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MHT-CET 2026 (17 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
    If tan⁡A\tan A and tan⁡B\tan B are the roots of the equation 5x2−4x+1=05x^2 - 4x + 1 = 0, then the value of A+BA + B is...

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  2. 2
    (cos⁡2θ+isin⁡2θ)7(cos⁡4θ+isin⁡4θ)3=\dfrac{(\cos 2\theta + i\sin 2\theta)^7}{(\cos 4\theta + i\sin 4\theta)^3} =

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  3. 3
    In a test, there are 5 questions of the 'true or false' type. No student has got all the answers correct and the sequence of answers for every student is unique. The maximum number of students who could have appeared for the test is....

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  4. 4
    The line y=2x+cy = 2x + c passes through a point that is equidistant from both the axes and lies in the first quadrant (x>0,y>0)(x > 0, y > 0). Then the value of c is...

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  5. 5
    If the equation ax2+4xy−2y2+4x+8y+1=0ax^2 + 4xy - 2y^2 + 4x + 8y + 1 = 0 represents a pair of straight lines, then the coordinates of their point of intersection are.........

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  6. 6
    If the circles x2+y2=16x^2 + y^2 = 16 and x2+y2+2ax+4y+4=0x^2 + y^2 + 2ax + 4y + 4 = 0 touch each other internally then a=a =

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  7. 7
    If the line 3x+4y+k=03x + 4y + k = 0 touches the ellipse 9x2+16y2=1449x^2 + 16y^2 = 144, then the value of k is.

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  8. 8
    Which of the following statements is/are False? S1:∃n∈NS_1 : \exists n \in N, such that n2+n+2n^2 + n + 2 is divisible by 4. S2:∃x∈NS_2 : \exists x \in N, such that x−17<20x - 17 < 20. S3:∀n∈N,x2+3x−10=0S_3 : \forall n \in N,\quad x^2 + 3x - 10 = 0. S4:∀n∈N,n2≥1S_4 : \forall n \in N,\quad n^2 \ge 1.

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  9. 9
    The negation of (p∧q)→((p∨r)→∼q)(p \wedge q) \to ((p \vee r) \to \sim q) is equivalent to ...

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  10. 10
    The statement pattern [(p∧q)→(∼p∨r)]∨[(∼p∨r)→(p∧q)][(p \wedge q) \to (\sim p \vee r)] \vee [(\sim p \vee r) \to (p \wedge q)] is

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  11. 11
    In △ABC\triangle ABC, with usual notation, if cot A, cot B, cot C are in arithmetic progression, then

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  12. 12
    If A=[1−tan⁡θ2tan⁡θ21]A = \begin{bmatrix} 1 & -\tan\dfrac{\theta}{2} \\ \tan\dfrac{\theta}{2} & 1 \end{bmatrix} and B=[1tan⁡θ2−tan⁡θ21]B = \begin{bmatrix} 1 & \tan\dfrac{\theta}{2} \\ -\tan\dfrac{\theta}{2} & 1 \end{bmatrix} then A−1BA^{-1}B is equal to

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  13. 13
    If A=[3000−50007];B=[−100020004]A = \begin{bmatrix} 3 & 0 & 0 \\ 0 & -5 & 0 \\ 0 & 0 & 7 \end{bmatrix}; B = \begin{bmatrix} -1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 4 \end{bmatrix} then, (2A+3B)−1=(2A + 3B)^{-1} = ______

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  14. 14
    If sin⁡−1(x−2)+cos⁡−1(x)+tan⁡−1(x+2)+cot⁡−1(x+4)=sec⁡−1(k)−π2\sin^{-1}(x - 2) + \cos^{-1}(x) + \tan^{-1}(x + 2) + \cot^{-1}(x + 4) = \sec^{-1}(\sqrt{k}) - \dfrac{\pi}{2}, then cos⁡(2 csc⁡−1k−1)=…\cos(2\,\mathrm{\csc}^{-1}\sqrt{k - 1}) = \ldots

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  15. 15
    sin⁡(3sin⁡−1(15))=\sin\left(3\sin^{-1}\left(\dfrac{1}{5}\right)\right) =

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  16. 16
    cos⁡−1(cos⁡4π3)+sin⁡−1(sin⁡4π3)=…\cos^{-1}\left(\cos\dfrac{4\pi}{3}\right) + \sin^{-1}\left(\sin\dfrac{4\pi}{3}\right) = \ldots

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  17. 17
    If f(x)=1−x1+xf(x) = \dfrac{1 - x}{1 + x}, then f(f(cos⁡x))=f(f(\cos x)) =

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  18. 18
    If the function f(x)=42(sin⁡3x+sin⁡x)2sin⁡2xsin⁡3x2+cos⁡5x2−cos⁡3x2f(x) = \dfrac{4\sqrt{2}(\sin 3x + \sin x)}{2\sin 2x\sin\dfrac{3x}{2} + \cos\dfrac{5x}{2} - \cos\dfrac{3x}{2}} for x≠π2x \neq \dfrac{\pi}{2} is continuous at x=π2x = \dfrac{\pi}{2}, then the value of f(π2)f\left(\dfrac{\pi}{2}\right) is equal to

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  19. 19
    If f(x)=cos⁡x cos⁡2x cos⁡4x cos⁡8x cos⁡16xf(x) = \cos x\ \cos 2x\ \cos 4x\ \cos 8x\ \cos 16x, then f′(π4)=f'\left(\dfrac{\pi}{4}\right) =

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  20. 20
    If y=13x+5y = \dfrac{1}{3x + 5}, then the value of d9ydx9\dfrac{d^9y}{dx^9} is..

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  21. 21
    The derivative of tan⁡−1(1+x2−1x)\tan^{-1}\left(\dfrac{\sqrt{1 + x^2} - 1}{x}\right) with respect to tan⁡−1(x1−x2)\tan^{-1}\left(\dfrac{x}{\sqrt{1 - x^2}}\right) at x=12x = \dfrac{1}{2} is

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  22. 22
    If y=cos⁡x2+cos⁡x2+cos⁡x2+…∞y = \sqrt{\cos x^2 + \sqrt{\cos x^2 + \sqrt{\cos x^2 + \ldots\infty}}} and dydx=f(x)2y−1\dfrac{dy}{dx} = \dfrac{f(x)}{2y - 1} then, ∫f(x) dx=…\displaystyle\int f(x)\,dx = \ldots

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  23. 23
    If f(x)=cot⁡−1(3x−x31−3x2)f(x) = \cot^{-1}\left(\dfrac{3x - x^3}{1 - 3x^2}\right) and g(x)=cos⁡−1(1−x21+x2)g(x) = \cos^{-1}\left(\dfrac{1 - x^2}{1 + x^2}\right), then lim⁡x→af(x)−f(a)g(x)−g(a), (0<a<12)\lim_{x \to a}\dfrac{f(x) - f(a)}{g(x) - g(a)},\ \left(0 < a < \dfrac{1}{2}\right) is

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  24. 24
    The coordinates of the points on the curve 4y=x24y = x^2 that are nearest to the point (0,5)(0, 5) are ...

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  25. 25
    The equation of the tangent to the curve y=9−3x2y = \sqrt{9 - 3x^2} at the point where the ordinate and abscissa equal is...

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  26. 26
    If the line y=4x−5y = 4x - 5 is tangent to the curve y2=ax3+by^2 = ax^3 + b at the point (2,3)(2, 3), then the value of 7a−2b7a - 2b is...

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  27. 27
    If ∫sin⁡xsin⁡4x dx=αlog⁡∣1+sin⁡x1−sin⁡x∣+βlog⁡∣1+2sin⁡x1−2sin⁡x∣+c\displaystyle\int\dfrac{\sin x}{\sin 4x}\,dx = \alpha\log\left|\dfrac{1 + \sin x}{1 - \sin x}\right| + \beta\log\left|\dfrac{1 + \sqrt{2}\sin x}{1 - \sqrt{2}\sin x}\right| + c, then the value of 32(α+β2)=32(\alpha + \beta^2) =

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  28. 28
    If ∫f(x) dx=g(x)\displaystyle\int f(x)\,dx = g(x) then ∫x3f(x2) dx\displaystyle\int x^3 f(x^2)\,dx is equal to

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  29. 29
    If ∫3x+7x2−3x+2 dx=mlog⁡(x−2x−1)+nlog⁡(x−2)+c\displaystyle\int\dfrac{3x + 7}{x^2 - 3x + 2}\,dx = m\log\left(\dfrac{x - 2}{x - 1}\right) + n\log(x - 2) + c, where m,n∈Rm, n \in R and c is an integration constant, then m+n=m + n =

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  30. 30
    If f(x)=sin⁡−1x1−x2f(x) = \dfrac{\sin^{-1}x}{\sqrt{1 - x^2}} and g(x)=esin⁡−1xg(x) = e^{\sin^{-1}x}, then the value of ∫f(x)g(x) dx=…\displaystyle\int f(x)g(x)\,dx = \ldots

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  31. 31
    The function f(x)=∫0xdt1+cos⁡tf(x) = \int_0^x\dfrac{dt}{1 + \cos t} satisfies which of the following differential equations?

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  32. 32
    If ∫ab(x2−x) dx=18, ∫abx3 dx=0\int_a^b(x^2 - x)\,dx = 18,\ \int_a^b x^3\,dx = 0, then a+ba + b is equal to

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  33. 33
    The value of the definite integral ∫0π15+4cos⁡x dx\int_0^\pi\dfrac{1}{5 + 4\cos x}\,dx is equal to ...

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  34. 34
    Let f(x)=x−[x]f(x) = x - [x] for every real number xx, where [x][x] is integral part of xx. then ∫−11f(x) dx\displaystyle\int_{-1}^{1} f(x)\,dx is

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  35. 35
    The area of the region bounded by curves y=sin⁡x,y=cos⁡xy = \sin x, y = \cos x and the lines x=0,x=π4x = 0, x = \dfrac{\pi}{4} is

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  36. 36
    The general solution of the differential equation sec⁡y+(x−esin⁡y)dydx=0\sec y + (x - e^{\sin y})\dfrac{dy}{dx} = 0 is...

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  37. 37
    The general solution of the differential equations dydx=(9x+y+5)2\dfrac{dy}{dx} = (9x + y + 5)^2 is...

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  38. 38
    Let a⃗\vec{a} and b⃗\vec{b} be linearly independent vectors such that ∣a⃗∣=3,∣b⃗∣=3|\vec{a}| = \sqrt{3}, |\vec{b}| = 3 and ∣a⃗−b⃗∣=4|\vec{a} - \vec{b}| = 4. If a⃗×(2i^+2j^−k^)=(2i^+2j^−k^)×b⃗\vec{a} \times (2\hat{i} + 2\hat{j} - \hat{k}) = (2\hat{i} + 2\hat{j} - \hat{k}) \times \vec{b} and ∣(a⃗+b⃗)⋅(3i^+4j^+2k^)∣=λ|(\vec{a} + \vec{b})\cdot(3\hat{i} + 4\hat{j} + 2\hat{k})| = \sqrt{\lambda}, then λ=…\lambda = \ldots

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  39. 39
    The value of bb such that the scalar product of the vector i^+j^+k^\hat{i} + \hat{j} + \hat{k} with the unit vector parallel to the sum of the vectors 2i^+4j^−5k^2\hat{i} + 4\hat{j} - 5\hat{k} and bi^+2j^+3k^b\hat{i} + 2\hat{j} + 3\hat{k} is one, is...

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  40. 40
    If ∣a⃗⋅b⃗∣=∣a⃗×b⃗∣,a⃗⋅b⃗<0|\vec{a}\cdot\vec{b}| = |\vec{a} \times \vec{b}|, \vec{a}\cdot\vec{b} < 0 and θ\theta is the angle between a⃗\vec{a} and b⃗\vec{b}, then the value of sin⁡θ+tan⁡θ\sin\theta + \tan\theta is...

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  41. 41
    Let a⃗=λi^+j^+k^,b⃗=2i^+4j^+4k^,c⃗=i^+μj^+k^\vec{a} = \lambda\hat{i} + \hat{j} + \hat{k}, \vec{b} = 2\hat{i} + 4\hat{j} + 4\hat{k}, \vec{c} = \hat{i} + \mu\hat{j} + \hat{k} If a⃗\vec{a} is parallel to b⃗\vec{b} and b⃗\vec{b} is perpendicular to c⃗\vec{c} then λ−μ=…\lambda - \mu = \ldots

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  42. 42
    The equation of the perpendicular line from the point (2,−3,1)(2, -3, 1) to the line x+12=y−33=z+2−1\dfrac{x + 1}{2} = \dfrac{y - 3}{3} = \dfrac{z + 2}{-1} is

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  43. 43
    The angle between the line x−1=2−y=2z−64x - 1 = 2 - y = \dfrac{2z - 6}{4} and the plane r⃗⋅(2i^+j^+k^)=10\vec{r}\cdot(2\hat{i} + \hat{j} + \hat{k}) = 10 is

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  44. 44
    The sum of the coordinates of one of the points on the line x−21=y+3−2=z+52\dfrac{x - 2}{1} = \dfrac{y + 3}{-2} = \dfrac{z + 5}{2} which is at a distance of 3 units from the point (2,−3,−5)(2, -3, -5) is ...

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  45. 45
    If A and B are the feet of the perpendiculars drawn from (1,2,3)(1, 2, 3) to planes YZYZ and ZXZX, then the equation of the plane passing through the points A, B and the origin is

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  46. 46
    If the distance of point B(2,1,−3)B(2, 1, -3) from the line passing through the point A(4,−2,2)A(4, -2, 2), and parallel to the vector c⃗=−4i^−6j^−2k^\vec{c} = -4\hat{i} - 6\hat{j} - 2\hat{k} is xx, then x4+x2+541=x^4 + x^2 + 541 =

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  47. 47
    In the following figure, the shaded region represents the system of constraints:

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  48. 48
    On average, one out of 10 persons is busy. If six persons are selected at random, then the probability that at least 5 of them will be busy is...

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  49. 49
    If a discrete random variable XX takes the values 1,2,3,41, 2, 3, 4 such that 2P(X=1)=3P(X=2)=P(X=3)=5P(X=4)2P(X = 1) = 3P(X = 2) = P(X = 3) = 5P(X = 4), then P(X=4)=…P(X = 4) = \ldots

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  50. 50
    Three numbers are chosen from 1 to 30. The probability that minimum is 10 and maximum is 26 is ______

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