Board Paper of Class 12-Science 2020 Math Delhi(Set 2) - Solutions

General Instructions:
Read the following instructions very carefully and strictly follow them :
(i) This question paper comprises four sections – A, B, C and D.
This question paper carries 36 questions. All questions are compulsory.
(ii) Section A – Question no. 1 to 20 comprises of 20 questions of one mark each.
(iii) Section B – Question no. 21 to 26 comprises of 6 questions of two marks each.
(iv) Section C – Question no. 27 to 32 comprises of 6 questions of four marks each.
(v) Section D – Question no. 33 to 36 comprises of 4 questions of six marks each.
(vi) There is no overall choice in the question paper. However, an internal choice has been provided in 3 questions of one mark, 2 questions of two marks, 2 questions of four marks and 2 questions of six marks. Only one of the choices in such questions have to be attempted.
(vii) In addition to this, separate instructions are given with each section and question, wherever necessary.
(viii) Use of calculators is not permitted.

Question 1
If $[\begin{matrix} x & 1 \end{matrix}] [\begin{matrix} 1 & 0 \\ - 2 & 0 \end{matrix}] = 0,$ then x equals
(a) 0
(b) –2
(c) –1
(d) 2 VIEW SOLUTION

Question 2
$\int 4^{x} 3^{x} d x$ equals

(a) $\frac{12^{x}}{\log 12} + C$

(b) $\frac{4^{x}}{\log 4} + C$

(c) $(\frac{4^{x} \cdot 3^{x}}{\log 4 \cdot \log 3}) + C$

(d) $\frac{3^{x}}{\log 3} + C$ VIEW SOLUTION

Question 3
A number is chosen randomly from numbers 1 to 60. The probability that the chosen number is a multiple of 2 or 5 is

(a) $\frac{2}{5}$

(b) $\frac{3}{5}$

(c) $\frac{7}{10}$

(d) $\frac{9}{10}$ VIEW SOLUTION

Question 4
ABCD is a rhombus whose diagonals intersect at E. Then $\vec{EA} + \vec{EB} + \vec{EC} + \vec{ED}$ equals

(a) $\vec{0}$

(b) $\vec{AD}$

(c) $2 \vec{BC}$

(d) $2 \vec{AD}$ VIEW SOLUTION

Question 5
If A is a square matrix of order 3, such that A (adj A) = 10 I, then |adj A| is equal to
(a) 1
(b) 10
(c) 100
(d) 101 VIEW SOLUTION

Question 6
A card is picked at random from a pack of 52 playing cards. Given that picked card is a queen, the probability of this card to be a card of spade is

(a) $\frac{1}{3}$

(b) $\frac{4}{13}$

(c) $\frac{1}{4}$

(d) $\frac{1}{2}$ VIEW SOLUTION

Question 7
If $\hat{i}, \hat{j}, \hat{k}$ are unit vectors along three mutually perpendicular directions, then
(a) $\hat{i} . \hat{j} = 1$
(b) $\hat{i} \times \hat{j} = 1$
(c) $\hat{i} . \hat{k} = 0$
(d) $\hat{i} \times \hat{k} = 0$ VIEW SOLUTION

Question 8

The graph of the inequality 2x + 3y > 6 is
(a) half plane that contains the origin.
(b) half plane that neither contains the origin nor the points of the line 2x + 3y = 6.
(c) whole XOY – plane excluding the points on the line 2x + 3y = 6.
(d) entire XOY plane.
VIEW SOLUTION

Question 9
The lines $\frac{x - 2}{1} = \frac{y - 3}{1} = \frac{4 - z}{k}$ and $\frac{x - 1}{k} = \frac{y - 4}{2} = \frac{z - 5}{- 2}$ are mutually perpendicular if the value of k is

(a) $- \frac{2}{3}$

(b) $\frac{2}{3}$

(c) –2

(d) 2 VIEW SOLUTION

Question 10
If y = Ae^5x + Be^–5x, then $\frac{d^{2} y}{d x^{2}}$ is equal to
(a) 25y
(b) 5y
(c) –25y
(d) 15y VIEW SOLUTION

Question 11
Fill in the blank.
A relation R on a set A is called ________, if (a₁, a₂) ∈ R and (a₂, a₃) ∈ R implies that (a₁, a₃)∈R, for a₁, a₂, a₃ ∈ A. VIEW SOLUTION

Question 12
Fill in the blank.
The integrating factor of the differential equation x $\frac{d y}{d x} + 2 y = x^{2}$ is _________.
OR

Fill in the blank.
The degree of the differential equation $1 + {(\frac{d y}{d x})}^{2} = x$ is _____________. VIEW SOLUTION

Question 13
Fill in the blank.
The vector equation of a line which passes through the points (3, 4, –7) and (1, –1, 6) is _________.
OR

Fill in the blank.
The line of shortest distance between two skew lines is ______ to both the lines. VIEW SOLUTION

Question 14
If $A + B = [\begin{matrix} 1 & 0 \\ 1 & 1 \end{matrix}]$ and $A - 2 B = [\begin{matrix} - 1 & 1 \\ 0 & - 1 \end{matrix}],$ then A = ________. VIEW SOLUTION

Question 15
Fill in the blank.
The least value of the function $f (x) = a x + \frac{b}{x} (a > 0, b > 0, x > 0)$ is __________. VIEW SOLUTION

Question 16
Evaluate : $\sin [\frac{π}{3} - \sin^{- 1} (- \frac{1}{2})] .$ VIEW SOLUTION

Question 17
Using differential, find the approximate value of $\sqrt{36.6}$ upto 2 decimal places.
OR

Find the slope of tangent to the curve y = 2 cos²(3x) at $x = \frac{π}{6} .$ VIEW SOLUTION

Question 18
Find the value of $\int_{1}^{4} |x - 5| d x .$ VIEW SOLUTION

Question 19
If the function f defined as

$f (x) = \{\begin{matrix} \frac{x^{2} - 9}{x - 3}, & x \neq 3 \\ k, & x = 3 \end{matrix}$

is continuous at x = 3, find the value of k. VIEW SOLUTION

Question 20
For $A = [\begin{matrix} 3 & - 4 \\ 1 & - 1 \end{matrix}]$ write A^–1. VIEW SOLUTION

Question 21
Find $\int \frac{x + 1}{x (1 - 2 x)} d x .$ VIEW SOLUTION

Question 22
Evaluate $\int \frac{x \sin^{- 1} (x^{2})}{\sqrt{1 - x^{4}}} d x .$ VIEW SOLUTION

Question 23
Find the value of $\int_{0}^{1} x {(1 - x)}^{n} d x .$ VIEW SOLUTION

Question 24
If x = a cos θ; y = b sin θ, then find $\frac{d^{2} y}{d x^{2}}$ .
OR

Find the differential of sin² x w.r.t. e^cosx. VIEW SOLUTION

Question 25
Given two independent events A and B such that P(A) = 0.3 and P(B) = 0.6, find P(A' ∩ B') VIEW SOLUTION

Question 26
If $f (x) = \frac{4 x + 3}{6 x - 4}, x \neq \frac{2}{3},$ then show that (fof) (x) = x; for all $x \neq \frac{2}{3} .$ Also, write inverse of f.
OR

Check if the relation R in the set ℝ of real numbers defined as R = {(a, b) : a < b} is (i) symmetric, (ii) transitive VIEW SOLUTION

Question 27
Prove that $\tan [2 \tan^{- 1} (\frac{1}{2}) - \cos^{- 1} 3] = \frac{9}{13} .$ VIEW SOLUTION

Question 28
If $y = {(\cos x)}^{x} + \tan^{- 1} \sqrt{x}, find \frac{d y}{d x} .$ VIEW SOLUTION

Question 29
If $\vec{a} = \hat{i} + 2 \hat{j} + 3 \hat{k} and \vec{b} = 2 \hat{i} + 4 \hat{j} - 5 \hat{k}$ represent two adjacent sides of a parallelogram, find unit vectors parallel to the diagonals of the parallelogram.
OR

Using vectors, find the area of the triangle ABC with vertices A(1, 2, 3), B(2, –1, 4) and C(4, 5, –1). VIEW SOLUTION

Question 30
A company manufactures two types of novelty souvenirs made of plywood. Souvenirs of type A requires 5 minutes each for cutting and 10 minutes each for assembling. Souvenirs of type B require 8 minutes each for cutting and and 8 minutes each for assembling. Given that total time for cutting is 3 hours 20 minutes and for assembling 4 hours. The profit for type A souvenir is ₹ 100 each and for type B souvenir, profit is ₹ 120 each. How many souvenirs of each type should the company manufacture in order to maximize the profit? Formulate the problem as an LPP and solve it graphically. VIEW SOLUTION

Question 31
Three rotten apples are mixed with seven fresh apples. Find the probability distribution of the number of rotten apples, if three apples are drawn one by one with replacement. Find the mean of the number of rotten apples.
OR

In a shop X, 30 tins of ghee of type A and 40 tins of ghee of type B which look alike, are kept for sale. While in shop Y, similar 50 tins of ghee of type A and 60 tins of ghee of type B are there. One tin of ghee is purchased from one of the randomly selected shop and is found to be of type B. Find the probability that it is purchased from shop Y. VIEW SOLUTION

Question 32
Solve the differential equation:

$x \sin (\frac{y}{x}) \frac{d y}{d x} + x - y \sin (\frac{y}{x}) = 0$

Given that x = 1 when $y = \frac{π}{2} .$ VIEW SOLUTION

Question 33
If a, b, c are p^th, q^th and r^th terms respectively of a G.P, then prove that $|\begin{matrix} \log a & p & 1 \\ \log b & q & 1 \\ \log c & r & 1 \end{matrix}| = 0$
OR

If $A = [\begin{matrix} 2 & - 3 & 5 \\ 3 & 2 & - 4 \\ 1 & 1 & - 2 \end{matrix}]$ , then find A^–1.
Using A^–1, solve the following system of equations :
2x – 3y + 5z = 11
3x + 2y – 4z = –5
x + y – 2z = –3
VIEW SOLUTION

Question 34
Find the vector and cartesian equations of the line which perpendicular to the lines with equations

$\frac{x + 2}{1} = \frac{y - 3}{2} = \frac{z + 1}{4} and \frac{x - 1}{2} = \frac{y - 2}{3} = \frac{z - 3}{4}$

and passes through the point (1, 1, 1). Also find the angle between the given lines. VIEW SOLUTION

Question 35
Using integration find the area of the region bounded between the two circles x² + y² = 9 and (x – 3)² + y² = 9.
OR

Evaluate the following integral as the limit of sums $\int_{1}^{4} (x^{2} - x) d x .$ VIEW SOLUTION

Question 36
Find the point on the curve y² = 4x which is nearest to the point (2, 1). VIEW SOLUTION

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