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MHT-CET 2026 (15 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
    Let f:R→Rf : R \rightarrow R be defined by f(x)=x2+2x+3x2+x+1f(x) = \dfrac{x^2 + 2x + 3}{x^2 + x + 1}. If the range of f(x)f(x) is [a,b][a, b], then a+b=…a + b = \ldots

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  2. 2
    The value of the expression cos⁡(π24)cos⁡(π12)cos⁡(π6)cos⁡(π3)\cos\left(\dfrac{\pi}{24}\right)\cos\left(\dfrac{\pi}{12}\right)\cos\left(\dfrac{\pi}{6}\right)\cos\left(\dfrac{\pi}{3}\right) is:

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  3. 3
    In △ABC\triangle ABC, if b=3+1b = \sqrt{3} + 1, c=3−1c = \sqrt{3} - 1 and A=π3A = \dfrac{\pi}{3}, then B−C=…B - C = \ldots

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  4. 4
    If n∈Zn \in Z, then the value of 2n(1+i)2n+2n(1−i)2n+(1+i)2n+(1−i)2n2n\dfrac{2^n}{(1+i)^{2n}} + \dfrac{2^n}{(1-i)^{2n}} + \dfrac{(1+i)^{2n} + (1-i)^{2n}}{2^n} is equal to

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  5. 5
    How many numbers (without repetition) between 400400 and 10001000 can be made with the digits 2,3,4,5,6,2, 3, 4, 5, 6, and 00?

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  6. 6
    The sum of the ordinates of the points on the line 4x+3y=104x + 3y = 10 whose distance from the line x+y−4=0x + y - 4 = 0 is 2\sqrt{2} is equal to ...

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  7. 7
    If the equation mx2−y2+2y+n=3mx^2 - y^2 + 2y + n = 3 represents a pair of perpendicular lines, then the values of mm and nn respectively are...

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  8. 8
    The equation of a circle whose center is at (3,−4)(3, -4) and the line 12x−5y−17=012x - 5y - 17 = 0 cuts the circle at AA and BB, where l(AB)=6l(AB) = 6 is ...

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  9. 9
    The tangent to the parabola x2=2yx^2 = 2y at point P makes an angle of 45∘45^\circ with the X-axis, then the coordinates of P are:

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  10. 10
    One of the tangents at the end of the latus rectum of the parabola y2=12xy^2 = 12x is

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  11. 11
    lim⁡h→0log⁡e(1+2h)−2log⁡e(1+h)h2\displaystyle\lim_{h \to 0} \dfrac{\log_e(1 + 2h) - 2\log_e(1 + h)}{h^2}

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  12. 12
    Let pp: switch S1S_1 is closed, qq: switch S2S_2 is closed. Then, the equivalent symbolic form for the following circuit is:

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  13. 13
    If p→(∼p∨q)p \rightarrow (\sim p \vee q) is false, then the truth values of p,qp, q are:

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  14. 14
    The logical expression [∼(p→q)∧(q→p)]∨∼(p↔q)[\sim(p \rightarrow q) \wedge (q \rightarrow p)] \vee \sim(p \leftrightarrow q) is equivalent to:

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  15. 15
    If sin⁡−1(y2)=tan⁡−1(x2)\sin^{-1}\left(\dfrac{y}{2}\right) = \tan^{-1}\left(\dfrac{x}{2}\right), ∣y∣<2|y| < 2, such that dxdy=k(4+x2)\dfrac{dx}{dy} = k(4 + x^2), then k=…k = \ldots

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  16. 16
    The solutions of the equation sin⁡[2cos⁡−1{cot⁡(2tan⁡−1x)}]=0\sin[2\cos^{-1}\{\cot(2\tan^{-1}x)\}] = 0 are

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  17. 17
    If tan⁡−1[5+261−6]=a+tan⁡−1(2)\tan^{-1}\left[\dfrac{\sqrt{5 + 2\sqrt{6}}}{1 - \sqrt{6}}\right] = a + \tan^{-1}(\sqrt{2}), then csc⁡(a)=…\csc(a) = \ldots

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  18. 18
    If cos⁡−1x+cos⁡−1y+cos⁡−1z=3π\cos^{-1}x + \cos^{-1}y + \cos^{-1}z = 3\pi, then xy+yz+zxxy + yz + zx is equal to

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  19. 19
    If A=[1−1102−3210]A = \begin{bmatrix} 1 & -1 & 1 \\ 0 & 2 & -3 \\ 2 & 1 & 0 \end{bmatrix}, B=adj AB = \mathrm{adj}\ A and C=5AC = 5A, then the value of ∣adj B∣∣C∣\dfrac{|\mathrm{adj}\ B|}{|C|} is:

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  20. 20
    If [1112−17312][xyz]=[172]\begin{bmatrix} 1 & 1 & 1 \\ 2 & -1 & 7 \\ 3 & 1 & 2 \end{bmatrix}\begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 1 \\ 7 \\ 2 \end{bmatrix} then [xyz]=\begin{bmatrix} x \\ y \\ z \end{bmatrix} =

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  21. 21
    Let f(x)=12x−4x−3x+11−cos⁡2xf(x) = \dfrac{12^x - 4^x - 3^x + 1}{1 - \cos 2x}, for x≠0x \ne 0. If f(x)f(x) is continuous at x=0x = 0 and f(0)=(log⁡a)⋅(log⁡3)4kf(0) = \dfrac{(\log a) \cdot (\log 3)}{4k}, then the value of aa in terms of kk is:

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  22. 22
    If f(x)=sin⁡(tan⁡−1x)f(x) = \sin(\tan^{-1}x), then ∫01xf′′(x)dx\int_0^1 x f''(x)dx is:

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  23. 23
    The derivative of f(cos⁡x)f(\cos x) with respect to g(tan⁡x)g(\tan x) at x=π6x = \dfrac{\pi}{6}, given that f′(32)=4f'\left(\dfrac{\sqrt{3}}{2}\right) = 4 and g′(13)=2g'\left(\dfrac{1}{\sqrt{3}}\right) = 2 is:

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  24. 24
    If y=xxx⋅⋅⋅∞y = x^{x^{x^{\cdot^{\cdot^{\cdot\infty}}}}}, then dydx\dfrac{dy}{dx} is:

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  25. 25
    If y=(log⁡x)log⁡xy = (\log x)^{\log x} and dydx=yx×f(x)\dfrac{dy}{dx} = \dfrac{y}{x} \times f(x) then f(x)=…f(x) = \ldots

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  26. 26
    A man of height 1.21.2 m is moving away from the lamp post at the rate 0.60.6 m/sec. If the height of the lamp post is 4.84.8 m, then the rate at which the shadow of the man is lengthening is:

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  27. 27
    The approximate surface area of a sphere of radius 2.012.01 cm is

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  28. 28
    ∫2x4−3x−x2 dx=…\int \dfrac{2x}{4 - 3x - x^2}\, dx = \ldots

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  29. 29
    The integral ∫csc⁡xcos⁡2[1+log⁡(tan⁡x2)] dx\int \dfrac{\csc x}{\cos^2[1 + \log(\tan\frac{x}{2})]}\, dx is equal to:

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  30. 30
    ∫ex(log⁡(sin⁡x)+1+cos⁡2x1−cos⁡2x)dx=…\int e^x \left(\log(\sin x) + \sqrt{\dfrac{1 + \cos 2x}{1 - \cos 2x}}\right) dx = \ldots

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  31. 31
    ∫xx5/2+4x3/2 dx=…\int \dfrac{x}{x^{5/2} + 4x^{3/2}}\, dx = \ldots

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  32. 32
    If ∫−14f(x)dx=9\int_{-1}^{4} f(x)dx = 9 and ∫24(7−f(x))dx=10\int_{2}^{4}(7 - f(x))dx = 10, then ∫2−1f(x)dx=\int_{2}^{-1} f(x)dx =

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  33. 33
    ∫π/6π/3sin⁡2xlog⁡(tan⁡x) dx=…\int_{\pi/6}^{\pi/3} \sin 2x \log(\tan x)\, dx = \ldots

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  34. 34
    The area of the region bounded by the line 4x+3y=124x + 3y = 12, the X-axis, and the lines x=−1x = -1 and x=2x = 2 (in sq units) is:

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  35. 35
    An ice ball melts. The rate at which it melts is proportional to the amount of the ice at that time. If half quantity of ice melts in 2020 minutes, then the time when (1/8)th(1/8)^{th} part of original amount of the ice is left is:

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  36. 36
    The solution of the differential equation dydx+1=ex+y\dfrac{dy}{dx} + 1 = e^{x+y} is

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  37. 37
    The order and degree of the differential equation edydx+dydx=xe^{\frac{dy}{dx}} + \dfrac{dy}{dx} = x is

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  38. 38
    Let aˉ=2i^+k^\bar{a} = 2\hat{i} + \hat{k}, bˉ=i^+j^+k^\bar{b} = \hat{i} + \hat{j} + \hat{k}, and cˉ=4i^−3j^+7k^\bar{c} = 4\hat{i} - 3\hat{j} + 7\hat{k}. If rˉ\bar{r} is a vector such that rˉ×bˉ=cˉ×bˉ\bar{r} \times \bar{b} = \bar{c} \times \bar{b} and rˉ⋅aˉ=0\bar{r} \cdot \bar{a} = 0, then the value of the scalar product rˉ⋅i^\bar{r} \cdot \hat{i} is:

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  39. 39
    If the scalar products of a vector aˉ=xi^+yj^+zk^\bar{a} = x\hat{i} + y\hat{j} + z\hat{k} with 2i^+j^+k^2\hat{i} + \hat{j} + \hat{k}, i^−j^+k^\hat{i} - \hat{j} + \hat{k}, i^+2j^+3k^\hat{i} + 2\hat{j} + 3\hat{k} are 5,3,5, 3, and 55 respectively, then x+y+z=…x + y + z = \ldots

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  40. 40
    If ∣aˉ∣=24|\bar{a}| = 24, ∣bˉ∣=7|\bar{b}| = 7, aˉ⋅bˉ=0\bar{a} \cdot \bar{b} = 0, aˉ+bˉ+cˉ=0ˉ\bar{a} + \bar{b} + \bar{c} = \bar{0}, then ∣cˉ∣=…|\bar{c}| = \ldots

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  41. 41
    If aˉ=i^+2j^+2k^\bar{a} = \hat{i} + 2\hat{j} + 2\hat{k}, bˉ=2i^−2j^+k^\bar{b} = 2\hat{i} - 2\hat{j} + \hat{k} and cˉ\bar{c} is a vector projection of aˉ\bar{a} on the line of bˉ\bar{b}, then the value of [2aˉ 3bˉ 3cˉ][2\bar{a}\ 3\bar{b}\ \sqrt{3}\bar{c}] is

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  42. 42
    If the angle between the line x=y−52=z+8kx = \dfrac{y - 5}{2} = \dfrac{z + 8}{k} and the plane 2x−3y+3z=92x - 3y + 3z = 9 is cos⁡−1(1322)\cos^{-1}\left(\sqrt{\dfrac{13}{22}}\right), then k=…k = \ldots

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  43. 43
    If the lines 1−x3=3y−12λ+1=4−z\dfrac{1 - x}{3} = \dfrac{3y - 1}{2\lambda + 1} = 4 - z and x+22μ=1−2y6=5−z7\dfrac{x + 2}{2\mu} = \dfrac{1 - 2y}{6} = \dfrac{5 - z}{7} are perpendicular, then 3μ+λ=…3\mu + \lambda = \ldots

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  44. 44
    Consider the lines L1:x2=y2=−zL_1: \dfrac{x}{2} = \dfrac{y}{2} = -z, L2:x−11=y−12=z−32L_2: \dfrac{x - 1}{1} = \dfrac{y - 1}{2} = \dfrac{z - 3}{2}. The unit vector perpendicular to both L1L_1 and L2L_2 is:

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  45. 45
    The coordinates of the foot of the perpendicular drawn from the point P(−1,3,2)P(-1, 3, 2) to the line formed by the intersection of the planes x+y+z=1x + y + z = 1 and 2x−y+z=22x - y + z = 2 are:

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  46. 46
    The Incenter of the triangle whose vertices are A(0,3,0)A(0, 3, 0), B(0,0,4)B(0, 0, 4), and C(0,3,4)C(0, 3, 4) is;

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  47. 47
    The L.P.P. Maximize Z=0.75x+0.5yZ = 0.75x + 0.5y subject to 3x+2y≤63x + 2y \le 6, x+4y≤4x + 4y \le 4, x≥0,y≥0x \ge 0, y \ge 0 has

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  48. 48
    If X∼B(10,0.4)X \sim B(10, 0.4), then the probability that XX takes a value at least 2 is:

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  49. 49
    The probability mass function (p.m.f.) of a random variable X is as follows: P(X=0)=3k3, P(X=1)=4k−10k2, P(X=2)=5k−1, P(X=x)=0P(X = 0) = 3k^3,\ P(X = 1) = 4k - 10k^2,\ P(X = 2) = 5k - 1,\ P(X = x) = 0 for any value of xx, then the value of kk is:

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  50. 50
    Two fair dice are tossed simultaneously. Let AA be the event that the sum of the numbers is prime, and BB be the event that at least one die shows a number 5. The odds against the occurrence of event BB given that event AA has already occurred is:

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