GUJCET PYQ’s 2025

SECTION A: Multiple Choice Questions

1. If $\int \tan ^{-1} x d x=\mathrm{A} x \cdot \tan ^{-1} x+\mathrm{B} \log \left(1+x^{2}\right)+\mathrm{C}$ then, $\mathrm{A}+\mathrm{B}=$ .

(A) $1 / 2$

(B) $-1 / 2$

(D) -1

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2. The area bounded by the curve $y=\sin x$ between $x=-\pi / 2$ and $x=\pi / 2$ is .

(A) 2

(B) 1

(D) 4

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3. Area of the region bounded by the curve $x^{2}=4 y$ and the line $y=3$ is

(A) $2 \sqrt{3}$

(B) $3 \sqrt{3}$

(D) $4 \sqrt{3}$

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4. Area of the region bounded by the curve $y=x^{3}$ , $x$ -axis and the ordinates $x=-1$ and $x=2$ is

(A) $19 / 4$

(B) $9 / 4$

(D) $17 / 4$

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5. The degree of the differential equation $\left(1+\frac{d y}{d x}\right)^{\frac{1}{2}}=\left(\frac{d^{2} y}{d x^{2}}\right)^{1 / 3}$ is

(A) 2

(B) 1

(D) 4

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6. The general solution of the differential equation $\frac{d y}{d x}=e^{y-x}$ is .

(A) $e^{x}-e^{y}=c$

(B) $e^{x}-e^{-y}=c$

(D) $e^{-x}-e^{-y}=c$

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7. The Integrating Factor of the differential equation $x \cdot \frac{d y}{d x}+2 y=x^{2},(x \neq 0)$ is

(A) $e^{-x}$

(B) $x^{2}$

(D) $\frac{1}{x^{2}}$

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8. $\hat{i} \cdot(\hat{k} \times \hat{j})+\hat{j} \cdot(\hat{i} \times \hat{k})+\hat{k} \cdot(\hat{i} \times \hat{j})=$

(A) 1

(B) 0

(D) -3

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9. A unit vector perpendicular to each of the vectors $(\vec{a}+\vec{b})$ and $(\vec{a}-\vec{b})$ is , where $\vec{a}=\hat{i}+\hat{j}+\hat{k}$ and $\vec{b}=\hat{i}+2 \hat{j}+3 \hat{k}$

(A) $-\frac{1}{\sqrt{12}} \hat{i}+\frac{2}{\sqrt{12}} \hat{j}-\frac{1}{\sqrt{12}} \hat{k}$

(B) $\frac{1}{\sqrt{6}} \hat{i}+\frac{2}{\sqrt{6}} \hat{j}+\frac{1}{\sqrt{6}} \hat{k}$

(D) $-\frac{1}{\sqrt{6}} \hat{i}+\frac{2}{\sqrt{6}} \hat{j}-\frac{1}{\sqrt{6}} \hat{k}$

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10. Area of a rectangle having vertices $A, B, C$ and $D$ with position vectors $-\hat{i}+\frac{1}{2} \hat{j}+4 \hat{k}$ , $\hat{i}+\frac{1}{2} \hat{j}+4 \hat{k}, \hat{i}-\frac{1}{2} \hat{j}+4 \hat{k}$ and $-\hat{i}-\frac{1}{2} \hat{j}+4 \hat{k}$ , respectively is .

(A) 1

(B) $\frac{1}{2}$

(D) 4

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11. The Cartesian equation of the line through the point $(5,-2,4)$ and which is parallel to the vector $3 \hat{i}-2 \hat{j}+8 \hat{k}$ is

(A) $\frac{x+5}{-3}=\frac{y-2}{2}=\frac{z+4}{8}$

(B) $\frac{x-5}{-3}=\frac{y+2}{2}=\frac{z-4}{8}$

(D) $\frac{x-5}{3}=\frac{y+2}{-2}=\frac{z-4}{8}$

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12. The shortest distance between the lines $\frac{x-1}{2}=\frac{y-2}{3}=\frac{z+4}{6}$ and $\frac{x-3}{2}=\frac{y-3}{3}=\frac{z+5}{6}$ is .

(A) $\sqrt{\frac{293}{49}}$

(B) $\sqrt{\frac{293}{7}}$

(D) $\sqrt{\frac{209}{49}}$

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13. The angle between the pair of lines $\vec{r}=-3 \hat{i}+\hat{j}+3 \hat{k}+\lambda(3 \hat{i}+5 \hat{j}+4 \hat{k})$ and $\vec{r}=-\hat{i}+4 \hat{j}+5 \hat{k}+\mu(\hat{i}+\hat{j}+2 \hat{k})$ is

(A) $\cos ^{-1}\left(\frac{6 \sqrt{2}}{15}\right)$

(B) $\sin ^{-1}\left(\frac{6 \sqrt{2}}{15}\right)$

(D) $\sin ^{-1}\left(\frac{8 \sqrt{3}}{15}\right)$

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14. The coordinates of the corner points of the bounded feasible region are $(0,0),(0,40),(20,40),(60,20),(60,0)$ . The maximum of the objective function $z=40 x+30 y$ is .

(A) 3400

(B) 3000

(D) 2000

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15. The maximum value of $z=5 x+3 y$ subject to constraints $3 x+5 y \leq 15, x \geq 0$ , $y \geq 0$ is :

(A) 25

(B) 9

(D) 10

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16. Two events $E$ and $F$ are independent. If $P(E)=\frac{3}{5}$ and $P(F)=\frac{3}{10}$ then $\mathrm{P}\left(\mathrm{E}^{\prime} / \mathrm{F}\right)+\mathrm{P}\left(\mathrm{F}^{\prime} / \mathrm{E}\right)=$

(A) $\frac{11}{10}$

(B) $\frac{10}{11}$

(D) $\frac{1}{10}$

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17. Let A and B be two events such that $\mathrm{P}(\mathrm{A})=\frac{3}{8}, \mathrm{P}(\mathrm{B})=\frac{5}{8}$ and $\mathrm{P}(\mathrm{A} \cup \mathrm{B})=\frac{3}{4}$ . Then $\mathrm{P}\left(\mathrm{A}^{\prime} \mid \mathrm{B}\right)-\mathrm{P}(\mathrm{A} \mid \mathrm{B})=$ .

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

(B) $\frac{4}{5}$

(D) $\frac{1}{5}$

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18. A man is known to speak truth 4 out of 5 times. He throws a die and reports that it is a six. The probability that actually there was a six is

(A) $\frac{4}{9}$

(B) $\frac{4}{35}$

(D) $\frac{5}{9}$

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19. Let $\mathrm{A}=\{1,2,3\}$ . Then number of relations containing $(1,2)$ which are symmetric and transitive but not reflexive is .

(A) 2

(B) 1

(D) 4

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20. Let $f: \mathrm{R} \rightarrow \mathrm{R}$ be defined as $f(x)=x^{3}$ . Then $f$ is .

(A) Many – one and onto

(B) One – one and onto

(D) Neither one – one nor onto

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21. $\tan ^{-1}\left[\frac{\sqrt{2}}{\sqrt{3}} \cos \left(5 \sin ^{-1} \frac{1}{\sqrt{2}}\right)\right]=$

(A) $\pi / 3$

(B) $\pi / 6$

(D) $-\pi / 3$

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22. If $y=3 \sin ^{-1} x+\sin ^{-1}\left(3 x-4 x^{3}\right)$ for all $x \in[-1 / 2,1 / 2]$ , then

(A) $-\pi / 3 \leq y \leq \pi / 3$

(B) $-\pi / 6 \leq y \leq \pi / 6$

(D) $-\pi \leq y \leq \pi$

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23. The number of real solutions of the equation $\tan ^{-1} \sqrt{x(x+1)}+\sin ^{-1} \sqrt{x^{2}+x+1}=\frac{\pi}{2}$ is

(A) 0

(B) 4

(D) 1

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24. $\left|\begin{array}{cc}\cos ^{2} \theta & -\sin ^{2} \theta \\ \sin ^{2} \theta & \cos ^{2} \theta\end{array}\right|=$ .

(A) $\frac{1}{4}(3+\cos 4 \theta)$

(B) $1+2 \sin ^{2} \theta \cdot \cos ^{2} \theta$

(D) $\frac{1}{2}-\frac{1}{2} \cos ^{2} 2 \theta$

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25. Let $A$ be an invertible square matrix of order $3 \times 3$ . Then $|(\operatorname{adj} \mathrm{A}) \cdot \mathrm{A}|$ is .

(A) $|\mathrm{A}|^{2}$

(B) $|\mathrm{A}|$

(D) $3|\mathrm{A}|$

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26. Find the area of a triangle given that midpoints of its sides are $(2,7),(1,1)$ and $(10,8)$ .

(A) 47

(B) $\frac{47}{2}$

(D) $\frac{47}{4}$

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27. If the matrix $\left[\begin{array}{ccc}x & x^{2}+3 x & 5 \\ -2 x-6 & x^{2} & -4 x-2 \\ 5 & x^{2}+2 & x^{3}\end{array}\right]$ is a symmetric matrix, then the value of $x$ is .

(A) 3,2

(B) -3,-2

(D) -2

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28. If $\mathrm{A}=\left[\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right]$ , then $(\mathrm{A}+\mathrm{I})^{3}+(\mathrm{A}-\mathrm{I})^{3}=$ .

(A) 8 I

(B) 6 I

(D) 8 A

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29. For matrix $A=\left[\begin{array}{ll}2 & 3 \\ 4 & 5\end{array}\right]$ , if $A^{2}-2 I=K A$ then $K=$ .

(A) 5

(B) 7

(D) -5

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30. $\frac{d}{d x}\left(5^{\log x}\right)=$

(A) $\log _{x} 5 \cdot 5^{\log x}$

(B) $\log 5 \cdot 5^{\log x}$

(D) $\log 5 \cdot x^{\log (5 e)}$

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31. If $x=a \cos \theta, y=a \sin \theta$ , then $\frac{d^{2} y}{d x^{2}}=$ $(a \neq 0 ; \theta \neq k \pi, k \in z)$

(A) $-\frac{1}{a} \operatorname{cosec}^{2} \theta \cdot \sec \theta$

(B) $\operatorname{cosec}^{2} \theta$

(D) $-\frac{1}{a} \operatorname{cosec}^{3} \theta$

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32. $\frac{d}{d x}\left[3 \sin \left(60^{\circ}-x^{\circ}\right)-4 \cos ^{3}\left(30^{\circ}+x^{\circ}\right)\right]=$ .

(A) $\frac{\pi}{60} \sin \left(3 x^{\circ}\right)$

(B) $-\frac{\pi}{60} \cos \left(3 x^{\circ}\right)$

(D) $-\frac{\pi}{60} \sin \left(3 x^{\circ}\right)$

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33. If $f(x)=\left\{\begin{array}{cl}\frac{x^{3}+x^{2}-16 x+20}{(x-2)^{2}} & , x \neq 2 \\ k & , x=2\end{array}\right.$ is continuous at $x=2$ then $k=$

(A) 7

(B) 5

(D) -7

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34. The total cost $C(x)$ in Rupees, associated with the production of $x$ units of an item is given by $\mathrm{C}(x)=0.05 x^{3}-0.2 x^{2}+3 x+500$ . The marginal cost, where $x=3$ is (in Ruppes)

(A) 30.15

(B) 30.015

(D) 3.15

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35. The function $f(x)=\tan x-4 x$ is strictly decreasing on .

(A) $\left(-\frac{\pi}{3}, \pi\right)$

(B) $\left(-\frac{\pi}{3}, \frac{\pi}{2}\right)$

(D) $\left(-\frac{\pi}{2}, \frac{\pi}{3}\right)$

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36. The absolute minimum value of the function $f(x)=x^{3}-18 x^{2}+96 x, x \in[0,9]$ is .

(A) 0

(B) 126

(D) -160

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37. If $\int \frac{3 e^{x}-5 e^{-x}}{4 e^{x}+5 e^{-x}} d x=p x+q \cdot \log \left|4 e^{x}+5 e^{-x}\right|+\mathrm{C}$ , then

(A) $p=1 / 8, q=7 / 8$

(B) $p=-1 / 8, q=7 / 8$

(D) $p=1 / 8, q=-7 / 8$

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38. $\int e^{\tan ^{-1} x}\left(\frac{1+x+x^{2}}{1+x^{2}}\right) d x=$

(A) $\frac{1+x^{2}}{x} \cdot e^{\tan ^{-1} x}$

(B) $\frac{x \cdot e^{\tan ^{-1} x}}{1+x^{2}}$

(D) $\frac{e^{\tan ^{-1} x}}{x}$

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39. $\int_{0}^{\pi / 4} \sqrt{1+\sin 2 x} d x=$

(A) 1

(B) 0

(D) 2

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40. $\int \frac{d x}{\sqrt{4 x-9 x^{2}}}=\quad+\mathrm{C}$

(A) $\frac{1}{9} \sin ^{-1}\left(\frac{3 x-2}{2}\right)$

(B) $\frac{1}{2} \sin ^{-1}\left(\frac{9 x-3}{2}\right)$

(D) $\frac{1}{3} \sin ^{-1}\left(\frac{9 x-2}{2}\right)$

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41. The order and the degree of the differential equation $\sqrt{\frac{d^2 y}{d x^2}}=\sqrt[3]{\left(\frac{d y}{d x}\right)^4+2}$ is respectively … and …

(A) 1,8

(B) 3,2

(D) 2,3

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