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# Board Paper of Class 12-Humanities 2020 Math Delhi(Set 3) - Solutions

General Instructions:
(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 A is a skew symmetric matrix of order 3, then the value of |A| is
(a) 3
(b) 0
(c) 9
(d) 27 VIEW SOLUTION

• Question 2
If  are unit vectors along three mutually perpendicular directions, then
(a)
(b)
(c)
(d) VIEW SOLUTION

• Question 3
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 4
If A is a 3 × 3 matrix such that |A| = 8, then |3A| equals.
(a) 8
(b) 24
(c) 72
(d) 216 VIEW SOLUTION

• Question 6
If $y={\mathrm{log}}_{e}\left(\frac{{x}^{2}}{{e}^{2}}\right),$ then $\frac{{d}^{2}y}{d{x}^{2}}$equals

(a) $-\frac{1}{x}$

(b) $-\frac{1}{{x}^{2}}$

(c) $\frac{2}{{x}^{2}}$

(d) $-\frac{2}{{x}^{2}}$ VIEW SOLUTION

• Question 7
A die is thrown once. Let A be the event that the number obtained is greater than 3. Let B be the event that the number obtained is less than 5. Then P(A∪B) is

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

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

(c) 0

(d) 1 VIEW SOLUTION

• Question 8
ABCD is a rhombus whose diagonals intersect at E. Then $\stackrel{\to }{\mathrm{EA}}+\stackrel{\to }{\mathrm{EB}}+\stackrel{\to }{\mathrm{EC}}+\stackrel{\to }{\mathrm{ED}}$ equals

(a) $\stackrel{\to }{0}$

(b) $\stackrel{\to }{\mathrm{AD}}$

(c) $2\stackrel{\to }{\mathrm{BC}}$

(d) $2\stackrel{\to }{\mathrm{AD}}$ VIEW SOLUTION

• Question 9
The distance of the origin (0, 0, 0) from the plane –2x + 6y – 3z = –7 is
(a) 1 unit
(b) $\sqrt{2}$ units
(c) $2\sqrt{2}$ units
(d) 3 units VIEW SOLUTION

• Question 10

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 11
Fill in the blank.
If A and B are square matrices each of order 3 and |A| = 5, |B| = 3, then the value of |3 AB| is __________ VIEW SOLUTION

• Question 12
Fill in the blank.
The least value of the function  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
Fill in the blank.
The integrating factor of the differential equation x $\frac{dy}{dx}+2y={x}^{2}$ is _________.

OR

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

• Question 15
Fill in the blank.
A relation in a set A is called ________  relation, if each element of A is related to itself. VIEW SOLUTION

• Question 16
Find the cofactors of all the elements of . VIEW SOLUTION

• Question 17
Let f(x) = x|x|, for all x ∈ R check its differentiability at x = 0. VIEW SOLUTION

• Question 19
Find the value of $\underset{1}{\overset{4}{\int }}\left|x-5\right|dx.$ VIEW SOLUTION

• Question 20
If f(x) = x4 – 10, then find the approximate value of f(2.1).

OR

Find the slope of the tangent to the curve y = 2 sin2 (3x) at $x=\frac{\mathrm{\pi }}{6}$. VIEW SOLUTION

• Question 22
If  then show that (fof) (x) = x; for all $x\ne \frac{2}{3}.$ Also, write inverse of f.

OR

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

• Question 23
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 25
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. ecosx. VIEW SOLUTION

• Question 26
Find the value of $\underset{0}{\overset{1}{\int }}{\mathrm{tan}}^{-1}\left(\frac{1-2x}{1+x-{x}^{2}}\right)dx.$ VIEW SOLUTION

• Question 27
Solve the equation x : ${\mathrm{sin}}^{-1}\left(\frac{5}{x}\right)+{\mathrm{sin}}^{-1}\left(\frac{12}{x}\right)=\frac{\mathrm{\pi }}{2}\left(x\ne 0\right)$. VIEW SOLUTION

• Question 28
Find the general solution of the differential equation
yex/y dx = (xex/y + y2) dy, y ≠ 0 VIEW SOLUTION

• Question 29
If then find $\frac{dy}{dx}.$ VIEW SOLUTION

• Question 30
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 31
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 32
If 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 33
Find the distance of the point P(3, 4, 4) from the point, where the line joining the points A(3, –4, –5) and B(2, –3, 1) intersects the plane 2x + y + z = 7. VIEW SOLUTION

• Question 34
Find the minimum value of (ax + by), where xy = c2. VIEW SOLUTION

• Question 35
If a, b, c are pth, qth and rth terms respectively of a G.P, then prove that

OR

If , 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 36
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 VIEW SOLUTION
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