MHT-CET 2025 (20 April Shift I), Paper I, Mathematics question paper

1. A.
   $y(x - 1) = x^{3} + c(x - 1)$, where $c$ is the constant of integration.
2. B.
   $y = x^{3}(x - 1) + c$, where c is the constant of integration.
3. C.
   $y(x - 1) = x^{3}(x - 1) + cx^{2}$, where $c$ is the constant of integration.
4. D.
   $y(x - 1) = x^{3}(x - 1) + c$, where c is the constant of integration.

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1. A.
   $\frac{- 3}{5\sqrt{2}},\frac{5}{5\sqrt{2}},\frac{4}{5\sqrt{2}}$
2. B.
   $\ \frac{3}{5\sqrt{2}},\frac{- 5}{5\sqrt{2}},\frac{4}{5\sqrt{2}}$
3. C.
   $\ \frac{3}{5\sqrt{2}},\frac{5}{5\sqrt{2}},\frac{4}{5\sqrt{2}}$
4. D.
   $\frac{3}{5\sqrt{2}},\frac{5}{5\sqrt{2}},\frac{- 4}{5\sqrt{2}}$

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1. A.
   15,5
2. B.
   16,4
3. C.
   12,8
4. D.
   14,6

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1. A.
   $\cot^{- 1}2$
2. B.
   $\cot^{- 1}\left( \frac{1}{3} \right)$
3. C.
   $\tan^{- 1}2$
4. D.
   $\tan^{- 1}\left( \frac{2}{3} \right)$

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1. A.
   $\frac{3\sqrt{2}}{8}$
2. B.
   $\frac{2\sqrt{3}}{8}$
3. C.
   $\frac{3\sqrt{2}}{5}$
4. D.
   $\frac{\sqrt{3}}{4}$

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1. A.
   $\ (x + 2)^{2} + (y - 2)^{2} = 4$
2. B.
   $\ (x - 3)^{2} + (y - 1)^{2} = 4$
3. C.
   $\ (x - 1)^{2} + (y - 2)^{2} = 1$
4. D.
   $(x + 1)^{2} + (y + 2)^{2} = 1$

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1. A.
   $\ \cos x + \frac{2}{3}\cos^{2}x - \frac{\cos^{5}x}{5} + c$, where c is the constant of integration
2. B.
   $\cos x + \frac{2}{3}\cos^{2}x + \frac{\cos^{5}x}{5} + c$, where $c$ is the constant of integration
3. C.
   $- \left( \cos x - \frac{2}{3}\cos^{2}x + \frac{\cos^{5}x}{5} + c \right)$, where $c$ is the constant of integration
4. D.
   $- \left( \cos x - \frac{2\cos^{3}x}{3} + \frac{\cos^{5}x}{5} \right) + c,$ where $c$ is the constant of integration

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1. A.
   $\frac{12}{11}$
2. B.
   $\frac{62}{55}$
3. C.
   $\frac{31}{11}$
4. D.
   $\frac{8}{5}$

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1. A.
   1
2. B.
   -1
3. C.
   2
4. D.
   -2

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1. A.
   $\ \cos^{- 1}\left( \frac{1}{\sqrt{6}} \right)$
2. B.
   $\ \cos^{- 1}\left( \frac{1}{\sqrt{5}} \right)$
3. C.
   $\sin^{- 1}\left( \frac{2}{\sqrt{5}} \right)$
4. D.
   $\cos^{- 1}\left( \frac{2}{\sqrt{6}} \right)$.

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1. A.
   4
2. B.
   $\frac{7}{2}$
3. C.
   6
4. D.
   $\frac{9}{2}$

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1. A.
   1
2. B.
   -1
3. C.
   0
4. D.
   $\frac{1}{2}$

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1. A.
   $\frac{\pi}{3}$
2. B.
   $\frac{\pi}{6}$
3. C.
   $\frac{3\pi}{4}$
4. D.
   $\frac{5\pi}{6}$

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1. A.
   $\frac{3}{2}$
2. B.
   $\frac{3}{4}$
3. C.
   $\frac{3}{8}$
4. D.
   2

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1. A.
   676 a
2. B.
   $676\overline{\text{ }b}$
3. C.
   $625\overline{a}$
4. D.
   $625\overline{\text{ }b}$

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1. A.
   $\frac{1}{4}$
2. B.
   $\frac{4}{7}$
3. C.
   $\frac{1}{3}$
4. D.
   $\frac{2}{5}$

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1. A.
   $p \land q$
2. B.
   $p \vee r$
3. C.
   $p \land r$
4. D.
   $(p \land r) \rightarrow (p \vee r)$

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1. A.
   $( \sim p) \land q$
2. B.
   $( \sim p) \vee ( \sim q)$
3. C.
   $( \sim p) \land ( \sim q)$
4. D.
   $p \vee q$

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1. A.
   $\frac{132}{221}$
2. B.
   $\frac{108}{221}$
3. C.
   $\frac{176}{221}$
4. D.
   $\frac{68}{221}$

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1. A.
   $\frac{5}{6}$
2. B.
   $\frac{6}{7}$
3. C.
   $\frac{7}{8}$
4. D.
   $\frac{8}{9}$

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1. A.
   $\frac{1}{2}(\pi - 2)$ sq. units
2. B.
   $\frac{3}{2}(\pi - 2)$ sq. units
3. C.
   $\frac{5}{4}(\pi - 2)$ sq. units
4. D.
   $\frac{2}{3}(\pi - 2)$ sq. units

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1. A.
   $m = 1,n = 0$
2. B.
   $\ m = \frac{n\pi}{2}$
3. C.
   $m = n = \frac{\pi}{2}$
4. D.
   $\ n = \frac{m\pi}{2}$

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1. A.
   $\frac{17}{3}$
2. B.
   $\frac{19}{3}$
3. C.
   19
4. D.
   17

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1. A.
   1
2. B.
   0
3. C.
   $\frac{1}{2}$
4. D.
   $\frac{1}{4}$

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1. A.
   100 sq. units
2. B.
   20 sq. units
3. C.
   40 sq. units
4. D.
   50 sq. units

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1. A.
   -1
2. B.
   0
3. C.
   1
4. D.
   2

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1. A.
   $\frac{1}{4}$
2. B.
   $\frac{7}{16}$
3. C.
   $\frac{1}{2}$
4. D.
   $\frac{9}{16}$

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1. A.
   only one
2. B.
   only onto
3. C.
   onto for $y \neq \frac{2}{3}$ and one-one
4. D.
   neither one-one nor onto

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1. A.
   Straight line passing through the origin and the first and third quadrant.
2. B.
   Straight line passing through the origin and the second and fourth quadrant.
3. C.
   Straight line passing through the point $(1, - 1)$ and having slope -1 .
4. D.
   Straight line passing through the point $(2,1)$ and having slope $\frac{1}{2}$.

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1. A.
   $\frac{5}{4}$
2. B.
   $- \frac{5}{4}$
3. C.
   $- \frac{5}{3}$
4. D.
   $\ - \frac{5}{6}$

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1. A.
   9050.88
2. B.
   18101.76
3. C.
   6788.16
4. D.
   12067.84

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1. A.
   $\frac{\pi}{3}$
2. B.
   $\frac{2\pi}{3}$
3. C.
   $\frac{\pi}{2}$
4. D.
   $\frac{\pi}{4}$

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1. A.
   1140
2. B.
   800
3. C.
   340
4. D.
   20

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1. A.
   $2\tan^{- 1}\left( \frac{1}{\sqrt{3}}\tan\frac{x}{2} \right) + c$, where $c$ is the constant of integration
2. B.
   $\frac{2}{\sqrt{3}}\tan^{- 1}\left( \frac{1}{\sqrt{3}}\tan\frac{x}{2} \right) + c$, where $c$ is the constant of integration
3. C.
   $\frac{1}{\sqrt{3}}\tan^{- 1}\left( \frac{1}{\sqrt{3}}\tan\frac{x}{2} \right) + c$, where c is the constant of integration
4. D.
   $\sqrt{3}\tan^{- 1}\left( \frac{1}{\sqrt{3}}\tan\frac{x}{2} \right) + c$, where $c$ is the constant of integration

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1. A.
   $\frac{1}{\sqrt{2}}$
2. B.
   $\frac{1}{2}$
3. C.
   $- \frac{1}{2}$
4. D.
   $- \frac{1}{\sqrt{2}}$

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1. A.
   $\sqrt{2}$
2. B.
   $4\sqrt{2}$
3. C.
   4
4. D.
   $2\sqrt{3}$

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1. A.
   $\frac{1}{\sqrt{5}}$ units
2. B.
   $\frac{1}{5}$ units
3. C.
   $\sqrt{5}$ units
4. D.
   5 units

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1. A.
   $\ x^{x}(1 + \log x) + x^{\frac{1}{x}}\frac{1}{x^{2}}(1 - \log x)$
2. B.
   $\left( x^{x} + x^{\frac{1}{x}} \right)\left\lbrack 1 + \log x + \frac{1}{x^{2}}(1 - \log x) \right\rbrack$
3. C.
   $\left( x^{x} + x^{\frac{1}{x}} \right)\left\lbrack (1 + \log x) - \frac{1}{x^{2}}(1 - \log x) \right\rbrack$
4. D.
   $x^{x}(1 + \log x) - x^{\frac{1}{x}}\frac{1}{x^{2}}(1 - \log x)$

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1. A.
   $\frac{\sqrt{61}}{2}$ sq. units
2. B.
   $2\sqrt{61}$ sq. units
3. C.
   $\sqrt{61}$ sq. units
4. D.
   $3\sqrt{61}$ sq. units

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1. A.
   0
2. B.
   2
3. C.
   $\ - 1$
4. D.
   4

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1. A.
   $\frac{\pi}{2}$
2. B.
   $\frac{\pi}{4}$
3. C.
   $\frac{\pi}{3}$
4. D.
   $\frac{\pi}{6}$

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1. A.
   9
2. B.
   12
3. C.
   15
4. D.
   33

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1. A.
   6
2. B.
   -6
3. C.
   $2\sqrt{3}$
4. D.
   1

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1. A.
   3
2. B.
   $\frac{1}{13}$
3. C.
   -3
4. D.
   $\frac{1}{3}$

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1. A.
   $\ \frac{(2n + 1)\pi}{8},n \in \mathbb{Z}$
2. B.
   $\frac{(2n + 1)\pi}{4},n \in \mathbb{Z}$
3. C.
   $\ \frac{(n + 2)\pi}{3},n \in \mathbb{Z}$
4. D.
   $n\pi,n \in \mathbb{Z}$

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1. A.
   $x - y \geq 0,x + y \geq 0$
2. B.
   $x - y \leq 0,x + y \geq 0$
3. C.
   $x - y \geq 0,x + y \leq 0$
4. D.
   $x - y \leq 0,x + y \leq 0$

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1. A.
   $acos\theta$
2. B.
   $bcos\theta$
3. C.
   $\cos\theta$
4. D.
   $ccos\theta$

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1. A.
   $\begin{bmatrix} \cos2x & - \sin2x \\ - \sin2x & \cos2x \end{bmatrix}$
2. B.
   $\begin{bmatrix} \cos2x & - \sin2x \\ \sin2x & \cos2x \end{bmatrix}$
3. C.
   $\begin{bmatrix} - \cos2x & \sin2x \\ \sin2x & \cos2x \end{bmatrix}$
4. D.
   $\begin{bmatrix} - \cos2x & \sin2x \\ - \sin2x & \cos2x \end{bmatrix}$

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1. A.
   1
2. B.
   3
3. C.
   5
4. D.
   7

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1. A.
   $\ x\frac{\text{ }dy}{\text{ }dx} + xy = 0$
2. B.
   $\ x^{2}\frac{\text{ }dy}{\text{ }dx} + 2xy - 4e^{x} = 0$
3. C.
   $\ x\frac{\text{ }dy}{\text{ }dx} + (x - 2)y = 0$
4. D.
   $x\frac{\text{ }dy}{\text{ }dx} + (2 - x)y = 0$

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