Board Paper of Class 12Commerce 2023 Math Delhi(Set 3)  Solutions
General Instructions:
Read the following instructions very carefully and follow them:
(i) This question paper contains 38 questions. All questions are compulsory.
(ii) Question paper is divided into FIVE SectionsSection A, B, C, D and E.
(ii) In Section A  Question Number 1 to 18 are Multiple Choice Questions (MCQ) type and Question Number 19 & 20 are AssertionReason based questions of 1 mark each.
(iv) In Section B  Question Number 21 to 25 are Very Short Answer (VSA) type questions of 2 marks each.
(v) In Section C  Question Number 26 to 31 are Short Answer (SA) type questions, carrying 3 marks each.
(vi) In Section D  Question Number 32 to 35 are Long Answer (LA) type questions carrying 5 marks each.
(vii) In Section E  Question Number 36 to 38 are case study based questions carrying 4 marks each where 2 VSA type questions are of 1 mark each and 1 SA type question is of 2 marks. Internal choice is provided in 2 marks question in each casestudy.
(viii) There is no overall choice. However, an internal choice has been provided in 2 questions in Section B  3 questions in Section C, 2 questions in
SectionD and 2 questions in Section  E.
(ix) Use of calculators is NOT allowed.
 Question 1
Let R be a relation in the set N given by
R = {a, b): a = b − 2, b > 6}.
Then
(a) (8, 7) ∊ R
(b) (6, 8) ∊ R
(c) (3, 8) ∊ R
(d) (2, 4) ∊ R VIEW SOLUTION
 Question 2
If $\mathrm{A}=\left[\begin{array}{cc}5& x\\ y& 0\end{array}\right]$ and A = A^{T}, where A^{T} is the transpose of the matrix A, then
(a) x = 0, y = 5
(b) x = y
(c) x + y = 5
(d) x = 5, y = 0 VIEW SOLUTION
 Question 3
$\mathrm{sin}\left[\frac{\mathrm{\pi}}{3}+{\mathrm{sin}}^{1}\left(\frac{1}{2}\right)\right]$ is equal to
(a) 1
(b) $\frac{1}{2}$
(c) $\frac{1}{3}$
(d) $\frac{1}{4}$ VIEW SOLUTION
 Question 4
If for a square matrix, A, A^{2} − A + I = O, then A^{−1} equals
(a) A
(b) A + I
(c) I − A
(d) A − I VIEW SOLUTION
 Question 5
If $\left\begin{array}{ccc}\mathrm{\alpha}& 3& 4\\ 1& 2& 1\\ 1& 4& 1\end{array}\right=0$, then the value of α is
(a) 1
(b) 2
(c) 3
(d) 4 VIEW SOLUTION
 Question 6
If f(x) =  cos x , then f$\left(\frac{3\mathrm{\pi}}{4}\right)\mathrm{is}$
(a) 1
(b) −1
(c) $\frac{1}{\sqrt{2}}$
(d) $\frac{1}{\sqrt{2}}$ VIEW SOLUTION
 Question 7
If x = A cos 4t + B sin 4t, then $\frac{{d}^{2}x}{d{t}^{2}}$ is equal to
(a) x
(b) −x
(c) 16x
(d) −16x VIEW SOLUTION
 Question 8
The function f(x) = [x], where [x] denotes the greatest integer less than or equal to x, is continuous at
(a) x = 1
(b) x = 1.5
(c) x = −2
(d) x = 4 VIEW SOLUTION
 Question 9
The function f(x) = x^{3} + 3x is increasing in interval
(a) (−∞, 0)
(b) (0, ∞)
(c) $\mathrm{\mathbb{R}}$
(d) (0, 1) VIEW SOLUTION
 Question 10
$\underset{1}{\overset{1}{\int}}\frac{\leftx2\right}{x2}dx,x\ne 2$ is equal to
(a) 1
(b) −1
(c) 2
(d) −2 VIEW SOLUTION
 Question 11
$\int \frac{\mathrm{sec}x}{\mathrm{sec}x\mathrm{tan}x}dx$ equals
(a) sec x − tan x + c
(b) sec x + tan x + c
(c) tan x − sec x + c
(d) −(sec x + tan x) + c VIEW SOLUTION
 Question 12
The order and the degree of the differential equation ${\left(1+3\frac{dy}{dx}\right)}^{2}=4\frac{{d}^{3}y}{d{x}^{3}}$ respectively are:
(a) $1,\frac{2}{3}$
(b) 3, 1
(c) 3, 3
(d) 1, 2 VIEW SOLUTION
 Question 13
If $\overrightarrow{a}\xb7\hat{i}=\overrightarrow{a}\xb7(\hat{i}+\hat{j})=\overrightarrow{a}\xb7(\hat{i}+\hat{j}+\hat{k})=1,\mathrm{then}\overrightarrow{\mathit{a}}\mathrm{is}$
$\left(a\right)\hat{k}\phantom{\rule{0ex}{0ex}}\left(b\right)\hat{i}\phantom{\rule{0ex}{0ex}}\left(c\right)\hat{j}\phantom{\rule{0ex}{0ex}}\left(d\right)\hat{i}+\dot{j}+\hat{k}$ VIEW SOLUTION
 Question 14
Five fair coins are tossed simultaneously. The probability of the events that atleast one head comes up is
(a) $\frac{27}{32}$
(b) $\frac{5}{32}$
(c) $\frac{31}{32}$
(d) $\frac{1}{32}$ VIEW SOLUTION
 Question 15
If for any two events A and B, P(A)$=\frac{4}{5}$ and P(A ∩ B)$=\frac{7}{10}$, then P(B/A) is equal to
(a) $\frac{1}{10}$
(b) $\frac{1}{8}$
(c) $\frac{7}{8}$
(d) $\frac{17}{20}$ VIEW SOLUTION
 Question 16
The angle between the lines 2x = 3y = −z and 6x = −y = −4z is
(a) 0°
(b) 30°
(c) 45°
(d) 90° VIEW SOLUTION
 Question 17
If a line makes angles of 90°, 135° and 45° with the x, y and z axes respectively, then its direction cosines are
(a) $0,\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}}$
(b) $\frac{1}{\sqrt{2}},0,\frac{1}{\sqrt{2}}$
(c) $\frac{1}{\sqrt{2}},0,\frac{1}{\sqrt{2}}$
(d) $0,\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}}$ VIEW SOLUTION
 Question 18
 Question 19
Assertion (A): $\underset{2}{\overset{8}{\int}}\frac{\sqrt{10x}}{\sqrt{x}+\sqrt{10x}}dx=3$
Reason (R): $\underset{a}{\overset{b}{\int}}f\left(x\right)dx=\underset{a}{\overset{b}{\int}}f(a+bx)dx$
(a) Both (A) and (R) are true and (R) is the correct explanation of (A).
(b) Both (A) and (R) are true, but (R) is not the correct explanation of (A).
(c) (A) is true and (R) is false.
(d) (A) is false, but (R) is true. VIEW SOLUTION
 Question 20
Assertion (A): Two coins are tossed simultaneously. The probability of getting two heads, if it is known that at least one head comes up, is $\frac{1}{3}$.
Reason (R): Let E and F be two events with a random experiment, then $\mathrm{P}(\mathrm{F}/\mathrm{E})=\frac{\mathrm{P}(\mathrm{E}\cap \mathrm{F})}{\mathrm{P}\left(\mathrm{E}\right)}.$
(a) Both (A) and (R) are true and (R) is the correct explanation of (A).
(b) Both (A) and (R) are true, but (R) is not the correct explanation of (A).
(c) (A) is true and (R) is false.
(d) (A) is false, but (R) is true. VIEW SOLUTION

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