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MHT-CET 2026 (20 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 ww is a complex cube root of unity , then the value of w10−w7+w5−w2+1w^{10} - w^7 + w^5 - w^2 + 1 is.

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  2. 2
    If a1,a2,a3,…,ana_1, a_2, a_3, \ldots, a_n are in arithmetic progression with common difference d, then tan⁡[tan⁡−1(d1+a1a2)+tan⁡−1(d1+a2a3)+…+tan⁡−1(d1+an−1an)]=\tan\left[\tan^{-1}\left(\dfrac{d}{1 + a_1a_2}\right) + \tan^{-1}\left(\dfrac{d}{1 + a_2a_3}\right) + \ldots + \tan^{-1}\left(\dfrac{d}{1 + a_{n-1}a_n}\right)\right] = ____

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  3. 3
    In a right-angled △ABC\triangle ABC, the measures of the angles are in an Arithmetic Progression (A.P.) If its smallest side is 44 units, then the area of △ABC\triangle ABC is.....

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  4. 4
    If sec⁡−1(x2+y2x2−y2)=2a\sec^{-1}\left(\dfrac{x^2 + y^2}{x^2 - y^2}\right) = 2a such that ydydx=x⋅f(a)y\dfrac{dy}{dx} = x \cdot f(a) then the value of f(2π3)f\left(\dfrac{2\pi}{3}\right) is

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  5. 5
    The value of cos⁡15∘−sin⁡15∘\cos 15^\circ - \sin 15^\circ is ......

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  6. 6
    The general solution of cos⁡x+sin⁡x=cos⁡2x+sin⁡2x\cos x + \sin x = \cos 2x + \sin 2x is x=npπx = np\pi or x=nqπ3+π6x = \dfrac{nq\pi}{3} + \dfrac{\pi}{6} for n∈Zn \in Z then p:q=p : q =

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  7. 7
    The number of common solutions of the pair of equations 2sin⁡2θ−2cos⁡2θ+1=02\sin^2\theta - 2\cos^2\theta + 1 = 0 and 2sin⁡2θ+3sin⁡θ−2=02\sin^2\theta + 3\sin\theta - 2 = 0 in the interval [0,2π][0, 2\pi], is

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  8. 8
    The area of a triangle formed by a line with the coordinate axes is 4949 sq. units. If the perpendicular drawn from the origin to this line makes an angle of 45∘45^\circ with the positive X-axis, then the equation of line is....

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  9. 9
    If the equation 16x2−24xy+9y2−8x+6y−35=016x^2 - 24xy + 9y^2 - 8x + 6y - 35 = 0 represents a pair of straight lines, then the equation of the locus of points equidistant from these two lines is..........

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  10. 10
    The equation of the circle which passes through the points (2,3)(2, 3) and (4,5)(4, 5) and whose centre lies on a straight line 4x−y−3=04x - y - 3 = 0, is

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  11. 11
    If θ\theta is the eccentric angle of a point on the ellipse x225+y29=1\dfrac{x^2}{25} + \dfrac{y^2}{9} = 1 such that the distance of the point from the center is 55, then θ=\theta =.......

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  12. 12
    lim⁡x→0[x⋅log⁡(1+4x)(e4x−1)2]=⋅s\lim_{x \to 0}\left[\dfrac{x \cdot \log(1 + 4x)}{(e^{4x} - 1)^2}\right] = \cdot s

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  13. 13
    Consider the following statements. pp: If 34>433^4 > 4^3, then 33>443^3 > 4^4 qq: The roots of the equation x2−2x+2=0x^2 - 2x + 2 = 0 are real if and only if Mumbai is in Maharashtra. rr: Statement pp is true or statement qq is false. Which of the following has truth value T (true)?

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  14. 14
    Which of the following is/are not true ? I) If 1 is not a prime number, then 2 is not a prime number. II) e is a vowel and 12×3=3612 \times 3 = 36 III) It is not true that 14 is a composite number and 3 is even number. IV) 5\sqrt{5} is an irrational number, but 3+53 + \sqrt{5} is a complex number.

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  15. 15
    The negation of the inverse of the statement ∼p∨∼q\sim p \vee \sim q is...

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  16. 16
    If A=[326112225],B=[101]A = \begin{bmatrix} 3 & 2 & 6 \\ 1 & 1 & 2 \\ 2 & 2 & 5 \end{bmatrix}, B = \begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix} such that XA=BTXA = B^T and A−1Y=BA^{-1}Y = B, then XY=XY =

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  17. 17
    Let A=[a11b]A = \begin{bmatrix} a & 1 \\ 1 & b \end{bmatrix}, where aa and bb are the roots of the equation x2−4x+2=0x^2 - 4x + 2 = 0. If A+A−1=kI2A + A^{-1} = kI_2, then the value of kk is ____

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  18. 18
    The domain of the function f(x)=x−1+6−xf(x) = \sqrt{x - 1} + \sqrt{6 - x} is...

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  19. 19
    The domain of the function f(x)=(10−x)C(x−6)f(x) = {}^{(10-x)}C_{(x-6)} is...

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  20. 20
    Let [x][x] denotes the greatest integer less than or equal to x and f(x)=[tan⁡2x]f(x) = [\tan^2 x], then which of the following is true ?

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  21. 21
    If y+x+y−x=c\sqrt{y + x} + \sqrt{y - x} = c and dydx=K−y2x2−1\dfrac{dy}{dx} = K - \sqrt{\dfrac{y^2}{x^2} - 1}, then the value of KK is

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  22. 22
    If y=(x2+1)sin⁡xy = (x^2 + 1)^{\sin x} for x>0x > 0 such that dydx=y[2xsin⁡xg(x)+cos⁡x⋅log⁡[g(x)]]\dfrac{dy}{dx} = y\left[\dfrac{2x\sin x}{g(x)} + \cos x \cdot \log[g(x)]\right], then the function 1g(x)\dfrac{1}{g(x)} is...

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  23. 23
    The derivative of f(sin⁡x)f(\sin x) with respect to g(sec⁡x)g(\sec x) at x=π4x = \dfrac{\pi}{4}, given that f′(12)=3f'\left(\dfrac{1}{\sqrt{2}}\right) = 3 and g′(2)=1g'(\sqrt{2}) = 1 is.....

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  24. 24
    A spherical iron ball 1010 cm in radius is coated with a layer of ice of uniform thickness that melts at a rate of 50 cm3/min50\ \mathrm{cm^3/min}. When the thickness of ice is 55 cm, the rate at which the thickness of ice decreases is...

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  25. 25
    If the line ax+by+5=0ax + by + 5 = 0 is a normal to the curve xy=1xy = 1 then .....

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  26. 26
    On the interval [0,1][0, 1], the function f(x)=x25(1−x)75f(x) = x^{25}(1 - x)^{75} attains its maximum value at the point x=x =....

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  27. 27
    If f(x)=∫x(x2+4)(x2+9) dxf(x) = \displaystyle\int\dfrac{x}{(x^2 + 4)(x^2 + 9)}\,dx and f(0)=15log⁡(23)f(0) = \dfrac{1}{5}\log\left(\dfrac{2}{3}\right), then f(1)=f(1) =

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  28. 28
    ∫x4(x10−1)x20+3x10+1 dx=⋅s\displaystyle\int\dfrac{x^4(x^{10} - 1)}{x^{20} + 3x^{10} + 1}\,dx = \cdot s

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  29. 29
    ∫(elog⁡(sin⁡x)+cos⁡x)x dx=\displaystyle\int\left(e^{\log(\sin x)} + \cos x\right)x\,dx =

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  30. 30
    The value of ∫sin⁡4xcos⁡3x dx\displaystyle\int\sin 4x\cos 3x\,dx is

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  31. 31
    If [x][x] denotes the greatest integer less than or equal to xx, then the value of the integral ∫02x2[x] dx\int_0^2 x^2[x]\,dx is equal to

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  32. 32
    If ∫01dxx+1−x=2k\int_0^1\dfrac{dx}{\sqrt{x + 1} - \sqrt{x}} = \sqrt{2}k, then the value of kk is...

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  33. 33
    The area of the region lying in the first quadrant and bounded by the curve y=4x2y = 4x^2, the Y-axis, and the lines y=2,y=4y = 2, y = 4 is...

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  34. 34
    The area bounded by the curves y=∣x∣−1y = |x| - 1 and y=−∣x∣+1y = -|x| + 1 is

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  35. 35
    If y=f(x)y = f(x) is a monotonically increasing function such that (dydx)2=6−dydx\left(\dfrac{dy}{dx}\right)^2 = 6 - \dfrac{dy}{dx} and y(0)=5y(0) = 5, then y(3)=⋅sy(3) = \cdot s

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  36. 36
    If tan⁡x\tan x is an integrating factor of the differential equation dydx+Py=Q\dfrac{dy}{dx} + Py = Q, then PP can be

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  37. 37
    The order and degree of the differential equation (d3ydx3)23−3d2ydx2+5dydx+4=0\left(\dfrac{d^3y}{dx^3}\right)^{\frac{2}{3}} - 3\dfrac{d^2y}{dx^2} + 5\dfrac{dy}{dx} + 4 = 0 are respectively

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  38. 38
    Let aˉ=i^+j^+k^,bˉ=i^−3j^+2k^\bar{a} = \hat{i} + \hat{j} + \hat{k}, \bar{b} = \hat{i} - 3\hat{j} + 2\hat{k} and cˉ=3i^−2k^\bar{c} = 3\hat{i} - 2\hat{k}. If a vector pˉ\bar{p} satisfies the conditions pˉ⋅cˉ=0\bar{p}\cdot\bar{c} = 0 and pˉ×aˉ=bˉ×aˉ\bar{p} \times \bar{a} = \bar{b} \times \bar{a}, then the value of ∣pˉ∣=|\bar{p}| =...

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  39. 39
    The volume of the tetrahedron whose vertices are A(−1,2,3),B(3,−2,1),C(p,1,3),D(−1,−2,4)\mathrm{A}(-1, 2, 3), \mathrm{B}(3, -2, 1), \mathrm{C}(p, 1, 3), \mathrm{D}(-1, -2, 4) is 163\dfrac{16}{3} cubic units then the value of p is

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  40. 40
    Let OD‾=i^+2j^+6k^,CB‾=−3i^−2k^\overline{OD} = \hat{i} + 2\hat{j} + 6\hat{k}, \overline{CB} = -3\hat{i} - 2\hat{k} be the diagonals of the parallelogram OBDC and OA‾=i^+2j^+3k^\overline{OA} = \hat{i} + 2\hat{j} + 3\hat{k} be another vector. Then the volume of a parallelopiped determined by vectors OA‾\overline{OA}, OB‾\overline{OB}, and OC‾\overline{OC} (in cubic units), is

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  41. 41
    If aˉ\bar{a} and bˉ\bar{b} are unit vectors perpendicular to each other, then [aˉ+(aˉ×bˉ)bˉ+(aˉ×bˉ)(aˉ×bˉ)]=⋅s\left[\bar{a} + (\bar{a} \times \bar{b})\quad \bar{b} + (\bar{a} \times \bar{b})\quad (\bar{a} \times \bar{b})\right] = \cdot s

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  42. 42
    If aˉ,bˉ\bar{a}, \bar{b} and cˉ\bar{c} are three vectors such that ∣aˉ+bˉ+cˉ∣=1,cˉ=λ(aˉ×bˉ)|\bar{a} + \bar{b} + \bar{c}| = 1, \bar{c} = \lambda(\bar{a} \times \bar{b}) and ∣aˉ∣=13,∣bˉ∣=12,∣cˉ∣=16|\bar{a}| = \dfrac{1}{\sqrt{3}}, |\bar{b}| = \dfrac{1}{\sqrt{2}}, |\bar{c}| = \dfrac{1}{\sqrt{6}}, then the angle between aˉ\bar{a} and bˉ\bar{b} is

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  43. 43
    The acute angle between the line r⃗=(i^+2j^+k^)+λ(i^+j^+k^)\vec{r} = (\hat{i} + 2\hat{j} + \hat{k}) + \lambda(\hat{i} + \hat{j} + \hat{k}) and the plane r⃗⋅(2i^+pj^+k^)=8\vec{r}\cdot(2\hat{i} + p\hat{j} + \hat{k}) = 8 is sin⁡−1(23)\sin^{-1}\left(\dfrac{\sqrt{2}}{3}\right), then the value of pp are...

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  44. 44
    The shortest distance between the lines x−33=y−8−1=z−31\dfrac{x - 3}{3} = \dfrac{y - 8}{-1} = \dfrac{z - 3}{1} and x+3−3=y+72=z−64\dfrac{x + 3}{-3} = \dfrac{y + 7}{2} = \dfrac{z - 6}{4} is

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  45. 45
    A plane meets the co-ordinate axes in A, B, C such that the centroid of the triangle ABC is the point (1,r,r2)(1, r, r^2), then the equation of the plane is,

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  46. 46
    If M denotes the midpoint of the line joining A(4,5,−10)A(4, 5, -10) and B(−1,2,1)B(-1, 2, 1), then the equation of the plane through M and perpendicular to AB is:

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  47. 47
    The difference between the maximum value and the minimum value of the objective function 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
    A random variable X∼B(n,p)X \sim B(n, p) follows a binomial distribution with n=6n = 6. If 9P(X=4)=P(X=2)9P(X = 4) = P(X = 2), then the probability of success pp is..

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  49. 49
    Let X be a continuous random variable with the probability density function(p.d.f.) given by f(x)={kx,0≤x<1k,1≤x<2−kx+3k,2≤x<30,otherwisef(x) = \begin{cases} kx, & 0 \le x < 1 \\ k, & 1 \le x < 2 \\ -kx + 3k, & 2 \le x < 3 \\ 0, & \text{otherwise} \end{cases} P(2<X≤3)=⋅sP(2 < X \le 3) = \cdot s

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  50. 50
    In a certain city, the ratio of men to women is 5:45 : 4. It is found that 80%80\% of men and 90%90\% of women are literate. If a person selected at random is found to be illiterate, then the probability that the person is a man is....

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