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MHT-CET 2026 (18 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
    The difference between the roots of the equation x2+2x+4=0x^2 + 2x + 4 = 0 is …\ldots (where i=−1i = \sqrt{-1})

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  2. 2
    In a test, there are 7 questions of the type 'True or False'. No student got all the answers correct. If the sequence of answers for every student is unique, then the maximum number of students who appeared for the test is …\ldots

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  3. 3
    Let fk(x)=1k(cos⁡kx+sin⁡kx)f_k(x) = \dfrac{1}{k}(\cos^k x + \sin^k x) where k∈Nk \in N, then f6(x)−f4(x)=…f_6(x) - f_4(x) = \ldots

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  4. 4
    The coordinates of the orthocenter of the triangle whose sides are represented by the lines 4x−7y+10=0,x+y=54x - 7y + 10 = 0, x + y = 5 and 7x+4y=157x + 4y = 15 are …\ldots

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  5. 5
    The combined equation of lines parallel to the coordinate axes and passing through the point of intersection of lines represented by x2−6xy+5y2+10x−14y+9=0x^2 - 6xy + 5y^2 + 10x - 14y + 9 = 0 is …\ldots

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  6. 6
    The number of circles passing through the origin and touching the lines x+y=1x + y = 1 and x−y=1x - y = 1 is …\ldots

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  7. 7
    A parabola has its focus on the positive X-axis and the Y-axis as its directrix. If P(α,4)P(\alpha, 4) is a point on this parabola such that the tangent to the parabola at point P passes through the origin, then the distance of P from origin is

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  8. 8
    If lim⁡x→kx3−k3x2−k2=lim⁡x→01−cos⁡(2x)xsin⁡x\lim_{x \to k}\dfrac{x^3 - k^3}{x^2 - k^2} = \lim_{x \to 0}\dfrac{1 - \cos(2x)}{x\sin x}, then the value of kk is …\ldots

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  9. 9
    In a switching circuit, if the combination (S1∧S2)(S_1 \wedge S_2) is connected in parallel to the combination (S1′∧S2′)(S_1' \wedge S_2'), then the room is lit only when …\ldots

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  10. 10
    If the truth value of the statement pattern (∼p∧q)∨(∼p∧∼q)∨(p∧∼q)(\sim p \wedge q) \vee (\sim p \wedge \sim q) \vee (p \wedge \sim q) is FF, then the truth values of (p∨∼q)(p \vee \sim q) and (p→q)(p \to q) are …\ldots respectively.

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  11. 11
    The minimum number of switches in the simplified form of the following switching circuit is

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  12. 12
    In △ABC\triangle ABC, with usual notations, if Δ\Delta denotes the area of triangle ABCABC then the value of 2s(b+c−a)tan⁡(A2)2s(b + c - a)\tan\left(\dfrac{A}{2}\right) is equal to …\ldots

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  13. 13
    If matrix A and its inverse A−1A^{-1} are given by A=[0121233x1]A = \begin{bmatrix} 0 & 1 & 2 \\ 1 & 2 & 3 \\ 3 & x & 1 \end{bmatrix} and A−1=[12−1212−43y52−3212]A^{-1} = \begin{bmatrix} \dfrac{1}{2} & -\dfrac{1}{2} & \dfrac{1}{2} \\ -4 & 3 & y \\ \dfrac{5}{2} & -\dfrac{3}{2} & \dfrac{1}{2} \end{bmatrix}, then the polar co-ordinates of the points whose Cartesian co-ordinates are (x,y)(x, y) are …\ldots

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  14. 14
    Let A=[−3214]A = \begin{bmatrix} -3 & 2 \\ 1 & 4 \end{bmatrix} and if A2−2A+I=[18pq11]A^2 - 2A + I = \begin{bmatrix} 18 & p \\ q & 11 \end{bmatrix}, then …\ldots

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  15. 15
    For x>0x > 0 and (xlog⁡x)<1(x\log x) < 1, if y=cot⁡−1(x−log⁡xx2log⁡ex2+log⁡xx)y = \cot^{-1}\left(\dfrac{x - \log x^{x^2}}{\log e^{x^2} + \log x^x}\right), then dydx=…\dfrac{dy}{dx} = \ldots

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  16. 16
    For x>0x > 0, if sin⁡(cos⁡−1x+tan⁡−1x)−cos⁡(sin⁡−1x+tan⁡−1x)=sin⁡(cot⁡−12)\sin(\cos^{-1}x + \tan^{-1}x) - \cos(\sin^{-1}x + \tan^{-1}x) = \sin(\cot^{-1}2) then x=x =

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  17. 17
    If tan⁡−1ax+tan⁡−13x=π4\tan^{-1}ax + \tan^{-1}3x = \dfrac{\pi}{4}, where 3ax2<13ax^2 < 1, then value of aa for x=16x = \dfrac{1}{6} is …\ldots

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  18. 18
    sec⁡2(tan⁡−13)−tan⁡2(sec⁡−13)=\sec^2(\tan^{-1}3) - \tan^2(\sec^{-1}3) =

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  19. 19
    The domain of the function f(x)=x1+xf(x) = \sqrt{\dfrac{x}{1 + x}} is …\ldots

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  20. 20
    If f(x)=3x+3−x−2tan⁡x⋅log⁡(1+x)f(x) = \dfrac{3^x + 3^{-x} - 2}{\tan x \cdot \log(1 + x)} for x≠0x \neq 0, is continuous at x=0x = 0, then the value of f(0)f(0) is equal to …\ldots

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  21. 21
    If ∫f′(x)⋅ex2 dx=(x−1)⋅ex2+k\displaystyle\int f'(x) \cdot e^{x^2}\,dx = (x - 1) \cdot e^{x^2} + k, where kk is constant of integration, then f(x)=…f(x) = \ldots

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  22. 22
    Let y=x3ypy = \sqrt[p]{x^3y}. If dydx=32\dfrac{dy}{dx} = \dfrac{3}{2} when y=1y = 1, then the value of pp is equal to …\ldots

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  23. 23
    If f′(x)=sin⁡2xf'(x) = \sin^2 x and y=f(2x−1x2+1)y = f\left(\dfrac{2x - 1}{x^2 + 1}\right), then dydx\dfrac{dy}{dx} at x=1x = 1 is

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  24. 24
    If f:R→Rf : R \to R is an even function then

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  25. 25
    Let g(x)=f(x)+f(1−x)g(x) = f(x) + f(1 - x) and f′′(x)<0,0≤x≤1f''(x) < 0, 0 \le x \le 1, then …\ldots

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  26. 26
    If the function f(x)=ax2+bx+sin⁡xf(x) = ax^2 + bx + \sin x satisfies all the conditions of Rolle's theorem on [0,π][0, \pi] and the slope of the tangent to the curve y=f(x)y = f(x) at x=π4x = \dfrac{\pi}{4} is zero, then a−b=a - b =

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  27. 27
    If a particle moves such that the displacement (s) is proportional to the square of the velocity (v), then its acceleration (a) is

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  28. 28
    If ∫ex+tan⁡−1x(x2+2sec⁡2(tan⁡−1x))dx=ef(x)+c\displaystyle\int e^{x + \tan^{-1}x}\left(\dfrac{x^2 + 2}{\sec^2(\tan^{-1}x)}\right)dx = e^{f(x)} + c, then …\ldots

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  29. 29
    Let f(x)=x,f1(x)=f(log⁡x),f2(x)=f1(log⁡x),f3(x)=f2(log⁡x),…f(x) = x, f_1(x) = f(\log x), f_2(x) = f_1(\log x), f_3(x) = f_2(\log x), \ldots and so on. Then ∫1f(x) f1(x) f2(x)…f2026(x) dx=…\displaystyle\int\dfrac{1}{f(x)\,f_1(x)\,f_2(x)\ldots f_{2026}(x)}\,dx = \ldots

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  30. 30
    If ∫02x(2−x)b dx=327\int_0^2 x(2 - x)^b\,dx = \dfrac{32}{7}, where b∈Nb \in N then b=b =

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  31. 31
    ∫02∣4x−5∣ dx=…\int_0^2|4x - 5|\,dx = \ldots

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  32. 32
    If the area bounded by y=x3+axy = x^3 + ax (where a>0a > 0), the xx-axis and the lines x=−2x = -2 and x=1x = 1 is 374\dfrac{37}{4} square units, then …\ldots

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  33. 33
    The rate of disintegration of a radioactive element at any time is proportional to its mass at that time, where k (k>0)k\ (k > 0) is the constant of proportionality. The time during which an original mass of 1.5 gm will disintegrate to a mass of 0.5 gm is …\ldots

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  34. 34
    The differential equation of the family of all parabolas whose axis is the yy-axis is …\ldots

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  35. 35
    The integrating factor of the differential equation xdydx+2y=x2log⁡xx\dfrac{dy}{dx} + 2y = x^2\log x is

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  36. 36
    A spherical raindrop evaporates at a rate proportional to its surface area. The differential equation involving the rate of change of its radius rr with time 't' is …\ldots (where kk is a positive constant)

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  37. 37
    The order and degree of the differential equation 1+1(dydx)2=(d2ydx2)32\sqrt{1 + \dfrac{1}{\left(\dfrac{dy}{dx}\right)^2}} = \left(\dfrac{d^2y}{dx^2}\right)^{\frac{3}{2}} are …\ldots

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  38. 38
    The equation of the plane passing through the points having position vectors (aˉ+bˉ),(bˉ+cˉ)(\bar{a} + \bar{b}), (\bar{b} + \bar{c}) and (aˉ+cˉ)(\bar{a} + \bar{c}) is …\ldots

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  39. 39
    If the volume of the tetrahedron whose coterminous edges are given by the vectors aˉ=−2i^+3j^−3k^,bˉ=4i^+5j^+(λ−10)k^,cˉ=6i^+2j^−3k^\bar{a} = -2\hat{i} + 3\hat{j} - 3\hat{k}, \bar{b} = 4\hat{i} + 5\hat{j} + (\lambda - 10)\hat{k}, \bar{c} = 6\hat{i} + 2\hat{j} - 3\hat{k} is 11 cubic units, then the sum of the possible values of λ\lambda is …\ldots

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  40. 40
    The value of θ∈(0,π2)\theta \in \left(0, \dfrac{\pi}{2}\right) for which vectors aˉ=(sin⁡θ)i^+(cos⁡θ)j^\bar{a} = (\sin\theta)\hat{i} + (\cos\theta)\hat{j} and bˉ=i^−3j^+2k^\bar{b} = \hat{i} - \sqrt{3}\hat{j} + 2\hat{k} are perpendicular is

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  41. 41
    If ∣aˉ∣=∣bˉ∣=1,∣cˉ∣=2|\bar{a}| = |\bar{b}| = 1, |\bar{c}| = 2 and aˉ×(aˉ×cˉ)+bˉ=0ˉ\bar{a} \times (\bar{a} \times \bar{c}) + \bar{b} = \bar{0}, then the acute angle between aˉ\bar{a} and cˉ\bar{c} is …\ldots

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  42. 42
    The value of ∣aˉ⋅bˉ∣2+∣aˉ×bˉ∣⋅∣aˉ×bˉ∣|\bar{a}\cdot\bar{b}|^2 + |\bar{a} \times \bar{b}| \cdot |\bar{a} \times \bar{b}| is …\ldots

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  43. 43
    If aˉ⋅bˉ=β\bar{a}\cdot\bar{b} = \beta and aˉ×bˉ=cˉ\bar{a} \times \bar{b} = \bar{c} then aˉ=\bar{a} =

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  44. 44
    If the equation of a plane passing through A(1,p,2),B(3,2,4)A(1, p, 2), B(3, 2, 4) and parallel to the z axis, is 3x−2y−q=03x - 2y - q = 0, then …\ldots

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  45. 45
    The angle between the line rˉ=(i^+2j^+k^)+λ(i^+j^+k^)\bar{r} = (\hat{i} + 2\hat{j} + \hat{k}) + \lambda(\hat{i} + \hat{j} + \hat{k}) and the plane rˉ⋅(2i^−j^+k^)=8\bar{r}\cdot(2\hat{i} - \hat{j} + \hat{k}) = 8 is …\ldots

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  46. 46
    The acute angle between the lines 2x=3y=−z2x = 3y = -z and 6x=−y=−4z6x = -y = -4z is …\ldots

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  47. 47
    The maximum value of z=4x+yz = 4x + y subject to the constraints x+y≤5,2x+y≤7,3x+2y≤11,x≥0,y≥0x + y \le 5, 2x + y \le 7, 3x + 2y \le 11, x \ge 0, y \ge 0 is …\ldots

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  48. 48
    A fair die is thrown 4 times. If getting a prime number on the die is considered as a success, then the probability of getting no success at all is …\ldots

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  49. 49
    If the p. d. f. of a continuous random variable X is given by f(x)={k(9+8x−x2),for −1≤x≤40,otherwisef(x) = \begin{cases} k(9 + 8x - x^2), & \text{for } -1 \le x \le 4 \\ 0, & \text{otherwise} \end{cases} then the value of kk is …\ldots

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  50. 50
    In an entrance test, there are multiple choice questions. There are four possible answers to each question, only one of which is correct. The probability that a student knows the answer to a question is 90%. If he gets the correct answer to a question, then the probability that he was guessing is

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