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MHT-CET 2026 (15 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 ω\omega is a complex cube root of unity, then the value of the expression 2(1+1ω)(1+1ω2)+3(2+1ω)(2+1ω2)+…+(n+1)(n+1ω)(n+1ω2)2\left(1+\dfrac{1}{\omega}\right)\left(1+\dfrac{1}{\omega^2}\right)+3\left(2+\dfrac{1}{\omega}\right)\left(2+\dfrac{1}{\omega^2}\right)+\ldots+(n+1)\left(n+\dfrac{1}{\omega}\right)\left(n+\dfrac{1}{\omega^2}\right) is...

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  2. 2
    The number of five-digit numbers formed using the digits 2,3,5,7,92, 3, 5, 7, 9 without repetition and which are greater than 2400024000 are

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  3. 3
    In △ABC\triangle ABC, with the usual notations, ∠C=90∘\angle C = 90^\circ, then sin⁡(A−B)\sin(A-B) is equal to....

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  4. 4
    If P=tan⁡20∘P = \tan 20^\circ, then the value of tan⁡160∘−tan⁡110∘1+tan⁡160∘tan⁡110∘\dfrac{\tan 160^\circ - \tan 110^\circ}{1 + \tan 160^\circ \tan 110^\circ} in terms of PP, is...

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  5. 5
    If cos⁡(pθ)+cos⁡(qθ)=0\cos(p\theta) + \cos(q\theta) = 0 and p≠qp \neq q, then the general value of θ\theta (where n is any integer) is...

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  6. 6
    If a triangle ABCABC has vertices A(1,−6),B(2,−3)A(1,-6), B(2,-3) and C(3,−2)C(3,-2), then the coordinates of its orthocenter are....

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  7. 7
    If θ\theta is the acute angle between the lines represented by the equation x2−3xy+2y2=0x^2 - 3xy + 2y^2 = 0, then 3sin⁡θ+2cos⁡θ3sin⁡θ−2cos⁡θ=\dfrac{3\sin\theta + 2\cos\theta}{3\sin\theta - 2\cos\theta} =

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  8. 8
    The triangle formed by the lines 2x2−3xy−2y2=02x^2 - 3xy - 2y^2 = 0 and 3x−y=73x - y = 7 is...

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  9. 9
    The centre and radius of the circle (a+1)x2+3y2−6x+9y+a+4=0(a+1)x^2 + 3y^2 - 6x + 9y + a + 4 = 0 are respectively ...

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  10. 10
    If the line 4x+3y=74x + 3y = 7 touches the hyperbola x2−y2=7x^2 - y^2 = 7, then the sum of the co-ordinates of the point of contact is...

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  11. 11
    If lim⁡x→0(4x−1)3tan⁡(x4)log⁡(1+x23)=96(log⁡a)b\displaystyle\lim_{x \to 0}\dfrac{(4^x - 1)^3}{\tan\left(\dfrac{x}{4}\right)\log\left(1 + \dfrac{x^2}{3}\right)} = 96(\log a)^b, then (a+b)=(a + b) =

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  12. 12
    The dual of the converse of the inverse of the logical statement p→(q→r)p \to (q \to r) is equivalent to...

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  13. 13
    The statements p, q and r have truth values True, False and False respectively. The truth values of a logical statement [∼(p∧∼q)∨(q∨∼r)][\sim(p \wedge \sim q) \vee (q \vee \sim r)] and its dual are, respectively.....

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  14. 14
    In △ABC\triangle ABC with usual notations, if 1+tan⁡(A2)tan⁡(B2)=ks1 + \tan\left(\dfrac{A}{2}\right)\tan\left(\dfrac{B}{2}\right) = \dfrac{k}{s} (where ss is the semi-perimeter), then the value of kk is...

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  15. 15
    Let A=[2k−11102k−11002k−1]A = \begin{bmatrix} 2k-1 & 1 & 1 \\ 0 & 2k-1 & 1 \\ 0 & 0 & 2k-1 \end{bmatrix} and B=[02k−111−2k0k−1−k0]B = \begin{bmatrix} 0 & 2k-1 & 1 \\ 1-2k & 0 & k \\ -1 & -k & 0 \end{bmatrix} where k is a real number. If det⁡(adjA)+det⁡(adjB)=116\det(\mathrm{adj}A) + \det(\mathrm{adj}B) = 11^6, then the value of k−5k - 5 is equal to...

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  16. 16
    If A=[cos⁡θ−sin⁡θsin⁡θcos⁡θ]A = \begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix}, then the matrix A−3A^{-3} when θ=π6\theta = \dfrac{\pi}{6} is equal to...

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  17. 17
    If tan⁡−1[5−261+6]=π3−tan⁡−1(k)\tan^{-1}\left[\dfrac{\sqrt{5 - 2\sqrt{6}}}{1 + \sqrt{6}}\right] = \dfrac{\pi}{3} - \tan^{-1}(k), then sec⁡−1(k)=…\sec^{-1}(k) = \ldots

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  18. 18
    If f(x−y)+f(x+y)=2f(x)f(y)f(x - y) + f(x + y) = 2f(x)f(y) for all x,y∈Rx, y \in \mathbb{R}, then f(x)f(x) is .....

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  19. 19
    If the function f is continuous at x=1x = 1, where f(x)=1+cos⁡(πx)π(1−x)2f(x) = \dfrac{1 + \cos(\pi x)}{\pi(1 - x)^2}, for x≠1x \neq 1, then the value of f(1)f(1) is....

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  20. 20
    If y=log⁡xxy = \log x^x, then the value of (dydx)2\left(\dfrac{dy}{dx}\right)^2 at x=e2x = e^2 is

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  21. 21
    If y=cos⁡−1(sin⁡x)y = \cos^{-1}(\sin x), where π2<x<π\dfrac{\pi}{2} < x < \pi, then dydx=…\dfrac{dy}{dx} = \ldots

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  22. 22
    If y=(ax+bcx+d)y = \left(\dfrac{ax + b}{cx + d}\right), then 2dydx⋅d3ydx32\dfrac{dy}{dx} \cdot \dfrac{d^3y}{dx^3} is equal to

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  23. 23
    Let f(x)=3x3+4ex2f(x) = 3x^3 + 4e^{\frac{x}{2}} and g(x)=f−1(x)g(x) = f^{-1}(x). Then the value of g′(4)g'(4) is equal to...

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  24. 24
    If the function f(x)=ax3−bx2−8x−4f(x) = ax^3 - bx^2 - 8x - 4 satisfies Roll's theorem in [1,3][1, 3], if f′(2)=0f'(2) = 0 then a−ba - b is equal to...

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  25. 25
    The co-ordinates of the point on the curve y=xlog⁡xy = x\log x at which the normal is parallel to the line 2x−2y=32x - 2y = 3 are...

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  26. 26
    If ∫1cos⁡3xsin⁡2x dx=p(tan⁡2x+q)tan⁡x+c\displaystyle\int\dfrac{1}{\cos^3 x\sqrt{\sin 2x}}\,dx = p(\tan^2 x + q)\sqrt{\tan x} + c, then the values of pp and qq respectively are

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  27. 27
    If ∫21+sin⁡x dx=−4cos⁡(ax+b)+c\displaystyle\int\sqrt{2}\sqrt{1 + \sin x}\,dx = -4\cos(ax + b) + c, then the value of a,ba, b respectively are...

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  28. 28
    ∫x−2x dx=\displaystyle\int\dfrac{\sqrt{x - 2}}{x}\,dx =

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  29. 29
    If f(x)f(x) and g(x)g(x) are integrable functions then [∫f(x)dx][∫g(x)dx]=\left[\displaystyle\int f(x)dx\right]\left[\displaystyle\int g(x)dx\right] =

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  30. 30
    If an antiderivative of f(x)f(x) is exe^x and an antiderivative of g(x)g(x) is cos⁡x\cos x, then ∫f(x)cos⁡x dx+∫g(x)ex dx=\displaystyle\int f(x)\cos x\,dx + \int g(x)e^x\,dx =

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  31. 31
    ∫0log⁡10[exex−1ex+8]dx=\displaystyle\int_0^{\log 10}\left[\dfrac{e^x\sqrt{e^x - 1}}{e^x + 8}\right]dx =

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  32. 32
    The value of the integral ∫−π/2π/2(x2cos⁡x1+ex)dx\displaystyle\int_{-\pi/2}^{\pi/2}\left(\dfrac{x^2\cos x}{1 + e^x}\right)dx is equal to (π2A)−B\left(\dfrac{\pi^2}{A}\right) - B. Then (AB)=\left(\dfrac{A}{B}\right) =

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  33. 33
    The area of the smaller region bounded by the circle x2+y2=4x^2 + y^2 = 4 and the line x=1x = 1 is...

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  34. 34
    If the area bounded by the curve x2=byx^2 = by and the lines y=1,y=4y = 1, y = 4 in the first quadrant is 2828 sq. units, then the value of b is...

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  35. 35
    The solution of the differential equation dydx=ex−y+x2e−y\dfrac{dy}{dx} = e^{x - y} + x^2e^{-y} is...

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  36. 36
    If the differential equation ∣f(x)f′(x)f′(x)f′′(x)∣=0\begin{vmatrix} f(x) & f'(x) \\ f'(x) & f''(x) \end{vmatrix} = 0 for all xx and f(0)=1,f′(0)=2f(0) = 1, f'(0) = 2, then...

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  37. 37
    The solution of the differential equation x2dydx−xy=1x^2\dfrac{dy}{dx} - xy = 1 is...

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  38. 38
    If aˉ\bar{a} and bˉ\bar{b} are unit vectors and θ (0<θ<π)\theta\ (0 < \theta < \pi) is the angle between them, then the value of ∣aˉ+bˉ∣∣aˉ−bˉ∣\dfrac{|\bar{a} + \bar{b}|}{|\bar{a} - \bar{b}|} is equal to...

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  39. 39
    If ABCDEF is a regular hexagon and AB‾+AC‾+AD‾+AE‾+AF‾=pAD‾=qAO‾\overline{AB} + \overline{AC} + \overline{AD} + \overline{AE} + \overline{AF} = p\overline{AD} = q\overline{AO}, where O is the center of the hexagon, then the values of pp and qq respectively are

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  40. 40
    The vector rˉ\bar{r} whose magnitude is 323\sqrt{2} units and which makes angles of π4\dfrac{\pi}{4} and π2\dfrac{\pi}{2} with the positive y- and z-axes respectively is....

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  41. 41
    Let (pˉ∧qˉ)(\bar{p} \wedge \bar{q}) denote the angle between pˉ\bar{p} and qˉ\bar{q}. If aˉ+bˉ+cˉ=0ˉ,∣aˉ∣=7,∣bˉ∣=5\bar{a} + \bar{b} + \bar{c} = \bar{0}, |\bar{a}| = 7, |\bar{b}| = 5 and ∣cˉ∣=3|\bar{c}| = 3 then (take π=227\pi = \dfrac{22}{7})

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  42. 42
    If the lines x−55m+2=2−y5=1−z−1\dfrac{x - 5}{5m + 2} = \dfrac{2 - y}{5} = \dfrac{1 - z}{-1} and x=2y+14m=1−z−3x = \dfrac{2y + 1}{4m} = \dfrac{1 - z}{-3} are perpendicular to each other, then the value of m is ...

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  43. 43
    For the line x+11=y−22=z+33\dfrac{x + 1}{1} = \dfrac{y - 2}{2} = \dfrac{z + 3}{3}, identify the incorrect statement among the following.

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  44. 44
    The direction cosines of a line which is perpendicular to the lines x−72=y+17−3=z−61\dfrac{x - 7}{2} = \dfrac{y + 17}{-3} = \dfrac{z - 6}{1} and x+51=y+32=z−6−2\dfrac{x + 5}{1} = \dfrac{y + 3}{2} = \dfrac{z - 6}{-2} are...

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  45. 45
    The equation of the plane containing the lines x−12=y+1λ=z2\dfrac{x - 1}{2} = \dfrac{y + 1}{\lambda} = \dfrac{z}{2} and x+15=y+12=zλ\dfrac{x + 1}{5} = \dfrac{y + 1}{2} = \dfrac{z}{\lambda} is

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  46. 46
    If a unit vector makes angles π4\dfrac{\pi}{4} with i^\hat{i}, π3\dfrac{\pi}{3} with j^\hat{j} and θ∈(0,π)\theta \in (0, \pi) with k^\hat{k}, then a value of θ\theta is equal to...

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  47. 47
    The minimum value of Z=3x+yZ = 3x + y, subject to the constraints 2x+3y≤6,x+y≥1,x≥0,y≥02x + 3y \le 6, x + y \ge 1, x \ge 0, y \ge 0 is....

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  48. 48
    The centers for disease control have determined that when a person is given a vaccine, the probability that the person will develop immunity to a virus is 0.80.8. If 88 people are given this vaccine, then the probability that exactly 44 will develop immunity is...

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  49. 49
    The Variance of the following probability distribution of X is
    X00112233
    P(X)q3q^33q2p3q^2p3qp23qp^2p3p^3
    where 0<p<1,q=1−p0 < p < 1, q = 1 - p

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  50. 50
    If A and B are two events such that P(A)=23P(A) = \dfrac{2}{3}, P(B)=12P(B) = \dfrac{1}{2} and P(A∣B)=23P(A \mid B) = \dfrac{2}{3}, then P(A′∪B)+P(A∪B′)=P(A' \cup B) + P(A \cup B') =

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