Board Paper of Class 12Commerce 2020 Math Delhi(Set 3)  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.
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 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 $\hat{i},\hat{j},\hat{k}$ 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 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 5
$\int {x}^{2}{e}^{{x}^{3}}dx\mathrm{equals}$
(a) $\frac{1}{3}{e}^{{x}^{3}}+\mathrm{C}$
(b) $\frac{1}{3}{e}^{{x}^{4}}+\mathrm{C}$
(c) $\frac{1}{2}{e}^{{x}^{3}}+\mathrm{C}$
(d) $\frac{1}{2}{e}^{{x}^{2}}+\mathrm{C}$ 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 $\overrightarrow{\mathrm{EA}}+\overrightarrow{\mathrm{EB}}+\overrightarrow{\mathrm{EC}}+\overrightarrow{\mathrm{ED}}$ equals
(a) $\overrightarrow{0}$
(b) $\overrightarrow{\mathrm{AD}}$
(c) $2\overrightarrow{\mathrm{BC}}$
(d) $2\overrightarrow{\mathrm{AD}}$ 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
VIEW SOLUTION
(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.
 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 $f\left(x\right)=ax+\frac{b}{x}\left(a>0,b0,x0\right)$ 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 $\left[\begin{array}{cc}1& 2\\ 4& 3\end{array}\right]$. VIEW SOLUTION
 Question 17
Let f(x) = xx, for all x ∈ R check its differentiability at x = 0. VIEW SOLUTION
 Question 18
Find the value of ${\mathrm{sin}}^{1}\left[\mathrm{sin}\left(\frac{17\mathrm{\pi}}{8}\right)\right]$. VIEW SOLUTION
 Question 19
Find the value of $\underset{1}{\overset{4}{\int}}\leftx5\rightdx.$ VIEW SOLUTION
 Question 20
If f(x) = x^{4} – 10, then find the approximate value of f(2.1).
OR
Find the slope of the tangent to the curve y = 2 sin^{2} (3x) at $x=\frac{\mathrm{\pi}}{6}$. VIEW SOLUTION
 Question 21
Find $\int \frac{x+1}{\left(x+2\right)\left(x+3\right)}dx.$ VIEW SOLUTION
 Question 22
If $f\left(x\right)=\frac{4x+3}{6x4},x\ne \frac{2}{3},$ 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 ℝ 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 24
Evaluate $\underset{1}{\overset{2}{\int}}\left[\frac{1}{x}\frac{1}{2{x}^{2}}\right]{e}^{2x}dx$. 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 sin^{2} x w.r.t. e^{cosx}. VIEW SOLUTION
 Question 26
Find the value of $\underset{0}{\overset{1}{\int}}{\mathrm{tan}}^{1}\left(\frac{12x}{1+x{x}^{2}}\right)dx.$ 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
ye^{x}^{/y} dx = (xe^{x}^{/y} + y^{2}) dy, y ≠ 0 VIEW SOLUTION
 Question 29
If $y={\left(\mathrm{log}x\right)}^{x}+{x}^{\mathrm{log}x},$ 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 $\overrightarrow{a}=\hat{i}+2\hat{j}+3\hat{k}\mathrm{and}\overrightarrow{b}=2\hat{i}+4\hat{j}5\hat{k}$ 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 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 = c^{2}. VIEW SOLUTION
 Question 35
If a, b, c are p^{th}, q^{th} and r^{th} terms respectively of a G.P, then prove that $\left\begin{array}{ccc}\mathrm{log}a& p& 1\\ \mathrm{log}b& q& 1\\ \mathrm{log}c& r& 1\end{array}\right=0$
OR
If $\mathrm{A}=\left[\begin{array}{ccc}2& 3& 5\\ 3& 2& 4\\ 1& 1& 2\end{array}\right]$, then find A^{–1}.
VIEW SOLUTION
Using A^{–1}, solve the following system of equations :
2x – 3y + 5z = 11
3x + 2y – 4z = –5
x + y – 2z = –3
 Question 36
Using integration find the area of the region bounded between the two circles x^{2} + y^{2} = 9 and (x – 3)^{2} + y^{2} = 9.
OR
Evaluate the following integral as the limit of sums $\underset{1}{\overset{4}{\int}}\left({x}^{2}x\right)dx.$ VIEW SOLUTION
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