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MHT-CET 2025 (21 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 a random variable X follows the Binomial distribution B(10,p)B(10,p) such that 5P(X=0)=P(X=1)5P(X = 0) = P(X = 1), then the value of P(X=5)P(X=6)\frac{P(X = 5)}{P(X = 6)} is equal to

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  2. 2
    A particular solution of dy dx=(x+9y)2\frac{dy}{\text{ }dx}= (x+ 9y)^{2}, when x=0,y=127x= 0,y=\frac{1}{27} is

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  3. 3
    lim⁡x→∞ex4−1ex4+1=\lim_{x\rightarrow \infty} \frac{e^{x^{4}}- 1}{e^{x^{4}}+ 1}=

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  4. 4
    The shortest distance between the lines r‾=(4i^−j^)+λ(i^+2j^−3k^)\overline{r}= (4\widehat{i}-\widehat{j}) +\lambda(\widehat{i}+ 2\widehat{j}- 3\widehat{k}) and r‾=(i^−j^+2k^)+μ(2i^+4j^−5k^)\overline{r}= (\widehat{i}-\widehat{j}+ 2\widehat{k}) +\mu(2\widehat{i}+ 4\widehat{j}- 5\widehat{k}) is

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  5. 5
    If f(x)=cos⁡(log⁡x)f(x) = \cos(\log x) then f(x2)⋅f(y2)−12[f(x2y2)+f(x2y2)]f\left( x^{2} \right)\cdot f\left( y^{2} \right)-\frac{1}{2}\left\lbrack f\left( \frac{x^{2}}{y^{2}} \right)+ f\left( x^{2}y^{2} \right) \right\rbrack has the value

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  6. 6
    The general solution of dydx=2xyex2\frac{dy}{dx}= 2xye^{x^{2}} is

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  7. 7
    Which of the following is not a homogeneous function?

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  8. 8
    The maximum value and minimum value of the volume of the parallelepiped having coterminous edges i^+xj^+k^,j^+xk^\widehat{i}+x\widehat{j}+\widehat{k},\widehat{j}+x\widehat{k} and xi^+k^x\widehat{i}+\widehat{k} are respectively

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  9. 9
    The following is the probability distribution of X
    XX0123
    P(X=x)P(X = x)1+p5\frac{1 + p}{5}2−2p5\frac{2 - 2p}{5}2−p5\frac{2 - p}{5}2p5\frac{2p}{5}
    For a minimum value of pp, the value of 5E(X)5E(X) is

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  10. 10
    Let aˉ\bar{a} and cˉ\bar{c} be unit vectors at an angle π3\frac{\pi}{3} with each other. If (aˉ×(bˉ×cˉ))⋅(aˉ×cˉ)=5(\bar{a}\times (\bar{b}\times\bar{c})) \cdot (\bar{a}\times\bar{c}) = 5, then [a‾b‾c‾]=\lbrack\overline{a}\overline{b}\overline{c}\rbrack =

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  11. 11
    A random variable X takes values 0,1,2,30,1,2,3, ____\_\_\_\_ with probabilities. P(X=x)=k(x+1)(12)x,kP(X =x) = k(x+ 1)\left( \frac{1}{2} \right)^{x},k is a constant, then P(X=1)=P(X= 1) =

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  12. 12
    The area inside the parabola y2=4axy^{2}= 4ax, between the lines x=ax= a and x=4ax= 4a is equal to

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  13. 13
    ∫24log⁡x2log⁡x2+log⁡(36−12x+x2)dx\int_{2}^{4} \frac{\log x^{2}}{\log x^{2}+ \log\left( 36 - 12x+x^{2} \right)}dx is equal to

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  14. 14
    Let a‾,b‾\overline{a},\overline{b}, and c‾\overline{c} be unit vectors. Suppose that aˉ⋅bˉ=aˉ⋅cˉ=0\bar{a}\cdot\bar{b}=\bar{a}\cdot\bar{c}= 0 and if the angle between bˉ\bar{b} and cˉ\bar{c} is π6\frac{\pi}{6}, then aˉ\bar{a} is

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  15. 15
    If a‾\overline{a} and b‾\overline{b} are unit vectors such that ∣a‾+b‾∣=3|\overline{a}+\overline{b}| =\sqrt{3}, then the angle between a‾\overline{a} and b‾\overline{b} is

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  16. 16
    The value of ∫02[x2]dx\int_{0}^{2} \left\lbrack x^{2} \right\rbrack dx is (where [x]\lbrack x\rbrack denotes the greatest integer function not greater than xx )

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  17. 17
    If ∫3sin⁡xcosx4sin⁡x+7 dx=Asinx−Blog(4sin⁡x+7)+c\int\frac{3\sin xcosx}{4\sin x+ 7}\text{ }dx= Asinx- Blog(4\sin x+ 7) + c where cc is the constant of integration, then the value of A+BA + B is equal to

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  18. 18
    Three vectors i^−k^,λi^+j^+(1−λ)k^\widehat{i}-\widehat{k},\lambda\widehat{i}+\widehat{j}+ (1 -\lambda)\widehat{k} and μi^+λj^+(1+λ−μ)k^\mu\widehat{i}+\lambda\widehat{j}+ (1 +\lambda-\mu)\widehat{k} represents coterminous edges of a parallelopiped, then the volume of the parallelopiped depends on.

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  19. 19
    ∫dxx(x2+1)=\int\frac{dx}{x\left( x^{2}+ 1 \right)}=

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  20. 20
    The joint equation of the bisectors of the angles between the lines x=5x= 5 and y=3y= 3 is

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  21. 21
    If sin⁡−113+sin⁡−135+sin⁡−1x=π2\sin^{- 1}\frac{1}{3}+\sin^{- 1}\frac{3}{5}+\sin^{- 1}x=\frac{\pi}{2} then x=x=

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  22. 22
    ∫x2−6x−16dx\int\sqrt{x^{2}- 6x- 16}dx equals

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  23. 23
    The value of 2tan⁡−112+tan⁡−1382\tan^{- 1}\frac{1}{2}+\tan^{- 1}\frac{3}{8} is

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  24. 24
    With usual notation, in triangle ABC , m∠A=30∘m\angle A =30^{\circ} then the value of (1+ac+bc)(1+cb−ab)\left( 1 +\frac{a}{c}+\frac{b}{c} \right)\left( 1 +\frac{c}{b}-\frac{a}{b} \right) is equal to

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  25. 25
    The greatest possible number of points of intersection of 8 distinct straight lines and 4 distinct circles is

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  26. 26
    In △ABC\bigtriangleup ABC, with usual notations, acosB=bcosA,acosC≠ccosAacosB = bcosA,acosC \neq ccosA then A(△ABC)A( \bigtriangleup ABC) ____\_\_\_\_ sq. units.

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  27. 27
    The radius of the base of a cone is increasing at the rate 3 cm/3\text{ }cm/ minute and the altitude is decreasing at the rate 4 cm/4\text{ }cm/ minute. The rate at which the lateral surface area is changing, when the radius is 7 cm and altitude is 24 cm is

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  28. 28
    In a triangle ABC , with usual notations if a=4, b=8,∠C=60∘a = 4,\text{ }b = 8,\angle C =60^{\circ}, then the value of ∠B\angle B and the ratio cos⁡A:cos⁡C\cos A:\cos C respectively are,

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  29. 29
    The third element in the second row of adjoint of a matrix A=[aij]3×3A =\left\lbrack a_{ij} \right\rbrack_{3 \times 3} (where aij=2i+ja_{ij}= 2i + j ) is

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  30. 30
    The function x5−5x4+5x3−10x^{5}- 5x^{4}+ 5x^{3}- 10 has a maximum, when xx is equal to

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  31. 31
    The function f defined by f(x)=(x+2)e−xf(x) = (x+ 2)e^{-x} is

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  32. 32
    If the function f(x)=x(x+3)e−x2f(x) =x(x+ 3)e^{-\frac{x}{2}} satisfies all the conditions of Rolle's theorem in [−3,0]\lbrack - 3,0\rbrack, then c is

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  33. 33
    If (a+bx)eyx=x(a + bx)e^{\frac{y}{x}}=x, then x3 d2y dx2x^{3}\frac{{\text{ }d}^{2}y}{\text{ }dx^{2}} is equal to

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  34. 34
    If x=log⁡t,t>0x= \log t,t> 0 and y=1ty=\frac{1}{t} then d2ydx2=\frac{d^{2}y}{dx^{2}}=

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  35. 35
    If y=Kcos⁡−1x1+Kcos⁡−1xy=\frac{K^{\cos^{- 1}x}}{1 +K^{\cos^{- 1}x}} and t=Kcos⁡−1xt =K^{\cos^{- 1}x}, then dydt=\frac{dy}{dt}=

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  36. 36
    The complex numbers sin⁡x+icos2x\sin x+icos2x and cos⁡x−isin2x,(i=−1)\cos x- isin2x,(i =\sqrt{- 1}) are conjugate to each other for,

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  37. 37
    The probability that in a random arrangement of the letters of the word 'UNIVERSITY', the two 'I' s do not come together is

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  38. 38
    The logical statement [∼(∼p∨q)∨(p∧r)∧(∼q∧r)]\left\lbrack \sim\left( \sim p\vee q \right)\vee\left( p\land r \right)\land\left( \sim q\land r \right) \right\rbrack is equivalent to

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  39. 39
    The L.P.P. , minimize z=30x+20y,x+y≤8z= 30x+ 20y,x+y\leq 8, x+2y≥4,6x+4y≥12,x⩾0,y⩾0x+ 2y\geq 4,6x+ 4y\geq 12,x\geqslant 0,y\geqslant 0 has

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  40. 40
    If the distance of the point P(1,−2,1)P(1, - 2,1) from the plane x+2y−2z=αx+ 2y- 2z=\alpha, where α>0\alpha> 0 is 5 units, then the foot of the perpendicular from P to the plane is

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  41. 41
    The equation of the plane containing the line x−23=y+12=z−4−2\frac{x- 2}{3}=\frac{y+ 1}{2}=\frac{z- 4}{- 2} and the point (0,5,0)(0,5,0) is

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  42. 42
    If the statement pattern (p∧q)→(r∨∼s)(p\land q) \rightarrow (r\vee \sim s) is false, then the truth values of p,q,rp,q,r and ss are respectively

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  43. 43
    If f(x)={8x−4x−2x+1x2, if x>0exsin⁡x+x+λlog⁡4, if x⩽0f(x) =\left\{ \begin{matrix} \frac{8^{x}-4^{x}-2^{x}+ 1}{x^{2}}, & \text{~if~}x> 0 \\ e^{x}\sin x+x+\lambda \log4, & \text{~if~}x\leqslant 0 \end{matrix} \right. is continuous at x=0x= 0 then the value of 1000eλ=1000e^{\lambda}=

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  44. 44
    The line of intersection of the planes r‾⋅(3i^−j^+k^)=1\overline{r}\cdot (3\widehat{i}-\widehat{j}+\widehat{k}) = 1 and  r‾⋅(i^+4j^−2k^)=2\ \overline{r}\cdot (\widehat{i}+ 4\widehat{j}- 2\widehat{k}) = 2 is parallel to the vector

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  45. 45
    The Y-intercept of the common tangent to the parabolas y2=32xy^{2}= 32x and x2=108yx^{2}= 108y is

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  46. 46
    If the line 2x+y=k2x+y= k passes though the point which divides the line segment joining the points (1,1)(1,1) and (2,4)(2,4) internally in the ratio 3:23:2 then, (k+1):(k−1)=(k+ 1):(k- 1) =

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  47. 47
    The minimum distance and maximum distance of the point P(2,−7)P(2, - 7) from the circle x2+y2−14x−10y−151=0x^{2}+y^{2}- 14x- 10y- 151 = 0 are respectively____\_\_\_\_ units

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  48. 48
    The lines r‾=a‾+λ(b‾×c‾)\overline{r}=\overline{a}+\lambda(\overline{b}\times\overline{c}) and r‾=c‾+λ(a‾×b‾)\overline{r}=\overline{c}+\lambda(\overline{a}\times\overline{b}) will intersect if

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  49. 49
    3tan⁡610∘−27tan⁡410∘+33tan⁡210∘=3\tan^{6}10^{\circ}- 27\tan^{4}10^{\circ}+ 33\tan^{2}10^{\circ}=

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  50. 50
    The assets of a person reduced in his business such that the rate of reduction is proportional to the square root of the existing assets. If the assets were initially ₹ 10 lakhs and due to loss they reduce to ₹ 10000 after 3 years, then the number of years required for the person to be bankrupt will be

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