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MHT-CET 2025 (21 April Shift II), Paper I, Mathematics question paper

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

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  1. 1
    Numbers are selected at random, one at a time from the two-digit numbers 00,01,02,……−00,01,02,\ldots\ldots -, 99 with replacement. An event EE occurs only if the product of the two digits of a selected number is 24. If four numbers are selected, then probability, that the event E occurs at least 3 times, is

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  2. 2
    A particular solution of 3extan⁡y dx+(1−ex)sec⁡2y dy=03e^{x}\tan y\text{ }dx+\left( 1 -e^{x} \right)\sec^{2}y\text{ }dy= 0 with y(1)=π4y(1) =\frac{\pi}{4} is

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  3. 3
    aˉ,bˉ,cˉ\bar{a},\bar{b},\bar{c} are nonzero vectors such that aˉ\bar{a} is perpendicular to b‾\overline{b} and c‾,∣a‾∣=1,∣ b‾∣=2,∣c‾∣=1\overline{c},|\overline{a}| = 1,|\overline{\text{ }b}| = 2,|\overline{c}| = 1 and b‾⋅c‾=1\overline{b}\cdot\overline{c}= 1. There is nonzero vector d‾\overline{d} coplanar with aˉ+bˉ\bar{a}+\bar{b} and 2bˉ−cˉ2\bar{b}-\bar{c}. If dˉ⋅aˉ=1\bar{d}\cdot\bar{a}= 1, then ∣dˉ∣2=|\bar{d}|^{2}=

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  4. 4
    If sin⁡−1x+sin⁡−1y=π3\sin^{- 1}x+\sin^{- 1}y=\frac{\pi}{3} and cot⁡−1(1x)−cot⁡−1(1y)=0\cot^{- 1}\left( \frac{1}{x} \right)-\cot^{- 1}\left( \frac{1}{y} \right)= 0 then 2x2+y2−xy=2x^{2}+y^{2}-xy= ____\_\_\_\_

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  5. 5
    The value of sin⁡[tan⁡−1(1−x22x)+cos⁡−1(1−x21+x2)]\sin\left\lbrack \tan^{- 1}\left( \frac{1 -x^{2}}{2x} \right)+\cos^{- 1}\left( \frac{1 -x^{2}}{1 +x^{2}} \right) \right\rbrack is

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  6. 6
    The area of the region bounded by the curve y=∣x−2∣y= |x- 2| between x=1,x=3x= 1,x= 3 and X -axis is ____\_\_\_\_

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  7. 7
    ∫−13(tan⁡−1(xx2+1)+tan⁡−1(x2+1x))dx=\int_{- 1}^{3} \left( \tan^{- 1}\left( \frac{x}{x^{2}+ 1} \right)+\tan^{- 1}\left( \frac{x^{2}+ 1}{x} \right) \right)dx=

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  8. 8
    Let A be a non-singular matrix of order n and ∣A∣=k|A| =k, then (adjA)−1(adjA)^{- 1} is

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  9. 9
    If f(x)={8x−4x−2x+1xx2, if x>0exsin⁡x+ix+λlog⁡4, if x⩽0,i∈Rf(x) =\left\{ \begin{matrix} \frac{8^{x}-4^{x}-2^{x}+1^{x}}{x^{2}}, & \text{~if~}x> 0 \\ e^{x}\sin x+ix+\lambda \log4, & \text{~if~}x\leqslant 0,i\in\mathbb{R} \end{matrix} \right. is continuous at x=0x= 0, then the value of 500eλ500e^{\lambda} is

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  10. 10
    If log⁡3x16+9log⁡273x3=5\sqrt{\log_{3}x^{16}}+ 9\log_{27}\sqrt[3]{\frac{3}{x}}= 5, then x=…x= \ldots.

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  11. 11
    The line passing through the point (5,1,a)(5,1,a) and (3, b,1)(3,\text{ }b,1) crosses the yzyz-plane at (0,172,−132)\left( 0,\frac{17}{2},\frac{- 13}{2} \right), then the value of 2a+3b2a+ 3b is

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  12. 12
    The direction ratios of the line of intersection of the planes x−y+z−5=0x-y+z- 5 = 0 and x−3y−6=0x- 3y- 6 = 0, are

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  13. 13
    The distance between the line rˉ=3i^−2j^+k^+λ(i^+j^+k^)\bar{r}= 3\widehat{i}- 2\widehat{j}+\widehat{k}+\lambda(\widehat{i}+\widehat{j}+\widehat{k}) and the plane rˉ⋅(2i^+j^+k^)=4\bar{r}\cdot (2\widehat{i}+\widehat{j}+\widehat{k}) = 4 is

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  14. 14
    If y=sin⁡2(cot⁡−1(1−x1+x))y=\sin^{2}\left( \cot^{- 1}\left( \sqrt{\frac{1 -x}{1 +x}} \right) \right), then dy dx=\frac{dy}{\text{ }dx}=

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  15. 15
    ∫sin⁡x5sin⁡2x+6cos⁡2x dx\int\frac{\sin x}{\sqrt{5\sin^{2}x+ 6\cos^{2}x}}\text{ }dx

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  16. 16
    If f(x)=ksinx+2cos⁡xsin⁡x+cos⁡xf(x) =\frac{ksinx+ 2\cos x}{\sin x+ \cos x} is strictly increasing for all real values of xx, then

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  17. 17
    The locus of the points represented by ∣z+3∣−∣z−3∣=6|z+ 3| - |z- 3| = 6, where zz is a complex number, is ....

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  18. 18
    The locus of point of intersection of the tangents to the circle x2+y2=16x^{2}+y^{2}= 16, such that the angle between them is 60∘60^{\circ}, is

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  19. 19
    A random variable X has p.m.f. P(X=x)= 4Cx24,x=0,1,2,3,4P(X =x) =\frac{\ ^{4}C_{x}}{2^{4}},x= 0,1,2,3,4 and μ\mu and σ2\sigma^{2} are mean and variance respectively of random variable X , then

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  20. 20
    The c.d.f. of a discrete random variable XX is
    XXF(X=x)F(X = x)
    −3-30.1
    −1-10.3
    00.5
    10.65
    30.75
    50.85
    70.90
    91
    Then P[X=−3]P[X<0]=\frac{P\lbrack X = - 3\rbrack}{P\lbrack X < 0\rbrack}=

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  21. 21
    The distance between the lines represented by 16x2+9y2+48x−24xy−36y+35=016x^{2}+ 9y^{2}+ 48x- 24xy- 36y+ 35 = 0 is ____\_\_\_\_ units

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  22. 22
    The value of tan⁡20∘tan⁡80∘cot⁡50∘=\tan20^{\circ}\tan80^{\circ}\cot50^{\circ}=

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  23. 23
    The equation of the curve passing through the origin and satisfying the equation (1+x2)dy dx+2xy=4x2\left( 1 +x^{2} \right)\frac{dy}{\text{ }dx}+ 2xy= 4x^{2}, is

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  24. 24
    The value of m∈Rm\in\mathbb{R}, when angle between the vectors pˉ=myi^−6j^+3k^\bar{p}=my\widehat{i}- 6\widehat{j}+ 3\widehat{k} and qˉ=yi^+2j^+2myk^\bar{q}=y\widehat{i}+ 2\widehat{j}+ 2my\widehat{k} is obtuse angle, is

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  25. 25
    The volume of the tetrahedron whose coterminous edges are represented by a‾=−12i^+pk^,b‾=3j^,−k^,c‾=2i^+j^−15k^\overline{a}= - 12\widehat{i}+ p\widehat{k},\overline{b}= 3\widehat{j}, -\widehat{k},\overline{c}= 2\widehat{i}+\widehat{j}- 15\widehat{k} is 570 cu. units, then p=p =

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  26. 26
    With usual notations, the perimeter of a triangle ABC is 6 times the arithmetic mean of sine of its angles. If a=1a = 1, then ∠A=\angle A =

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  27. 27
    The angle between the lines whose direction cosines are −34,14,−32\frac{-\sqrt{3}}{4},\frac{1}{4},\frac{-\sqrt{3}}{2} and −34,14,32\frac{-\sqrt{3}}{4},\frac{1}{4},\frac{\sqrt{3}}{2} is

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  28. 28
    If sin⁡(π4cot⁡θ)=cos⁡(π4tan⁡θ)\sin\left( \frac{\pi}{4}\cot\theta \right)= \cos\left( \frac{\pi}{4}\tan\theta \right), then the general solution of θ\theta is

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  29. 29
    The differential equation of all circles having their centres on the line y=5y= 5 and touching ( X -axis) is ____\_\_\_\_

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  30. 30
    ∫0π4cos⁡2xsin⁡2x(cos⁡3x+sin⁡3x)2 dx=\displaystyle\int_{0}^{\frac{\pi}{4}} \frac{\cos^{2}x\sin^{2}x}{\left(\cos^{3}x+\sin^{3}x\right)^{2}}\,dx=

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  31. 31
    If [(p∧∼q)∧(p∧r)]→(∼p∨q)[(p\land \sim q) \land (p\land r)] \to (\sim p\vee q) has truth value false then truth values of the statements p,q,rp,q,r are respectively

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  32. 32
    In a triangle ABC , with usual notations, tan⁡(A2)=56,tan⁡(C2)=25\tan\left( \frac{A}{2} \right)=\frac{5}{6},\tan\left( \frac{C}{2} \right)=\frac{2}{5}, then

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  33. 33
    If  n+4Cn+1− n+3Cn=15(n+2)\ ^{n+ 4}C_{n+ 1}-\ ^{n+ 3}C_{n}= 15(n+ 2), then n=n=

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  34. 34
    The correct simplified circuit diagram for the logical statement [{q∧(∼q∨r)}∧{∼p∨(p∧∼r)}]∨(p∧r)\mathbf{\lbrack\{ q \land ( \sim q \vee r)\} \land \{ \sim p \vee (p \land \sim r)\}\rbrack \vee (p \land r)} Where p,q,rp,q,r represents switches s1,s2,s3s_{1},s_{2},s_{3} respectively.

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  35. 35
    lim⁡n→∞[11−n4+81−n4+…………+n31−n4]=\lim_{n\rightarrow \infty} \left\lbrack \frac{1}{1 -n^{4}}+\frac{8}{1 -n^{4}}+ \ldots\ldots\ldots\ldots +\frac{n^{3}}{1 -n^{4}} \right\rbrack=

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  36. 36
    If f(θ)=cos⁡θ1⋅cos⁡θ2⋅cos⁡θ3f(\theta) = \cos\theta_{1}\cdot \cos\theta_{2}\cdot \cos\theta_{3} ____\_\_\_\_ cos⁡θn\cos\theta_{n}, then tan⁡θ1+tan⁡θ2+tan⁡θ3+………+tan⁡θn=\tan\theta_{1}+ \tan\theta_{2}+ \tan\theta_{3}+ \ldots\ldots\ldots + \tan\theta_{n}=

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  37. 37
    The length of the altitude through the point D of tetrahedron where the vertices of the tetrahedron are A(2,3,1),B(4,1,−2),C(6,3,7),D(−5,−4,8)A(2,3,1),B(4,1, - 2),C(6,3,7),D( - 5, - 4,8), is

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  38. 38
    The angle between the lines x−1l=y+1m=zn\frac{x- 1}{l}=\frac{y+ 1}{m}=\frac{z}{n} and x+1 m=y−3n=z−1l\frac{x+ 1}{\text{ }m}=\frac{y- 3}{n}=\frac{z - 1}{l}, where l>m>nl> m > n and 1, m,n1,\text{ }m,n are roots of the equation x3+x2−4x−4=0x^{3}+x^{2}- 4x- 4 = 0, is

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  39. 39
    The distance of the point P(3,8,2)P(3,8,2) from the line x−12=y−34=z−23\frac{x- 1}{2}=\frac{y- 3}{4}=\frac{z- 2}{3} measured parallel to the plane 3x+2y−2z+15=03x+ 2y- 2z+ 15 = 0 is

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  40. 40
    The solution set for minimizing the function z=x+yz =x+y with constraints x+y⩾2,x+2y⩽8,y⩽3,x,y⩾0x+y\geqslant 2,x+ 2y\leqslant 8,y\leqslant 3,x,y\geqslant 0 contains

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  41. 41
    The abscissae of the points of the curve y=x3y=x^{3} are in the interval [−2,2]\lbrack - 2,2\rbrack, where the slope of the tangents can be obtained by mean value theorem for the interval [−2,2]\lbrack - 2,2\rbrack are

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  42. 42
    Let xx be the length of each of the equal sides of an isosceles triangle and θ\theta be the angle between these sides. If  x\ x is increasing at the rate 112 m\frac{1}{12}\text{ }m / hour and θ\theta is increasing at the rate π180rad/\frac{\pi}{180}rad/ hour, then the rate at which area of the triangle is increasing when x=12 mx= 12\text{ }m and θ=π4\theta=\frac{\pi}{4} is

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  43. 43
    ∫cos⁡(x16)⋅cos⁡(x8)⋅cos⁡(x4)⋅sin⁡(x16)dx=\int \cos\left( \frac{x}{16} \right)\cdot \cos\left( \frac{x}{8} \right)\cdot \cos\left( \frac{x}{4} \right)\cdot \sin\left( \frac{x}{16} \right)dx=

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  44. 44
    ∫x3(x+1)2dx=\int\frac{x^{3}}{(x+ 1)^{2}}dx=

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  45. 45
    A wire of length 8 units is cut into two parts which are bent respectively in the form of a square and a circle. The least value of the sum of the areas so formed is

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  46. 46
    If A,B,CA,B,C are mutually exclusive and exhaustive events of a sample space SS such that P(B)=32P(A)P(B) =\frac{3}{2}P(A) and P(C)=12P(B)P(C) =\frac{1}{2}P(B), then P(A)=P(A) =

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  47. 47
    Angle between the parabola y2=4(x−1)y^{2}= 4(x- 1) and x2+4(y−3)=0x^{2}+ 4(y- 3) = 0 at the common end of their latus rectum is

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  48. 48
    The points (1,3)(1,3) and (5,1)(5,1) are two opposite vertices of a rectangle. The other two vertices are lie on the line y=2x+cy= 2x+ c where c is the constant, then co-ordinates of other two vertices are

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  49. 49
    If y=ax⋅b2x−1y=a^{x}\cdot b^{2x- 1}, then d2ydx2\frac{d^{2}y}{dx^{2}} is equal to

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  50. 50
    In a culture bacteria count is 1,00,0001,00,000 initially. The number increases by 10%10\% in first 2 hours. In how many hours will the count reach 2,00,0002,00,000, if the rate of growth of bacteria is proportional to the number present?

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