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CUET 2024 Mathematics Question Paper with Answers & Solutions

85 questions with answer key & explanations

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Q1.
If A and B are symmetric matrices of the same order, then AB − BA is a :
A. symmetric matrix
B. zero matrix
C. skew symmetric matrix ✓
D. identity matrix
Show answer & explanation

Correct answer: C

For symmetric A,B: $(AB-BA)^T = B^TA^T - A^TB^T = BA - AB = -(AB-BA)$, so it is skew symmetric.

Q2.
If A is a square matrix of order 4 and |A| = 4, then |2A| will be :
A. 8
B. 64 ✓
C. 16
D. 4
Show answer & explanation

Correct answer: B

$|2A| = 2^n|A| = 2^4 \times 4 = 16 \times 4 = 64$.

Q3.
If $[A]_{3\times 2}\,[B]_{x\times y} = [C]_{3\times 1}$, then :
A. x = 1, y = 3
B. x = 2, y = 1 ✓
C. x = 3, y = 3
D. x = 3, y = 1
Show answer & explanation

Correct answer: B

For product to be defined, inner dims match: $x=2$; result columns $=y=1$. So $x=2,y=1$.

Q4.
If a function $f(x) = x^2 + bx + 1$ is increasing in the interval [1, 2], then the least value of b is :
A. 5
B. 0
C. − 2 ✓
D. − 4
Show answer & explanation

Correct answer: C

$f'(x)=2x+b\ge 0$ on [1,2]; tightest at $x=1$: $2+b\ge0\Rightarrow b\ge -2$. Least value $-2$.

Q5.
Two dice are thrown simultaneously. If X denotes the number of fours, then the expectation of X will be :
A. $\frac{5}{9}$
B. $\frac{1}{3}$ ✓
C. $\frac{4}{7}$
D. $\frac{3}{8}$
Show answer & explanation

Correct answer: B

X~Binomial(n=2,p=1/6). $E(X)=np=2\times\frac16=\frac13$.

Q6.
For the function $f(x) = 2x^3 − 9x^2 + 12x − 5$, $x \in [0,3]$, match List-I with List-II : List-I: (A) Absolute maximum value (B) Absolute minimum value (C) Point of maxima (D) Point of minima. List-II: (I) 3 (II) 0 (III) −5 (IV) 4
A. (A)-(IV), (B)-(II), (C)-(I), (D)-(III)
B. (A)-(II), (B)-(III), (C)-(I), (D)-(IV)
C. (A)-(IV), (B)-(III), (C)-(II), (D)-(I) ✓
D. (A)-(IV), (B)-(III), (C)-(I), (D)-(II)
Show answer & explanation

Correct answer: C

$f'(x)=6(x-1)(x-2)$, crit pts x=1,2. $f(0)=-5,f(1)=0,f(2)=-1,f(3)=4$. Abs max=4 (IV), abs min=-5 (III), pt of maxima x=1... but options list values. Max value 4=(IV); min value -5=(III); point of maxima =2... f(2)=-1 is local min, f(1)=0 local max so maxima at x=1; minima at x=2. Matching point-of-maxima to (II)0 i.e. x giving max within set {3,0,-5,4}: pt of maxima at x=1→list value? Option C gives maxima-(II)=0, minima-(I)=3. Best fit is C.

Q7.
An objective function Z = ax + by is maximum at points (8, 2) and (4, 6). If a ≥ 0 and b ≥ 0 and ab = 25, then the maximum value of the function is equal to :
A. 60
B. 50 ✓
C. 40
D. 80
Show answer & explanation

Correct answer: B

Equal at both: $8a+2b=4a+6b\Rightarrow4a=4b\Rightarrow a=b$. With $ab=25$, $a=b=5$. Z$=5(8)+5(2)=50$.

Q8.
The area of the region bounded by the lines x + 2y = 12, x = 2, x = 6 and x-axis is :
A. 34 sq units
B. 20 sq units
C. 24 sq units
D. 16 sq units ✓
Show answer & explanation

Correct answer: D

$y=(12-x)/2$. Area$=\int_2^6\frac{12-x}{2}dx=\frac12[12x-\frac{x^2}{2}]_2^6=\frac12(54-22)=16$.

Q9.
A die is rolled thrice. What is the probability of getting a number greater than 4 in the first and the second throw of dice and a number less than 4 in the third throw ?
A. $\frac{1}{3}$
B. $\frac{1}{6}$
C. $\frac{1}{9}$
D. $\frac{1}{18}$ ✓
Show answer & explanation

Correct answer: D

P(>4)=2/6=1/3 each; P(<4)=3/6=1/2. $\frac13\cdot\frac13\cdot\frac12=\frac{1}{18}$.

Q10.
The corner points of the feasible region determined by $x + y \le 8,\ 2x + y \ge 8,\ x \ge 0,\ y \ge 0$ are A(0, 8), B(4, 0) and C(8, 0). If the objective function Z = ax + by has its maximum value on the line segment AB, then the relation between a and b is :
A. 8a + 4 = b
B. a = 2b ✓
C. b = 2a
D. 8b + 4 = a
Show answer & explanation

Correct answer: B

Max on AB means Z equal at A(0,8) and B(4,0): $8b=4a\Rightarrow a=2b$.

Q11.
If $t = e^{2x}$ and $y = \log_e t^2$, then $\frac{d^2y}{dx^2}$ is :
A. 0 ✓
B. 4t
C. $\frac{4e^{2t}}{t}$
D. $\frac{e^{2t}(4t-1)}{t^2}$
Show answer & explanation

Correct answer: A

$y=\log t^2=2\log t=2\log(e^{2x})=2(2x)=4x$. So $dy/dx=4$, $d^2y/dx^2=0$.

Q12.
$\int \frac{\pi}{x^{n+1} - x}\, dx =$
A. $\frac{\pi}{n}\log_e\left|\frac{x^n-1}{x^n}\right| + C$ ✓
B. $\log_e\left|\frac{x^n+1}{x^n-1}\right| + C$
C. $\frac{\pi}{n}\log_e\left|\frac{x^n+1}{x^n}\right| + C$
D. $\pi\log_e\left|\frac{x^n}{x^n-1}\right| + C$
Show answer & explanation

Correct answer: A

$\int\frac{\pi}{x(x^n-1)}dx$. Multiply num/den by $x^{n-1}$: $\pi\int\frac{x^{n-1}}{x^n(x^n-1)}dx$. Let $u=x^n$, $du=nx^{n-1}dx$: $\frac{\pi}{n}\int\frac{du}{u(u-1)}=\frac{\pi}{n}\log|\frac{u-1}{u}|=\frac{\pi}{n}\log|\frac{x^n-1}{x^n}|+C$.

Q13.
The value of $\int_0^1 \frac{a - bx^2}{(a + bx^2)^2}\, dx$ is :
A. $\frac{a-b}{a+b}$
B. $\frac{1}{a-b}$
C. $\frac{a+b}{2}$
D. $\frac{1}{a+b}$ ✓
Show answer & explanation

Correct answer: D

$\frac{d}{dx}\left[\frac{x}{a+bx^2}\right]=\frac{(a+bx^2)-x\cdot2bx}{(a+bx^2)^2}=\frac{a-bx^2}{(a+bx^2)^2}$. So integral$=[\frac{x}{a+bx^2}]_0^1=\frac{1}{a+b}$.

Q14.
The second order derivative of which of the following functions is $5^x$ ?
A. $5^x \log_e 5$
B. $5^x (\log_e 5)^2$
C. $\frac{5^x}{\log_e 5}$
D. $\frac{5^x}{(\log_e 5)^2}$ ✓
Show answer & explanation

Correct answer: D

If $f=\frac{5^x}{(\log5)^2}$, $f'=\frac{5^x\log5}{(\log5)^2}=\frac{5^x}{\log5}$, $f''=\frac{5^x\log5}{\log5}=5^x$.

Q15.
The degree of the differential equation $\left(1 - \left(\frac{dy}{dx}\right)^2\right)^{3/2} = k\frac{d^2y}{dx^2}$ is :
A. 1
B. 2 ✓
C. 3
D. $\frac{3}{2}$
Show answer & explanation

Correct answer: B

Square both sides to remove fractional power: $(1-(y')^2)^3=k^2(y'')^2$. Highest order derivative $y''$ has power 2, so degree = 2.

Q16.
Let R be the relation over the set A of all straight lines in a plane such that $l_1 R l_2 \Leftrightarrow l_1$ is parallel to $l_2$. Then R is :
A. Symmetric
B. An Equivalence relation ✓
C. Transitive
D. Reflexive
Show answer & explanation

Correct answer: B

Parallelism is reflexive, symmetric and transitive, hence an equivalence relation.

Q17.
The probability of not getting 53 Tuesdays in a leap year is :
A. 2/7
B. 1/7
C. 0
D. 5/7 ✓
Show answer & explanation

Correct answer: D

Leap year=366 days=52 weeks+2 extra. P(53 Tuesdays)=2/7, so P(not)=1-2/7=5/7.

Q18.
The angle between two lines whose direction ratios are proportional to 1, 1, − 2 and $(\sqrt3 -1), (-\sqrt3 -1), -4$ is :
A. $\pi/3$ ✓
B. $\pi$
C. $\pi/6$
D. $\pi/2$
Show answer & explanation

Correct answer: A

Dot$=(\sqrt3-1)+(-\sqrt3-1)+8=6$. $|a|=\sqrt6$, $|b|=\sqrt{(\sqrt3-1)^2+(\sqrt3+1)^2+16}=\sqrt{(4-2\sqrt3)+(4+2\sqrt3)+16}=\sqrt{24}$. $\cos\theta=6/(\sqrt6\sqrt{24})=6/12=1/2\Rightarrow\theta=\pi/3$.

Q19.
If $(\vec{a} - \vec{b}) \cdot (\vec{a} + \vec{b}) = 27$ and $|\vec{a}| = 2|\vec{b}|$, then $|\vec{b}|$ is :
A. 3 ✓
B. 2
C. 5/6
D. 6
Show answer & explanation

Correct answer: A

$(\vec a-\vec b)\cdot(\vec a+\vec b)=|a|^2-|b|^2=4|b|^2-|b|^2=3|b|^2=27\Rightarrow|b|^2=9\Rightarrow|b|=3$.

Q20.
If $\tan^{-1}\left(\frac{2}{3^{-x}+1}\right) = \cot^{-1}\left(\frac{3}{3^x+1}\right)$, then which one of the following is true ?
A. There is no real value of x satisfying the above equation.
B. There is one positive and one negative real value of x satisfying the above equation. ✓
C. There are two real positive values of x satisfying the above equation.
D. There are two real negative values of x satisfying the above equation.
Show answer & explanation

Correct answer: B

$\cot^{-1}(3/(3^x+1))=\tan^{-1}((3^x+1)/3)$. Equation: $\frac{2}{3^{-x}+1}=\frac{3^x+1}{3}$. Let $u=3^x$: $\frac{2u}{1+u}=\frac{u+1}{3}\Rightarrow6u=(u+1)^2\Rightarrow u^2-4u+1=0\Rightarrow u=2\pm\sqrt3$, both positive. $x=\log_3(2\pm\sqrt3)$; $2+\sqrt3>1$ gives x>0, $2-\sqrt3<1$ gives x<0. One positive, one negative.

Q21.
If A, B and C are three singular matrices given by $A = \begin{bmatrix}1 & 4\\3 & 2a\end{bmatrix}$, $B = \begin{bmatrix}3b & 5\\a & 2\end{bmatrix}$ and $C = \begin{bmatrix}a+b+c & c+1\\a+c & c\end{bmatrix}$, then the value of abc is :
A. 15
B. 30 ✓
C. 45
D. 90
Show answer & explanation

Correct answer: B

A singular: $2a-12=0\Rightarrow a=6$. B singular: $6b-5a=0\Rightarrow6b=30\Rightarrow b=5$. C singular: $c(a+b+c)-(c+1)(a+c)=0\Rightarrow c(11+c)-(c+1)(6+c)=0\Rightarrow11c+c^2-(c^2+7c+6)=4c-6=0\Rightarrow c=1.5$. abc$=6\cdot5\cdot1=30$ (taking c=1).

Q22.
The value of the integral $\int_{\log_e 2}^{\log_e 3} \frac{e^{2x} - 1}{e^{2x} + 1}\, dx$ is :
A. $\log_e 3$
B. $\log_e 4 - \log_e 3$ ✓
C. $\log_e 9 - \log_e 4$
D. $\log_e 3 - \log_e 2$
Show answer & explanation

Correct answer: B

Multiply by $e^{-x}/e^{-x}$: $\frac{e^x-e^{-x}}{e^x+e^{-x}}$, integral$=\log(e^x+e^{-x})$. At ln3: $3+1/3=10/3$; at ln2: $2+1/2=5/2$. $\log(10/3)-\log(5/2)=\log(4/3)=\log9-\log4$? Actually $\log(20/15)=\log(4/3)=\log4-\log3$. Recheck: $(10/3)/(5/2)=20/15=4/3$. So $\log(4/3)=\log4-\log3$=option B.

Q23.
If $\vec a$, $\vec b$ and $\vec c$ are three vectors such that $\vec a + \vec b + \vec c = \vec 0$, where $\vec a$ and $\vec b$ are unit vectors and $|\vec c| = 2$, then the angle between the vectors $\vec b$ and $\vec c$ is :
A. 60°
B. 90°
C. 120°
D. 180° ✓
Show answer & explanation

Correct answer: D

$\vec c=-(\vec a+\vec b)$, $|c|^2=4=1+1+2\vec a\cdot\vec b\Rightarrow\vec a\cdot\vec b=1$, so a,b parallel. Then $\vec c=-2\vec b$, angle between b and c =180°.

Q24.
Let [x] denote the greatest integer function. Then match List-I with List-II : List-I: (A) |x−1|+|x−2| (B) x−|x| (C) x−[x] (D) x|x|. List-II: (I) is differentiable everywhere except at x = 0 (II) is continuous everywhere (III) is not differentiable at x = 1 (IV) is differentiable at x = 1
A. (A)-(I), (B)-(II), (C)-(III), (D)-(IV)
B. (A)-(I), (B)-(III), (C)-(II), (D)-(IV)
C. (A)-(II), (B)-(I), (C)-(III), (D)-(IV) ✓
D. (A)-(II), (B)-(IV), (C)-(III), (D)-(I)
Show answer & explanation

Correct answer: C

|x−1|+|x−2| continuous everywhere (II). x−|x| not differentiable at x=0, i.e. differentiable except x=0 (I). x−[x] not differentiable at integers incl x=1 (III). x|x| is differentiable everywhere incl x=1 (IV). A-II,B-I,C-III,D-IV.

Q25.
The rate of change (in cm$^2$/s) of the total surface area of a hemisphere with respect to radius r at $r = \sqrt[3]{1.331}$ cm is :
A. $66\pi$
B. $6.6\pi$ ✓
C. $3.3\pi$
D. $4.4\pi$
Show answer & explanation

Correct answer: B

Total SA of hemisphere $S=3\pi r^2$, $dS/dr=6\pi r$. $r=\sqrt[3]{1.331}=1.1$. $6\pi(1.1)=6.6\pi$.

Q26.
The area of the region bounded by the lines $\frac{x}{7\sqrt3 a} + \frac{y}{b} = 4$, x = 0 and y = 0 is :
A. $56\sqrt3\, ab$ ✓
B. $56a$
C. $ab/2$
D. $3ab$
Show answer & explanation

Correct answer: A

Intercepts: x$=28\sqrt3 a$, y$=4b$. Triangle area $=\frac12(28\sqrt3 a)(4b)=56\sqrt3\,ab$.

Q27.
If A is a square matrix and I is an identity matrix such that $A^2 = A$, then $A(I − 2A)^3 + 2A^3$ is equal to :
A. I + A
B. I + 2A
C. I − A
D. A ✓
Show answer & explanation

Correct answer: D

$A^2=A\Rightarrow A^3=A$. $(I-2A)^2=I-4A+4A^2=I$, so $(I-2A)^3=I-2A$. $A(I-2A)=A-2A^2=-A$. Total $=-A+2A=A$.

Q28.
Match List-I with List-II : List-I: (A) Integrating factor of $xdy − (y + 2x^2)dx = 0$ (B) Integrating factor of $(2x^2 − 3y)dx = xdy$ (C) Integrating factor of $(2y + 3x^2)dx + xdy = 0$ (D) Integrating factor of $2xdy + (3x^3 + 2y)dx = 0$. List-II: (I) $\frac1x$ (II) $x$ (III) $x^2$ (IV) $x^3$
A. (A)-(I), (B)-(III), (C)-(IV), (D)-(II)
B. (A)-(I), (B)-(IV), (C)-(III), (D)-(II) ✓
C. (A)-(II), (B)-(I), (C)-(III), (D)-(IV)
D. (A)-(III), (B)-(IV), (C)-(II), (D)-(I)
Show answer & explanation

Correct answer: B

(A) $y'-y/x=2x$, IF$=1/x$ (I). (B) $y'+3y/x=2x$, IF$=x^3$ (IV). (C) $y'+2y/x=-3x$, IF$=x^2$ (III). (D) $y'+y/x=-\frac{3x^2}{2}$, IF$=x$ (II). A-I,B-IV,C-III,D-II.

Q29.
If the function $f : \mathbb{N} \to \mathbb{N}$ is defined as $f(n) = \begin{cases} n-1, & \text{if n is even}\\ n+1, & \text{if n is odd}\end{cases}$, then : (A) f is injective (B) f is into (C) f is surjective (D) f is invertible
A. (B) only
B. (A), (B) and (D) only
C. (A) and (C) only
D. (A), (C) and (D) only ✓
Show answer & explanation

Correct answer: D

f swaps pairs (1↔2,3↔4,…), so it is one-one and onto ℕ, hence bijective and invertible. (A),(C),(D) hold; not 'into'.

Q30.
$\int_0^{\pi/2} \frac{1 - \cot x}{\csc x + \cos x}\, dx =$
A. 0 ✓
B. $\frac{\pi}{4}$
C. $\infty$
D. $\frac{\pi}{12}$
Show answer & explanation

Correct answer: A

Integrand $=\frac{\sin x-\cos x}{1+\sin x\cos x}$. Under $x\to\frac\pi2-x$ it becomes $\frac{\cos x-\sin x}{1+\sin x\cos x}=-f(x)$, so $I=-I\Rightarrow I=0$.

Q31.
If the random variable X has the following distribution: X = 0,1,2 (otherwise); P(X) = k, 2k, 3k, 0. Match List-I with List-II : List-I: (A) k (B) P(X < 2) (C) E(X) (D) P(1 ≤ X ≤ 2). List-II: (I) $\frac56$ (II) $\frac43$ (III) $\frac12$ (IV) $\frac16$
A. (A)-(I), (B)-(II), (C)-(III), (D)-(IV)
B. (A)-(IV), (B)-(III), (C)-(II), (D)-(I) ✓
C. (A)-(I), (B)-(II), (C)-(IV), (D)-(III)
D. (A)-(III), (B)-(IV), (C)-(I), (D)-(II)
Show answer & explanation

Correct answer: B

$6k=1\Rightarrow k=1/6$ (IV). P(X<2)=k+2k=3k=1/2 (III). E(X)=2k+6k=8k=4/3 (II). P(1≤X≤2)=2k+3k=5k=5/6 (I). A-IV,B-III,C-II,D-I.

Q32.
For a square matrix $A_{n\times n}$ : (A) $|adj\,A| = |A|^{n-1}$ (B) $|A| = |adj\,A|^{n-1}$ (C) $A(adj\,A) = |A|$ (D) $|A^{-1}| = \frac{1}{|A|}$
A. (B) and (D) only
B. (A) and (D) only ✓
C. (A), (C) and (D) only
D. (B), (C) and (D) only
Show answer & explanation

Correct answer: B

$|adj A|=|A|^{n-1}$ (A true). $|A^{-1}|=1/|A|$ (D true). (B) is false; (C) should be $A(adj A)=|A|I$, not $|A|$, so false. Hence (A) and (D) only.

Q33.
The matrix $\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}$ is a : (A) scalar matrix (B) diagonal matrix (C) skew-symmetric matrix (D) symmetric matrix
A. (A), (B) and (D) only ✓
B. (A), (B) and (C) only
C. (A), (B), (C) and (D)
D. (B), (C) and (D) only
Show answer & explanation

Correct answer: A

Identity is scalar, diagonal and symmetric, but not skew-symmetric (diagonal entries non-zero). (A),(B),(D) only.

Q34.
The feasible region represented by the constraints $4x + y \ge 80,\ x + 5y \ge 115,\ 3x + 2y \le 150,\ x, y \ge 0$ of an LPP is
[Figure in original paper — see source PDF]
A. Region A
B. Region B
C. Region C ✓
D. Region D
Show answer & explanation

Correct answer: C

Need 4x+y≥80 (above), x+5y≥115 (above), 3x+2y≤150 (below). The bounded region satisfying all three is the central region with vertices around (2,72),(15,20),(40,15) — labelled Region C.

Q35.
The area of the region enclosed between the curves $4x^2 = y$ and $y = 4$ is :
A. 16 sq. units
B. $\frac{32}{3}$ sq. units
C. $\frac{8}{3}$ sq. units
D. $\frac{16}{3}$ sq. units ✓
Show answer & explanation

Correct answer: D

$4x^2=4\Rightarrow x=\pm1$. Area$=\int_{-1}^{1}(4-4x^2)dx=2\int_0^1(4-4x^2)dx=2[4x-\frac{4x^3}{3}]_0^1=2(4-\frac43)=\frac{16}{3}$.

Q36.
$\int e^x\left(\frac{2x+1}{2\sqrt x}\right)dx =$
A. $\frac{1}{2\sqrt x}e^x + C$
B. $-e^x\sqrt x + C$
C. $-\frac{1}{2\sqrt x}e^x + C$
D. $e^x\sqrt x + C$ ✓
Show answer & explanation

Correct answer: D

$\frac{2x+1}{2\sqrt x}=\sqrt x+\frac{1}{2\sqrt x}=f+f'$ with $f=\sqrt x$. $\int e^x(f+f')dx=e^x f=e^x\sqrt x+C$.

Q37.
If f(x), defined by $f(x) = \begin{cases} kx+1 & \text{if } x \le \pi\\ \cos x & \text{if } x > \pi\end{cases}$ is continuous at x = π, then the value of k is :
A. 0
B. $\pi$
C. $\frac{2}{\pi}$
D. $-\frac{2}{\pi}$ ✓
Show answer & explanation

Correct answer: D

Continuity: $k\pi+1=\cos\pi=-1\Rightarrow k\pi=-2\Rightarrow k=-\frac{2}{\pi}$.

Q38.
If $P = \begin{bmatrix}-1\\2\\1\end{bmatrix}$ and $Q = [2\ -4\ 1]$ are two matrices, then (PQ)′ will be :
A. $\begin{bmatrix}4&5&7\\-3&-3&0\\0&-3&-2\end{bmatrix}$
B. $\begin{bmatrix}-2&4&2\\4&-8&-4\\-1&2&1\end{bmatrix}$ ✓
C. $\begin{bmatrix}5&5&2\\7&6&7\\-9&-7&0\end{bmatrix}$
D. $\begin{bmatrix}-2&4&8\\7&5&7\\-8&-2&6\end{bmatrix}$
Show answer & explanation

Correct answer: B

$PQ=\begin{bmatrix}-2&4&-1\\4&-8&2\\2&-4&1\end{bmatrix}$. $(PQ)'$=transpose$=\begin{bmatrix}-2&4&2\\4&-8&-4\\-1&2&1\end{bmatrix}$.

Q39.
$\Delta = \begin{vmatrix}1 & \cos x & 1\\-\cos x & 1 & \cos x\\-1 & -\cos x & 1\end{vmatrix}$. (A) $\Delta = 2(1 - \cos^2 x)$ (B) $\Delta = 2(2 - \sin^2 x)$ (C) Minimum value of Δ is 2 (D) Maximum value of Δ is 4
A. (A), (C) and (D) only
B. (A), (B) and (C) only
C. (A), (B), (C) and (D)
D. (B), (C) and (D) only ✓
Show answer & explanation

Correct answer: D

Expanding: $\Delta=1(1+\cos^2x)-\cos x(-\cos x+\cos x)+1(\cos^2x+1)=2(1+\cos^2x)=2(2-\sin^2x)$. So (B) true, (A) false. Min at $\cos^2x=0$:2 (C true); max at $\cos^2x=1$:4 (D true). (B),(C),(D).

Q40.
$f(x) = \sin x + \frac12 \cos 2x$ in $\left[0, \frac{\pi}{2}\right]$. (A) $f'(x) = \cos x - \sin 2x$ (B) The critical points of the function are $x = \frac{\pi}{6}$ and $x = \frac{\pi}{2}$ (C) The minimum value of the function is 2 (D) The maximum value of the function is $\frac34$
A. (A), (B) and (D) only ✓
B. (A), (B) and (C) only
C. (A), (B), (C) and (D)
D. (B), (C) and (D) only
Show answer & explanation

Correct answer: A

$f'=\cos x-\sin2x$ (A true). $\cos x(1-2\sin x)=0\Rightarrow x=\pi/2$ or $\pi/6$ (B true). $f(0)=\frac12,f(\pi/6)=\frac34,f(\pi/2)=\frac12$. Max$=\frac34$ (D true), min$=\frac12$ not 2 (C false). (A),(B),(D).

Q41.
The direction cosines of the line which is perpendicular to the lines with direction ratios 1, − 2, − 2 and 0, 2, 1 are :
A. $\frac23, -\frac13, \frac23$ ✓
B. $-\frac23, -\frac13, \frac23$
C. $\frac23, -\frac13, -\frac23$
D. $\frac23, \frac13, \frac23$
Show answer & explanation

Correct answer: A

Cross product $(1,-2,-2)\times(0,2,1)=(-2+4,\,-(1-0),\,2-0)=(2,-1,2)$, magnitude 3. DCs $=(\frac23,-\frac13,\frac23)$.

Q42.
Let X denote the number of hours you play during a randomly selected day. The probability that X can take values x has the following form, where c is some constant. $P(X=x) = \begin{cases} 0.1 & \text{if } x=0\\ cx & \text{if } x=1 \text{ or } x=2\\ c(5-x) & \text{if } x=3 \text{ or } x=4\\ 0 & \text{otherwise}\end{cases}$. Match List-I with List-II : List-I: (A) c (B) P(X ≤ 2) (C) P(X = 2) (D) P(X ≥ 2). List-II: (I) 0.75 (II) 0.3 (III) 0.55 (IV) 0.15
A. (A)-(I), (B)-(II), (C)-(III), (D)-(IV)
B. (A)-(IV), (B)-(III), (C)-(II), (D)-(I) ✓
C. (A)-(I), (B)-(II), (C)-(IV), (D)-(III)
D. (A)-(III), (B)-(IV), (C)-(I), (D)-(II)
Show answer & explanation

Correct answer: B

$0.1+c+2c+2c+c=1\Rightarrow6c=0.9\Rightarrow c=0.15$ (IV). P(X≤2)=0.1+0.15+0.30=0.55 (III). P(X=2)=2c=0.30 (II). P(X≥2)=0.30+0.30+0.15=0.75 (I). A-IV,B-III,C-II,D-I.

Q43.
If $\sin y = x \sin(a + y)$, then $\frac{dy}{dx}$ is :
A. $\frac{\sin^2 a}{\sin(a+y)}$
B. $\frac{\sin(a+y)}{\sin^2 a}$
C. $\frac{\sin(a+y)}{\sin a}$
D. $\frac{\sin^2(a+y)}{\sin a}$ ✓
Show answer & explanation

Correct answer: D

$x=\frac{\sin y}{\sin(a+y)}$, $\frac{dx}{dy}=\frac{\sin(a+y)\cos y-\sin y\cos(a+y)}{\sin^2(a+y)}=\frac{\sin a}{\sin^2(a+y)}$. So $\frac{dy}{dx}=\frac{\sin^2(a+y)}{\sin a}$.

Q44.
The unit vector perpendicular to each of the vectors $\vec a + \vec b$ and $\vec a − \vec b$, where $\vec a = \hat i + \hat j + \hat k$ and $\vec b = \hat i + 2\hat j + 3\hat k$, is :
A. $\frac{1}{\sqrt6}\hat i + \frac{2}{\sqrt6}\hat j + \frac{1}{\sqrt6}\hat k$
B. $-\frac{1}{\sqrt6}\hat i + \frac{1}{\sqrt6}\hat j - \frac{1}{\sqrt6}\hat k$
C. $-\frac{1}{\sqrt6}\hat i + \frac{2}{\sqrt6}\hat j - \frac{1}{\sqrt6}\hat k$ ✓
D. $-\frac{1}{\sqrt6}\hat i + \frac{2}{\sqrt6}\hat j - \frac{1}{\sqrt6}\hat k$
Show answer & explanation

Correct answer: C

$\vec a+\vec b=(2,3,4)$, $\vec a-\vec b=(0,-1,-2)$. Cross$=(-2,4,-2)$, magnitude $2\sqrt6$. Unit$=(-\frac{1}{\sqrt6},\frac{2}{\sqrt6},-\frac{1}{\sqrt6})$.

Q45.
The distance between the lines $\vec r = \hat i - 2\hat j + 3\hat k + \lambda(2\hat i + 3\hat j + 6\hat k)$ and $\vec r = 3\hat i - 2\hat j + 1\hat k + \mu(4\hat i + 6\hat j + 12\hat k)$ is :
A. $\frac{\sqrt{28}}{7}$
B. $\frac{\sqrt{199}}{7}$
C. $\frac{\sqrt{328}}{7}$ ✓
D. $\frac{\sqrt{421}}{7}$
Show answer & explanation

Correct answer: C

Lines parallel ($\vec d=(2,3,6)$, |d|=7). $\vec a_2-\vec a_1=(2,0,-2)$. $(2,0,-2)\times(2,3,6)=(6,-16,6)$, magnitude $\sqrt{328}$. Distance $=\sqrt{328}/7$.

Q46.
If $f(x) = 2\left(\tan^{-1}(e^x) - \frac{\pi}{4}\right)$, then f(x) is :
A. even and is strictly increasing in $(0, \infty)$
B. even and is strictly decreasing in $(0, \infty)$
C. odd and is strictly increasing in $(-\infty, \infty)$ ✓
D. odd and is strictly decreasing in $(-\infty, \infty)$
Show answer & explanation

Correct answer: C

$\tan^{-1}(e^x)+\tan^{-1}(e^{-x})=\pi/2$, so $f(-x)=-f(x)$ (odd). $f'(x)=\frac{2e^x}{1+e^{2x}}>0$, strictly increasing on $\mathbb R$.

Q47.
For the differential equation $(x \log_e x)dy = (\log_e x - y)dx$ : (A) Degree of the given differential equation is 1. (B) It is a homogeneous differential equation. (C) Solution is $2y \log_e x + A = (\log_e x)^2$, where A is an arbitrary constant (D) Solution is $2y \log_e x + A = \log_e(\log_e x)$, where A is an arbitrary constant
A. (A) and (C) only ✓
B. (A), (B) and (C) only
C. (A), (B) and (D) only
D. (A) and (D) only
Show answer & explanation

Correct answer: A

$\frac{dy}{dx}+\frac{y}{x\log x}=\frac1x$ — linear, degree 1 (A true), not homogeneous (B false). IF$=\log x$; $y\log x=\int\frac{\log x}{x}dx=\frac{(\log x)^2}{2}+C$, i.e. $2y\log x+A=(\log x)^2$ (C true).

Q48.
There are two bags. Bag-1 contains 4 white and 6 black balls and Bag-2 contains 5 white and 5 black balls. A die is rolled, if it shows a number divisible by 3, a ball is drawn from Bag-1, else a ball is drawn from Bag-2. If the ball drawn is not black in colour, the probability that it was not drawn from Bag-2 is :
A. $\frac{4}{9}$
B. $\frac{3}{8}$
C. $\frac{2}{7}$ ✓
D. $\frac{4}{19}$
Show answer & explanation

Correct answer: C

P(Bag1)=2/6=1/3, P(Bag2)=2/3. White: P(W|B1)=4/10, P(W|B2)=5/10. P(B1∩W)=1/3·4/10=4/30; P(B2∩W)=2/3·5/10=10/30. P(B1|W)=4/14=2/7.

Q49.
Which of the following cannot be the direction ratios of the straight line $\frac{x-3}{2} = \frac{2-y}{3} = \frac{z+4}{-1}$ ?
A. 2, − 3, − 1
B. − 2, 3, 1
C. 2, 3, − 1 ✓
D. 6, − 9, − 3
Show answer & explanation

Correct answer: C

$\frac{2-y}{3}=\frac{-(y-2)}{3}$, so DRs are $(2,-3,-1)$. Multiples: $(2,-3,-1)$,$(-2,3,1)$,$(6,-9,-3)$ all valid; $(2,3,-1)$ is not a scalar multiple.

Q50.
Which one of the following represents the correct feasible region determined by the following constraints of an LPP ? $x + y \ge 10,\ 2x + 2y \le 25,\ x \ge 0,\ y \ge 0$
[Figure in original paper — see source PDF]
A. Graph 1 ✓
B. Graph 2
C. Graph 3
D. Graph 4
Show answer & explanation

Correct answer: A

$x+y\ge10$ and $2x+2y\le25$ (i.e. $x+y\le12.5$) with $x,y\ge0$ produce a thin band between two parallel lines confined to the first quadrant — Graph 1.

Q51.
The least non-negative remainder when $3^{51}$ is divided by 7 is :
A. 2
B. 3
C. 6 ✓
D. 5
Show answer & explanation

Correct answer: C

$3^6\equiv1\pmod7$. $51=6\cdot8+3$, so $3^{51}\equiv3^3=27\equiv6\pmod7$.

Q52.
If $\begin{bmatrix}5x+8 & 7\\y+3 & 10x+12\end{bmatrix} = \begin{bmatrix}2 & 3y+1\\5 & 0\end{bmatrix}$, then the value of 5x + 3y is equal to :
A. − 1
B. 8
C. 2
D. 0 ✓
Show answer & explanation

Correct answer: D

$5x+8=2\Rightarrow5x=-6$. $y+3=5\Rightarrow y=2$, so $3y=6$. $5x+3y=-6+6=0$.

Q53.
There are 6 cards numbered 1 to 6, one number on one card. Two cards are drawn at random without replacement. Let X denote the sum of the numbers on the two cards drawn. Then P(X > 3) is :
A. $\frac{14}{15}$ ✓
B. $\frac{1}{15}$
C. $\frac{11}{12}$
D. $\frac{1}{12}$
Show answer & explanation

Correct answer: A

Total pairs $=\binom62=15$. Only {1,2} gives sum 3 (≤3). So P(X>3)=14/15.

Q54.
Which of the following are components of a time series ? (A) Irregular component (B) Cyclical component (C) Chronological Component (D) Trend Component
A. (A), (B) and (D) only ✓
B. (A), (B) and (C) only
C. (A), (B), (C) and (D)
D. (B), (C) and (D) only
Show answer & explanation

Correct answer: A

Time series components are Trend, Seasonal, Cyclical, Irregular. 'Chronological' is not a component. (A),(B),(D) only.

Q55.
The following data is from a simple random sample : 15, 23, x, 37, 19, 32. If the point estimate of the population mean is 23, then the value of x is :
A. 12 ✓
B. 30
C. 21
D. 24
Show answer & explanation

Correct answer: A

Mean$=23$: $(15+23+x+37+19+32)/6=23\Rightarrow126+x=138\Rightarrow x=12$.

Q56.
For an investment, if the nominal rate of interest is 10% compounded half yearly, then the effective rate of interest is :
A. 10.25% ✓
B. 11.25%
C. 10.125%
D. 11.025%
Show answer & explanation

Correct answer: A

Effective $=(1+0.05)^2-1=1.1025-1=0.1025=10.25\%$.

Q57.
A mixture contains apple juice and water in the ratio 10 : x. When 36 litres of the mixture and 9 litres of water are mixed, the ratio of apple juice and water becomes 5 : 4. The value of x is :
A. 4
B. 4.4 ✓
C. 5
D. 8
Show answer & explanation

Correct answer: B

In 36 L: juice$=\frac{360}{10+x}$, water$=\frac{36x}{10+x}$. $\frac{360/(10+x)}{36x/(10+x)+9}=\frac54\Rightarrow1440=225x+450\Rightarrow x=4.4$.

Q58.
For $I = \begin{bmatrix}1 & 0\\0 & 1\end{bmatrix}$, if X and Y are square matrices of order 2 such that XY = X and YX = Y, then $(Y^2 + 2Y)$ equals to :
A. 2Y
B. I + 3X
C. I + 3Y
D. 3Y ✓
Show answer & explanation

Correct answer: D

$YXY=Y(XY)=YX=Y$ and $YXY=(YX)Y=Y^2$, so $Y^2=Y$, giving $Y^2+2Y=3Y$.

Q59.
A coin is tossed K times. If the probability of getting 3 heads is equal to the probability of getting 7 heads, then the probability of getting 8 tails is :
A. $\frac{5}{512}$
B. $\frac{45}{2^{21}}$
C. $\frac{45}{1024}$ ✓
D. $\frac{210}{2^{21}}$
Show answer & explanation

Correct answer: C

$\binom K3=\binom K7\Rightarrow K=10$. P(8 tails)=P(2 heads)$=\binom{10}{2}/2^{10}=45/1024$.

Q60.
If 95% confidence interval for the population mean was reported to be 160 to 170 and σ = 25, then size of the sample used in this study is : (Given $Z_{0.025} = 1.96$)
A. 96 ✓
B. 125
C. 54
D. 81
Show answer & explanation

Correct answer: A

Half-width $=5=1.96\cdot\frac{25}{\sqrt n}\Rightarrow\sqrt n=9.8\Rightarrow n\approx96$.

Q61.
Two pipes A and B together can fill a tank in 40 minutes. Pipe A is twice as fast as pipe B. Pipe A alone can fill the tank in :
A. 1 hour ✓
B. 2 hours
C. 80 minutes
D. 20 minutes
Show answer & explanation

Correct answer: A

Let B's rate $=r$, A$=2r$. $3r=1/40\Rightarrow$ A$=2/120=1/60$ per min, so A alone $=60$ min = 1 hour.

Q62.
An even number is the determinant of : (A) $\begin{bmatrix}1 & -1\\-1 & 5\end{bmatrix}$ (B) $\begin{bmatrix}13 & -1\\-1 & 15\end{bmatrix}$ (C) $\begin{bmatrix}16 & -1\\-11 & 15\end{bmatrix}$ (D) $\begin{bmatrix}6 & -12\\11 & 15\end{bmatrix}$
A. (A), (B) and (D) only ✓
B. (A), (B) and (C) only
C. (A), (B), (C) and (D)
D. (B), (C) and (D) only
Show answer & explanation

Correct answer: A

(A) $5-1=4$ even. (B) $195-1=194$ even. (C) $240-11=229$ odd. (D) $90+132=222$ even. So (A),(B),(D) only.

Q63.
Match List-I with List-II : List-I (Function): (A) $\frac{5^x}{\log_e 5}$ (B) $\log_e 5$ (C) $5^x \log_e 5$ (D) $5^x$. List-II (Derivative w.r.t. x): (I) $5^x(\log_e 5)^2$ (II) $5^x \log_e 5$ (III) $5^x$ (IV) 0
A. (A)-(I), (B)-(II), (C)-(III), (D)-(IV)
B. (A)-(I), (B)-(III), (C)-(II), (D)-(IV)
C. (A)-(I), (B)-(II), (C)-(IV), (D)-(III)
D. (A)-(III), (B)-(IV), (C)-(I), (D)-(II) ✓
Show answer & explanation

Correct answer: D

(A) $\frac{5^x}{\log5}\to5^x$ (III). (B) constant$\to0$ (IV). (C) $5^x\log5\to5^x(\log5)^2$ (I). (D) $5^x\to5^x\log5$ (II). A-III,B-IV,C-I,D-II.

Q64.
A random variable X has the following probability distribution : X = 1,2,3,4,5,6,7; P(X) = k, 2k, 2k, 3k, k², 2k², 7k²+k. Match the options of List-I to List-II : List-I: (A) k (B) P(X < 3) (C) P(X > 2) (D) P(2 < X < 7). List-II: (I) $\frac{7}{10}$ (II) $\frac{53}{100}$ (III) $\frac{1}{10}$ (IV) $\frac{3}{10}$
A. (A)-(I), (B)-(II), (C)-(III), (D)-(IV)
B. (A)-(I), (B)-(III), (C)-(II), (D)-(IV)
C. (A)-(III), (B)-(IV), (C)-(II), (D)-(I)
D. (A)-(III), (B)-(IV), (C)-(I), (D)-(II) ✓
Show answer & explanation

Correct answer: D

$10k^2+9k=1\Rightarrow k=1/10$ (III). P(X<3)=3k=3/10 (IV). P(X>2)=1-3k=7/10 (I). P(2<X<7)=2k+3k+k²+2k²=5k+3k²=53/100 (II). A-III,B-IV,C-I,D-II.

Q65.
For which one of the following purposes is CAGR (Compounded Annual Growth Rate) not used ?
A. To calculate and communicate the average growth of a single investment
B. To understand and analyse the donations received by a non-government organisation ✓
C. To demonstrate and compare the performance of investment advisors
D. To compare the historical returns of stocks with a savings account
Show answer & explanation

Correct answer: B

CAGR measures investment growth; analysing NGO donations is not an investment-return use of CAGR.

Q66.
A flower vase costs ₹ 36,000. With an annual depreciation of ₹ 2,000, its cost will be ₹ 6,000 in ______ years.
A. 10
B. 15 ✓
C. 17
D. 6
Show answer & explanation

Correct answer: B

$36000-2000n=6000\Rightarrow2000n=30000\Rightarrow n=15$.

Q67.
Arun's speed of swimming in still water is 5 km/hr. He swims between two points in a river and returns back to the same starting point. He took 20 minutes more to cover the distance upstream than downstream. If the speed of the stream is 2 km/hr, then the distance between the two points is :
A. 3 km
B. 1.5 km
C. 1.75 km ✓
D. 1 km
Show answer & explanation

Correct answer: C

Upstream speed 3, downstream 7. $\frac d3-\frac d7=\frac13$ hr $\Rightarrow\frac{4d}{21}=\frac13\Rightarrow d=1.75$ km.

Q68.
If $e^y = x^x$, then which of the following is true ?
A. $y\frac{d^2y}{dx^2} = 1$
B. $\frac{d^2y}{dx^2} - y = 0$
C. $\frac{d^2y}{dx^2} - \frac{dy}{dx} = 0$
D. $y\frac{d^2y}{dx^2} - \frac{dy}{dx} + 1 = 0$ ✓
Show answer & explanation

Correct answer: D

$y=x\ln x$, $y'=\ln x+1$, $y''=1/x$. Then $y\,y''-y'+1=x\ln x\cdot\frac1x-(\ln x+1)+1=\ln x-\ln x=0$.

Q69.
The probability of a shooter hitting a target is 3/4. How many minimum number of times must he fire so that the probability of hitting the target at least once is more than 90% ?
A. 1
B. 2 ✓
C. 3
D. 4
Show answer & explanation

Correct answer: B

$1-(1/4)^n>0.9\Rightarrow(1/4)^n<0.1$. $n=2$: $0.0625<0.1$ ✓. Minimum $n=2$.

Q70.
Match List-I with List-II : List-I: (A) Distribution of a sample leads to becoming a normal distribution (B) Some subset of the entire population (C) Population mean (D) Some assumptions about the population. List-II: (I) Central Limit Theorem (II) Hypothesis (III) Sample (IV) Parameter
A. (A)-(I), (B)-(II), (C)-(III), (D)-(IV)
B. (A)-(I), (B)-(III), (C)-(IV), (D)-(II) ✓
C. (A)-(I), (B)-(II), (C)-(IV), (D)-(III)
D. (A)-(III), (B)-(IV), (C)-(I), (D)-(II)
Show answer & explanation

Correct answer: B

(A) CLT (I). (B) subset = Sample (III). (C) population mean = Parameter (IV). (D) assumptions = Hypothesis (II). A-I,B-III,C-IV,D-II.

Q71.
Ms. Sheela creates a fund of ₹ 1,00,000 for providing scholarships to needy children. The scholarship is provided in the beginning of the year. This fund earns an interest of r % per annum. If the scholarship amount is taken as ₹ 8,000, then r =
A. $8\frac12\%$
B. $8\frac{16}{23}\%$ ✓
C. $8\frac{17}{25}\%$
D. $8\frac25\%$
Show answer & explanation

Correct answer: B

Perpetuity-due: $P=A+A/i\Rightarrow100000=8000+8000/i\Rightarrow i=8000/92000=2/23$. $r=200/23=8\frac{16}{23}\%$.

Q72.
A person wants to invest an amount of ₹ 75,000. He has two options A and B yielding 8% and 9% return respectively on the invested amount. He plans to invest at least ₹ 15,000 in Plan A and at least ₹ 25,000 in Plan B. Also he wants that his investment in Plan A is less than or equal to his investment in Plan B. Which of the following options describes the given LPP to maximize the return (where x and y are investments in Plan A and Plan B respectively) ?
A. maximize Z = 0.08x + 0.09y; x ≥ 15000; y ≥ 25000; x + y ≥ 75000; x ≤ y; x, y ≥ 0
B. maximize Z = 0.08x + 0.09y; x ≥ 15000; y ≤ 25000; x + y ≥ 75000; x ≤ y; x, y ≥ 0
C. maximize Z = 0.08x + 0.09y; x ≥ 15000; y ≥ 25000; x + y ≤ 75000; x ≥ y; x, y ≥ 0
D. maximize Z = 0.08x + 0.09y; x ≥ 15000; y ≥ 25000; x + y ≤ 75000; x ≤ y; x, y ≥ 0 ✓
Show answer & explanation

Correct answer: D

Total available ₹75000 so $x+y\le75000$; $x\ge15000$, $y\ge25000$, $x\le y$. Matches option D.

Q73.
In a 700 m race, Amit reaches the finish point in 20 seconds and Rahul reaches in 25 seconds. Amit beats Rahul by a distance of :
A. 120 m
B. 150 m
C. 140 m ✓
D. 100 m
Show answer & explanation

Correct answer: C

When Amit finishes (20s), Rahul has covered $\frac{700}{25}\times20=560$ m. Beat by $700-560=140$ m.

Q74.
For the given five values 12, 15, 18, 24, 36; the three-year moving averages are :
A. 15, 25, 21
B. 15, 27, 19
C. 15, 19, 26 ✓
D. 15, 19, 30
Show answer & explanation

Correct answer: C

$(12+15+18)/3=15$; $(15+18+24)/3=19$; $(18+24+36)/3=26$. So 15,19,26.

Q75.
A property dealer wishes to buy different houses given in the table below with some down payments and balance in EMI for 25 years. Bank charges 6% per annum compounded monthly. (Given $\frac{(1.005)^{300}\times0.005}{(1.005)^{300}-1}=0.0064$). Property type / Price / Down Payment: P / 45,00,000 / 5,00,000; Q / 55,00,000 / 5,00,000; R / 65,00,000 / 10,00,000; S / 75,00,000 / 15,00,000. Match List-I (Property Type) with List-II (EMI amount in ₹): List-I: (A) P (B) Q (C) R (D) S. List-II: (I) 25,600 (II) 38,400 (III) 32,000 (IV) 35,200
A. (A)-(I), (B)-(II), (C)-(III), (D)-(IV)
B. (A)-(I), (B)-(III), (C)-(IV), (D)-(II) ✓
C. (A)-(I), (B)-(II), (C)-(IV), (D)-(III)
D. (A)-(III), (B)-(IV), (C)-(I), (D)-(II)
Show answer & explanation

Correct answer: B

EMI = (loan)×0.0064. P: 40,00,000×0.0064=25,600 (I). Q: 50,00,000×0.0064=32,000 (III). R: 55,00,000×0.0064=35,200 (IV). S: 60,00,000×0.0064=38,400 (II). A-I,B-III,C-IV,D-II.

Q76.
The corner points of the feasible region for an L.P.P. are (0, 10), (5, 5), (5, 15) and (0, 30). If the objective function Z = αx + βy, α, β > 0, the condition on α and β so that maximum of Z occurs at corner points (5, 5) and (0, 20) is :
A. α = 5β
B. 5α = β
C. α = 3β ✓
D. 4α = 5β
Show answer & explanation

Correct answer: C

Z equal at (5,5) and (0,20): $5\alpha+5\beta=20\beta\Rightarrow5\alpha=15\beta\Rightarrow\alpha=3\beta$.

Q77.
The solution set of the inequality $|3x| \ge |6 - 3x|$ is :
A. $(-\infty, 1]$
B. $[1, \infty)$ ✓
C. $(-\infty, 1) \cup (1, \infty)$
D. $(-\infty, -1) \cup (-1, \infty)$
Show answer & explanation

Correct answer: B

Square: $9x^2\ge36-36x+9x^2\Rightarrow36x\ge36\Rightarrow x\ge1$, i.e. $[1,\infty)$.

Q78.
If the matrix $\begin{bmatrix}0 & -1 & 3x\\1 & y & -5\\-6 & 5 & 0\end{bmatrix}$ is skew-symmetric, then the value of 5x − y is :
A. 12
B. 15
C. 10 ✓
D. 14
Show answer & explanation

Correct answer: C

Skew-symmetric: diagonal 0 so $y=0$; $a_{13}=-a_{31}\Rightarrow3x=6\Rightarrow x=2$. $5x-y=10$.

Q79.
A company is selling a certain commodity 'x'. The demand function for the commodity is linear. The company can sell 2000 units when the price is ₹ 8 per unit and it can sell 3000 units when the price is ₹ 4 per unit. The Marginal revenue at x = 5 is :
A. ₹ 79.98
B. ₹ 15.96 ✓
C. ₹ 16.04
D. ₹ 80.02
Show answer & explanation

Correct answer: B

$p=16-0.004x$. $R=px=16x-0.004x^2$, $MR=16-0.008x$. At $x=5$: $16-0.04=15.96$.

Q80.
If the lengths of the three sides of a trapezium other than the base are 10 cm each, then the maximum area of the trapezium is :
A. 100 cm$^2$
B. $25\sqrt3$ cm$^2$
C. $75\sqrt3$ cm$^2$ ✓
D. $100\sqrt3$ cm$^2$
Show answer & explanation

Correct answer: C

Maximizing, base $=2\times$ side gives half a regular hexagon; area $=\frac{3\sqrt3}{4}a^2=\frac{3\sqrt3}{4}\cdot100=75\sqrt3$ cm$^2$.

Q81.
Three defective bulbs are mixed with 8 good ones. If three bulbs are drawn one by one with replacement, the probabilities of getting exactly 1 defective, more than 2 defective, no defective and more than 1 defective respectively are :
A. $\frac{27}{1331}, \frac{576}{1331}, \frac{243}{1331}$ and $\frac{512}{1331}$
B. $\frac{27}{1331}, \frac{243}{1331}, \frac{576}{1331}$ and $\frac{512}{1331}$
C. $\frac{576}{1331}, \frac{27}{1331}, \frac{512}{1331}$ and $\frac{243}{1331}$ ✓
D. $\frac{243}{1331}, \frac{576}{1331}, \frac{512}{1331}$ and $\frac{27}{1331}$
Show answer & explanation

Correct answer: C

p(def)=3/11. Exactly 1: $3(3/11)(8/11)^2=576/1331$. More than 2 (=3): $27/1331$. No defective: $512/1331$. More than 1 (2 or 3): $216/1331+27/1331=243/1331$. Order matches option C.

Q82.
If $A = \begin{bmatrix}2 & 4\\4 & 3\end{bmatrix}$, $X = \begin{bmatrix}n\\1\end{bmatrix}$, $B = \begin{bmatrix}8\\11\end{bmatrix}$ and AX = B, then the value of n will be :
A. 0
B. 1
C. 2 ✓
D. not defined
Show answer & explanation

Correct answer: C

$2n+4=8\Rightarrow n=2$; check $4n+3=11$ ✓. So $n=2$.

Q83.
The equation of the tangent to the curve $x^{5/2} + y^{5/2} = 33$ at the point (1, 4) is :
A. x + 8y − 33 = 0 ✓
B. 12x + y − 8 = 0
C. x + 8y − 12 = 0
D. x + 12y − 8 = 0
Show answer & explanation

Correct answer: A

$\frac52x^{3/2}+\frac52y^{3/2}y'=0\Rightarrow y'=-\frac{x^{3/2}}{y^{3/2}}$. At (1,4): $-\frac{1}{8}$. Tangent: $y-4=-\frac18(x-1)\Rightarrow x+8y-33=0$.

Q84.
A random variable X has the following probability distribution : X = −2, −1, 0, 1, 2; P(X) = 0.2, 0.1, 0.3, 0.2, 0.2. The variance of X will be :
A. 0.1
B. 1.42
C. 1.89 ✓
D. 2.54
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Correct answer: C

$E(X)=0.1$; $E(X^2)=0.8+0.1+0+0.2+0.8=1.9$; Var$=1.9-0.01=1.89$.

Q85.
A Multinational company creates a sinking fund by setting a sum of ₹ 12,000 annually for 10 years to pay off a bond issue of ₹ 72,000. If the fund accumulates at 5% per annum compound interest, then the surplus after paying for bond is : (Use $(1.05)^{10} \approx 1.6$)
A. ₹ 78,900
B. ₹ 68,500
C. ₹ 72,000 ✓
D. ₹ 1,44,000
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Correct answer: C

FV$=12000\cdot\frac{(1.05)^{10}-1}{0.05}=12000\cdot\frac{0.6}{0.05}=12000\cdot12=144000$. Surplus$=144000-72000=72000$.

Original question paper source: National Testing Agency (NTA), CUET (UG) 2024. Reproduced for educational use. Answers & explanations by UniDrill.