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MHT-CET 2026 (19 May 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 f(x)={x}f(x) = \{x\} represents the fractional part function, then for any x∉Zx \notin \mathbb{Z}, the expression f(x)+f(−x)f(x) + f(-x) is equal to...

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  2. 2
    If cos⁡A=34\cos A = \dfrac{3}{4}, then the value of 2sin⁡A2⋅sin⁡5A2=…2\sin\dfrac{A}{2} \cdot \sin\dfrac{5A}{2} = \ldots

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  3. 3
    The general solution of the equation 4sin⁡xcos⁡x+2sin⁡x+2cos⁡x+1=04\sin x\cos x + 2\sin x + 2\cos x + 1 = 0, where n∈Zn \in Z is:

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  4. 4
    If PP and P′P' denote respectively the circumcenter and the orthocenter of △ABC\triangle ABC, then PA‾+PB‾+PC‾=\overline{PA} + \overline{PB} + \overline{PC} =

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  5. 5
    The adjacent sides of a cyclic quadrilateral are 2 and 5, and the angle between them is 60∘60^\circ. If the area of the quadrilateral is 434\sqrt{3} sq. units then the perimeter of quadrilateral (in units) is:

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  6. 6
    In ΔABC\Delta ABC, with the usual notations, if a+b7=b+c8=c+a6=1\dfrac{a + b}{7} = \dfrac{b + c}{8} = \dfrac{c + a}{6} = 1, then sin⁡Asin⁡B=…\dfrac{\sin A}{\sin B} = \ldots

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  7. 7
    If w=−12+i32w = -\dfrac{1}{2} + \dfrac{i\sqrt{3}}{2}, then the value of ∣1111−1−w2w21w2w4∣\begin{vmatrix} 1 & 1 & 1 \\ 1 & -1 - w^2 & w^2 \\ 1 & w^2 & w^4 \end{vmatrix} is...

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  8. 8
    Everyone in the room handshakes with everyone else. If the total number of handshakes is 66, then the number of persons in the room is...

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  9. 9
    There are 4 boys and 6 girls to be seated in a circle for a photograph. The probability that no two boys are sitting together is...

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  10. 10
    The equation of the line which passes through the point of intersection of lines x+2y+6=0x + 2y + 6 = 0, 2x−y=22x - y = 2 and which makes intercept 5 on Y - axis is

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  11. 11
    If one of the lines given by 6x2−xy+4ky2=06x^2 - xy + 4ky^2 = 0 is 3x+4y=03x + 4y = 0, then kk lies in the interval:

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  12. 12
    The equation of the director circle of the circle x2+y2+3x−8y−3=0x^2 + y^2 + 3x - 8y - 3 = 0 is:

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  13. 13
    The area (in sq.units) of a triangle formed by joining the vertex and the endpoints of the latus rectum of the parabola x2=−20yx^2 = -20y is...

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  14. 14
    lim⁡x→π2[1−tan⁡(x2)](1−sin⁡x)[1+tan⁡(x2)](π−2x)3=\displaystyle\lim_{x \to \frac{\pi}{2}} \dfrac{[1 - \tan(\frac{x}{2})](1 - \sin x)}{[1 + \tan(\frac{x}{2})](\pi - 2x)^3} =

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  15. 15
    The statement pattern p→(q→p)≡p \rightarrow (q \rightarrow p) \equiv

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  16. 16
    The negation of the inverse of ∼p→q\sim p \rightarrow q is equivalent to:

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  17. 17
    If α≤2sin⁡−1x+cos⁡−1x≤β\alpha \le 2\sin^{-1}x + \cos^{-1}x \le \beta, then the values of α\alpha and β\beta are

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  18. 18
    If A=[−12−3232−12]A = \begin{bmatrix} -\dfrac{1}{2} & -\dfrac{\sqrt{3}}{2} \\ \dfrac{\sqrt{3}}{2} & -\dfrac{1}{2} \end{bmatrix}, the value of ∣Adj(A2026)∣|\mathrm{Adj}(A^{2026})| is equal to:

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  19. 19
    Let A=[12−51]A = \begin{bmatrix} 1 & 2 \\ -5 & 1 \end{bmatrix} and A−1=xA+yIA^{-1} = xA + yI, then the pair (x,y)=⋅s(x, y) = \cdot s

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  20. 20
    If A=[2−35−7]A = \begin{bmatrix} 2 & -3 \\ 5 & -7 \end{bmatrix}, then A−A−1=A - A^{-1} =

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  21. 21
    If f(x)={1−sin⁡5x5cos⁡2x,x<π2a,at x=π2b(1−sin⁡x)8(π−2x)2,x>π2f(x) = \begin{cases} \dfrac{1 - \sin^5 x}{5\cos^2 x} & , x < \dfrac{\pi}{2} \\ a & , \text{at } x = \dfrac{\pi}{2} \\ \dfrac{b(1 - \sin x)}{8(\pi - 2x)^2} & , x > \dfrac{\pi}{2} \end{cases} is continuous at x=π2x = \dfrac{\pi}{2}, then (ba)53=…\left(\dfrac{b}{a}\right)^{\frac{5}{3}} = \ldots

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  22. 22
    Let 3f(x)−2f(1x)=x3f(x) - 2f\left(\dfrac{1}{x}\right) = x for x≠0x \neq 0. Then the value of f′(2)f'(2) is:

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  23. 23
    If gg is a differentiable function of xx such that g(x5)=3x7g(x^5) = 3x^7, for x≠0,x∈Rx \neq 0, x \in \mathbb{R} then g′(3125)=…g'(3125) = \ldots

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  24. 24
    If y=sin⁡−1(y⋅cos⁡2x)y = \sin^{-1}(y \cdot \cos 2x) such that dydx=2y⋅f(x)g(x)−cos⁡y\dfrac{dy}{dx} = \dfrac{2y \cdot f(x)}{g(x) - \cos y} then {f(x)}2−{g(x)}2=\{f(x)\}^2 - \{g(x)\}^2 =

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  25. 25
    Let f(x)f(x) be a twice differentiable function such that f′′(x)=−f(x)f''(x) = -f(x), and f′(x)=g(x)f'(x) = g(x). Let h(x)={f(x)}2+{g(x)}2h(x) = \{f(x)\}^2 + \{g(x)\}^2. If h(2)=7h(2) = 7, then h(8)h(8) is equal to:

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  26. 26
    The tangent to the curve y=e2xy = e^{2x} is drawn at the point P(0,1)P(0, 1). If this tangent intersects the xx-axis at point AA and the yy-axis at point BB, then the area of △OAB\triangle OAB (where OO is the origin) in square units is:

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  27. 27
    The equation of the line tangent to the curve y2=4xy^2 = 4x with an inclination of 60∘60^\circ is:

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  28. 28
    A student learned 100 Math formulae for an examination. The number of formulae the student can remember in 'tt' days after learning is given by f(t)=100(1−0.1t)2f(t) = 100(1 - 0.1t)^2. Then, the magnitude of the rate at which the student forgets the formula 3 days after learning is:

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  29. 29
    ∫x3+x2+2x+1x2−x+1dx=…\displaystyle\int \dfrac{x^3 + x^2 + 2x + 1}{x^2 - x + 1}dx = \ldots

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  30. 30
    ∫tan⁡−1x dx=…\displaystyle\int \tan^{-1}\sqrt{x}\,dx = \ldots

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  31. 31
    If ∫[sin⁡(log⁡x)−cos⁡(log⁡x)]dx=xf(x)+c\displaystyle\int[\sin(\log x) - \cos(\log x)]dx = xf(x) + c, then the value of f(x)f(x) is:

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  32. 32
    ∫sec⁡22x⋅csc⁡⁡22x dx=…\displaystyle\int \sec^2 2x \cdot \operatorname{\csc}^2 2x\,dx = \ldots

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  33. 33
    Let f(x)f(x) be odd and continuous on [−a,a][-a, a]. Then the value of ∫−aaxf(x2)dx\int_{-a}^{a} xf(x^2)dx is:

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  34. 34
    The value of lim⁡n→∞(n(n+1)2n+1+n(n+2)2(2n+2)+n(n+3)3(2n+3)+⋅s+12n3)\displaystyle\lim_{n \to \infty}\left(\dfrac{n}{(n + 1)\sqrt{2n + 1}} + \dfrac{n}{(n + 2)\sqrt{2(2n + 2)}} + \dfrac{n}{(n + 3)\sqrt{3(2n + 3)}} + \cdot s + \dfrac{1}{2n\sqrt{3}}\right) is:

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  35. 35
    If ∫0f(x)dt=xcos⁡(πx)\displaystyle\int_0^{f(x)} dt = x\cos(\pi x) then f′(4)=…f'(4) = \ldots

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  36. 36
    The value of ∫0π/2cos⁡3x+12cos⁡x−1dx\displaystyle\int_0^{\pi/2} \dfrac{\cos 3x + 1}{2\cos x - 1}dx is

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  37. 37
    A body cools from 100∘C100^\circ\mathrm{C} to 80∘C80^\circ\mathrm{C} in 5 minutes in air at 20∘C20^\circ\mathrm{C}. The time taken to cool from 80∘C80^\circ\mathrm{C} to 60∘C60^\circ\mathrm{C} will be:

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  38. 38
    If the general solution of the differential equation dydx=cos⁡(2x−3y)−cos⁡(2x+3y)\dfrac{dy}{dx} = \cos(2x - 3y) - \cos(2x + 3y) is cos⁡2x=g(y)+c\cos 2x = g(y) + c, then g(y)=…g(y) = \ldots

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  39. 39
    The solution of the differential equation dydx=xy+yxy+x\dfrac{dy}{dx} = \dfrac{xy + y}{xy + x} is

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  40. 40
    The scalar triple product of the vectors [2aˉ+3bˉ  3bˉ+cˉ  cˉ+2aˉ][2\bar{a} + 3\bar{b}\ \ 3\bar{b} + \bar{c}\ \ \bar{c} + 2\bar{a}] is equal to:

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  41. 41
    If the vector rˉ\bar{r} is perpendicular to i^−k^\hat{i} - \hat{k} and rˉ×(i^+xj^−k^)=i^+2j^+3k^\bar{r} \times (\hat{i} + x\hat{j} - \hat{k}) = \hat{i} + 2\hat{j} + 3\hat{k}, then the value of rˉ⋅(i^+xj^−k^)=\bar{r} \cdot (\hat{i} + x\hat{j} - \hat{k}) =

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  42. 42
    A(1,2)A(1, 2), B(1,0)B(1, 0) and C(0,2)C(0, 2) are vertices of ΔABC\Delta ABC. The position vector of the point in which the bisector of angle AA cuts BCBC is .....

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  43. 43
    The equation of the plane that perpendicularly bisects the line segment joining the points A(2,−1,3)A(2, -1, 3) and B(3,1,0)B(3, 1, 0) is:

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  44. 44
    The angle between the line x+11=y−22=z−33\dfrac{x + 1}{1} = \dfrac{y - 2}{2} = \dfrac{z - 3}{3} and the plane 3x+4y+z+5=03x + 4y + z + 5 = 0 is:

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  45. 45
    The equation of the plane containing the line x+14=y−2−1=z+53\dfrac{x + 1}{4} = \dfrac{y - 2}{-1} = \dfrac{z + 5}{3} and parallel to the vector i^+j^−k^\hat{i} + \hat{j} - \hat{k} is ax+by+cz=9ax + by + cz = 9. Then, the value of a+b−ca + b - c is:

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  46. 46
    The distance of the point (1,1,1)(1, 1, 1) from the line passing through the point (2,3,4)(2, 3, 4) whose direction ratios are 2,−3,62, -3, 6 is:

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  47. 47
    The position vector of a point PP, such that OPOP inclined to the X axis at 45∘45^\circ and to the Y axis at 60∘60^\circ and ∣OP∣=12|OP| = 12 units is:

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  48. 48
    The minimum value of Z=4x+6yZ = 4x + 6y, subject to constraint x+2y≥80,3x+y≥75,x≥0,y≥0x + 2y \ge 80, 3x + y \ge 75, x \ge 0, y \ge 0 is:

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  49. 49
    The sum and product of mean and variance of a binomial distribution are 3 and 2 respectively. Then, P(X>0)=…P(X > 0) = \ldots

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  50. 50
    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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