Plus Two

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

Kerala Plus Two Maths Previous Year Question Paper March 2018 with Answers

Board	SCERT
Class	Plus Two
Subject	Maths
Category	Plus Two Previous Year Question Papers

Time : 2 1/2 Hours
Cool off time : 15 Minutes
Maximum : 80 Score

General Instructions to Candidates :

• There is a ‘Cool off time’ of 15 minutes in addition to the writing time.
• Use the ‘Cool off time’ to get familiar with questions and to plan your answers.
• Read questions carefully before you answering.
• Read the instructions carefully.
• 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.

Question 1 to 7 carry 3 scores each. Answer any six questions. (6 × 3 = 18)

Question 1.
If f(x) = \(\frac{x}{x-1}\), x ≠ 1
i) Find fof (x)
ii) Find the inverse of f.
Answer:

Question 2.
Using elementary row operations, find the inverse of the matrix \(\left[\begin{array}{cc}
1 & 2 \\
2 & -1
\end{array}\right]\)
Answer:

Question 3.
i) f(x) is a strictly increasing function, if f'(x) is ………..
a) positive
b) negative
c) 0
d) None of these,
ii) Show that the function f given by f(x) = x³ – 3x² + 4x, x ∈ R is strictly increasing.
Answer:
i) a) Positive.
ii) f(x) = x³ – 3x² + 4x
= 3 (x² – 2x + 1 – 1 + \(\frac{4}{3}\)) = 3((x + 1)2 + \(\frac{1}{3}\)) > 0

Question 4.
i) \(\int_{0}^{a}\)f(a – x)dx = …………

Answer:

Question 5.
Find the area of the region bounded by the curve y² = x’, x-axis and the lines x = 1 and x = 4.
Answer:

Question 6.
Find the general solution of the differential equation x \(\frac{d y}{d x}\) + 2y = x² log x

Solution is

Question 7.
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 1 hour on Machine B to produce a package of bolts. He earns profit of Rs. 17.50 per package on nuts and Rs. 7.00 per package on bolts. Formulate the above LPp if the Machine operates for at most 12 hours a day.
Answer:
Let x packet of nuts and y packets of bolts.
Maximise: Z=17. 5x + 7y
Subject to
x + 3y ≤ 12; 3x + y ≤ 12; x, y ≥ O

Questions 8 to 17 carry 4 scores each. Answer any 8. (8 × 4 = 32)

Question 8.
Let A = N × N and ‘*’ be a binary operation on A defined by
(a, b)*(c, d)=(a + c, b + d)
i) Find (1, 2) * (2, 3)
ii) Prove that * is commutative.
iii) Prove that * is associative.
Answer:
i) (1, 2)*(2, 3)=(1 + 2, 2 + 3) = (3, 5)
ii) (c, d)*(a, b) = (c + a, d + b)
= (a + c, b + d) = (a, b)* (c, d)
iii) (a, b)*[(c, d)*(e, f)] = (a, b)*(c + e, d + f)
= (a + c + e, b + d + f)
[(a, b) * (c, d)] * (e, f) = (a + c, b + d) * (e, f)
= (a + c + e, b + d + f)

Question 9.
i) Identify the function from the above graph

a) tan^-1x
b) sin^-1x
c) cos^-1x
d) cos ec^-1x
ii) Find the domain and range of the function represented in above graph.
iii) Prove that \(\tan ^{-1} \frac{1}{2}+\tan ^{-1} \frac{2}{11}=\tan ^{-1} \frac{3}{4}\)
Answer:
i) sin^-1x

Question 10.
i) \(\frac{d\left(a^{x}\right)}{d x}\) = ………..
a) a^x
b) log(a^x)
c) a^x log a
d) xa^x-1
ii) Find \(\frac{d y}{d x}\) if x^y = y^x
Answer:
i) a^x loga
ii) Given; y^x = x^y, taking log on both sides;
x log y = y log x,
Differentiating with respect to x;

Question 11.
i) Find the slope of the tangent to the curve y = (x – 2)² at x = 1.
ii) Find a point at which the tangent to the curve y = (x – 2)² is parallel to the chord joining the point A (2, 0) and B (4, 4)
iii) Find the equation of the tangent to the above curve and parallel to the line AB.
y² = 4ax, a > 0 and x² = 4ay, a > 0
Answer:
i) \(\frac{d y}{d x}\) = 2(x – 2) ⇒ Slope = 2(1 – 2) = -2
ii) Slope of AB = \(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}=\frac{4-0}{4-2}=2\)
Here 2(x – 2) = 2 ⇒ x = 3, y = 1
iii) Equation of the tangent line is
y – 1 = 2 (x – 3) ⇒ 2x – y = 5

Question 12.
\(\int_{0}^{2}\)(x² +1)dx as the limit of a sum.
Answer:
Here; f(x) = x² + 1; a = 0, b = 2 ⇒ h = \(\frac{2}{n}\)
\(\int_{0}^{2}\)(x² +1)dx
\(\lim _{h \rightarrow 0}\) h[f(0) + f(0 + h) + f(0 + 2h)+…+ f(0 + (n – 1)h)]
\(\lim _{h \rightarrow 0}\) h (1 + (h² + 1)+((2h) ² + 1)+…+[(n – 1)h²] + 1]

Question 13.
Consider the following figure:

i) Find the point of intersection ‘P’ of the circle x² + y² = 50 and the line y = x
ii) Find the area of the shaded region.
Answer:
i) point of intersection is x² + x² = 50 ⇒ x² = 25 ⇒ ±5; y = ±5
point P is (5, 5)

Question 14.
i) The degree of the differential equation \(x y\left(\frac{d^{2} y}{d x^{2}}\right)^{2}+x^{4}\left(\frac{d y}{d x}\right)^{3}-y \frac{d y}{d x}=0\) is ………
a) 4
b)3
c)2
d)1
ii) Find the general solution of the differential equation
sec² x tan ydx + sec² y tan xdy = 0
Answer:
i) c
ii) Given; sec²x tan ydx + sec²y tan xdy = 0

log tanx = -log tany + c
log tan x + log tan y = c

Question 15.
i) Prove that for any vector \(\bar{a}, \bar{b}, \bar{c}\)
\([\bar{a}+\bar{b}, \bar{b}+\bar{c}, \bar{c}+\bar{a}]=2\left[\begin{array}{lll}
\bar{a} & \bar{b} & \bar{c}
\end{array}\right]\)
ii) Show that if \(\bar{a}+\bar{b}, \bar{b}+\bar{c}, \bar{c}+\bar{a}\) are coplanar then \(\bar{a}, \bar{b}, \bar{c}\) are also coplanar.
Answer:

Question 16.
i) Find the equation of a plane which makes x,y,z intercepts respectively as 1, 2, 3.
ii) Find the equation of a plane passing through the point (1, 2, 3) which is parallel to above plane.
Answer:
\(\frac{x}{1}+\frac{y}{2}+\frac{z}{3}\) ⇒ 6x + 3y + 2z = 6
ii) Equation of the parallel plane passing
6(x – 1) + 3( y – 2) + 2(z – 3) = 0
⇒ 6x + 3y + 2z – 18 = 0

Question 17.
Solve the LPP graphically:
Minimise z = -3x + 4y
Subject to constraints:
x + 2y ≤ 8; 3x + 2y ≥ l2; x, y ≥ 0
Answer:
x + 2y = 8

X	0	8
Y	4	0

3x + 12y = 12

X	0	4
Y	6	0

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

Corner point	Z = -3x + 4y
O(0, 0)	Z = 0 + 0 = 0
A(4, 0)	Z = -12 + 0 = -12
B(2, 3)	Z = -6 + 12 = 6
C(0, 4)	Z = 0 + 16 = 16

Z attains minimum at (4,0).

Questions from 18 to 24 carry 6 scores each. Answer any 5. (5 × 6 = 30)

Question 18.
i) Find x and y if \(x\left[\begin{array}{l}
2 \\
3
\end{array}\right]+y\left[\begin{array}{c}
-1 \\
1
\end{array}\right]=\left[\begin{array}{c}
10 \\
5
\end{array}\right]\)
ii) Express the matrix \(\left[\begin{array}{ccc}
2 & -2 & -4 \\
-1 & 3 & 4 \\
1 & -2 & -3
\end{array}\right]\) as the sum of a symmetric and a skew symmetric matrices.
Answer:
Given; 2x – y = 10; 3x + y = 5
5x = 15 ⇒ x = 3; y = -4

Question 19.
i) Prove that \(\left|\begin{array}{ccc}
a & b & c \\
a+2 x & b+2 y & c+2 z \\
x & y & z
\end{array}\right|=0\)
ii) If A = \(\left[\begin{array}{ccc}
1 & -1 & 2 \\
0 & 2 & -3 \\
3 & -2 & 4
\end{array}\right]\), B = \(\left[\begin{array}{ccc}
-2 & 0 & 1 \\
9 & 2 & -3 \\
6 & 1 & -2
\end{array}\right]\)
a) Prove that B = A^-1
b) Using Asolve the system of linear equations given below:
x – y + 2z = 1; 2y – 3z = 1; 3x – 2y + 4z = 2
Answer:

iii) Matrix form of the system of linear equation is

Question 20.
i) Prove that the function defined by f(x) = cos x² is a continuous function.
ii) a) If y = \(e^{a \cos ^{-1} x}\), -1 ≤ x ≤ 1, show that
\(\frac{d y}{d x}=\frac{-a e^{a \cos ^{-1} x}}{\sqrt{1-x^{2}}}\)
b) Hence prove that
\(\left(1-x^{2}\right) \frac{d^{2} y}{d x^{2}}-x \frac{d y}{d x}-a^{2} y=0\)
Answer:
i) f(x) = cosx; g(x) = x² both are continuous.
Composition two continuous function is again continuous.
fog(x) = f(g(x)) = cos(x²)

Question 21.
Evaluate the following:

Answer:

Question 22.
i) lf \(\bar{a}\) = 3i + 2j + 2k, \(\bar{b}\) = i + 2j – 2k
a) Find \(\bar{a}\) + \(\bar{b}\), \(\bar{a}\) – \(\bar{b}\)
b) Find a unit vector perpendicular to both \(\bar{a}\) + \(\bar{b}\), \(\bar{a}\) – \(\bar{b}\)
ii) Consider the points A(1, 2, 7); B(2, 6, 3); C (3, 10, -1)
a) Find \(\overline{A B}, \overline{B C}\)
b) Prove that A, B, C are collinear points.
Answer:

Question 23.
i) Find the angle between the lines \(\frac{x-2}{2}=\frac{y-1}{5}=\frac{z+3}{-3}\) and \(\frac{x+2}{-1}=\frac{y-4}{8}=\frac{z-5}{4}\)
ii) Find the shortest distance between the pair of lines
\(\bar{r}\) = (i + 2j + 3k) + λ(i – 3j + 2k)
\(\bar{r}\) = (4i + 5j +6k) + μ(2i + 3j + k)
Answer:

Question 24.
i) The probability distribution of a random variable is given by P(x). What is Σp(x)?
ii) The following is a probability distribution function of a random variable.

X	-5	-4	-3	-2	-1	0
P(x)	k	2k	3k	4k	5k	7k
X	1	2	3	4	5
P(x)	8k	9k	10k	11k	12k

a) Find k
b) P(x > 3)
c) P(-3 < x < 4)
d) P(x < -3)
Answer:
Σp(x) = 1

ii) a) k + 2k + 3k + 4k + 5k + 7k
+ 8k + 9k + 10k + 11k + 12k = 1
72k = 1 ⇒ k = \(\frac{1}{72}\)
b) P(x > 3) = P(4) + P(5) = 23k = \(\frac{23}{72}\)
c) P(-3 < x < 4) = P(-2) + P(-1) + P(0) + P(1) + P > (2)+ P > (3) = 43k = \(\frac{43}{72}\)
d) P(x < -3) = P(-5) + P(-4) = 3k = \(\frac{3}{72}\)

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Kerala Plus Two Maths Model Question Paper 2

Time : 2 1/2 Hours
Cool off time : 15 Minutes
Maximum : 80 Score

General Instructions to Candidates :

• There is a ‘Cool off time’ of 15 minutes in addition to the writing time.
• Use the ‘Cool off time’ to get familiar with questions and to plan your answers.
• Read questions carefully before you answering.
• Read the instructions careully.
• 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.

QUESTIONS

Question 1 to 7 carry 3 scores each. Answer any six questions only

Question 1.

Question 2.

Question 3.
Consider the differential equation \(\frac { { d }^{ 2 }{ y } }{ { dx }^{ 2 } } +y=0\)
a. write its order and degree
b. Verify that y = a cos x + b sin x, where a, b ∈ R is a solution of the given differential equation

Question 4.
Consider the differential equation x \(\frac { dy }{ dx } \) + 2y = x2 ; x ≠ 0
a. What is its integrating factor?
b. Obtain its general solution.

Question 5.
The foot of the perpendicular drawn from origin to a plane is (4,-2, 5).
a. How far is the plane from the origin?
b. Find a unit vector perpendicular to that plane.
c. Obtain the equation of the plane in general form.

Question 6.

Question 7.

Question 8 to 17 carry 4 scores each. Answer any eight questions only

Question 8.

Question 9.

Question 10.

Question 11.
Consider the points A (2, 2, -1), B (3, 4, 2) and C (7, 0, 6). Find the vector and Cartesian equation of the plane passing through these points.

Question 12.
a. Find the cartesian equation of the plane through the point (1,2,-3) and perpen-dicular to the vector 2\(\widehat { i } \) – \(\widehat { j } \) + 2\(\widehat { k } \).
b. Find the angle between the above plane and the line \(\frac { x-1 }{ 2 } \) = \(\frac { y-3 }{ 3 } \) = \(\frac { z }{ 6 } \)

Question 13.
a. Let R be the relation on the set N of . natural numbers given by R = {(a,b): a – b > 2, b > 3}.
Choose the correct answer.
A. (4,1) ∈ R
B. (5,8) ∈ R
C. (8,7) ∈ R
D.(10,6) ∈ R

b. If f (x) = 8x³ and g (x) = x^1/3, find g (f(x)) and f (g (x)).

c. Let * be a binary operation on the set Q of rational numbers defined by a*b= \(\frac { ab }{ 3 } \).

Check whether * is commutative and associative?

Question 14.

X	1	2	3	4	5
P(X)	1/2	1/4	1/8	1/16	P

The probability distribution of a random variable X taking values 1, 2, 3, 4, 5 is given.
a. Find the value of P.
b. Find the mean of X.
c. Find the variance of X.

X	1	2	3	4	5
P(X)	1/2	1/4	1/8	1/16	P

Question 15.
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,

Question 16.
Consider the functions: f (x) = |x|-1 and g(x) = 1- | x|
a. Sketch their graphs and shade the closed region between them.
b. Find the area of their shaded region.

Question 17.
a. What is the value of sin^-1 (sin 160°)?

Question 18 to 24 carry 6 scores each. Answer any 5 questions only

Question 18.

Question 19.
integrate the following :

Question 20.
a. Choose the correct statement related to the matrices A = \(\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}\) and B = \(\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}\)

Question 21.
a. Which of the following functions is always increasing?
i. x + sin 2x
ii. x – sin 2x
iii. 2x + sin 3x
iv. 2x – sin x
b. The radius of a cylinder increases at a rate of 1 cm/s and its height decreases at a rate of 1 cm/s. Find the rate of change of its volume when the radius is 5 cm and the height is 15 cm.
If the volume should not change even when the radius and height are changed, what is the relation between the radius and height?
c. Write the equation of tangent at (1,1) on the curve 2x² + 3y² = 5.

Question 22.
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
a. Draw its feasible region.
b. Find the vertices of the feasible region.
c. Find the minimum value of Z subject to the given constraints.
Minimize Z = 3x + 9y
Subject to the constraints :
x + 3y < 60 ; x + y > 10 ; x < y ; x > 0, y > 0

Question 23.
Consider the following figure :

a. Find the point of intersection P, of the circle, x²+ y² = 32 and the line y = x.
b. Express the area of the shaded portion as a Sum of two definite integrals.
c. Find the area of the shaded portion.

Question 24.

ANSWERS

Answer 1.

Answer 2.

Answer 3.

Answer 4.

Answer 5.

Answer 6.

Answer 7.

Answer 8.

Answer 9.

Answer 10.

Answer 11.

Answer 12.

Answer 13.

Answer 14.

Answer 15.

Answer 16.

Answer 17.

Answer 18.

Answer 19.

Answer 20.

Answer 21.

Answer 22.

Answer 23.

Answer 24.

c. 1. f (x) is continuous at (-2, 2)
2. f (x) is differentiable at (-2, 2)
3. If f (b) = f (a) then there exist a point at (-2, 2) such that f¹ (c) = 0
f (b) = f (-2) = (-2)² + 2 = 6
f (a) = f (2) = 2² + 2 = 6
∴ Then there exist a point C such that
f¹ (c) = 0
f¹ (x) = 2x
f¹ (x) = 2c = 0
∴ c = 0 at (-2,2)
∴ Rolle’s theorem verified.

Plus Two Maths Previous Year Question Papers and Answers

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