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MHT-CET 2026 (13 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
    The polar form of the complex number z=11+iz = \dfrac{1}{1 + i}, (where i=−1i = \sqrt{-1}) is

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  2. 2
    The number of cubic equations that can be formed with the coefficients 0, 3, 2, 4, 5, 6 when repetition of coefficients is allowed, is....

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  3. 3
    tan⁡105∘=……\tan 105^\circ = \ldots\ldots

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  4. 4
    The number of solutions in [0,π2)\left[0, \dfrac{\pi}{2}\right) of the equation cos⁡3x⋅tan⁡5x=sin⁡7x\cos 3x \cdot \tan 5x = \sin 7x is

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  5. 5
    The area of the triangle formed by the co-ordinate axes and a line px+qy=rpx + qy = r (where p, q and r are positive real numbers) is 154\dfrac{1}{54} sq.units, then ....

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  6. 6
    The joint equation of the pair of lines through the origin and making an angle of π4\dfrac{\pi}{4} with the line 3x+y−6=03x + y - 6 = 0 is

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  7. 7
    If a circle with center (−1,1)(-1, 1) touches the line x+2y+4=0x + 2y + 4 = 0, then the co-ordinates of the point of contact are

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  8. 8
    A point on the parabola y2=365xy^2 = \dfrac{36}{5}x at which the ordinate increases at thrice the rate of the abscissa is .....

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  9. 9
    The area of the triangle formed by the lines joining the vertex of the parabola x2=48yx^2 = 48y to the ends of its latus rectum, is .....

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  10. 10
    lim⁡x→0[log⁡∣2+x∣−log⁡∣2−x∣tan⁡x]=…….\displaystyle\lim_{x \to 0}\left[\dfrac{\log|2 + x| - \log|2 - x|}{\tan x}\right] = \ldots\ldots.

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  11. 11
    Consider the statement patterns A. (q→p)∨(p→q)(q \to p) \vee (p \to q) B. (∼p∨∼q)↔∼(p∧q)(\sim p \vee \sim q) \leftrightarrow \sim(p \wedge q) C. [(p∨q)∧∼p]∧∼q[(p \vee q) \wedge \sim p] \wedge \sim q D. (p∧q)∧(∼p∨∼q)(p \wedge q) \wedge (\sim p \vee \sim q) then statement patterns

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  12. 12
    The statement pattern (p∧q)→(r∨∼s)(p \wedge q) \to (r \vee \sim s) is false. Then the truth values of p,q,rp, q, r and ss are respectively

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  13. 13
    In a triangle ABC, with usual notations if s−a11=s−b12=s−c13\dfrac{s - a}{11} = \dfrac{s - b}{12} = \dfrac{s - c}{13} and λtan⁡2A2=455\lambda\tan^2\dfrac{A}{2} = 455, then λ=\lambda =

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  14. 14
    With usual notations, in △ABC\triangle ABC, if 2a2=b2+c22a^2 = b^2 + c^2, then cos⁡3Acos⁡A+2=\dfrac{\cos 3A}{\cos A} + 2 =

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  15. 15
    With usual notations, in △ABC\triangle ABC, (b−c)2cos⁡2A2+(b+c)2sin⁡2A2=(b - c)^2\cos^2\dfrac{A}{2} + (b + c)^2\sin^2\dfrac{A}{2} =

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  16. 16
    Let A=[−5−321]A = \begin{bmatrix} -5 & -3 \\ 2 & 1 \end{bmatrix}. The Row transformation R1→R1+3R2R_1 \to R_1 + 3R_2 will transform matrix A into

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  17. 17
    If A=[20−1510013]3×3A = \begin{bmatrix} 2 & 0 & -1 \\ 5 & 1 & 0 \\ 0 & 1 & 3 \end{bmatrix}_{3 \times 3}, and A−1=[γ−11α6−5β−22]3×3A^{-1} = \begin{bmatrix} \gamma & -1 & 1 \\ \alpha & 6 & -5 \\ \beta & -2 & 2 \end{bmatrix}_{3 \times 3}, then ∣α⋅β⋅γ∣=|\alpha \cdot \beta \cdot \gamma| = _____ (where ∣⋅∣|\cdot| denotes the absolute value)

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  18. 18
    If y=tan⁡−1{1+x2−1−x21+x2+1−x2}y = \tan^{-1}\left\{\dfrac{\sqrt{1 + x^2} - \sqrt{1 - x^2}}{\sqrt{1 + x^2} + \sqrt{1 - x^2}}\right\}, where ∣x∣<1|x| < 1, then dydx\dfrac{dy}{dx} is equal to

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  19. 19
    The value of tan⁡−1(3)+sec⁡−1(−2)−sin⁡−1(−12)\tan^{-1}(\sqrt{3}) + \sec^{-1}(-2) - \sin^{-1}\left(-\dfrac{1}{2}\right) is

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  20. 20
    The domain of the function f(x)=e5x−3−2x2f(x) = e^{\sqrt{5x - 3 - 2x^2}} is

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  21. 21
    If f(x)={(8−2x)13−23−(243+5x)15,if x≠0k,if x=0f(x) = \begin{cases} \dfrac{(8 - 2x)^{\frac{1}{3}} - 2}{3 - (243 + 5x)^{\frac{1}{5}}}, & \text{if } x \neq 0 \\ k, & \text{if } x = 0 \end{cases} is continuous at x=0x = 0, then k=k =

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  22. 22
    If y=x+exy = x + e^x, then d2xdy2\dfrac{d^2x}{dy^2} is equal to

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  23. 23
    Let f(x)=x−5f(x) = x - 5. If g(x)=[f(4h(x)+3)]2g(x) = [f(4h(x) + 3)]^2 and h(1)=4,h′(1)=−2h(1) = 4, h'(1) = -2, then g′(1)=g'(1) = .....

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  24. 24
    The equation of the normal to the curve xy+7=0xy + 7 = 0 is Ax+By+C=0Ax + By + C = 0, then

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  25. 25
    A wire 40 metre in length is to be cut into two pieces. One piece is formed into a square and the other piece into a circle. The lengths of the two pieces so that the combined area of the square and the circle is minimum, are respectively

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  26. 26
    If a spherical balloon has a variable diameter 3x+923x + \dfrac{9}{2} units, then the rate of change of its volume with respect to xx is

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  27. 27
    ∫30x14+15xlog⁡225x15+15x dx=\displaystyle\int\dfrac{30x^{14} + 15^x\log 225}{x^{15} + 15^x}\,dx =

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  28. 28
    If ∫sin⁡2xsin⁡4x+cos⁡4x dx=f(x)+c\displaystyle\int\dfrac{\sin 2x}{\sin^4 x + \cos^4 x}\,dx = f(x) + c where c is constant of integration, then the value of tan⁡(f(x))\tan(f(x)) is

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  29. 29
    ∫e2x2(sin⁡2xcos⁡2x−1)2sin⁡22x dx=Ae2xcot⁡2x+c\displaystyle\int e^{2x}\dfrac{2(\sin 2x\cos 2x - 1)}{2\sin^2 2x}\,dx = Ae^{2x}\cot 2x + c (Where c is the constant of integration.), then A3=A^3 =

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  30. 30
    If ∫0adx1+4x2=π8\displaystyle\int_0^a\dfrac{dx}{1 + 4x^2} = \dfrac{\pi}{8}, then a=a =

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  31. 31
    The value of the integral ∫−33(2x9+3x8−5x7+9x6+4x3−x+3)x2+3 dx\displaystyle\int_{-\sqrt{3}}^{\sqrt{3}}\dfrac{(2x^9 + 3x^8 - 5x^7 + 9x^6 + 4x^3 - x + 3)}{x^2 + 3}\,dx is

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  32. 32
    The area of the region (in sq.units) bounded by the curve y=21−x2y = 2\sqrt{1 - x^2} and the X-axis is _____

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  33. 33
    If dydx=y+5\dfrac{dy}{dx} = y + 5 and y(0)=4y(0) = 4 then y(log⁡2)y(\log 2) is equal to

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  34. 34
    The solution of the differential equation dydx=x+yx−y\dfrac{dy}{dx} = \dfrac{x + y}{x - y} is

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  35. 35
    A spherical balloon expands at a rate proportional to its surface area. Initially its radius is 2 cm and 5 minutes later it increases upto 7 cm, then surface area of spherical balloon after 12 minutes will be

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  36. 36
    Let aˉ,bˉ,cˉ\bar{a}, \bar{b}, \bar{c} be three vectors having magnitudes 1, 1 and 2 respectively. If 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

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  37. 37
    The volume of a tetrahedron with vertices 5i^−j^+k^,7i^−4j^+pk^,i^−6j^+10k^5\hat{i} - \hat{j} + \hat{k}, 7\hat{i} - 4\hat{j} + p\hat{k}, \hat{i} - 6\hat{j} + 10\hat{k} and −i^−3j^+7k^-\hat{i} - 3\hat{j} + 7\hat{k} is 11 cubic units, then one of the values of p is

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  38. 38
    Let aˉ,bˉ,cˉ\bar{a}, \bar{b}, \bar{c} be unit vectors such that aˉ\bar{a} is perpendicular to the plane of bˉ\bar{b} and cˉ\bar{c}. If the angle between bˉ\bar{b} and cˉ\bar{c} is π3\dfrac{\pi}{3}, then ∣aˉ+bˉ+cˉ∣=|\bar{a} + \bar{b} + \bar{c}| =

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  39. 39
    If a parallelogram is constructed on the vectors aˉ=3pˉ−qˉ,bˉ=pˉ+3qˉ\bar{a} = 3\bar{p} - \bar{q}, \bar{b} = \bar{p} + 3\bar{q} and ∣pˉ∣=3,∣qˉ∣=2|\bar{p}| = 3, |\bar{q}| = 2 and angle between pˉ\bar{p} and qˉ\bar{q} is π3\dfrac{\pi}{3}, then the ratio of the lengths of adjacent sides aˉ\bar{a} and bˉ\bar{b} of the parallelogram is

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  40. 40
    A vector rˉ\bar{r} of magnitude 323\sqrt{2} units which makes angles of π4\dfrac{\pi}{4} and π2\dfrac{\pi}{2} respectively with Y and Z axes is

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  41. 41
    The point having position vector 4i^−11j^+2k^4\hat{i} - 11\hat{j} + 2\hat{k} lies on the line

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  42. 42
    The shortest distance between the lines, where the first line passes through (0,0,0)(0, 0, 0) and (2,0,3)(2, 0, 3) and the second line passes through (2,5,0)(2, 5, 0) and (0,4,0)(0, 4, 0) is

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  43. 43
    The co-ordinates of the foot of the perpendicular from the origin to the plane 2x−3y−6z=492x - 3y - 6z = 49 are...

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  44. 44
    Let a plane P pass through the point (3,7,−7)(3, 7, -7) and contain the line x−2−3=y−32=z+21\dfrac{x - 2}{-3} = \dfrac{y - 3}{2} = \dfrac{z + 2}{1}. If the distance of the plane P from the origin is d, then d2d^2 is

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  45. 45
    The plane x2+y3+z4=1\dfrac{x}{2} + \dfrac{y}{3} + \dfrac{z}{4} = 1 cuts the co-ordinate axes at the points A, B, C respectively. Then the area of triangle ABC is

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  46. 46
    In L.P.P., the corner points of the feasible region for the constraints 3x−y≥6,x≤3,y≤2,y≥0,x≥03x - y \ge 6, x \le 3, y \le 2, y \ge 0, x \ge 0 are .....

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  47. 47
    The odds in favour of A winning a game of table tennis against B are 1:2. If 3 games are to be played, then the probability of A winning at least two games out of the three is.....

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  48. 48
    A random variable X has the following probability distribution
    X=xX = x1122334455667788
    P(X=x)P(X = x)0.150.150.230.230.100.100.120.120.200.200.080.080.070.070.050.05
    For the events E={XE = \{X is a prime number}\}, F={X<4}F = \{X < 4\}, P(E∪F)P(E \cup F) is

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  49. 49
    A random variable X takes the values 0, 1, 2, 3 and its mean is 1.31.3. If P(X=3)=2P(X=1)P(X = 3) = 2P(X = 1) and P(X=2)=0.3P(X = 2) = 0.3, then P(X=0)P(X = 0) is

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  50. 50
    Let X∼B(6,12)X \sim B\left(6, \dfrac{1}{2}\right). Then the maximum probability occurs at

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