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MHT-CET 2025 (23 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
    A fair coin is tossed 100 times. The chance of getting a head even number of times is

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  2. 2
    Let a‾,b‾,c‾,d‾\overline{a},\overline{b},\overline{c},\overline{d} are vectors such that a‾×b‾=2i‾+3j‾−k‾\overline{a}\times\overline{b}= 2\overline{i}+ 3\overline{j}-\overline{k} and c‾×d‾=3i^+2j^+λk^\overline{c}\times\overline{d}= 3\widehat{i}+ 2\widehat{j}+\lambda\widehat{k} and if ∣aˉ⋅cˉbˉ⋅cˉaˉ⋅dˉbˉ⋅dˉ∣=0\left| \begin{matrix} \bar{a}\cdot\bar{c} & \bar{b}\cdot\bar{c} \\ \bar{a}\cdot\bar{d} & \bar{b}\cdot\bar{d} \end{matrix} \right|= 0, then λ=\lambda=

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  3. 3
    In △ABC\bigtriangleup ABC, with usual notations, if cos⁡B2=c+a2a\cos\frac{B}{2}=\sqrt{\frac{c+a}{2a}}, then a2=a^{2}=

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  4. 4
    The joint equation of two lines passing through (−2,3)( - 2,3) and parallel to the bisectors of the angle between the co-ordinate axes is

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  5. 5
    Let the circle with center at origin pass through the vertices of an equilateral triangle ABC . If A≡(2,4)A\equiv (2,4), then the length of the median through A is

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  6. 6
    Let aˉ=i^+j^+k^,bˉ\bar{a}=\widehat{i}+\widehat{j}+\widehat{k},\bar{b} and cˉ=j^−k^\bar{c}=\widehat{j}-\widehat{k} be three vectors such that a‾×b‾=c‾\overline{a}\times\overline{b}=\overline{c} and a‾⋅c‾=1\overline{a}\cdot\overline{c}= 1. If the length of projection vector of the vector b‾\overline{b} on the vector a‾×c‾\overline{a}\times\overline{c} is ll, then the value of 3l23l^{2} is

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  7. 7
    The distance of the point (1,2)(1,2) from the line x+y=0x+y= 0 measured parallel to the line 3x−y=23x-y= 2 is

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  8. 8
    lim⁡x→2x+3x2+5x3+7x4−166x−2=\lim_{x\rightarrow 2} \frac{x+ 3x^{2}+ 5x^{3}+ 7x^{4}- 166}{x- 2}=

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  9. 9
    If f(x)=10x+7x−14x−5x1−cos⁡x,x≠0f(x) =\frac{10^{x}+7^{x}-14^{x}-5^{x}}{1 - \cos x},x\neq 0 is continuous at x=0x= 0, then the value of f(0)f(0) is

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  10. 10
    If A and B are non-singular matrices of order 2 such that (AB)−1=16[−7−323](AB)^{- 1}=\frac{1}{6}\begin{bmatrix} - 7 & - 3 \\ 2 & 3 \end{bmatrix} and A−1=13[43−10]A^{- 1}=\frac{1}{3}\begin{bmatrix} 4 & 3 \\ - 1 & 0 \end{bmatrix} then B−1=B^{- 1}=

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  11. 11
    If sin⁡A+sin⁡B=x\sin A + \sin B =x and cos⁡A+cos⁡B=y\cos A + \cos B =y, then sin⁡(A+B)=\sin(A + B) =

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  12. 12
    Let mean and standard deviation of probability distribution
    X=xX = x−3-301α\alpha
    P(X=x)P(X = x)14\frac{1}{4}KK14\frac{1}{4}13\frac{1}{3}
    be μ\mu and σ\sigma respectively and if σ−μ=2\sigma-\mu= 2 then σ=\sigma=

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  13. 13
    In a game a man wins ₹ 40 if he gets 5 or 6 on a throw of a fair die and loses ₹ 20 for getting any other number on the die. If he decides to throw the die either till, he gets a five or six or to a maximum of three throws, then his expected gain/loss (in rupees) is

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  14. 14
    The rate at which a substance cools in moving air, is proportional to the difference between the temperature of the substance and that of air. The temperature of air is 290 K and the substance cools from 370 K to 330 K in 10 minutes. Then the time to cool the substance up to 295 K is

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  15. 15
    If x dy dx=y(log⁡y−log⁡x+1)x\frac{\text{ }dy}{\text{ }dx}=y(\log y- \log x+ 1), then the solution of the equation is

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  16. 16
    The order and degree of the differential equation dy dx−4 dy dx−7x=0\sqrt{\frac{dy}{\text{ }dx}}- 4\frac{\text{ }dy}{\text{ }dx}- 7x= 0 is respectively

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  17. 17
    The differential equation of all circles touching the Y -axis at the origin and center on the X -axis is

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  18. 18
    The area bounded by the curve x2=8yx^{2}= 8y and the straight line x−8y+2=0x- 8y+ 2 = 0 is

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  19. 19
    In a triangle ABC with usual notations, cot⁡A2+cot⁡B2+cot⁡C2=\cot\frac{A}{2}+ \cot\frac{B}{2}+ \cot\frac{C}{2}=

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  20. 20
    The area of the rectangle having vertices P,Q,R,SP,Q,R,S with position vectors −i^+j^+k^,i^+j^+k^,i^−j^+k^,−i^−j^+k^-\widehat{i}+\widehat{j}+\widehat{k},\widehat{i}+\widehat{j}+\widehat{k},\widehat{i}-\widehat{j}+\widehat{k}, -\widehat{i}-\widehat{j}+\widehat{k} respectively is

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  21. 21
    The value of ∫01tan⁡−1(1−x+x2)dx\int_{0}^{1} \tan^{- 1}\left( 1 -x+x^{2} \right)dx is

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  22. 22
    ∫35x dx8−x+x=\int_{3}^{5} \frac{\sqrt{x}\text{ }dx}{\sqrt{8 -x}+\sqrt{x}}=

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  23. 23
    ∫dxx+x=\int\frac{dx}{\sqrt{x}+x}=

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  24. 24
    With usual notations, in △ABC\bigtriangleup ABC, the lengths of two sides are 10 cm and 9 cm respectively. If angles A,B,CA,B,C are in A.P. then perimeter of △ABC\bigtriangleup ABC is

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  25. 25
    ∫dxex−1=\int\frac{dx}{e^{x}- 1}=

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  26. 26
    The maximum value of x2/3+(x−2)2/3x^{2/3}+ (x- 2)^{2/3} is

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  27. 27
    ∫(x−3x2+9)2dx=\int\left( \frac{x- 3}{x^{2}+ 9} \right)^{2}dx=

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  28. 28
    The point on the curve 4y2−4y+2x−1=04y^{2}- 4y+ 2x- 1 = 0 at which the tangent becomes parallel toY-axis is

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  29. 29
    The value of tan⁡2(sec⁡−14)+cot⁡2(csc⁡−13)\tan^{2}\left( \sec^{- 1}4 \right)+\cot^{2}\left( \csc^{- 1}3 \right) is

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  30. 30
    In a triangle ABC , with usual notations, if a=5, b=4,cos⁡( A−B)=3132a = 5,\text{ }b = 4,\cos(\text{ }A - B) =\frac{31}{32}, then c=c =

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  31. 31
    Which of the following is the negation of the statement " For all M>0M > 0, there exist x∈ sx\in\text{ }s such that x⩾M′′x\geqslant M^{''}

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  32. 32
    The equation of the line passing through the point of intersection of x−12=y−23=z−34\frac{x- 1}{2}=\frac{y- 2}{3}=\frac{z- 3}{4} and x−45=y−12=z\frac{x- 4}{5}=\frac{y- 1}{2}=z and also through the point ( 2,1,−22,1, - 2 ) is

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  33. 33
    A particle moves along a curve y=2x3−13y=\frac{2x^{3}- 1}{3}. The points on the curve at which the yy co-ordinate is changing 18 times the xx co-ordinate are

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  34. 34
    The equation of motion of the particle is s=at2+bt+cs =at^{2}+ bt + c. If the displacement after 1 second is 20 m , velocity after 2 seconds is 30 m/30\text{ }m/ seconds and the acceleration is 10 m/10\text{ }m/ seconds  2\ ^{2}, then

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  35. 35
    If y=tan⁡−1[4sin⁡2xcos⁡2x−6sin⁡2x]y=\tan^{- 1}\left\lbrack \frac{4\sin2x}{\cos2x- 6\sin^{2}x} \right\rbrack, then dy dx\frac{dy}{\text{ }dx} at x=0x= 0 is

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  36. 36
    If x=asin2t (1+cos⁡2t),y=bcos2t (1−cos⁡2t)x=asin2t\ (1 + \cos2t),y=bcos2t\ (1 - \cos2t) then dy dx\frac{dy}{\text{ }dx} is equal to

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  37. 37
    The contrapositive of the statement ∼p∨(q∧∼r)\sim p\vee (q\land \sim r) is

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  38. 38
    If the lines x−12=y+13=z−14\frac{x- 1}{2}=\frac{y+ 1}{3}=\frac{z- 1}{4} and x−31=y−k2=z1\frac{x- 3}{1}=\frac{y- k}{2}=\frac{z}{1}. intersect, then the value of k is

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  39. 39
    If y=x+y+x+y+..................∞y=\sqrt{x+\sqrt{y+\sqrt{x+\sqrt{y+ ..................\infty}}}}, then dy dx=\frac{dy}{\text{ }dx}=

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  40. 40
    The graph with correct feasible region of L.P.P. for the constraints 2x+y⩽10,y⩽x,y⩽2,x,y⩾02x+y\leqslant 10,y\leqslant x,y\leqslant 2,x,y\geqslant 0 is ...

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  41. 41
    The distance of the point P(3,4,4)P(3,4,4) from the point of intersection of the line joining the points Q(3,−4,−5),R(2,−3,1)Q(3, - 4, - 5),R(2, - 3,1) and the plane 2x+y+z=72x+y+z= 7 is

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  42. 42
    If f(x)=3x+10, g(x)=x2−1f(x) = 3x+ 10,\text{ }g(x) =x^{2}- 1, then (fog)−1(x)=(fog)^{- 1}(x) =

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  43. 43
    A family consisting of a mother, father and their 8 children ( 4 boys and 4 girls) are to be seated at a round table in a party. How many ways can this be done if the mother and father sit together and the males and females alternate?

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  44. 44
    If aˉ,bˉ,cˉ\bar{a},\bar{b},\bar{c} are three vectors such that ∣a‾∣=31,4∣ b‾∣=∣c‾∣=2|\overline{a}| =\sqrt{31},4|\overline{\text{ }b}| = |\overline{c}| = 2 and 2(a‾×b‾)=3(c‾×a‾)2(\overline{a}\times\overline{b}) = 3(\overline{c}\times\overline{a}) and if the angle between bˉ\bar{b} and cˉ\bar{c} is 2π3\frac{2\pi}{3} then ∣a‾×c‾a‾⋅b‾∣2=\left| \frac{\overline{a}\times\overline{c}}{\overline{a}\cdot\overline{b}} \right|^{2}=

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  45. 45
    Let aˉ=2i^+j^+k^,bˉ=i^+2j^−k^\bar{a}= 2\widehat{i}+\widehat{j}+\widehat{k},\bar{b}=\widehat{i}+ 2\widehat{j}-\widehat{k} and vector cˉ\bar{c} be coplanar. If cˉ\bar{c} is perpendicular to aˉ\bar{a}, then cˉ\bar{c} is

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  46. 46
    If x=−2+−3x= - 2 +\sqrt{- 3}, then the value of 2x4+5x3+7x2−x+382x^{4}+ 5x^{3}+ 7x^{2}-x+ 38 is equal to

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  47. 47
    Bag I contain 3 red and 2 green balls and Bag II contains 5 red and 3 green balls. A ball is drawn from one of the bag at random and it is found to be green. Then the probability that it is drawn from Bag I is

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  48. 48
    If the tangent at the point (2secθ,3tan⁡θ)(2sec\theta,3\tan\theta) to the hyperbola x24−y29=1\frac{x^{2}}{4}-\frac{y^{2}}{9}= 1 is parallel to 3x−y+4=03x-y+ 4 = 0, then the value of θ\theta is

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  49. 49
    The equation of the plane containing the line x1=y2=z3\frac{x}{1}=\frac{y}{2}=\frac{z}{3} and perpendicular to the plane containing the lines x2=y3=z1\frac{x}{2}=\frac{y}{3}=\frac{z}{1} and x3=y2=z1\frac{x}{3}=\frac{y}{2}=\frac{z}{1} is

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  50. 50
    The equation of the plane containing the line x+12=y+21=z−23\frac{x+ 1}{2}=\frac{y+ 2}{1}=\frac{z- 2}{3} and the point (1,−1,3)(1, - 1,3) is

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