2nd PUC Maths Previous Year Question Paper June 2018

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Karnataka 2nd PUC Maths Previous Year Question Paper June 2018

Time: 3 Hrs 15 Min
Max. Marks: 100

Instructions

  • The question paper has five parts namely A, B, C, D, and E. Answer all the parts.
  • Use the graph sheet for the question on Linear programming in Part – E

Part – A

Answer ALL the following questions: (10 × 1 = 10)

Question 1.
The relation R on set A = {1, 2, 3} is defined as R {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3)} is not transitive. Why?
Solution:
(1, 2) ∈ R, (2, 3) ∈ R. But (1, 3) ∉ R

Question 2.
Write the range of y = cos-1 x
Solution:
[0, π]

Question 3.
If a matrix has 5 elements, what are the possible orders it can have?
Solution:
1 × 5, 5 × 1

2nd PUC Maths Previous Year Question Paper June 2018

Question 4.
Find the values of x for which \left|\begin{array}{cc}x & 2 \\18 & x\end{array}\right|=\left|\begin{array}{cc}6 & 2 \\18 & 6\end{array}\right|
Solution:
x2 = 36 ⇒ x = ±6

Question 5.
Find \frac{d y}{d x}, if y = sin(ax + b)
Solution:
\frac{d y}{d x} = a cos(ax + b)

Question 6.
Evaluate: ∫sec x (sec x + tan x) dx
Solution:
I = ∫sec2 x dx + ∫ sec x tan x dx = tan x + sec x + c

Question 7.
Define the negative of a vector.
Solution:
Negative of the vector is obtained by changing the direction of the given vector into the opposite direction or by multiplying the given vector by -1.
Say \vec{a} be the given vector so negative of \vec{a} is –\vec{a}
\vec{a} and –\vec{a} will have the same magnitude but opposite direction.

Question 8.
The Cartesian equation of a line is \frac{x-5}{3}=\frac{y-4}{7}=\frac{z-6}{2}. Write its vector form.
Solution:
2nd PUC Maths Previous Year Question Paper June 2018 Q8

Question 9.
Define optimal solution in a linear programming problem.
Solution:
Any feasible solution of LPP which maximizes or minimizes the objective function is called an optimal solution.

2nd PUC Maths Previous Year Question Paper June 2018

Question 10.
Find P(A/B), if P(B) = 0.5 and P(A∩B) = 0.32.
Solution:
2nd PUC Maths Previous Year Question Paper June 2018 Q10

Part – B

Answer any TEN questions: (10 × 2 = 20)

Question 11.
Define binary operation on a set. Verify whether the operation * is defined on Q set of rational numbers by a * b = ab + 1, ∀ a, b ∈ Q is binary or not.
Solution:
A binary operation on a set A is a function
* : A × A → A
a * b = ab + 1 ∈ Q
∴ * is a Binary Operation on Q.

Question 12.
Write \tan ^{-1}(\sqrt{\frac{1-\cos x}{1+\cos x}}), 0 < x < π in the simplest form.
Solution:
2nd PUC Maths Previous Year Question Paper June 2018 Q12

Question 13.
Find the value of \cos ^{-1}\left(\cos \frac{13 \pi}{6}\right)
Solution:
2nd PUC Maths Previous Year Question Paper June 2018 Q13

Question 14.
If the area of the triangle with vertices (2, -6), (5, 4) and (K, 4) is 35 sq. units, then find the values of K, using determinants.
Solution:
2nd PUC Maths Previous Year Question Paper June 2018 Q14

Question 15.
2nd PUC Maths Previous Year Question Paper June 2018 Q15
Solution:
2nd PUC Maths Previous Year Question Paper June 2018 Q15.1

Question 16.
Differentiate (sin x)cos x with respect to x.
Solution:
2nd PUC Maths Previous Year Question Paper June 2018 Q16

Question 17.
If the radius of a sphere is measured as 7 m with an error of 0.02 m, then find the approximate error in calculating its volume.
Solution:
Let r be the radius of the sphere and ∆r be the error in measuring radius.
Then, r = 7 m and ∆r = 0.02 m
Now, volume of a sphere is given by V = \frac{4}{3} \pi r^{3}
On differentiate w.r.t r, we get \frac{d V}{d r}=\left(\frac{4}{3} \pi\right)\left(3 r^{2}\right)=4 \pi r^{2}
ΔV = (\frac{d V}{d r}) Δr
= (4πr2) Δr
= 4π × 72 × 0.002
= 3.92π m3
Hence, the approximate error in calculating the volume is 3.92π m3.

2nd PUC Maths Previous Year Question Paper June 2018

Question 18.
Evaluate: \int \cos 6 x \sqrt{1+\sin 6 x} d x
Solution:
2nd PUC Maths Previous Year Question Paper June 2018 Q18

Question 19.
Integrate \frac{x e^{x}}{(1+x)^{2}} with respect to x.
Solution:
2nd PUC Maths Previous Year Question Paper June 2018 Q19
2nd PUC Maths Previous Year Question Paper June 2018 Q19.1

Question 20.
Find the order and degree, if defined, of the differential equation
2nd PUC Maths Previous Year Question Paper June 2018 Q20
Solution:
Order = 3, Degree = 1

Question 21.
Find the projection of the vector \vec{a}=\hat{i}-\hat{j}+3 \hat{k} on the vector \vec{b}=2 \hat{i}+3 \hat{j}+2 \hat{k}.
Solution:
2nd PUC Maths Previous Year Question Paper June 2018 Q21

Question 22.
Find the area of the parallelogram whose adjacent sides are given by the vectors \vec{a}=\hat{i}-\hat{j}+3 \hat{k} and \vec{b}=2 \hat{i}-7 \hat{j}+\hat{k}
Solution:
2nd PUC Maths Previous Year Question Paper June 2018 Q22

Question 23.
Find the angle between the line \frac{x+1}{2}=\frac{y}{3}=\frac{z-3}{6} and the plane 10x + 2y – 11z = 3
Solution:
2nd PUC Maths Previous Year Question Paper June 2018 Q23

Question 24.
The random variable X has a probability distribution P(X) of the following form, where K is some number.
2nd PUC Maths Previous Year Question Paper June 2018 Q24
(a) Determine the value of K.
(b) Find P(X < 2).
Solution:
(a) ΣP(X) = 1
⇒ 6K = 1
⇒ K = \frac{1}{6}
(b) P(X < 2) = P(X = 0) + P(X = 1)
= K + 2K
= 3K
= 3(\frac{1}{6})
= \frac{1}{2}

Part – C

Answer any TEN questions: (10 × 3 = 30)

Question 25.
If f : R → R and g : R → R are given by f(x) = cos x and g(x) = 3x2, then show that gof ≠ fog.

Question 26.
Solve : tan-1 2x + tan-1 3x = \frac{\pi}{4}
Solution:
2nd PUC Maths Previous Year Question Paper June 2018 Q26
Since x = -1 does not satisfy the equation, x = \frac{1}{6} is the only solution of the given equation.

Question 27.
By using elementary operations, find the inverse of the matrix A = \left[\begin{array}{cc}3 & -1 \\-4 & 2\end{array}\right]
Solution:
A = I A
2nd PUC Maths Previous Year Question Paper June 2018 Q27

Question 28.
If x = a(θ – sin θ) and y = a(1 + cos θ), then prove that \frac{d y}{d x}=-\cot (\theta / 2)
Solution:
Given x = a(θ – sin θ) and y = a(1 + cos θ)
Differentiating w.r.t θ, we get
2nd PUC Maths Previous Year Question Paper June 2018 Q28

Question 29.
Verify Mean Value Theorem if f(x) = x2 – 4x + 3 in the interval x ∈ [a, b], where a = 1 and b = 4.
Solution:
f(x) is a polynomial in x.
It is continuous in {1, 4} and differentiable in {1, 4) and f'(x) = 2x – 4.
There exists at least one value c ∈ (1, 4) such that f'(c) = \frac{f(b)-f(a)}{b-a}
a = 1, f(a) = f(1) = 12 – 4(1) – 3 = 1 – 4 – 3 = 1 – 7 = -6
b = 4, f(b) = f(4) = 42 – 4(4) – 3 = 16 – 16 – 3 = -3
f'(c) = 2c – 4
2nd PUC Maths Previous Year Question Paper June 2018 Q29
∴ Mean Value theorem is verified.

2nd PUC Maths Previous Year Question Paper June 2018

Question 30.
Find two positive numbers whose sum is 15 and the sum of whose squares is minimum.

Question 31.
Evaluate: \int_{0}^{1} \frac{\tan ^{-1} x}{1+x^{2}} d x
Solution:
2nd PUC Maths Previous Year Question Paper June 2018 Q31

Question 32.
Integrate \frac{d x}{x\left(x^{2}+1\right)} with respect to x.
Solution:
2nd PUC Maths Previous Year Question Paper June 2018 Q32
2nd PUC Maths Previous Year Question Paper June 2018 Q32.1

Question 33.
Find the area of the parabola y2 = 4ax bounded by its latus rectum.
Solution:
2nd PUC Maths Previous Year Question Paper June 2018 Q33
2nd PUC Maths Previous Year Question Paper June 2018 Q33.1

Question 34.
Find the differential equation representing the family of curves y = a sin(x + b), where a, b are arbitrary constants.

2nd PUC Maths Previous Year Question Paper June 2018

Question 35.
Find a unit vector perpendicular to each of the vectors (\vec{a}+\vec{b}) and (\vec{a}-\vec{b}) where \vec{a}=3 \hat{i}+2 \hat{j}+2 \hat{k} and \vec{b}=\hat{i}+2 \hat{j}-2 \hat{k}
Solution:
2nd PUC Maths Previous Year Question Paper June 2018 Q35
2nd PUC Maths Previous Year Question Paper June 2018 Q35.1

Question 36.
Prove that [\vec{a}+\vec{b}, \vec{b}+\vec{c}, \vec{c}+\vec{a}]=2[\vec{a}, \vec{b}, \vec{c}]
Solution:
2nd PUC Maths Previous Year Question Paper June 2018 Q36

Question 37.
Find the equation of the plane through the intersection of the planes 3x – y + 2z = 0 and x + y + z – 2 = 0 and the point (2, 2, 1)

Question 38.
A man is known to speak truth 4 out of 5 times. He tossed a coin and reports that it is head. Find the probability that it is actually head.
Solution:
E1 : coin shows a head
E2 : coins shows a tail
P(E1) = P(E2) = \frac{1}{2}
S = {H, T}
E1 = {H}, E2 = {T}
Let E : A reports that a head appears
2nd PUC Maths Previous Year Question Paper June 2018 Q38

Part – D

Answer any SIX questions: (6 × 5 = 30)

Question 39.
Let R+ be the set of all non-negative real numbers. Show that the function f : R+ → [4, ∞] given by f(x) = x2 + 4 is invertible and write the inverse of f.
Solution:
2nd PUC Maths Previous Year Question Paper June 2018 Q39
2nd PUC Maths Previous Year Question Paper June 2018 Q39.1

Question 40.
If A = \left[\begin{array}{c}1 \\-4 \\3\end{array}\right], B = \left[\begin{array}{lll}-1 & 2 & 1\end{array}\right], verify that (AB)’ = B’A’. Calculate AC, BC and (A + B)C. Also, verify that (A + B)C = AC + BC.
Solution:
2nd PUC Maths Previous Year Question Paper June 2018 Q40
2nd PUC Maths Previous Year Question Paper June 2018 Q40.1

Question 41.
Solve the following system of equations by matrix method.
4x + 3y + 2z = 60
2x + 4y + 6z = 90
6x + 2y + 3z = 70
Solution:
This system of equations can be written as AX = B, where
2nd PUC Maths Previous Year Question Paper June 2018 Q41
Thus, A is non-singular, Therefore, its inverse exists.
Therefore, the given system is consistent and has a unique solution given by X = A-1 B.
Cofactors of A are
A11 = 12 – 12 = 0
A12 = -(6 – 36) = 30
A13 = 4 – 24 = -20
A21 = -(9 – 4) = -5
A22 = 12 – 12 = 0
A23 = -(8 – 18) = 10
A31 = (18 – 8) = 10
A32 = -(24 – 4) = -20
A33 = 16 – 6 = 10
2nd PUC Maths Previous Year Question Paper June 2018 Q41.1
2nd PUC Maths Previous Year Question Paper June 2018 Q41.2

Question 42.
If Y = A emx + B enx show that \frac{d^{2} y}{d x^{2}}-(m+n) \frac{d y}{d x}+m n y=0
Solution:
2nd PUC Maths Previous Year Question Paper June 2018 Q42

2nd PUC Maths Previous Year Question Paper June 2018

Question 43.
A particle moves along the curve 6y = x3 + 2. Find the points on the curve at which y-coordinate is changing 8 times as fast as the x-coordinate.
Solution:
2nd PUC Maths Previous Year Question Paper June 2018 Q43
2nd PUC Maths Previous Year Question Paper June 2018 Q43.1

Question 44.
Find the integral of \frac{1}{\sqrt{a^{2}-x^{2}}} with respect to x and hence find \int \frac{1}{\sqrt{7-6 x-x^{2}}} d x
Solution:
2nd PUC Maths Previous Year Question Paper June 2018 Q44

Question 45.
Find the area eiiclosed by the ellipse \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 by the method of integration.
Solution:
2nd PUC Maths Previous Year Question Paper June 2018 Q45
2nd PUC Maths Previous Year Question Paper June 2018 Q45.1

Question 46.
Find the general solution of the differential equation x \frac{d y}{d x} + 2y = x2, (x ≠ 0)
Solution:
2nd PUC Maths Previous Year Question Paper June 2018 Q46

Question 47.
Derive the equation of a line in space passing through two given points both in vector and Cartesian form.
Solution:
Vector Form
Let \overrightarrow{\mathrm{a}} and \overrightarrow{\mathrm{b}} be the position vectors of two points A(x1, y1, z1) and B(x2, y2, z2) respectively that are lying on a line.
Let \overrightarrow{\mathrm{r}} be the position vector of an arbitrary point P (x, y, z), then P is a point on the line if and only if \overrightarrow{\mathrm{AP}}=\overrightarrow{\mathrm{r}}-\overrightarrow{\mathrm{a}} and \overrightarrow{\mathrm{AB}}=\overrightarrow{\mathrm{b}}-\overrightarrow{\mathrm{a}} are collinear vectors. Therefore, P is on the line if an only
2nd PUC Maths Previous Year Question Paper June 2018 Q47
2nd PUC Maths Previous Year Question Paper June 2018 Q47.1
2nd PUC Maths Previous Year Question Paper June 2018 Q47.2

2nd PUC Maths Previous Year Question Paper June 2018

Question 48.
A person buys a lottery ticket in 50 lotteries, in each of which his chance of winning a prize is \frac{1}{100}. What is the probability that he will win a prize
(a) at least once
(b) exactly once
Solution:
Let X denote the number of wins
2nd PUC Maths Previous Year Question Paper June 2018 Q48

Part – E

Answer any ONE question: (1 × 10 = 10)

Question 49(a).
2nd PUC Maths Previous Year Question Paper June 2018 Q49(a)
Solution:
2nd PUC Maths Previous Year Question Paper June 2018 Q49(a).1
2nd PUC Maths Previous Year Question Paper June 2018 Q49(a).2
2nd PUC Maths Previous Year Question Paper June 2018 Q49(a).3

Question 49(b).
Find the value of K, if 2nd PUC Maths Previous Year Question Paper June 2018 Q49(b)
Solution:
2nd PUC Maths Previous Year Question Paper June 2018 Q49(b).1

2nd PUC Maths Previous Year Question Paper June 2018

Question 50(a).
Solve the following problem graphically:
Maximize and minimize
Z = 10500x + 9000y
Subject to the constraints
x + y ≤ 50
2x + y ≤ 80
x ≥ 0, y ≥ 0
Solution:
x + y = 50
y = 0 ⇒ x = 50
∴ A = (50, 0), B = (0, 50)
(0, 0) lies on x + y ≤ 50
2x + y = 80
y = 0, x = 40 ⇒ C(40, 0)
x = 0, y = 80 ⇒ D(0, 80)
Origin lies on 2x + y ≤ 80
Solving x + y = 50
2x + y = 80 we get x = 30, y = 20
E = (30, 20)
2nd PUC Maths Previous Year Question Paper June 2018 Q50(a)
OBEC is the feasible region.
2nd PUC Maths Previous Year Question Paper June 2018 Q50(a).1

Question 50(b).
2nd PUC Maths Previous Year Question Paper June 2018 Q50(b)
Solution:
2nd PUC Maths Previous Year Question Paper June 2018 Q50(b).1

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