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MHT-CET 2025 (22 April Shift I), Paper I, Mathematics question paper

50 questions · 100 marks · 90 minutes · +2 per correct answer, no negative marking

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  1. 1
    If y=y(x)y=y(x) and (2+sin⁡xy+1)dy dx=−cos⁡x,y(0)=1\left( \frac{2 + \sin x}{y+ 1} \right)\frac{dy}{\text{ }dx}= - \cos x,y(0) = 1, then y(π2)=y\left( \frac{\pi}{2} \right)=

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  2. 2
    If A=[1cot⁡θ2−cot⁡θ21]A =\begin{bmatrix} 1 & \cot\frac{\theta}{2} \\ - \cot\frac{\theta}{2} & 1 \end{bmatrix} then A−1=A^{- 1}=

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  3. 3
    If a‾,b‾,c‾\overline{a},\overline{b},\overline{c} are three unit vectors such that ∣a‾+b‾∣2+∣a‾+c‾∣2=8|\overline{a}+\overline{b}|^{2}+ |\overline{a}+\overline{c}|^{2}= 8, then ∣a‾+3 b‾∣2+∣a‾+3c‾∣2=|\overline{a}+ 3\overline{\text{ }b}|^{2}+ |\overline{a}+ 3\overline{c}|^{2}=

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  4. 4
    The probability that a certain kind of component will survive a given test is 23\frac{2}{3}. The probability that at most 2 components out of 4 tested, will survive is

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  5. 5
    From the following options, the nearest line to the origin is ....

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  6. 6
    If the truth value of the statement pattern [p∧∼r]→∼r∧q\lbrack p \land \sim r\rbrack \rightarrow \sim r \land q is False, then which of the following has truth value False?

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  7. 7
    a‾=i^−j^,b‾=j^−k^,c‾=k^−i^\overline{a}=\widehat{i}-\widehat{j},\overline{b}=\widehat{j}-\widehat{k},\overline{c}=\widehat{k}-\widehat{i} then a unit vector d‾\overline{d} such that a‾⋅d‾=0=[b‾c‾d‾]\overline{a}\cdot\overline{d}= 0 = \lbrack\overline{b}\overline{c}\overline{d}\rbrack is

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  8. 8
    In △ABC\bigtriangleup ABC, with usual notations, if a4+b4+c4−2a2c2−2c2b2=0a^{4}+b^{4}+c^{4}- 2a^{2}c^{2}- 2c^{2}b^{2}= 0, then ∠C=…\angle C= \ldots.

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  9. 9
    If the planes rˉ⋅(2i^−λj^+k^)=3\bar{r}\cdot (2\widehat{i}-\lambda\widehat{j}+\widehat{k}) = 3 and r‾⋅(4i^−j^+μk^)=5\overline{r}\cdot (4\widehat{i}-\widehat{j}+\mu\widehat{k}) = 5 are parallel, then λ+μ=\lambda+\mu=

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  10. 10
    If xx is real, then the difference between the greatest and least values of x2−x+1x2+x+1\frac{x^{2}-x+ 1}{x^{2}+x+ 1} is

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  11. 11
    The perimeter of a square whose two sides have equations x−12=y+23=z−34\frac{x- 1}{2}=\frac{y+ 2}{3}=\frac{z- 3}{4} and x2=y−13=z+14\frac{x}{2}=\frac{y- 1}{3}=\frac{z+ 1}{4} is

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  12. 12
    In a triangle ABC with usual notations if, tan⁡(B−C2)=xcot A2\tan\left( \frac{B - C}{2} \right)=xcot\frac{\text{ }A}{2}, then x=x=

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  13. 13
    A coin is tossed until one head appears or a tail appears 4 times in succession. The probability distribution of the number of tosses is

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  14. 14
    The degree of the differential equation d2y dx2+3( dy dx)2=x2log⁡( d2y dx2)\frac{d^{2}y}{\text{ }dx^{2}}+ 3\left( \frac{\text{ }dy}{\text{ }dx} \right)^{2}=x^{2}\log\left( \frac{{\text{ }d}^{2}y}{\text{ }dx^{2}} \right) is

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  15. 15
    If the sum of the squares of the distances of a point P(x,y,z)P(x,y,z) from the three co-ordinate axes is 324 , then the distance of point P from the origin is ...

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  16. 16
    For a real number x,[x]x,\lbrack x\rbrack denotes the greatest integer less than or equal to xx. Then the value of [12]+[12+1100]+[12+2100]+[12+3100]+\left\lbrack \frac{1}{2} \right\rbrack+\left\lbrack \frac{1}{2}+\frac{1}{100} \right\rbrack+\left\lbrack \frac{1}{2}+\frac{2}{100} \right\rbrack+\left\lbrack \frac{1}{2}+\frac{3}{100} \right\rbrack+.........+[12+99100]=+\left\lbrack \frac{1}{2}+\frac{99}{100} \right\rbrack=

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  17. 17
    The angle between the lines 3x=2y=−z3x= 2y= -z and −x=6y=−4z-x= 6y= - 4z is

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  18. 18
    If y+ddx(xy)=x(sin⁡x+log⁡x)y+\frac{d}{dx}(xy) =x(\sin x+ \log x) then

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  19. 19
    If the pair of straight lines xy−x+y−1=0xy-x+y- 1 = 0 and the line x+ky−3=0x+ky- 3 = 0 are concurrent, then the value of kk is equal to

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  20. 20
    If the function f(x)={x+a2sin⁡x if 0≤x≤π42xcotx+b if π4<x≤π2acos2x−bsinx if π2<x≤πf(x) =\left\{ \begin{matrix} x+a\sqrt{2}\sin x & \text{~if~}0 \leq x\leq\frac{\pi}{4} \\ 2xcotx+b & \text{~if~}\frac{\pi}{4}<x\leq\frac{\pi}{2} \\ acos2x-bsinx & \text{~if~}\frac{\pi}{2}<x\leq\pi \end{matrix} \right. is continuous in [0,π]\lbrack 0,\pi\rbrack then a−b=a - b =

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  21. 21
    Which of the following statements has the truth value T? (A) cube roots of unity are in Geometric progression and their sum is 1 (B)  4+7>10\ 4 + 7 > 10 iff 2+8<102 + 8 < 10 (C)  ∃x∈N\ \exists x \in \mathbb{N} such that x2−3x+2=0x^{2} - 3x + 2 = 0 and ∃n∈N\exists n \in \mathbb{N} such that n is an odd number (D) 3+i3 + i is a complex number or 2+3=5\sqrt{2} + \sqrt{3} = \sqrt{5}

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  22. 22
    The p.d.f. of a continuous random variable X is f(x)={x218 if −3<x<30 otherwise f(x) =\left\{ \begin{matrix} \frac{x^{2}}{18} & \text{~if~}- 3 <x< 3 \\ 0 & \text{~otherwise~} \end{matrix} \right. Then P[∣X∣<2]=P\lbrack|X| < 2\rbrack =

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  23. 23
    The population of a town increases at a rate proportional to the population at that time. If the population increases from forty thousand to eighty thousand in 20 years, then the population in another 40 years will be

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  24. 24
    If the vectors mi^+mj^+nk^,i^+k^,ni^+nj^+pk^m\widehat{i}+m\widehat{j}+n\widehat{k},\widehat{i}+\widehat{k},n\widehat{i}+n\widehat{j}+p\widehat{k} lie in a plane then...

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  25. 25
    The value of sin⁡25∘+sin⁡210∘+sin⁡215\sin^{2}5^{\circ}+\sin^{2}10^{\circ}+\sin^{2}15 +.........+sin⁡285∘+sin⁡290∘=+ ......... +\sin^{2}85^{\circ}+\sin^{2}90^{\circ}=

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  26. 26
    The area bounded by the curve x=2−y−y2x= 2 -y-y^{2} and the Y -axis is

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  27. 27
    If y=sin⁡−1(2x1+x2)+sec⁡−1(1+x21−x2)y=\sin^{- 1}\left( \frac{2x}{1 +x^{2}} \right)+\sec^{- 1}\left( \frac{1 +x^{2}}{1 -x^{2}} \right) then the value of dy dx\frac{dy}{\text{ }dx} at x=3x=\sqrt{3} is

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  28. 28
    ∫π32π3x1+sin⁡x dx=\int_{\frac{\pi}{3}}^{\frac{2\pi}{3}} \frac{x}{1 + \sin x}\text{ }dx=

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  29. 29
    ∫023x+1x2+4dx=\int_{0}^{2} \frac{3x+ 1}{x^{2}+ 4}dx=

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  30. 30
    ∫sin⁡2xcos2x9−cos⁡42xdx=\int\frac{\sin2xcos2x}{\sqrt{9 -\cos^{4}2x}}dx=

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  31. 31
    ∫cos⁡2x−cos⁡2αcos⁡x−cos⁡αdx=\int\frac{\cos2x- \cos2\alpha}{\cos x- \cos\alpha}dx=

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  32. 32
    21 friends were invited for a party. Two round tables can accommodate 12 and 9 friends each. The number of ways of the seating arrangements of friends is

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  33. 33
    The area of a parallelogram whose diagonals are the vectors 2aˉ−bˉ2\bar{a}-\bar{b} and 4aˉ−5bˉ4\bar{a}- 5\bar{b}, where aˉ\bar{a} and bˉ\bar{b} are unit vectors forming an angle of 45∘45^{\circ} is

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  34. 34
    lim⁡x→∞(x+8x+1)x+5=…\lim_{x\rightarrow \infty} \left( \frac{x+ 8}{x+ 1} \right)^{x+ 5}= \ldots

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  35. 35
    cot⁡−1(2cos⁡(2csc⁡−1(2)))=…\cot^{- 1}\left( 2\cos\left( 2\csc^{- 1}(\sqrt{2}) \right) \right)= \ldots

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  36. 36
    If z=x+iyz =x+ iy is a complex number, then the equation ∣z+iz−i∣=3\left| \frac{z+i}{z-i} \right|=\sqrt{3} represents the

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  37. 37
    If ∫2x2+3(x2−1)(x2−4)dx=log⁡[(x−2x+2)a⋅(x+1x−1)b]+c\int\frac{2x^{2}+ 3}{\left( x^{2}- 1 \right)\left( x^{2}- 4 \right)}dx= \log\left\lbrack \left( \frac{x- 2}{x+ 2} \right)^{a}\cdot\left( \frac{x+ 1}{x- 1} \right)^{b} \right\rbrack+ c, (where c is the constant of integration) then the value of a+ba + b is equal to

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  38. 38
    For N∈N,dndxn(log⁡x)=N \in\mathbb{N},\frac{d^{n}}{dx^{n}}(\log x) =

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  39. 39
    In a single toss of a fair die, the odds against the event that number 4 or 5 turns up is

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  40. 40
    If f(x)=x⋅ex(1−x)f(x) =x\cdot e^{x(1 -x)}, then f(x)f(x) is

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  41. 41
    The approximate value of 64⋅043\sqrt[3]{64 \cdot 04} is

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  42. 42
    The X and Y intercepts of the tangent to the hyperbola x220−y25=1\frac{x^{2}}{20}-\frac{y^{2}}{5}= 1 which is perpendicular to the line 4x+3y=74x+ 3y= 7, are respectively

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  43. 43
    If the lines x−12=y+1k=z2\frac{x- 1}{2}=\frac{y+ 1}{k}=\frac{z}{2} and x+15=y+12=zk\frac{x+ 1}{5}=\frac{y+ 1}{2}=\frac{z}{k} are coplanar, then the equation of the plane containing these lines are

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  44. 44
    The line y=mx+3y=mx+ 3 is tangent to the parabola y2=4xy^{2}= 4x, if the value of m is

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  45. 45
    If the tangent and the normal at the point (3,1)(\sqrt{3},1) to the circle x2+y2=4x^{2}+y^{2}= 4, and the XX-axis form a triangle, then the area (in sq.units) of this triangle is

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  46. 46
    If 3sin⁡−1(2x1+x2)−4cos⁡−1(1−x21+x2)+2tan⁡−1(2x1−x2)=π33\sin^{- 1}\left( \frac{2x}{1 +x^{2}} \right)- 4\cos^{- 1}\left( \frac{1 -x^{2}}{1 +x^{2}} \right)+ 2\tan^{- 1}\left( \frac{2x}{1 -x^{2}} \right)=\frac{\pi}{3} then the value of x=x=

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  47. 47
    If x=acos3θy=asin3θx=acos^{3}\theta y=asin^{3}\theta Then 1+(dy dx)2=\sqrt{1 +\left( \frac{dy}{\text{ }dx} \right)^{2}}=

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  48. 48
    If the difference between the maximum and minimum values of the objective function z=7x−8yz = 7x- 8y subject to the constraints x+y⩽20,y⩾5,x,y⩾0x+y\leqslant 20,y\geqslant 5,x,y\geqslant 0 is 5k+2005k + 200, then the value of k is

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  49. 49
    The number of solutions of 16sin⁡2x+16cos⁡2x=1016^{\sin^{2}x}+16^{\cos^{2}x}= 10 in 0⩽x⩽2π0 \leqslant x\leqslant 2\pi are

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  50. 50
    Let the line x−23=y−1−5=z+22\frac{x- 2}{3}=\frac{y- 1}{- 5}=\frac{z+ 2}{2} lie in the plane x+3y−αz+β=0x+ 3y-\alpha z+\beta= 0, then the value of (β−α)(\beta-\alpha) is equal to

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