## Plus Two Maths Previous Year Question Paper March 2018

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.
• 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.

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

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.
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 = …………

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

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.
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.
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 $$\tan ^{-1} \frac{1}{2}+\tan ^{-1} \frac{2}{11}=\tan ^{-1} \frac{3}{4}$$
i) sin-1x

Question 10.
i) $$\frac{d\left(a^{x}\right)}{d x}$$ = ………..
a) ax
b) log(ax)
c) ax log a
d) xax-1
ii) Find $$\frac{d y}{d x}$$ if xy = yx
i) ax loga
ii) Given; yx = xy, 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
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.
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) + 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.
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
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.

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.
$$\frac{x}{1}+\frac{y}{2}+\frac{z}{3}$$ ⇒ 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
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.
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

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
$$\left(1-x^{2}\right) \frac{d^{2} y}{d x^{2}}-x \frac{d y}{d x}-a^{2} y=0$$
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:

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.

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)

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)
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 = $$\frac{3}{72}$$

ICICIGI Pivot Point Calculator

## 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.
• 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}.
A. (4,1) ∈ R
B. (5,8) ∈ R
C. (8,7) ∈ R
D.(10,6) ∈ R

b. If f (x) = 8x3 and g (x) = x1/3, 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.

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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Unit 4 Live and Let Live

Unit 5 The Lighter Side

Plus Two English Guide has been designed with a view to improve the skill of answering all types of questions quickly and precisely. Plus Two English material will boost your confidence in the subject and help you face examinations with ease.

Salient Features of Plus Two English Material

• A simple and brief summary of each lesson is given with its Malayalam translation.
• All the activities and questions given after each lesson have been fully answered.
• The answers are given immediately after the questions. Malayalam translation of the questions is also given.
• Besides the questions in the text, Exam oriented questions and their answers in the new pattern are given.
• Questions on grammar and discourse are also included in this guide.

We hope that HSSLive Plus Two English Study Material will help you improve your quality of learning as per the new grading system and to get an A+ grade in English. Suggestions for further improvement of this material are always appreciated.

Plus Two English Previous Year Question Papers and Answers

## Plus Two Computer Application Chapter Wise Questions and Answers Kerala

HSE Kerala Board Syllabus HSSLive Plus Two Computer Application Chapter Wise Questions and Answers Pdf Free Download in both English Medium and Malayalam Medium are part of SCERT Kerala Plus Two Chapter Wise Questions and Answers. Here HSSLive.Guru has given Higher Secondary Kerala Plus Two Computer Application Chapter Wise Questions and Answers based on CBSE NCERT syllabus.

 Board SCERT, Kerala Text Book NCERT Based Class Plus Two Subject Computer Application Chapter All Chapters Category Kerala Plus Two

## Kerala Plus Two Computer Application Chapter Wise Questions and Answers

We hope the given HSE Kerala Board Syllabus HSSLive Plus Two Computer Application Chapter Wise Questions and Answers Pdf Free Download in both English Medium and Malayalam Medium will help you. If you have any query regarding Higher Secondary Kerala Plus Two Computer Application Chapter Wise Questions and Answers based on CBSE NCERT syllabus, drop a comment below and we will get back to you at the earliest.