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MHT-CET 2026 (15 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 xx, y, z are the sides of a right angled triangle, where z is the largest side, then 1log⁡x+zy+1log⁡z−xy=\dfrac{1}{\log_{x+z}y} + \dfrac{1}{\log_{z-x}y} =

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  2. 2
    If ω\omega is a complex cube root of unity, then the value of sin⁡[π(ω10+ω23)−π4]=\sin\left[\pi(\omega^{10} + \omega^{23}) - \dfrac{\pi}{4}\right] =

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  3. 3
    If 6 boys and 3 girls are to be seated in a row for a photograph, then the probability that the end seats are occupied by the girls and no two girls are side by side is

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  4. 4
    A bag contains 23 balls, of which 7 are identical. The number of ways of selecting 12 balls from the bag is....

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  5. 5
    The approximate value of (0.007)13(0.007)^{\frac{1}{3}} is

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  6. 6
    If 33θ=π33\theta = \pi, then the value of cos⁡θcos⁡2θcos⁡4θcos⁡8θcos⁡16θ\cos\theta\cos 2\theta\cos 4\theta\cos 8\theta\cos 16\theta is

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  7. 7
    If sec⁡4θ−sec⁡2θ=2\sec 4\theta - \sec 2\theta = 2, then θ=\theta =

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  8. 8
    The equation of the line passing through the point of intersection of the lines x+2y+6=0x + 2y + 6 = 0 and 2x−y=22x - y = 2 and making an intercept 5 on the y-axis is

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  9. 9
    The line x+y=3x + y = 3 intersects the pair of straight lines x2−3xy+y2=0x^2 - 3xy + y^2 = 0 at points A and B. Then the co-ordinates of the mid-point of AB are

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  10. 10
    The angle between the tangents drawn from the origin to the circle (x−7)2+(y+1)2=25(x - 7)^2 + (y + 1)^2 = 25 is

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  11. 11
    The equation of the common tangent touching the circle (x−3)2+y2=9(x - 3)^2 + y^2 = 9 and the parabola y2=4xy^2 = 4x above the X-axis is

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  12. 12
    If lim⁡x→∞(2x−1)19⋅(3x+2)11(6x−5)30=2a⋅3b\displaystyle\lim_{x \to \infty}\dfrac{(2x - 1)^{19}\cdot(3x + 2)^{11}}{(6x - 5)^{30}} = 2^a \cdot 3^b, then a+b=a + b =

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  13. 13
    The dual of the statement pattern (p∧∼q)⟶(q∧∼p)(p \wedge \sim q) \longrightarrow (q \wedge \sim p) is equivalent to

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  14. 14
    Negation of the statement "If an integer is greater than 4 and less than 5, then it is a multiple of 3", is

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  15. 15
    In a triangle ABC, with the usual notations, ∠B=π3,∠C=π4\angle B = \dfrac{\pi}{3}, \angle C = \dfrac{\pi}{4}. If D divides BC internally in the ratio 1:3, then sin⁡∠BADsin⁡∠CAD=\dfrac{\sin\angle BAD}{\sin\angle CAD} =

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  16. 16
    Let P1,P2,P3P_1, P_2, P_3 be the altitudes of a triangle ABC from the vertices A, B, C respectively. If △\triangle denotes the area of the triangle and s is the semi-perimeter of the triangle, then cos⁡AP1+cos⁡BP2+cos⁡CP3=\dfrac{\cos A}{P_1} + \dfrac{\cos B}{P_2} + \dfrac{\cos C}{P_3} =

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  17. 17
    If A=[31−12],C=[7306]A = \begin{bmatrix} 3 & 1 \\ -1 & 2 \end{bmatrix}, C = \begin{bmatrix} 7 & 3 \\ 0 & 6 \end{bmatrix} and AB=CAB = C, then the inverse of matrix B is

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  18. 18
    The element in the third row and the second column in the inverse matrix of a matrix [133313334]\begin{bmatrix} 1 & 3 & 3 \\ 3 & 1 & 3 \\ 3 & 3 & 4 \end{bmatrix} is

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  19. 19
    If y=tan⁡−11+sin⁡2x1−sin⁡2xy = \tan^{-1}\sqrt{\dfrac{1 + \sin 2x}{1 - \sin 2x}}, then dydx\dfrac{dy}{dx} at x=π6x = \dfrac{\pi}{6} is

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  20. 20
    The value of tan⁡−1(cos⁡(19π4)−1sin⁡(π4))\tan^{-1}\left(\dfrac{\cos\left(\frac{19\pi}{4}\right) - 1}{\sin\left(\frac{\pi}{4}\right)}\right) is equal to

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  21. 21
    The value of f(0) so that the function f(x)=(256−8x)14−416−4(64+3x)13,x≠0f(x) = \dfrac{(256 - 8x)^{\frac{1}{4}} - 4}{16 - 4(64 + 3x)^{\frac{1}{3}}}, x \neq 0 is continuous at x=0x = 0, is

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  22. 22
    The derivative of sin⁡(log⁡(x+3x))\sin\left(\log\left(\dfrac{x + 3}{x}\right)\right) is

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  23. 23
    If ey+xy=ee^y + xy = e, then the ordered pair (dydx,d2ydx2)\left(\dfrac{dy}{dx}, \dfrac{d^2y}{dx^2}\right) at x=0x = 0 is equal to

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  24. 24
    The surface area of a spherical ball is increasing at the rate of 4π cm24\pi\ \mathrm{cm^2}/second. The rate at which the radius is increasing when the surface area is 16π cm216\pi\ \mathrm{cm^2} is

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  25. 25
    The maximum value of (1x)x,x>0\left(\dfrac{1}{x}\right)^x, x > 0 is

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  26. 26
    The point on the curve 9y2=x39y^2 = x^3 where the normal to the curve makes equal intercepts with the co-ordinate axes is

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  27. 27
    A spherical mothball has initial radius 3 cm. Due to evaporation, the radius of the ball reduces to 1 cm in 4 months. In how many months would the mothball evaporate completely if the volume is lost at a rate proportional to the surface area ?

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  28. 28
    ∫x2 dx(x2+2)(x2+5)=\displaystyle\int\dfrac{x^2\,dx}{(x^2 + 2)(x^2 + 5)} =

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  29. 29
    The value of ∫sin⁡6x+cos⁡6xsin⁡2x⋅cos⁡2x dx\displaystyle\int\dfrac{\sin^6 x + \cos^6 x}{\sin^2 x\cdot\cos^2 x}\,dx is

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  30. 30
    ∫csc⁡−1(a+xx)dx=\displaystyle\int\mathrm{\csc}^{-1}\left(\sqrt{\dfrac{a + x}{x}}\right)dx =

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  31. 31
    ∫0π∣sin⁡2x∣ dx=\displaystyle\int_0^\pi|\sin 2x|\,dx =

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  32. 32
    If f(x)f(x) is an even function, then ∫−22(∣x∣+f(x)sin⁡x) dx\displaystyle\int_{-2}^{2}(|x| + f(x)\sin x)\,dx is

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  33. 33
    The area of the region bounded by the curves y=∣x−4∣,x=3y = |x - 4|, x = 3 and x=5x = 5, and the X-axis is

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  34. 34
    The solution of dydx=sin⁡(x+y)+cos⁡(x+y)\dfrac{dy}{dx} = \sin(x + y) + \cos(x + y) is

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  35. 35
    The order and degree of the differential equation 3−(d3ydx3)73=(dydx)53 - \left(\dfrac{d^3y}{dx^3}\right)^{\frac{7}{3}} = \left(\dfrac{dy}{dx}\right)^5 are respectively

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  36. 36
    The solution of the differential equation e−x(y+1)dy+(cos⁡2x−sin⁡2x)y dx=0e^{-x}(y + 1)dy + (\cos^2 x - \sin 2x)y\,dx = 0, given that y=1y = 1 when x=0x = 0 is

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  37. 37
    A unit vector coplanar with i^+j^+2k^\hat{i} + \hat{j} + 2\hat{k} and i^+2j^+k^\hat{i} + 2\hat{j} + \hat{k} and perpendicular to i^+j^+k^\hat{i} + \hat{j} + \hat{k} is

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  38. 38
    If a⃗+b⃗+c⃗=0⃗,∣a⃗∣=∣b⃗∣=∣c⃗∣=3\vec{a} + \vec{b} + \vec{c} = \vec{0}, |\vec{a}| = |\vec{b}| = |\vec{c}| = 3 and θ\theta is the angle between b⃗\vec{b} and c⃗\vec{c} then tan⁡2θ+cot⁡2θ=\tan^2\theta + \cot^2\theta =

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  39. 39
    The altitude of the parallelopiped, whose coterminous edges are the vectors a⃗=i^+j^+k^\vec{a} = \hat{i} + \hat{j} + \hat{k}, b⃗=2i^+4j^−k^\vec{b} = 2\hat{i} + 4\hat{j} - \hat{k}, c⃗=i^+j^+3k^\vec{c} = \hat{i} + \hat{j} + 3\hat{k}, where a⃗,b⃗\vec{a}, \vec{b} are the sides of the base of parallelopiped, is

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  40. 40
    Let ABCD be a quadrilateral with AB‾=a⃗,AD‾=b⃗\overline{AB} = \vec{a}, \overline{AD} = \vec{b} and AC‾=3a⃗+2b⃗\overline{AC} = 3\vec{a} + 2\vec{b}. If its area is α\alpha times the area of the parallelogram with AB, AD as adjacent sides, then the value of α\alpha is equal to

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  41. 41
    Let a⃗=i^+2j^−2k^\vec{a} = \hat{i} + 2\hat{j} - 2\hat{k} and b⃗=i^−j^+k^\vec{b} = \hat{i} - \hat{j} + \hat{k}. If c⃗\vec{c} is a vector such that a⃗⋅c⃗=∣c⃗∣,∣c⃗−a⃗∣=22\vec{a}\cdot\vec{c} = |\vec{c}|, |\vec{c} - \vec{a}| = 2\sqrt{2} and the angle between a⃗×b⃗\vec{a} \times \vec{b} and c⃗\vec{c} is 60∘60^\circ, then ∣(a⃗×b⃗)×c⃗∣|(\vec{a} \times \vec{b}) \times \vec{c}| is equal to

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  42. 42
    The co-ordinates of the point where the line joining the points (3,5,−7)(3, 5, -7) and (−2,1,8)(-2, 1, 8) is intersected by the YOZ plane are

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  43. 43
    If the lines r⃗=(i^+mj^+3k^)+λ(2i^+3j^+4k^)\vec{r} = (\hat{i} + m\hat{j} + 3\hat{k}) + \lambda(2\hat{i} + 3\hat{j} + 4\hat{k}) and r⃗=(4i^+j^)+μ(5i^+mj^+k^)\vec{r} = (4\hat{i} + \hat{j}) + \mu(5\hat{i} + m\hat{j} + \hat{k}) intersect each other, then m =

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  44. 44
    If the lines x=−1+s,y=3−λs,z=1+λsx = -1 + s, y = 3 - \lambda s, z = 1 + \lambda s and x=t2,y=1+t,z=2−tx = \dfrac{t}{2}, y = 1 + t, z = 2 - t with parameters s and t, are coplanar, then λ=\lambda =

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  45. 45
    The equation of the plane passing through the point (1,2,1)(1, 2, 1) and perpendicular to the planes x+2y+2z−7=0x + 2y + 2z - 7 = 0 and 3x+3y+2z−5=03x + 3y + 2z - 5 = 0 is

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  46. 46
    The distance of the point (1,0,−3)(1, 0, -3) from the plane x−y−z=9x - y - z = 9 measured parallel to the line x−22=y+23=z−6−6\dfrac{x - 2}{2} = \dfrac{y + 2}{3} = \dfrac{z - 6}{-6} is

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  47. 47
    The difference between the maximum and minimum values of the objective function Z=3x+5yZ = 3x + 5y, subject to the constraints x+3y≤60,x+y≥10,x−y≤0,x,y≥0x + 3y \le 60, x + y \ge 10, x - y \le 0, x, y \ge 0 is

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  48. 48
    For the following probability distribution, the standard deviation of the random variable X is
    X234
    p(X=xx)0.20.50.3

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  49. 49
    A random variable X has the following probability distribution
    X12345678
    P(X=xx)0.150.230.120.100.200.080.070.05
    For the event E = { X is a prime number } and F = { X < 4 }, P(E∪\cupF) =

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  50. 50
    The probability that event A happens in a trial is 0.4. If three independent trials are made, then the probability that A happens at least once is.......

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