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MHT-CET 2025 (26 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
    The general solution of the differential equation dy dx=cot⁡x⋅cot⁡y\frac{dy}{\text{ }dx}= \cot x\cdot \cot y is

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  2. 2
    The probability that a person is not a sportsperson is 16\frac{1}{6}. Then the probability that out of the 6 members of the family, 5 are sportspersons is

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  3. 3
    If tan⁡(πcos⁡θ)=cot⁡(πsin⁡θ)\tan(\pi\cos\theta) = \cot(\pi\sin\theta), then sin⁡(π4+θ)=\sin\left( \frac{\pi}{4}+\theta \right)=

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  4. 4
    The line MN whose equation is x−y−2=0x-y- 2 = 0 cuts the X -axis at M and co-ordinates of N are (4, 2). The line MN is rotated about M through 45∘45^{\circ} in anticlockwise direction. The equation of the line MN in the new position is

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  5. 5
    The money invested in a company is compounded continuously. If ₹400 invested today becomes ₹ 800 in 6 years, then at the end of 30 years, it will become (in ₹)

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  6. 6
    The least distance of the point A(10,7)A(10,7) from the circle x2+y2−4x−2y−20=0x^{2}+y^{2}- 4x- 2y- 20 = 0 is length of seg AM. If MM′MM^{'} is the diameter of the circle, then the lengths of AMAM and AM′AM^{'} are respectively ____\_\_\_\_ , ____\_\_\_\_ units

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  7. 7
    If the foot of the perpendicular drawn from the origin to a plane is P(−1,−1,2)P( - 1, - 1,2), then equation of the plane is

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  8. 8
    In a triangle ABC, with usual notations, if a=5a=5, b=7b=7, sin⁡A=34\sin A=\frac{3}{4}, then total number of triangles possible are

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  9. 9
    If sin⁡−1(4x)+sin⁡−1(43x)=−π2\sin^{- 1}(4x) +\sin^{- 1}(4\sqrt{3}x) = -\frac{\pi}{2}, then the value of xx is

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  10. 10
    Let aˉ=αi^+3j^−k^,bˉ=3i^−j^+βk^\bar{a}=\alpha\widehat{i}+ 3\widehat{j}-\widehat{k},\bar{b}= 3\widehat{i}-\widehat{j}+\beta\widehat{k} and c‾=i^+2j^−2k^\overline{c}=\widehat{i}+ 2\widehat{j}- 2\widehat{k} where α,β∈R\alpha,\beta\in\mathbb{R}, be three vectors. If the projection of aˉ\bar{a} on cˉ\bar{c} is 103\frac{10}{3} and b‾×c‾=−6i^+10j^+7k^\overline{b}\times\overline{c}= - 6\widehat{i}+ 10\widehat{j}+ 7\widehat{k}, then the value of (α+β)(\alpha+\beta) is equal to

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  11. 11
    The equation of the directrix of the parabola y2+4y+4x+2=0y^{2}+ 4y+ 4x+ 2 = 0 is

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  12. 12
    If f(x)=1+cos⁡2(x2)f(x) =\sqrt{1 +\cos^{2}\left( x^{2} \right)}, then f′(π2)f^{'}\left( \frac{\sqrt{\pi}}{2} \right) is

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  13. 13
    The modulus of the square root of the complex number 6+8i6 + 8i (where i=−1i=\sqrt{- 1} ) is

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  14. 14
    If [x]2−5[x]+6=0\lbrack x\rbrack^{2}- 5\lbrack x\rbrack + 6 = 0, where [⋅]\lbrack \cdot \rbrack denotes the greatest integer function, then

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  15. 15
    The area of smaller part between the circle x2+y2=4x^{2}+y^{2}= 4 and the line x=1x= 1 is ____\_\_\_\_ sq.units.

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  16. 16
    If the function f(x)={cos⁡ax−cos⁡bxcos⁡cx−cos⁡bx, if x≠0 −1, if x=0f(x) =\left\{ \begin{matrix} \dfrac{\cos ax- \cos bx}{\cos cx- \cos bx} & ,\ \text{if } x\neq 0 \ -1 & ,\ \text{if } x = 0 \end{matrix} \right. is continuous at x=0x = 0, then a2,b2,c2a^{2}, b^{2}, c^{2} are in

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  17. 17
    ∫01tan⁡−1x dx=\int_{0}^{1} \tan^{- 1}x\text{ }dx=

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  18. 18
    If the truth value of the expression [(p∨q)∧(q→r)∧(∼r)]→(p∧q)\lbrack(p\vee q) \land (q\rightarrow r) \land ( \sim r)\rbrack \rightarrow (p\land q) is False, then truth values of p,q,rp,q,r are respectively.

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  19. 19
    Consider statements p:S1p:S_{1} is closed; q:S2q:S_{2} is closed; r:S3r:S_{3} is closed. The simplified equivalent circuit diagram and its logical statement for the switching circuit is respectively.

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  20. 20
    In a triangle ABC , with usual notations, cot⁡(A+B2)⋅tan⁡(A−B2)=\cot\left( \frac{A + B}{2} \right)\cdot \tan\left( \frac{A - B}{2} \right)=

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  21. 21
    In a triangle ABC , with usual notations, (a+b+c)(a+b−c)=3ab(a + b + c)(a + b - c) = 3ab, then ∠C=\angle C =

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  22. 22
    If aˉ,bˉ,cˉ\bar{a},\bar{b},\bar{c} are three coplanar vectors such that ∣a‾∣=1,∣ b‾∣=2, b‾⋅c‾=8|\overline{a}| = 1,|\overline{\text{ }b}| = 2,\overline{\text{ }b}\cdot\overline{c}= 8, the angle between b‾\overline{b} and c‾\overline{c} is 45∘45^{\circ}, then ∣a‾×(b‾×c‾)∣=|\overline{a}\times (\overline{b}\times\overline{c})| =

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  23. 23
    Let aˉ,bˉ\bar{a},\bar{b} and cˉ\bar{c} be vectors of magnitude 2,3 and 4 respectively. If aˉ\bar{a} is perpendicular to (b‾+c‾\overline{b}+\overline{c}), b‾\overline{b} is perpendicular to (c‾+a‾\overline{c}+\overline{a}) and c‾\overline{c} is perpendicular to (aˉ+bˉ)(\bar{a}+\bar{b}), then the magnitude of a‾+b‾+c‾\overline{a}+\overline{b}+\overline{c} is

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  24. 24
    If ∫(x4+1)x(x2+1)2 dx=Alog∣x∣+B1+x2+c\int\frac{\left( x^{4}+ 1 \right)}{x\left( x^{2}+ 1 \right)^{2}}\text{ }dx= Alog|x| +\frac{B}{1 +x^{2}}+ c, then A−BA - B is (where c is the constant of integration)

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  25. 25
    The angle between the tangents drawn from the point (1,4)(1,4) to the parabola y2=4xy^{2}= 4x, is

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  26. 26
    The solution for minimizing the function z=x+yz =x+y under an L.P.P. with constraints x+y≥2,x+2y≤8,y≤3,x,y≥0x+y\geq 2,x+ 2y\leq 8,y\leq 3,x,y\geq 0 is

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  27. 27
    The angle between lines whose direction cosines satisfy the equation l+m+n=0l+ m + n = 0 and l2−m2−n2=0l^{2}-m^{2}-n^{2}= 0, is

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  28. 28
    The circumcenter of the triangle formed by lines xy+2x+2y+4=0xy+ 2x+ 2y+ 4 = 0 and x+y+2=0x+y+ 2 = 0 is

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  29. 29
    A triangle ABC is formed by A(1,−1,0)A(1, - 1,0), B(3,5,3),C(−11,−5,6)B(3,5,3),C( - 11, - 5,6). The equation of internal angle bisector of angle A is

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  30. 30
    If a‾=λxi^+yj^+4zk^\overline{a}=\lambda x\hat{i}+y\hat{j}+4z\hat{k}, b‾=yi^+xj^+3yk^\overline{b}=y\hat{i}+x\hat{j}+3y\hat{k} and c‾=−zi^−2zj^−(λ+1)xk^\overline{c}=-z\hat{i}-2z\hat{j}-(\lambda+1)x\hat{k} are the sides of the triangle ABC, where x,y,zx,y,z are not all zero, such that a‾+b‾−c‾=0‾\overline{a}+\overline{b}-\overline{c}=\overline{0}, then the value of λ\lambda is

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  31. 31
    The cumulative distribution function of a discrete random variable X is then P(X⩽0)P(X>0)=\frac{P(X \leqslant 0)}{P(X > 0)}=
    X=xX = xF(X=x)F(X = x)
    −4-40.1
    −2-20.3
    00.5
    20.65
    40.75
    60.85
    80.90
    101

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  32. 32
    The mirror image of the point P(−1,2,−4)P( - 1,2, - 4) in the plane x−y−2z+1=0x-y- 2z+ 1 = 0 is

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  33. 33
    A manufacturer sells xx items at a price of rupees (6−x40)\left( 6 -\frac{x}{40} \right) each. The cost price of xx items is ₹ (x5+193)\left( \frac{x}{5}+ 193 \right). The maximum profit in ₹ ____\_\_\_\_ is

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  34. 34
    The following is p.d.f. of continuous random variable X f(x)={x8, if 0<x<40, otherwise f(x) =\left\{ \begin{matrix} \frac{x}{8} & ,\text{~if~}0 <x< 4 \\ 0 & ,\text{~otherwise~} \end{matrix} \right..Then F(0.5),F(1.7)F(0.5),F(1.7) and F(5)F(5) is respectively

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  35. 35
    The equation of the tangent to the curve (1+x2)y=2−x\left( 1 +x^{2} \right)y= 2 -x, where it crosses the X -axis, is

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  36. 36
    If y=αlog⁡x+βx3−xy=\alpha \log x+\beta x^{3}-x has extreme values at x=−1x= - 1 and x=1x= 1, then α\alpha and β\beta are respectively

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  37. 37
    The equation of a curve passing through (1,0)(1,0) and having slope of tangent at any point (x,y)(x,y) of the curve as y−1x2+x\frac{y- 1}{x^{2}+x} is

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  38. 38
    A box contains 8 red and xx number of green balls. 3 balls are drawn at random, if the probability that 3 balls being red is 715\frac{7}{15}, then number of green balls is...

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  39. 39
    The number of ways in which a team of 11 players can be formed out of 25 players, if 6 out of them are always to be included and 5 of them are always to be excluded, is

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  40. 40
    lim⁡x→∞(2x+1)50+(2x+2)50+(2x+3)50+...+(2x+100)50(2x)50+(10)50=\lim_{x\rightarrow \infty} \frac{(2x+ 1)^{50}+ (2x+ 2)^{50}+ (2x+ 3)^{50}+ ... + (2x+ 100)^{50}}{(2x)^{50}+ (10)^{50}}=

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  41. 41
    The differential equation which represents the family of curves y=c1ec2xy=c_{1}e^{c_{2}x}, where c1,c2c_{1},c_{2} are arbitrary constants is

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  42. 42
    ∫1221xcsc⁡101(x−1x)dx=\int_{\frac{1}{2}}^{2} \frac{1}{x}\csc^{101}\left( x-\frac{1}{x} \right)dx=

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  43. 43
    Matrix A is non-singular matrix and (A−3I)(A−5I)=0(A- 3I)(A- 5I) = 0, then 158A−1=\frac{15}{8}A^{- 1}= ____\_\_\_\_

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  44. 44
    If the plane x2+y3+z6=1\frac{x}{2}+\frac{y}{3}+\frac{z}{6}= 1 cuts the co-ordinate axes at points A,B,CA,B,C respectively, then area of the triangle ABC is

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  45. 45
    The number of positive integral solutions of tan⁡−1x+cos⁡−1(y1+y2)=sin⁡−1(310)\tan^{- 1}x+\cos^{- 1}\left( \frac{y}{\sqrt{1 +y^{2}}} \right)=\sin^{- 1}\left( \frac{3}{\sqrt{10}} \right) are

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  46. 46
    If ∫dxx4+5x2+4=Atan⁡−1x+Btan⁡−1x2+c\int\frac{dx}{x^{4}+ 5x^{2}+ 4}=A\tan^{- 1}x+B\tan^{- 1}\frac{x}{2}+c where cc is a constant of integration, then

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  47. 47
    ∫tan⁡xsin⁡x⋅cos⁡x dx=\displaystyle\int\dfrac{\sqrt{\tan x}}{\sin x\cdot \cos x}\,dx=

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  48. 48
    If f(x)=sin⁡2x1+cot⁡x+cos⁡2x1+tan⁡xf\left( x \right)=\frac{\sin^{2}x}{1 + \cot x}+\frac{\cos^{2}x}{1 + \tan x}, then the value of f′(π6)f^{'}\left( \frac{\pi}{6} \right) is equal to

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  49. 49
    If y=tan⁡−1(1+sin⁡x1−sin⁡x),0⩽x<π2y=\tan^{- 1}\left( \sqrt{\frac{1 + \sin x}{1 - \sin x}} \right),0 \leqslant x<\frac{\pi}{2}, then y′(π6)=y^{'}\left( \frac{\pi}{6} \right)=

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  50. 50
    The volume of tetrahedron with co-terminus edges aˉ,bˉ,cˉ\bar{a},\bar{b},\bar{c} is 643\frac{64}{3} cubic units, then volume of parallelopiped considering co-terminus edges given by the vectors aˉ+bˉ,bˉ+cˉ,cˉ+aˉ\bar{a}+\bar{b},\bar{b}+\bar{c},\bar{c}+\bar{a} is ____\_\_\_\_ cubic units.

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