Board Paper of Class 12 2020 Mathematics - Solutions

General Instructions:
The question paper is divided into FOUR sections.
(1) Section A:
Q.1 contains Eight multiple choice type of questions, each carrying Two marks.
Q.2 contains Four sub-questions, each carrying one mark.
(2) Section B: Q.3 to Q.14 each carries Two marks. (Attempt any Eight).
(3) Section C: Q.15 to Q.26 each carries Three marks. (Attempt any Eight).
(4) Section D: Q.27 to Q.34 each carries Four marks. (Attempt any Five).
(5) Use of log table is allowed. Use of calculator is not allowed.
(6) Figures to the right indicate full marks.
(7) Use of graph paper is not necessary. Only rough sketch of graph is expected.
(8) For each MCQ, correct answer must be written along with its alphabet:

e.g. (a) _______ (b) _______ (c) _______ (d) _______, etc.

(9) Start answers to each section on a new page.

Question 1
(a) In ∆ABC, if a = 2, b = 3 and $\sin (A) = \frac{2}{3},$ then ∠B = _________.

(a) $\frac{π}{4}$

(b) $\frac{π}{2}$

(c) $\frac{π}{3}$

(d) $\frac{π}{6}$

(b) If $\bar{a} = 3 i - j + 4 k,$ b = 2i + 3j – k and c = –5i + 2j + 3k, then $\bar{a} \cdot (\bar{b} \times \bar{c})$ is _________.

(a) 100

(b) 110

(c) 109

(d) 108

(c) The cartesian equation of the line passing through the points A(4, 2, 1) and B(2, –1, 3) is

(a) $\frac{x + 4}{2} = \frac{y - 2}{3} = \frac{z - 1}{- 2}$

(b) $\frac{x - 4}{- 2} = \frac{y - 2}{- 3} = \frac{z - 1}{- 2}$

(c) $\frac{x - 4}{2} = \frac{y - 2}{3} = \frac{z - 1}{- 2}$

(d) $\frac{x - 4}{- 2} = \frac{y - 2}{3} = \frac{z - 1}{- 2}$

(d) If the line $\bar{r} = (\hat{i} - 2 \hat{j} + 3 \hat{k}) + λ (2 \hat{i} + \hat{j} + 2 \hat{k})$ is parallel to the plane $\bar{r} \cdot (3 \hat{i} - 2 \hat{j} + m \hat{k}) = 10$ then value of m is _________.

(a) –2

(b) 2

(c) ±2

(d) 0

(e) If f(x) = 1 – x, for 0 < x ≤ 1 and k, for x = 0 is continuous at x = 0, then k = _____.

(a) 0

(b) –1

(c) 2

(d) 1

(f) The function f(x) = x^x is minimum at x = _____.

(a) e

(b) –e

(c) $\frac{1}{e}$

(d) $- \frac{1}{e}$

(g) If $\int_{0}^{k} 4 x^{3} d x = 16,$ then the value of k is ____.

(a) 1

(b) 2

(c) 3

(d) 4 VIEW SOLUTION

Question 2
(a) Write the dual of p ∧ ~ p ≡ F

(b) Find the general solution of tan 2x = 0

(c) Differentiate sin(x² + x) w.r.t. x

(d) If X ~ B(n, p) and n = 10, E(X) = 5, then find the value of p. VIEW SOLUTION

Question 3
Attempt any EIGHT of the following questions:
Using truth table verify that ~ (p ∨ q) ≡ ~ p ∧ ~ q VIEW SOLUTION

Question 4
Find the matrix of co-factors for the matrix $[\begin{matrix} 1 & 3 \\ 4 & - 1 \end{matrix}]$ VIEW SOLUTION

Question 5
Find the angle between the lines represented by 3x² + 4xy – 3y² = 0 VIEW SOLUTION

Question 6
$\bar{a}$ and $\bar{b}$ are non-collinear vectors. If $\bar{c} = (x - 2) \bar{a} + \bar{b}$ and $\bar{d} = (2 x + 1) \bar{a} - \bar{b}$ are collinear, then find the value of x. VIEW SOLUTION

Question 7
If a line makes angles 90°, 135°, 45° with X, Y and Z axes respectively, then find its direction cosines. VIEW SOLUTION

Question 8
Express the following circuit in symbolic form:
VIEW SOLUTION

Question 9
Differentiate log(sec x + tan x) w.r.t x. VIEW SOLUTION

Question 10
Evaluate: $\int \frac{d x}{x^{2} + 4 x + 8}$ VIEW SOLUTION

Question 11
Evaluate: $\int_{0}^{\frac{π}{2}} \cos^{2} x d x$ VIEW SOLUTION

Question 12
Solve the differential equation $\frac{d y}{d x} = x^{2} y + y$ VIEW SOLUTION

Question 13
Find the expected value of the random variable X whose probability mass function is:

X = x
1 2 3

P(X = x)
$\frac{1}{5}$ $\frac{2}{5}$ $\frac{2}{5}$

VIEW SOLUTION

Question 14
If y = x log x, then find $\frac{d^{2} y}{d x^{2}} .$ VIEW SOLUTION

Question 15
State the converse, inverse and contrapositive of the conditional statement: ‘If a sequence is bounded, then it is convergent’. VIEW SOLUTION

Question 16
Show that: $\sin^{- 1} (\frac{8}{17}) + \sin^{- 1} (\frac{3}{5}) = \sin^{- 1} (\frac{77}{85})$ VIEW SOLUTION

Question 17
Show that the points A(2, 1, –1), B(0, –1, 0), C(4, 0, 4) and D(2, 0, 1) are coplanar. VIEW SOLUTION

Question 18
If ΔABC is right-angled at B, where A(5, 6, 4), B(4, 4, 1) and C(8, 2, x), then find the value of x. VIEW SOLUTION

Question 19
Find the equation of the line passing through the point (3, 1, 2) and perpendicular to the lines $\frac{x - 1}{1} = \frac{y - 2}{2} = \frac{z - 3}{3}$ and $\frac{x}{- 3} = \frac{y}{2} = \frac{z}{5}$ VIEW SOLUTION

Question 20
Find the distance of the point $\hat{i} + 2 \hat{j} - \hat{k}$ from the plane $\vec{r} \cdot (\hat{i} - 2 \hat{j} + 4 \hat{k}) = 10$ VIEW SOLUTION

Question 21
Form the differential equation if e^x + e^y = e^{x + y} VIEW SOLUTION

Question 22
The surface area of a spherical balloon is increasing at the rate of 2cm²/sec. At what rate the volume of the balloon is increasing when the radius of the balloon is 6 cm? VIEW SOLUTION

Question 23
Find the approximate value of e^1.005; given e = 2.7183. VIEW SOLUTION

Question 24
Evaluate: $\int \frac{x^{2} \cdot \tan^{- 1} (x^{3})}{1 + x^{6}} d x$ VIEW SOLUTION

Question 25
Solve the differential equation $\frac{d y}{d x} + y = e^{- x} .$ VIEW SOLUTION

Question 26
If f(x) = kx, 0 < x < 2 = 0, otherwise,
Is a probability density function of a random variable x, then find :
(i) Value of k
(ii) P(1 < x < 2) VIEW SOLUTION

Question 27
Prove that a homogeneous equation of degree two in x and y i.e. ax² + 2hxy + by² = 0 represent a pair of lines passing through the origin, if h² – ab ≥ 0. VIEW SOLUTION

Question 28
Solve the following linear programming problem:
Maximise: z = 150x + 250y
Subject to: 4x + y ≤ 40
3x + 2y ≤ 60
x ≥ 0
y ≥ 0 VIEW SOLUTION

Question 29
Solve the following equations by the method of reduction:
x + 3y + 3z = 12
x + 4y + 4z = 15
x + 3y + 4z = 13 VIEW SOLUTION

Question 30
In then prove that ∆ABC, if a + b + c = 2s, $\sin (\frac{A}{2}) = \sqrt{\frac{(s - b) (s - c)}{b c}},$ with usual notations. VIEW SOLUTION

Question 31
Function f(x) is continuous on its domain [–2, 2], where

$\begin{array}{rcl} f (x) & = & \frac{\sin a x}{x} + 2, for - 2 \leq x < 0 \\ = & 3 x + 5, for 0 \leq x \leq 1 \\ = & \sqrt{x^{2} + 8} - b, for 1 \leq x \leq 2 \end{array}$

Find the value of a + b + 2. VIEW SOLUTION

Question 32
Prove that:

$\int \sqrt{x^{2} + a^{2}} \cdot d x = \frac{x}{2} \sqrt{x^{2} + a^{2}} + \frac{a^{2}}{2} \log |x + \sqrt{x^{2} + a^{2}}| + c$ VIEW SOLUTION

Question 33
A fair coin is tossed 8 times. Find the probability that:
(i) it shows no head
(ii) it shows head at least once. VIEW SOLUTION

Question 34
Prove that:

$\int_{0}^{2 a} f (x) d x = \int_{0}^{a} f (x) d x + \int_{0}^{a} f (2 a - x) d x$ VIEW SOLUTION

X = x	1	2	3
P(X = x)	$\frac{1}{5}$	$\frac{2}{5}$	$\frac{2}{5}$