (a-z & A-Z), numbers, and special characters.
• Do not use bad or rude language in social media and emails.
• Untick ‘Remember Me’ before login.

Question 27.
Briefly explain any three applications of computational intelligence.
Answer:
Computational Intelligence is the ability to make a computer to face and solve real-life problems just like an intelligent man do it. It includes Artificial Neural Networks (ANN), Evolutionary Computation (EC), Swarm Intelligence (SI), and Fuzzy Systems (FS).

A) Artificial Neural Networks (ANN): The brain is a complex, nonlinear and parallel computer with the ability to perform tasks such as recognise the pattern, perception, and motor control. ANN is the method of simulate biological neural systems to learn, memorise, and generalize like human beings. A human brain cortex consists of 10-500 billion neurons with 60 trillion synapses(a synapse is a structure that permits a neuron to pass electrical).

B) Evolutionary Computation (EC): It is the simulation of the natural evolution, i.e. survival of the fittest. In the surrounding, we can see that the stronger must win and others will lose. EC applied for data mining, fault diagnosis, classification, scheduling, etc.

C) Swarm Intelligence (SI): Swarm Intelligence is the study of behaviour of colonies or groups of social animals, birds, insects, ants, etc. How they communicate and create and manage their own colonies beautifully.

D) Fuzzy Systems: Human beings use common sense while facing a problem, just like human beings fuzzy systems can also use common sense and behave like a human being. Fuzzy systems is used to control gear transmission and raking systems, control lifts, home appliances, controlling traffic signals, etc.

Answer any 2 questions from 28 to 30. Each carries 5 scores. (2 × 5 = 10)

Question 28.
(a) Name and explain any two attributes of the FORM tag. (2)
(b) Check the given HTML code. Fill the missing code to generate an output as shown in the figure. (3)
<HTML>
<body>
<form name= ‘loginForm’>
username:
<input type = “text”>
password:
………………………………
<input type = “………… ” value = “Login”>
………………………………
</form>
</body>
</html>
Answer:
a) <Form> attributes
1. Action – Here we give the name of the program (including the path) stored in the Webserver.
2. Method – There are 2 types of methods get and post.
3. Target – Specifies the target window for displaying the result. Values are given below.
_blank – Opens in a new window
_self – Opens in the same frame
_parent – Opens in the parent frameset
_top – Opens in the main browser window
name – Opens in the window with the specified name.

b) <inputtype=”password”>
<inputtype=”submit”>
<inputtype=”reset”>

Question 29.
(a) What are scripts in web programming? (2)
(b) Differentiate Client-side Scripting and Server-side Scripting. (3)
Answer:
a) Scripts are small programs embedded in the HTML pages, to write scripts <SCRIPT> tag is used.
Two types of scripts
1. Client scripts: These are scripts executed by the browser(client) hence reduces network traffic and workload on the server.
2. Server scripts: These are scripts executed by the server and the results as a webpage returned to the client browser.

b)

Question 30.
Write SQL queries based on the table PRODUCT given below:

1. Set PID as a primary key.
2. Display the Name, Price of the product having the highest price.
3. Change the Name of Supplier ‘Exotic Liquids’ to ‘Singapore Foods.’
4. Delete all products of supplier ‘Tokyo Traders’.
5. Display Pname and Supplier of all products in the ascending order of price.

Answer:
create table student(name varchar(20) primarykey, roll no int, marks int);

1. Create table PRODUCT (PID varchar(2) primarykey, Name varchar(20), Product varchar(20), Price dec(8, 2));
2. Select Name, Price from PRODUCT where Price=(Select max(Price) from PRODUCT);
3. Update PRODUCT set Name=”Singapore Foods” where Name=”Exotic Liquids”;
4. Delete from PRODUCT where-Name=”Tokyo Traders”;
5. Select Name, Product from PRODUCT order by Price asc;

Kerala State Board New Syllabus Plus Two Computer Science Previous Year Question Papers and Answers.

Kerala Plus Two Computer Science Previous Year Question Paper March 2019 with Answers

Time: 2 Hours
Cool off time : 15 Minutes

General Instructions to candidates

• There is a ‘cool off time’ of 15 minutes in addition to the writing time of 2 hrs.
• You are not allowed to write your answers nor to discuss anything with others during the ‘cool off time’.
• Use the ‘cool off time’ to get familiar with the questions and to plan your answers.
• Read questions carefully before you answering.
• All questions are compulsory and the only internal choice is allowed.
• When you select a question, all the sub-questions must be answered from the same question itself.
• Calculations, figures, and graphs should be shown in the answer sheet itself.
• Malayalam version of the questions is also provided.
• Give equations wherever necessary.
• Electronic devices except non-programmable calculators are not allowed in the Examination Hall.

Part – A

Answer all questions from 1 to 5. Each carries 1 score. (5 × 1 = 5)

Question 1.
The wrapping up of data end functions into a single unit is called ______
Answer:
Data Encapsulation

Question 2.
In a linked list, the linked part of the last node contains ________ data.
Answer:
Null Pointer

Question 3.
Give the full form of VPS.
Answer:
Virtual Private Server

Question 4.
The number of rows in a relation is called ________
Answer:
Cardinality

Question 5.
In PHP, arrays that use string keys are called ________
Answer:
Associative array

Answer any 9 questions from 6 to 16. Each carries 2 scores. (9 × 2 = 18)

Question 6.
Define a structure named ‘Time’ with elements hour, minute, and second.
Answer:
struct time
{
int h, m, s;
};

Question 7.
Read the following C++code:
int a[5] = {10, 15, 20, 25, 30};
int *p = a;
Write the output of the following statements:
(a) cout < < * (p + 2); .
(b) cout << * p + 3;
Answer:
(a) 20(Third element)
(b) *p is 10(First element)
So *p+3 = 10 + 3 = 13.

Question 8.
List the different operations performed on data structures.
Answer:
The operations are Traversing, Searching, Inserting(Push), Deleting(Pop), Sorting and Merging

Question 9.
Write HTML tag for the following:

1. Hyperlink to the website http://WWW.dhsekerala.gov.in
2. [email protected]

Answer:

1. <a href=”http://www.dhsekerala.gov.in”>
2. <a href=mailto:”[email protected]”>

Question 10.
Describe the use of ‘action’ and ‘method’ attributes of <FQRM> tag.
Answer:
<Form> attributes

1. Action – Here we give the name of the program (including the path) stored in the Webserver.
2. Method – There are 2 types of methods get and post.

Question 11.
Explain the two purposes of the ‘+’ operator used in JavaScript.
Answer:
‘+’ if the operands are numbers then it adds and if one of the operands is a string then it concatenates the string.
eg. 5+3=8
“BVM”+8172=”BVM8172”

Question 12.
The Javascript function given below is used to display the sum of digits of a given number. Fill in the blanks to complete the function.
<Script language = “JavaScript”>
………. sum digit( )
{var n, s;
n = document.frm.txt1. …………;
for (s = 0; ……….; n = n/10)
s = s + ………..;
document.frm.txt2.value = s;
}
</script>
Answer:
function sum digit()
n=document.frm.txt1.value;
for(s=0; n>0: n=n/10)
s=s+n%10;

Question 13.
Explain any two constraints used in SQL.
Answer:
Constraints are used to ensure database integrity.

1. Not Null – It ensures that a column can never have NULL values.
2. Unique – It ensures that no two rows have the same value in a column.
3. Primary key – Similar to unique but it can be used only once in a table.
4. Default – We can set a default value.
5. Auto_increment – This constraint is used to perform auto_increment the values in a column. That automatically generates serial numbers. Only one auto_increment column per table is allowed.

Question 14.
List the core data types in PHP.
Answer:
Data types are Integer, Float/Double, Boolean, and String.

Question 15.
What is meant by GIS? Give an example.
Answer:
Geographic Information System (GIS) technology is developed from the digital cartography and Computer-Aided Design (CAD) database management system. GIS as the name implies capturing, storing for future reference, checking, and displaying data related to various positions on the earth’s surface. GIS can be applied in many areas such as soil mapping, agricultural mapping, forest mapping, e-Governance, etc.

Question 16.
Define Infringement.
Answer:
Infringement (Violation): Unauthorized copying or use of Intellectual property rights such as Patents, Copyrights, and Trademarks are called intellectual property Infringement (violation). It is a punishable offense.

• Patent Infringement
• Trademark Infringement
• Copy right Infringement.

Answer any 9 questions from 17 to 27. Each carries 3 scores. (9 × 3 = 27)

Question 17.
What are the different memory allocations used in C++? Explain.
Answer:
The main memory can be allocated in two methods.

1. Static memory allocation
2. Dynamic memory allocation

When the amount of memory to be allocated is known in advance and memory is allocated during compilation itself, it is referred to as static memory allocation.

When the amount of memory to be allocated is not known in advance and it is required to allocate memory as and when required during run time, it is known as dynamic memory allocation.

‘new’ operator is used for dynamic allocation of memory syntax,
datatype*pointer variable = new datatype;
eg: int *ptr = new int;

Question 18.
What is polymorphism? Give an example.
Answer:
Polymorphism is the ability for a message or data to be processed in more than one form. This is achieved by function overloading, operator overloading, and dynamic binding.
There are two types of polymorphism.

a) Compile time (early binding/static) polymorphism. It is the ability of the compiler to relate or bind a function call with the function definition during compilation time itself.
Examples are Function overloading and operator overloading.
Function overloading: Functions with the same name and different signatures (the number of parameters or data types are different).

Operator overloading: It gives new meaning to an existing C++ operator.
Eg: we know that + is used to add two numbers, Operator overloading assigns + to a new job such as it concatenates two strings into one string.

b) Run time (late binding/dynamic) polymorphism. It is the ability of the compiler to relate or bind a function call with the function definition during run time. It uses the concept of pointers and inheritance.

Question 19.
Write an algorithm to insert a new item into a Queue.
Answer:
The algorithm is given below:
Step 1: If front = 1 and rear=N or front = rear + 1.
Then print “OVERFLOW” and return
Step 2: If front = Null then
Set front = 1 and rear = 1
Else if rear = N then set rear = 1
Else
Set rear = rear+1
End if
Step 3: Set Queue[rear] = item
Step 4: stop

Question 20.
Differentiate between a static web page and a dynamic web page.
Answer:

Question 21.
Briefly explain the different ways in which a JavaScript code can be inserted into a web page.
Answer:
Ways to add scripts to a web page.
1. Inside <BODY> section
Scripts can be placed inside the <BODY> section.

2. Inside<HEAD> section: Scripts can be placed inside the <HEAD> section. This method is a widely accepted method.

3. External (another) JavaScript file: We can write scripts in a file and save it as a separate file with the extension js. The advantage is that this file can be used across multiple HTML files and can be enhanced the speed of the page loading.

Question 22.
Distinguish between shared hosting and dedicated hosting.
Answer:
1. Shared Hosting: This type of hosting shares resources, like memory, disk space, and CPU hence the name shared. Several websites share the same server. This is suitable for small websites that have less traffic and it is not suitable for large websites that have large bandwidth, large storage space, and have a large volume of traffic.

E.g.: Shared hosting is very similar to living in an Apartment (Villas) complex. All residents are in the same location and must share the available resources (Car parking area, Swimming pool, Gymnasium, playground, etc) with everyone.

2. Dedicated Hosting: A web server and its resources are exclusively for one website that has a large volume of traffic mean a large volume of requests by the visitors. Some Govt, departments, or large organizations require uninterrupted services for that round a clock power supply is needed. It is too expensive but it is more reliable and provides good service to the public.

E.g.: It is similar to living in an Our own house. All the resources in your house is only for you. No one else’s account resides on the computer and would not be capable of tapping into your resources.

Question 23.
Explain different levels of data abstraction in DBMS.
Answer:
Database Abstraction: Abstraction means hiding, it hides certain details of how data is stored and maintained.
Levels of Database Abstraction

1. Physical Level (Lowest Level) – It describes how the data is actually stored in the storage medium.
2. Logical Level (Next Higher Level) – It describes what data are stored in the database.
3. View Level (Highest level) – It is closest to the users. It is concerned with the way in which the individual users view the data.

Question 24.
Describe the ‘union’ and ‘intersection’ operations in relational algebra with suitable examples.
Answer:
Union Operation (∪) – All tuples appearing in either or both of two relations.
Intersection operation (∩) – All tuples appearing in both relations.

Question 25.
Write PHP code to display all even numbers below 100.
Answer:
<?php
echo”All even numbers below 100″;
echo”<br>=================”;
for($i=2; $i<=100; $i=$i+2)
{
echo”<br>”.$i; ‘
}

Question 26.
Explain the cloud service models.
Answer:
Cloud service models (3 major services)
1. Software as a Service (SaaS): A SaaS provider company provides more services on demand such as they allow to access both resources and applications.
Examples are Google Docs, Adobe creative cloud, Microsoft Office 365, Facebook.com, etc.

2. Platform as a Service (PaaS): A PaaS provider company provides subscribers access to the components that they require to develop and operate applications over the Internet.
Example: LAMP platform (Linux, Apache Server, MySQL, and PHP), ASP.NET, PHP, and Python, Google’s App Engine, Microsoft Azure, Force.com, etc.

3. Infrastructure as a Service (IaaS): It provides basic storage devices and computing capabilities as standardized services over the network.
Example: Amazon Web Services, Joyent, AT&T, GoGrid, etc.

Question 27.
List and explain any three e-learning tools.
Answer:
e-Learning tools
a) Electronic books reader (eBooks): With the help of a tablet or portable computer or arty another device we can read digital files by using an s/w is called electronic books reader.

b) e-text: The electronic format of textual data is called e-Text.

c) Onlinechat: Real-time exchange of text or audio or video messages between two or more people over the Internet.

d) e-Content: The data or information such as text, audio, video, presentations, images, animations, etc, are stored in electronic format.

e) Educational TV channels: TV channels dedicated only for the e-Learning purpose.
Eg.: VICTERS (Virtual Classroom Technology on Edusat for Rural Schools OR Versatile ICT Enabled Resources for Students).

Answer any 2 questions from 28 to 30. Each carries 5 scores. (2 × 5 = 10)

Question 28.
Explain the various attributes of the <BODY> tag.
Answer:
Attributes of <body> tag
1. BGCOLOR – Specifies the background color for the document body
Eg. <BODY BGCOLOR = ”RED”>

2. BACKGROUND – Sets the image as the background for the document body
Eg. <BODY BACKGROUNG= “C:\result.jpg”>.

3. TEXT-Specifies the color of the text content of the page.
Eg. <BODY TEXT= “Red”>

4. Link – Specifies colour of the hyperlinks that are not visited by the user.

5. ALINK – Specifies the colour of hyperlinks.

6. VLINK – Specifies the color of hyperlinks which are already visited by the viewer. ‘
Eg. < BODY ALINK= “Cyan” LINK-’ Magenta”
VLINK= “Orange”>.

7. Leftmargin and Right margin-Sets margin from the left and top of the document window.

Question 29.
Write HTML code to display the following table in a web page:
Result of ABC school

Answer:

<html>
<head>
<title>Table
</title>
<body bgcolor=”cyan”>
<h1 align=”center”>Result of ABC school</h1>
<tableborder=”1"align=”center">
<tralign=”center”> •
<th rowspan=”2">Year</th>
<th colspan=”2">Students</th>
<th rowspan=”2">Pass<br>Percentage</th>
</tr>
<tr align=”center”>
<th>Registered</th>
<th>Passed</th>
</tr>
<tr align="center”>
<td>2014</td>
<td>200</td>
<td> 130</td >
<td>65</td>
</tr>
<tralign=”center”>
<td>2015</td>
<td>200</td>
<td>150</td>
<td>75</td>
</tr><tralign=”center”>
<td>2016</td>
<td>200</td>
<td > 160</td >
<td>80</td>
</tr>
</table>
</body>
</html>

Question 30.
A table named ‘student’ with fields Roll no, Name, Batch, Mark, Grade is given. Write SQL statements for the following:

1. To display the details of all students in the ‘Science’ batch.
2. To display the details of these students having grade A or A+.
3. To count the number of students in each batch.
4. To change the grade of the student to A+ whose Roll no. is 50.
5. Remove the details of the student whose Roll no. is 10.

Answer:

1. select * from student where Batch=”Science”;
2. select * from student where Grade in(“A”,”A+”);
3. select Batch,count(*) from student group by Batch;
4. update student set Grade =”A+”; where Roll_no=50;
5. delete from student where Roll_no =10;

Kerala State Board New Syllabus Plus Two Maths Chapter Wise Previous Questions and Answers Chapter 10 Vector Algebra.

Kerala Plus Two Maths Chapter Wise Previous Questions Chapter 10 Vector Algebra

Plus Two Maths Vector Algebra 3 Marks Important Questions

Question 1.
(i) With help of a suitable figure for any three vectorsa,bandc show that \((\bar{a}+\bar{b})+\bar{c}=\bar{a}+(\vec{b}+\bar{c})\)
(ii) If \(\bar{a}\) = i – j + k and \(\bar{b}\) = 2i – 2j – k. What is the projection of a on b? (March – 2011)
Answer:
(i) Answered in previous years questions
No. 1(ii) (6 Mark question)
(ii) Projection of \(\bar{a} \text { on } \bar{b}=\frac{\bar{a} \cdot \bar{b}}{|\bar{b}|}=\frac{2+2-1}{\sqrt{4+4+1}}=1\)

Question 2.
(i) If \(\bar{a}\) = 3i – j – 5k and \(\bar{b}\) = i – 5j + 3k Show that \(\bar{a}\) + \(\bar{b}\) and a bare perpendicular.
(ii) Given the position vectors of three points as A(i – j + k); B(4i + 5j + 7k) C(3i + 3j + 5k)
(a)Find \(\bar{AB}\) and \(\bar{BC}\)
(b) Prove that A,B and C are collinear points. (March – 2011)
Answer:

Question 3.
(i) Write the unit vector in direction of i + 2j – 3k.
(ii) If \(\overline{P Q}\) = 31 + 2j — k and the coordinate of P are(1, -1,2) , find the coordinates of Q. (May – 2012)
Answer:

Question 4.
(a) The angle between the vectors \(\bar{a}\) and \(\bar{b}\) such that \(|\vec{a}|=|\bar{b}|=\sqrt{2}\)
\(\bar{a}\).\(\bar{b}\) = 1 is
\(\begin{array}{lll}
\text { (i) } \frac{\pi}{2} & \text { (ii) } \frac{\pi}{3} & \text { (iii) } \frac{\pi}{4} & \text { (iv) } 0
\end{array}\)
(b) Find the unit vector along \(\bar{a}-\bar{b}\) where \(\bar{a}\) = i + 3j – k and \(\bar{b}\) 3i + 2j + k (March -2016)
Answer:

Plus Two Maths Vector Algebra 4 Marks Important Questions

Question 1.
Consider the vectors \(\bar{a}\) = 21+ j – 2k and \(\bar{b}\) = 6i – 3j + 2k.
(i) Find \(\bar{a} \bar{b}\) and \(\bar{a} \times \bar{b}\).
(ii) Verity that \(|\bar{a} \times \bar{b}|=|\vec{a}|^{2}|\bar{b}|^{2}-(\bar{a} \cdot \bar{b})^{2}\) (March – 2012)
Answer:

Question 2.
(i) For any three vectors \(\bar{a}, \bar{b}, \bar{c}\), show that \(\bar{a} \times(\bar{b}+\bar{c})+\bar{b} \times(\bar{c}+\bar{a})+\bar{c} \times(\bar{a}+\bar{b})=0\)
(ii) Given A (1, 1, 1), B (1, 2, 3), C (2, 3, 1) are the vertices of MBCa triangle. Find the area of the ∆ABC (May – 2012)
Answer:

Question 3.
Consider A (2, 3, 4) , B (4, 3, 2) and C (5, 2, -1) be any three points
(i) Find the projection of \(\overline{B C}\) on \(\overline{A B}\)
(ii) Find the area of triangle ABC (March – 2013)
Answer:

Question 4.
(i) Find the angle between the vectors \(\bar{a}\) =3i + 4j + k and \(\bar{b}\) = 2i + 3j – k
(ii) The adjacent sides of a parallelogram are \(\bar{a}\) = 3i + λj + 4k and \(\bar{b}\) = i – λj + k
(a) Find \(\bar{a} \times \bar{b}\)
(b) If the area of the parallelogram is square units, find the value of A (May – 2013)
Answer:

Question 5.
Let \(\bar{a}\) = 2i – j + 2k and \(\bar{b}\) = 6i + 2j + 3k
(i) Find a unit vector in the direction of \(\bar{a}\) + \(\bar{b}\)
(ii) Find the angle between a and b (March – 2014)
Answer:

Question 6.
Consider the triangle ABC with vertices A(1, 1, 1) , B (1, 2, 3) and C (2, 3, 1)
(i) Find \(\overline{A B}\) and \(\overline{A C}\)
(ii) Find \(\overline{A B}\) x \(\overline{A C}\)
(iii) Hence find the area of the triangle (March – 2014)
Answer:

Question 7.
Consider the vectors \(\bar{a}\) = i – 7j + 7k; \(\bar{b}\) = 3i – 2j + 2k
(a) Find \(\bar{a b}\).
(b) Find the angle between \(\bar{a}\) and \(\bar{b}\).
(c) Find the area of parallelogram with adjacent sides \(\bar{a}\) and \(\bar{b}\). (May – 2014)
Answer:

Question 8.
(a) If the points A and B are (1, 2, -1) and (2, 1, -1) respectively, then is
(i) i + J
(ii) i – J
(iii) 2i + j – k
(iv) i + j + k
(b) Find the value of for which the vectors 2i – 4j + 5k, i – λj + k and 3i + 2j – 5k are coplanar.
(c) Find the angle between the vectors a = 2i + j – k and b = i – j + k (March – 2016)
Answer:

Question 9.
(i) \((\bar{a}-\bar{b}) \times(\bar{a}+\bar{b})\) is equaito
\(\begin{array}{lll}
\text { (a) } \bar{a} & \text { (b) }|\bar{a}|^{2}-|\bar{b}|^{2} & \text { (c) } \bar{a} \times \bar{b} \text { (d) } 2(\bar{a} \times \bar{b})
\end{array}\)

(ii) If \(\bar{a}\) and \(\bar{b}\) are any two vectors, then
\((\bar{a} \times \bar{b})^{2}=\left|\begin{array}{ll}
\bar{a} \cdot \bar{a} & \bar{a} \cdot \bar{b} \\
\bar{a} \cdot \bar{b} & \bar{b} \bar{b}
\end{array}\right|\)

(iii) Using vectors, show that the points A(1, 2, 7), B(2, 6, 3), C(3, 10, -i) are collinear. (May – 2016)
Answer:

Plus Two Maths Vector Algebra 6 Marks Important Questions

Question 1.
(i) Find a vector in the direction of \(\bar{r}\) = 3E – 4j that has a magnitude of 9.
(ii) For any three vectors \(\bar{a,b}\) and \(\bar{c}\), and Prove that \((\bar{a}+\bar{b})+\bar{c}=\bar{a}+(\bar{b}+\bar{c})\).
(iii) Find a unit vector perpendicular to \(\bar{a}+\bar{b}\) and \(\bar{a}-\bar{b}\), where \(\bar{a}\) = i – 3j + 3k and \(\bar{b}\) and \(/bar{c}\) = 3E—3j+2k. (March – 2010)
Answer:
(i) Unit vector of magnitude 9

Question 2.
Let A(2, 3, 4), B(4, 3, 2) and C(5, 2, -1) be three points
(i) Find \(\overline{A B}\) and \(\overline{B C}\)
(ii) Find the projection of \(\overline{B C}\) on \(\overline{A B}\)
(iii) Fiñd the area of the triangle ABC. (May – 2010)
Answer:

Question 3.
ABCD s a parallelogram with A as the origin, \(\bar{b}\) and \(\bar{d}\) are the position vectors of B and D respectively.

(i) What is the position vector of C?
(ii) What is the angle between \(\bar{AB}\) and \(\bar{AD}\)?
(iii) If \(|\overrightarrow{A C}|=|\overrightarrow{B D}|\), show that ABCD is a rectangle. (May – 2011)
Answer:
(i) Since ABCD is a parallelogram with A as the

Question 4.
(a) If \(\bar{a}, \bar{b}, \bar{c}, \bar{d}\) respectively are the position vectors representing the vertices A,B,C,D of a parallelogram, then write \(/bar{d}\) in terms of \(\bar{a}, \bar{b}, \bar{c}\).
(b) Find the projection vector of \(/bar{b}\) = i + 2j + k along the vector \(/bar{a}\) = 21 – i – j + 2k. Also write \(/bar{b}\) as the sum of a vector along \(/bar{a}\) and a perpendicular to \(/bar{a}\).
(C) Find the area of a parallelogram for which the vectors 21 + j, 31 + j +4k are adjacent sides. (March – 2015)
Answer:

Question 5.
(a) Write the magnitude of a vector \(/bar{a}\) in terms of dot product.
(b) If \(\bar{a}, \bar{b}, \bar{a}+\bar{b}\) are unit vectors, then prove that the angle between \(/bar{a}\) and \(/bar{b}\) is \(\frac{2 \pi}{3}\)
(c) If 2i + j – 3k and mi + 3j – k are perpendicular to each other, then find ‘m’.
Also find the area of the rectangle having these two vectors as sides. (March – 2015)
Answer:

Question 6.
Consider the triangle ABC with vertices A(1, 2, 3), B(-1, 0, 4), C(0, 1, 2)
(a) Find \(\overline{A B}\) and \(\overline{A C}\)
(b) Find \(\angle A\)
(c) Find the area of triangle ABC. (May – 2015)
Answer:

Kerala State Board New Syllabus Plus Two Maths Chapter Wise Previous Questions and Answers Chapter 9 Differential Equations.

Kerala Plus Two Maths Chapter Wise Previous Questions Chapter 9 Differential Equations

Plus Two Maths Differential Equations 3 Marks Important Questions

Question 1.
Form a differential equation of the family of circles having a centre on y-axis and radius 3 units. (May -2013)
Answer:
The equation of the circle passing through the point (O,k)and radius 3 is of the form

Question 2.
Consider the Differential equation
\(\frac{d^{2} y}{d x^{2}}+y=0\)
(i) Write the order and degree.
(ii) Verify that y = a cos x + b sin x where a,b ∈ R is a solution of the given DE. (March – 2014)
Answer:
(i) Order = 2; Degree = I
(ii) Given; y = acosx + bsin x
y₁ = – asin x + bcos x
y₂ = – acos x – bsin x
We have; y₂ = – (a cos x + b sin x)
⇒ y₂ = -y ⇒ y₂ + y = 0

Plus Two Maths Differential Equations 4 Marks Important Questions

Question 1.
If cosx\(\frac{d y}{d x}\) + y sin x = tan² x is a DE then
(i) Find its order and degree.
(ii) Find its general solution. (May -2010)
Answer:

Question 2.
(i) Write the order and degree of the DE
\(\left[\frac{d y}{d x}\right]^{2}+\frac{d y}{d x}-\sin ^{2} y=0\)
(ii) Solve the \(\frac{d y}{d x}+2 y \tan x=\sin x\) (May-2011)
Answer:

Question 3.
(i) The general solution of the DE \(\frac{d y}{d x}=e^{x-y} \text { is }\)
(a) e^y + e^x = c
(b) e^y e^x = c
(c) e^-y + e^-x = c
(d) e^-y – e^x = c
(ii) Solve the DE \(\frac{d y}{d x}=\frac{2 x y}{1+x^{2}}+x^{2}+2\) (March – 2013)
Answer:

Question 4.
(a) Consider the family of all circles having their centre at the point (1,2). Write the equation of the family. Write the corresponding differential equation.
(b) Write the integrating factor of the differential equation
\(\cos x \frac{d y}{d x}+y=\sin x, \quad 0 \leq x<\frac{\pi}{2}\) (March – 2015)
Answer:
(a) The equation of the circle ¡s

Question 5.
(a) Write the order and degree of the differential equations.
\(x y\left(\frac{d^{2} y}{d x^{2}}\right)^{2}+x\left(\frac{d y}{d x}\right)^{3}-y \frac{d y}{d x}=0\)

(b) Find the general solution of the differential equation ylog ydx – xdy = 0
(c) Find the integrating factor of the differential equation \(x \frac{d y}{d x}-y=2 x^{2}\) (May -2015)
Answer:

Question 6.
(a) y = a cosx +b sin x is the solution of the differential equation

(b) Find the solution of the differential equation \(x \frac{d y}{d x}+2 y=x^{2}, \quad(x \neq 0)\) given that y = 0 when x=1. (March – 2016)
Answer:

Plus Two Maths Differential Equations 6 Marks Important Questions

Question 1.
(i) Form the DE corresponding to the Function y = ae^x + be^2x
(ii) State the order and degree of the above DE.
(iii) Solve \(x \frac{d y}{d x}=x+y\) (March – 2009)
Answer:

Question 2.
(i) Form the DE corresponding to the function Xy = C²
(ii) Consider the DE (x² + y² ) dx = 2xydy
(a) Write the DE in the ton \(\frac{d y}{d x}=g\left[\frac{y}{x}\right]\)
(b) Solve the DE completely (May -2009, May -2013)
Answer:

Question 3.
(i) Equation of a circle touching the y-axis at origin is x² + y² – 2ax = 0. Find the DE of all such circles.
(ii) SolvetheDE \(\left(1+x^{2}\right) \frac{d y}{d x}+y=\tan ^{-1} x\) (March – 2010)
Answer:

Question 4.
(i) Solution of the DE y^‘ – y = 0 is y = ………….
(ii) Solve the DE \(\Rightarrow \frac{d y}{d x}+y \sec x=\tan x\)
(iii) Form the DE of the family of ellipse having foci on the x-axis and centre at the origin. (March-2011)
Answer:

Question 5.
Consider the DE \(x d y-y d x=\sqrt{x^{2}+y^{2}} d x\)
(i) Express it in the form \(\frac{d y}{d x}\) = F(x, y)
(ii) Find the general solution. (March -2012; Edumate -2017)
Answer:

Question 6.
(i) Prove that the DE is (3xy + y²) dx + (x² + xy) dy = 0 a homogeneous DE of degree 0.
(ii) Solve the DE (3xy + y²) dx + (x² + xy) dy = 0 (May —2012, Edumate -2017)
Answer:

Question 7.
Consider the differential equation \(\frac{d y}{d x}-3 \cot x y=\sin 2 x\)
(a) Find its integrating factors.
(b) Fînd its solution, given that y = 2 When x = \(\frac{\pi}{2}\). (May-2014)
Answer:

Kerala State Board New Syllabus Plus Two Maths Chapter Wise Previous Questions and Answers Chapter 8 Application of Integrals.

Kerala Plus Two Maths Chapter Wise Previous Questions and Answers Chapter 8 Application of Integrals

Plus Two Maths Application of Integrals 4 Marks Important Questions

Question 1.
(i) The area bounded by the curve y = f(x) x-axis and the line x= a and x= b is……
(ii) Find the area enclosed between the Parabola y = x² and the straight line 2x – y + 3 = 0 (March – 2010)
Answer:

Question 2.
Find the area enclosed between the curve x² = 4y and the line x = 4y – 2 (March -2011)
Answer:

Question 3.
(i) Area of the shaded portion in the figure is equal to

(ii) Consider the curves y = x², x = 0, y = 1, y = 4.
Draw a rough sketch and shade the region bounded by these curves, Find area of the shaded region.
Answer:

Question 4.
Consider the following figure:

(i) Find the point of Intersection P of the circle x² + y² = 32 and the line y = x.
(ii) Find the area of the shaded region. (EDUCATE – 2017; March – 2013; March – 2014)
Answer:

Question 5.
(a) The area bounded by the curve, above the x-axis, between x = a and x = b is

(b) Find the area of the circle x² + y² = 4 using integration. (March – 2016)
Answer:

Question 6.
(i) The area bounded by y = 2cosx , the x-axis from x = 0 to x = \(\frac{\pi}{2}\) is
(a) 0
(b) 1
(c) 2
(d) -1
(ii) Find the area of the region bounded by the y² = 4ax and x² = 4ay, a > 0 (March – 2017)
Answer:

When x = 4a, y = 4a and x = 0, y = 0.
Therefore the point is (0, 0) and (4a, 4a).

Plus Two Maths Application of Integrals 6 Marks Important Questions

Question 1.
Consider the circle x² + y² = 16 and the straight line \(y=\sqrt{3} x\) as shown ¡n the figure

(i) Find the points A and B as shown in the figure.
(ii) Find the area of the shaded region in the figure using definite integral. (May -2010)
Answer:
(i) The point of intersection of x² + y² = l6 and

Question 2.
(i) Draw the rough sketch of \(\frac{x^{2}}{4}+\frac{y^{2}}{9}=1\)
(ii) Find the area bounded by the above curve using integration. (May – 2011)
Answer:
(i) The curve is an ellipse.

Question 3.
(i) Find the area enclosed between the curve y² = x, x = 1, x = 4 and x-axis.
(ii) Using ntegration, find the area of the region bounded by the triangle whose vertices are (1, 0), (2, 2) and (3, 1). (March-2012)
Answer:

Required area ΔABC = Area ΔAB2
+ Area 2BC3 – Area ΔAC3
Equation AC is y = 2(x – 1)
Equation BC is y = 4 – x
Equation AB is y = 1/2 (x – 1)

Question 4.

Using the above figure
Find the equation of AB.
Findthe point P.
Find the area of the shaded region by integration. (May – 2013)
Answer:
(i) The equation of a line passing through (0,2)

Question 5.
Consider the ellipse \(\frac{x^{2}}{9}+\frac{y^{2}}{4}=1\) and the line \(\frac{x}{3}+\frac{y}{2}=1\).
(a) Find the points where the line intersects the ellipse?
(b) Shade the smaller region bounded by the ellipse and the line.
(c) Find the area of the shaded region. (May – 2014)
Answer:

Question 6.
Consider the function f(x) = |x| + 1; g(x) = 1 – |x|
(a) Sketch the graph and shade the enclosed region between them.
(b) Find the area of the shaded region. (March – 2015)
Answer:

(b) The equation of the line through AB is

Question 7.
Using the given figure answer the following questions.

Define the equation of the given curve.
Find the area of the enclosed region.
Find the area when a = lo and b = 5. (March – 2011; May – 2015; March – 2017)
Answer:
(i) The figure represents an ellipse with equation

Ellipse is symmetric w.r.t coordinate axles. Therefore the area of the enclosed region is same as four times area enclosed by the curve in first quadrant.

Kerala State Board New Syllabus Plus Two Maths Chapter Wise Previous Questions and Answers Chapter 7 Integrals.

Kerala Plus Two Maths Chapter Wise Previous Questions and Answers Chapter 7 Integrals

Plus Two Maths Application of Derivatives 3 Marks Important Questions

Question 1.
Find the following integrals. (May -2011)
\(\begin{array}{l}
\text { (i) } \int x^{2} e^{2 x} d x \\
\text { (ii) } \int e^{x} \sin x d x
\end{array}\)
Answer:

Question 2.
(i) \(\int e^{x} \sec x(1+\tan x) d x=\ldots \ldots\)
(a) ex cosx + c (b) ex sec x + c
(C) ex tanx + c (d) ex sin x + c
(ii) Find \(\int \sin 2 x \cos 3 x d x\) (March – 2014)
Answer:

Question 3.
Find the following integrals.
(i) \(\begin{array}{l}
\text { (i) } \int \frac{1}{(x+1)(x+2)} d x \\
\text { (ii) } \int \frac{2 x-1}{(x-1)(x+2)^{2}} d x
\end{array}\) (March – 2014)
Answer:

Plus Two Maths Application of Derivatives 4 Marks Important Questions

Question 1.
Consider the integral \(I=\int_{0}^{\pi} \frac{x \sin x}{1+\cos ^{2} x} d x\)
(i) Express \(I=\frac{\pi}{2} \int_{0}^{\pi} \frac{\sin x}{1+\cos ^{2} x} d x\)
(ii) Show that \(I=\frac{\pi^{2}}{4}\) (March – 2012)
Answer:

Question 2.
(i) Evaluate: \(\int_{2}^{3} \frac{x}{x^{2}+1} d x\)
(ii) Evaluate: \(\int_{0}^{\pi} \frac{x}{1+\sin x} d x\) (March – 2014)
Answer:

Question 3.
(a) What is the value of \(\int_{0}^{1} x(1-x)^{9} d x\) If the
\(\begin{array}{llll}
\text { (i) } \frac{1}{10} & \text { (ii) } \frac{1}{11} & \text { (iii) } \frac{1}{90} & \text { (iv) } \frac{1}{110}
\end{array}\)
(b) Find \(\int_{0}^{1}(2 x+3) d x\) of a sum. (March – 2015)
Answer:

Question 4.
Evaluate \(\int_{0}^{x} \log (1+\cos x) d x\)
Answer:

Question 5.
Find \(\int_{0}^{5}(x+1) d x \text { as limit of a sum. }\)
Answer:

Question 6.
Evaluate \(\int_{0}^{4} x^{2} d x\) as the limit of a sum. (March – 2017)
Answer:

Plus Two Maths Application of Derivatives 6 Marks Important Questions

Question 1.
(i) Fill in the blanks; \(\int \frac{1}{x} d x=\)_____
(ii) Evaluate \(\int \frac{5 x+1}{x^{2}-2 x-35} d x\)
(iii) Integrate with respect to x. \(\sqrt{x^{2}+4 x+8}\) (March – 2010)
Answer:

Question 2.
(i) Evaluate \(\int-\frac{\cos e c^{2} x}{\sqrt{\cot ^{2} x+9}} d x\)
(ii) Evaluate \(\int\left(\cos ^{-1} x\right)^{2} d x\) (May -2010)
Answer:

Question 3.
(i) Evaluate \(\int_{0}^{\pi} \frac{x \sin x}{1+\cos ^{2} x} d x\)
(ii) Evaluate \(\int_{0}^{2} e^{x} d x \text { as limit of a sum. }\) (May -2010)
Answer:

Question 4.
(i) Fill in the blanks \(\int \cot x d x=\)_____
(ii) Evaluate the integrals
\(\begin{array}{l}
\text { (a) } \int \sin 2 x \cos 4 x d x \\
\text { (b) } \int \frac{x}{(x+1)(x+2)} d x
\end{array}\) (March -2011)
Answer:

Question 5.
(i) Evaluate \(\int_{0}^{1} x d x\) as the limit of a sum.
(ii) Evaluate \(\int_{0}^{1} x(1-x)^{n} d x\) (March – 2011)
Answer:

Question 6.
(i) Evaluate \(\int_{1}^{2} \frac{1}{x(1+\log x)^{2}} d x\)
(ii) Evaluate \(\int_{0}^{3}\left(2 x^{2}+3\right) d x\) as the limit of a sum. (May – 2011)
Answer:

Question 7.
(i) What is \(\int \frac{1}{9+x^{2}} d x=?\)
(ii) Evaluate the integrals \(\int \frac{1}{1+x+x^{2}+x^{3}} d x\) (May – 2012)
Answer:

Question 8.
(i) Evaluate \(\int_{0}^{3} f(x) d x\)
where \(f(x)=\left\{\begin{array}{ll}
x+3, & 0 \leq x \leq 2 \\
3 x, & 2 \leq x \leq 3
\end{array}\right.\)

(ii) Prove that \(\int_{0}^{1} \log \left(\frac{x}{1-x}\right) d x=\int_{0}^{1} \log \left(\frac{1-x}{x}\right) d x\) Find the value of \(\int_{0}^{1} \log \left(\frac{x}{1-x}\right) d x\) (May – 2012)
Answer:

Question 9.
(i) Find \(\int \cot x d x=\ldots \ldots\)
(ii) Find \(\int x \log x d x\)
(iii) Find \(\int \frac{x-1}{(x-2)(x-3)} d x\) (March – 2013)
Answer:

Question 10.
Evaluate
\(\text { (i) } \int \frac{x+3}{\sqrt{5-4 x-x^{2}}} d x\)
\(\text { (ii) } \int_{\pi / 6}^{\pi / 3} \frac{d x}{1+\sqrt{\tan x}}\) (May – 2013)
Answer:

Question 11.
Evaluate
\(\begin{array}{l}
\text { (i) } \int x^{2} \tan ^{-1} x d x \\
\text { (ii) } \int_{-1}^{2} x^{3}-x d x
\end{array}\) (May – 2013)
Answer:

Question 12.
Evaluate \(\int_{0}^{\pi / 4} \log (1+\tan x) d x\) (March – 2013)
Answer:

Question 13.
(a) The value of \(\int_{\frac{-\pi}{2}}^{\frac{\pi}{2}} \cos x d x\) (May – 2014)
(b) Prove that \(\int_{0}^{\pi} \frac{x}{a^{2} \cos ^{2} x+b^{2} \sin ^{2} x} d x=\frac{\pi^{2}}{2 a b}\)
Answer:

Question 14.
(a) \(\int \frac{1}{x^{2}+a^{2}} d x=\)
(b) Find \(\int \frac{1}{9 x^{2}+6 x+5} d x\)
(c) Find \(\int \frac{x}{(x-1)^{2}(x+2)} d x\) (May – 2014)
Answer:

Question 15.
Integrate the following
\(\begin{array}{l}
\text { (a) } \frac{x-1}{x+1} \\
\text { (b) } \frac{\sin x}{\sin (x-a)} \\
\text { (c) } \frac{1}{\sqrt{3-2 x-x^{2}}}
\end{array}\) (March – 2015)
Answer:

Question 16.
(a) Prove that \(\int \cos ^{2} x d x=\frac{x}{2}+\frac{\sin 2 x}{4}+c\)
(b)Find \(\int \frac{1}{\sqrt{2 x-x^{2}}} d x\)
(c) Find \(\int x \cos x d x\) (May – 2015)
Answer:

Question 17.
Find the following:
\(\begin{array}{l}
\text { (i) } \int \frac{1}{x\left(x^{7}+1\right)} d x \\
\text { (ii) } \int_{1}^{4}|x-2| d x
\end{array}\)
Answer:

Question 18.
Find \(\int_{0}^{\frac{\pi}{2}} \log \sin x d x\)
Answer:

Question 19.
Find the following: \(\begin{array}{l}
\text { (i) } \int \cot x \log (\sin x) d x \\
\text { (ii) } \int \frac{1}{x^{2}+2 x+2} d x \\
\text { (iii) } \int x e^{9 x} d x
\end{array}\) (May – 2017)
Answer:

Kerala State Board New Syllabus Plus Two Maths Chapter Wise Previous Questions and Answers Chapter 6 Application of Derivatives.

Kerala Plus Two Maths Chapter Wise Previous Questions and Answers Chapter 6 Application of Derivatives

Plus Two Maths Application of Derivatives 4 Marks Important Questions

Question 1.
(a) Find the equation of the tangent to the curve \(x^{\frac{2}{3}}+y^{\frac{2}{3}}=2\) at (1,1).
(b) Find two positive numbers whose sum is 15 and the sum of whose squares is minimum. (May – 2015)
Answer:

Question 2.
(a) The slope of the tangent to the curve given
\(x=1-\cos \theta, y=\theta-\sin \theta \text { by at } \theta=\frac{\pi}{2}\)
(i) 0
(ii) – 1
(iii) 1
(iv) Not defined.

(b) Find the intervals in which the function f(x) = x² – 4x + 6 is strictly decreasing.
(C) Find the minimum and maximum value, if any, of the function f(x) = (2x – 1)² + 3 (March – 2016)
Answer:
(a) (iii) 1
(b) Given; f(x) = x² – 4x + 6 ⇒ f’(x) = 2x – 4
For turning points; f’(x) = 2x – 4 0 ⇒ x = 2
So volurn.,e is niaxirnum when h = 2r
The intervals are (- ∞, 2); (2, ∞)
f’(0) = 2 x 0 – 4 = -4
Therefore f(x) is decreasing in (- ∞, 2)
(c) f(x) = (2 x 1)² + 3
⇒ f’(x) 2(2x – 1) x 2 f”(x) = 8
For tuming points; f’(x) = 8x – 4 = 0 ⇒ x = 1/2
f(x) has minimum value at x = 1/2 minimum value is \(f\left(\frac{1}{2}\right)=3\)
2)

Question 3.
(a) Which of the following function has neither local maxima nor local minima?
(i) f(x) = x² + x
(ii) f(x) = logx
(iii) f(x) = x³ – 3x + 3
(iv) f(x) = 3 + |x|
(b) Find the equation of the tangent to the curve y = 3x² at (1,1). (March – 2016)
Answer:

Question 4.
(i) The slope of the normal to the curve, y = x³ – x² at (1, -1) is
(a) 1
(b) – 1
(c) 2
(d)0

(ii) Find the intervals in which the function f(x) = 2x³ – 24x + 25 is increasing or decreasing. (May – 2016)
Answer:
(i) (b) – 1
(ii) f(x) = 2x³ – 24x + 25
f’(x) = 6x² – 24
f’(x) = O
⇒ 6x² – 24 = 0 ⇒ x² = 4 ⇒ x = – 2,2
Therefore the intervals are (-∞, -2); (-2, 2); (2, ∞)
f(x) is increasing in the intervals (-∞, -2); (2, ∞)
f(x) is decreasing in the intervals (-2, 2)

Question 5.
(i) The slope of the normal to the curve, y² – 4x at (1,2) is
(a) 1
(b) 1/2
(c) 2
(d) – 1

(ii) Find the intervals in which the function 2x³ + 9x² + 12x – 1 is strictly increasing. (March – 2017)
Answer:
(i) (b) – 1
(ii) f(x) = 2x³ + 9x² + 12x – 1
f’(x) = 6x² + 18x + 12
= 6(x² + 3x + 2) = 6(x + 1) (x + 2)
f’(x) = O
⇒ 6(x + 1)(x + 2) = 0 ⇒ x = – 1 – 2
Therefore the intervals are
(- ∞, – 2); (- 2, – 1); (- 1, ∞)
In the ¡nterval (- ∞, – 2)
f’( – 3) = 6(- 3 + 1) (- 3 + 2) > 0
Therefore increasing In the interval (- 2, – 1)
f’(- 1.5) = 6(- 1.5 + 1)(- 1.5 + 2) < 0
Therefore decreasing In the interval (- 1, ∞)
f’(0) = 6(0 + 1)(0 + 2) > 0
Therefore increasing

Question 6.
Find two positive numbers whose sum is 16 and sum of whose cubes is minimum. (March – 2017)
Answer:
Let the numbers be x and 16 – x. Then,
S = x³ + (16 – x)³
= S’ = 3x² + 3(16 – x)²(- 1)
⇒ S” = 6x + 6(16 – x)………..(1)
For turning points S’ = 0 ⇒ 3 x² – 3(16 – x)² = 0
⇒ x² – 16 + 32x – x² =0
⇒ – 16² + 32x = 0 = x² = \(\frac{16 \times 16}{32}\) =8
(1) ⇒ S” = 6(8) + 6(16 – 8) > 0
TherefocemrnimumM x = 8
Thusthe numbers are8 and 16 – 8 = 8

Plus Two Maths Application of Derivatives 6 Marks Important Questions

Question 1.
(i) Show that the function x³ – 6x² + 15x + 4 is strictly increasing in R.
(ii) Find the approximate change in volume of a cube of side x meters caused by an increase in the side by 3%.
(iii) Find the equation of the tangent and normal at the point (1,2) on the parabola y² = 4x. (March – 2010)
Answer:
(i) Given; f(x) = x³ – 6x² + 15x + 4
f’(x) = 3x² – 12x + 15 = 3(x² – 4x +5)
= 3(x² – 4x + 4 + 1) = 3(x – 2)² + 1) > 0
For any value of x, f(x) is a strKly ¡ncreasing.

(ii) We have; V = x³ and Δx = 3% of x = 0.03x
\(d V=\frac{d V}{d x} \Delta x=3 x^{2} \Delta x\)
= 3x² x 0.03x = 0.09x³ = 0.09V
\(\Rightarrow \frac{d V}{V}=0.09\)

Therefore 9% is the approximate increase In volume.

(iii) Given; y²….4x ⇒ 2y \(\frac{d y}{d x}\) = 4 ⇒ \(\frac{d y}{d x}=\frac{2}{y}\)
Slope at (1,2) = \(\frac{2}{2}\) = 1
Equation of tangent at (1,2) is; y – 2 = 1(x – 1)
⇒ x – y + 1 = 0
Equation of normal at (1,2) is; y – 2 = – 1(x – 1)
⇒ x + y – 3 = 0

Question 2.
Consider the parametric forms
x = 1 + \(\frac{1}{t}\) – and y = t – \(\frac{1}{t}\) ofa curve
(i) Find \(\frac{d y}{d x}\)
(ii) Find the equation of the tangent at t = 2.
(iii) Find the equation of the normal at t = 2. (May – 2010)
Answer:

Question 3.
(i) The radius of a circle is increasing at the rate of 2cmls. Find the rate at which area of the circle is increasing when radius is
6cm.
(ii) Prove that the function f(x) = log sin x is strictly increasing in \(\left(0, \frac{\pi}{2}\right)\) and strictly decreasing in \(\left(\frac{\pi}{2}, \pi\right)\)
(iii) Find the maximum and minimum value of the function f(x) = x³ – 6x² + 9x + 15. (March – 2011)
Answer:

Question 4.
(i) Find the approximate value of (82)^1/4 up to three places of decimals using differentiation.
(ii) Find two positive numbers such that Their sum is 8 and the sum of their squares is minimum. (May – 2011)
Answer:

(ii) Let the numbers be x and 8 – x. Then,
S = x² + (8 – x)²
⇒ S’ = 2x + 2(8 – x)( – 1)
⇒ S” = 2 + 2 = 4 ………..(1)
For turning points S’ = 0 = 2x – 2(8 – x) = 0
⇒ 4x – 16 = 0 ⇒ x = 4
(1) ⇒ S” = 4 > 0
Therefore minimum at x = 4
Thus the numbers are 4 and 8 – 4 = 4.

Question 5.
(i) The slope of the tangent to the curve y = x³ – 1 at x = 2 is ……….
(ii) Use differentiation to approximate \(\sqrt{36.6}\)
(iii) Find two numbers whose sum is 24 and whose product as large as possible. (March – 2012, March – 2016)
Answer:

Therefore minimum at x =12
Thus the numbers are 12 and 24 – 12 = 12.

Question 6.
(i) Show that the function x³ – 3x² + 6x – 5 is strictly increasing on R.
(ii) Find the interval in which the function f(x) = sin x + cosx; 0 < x < 2π is strictly increasing or strictly decreasing. (May – 2012)
Answer:
(i) Given; f(x) = x³ – 3x² + 6x – 5
f’(x) = 3x² – 6x + 6 = 3(x² – 2x +2)
= 3(x² – 2x + 1 + 1) 3(x – 1)² + 1) > 0
For any value cit x, f(x) is a strictly increasing.

Question 7.
(i) Find the slope of the normal to the curve y = sinθ at θ = π/4
(ii) Show that the function f(x) = x³ – 6x² + 15x + 4 is strictly increasing in R.
(iii) Show that all rectangles with a given perimeter, the square has the maximum area. (March – 2013)
Answer:

(ii) f(x) = x³ – 6x² + 15x + 4
Differentiating w.r.t x;
f(x) = 3x² – 12x + 15 = 3(x² – 4x + 5)
= 3 (x² – 4x + 4 + 1)
= 3 ((x – 2)² + 1) > 0, ∀x∈R
Therefore fis strictly increasing in R.

(iii) Let x and ybe the length and breadth of a rectangle with area A and perimeter P.

Question 8.
A right circular cylinder is inscribed in a given cone of radius R cm and height H cm as shown in the figure.

(i) Find the Surface Area S of the circular cylinder as a function of x.
(ii) Find a relation connecting x and R when S is a maximum. (May – 2013)
Answer:
(i) There are two similar triangles ΔDJB and ΔDHF

Question 9.
(i) Which of the following function is Increasing for all values of x in its domain?
(a) sin x
(b) log x
(c) x²
(d) |x|

(ii) Find a point on the curve y = (x – 2)² at which the tangent is parallel to the chord joining the points (2,0) and (4,4).
(iii) Find the maximum profit that a company can make, if the profit function is given by p(x) = 41 – 24x – 6x². (March – 2014)
Answer:
(i) (b) log x
(ii) Given; y = (x – 2)² ⇒ \(\frac{d y}{d x}\) = 2(x – 2)
Slope of the chord = \(\frac{4-0}{4-2}=2\)
\(\Rightarrow 2=2(x-2) \Rightarrow x=3 \Rightarrow y=(3-2)^{2}=1\)
Therefore the required point is (3, 1)

(iii) Given; p(x) = 41 – 24x – 6x²
p’(x) = – 24 – 12x
p”(x) = – 12
For turning points p’(x) = – 24 – 12x = 0
⇒ x = -2
Since p”(x) = – 12 always maximum Therefore maximum value p(- 2) = 41 – 24(- 2) 6(- 2)² = 65

Question 10.
(a) Find the slope of the tangent to the parabola y² = 4ax at (at², 2at).
(b) Find the intervals in which the function x² – 2x + 5 is strictly increasing.
(c) A spherical bubble volume at the rate of which the diminishing when the is decreasing in 2cm³/sec. Find the surface area is radius is 3cm. (May – 2014)
Answer:

Question 11.
(a) Which of the following function is always increasing?
(i) x + sin 2x
(ii) x – sin 2x
(ill) 2x + sin 3x
(iv) 2x – sin 2x
(b) The radius of a cylinder is increasing at a rate of 1cm/s and its height decreasing at a rate of 1cm/s. Find the rate of change of its volume when the radius is 5cm and the height is 5cm.
(c) Write the equation of tangent at (1,1) on the curve 2x² + 3y² = 5. (March – 2015)
Answer:

Kerala State Board New Syllabus Plus Two Maths Chapter Wise Previous Questions and Answers Chapter 5 Continuity and Differentiability.

Kerala Plus Two Maths Chapter Wise Previous Questions Chapter 5 Continuity and Differentiability

Plus Two Maths Continuity and Differentiability 3 Marks Important Questions

Question 1.
Consider \(f(x)=\left\{\begin{array}{ll}
\frac{x^{2}-x-6}{x+2}, & x \neq-2 \\
-5, & x=-2
\end{array}\right.\)

(i) Find f(-2)
(ii) Check whether the function f(x) is continuous at x= -2. (March – 2009)
Answer:

Question 2.
If f(x) = sin(Log x), prove that x² y₂ + xy₁ + y = 0 (May -2009)
Answer:
Given; y sin(Iogx)
Differentiating with respect to X;
\(y_{1}=\cos (\log x) \frac{1}{x} \Rightarrow x y_{1}=\cos (\log x)\)
Again differentiating with respect to x
\(\begin{array}{l}
\Rightarrow x y_{2}+y_{1}=-\sin (\log x) \frac{1}{x} \\
\Rightarrow x^{2} y_{2}+x y_{1}=-y \Rightarrow x^{2} y_{2}+x y_{1}+y=0
\end{array}\)

Question 3.
(i) Establish that g(x) =1 – x + |x| is continuous at origin.
(ii) Check whether h(x) = |l – x + |x|| is continuous at origin. (March – 2010)
Answer:
(i) Given; g(x) = 1 – x + |x| ⇒ g(x) (1 – x) + |x|
Here g(x) is the sum of two functions continuous functions hence continuous.
(ii) We have;
\(\begin{array}{l}
f o g(x)=f(g(x)) \\
=\quad f(1-x+|x|)=|1-x+| x \mid=h(x)
\end{array}\)
The composition of two continuous functions is again continuous. Therefore h(x) continuous.

Question 4.
Find \(\frac{d y}{d x}\) of the following
\(\begin{array}{l}
\text { (i) } x=\sqrt{a^{\sin ^{4} 4}} \quad y=\sqrt{a^{\cos ^{-1} t}} \\
\text { (ii) } y=\cos ^{-1} \frac{\left(1-x^{2}\right)}{\left(1+x^{2}\right)}, 0<x<1 \\
\text { (iii) } y=\sin ^{-1} 2 x \sqrt{1-x^{2}}, y_{\sqrt{2}}<x<y_{\sqrt{2}}
\end{array}\)
Answer:

Question 5.
Find \(\frac{d y}{d x} \text { if } x^{3}+2 x^{2} y+3 x y^{2}+4 y^{3}=5\) (March – 2015)
Answer:

Question 6.
Find all points of discontinuity of f where f is defined by \(f(x)=\left\{\begin{array}{ll}
2 x+3, & x \leq 2 \\
2 x-3, & x>2
\end{array}\right.\) (March – 2016)
Answer:
In both the intervals x \(\leq[latex] 2 and x > 2 the function f(x) is a polynomial so continuous. So we havetocheckthe continuity at x = 2.

Question 7.
If e^x-y = x^y, then prove that [latex]\frac{d y}{d x}=\frac{\log x}{[\log \operatorname{ex}]}\) (May – 2014; March – 2016)
Answer:

Plus Two Maths Continuity and Differentiability 4 Marks Important Questions

Question 1.
Find \(\frac{d y}{d x}\) of the following (March – 2009)
\(\begin{array}{l}
\text { (i) } y=\sin ^{-1}\left(3 x-4 x^{3}\right)+\cos ^{-1}\left(4 x^{3}-3 x\right) \\
\text { (ii) } y=\tan ^{-1}\left(\sqrt{\frac{1-\cos x}{1+\cos x}}\right)
\end{array}\)
Answer:

Question 2.
Consider the function f(x) = |x| x ∈ R
(i) Draw the graph of f(x) =|x|
(ii) Show that the function is continuous at x = 0. (March – 2010)
Answer:

f(0^–) f(0^–) = 0. therefore continuous at x = 0.
AIso from the figure we can see that the graph does not have a break or jump.

Question 3.
(i) Find the derivative of y = x^a + a^x with respect to x.
(ii) If e^y (x + 1) = 1 , showthat \(\frac{d^{2} y}{d x^{2}}=\left(\frac{d y}{d x}\right)^{2}\) (May – 2011)
Answer:

Question 4.
(i) Check the continuity of the function given by f(x) \(f(x)=\left\{\begin{array}{ll}
x \sin \frac{1}{x}, & x \neq 0 \\
1, & x=0
\end{array}\right.\)

(ii) Verify Mean Value Theorem for the function f(x) = x + 1/x in the interval [1,3]. (May – 2011)
Answer:


Hence Mean Value Theorem ¡s verified.

Question 5.
(i) Determine the value of k so that the function (May – 2012)

Answer:

Question 6.
Consider a fUnction f: R → R defined by
\(f(x)=\left\{\begin{array}{cc}
a+x, & x \leq 2 \\
b-x, & x>2
\end{array}\right.\)

(i) Find a relation between a and b if f is continuous at x = 2.
(ii) Find a and b, if f is continuous at x2 and a + b = 2. (May – 2013)
Answer:
(i) Since fis continuous at x = 2, we have;

(ii) Given a = 2 …(2) Solving (1) and (2) we have;
⇒ 2a = – 2 ⇒ a = – 1
⇒ b = 2 – a = 2 + 1 = 3

Question 7.
(i) Find if x = a(t – sin t) y = a(1 + cos t)
(ii) Verify Rolles theorem for the function f(x) = x2 + 2 in the interval [-2, 2] (March – 2014)
Answer:

Question 8.
(a) Find the relationship between a and b so that the function f defined by
\(f(x)=\left\{\begin{array}{ll}
a x^{2}-1, & x \leq 2 \\
b x+3, & x>2
\end{array}\right.\) is continuous.

(b) Verify mean value theorem for the function f(x) = x2 – 4x -3 ¡n the interval [1, 4]. (May – 2014)
Answer:
(a) Since fis continuous

Hence mean value theorem satisfies for the funcion.

Question 9.
(a) Find ‘a’ and ‘b’ if the function
\(f(x)=\left\{\begin{array}{ll}
\frac{\sin x}{x}, & -2 \leq x \leq 0 \\
a \times 2^{x}, & 0 \leq x \leq 1 \\
b+x, & 1<x \leq 2
\end{array}\right.\) is continous on [-2, 2]

(b) How many of the functions
f(x) = |x|, g(x) = |x|², h(x) = |x|³ are not differentiable at x = 0?
(i) 0
(ii) 1
(iii) 2
(iv) 3 (March – 2015)
Answer:
(a) Since f(x) is continuous on [-2, 2]

Question 10.
(a) Find the relation between ‘a’ and ‘b’ if the function f defined by
\(f(x)=\left\{\begin{array}{l}
a x+1, x \leq 3 \\
b x+3, x>3
\end{array}\right.\) is continuous.
lbx+3.x>3
(b) If e^y (x + 1) = 1, show thats \(\frac{d^{2} y}{d x^{2}}=\left(\frac{d y}{d x}\right)^{2}\) (May -2015)
Answer:

Question 11.
Find the value of a and b such that the function \(f(x)=\left\{\begin{array}{cc}
5 a & x \leq 0 \\
a \sin x+\cos x & 0<x<\frac{\pi}{2} \\
b-\frac{\pi}{2} & x \geq \frac{\pi}{2}
\end{array}\right.\) is continuous. (March – 2010)
Answer:

Question 12.
(i) Find \(\frac{d y}{d x}, \text { if } x=a \cos ^{2} \theta ; y=b \sin ^{2} \theta\)
(ii) Find the second derivative of the function y = e^x sinx. (May – 2017)
Answer:

Question 13.
Find \(\frac{d y}{d x}\) of the following (4 score each)
(i) y^x = x^y (May – 2015)
(ii) (COSx)^y = (cosy)^x (March – 2017)
Answer:

Plus Two Maths Continuity and Differentiability 6 Marks Important Questions

Question 1.
Find \(\frac{d y}{d x}\) if
(i) sinx + cosy = xy
(ii) x = acos³t, y = asin³t
(iii) y = x^x + (logx)^x (May -2009; May -2011; March -2015)
Answer:
(i) Given; sinx + cosy = xy
Differentiating with respect to x;

Question 2.
(i) Let y =3 cos(log x) + 4 sin (log x)
(a) Find \(\frac{d y}{d x}\)
(b) Prove that x² y₂ + xy₁ + y = 0

(ii) (a) Find the derivative of y = e^2x+logx
(b) Find \(\frac{d y}{d x}\)
if x = a (θ – sinθ), y = a(1 – cosθ) (March – 2009)
Answer:

Question 3.
(i) Show that the function f (x) defined by f(x) = sin (cosx) is a continuous function.
(ii) If \(\frac{d y}{d x}=\frac{1}{\frac{d x}{d y}}\), Show that \(\frac{d^{2} y}{d x^{2}}=\frac{-\frac{d^{2} x}{d y^{2}}}{\left(\frac{d x}{d y}\right)^{3}}\) (May -2010)
Answer:
Given; f(x) = sin(cos x)
Let g(x) = sin(x) and h(x) = cos x
Both these function are trigonometric functions hence continuous.
goh(x) = g(h(x)) = g(cos x) = sin(cos x) = f(x)

Since f(x) is the composition of two continuous functions, hence continuous.

Question 4.
(i) Let y = x^{sin x} + (sinx)^x. Find \(\frac{d y}{d x}\)
(ii) Given; \(y=\sqrt{\tan ^{-1} x}\)
(a) \(2\left(1+x^{2}\right) y \frac{d y}{d x}=1\)
(b) \(\left(1+x^{2}\right) y \frac{d^{2} y}{d x^{2}}+\left(1+x^{2}\right)\left(\frac{d y}{d x}\right)^{2}+2 x y \frac{d y}{d x}=0\) (May – 2010; Onam – 2017)
Answer:

Question 5.
(i) The function \(f(x)=\left\{\begin{array}{ll}
5, & x \leq 2 \\
a x+b, 2< & x<10 \text { is } \\
21, & x \geq 10
\end{array}\right.\) continuous. Find a and b
(ii) Find \(\frac{d y}{d x}\) (a) if y = Sin (xsinx)
(iii) If y = ae” + be’; show that \(\frac{d^{2} y}{d x^{2}}-(m+n) \frac{d y}{d x}+m n y=0\) (March – 2011)
Answer:

Question 6.
(i) Match the following.

(ii) If y = sin^-1 x, prove that (1 – x²) y₂ – xy₁ = 0 (March – 2012; May -2017)
Answer:

Question 7.
(i) Consider \(f(x)=\left\{\begin{array}{ll}
3 x-8, & x \leq 5 \\
2 k, & x>5
\end{array}\right.\) Find the value of k if f(x) is continuous at x = 5.
(ii) Find \(\frac{d y}{d x}, \text { if } y=(\sin x)^{\log x}, \sin x>0\)
(iii) If y = (sin-1 x)2, then show that \(\left(1-x^{2}\right) \frac{d^{2} y}{d x^{2}}-x \frac{d y}{d x}=2\). (March -2013)
Answer:

Question 8.
(i) Find, if y = 1ogx, x>0
(ii) Is f(x) = |x| differentiable at x = 0?
(iii) Find if x = sin θ – cos θ and y= sinθ + cosθ (May – 2013)
Answer:

Kerala State Board New Syllabus Plus Two Maths Chapter Wise Previous Questions and Answers Chapter 4 Determinants.

Kerala Plus Two Maths Chapter Wise Previous Questions Chapter 4 Determinants

Plus Two Maths Determinants 3 Marks Important Questions

Question 1.
Prove that \(\begin{array}{|lll|}
1 ! & 2 ! & 3 ! \\
2 ! & 3 ! & 4 ! \\
3 ! & 4 ! & 5 !
\end{array}\) (March – 2015)
Answer:

Question 2.
Using properties of determinants prove the following. (March – 2010; Christmas -2017)
Answer:

Plus Two Maths Determinants 4 Marks Important Questions

Question 1.
Consider the matrix \(A=\left[\begin{array}{ll}
2 & 5 \\
3 & 2
\end{array}\right]\)

(i) Find adj (A)
(ii) Find A1
(iii) Using A^-1 solve the system of linear equations 2x + 5y = 13x + 2y = 7 (March – 2010)
Answer:

Plus Two Maths Determinants 6 Marks Important Questions

Question 1.
Consider the matrix \(A=\left[\begin{array}{lll}
a & b & c \\
b & c & a \\
c & a & b
\end{array}\right]\)

(i) Using the column operat,on
C₁ → C₁ + C₂ + C₃,
show that \(|A|=(a+b+c)\left|\begin{array}{ccc}
1 & b & c \\
1 & c & a \\
1 & a & b
\end{array}\right|\)
(ii) Show that |A| = – (a3 + b3 + e3 — 3abc)
(iii) Find A x adj(A) if a = 1,b = 10,c = 100 (May – 2010)
Answer:

Question 2.
(i) (a) If \(A=\left[\begin{array}{ccc}
1 & 1 & 5 \\
0 & 1 & 3 \\
0 & -1 & -2
\end{array}\right]\)

What is the value of |3A|?
(b) Find the equation of the line joining the points (1,2) and (-3,-2) using determinants.
(ii) Show that \(\left|\begin{array}{lll}
1 & a & a^{2} \\
1 & b & b^{2} \\
1 & c & c^{2}
\end{array}\right|=(a-b)(b-c)(c-a)\)
Answer:

(b) Let (x,y) be the coordinate of any point on The line, then (1,2), (-3, -2) and (x, y) are collinear.

Hence the area formed will be zero.

Question 3.
Consider the following system of linear equations; x + y + z = 6, x – y + z = 2, 2x + y + z = 1
(i) Express this system of equations in the Standard form AXB
(ii) Prove that A is non-singular.
(iii) Find the value of x, y and z satisfying the above equation. (May – 2011)
Answer:

Question 4.
(i) lf \(\left|\begin{array}{ll}
x & 3 \\
5 & 2
\end{array}\right|=5\), then x = ………..
(ii) Prove that
\(\left|\begin{array}{ccc}
y+k & y & y \\
y & y+k & y \\
y & y & y+k
\end{array}\right|=k^{2}(3 y+k)\)
(iii) Solve the following system of linear Equations, using matrix method; 5x + 2y = 3, 3x + 2y = 5 (March – 2012)
Answer:

Question 5.
(i) Let B is a square matrix of order 5, then |kB| is equal to ………..
(a) |B|
(b) k|B|
(c) k⁵|B|
(d) 5|B|

(ii) Prove that \(\left|\begin{array}{lll}
1 & x & x^{2} \\
1 & y & y^{2} \\
1 & z & z^{2}
\end{array}\right|=(x-y)(y-z)(z-x)\)
(iii) Check the consistency of the following equations; 2x + 3y + z = 6, x + 2y – z = 2, 7x + y + 2z =10 (May – 2012)
Answer:

Therefore the system is consistent and has unique solutions.

Question 6.
(i) Find the values of x in which \(\left|\begin{array}{ll}
3 & x \\
x & 1
\end{array}\right|=\left|\begin{array}{ll}
3 & 2 \\
4 & 1
\end{array}\right|\)

(ii) Using the property of determinants, show that the points A(a,b + c), B(b,c + a), C(c,a + b) are collinear.
(iii) Examine the consistency of system of following equations: 5x – 6y + 4z = 15, 7x + y – 3z = 19, 2x + y + 6z = 46 (EDUMATE – 2017; March – 2013)
Answer:

Since, the system is consistent and has unique solutions.

Question 7.
Consider a system of linear equations which is given below;
\(\begin{array}{l}
\frac{2}{x}+\frac{3}{y}+\frac{10}{z}=4 ; \frac{4}{x}-\frac{6}{y}+\frac{5}{z}=1 \\
\frac{6}{x}+\frac{9}{y}-\frac{20}{z}=2
\end{array}\)

(i) Express the above equation in the matrix form AX = B.
(ii) Find A^-1, the inverse of A.
(iii) Find x,y and z. (May – 2013)
Answer:

Question 8.
Consider the matrices \(A=\left[\begin{array}{ll}
2 & 3 \\
4 & 5
\end{array}\right]\)

(i) Find A² – 7A – 21 = 0
(ii) Hence find A-1
(iii) Solve the following system of equations using matrix method 2x + 3y = 4; 4x + 5y = 6 (March – 2014)
Answer:

(iii) The given system of equations can be converted into matrix form AX = B

Question 9.
(i) Let A be a square matrix of order 2 x 2 then |KA| is equal to
(a) K|A|
(b) K²|A|
(c) K³|A|
(d) 2K|A|

(ii) Prove that
\(\left|\begin{array}{ccc}
\mathbf{a}-\mathbf{b}-\mathbf{c} & \mathbf{2 a} & 2 \mathbf{a} \\
2 \mathrm{~b} & \mathrm{~b}-\mathrm{c}-\mathrm{a} & 2 \mathrm{~b} \\
2 \mathrm{c} & 2 \mathrm{c} & \mathrm{c}-\mathrm{a}-\mathrm{b}
\end{array}\right|=(\mathrm{a}+\mathrm{b}+\mathrm{c})^{3}\)

(iii) Examine the consistency of the system of Equations. 5x + 3y = 5; 2x + 6y = 8 (May- 2014)
Answer:

(iii) The given system of equation can be written in matrix form as

solution exist and hence it is consistent.

Question 10.
(a) Choose the correct statement related to the matnces \(A=\left[\begin{array}{ll}
1 & 0 \\
0 & 1
\end{array}\right], B=\left[\begin{array}{ll}
0 & 1 \\
1 & 0
\end{array}\right]\)
\(\begin{array}{l}
\text { (i) } A^{3}=A, B^{3} \neq B \\
\text { (ii) } A^{3} \neq A, B^{3}=B \\
\text { (iii) } A^{3}=A, B^{3}=B \\
\text { (iv) } A^{3} \neq A, B^{3} \neq B
\end{array}\)

(b) lf \(M=\left[\begin{array}{ll}
7 & 5 \\
2 & 3
\end{array}\right]\) then verity the equation M² – 10M + 11 I₂ = O

(c) Inverse of the matrix \(\left[\begin{array}{lll}
0 & 1 & 2 \\
0 & 1 & 1 \\
1 & 0 & 2
\end{array}\right]\) (March – 2015)
Answer:

Question 11.
Solve the system of Linear equations x + 2y + z = 8; 2x + y – z = 1; x – y + z = 2 (March – 2015)
Answer:

Question 12.
(a) If \(\left|\begin{array}{ll}
x & 1 \\
1 & x
\end{array}\right|=15\) then find the value of X.

(b)Solve the following system of equations 3x – 2y + 3z = ?, 2x + y – z = 1 4x – 3y + 2z = 4 (May – 2015)
Answer:

Question 13.
(i)The value of the determinant \(\left|\begin{array}{ccc}
1 & 1 & 1 \\
1 & -1 & -1 \\
1 & 1 & -1
\end{array}\right|\) is
(a) -4
(b) 0
(c) 1
(d) 4

(ii) Using matrix method, solve the system of linear equations x + y + 2z = 4; 2x – y + 3z = 9; 3x – y – z = 2 (May – 2016)
Answer:
(i) (d) 4
(ii) Express the given equation in the matrix form as AX = B

Question 14.
(i) If \(A=\left[\begin{array}{ll}
a & 1 \\
1 & 0
\end{array}\right]\) is such that A² = I then a equals
(a) 1
(b) -1
(c) 0
(d) 2

(ii)Solve the system of equations x – y + z = 4; 2x + y – 3z = 0; x + y + z = 2 Using matrix method. (March – 2017)
Answer:

Question 15.
(i) IfA is a 2 x 2 matrix with |A| = 5, then |adjA| is
(a) 5
(b) 25
(c) 1/5
(d) 1/25

(ii) Solve the system of equations using matrix method.
x + y + z = 1; 2x + 3y – z = 6; x – y + z = -1 (May – 2017)
Answer:
(i) (a) 5
(ii) LetA X=B,

Kerala State Board New Syllabus Plus Two Maths Chapter Wise Previous Questions and Answers Chapter 3 Matrices.

Kerala Plus Two Maths Chapter Wise Previous Questions Chapter 3 Matrices

Plus Two Maths Matrices 3 Marks Important Questions

Question 1.
Write A as the sum of a symmetric and a skew-symmetric matrix. \(A=\left[\begin{array}{ccc}
1 & 4 & -1 \\
2 & 5 & 4 \\
-1 & -6 & 3
\end{array}\right]\) (March – 2010)
Answer:

Question 2.
Consider the matrices
\(A=\left[\begin{array}{lll}
2 & 1 & 3 \\
2 & 3 & 1 \\
1 & 1 & 1
\end{array}\right] \text { and } B=\left[\begin{array}{ccc}
-1 & 2 & 3 \\
-2 & 3 & 1 \\
-1 & 1 & 1
\end{array}\right]\)
(i) Find A+B
(ii) Find (A + B) (A-B) (May -2010)
Answer:

Question 3.
Given \(P=\left[\begin{array}{cc}
2 & -3 \\
-1 & 2
\end{array}\right]\) Find the inverse of P by elementary row operation. (March 2011)
Answer:

Question 4.
Let \(A=\left[\begin{array}{lll}
3 & 6 & 5 \\
6 & 7 & 8
\end{array}\right] \text { and } C=\left[\begin{array}{ccc}
1 & 2 & -3 \\
4 & 5 & 6
\end{array}\right]\)

(i) Find 2A
(ii) Find the matrix B such that 2A + B = 3C (May 2011)
Answer:

Question 5.
Let \(A=\left[\begin{array}{cc}
2 & 4 \\
-1 & 1
\end{array}\right]\)
(i) Apply elementary transformation R → R R1/2 in the matrix A.
(ii) Find the inverse of A by the elementary transformation. (May 2011)
Answer:

Question 6.
Consider the matrix \(A=\left[\begin{array}{cc}
3 & 1 \\
-1 & 2
\end{array}\right]\)
(i) Find A²
(ii) Find ksothat A2 = kA – 7I (March – 2012)
Answer:

Question 7.
Consider a 2×2 matrix
\(A=\left[a_{i j}\right]$ where $a_{i j}=|2 i-3 j|\)
(i) Write A
(ii) Find A + A^T (March – 2012)
Answer:

Question 8.
If \(A=\left[\begin{array}{cc}
3 & 1 \\
-1 & 2
\end{array}\right]\) then
(i) Find A²
(ii) Hence show that A² – 5A + 7I = 0. (March 2013)
Answer:

Question 9.
If a matrix \(A=\left[\begin{array}{ll}3 x & x \\ -x & 2 x\end{array}\right]\) is a solution of the equation x² – 5x + 7 = 0, find any one value of X. (May 2013)
Answer:

Question 10.
Consider the matrices \(A=\left[\begin{array}{cc}1 & -2 \\ -1 & 3\end{array}\right]$ and $B=\left[\begin{array}{ll}a & b \\ c & d\end{array}\right]$\) \(A B=\left[\begin{array}{ll}2 & 9 \\ 5 & 6\end{array}\right]\), find the values of a,b,c,d (March – 2014)
Answer:

Question 11.
Consider a 2 x 2 matrix A=[a_ij] Where \(a_{i j}=\frac{(i+2 j)^{2}}{2}\)
(i) Write A
(ii) Find A + A^T (March – 2014)
Answer:

Question 12.
If X + Y = \(\left[\begin{array}{ll}7 & 0 \\ 2 & 5\end{array}\right]\) and X – Y = \(\left[\begin{array}{ll}3 & 0 \\ 0 & 3\end{array}\right]\) then
(i) Find X and Y.
(ii) Find 2X + Y. (May – 2014)
Answer:

Question 13.
i) If A, B are symmetric matrices of same order then AB – BA is always a ………….
A) Skew-Symmetric matrix
B) Symmetric matrix
C) Identity matrix
D) Zero matrix
(ii) For the matrix \(A=\left[\begin{array}{ll}2 & 4 \\ 5 & 6\end{array}\right]\), verify that A + A^T is a symmetric matrix. (March – 2015)
Answer:

Question 14.
Consider the matrix \(A=\left[\begin{array}{ll}3 & -2 \\ 4 & -2\end{array}\right]\)
(i) Find A²
(ii) Find k so that A² = kA – 21 (May – 2015)
Answer:

Plus Two Maths Matrices 4 Marks Important Questions

Question 1.
(i) Find the value of x and y from the equations \(a\left[\begin{array}{cc}x & 5 \\ 7 & y-3\end{array}\right]+\left[\begin{array}{cc}3 & -4 \\ 1 & 2\end{array}\right]=\left[\begin{array}{cc}7 & 6 \\ 15 & 14\end{array}\right]\)
(ii) Given \(A=\left[\begin{array}{cc}1 & 2 \\ 3 & -1 \\ 4 & 2\end{array}\right], B=\left[\begin{array}{ccc}-1 & 4 & -5 \\ 2 & 1 & 0\end{array}\right]\) Show that AB ≠ BA (March – 2011)
Answer:

Question 2.
(i) Find a, b matrix \(\left[\begin{array}{ccc}0 & 3 & a \\ b & 0 & -2 \\ 5 & 2 & 0\end{array}\right]\) is skew symmetric matrix.
(ii) Express \(A=\left[\begin{array}{ccc}7 & 3 & -5 \\ 0 & 1 & 5 \\ -2 & 7 & 3\end{array}\right]\) sum of a symmetric and a skew symmetric matrix. (May – 2012)
Answer:

Question 3.
Consider the matrices \(A=\left[\begin{array}{cc}2 & -6 \\ 1 & 2\end{array}\right]$ and $A+3 B=\left[\begin{array}{cc}5 & -3 \\ -2 & -1\end{array}\right]\)
(i) Find matrix B
(il) Find matrix AB.
(iii) Find the transpose of B. (May – 2013)
Answer:

Question 4.
(i) The value of k such that matrix \(\left[\begin{array}{cc} 1 & k \\ -k & 1 \end{array}\right]\) is symmetric if
(a) 0
(b) 1
(c) – 1
(d) 2

(ii) If \(A=\left[\begin{array}{cc}
\cos \theta & \sin \theta \\
-\sin \theta & \cos \theta
\end{array}\right]\) then prove that \(A^{2}=\left[\begin{array}{cc}
\cos 2 \theta & \sin 2 \theta \\
-\sin 2 \theta & \cos 2 \theta
\end{array}\right]\)
\(\text { (iii) if } A=\left[\begin{array}{ll}
1 & 3 \\
4 & 1
\end{array}\right], \text { then find }\left|3 A^{T}\right|\) (March – 2017)
Answer:

Plus Two Maths Matrices 6 Marks Important Questions

Question 1.
Let A be a matrix of order 3 x 3 whose elements are given by a_ij = 21 – j
(i) Obtain the matrix A.
(ii) Find AT Also express A as the sum of symmetric and skew-symmetric matrix. (March – 2010)
Answer:

Question 2.
Consider a 2 x 2 matrix \(A=\left[a_{\theta}\right]\) with a_ij = 2ⁱ + j
(i) Construct A.
(ii) Find A + A^T, A – A^T
(iii) Express A as sum of a symmetric and skew-symmetric matrix. (May -2015)
Answer:

Question 3.
(i) \(A=\left[\begin{array}{ll}
0 & 1 \\
0 & 0
\end{array}\right], B=\left[\begin{array}{ll}
1 & 0 \\
0 & 0
\end{array}\right]\) then BA = _____
\(\begin{array}{l}
\text { (a) }\left[\begin{array}{ll}
1 & 0 \\
0 & 1
\end{array}\right] & \text { (b) }\left[\begin{array}{ll}
0 & 1 \\
1 & 0
\end{array}\right] \\
\text { (c) }\left[\begin{array}{ll}
0 & 1 \\
0 & 0
\end{array}\right] & \text { (d) }\left[\begin{array}{ll}
0 & 0 \\
0 & 0
\end{array}\right]
\end{array}\)

(ii) Write \(A=\left[\begin{array}{cc}
3 & 5 \\
1 & -1
\end{array}\right]\) as the sum of a symmetric and a skew symmetric matrix.
(iii) Find the inverse of \(A=\left[\begin{array}{ll}
2 & -6 \\
1 & -2
\end{array}\right]\) (March 2016)
Answer:

Question 4.
(i) If the matrix A is both symmetric and skew-symmetric, then A is a
(a) diagonal matrix
(b) zero matrix
(c) square matrix
(d) scalar matrix

(ii) If \(A=\left[\begin{array}{cc}
1 & 3 \\
-2 & 4
\end{array}\right]\), then show that

(iii) Hence find A^-1 (May 2016)
Answer:

Question 5.
(i) The number of all possible 2 x 2 matrices with entries O or 1 is
(a) 8
(b) 9
(c) 16
(d) 25

(ii) If the area of a triangle whose vertices are (k,0), (5,0), (0,1) is 10 square units the find k.
(iii) Using elementary transformations find the inverse of the matrix \(\left[\begin{array}{ll}
2 & 1 \\
1 & 1
\end{array}\right]\) (May 2017)
Answer:

Kerala State Board New Syllabus Plus Two Maths Chapter Wise Previous Questions and Answers Chapter 2 Inverse Trigonometric Functions.

Kerala Plus Two Maths Chapter Wise Previous Questions Chapter 2 Inverse Trigonometric Functions

Plus Two Maths Inverse Trigonometric Functions 3 Marks Important Questions

Question 1.
(i) Find the principal value of \(\cos ^{-1}\left(\frac{\sqrt{3}}{2}\right)\)
(ii) Prove that \(2 \sin ^{-1}\left(\frac{3}{5}\right)=\tan ^{-1}\left(\frac{24}{7}\right)\) (March – 2010)
Answer:

Question 2.
Prove \(2 \tan ^{-1} \frac{1}{2}+\tan ^{-1} \frac{1}{7}=\tan ^{-1} \frac{31}{17}\) (March – 2015; March – 2016)
Answer:

Plus Two Maths Inverse Trigonometric Functions 4 Marks Important Questions

Question 1.
(i) Find the principal value of \(\cos ^{-1}\left(-\frac{1}{2}\right)\)
(ii) Show that \(\left(\frac{\cos x}{1-\sin x}\right)=\frac{\pi}{4}+\frac{x}{2}\) (March-2011)
Answer:

Question 2.
(i) The principal value of \(\cos ^{-1}\left(-\frac{1}{2}\right)\)
(ii)Expresstan \(\tan ^{-1}\left(\frac{\cos x}{1-\sin x}\right)\) in the simplest Form. (May – 2012)
Answer:

Question 3.
(i) Write the principal value of \(\sin ^{-1}\left(\frac{1}{2}\right)\)
(ii) Show that \(\sin ^{-1}\left(\frac{3}{5}\right)-\sin ^{-1}\left(\frac{8}{17}\right)=\cos ^{-1}\left(\frac{84}{85}\right)\) (March – 2013, Onam – 2017)
Answer:

Question 4.
(a) The principal value of tan’1 (-1) is \(\left[\frac{\pi}{4},-\frac{\pi}{4}, \quad \pi-\frac{\pi}{4}, \quad \pi+\frac{\pi}{4}\right]\)
(b) If \(\tan ^{-1}\left(\frac{x-1}{x-2}\right)+\tan ^{-1}\left(\frac{x+1}{x+2}\right)=\frac{\pi}{4}\) then find the value of x. (May 2014)
Answer:

Plus Two Maths Inverse Trigonometric Functions 6 Marks Important Questions

Question 1.
Match the following. (1 + 1 + 1 + 3 = 6) (May 2010)

Answer:

Question 2.
(i) Give an expression for tan(x + y)
(ii) Prove that xy < 1, \(\tan ^{-1} x+\tan ^{-1} y=\tan ^{-1}\left(\frac{x+y}{1-x y}\right)\)
(iii) Using the above result prove that \(\tan ^{-1} \frac{1}{2}+\tan ^{-1} \frac{1}{3}=\frac{\pi}{4}\) (May 2011)
Answer:

Question 3.
(i) Show that \(\tan ^{-1} \frac{1}{5}+\tan ^{-1} \frac{1}{7}+\tan ^{-1} \frac{1}{3}+\tan ^{-1} \frac{1}{8}=\frac{\pi}{4}\)
(ii) Given that \(\cot 3 \theta=\frac{3 \cot ^{2} \theta-1}{\cot ^{3} \theta-3 \cot \theta}\), Show that \(\cot ^{-1} \frac{3 x^{2}-1}{x^{3}-3 x}, \quad|x|<\sqrt{3}$ is $3 \cot ^{1} x\) (May 2013 )
Answer: