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# Board Paper of Class 12-Commerce 2023 Math Delhi(Set 1) - Solutions

General Instructions:
(i) This question paper contains 38 questions. All questions are compulsory.
(ii) Question paper is divided into FIVE Sections-Section 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 Assertion-Reason 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 case-study.
(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
Section-D and 2 questions in Section - E.
(ix) Use of calculators is NOT allowed.

• Question 1
Let A = {3, 5}. Then number of reflexive relations on A is
(a) 2
(b) 4
(c) 0
(d) 8 VIEW SOLUTION

• Question 2
$\mathrm{sin}\left[\frac{\mathrm{\pi }}{3}+{\mathrm{sin}}^{-1}\left(\frac{1}{2}\right)\right]$ is equal to
(a) 1
(b) $\frac{1}{2}$
(c) $\frac{1}{3}$
(d) $\frac{1}{4}$ VIEW SOLUTION

• Question 3
If for a square matrix, A, A2 − A + I = O, then A−1 equals
(a) A
(b) A + I
(c) I − A
(d) A − I VIEW SOLUTION

• Question 4
If  and A = B2, then x equals
(a) $±1$
(b) −1
(c) 1
(d) 2
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
The derivative of x2x w.r.t. x is
(a) x2−1
(b) 2x2x log x
(c) 2x2x(1 + log x)
(d) 2x2x(1 − log x) VIEW SOLUTION

• Question 7
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 8
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 9
The interval in which the function f(x) = 2x3 + 9x2 + 12x − 1 is decreasing, is
(a)
(b)
(c)
(d)  VIEW SOLUTION

• Question 10
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 11
is equal to
(a) 1
(b) −1
(c) 2
(d) −2 VIEW SOLUTION

• Question 12
The sum of the order and the degree of the differential equation $\frac{d}{dx}\left({\left(\frac{dy}{dx}\right)}^{3}\right)$ is
(a) 2
(b) 3
(c) 5
(d) 0 VIEW SOLUTION

• Question 13
Two vectors  are collinear if
(a) ${a}_{1}{b}_{1}+{a}_{2}{b}_{2}+{a}_{3}{b}_{3}=0$
(b) $\frac{{a}_{1}}{{b}_{1}}=\frac{{a}_{2}}{{b}_{2}}=\frac{{a}_{3}}{{b}_{3}}$
(c)
(d) ${a}_{1}+{a}_{2}+{a}_{3}={b}_{1}+{b}_{2}+{b}_{3}$ VIEW SOLUTION

• Question 14
The magnitude of the vector $6\stackrel{^}{i}-2\stackrel{^}{j}+3\stackrel{^}{k}$ is
(a) 1
(b) 5
(c) 7
​(d) 12 VIEW SOLUTION

• Question 15
If a line makes angles of 90°, 135° and 45° with the x, y and z axes respectively, then its direction cosines are
(a)
(b)
(c)
(d)  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 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 18
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 19
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 $\mathrm{P}\left(\mathrm{F}/\mathrm{E}\right)=\frac{\mathrm{P}\left(\mathrm{E}\cap \mathrm{F}\right)}{\mathrm{P}\left(\mathrm{E}\right)}.$
​(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

• Question 20
Assertion (A): $\underset{2}{\overset{8}{\int }}\frac{\sqrt{10-x}}{\sqrt{x}+\sqrt{10-x}}dx=3$
Reason (R): $\underset{a}{\overset{b}{\int }}f\left(x\right)dx=\underset{a}{\overset{b}{\int }}f\left(a+b-x\right)dx$

(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 21
Write the domain and range (principle value branch) of the following functions:
f(x) = tan –1x VIEW SOLUTION

• Question 22
Find the value (s) of $\text{'}\lambda \text{'}$, if the function

OR

If f(x) = then show that​ f is not differentiable at x = 1. VIEW SOLUTION

• Question 23
Sketch the region bounded by the lines 2x + y = 8, y = 2, y = 4 and the y-axis. Hence, obtain its area using integration. VIEW SOLUTION

• Question 24
If the vectors $\stackrel{\to }{a}$ and $\stackrel{\to }{b}$ are such that  and $\stackrel{\to }{a}×\stackrel{\to }{b}$ is a unit vector, then  find the angle between $\stackrel{\to }{a}$ and $\stackrel{\to }{b}.$

OR

Find the area of a parallelogram whose adjacent sides are determined by the vectors  VIEW SOLUTION

• Question 25
Find the vector and the cartesian equations of a line that passes through the point A(1, 2, –1) and parallel to the line 5x – 25 = 14 – 7= 35z. VIEW SOLUTION

• Question 26
If $\mathrm{A}=\left[\begin{array}{ccc}1& 2& 3\\ 3& -2& 1\\ 4& 2& 1\end{array}\right],$ then show that A3 – 23A – 40I = 0.   VIEW SOLUTION

• Question 27
If y = tan x + sec x, then prove that $\frac{{d}^{2}y}{d{x}^{2}}=\frac{\mathrm{cos}x}{{\left(1-\mathrm{sin}x\right)}^{2}}.$

OR

Differentiate sec–1  VIEW SOLUTION

• Question 28
Evaluate : ${\int }_{0}^{2\pi }\frac{1}{1+{\mathrm{e}}^{\mathrm{sin}x}}\mathrm{d}x$

OR

Find : $\int \frac{{x}^{4}}{\left(x–1\right)\left({x}^{2}+1\right)}\mathrm{d}x$ VIEW SOLUTION

• Question 29
Find the area of the following region using integration :
{(x, y) : y2 ≤ 2x and y ≥ x – 4} VIEW SOLUTION

• Question 30
Three vectors  satisfy the condition $\stackrel{\to }{\mathrm{a}}+\stackrel{\to }{\mathrm{b}}+\stackrel{\to }{\mathrm{c}}=\stackrel{\to }{0}.$
Evaluate the quantity

OR

Find the coordinates of the foot of the perpendicular drawn from the point P(0, 2, 3) to the line $\frac{x+3}{5}=\frac{y–1}{2}=\frac{z+4}{3}.$ VIEW SOLUTION

• Question 31
Find the distance between the lines :
VIEW SOLUTION

• Question 32
The median of an equilateral triangle is increasing at the rate of $2\sqrt{3}$ cm/s. Find the rate at which its side is increasing.

OR

Sum of two numbers is 5. If the sum of the cubes of these numbers is least, then find the sum of the squares of these numbers. VIEW SOLUTION

• Question 34
Solve the following Linear Programming Problem graphically:
Maximize : P = 70x + 40y
subject to: 3x + 2y $\le$ 9,
3x + y $\le$ 9,
$\ge$ 0, y $\ge$0 VIEW SOLUTION

• Question 35
In answering a question on a multiple choice test, a student either knows the answer or guesses. Let be the probability that he knows the answer and be the probability that he guesses. Assuming that a student who guesses at the answer will be correct with probability . What is the probability that the student knows the answer, given that he answered it correctly ?

OR

A box contains 10 tickets, 2 of which carry a prize of ₹ 8 each, 5 of which carry a prize of ₹ 4 each, and remaining 3 carry a prize of ₹ 2 each. If one ticket is drawn at random, find the mean value of the prize. VIEW SOLUTION

• Question 36
Case Study
An organization conducted bike race under two different categories - Boys and Girls. There were 28 participants in all. Among all of them, finally three from category 1 and two from category 2 were selected for the final race. Ravi forms two sets B and G with these participants for his college project.
Let B = {b1, b2, b3} and G = {g1, g2}, where B represents the set of Boys selected and G the set of Girls selected for the final race. Based on the above information, answer the following questions
(I) How many relations are possible from B to G ?
(II) Among all the possible relations from B to G, how many functions can be formed from B to G ?
(III) Let R : B → B be defined by R = {(x, y) : x and y are students of the same sex}. Check if R is an equivalence relation.

OR

(III) A function f : B → G be defined by f = {b1, g1). (b2, g2), (b3, g1)}. Check if f is bijective. Justify your answer. VIEW SOLUTION

• Question 37
Case Study
Gautam buys 5 pens, 3 bags and 1 instrument box and pays a sum of ₹160. From the same shop, Vikram buys 2 pens, 1 bag and 3 instrument boxes and pays a sum of ₹190. Also Ankur buys 1 pen, 2 bags and 4 instrument boxes and pays a sum of ₹ 250.

Based on the above information, answer the following questions:
(I) Convert the given above situation into a matrix equation of the form AX = B.
(II) Find |A|.
(III) Find A–1.

OR

(III) Determine P = A2 – 5A. VIEW SOLUTION

• Question 38
Case Study
An equation involving derivatives of the dependent variable with respect to the independent variables is called a differential equation. A differential equation of the form  is said to be homogeneous if F(x, y) is a homogeneous function of degree zero, whereas a function F(x, y) is a homogenous function of degree n if . To solve a homogeneous differential equation of the type  , we make the substitution = vx and then separate the variables.

Based on the above, answer the following questions:
(I) Show that (x2y2) dx + 2xy dy = 0 is a differential equation of the type $\frac{dy}{dx}=g\left(\frac{y}{x}\right)$.
(II) Solve the above equation to find its general solution. VIEW SOLUTION
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