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MHT-CET 2026 (15 May 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 f(x)+2f(1−x)=x2+5f(x) + 2f(1 - x) = x^2 + 5 for all real xx, then f(x)f(x) is equal to...

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  2. 2
    If α\alpha and β\beta are the solutions of 2cos⁡θ=12\cos\theta = 1 in the interval [0,2π)[0, 2\pi), then the solutions of 2cos⁡θ+cos⁡(α+β)=02\cos\theta + \cos(\alpha + \beta) = 0 in the interval [0,2π)[0, 2\pi) are...

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  3. 3
    If x∈(0,2π)x \in (0, 2\pi) satisfies the equation cos⁡x+sin⁡x=12\cos x + \sin x = \frac{1}{2}, then the sum of all distinct values of tan⁡x\tan x is...

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  4. 4
    In a triangular plot ABCABC, the side lengths are BC=7 mBC = 7\,m, CA=8 mCA = 8\,m, and AB=9 mAB = 9\,m. A vertical lamp post is located at the midpoint EE of side ACAC. If the angle of elevation of the top of the lamp post from point BB is tan⁡−1(4)\tan^{-1}(4), then the height of the lamp post is...

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  5. 5
    In a triangle, the lengths of the two largest sides are 1010 cm. and 99 cm. If the angles of the triangle are in arithmatic progression (A. P.), then the length of the third side (in cm.) is...

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  6. 6
    If ω\omega is a complex cube root of unity, then the value of the expression (1−ω+ω2)7+(1+ω−ω2)7(1 - \omega + \omega^2)^7 + (1 + \omega - \omega^2)^7 is equal to...

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  7. 7
    If all permutations of the letters of the word THREE are arranged as in dictionary, then 47th47^{th} word is...

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  8. 8
    If θ1\theta_1 and θ2\theta_2 are the acute angles between the lines 3x2−7xy+4y2=03x^2 - 7xy + 4y^2 = 0 and 6x2−5xy+y2=06x^2 - 5xy + y^2 = 0 respectively, then ...

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  9. 9
    The coordinates of the point on the curve x2+y2=8x^2 + y^2 = 8 which are nearest to the point (2,0)(2, 0) are:

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  10. 10
    The angle between the tangents drawn from the point (1,1)(1, 1) to the circle 2x2+2y2−8x+4y+5=02x^2 + 2y^2 - 8x + 4y + 5 = 0 is …\ldots

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  11. 11
    The equations of the lines passing through the center of circle x2+y2−4x−10y−7=0x^2 + y^2 - 4x - 10y - 7 = 0 and intersecting the line 3x−y−1=03x - y - 1 = 0 at an angle π4\dfrac{\pi}{4} is...

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  12. 12
    The equation of the directrix of the parabola x2−4x−5y−1=0x^2 - 4x - 5y - 1 = 0 is...

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  13. 13
    If lim⁡x→0(f(x)x2)\lim_{x \to 0}\left(\dfrac{f(x)}{x^2}\right) exists finitely and lim⁡x→0(1+x+f(x)x)1/x=e3\lim_{x \to 0}\left(1 + x + \dfrac{f(x)}{x}\right)^{1/x} = e^3, then ∫f(x)⋅log⁡x dx\int f(x) \cdot \log x\, dx is equal to:

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  14. 14
    Let f(x)f(x) be a function continuous at x=0x = 0, defined for x≠0x \ne 0 as f(x)=1−cos⁡(1−cos⁡(6x))x2sin⁡2(3x)f(x) = \dfrac{1 - \cos(1 - \cos(6x))}{x^2\sin^2(3x)}. Then the value of f(0)f(0) is...

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  15. 15
    If k>5k > 5 and lim⁡x→0k2x+25x−2⋅(5k)xtan⁡3x⋅ln⁡(1+3x)=lim⁡n→∞(n+1)!+n!(n+1)!−n!\lim_{x \to 0}\dfrac{k^{2x} + 25^x - 2\cdot(5k)^x}{\tan 3x \cdot \ln(1 + 3x)} = \lim_{n \to \infty}\dfrac{(n + 1)! + n!}{(n + 1)! - n!}, then value of kk is...

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  16. 16
    The statement (p∧(∼q))∨((∼p)∧q)∨((∼p)∧(∼q))(p \wedge (\sim q)) \vee ((\sim p) \wedge q) \vee ((\sim p) \wedge (\sim q)) is equivalent to :

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  17. 17
    Let A={1,2,3,…,10}A = \{1, 2, 3, \ldots, 10\}. Consider the following statements: p:∀x∈A, x2−x>0p : \forall x \in A,\ x^2 - x > 0 q:∃x∈Aq : \exists x \in A, such that x+10=20x + 10 = 20 Then the truth values of pp and qq are respectively ____

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  18. 18
    If the compound statement (p∧∼q)→(q∨∼r)(p \wedge \sim q) \rightarrow (q \vee \sim r) is false then which of the following statements is a true ?

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  19. 19
    If y=tan⁡−1(2x5/21−x5)y = \tan^{-1}\left(\dfrac{2x^{5/2}}{1 - x^5}\right), then dydx\dfrac{dy}{dx} is equal to ....

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  20. 20
    If x∈[0,1)x \in [0, 1) then the value of tan⁡{12sin⁡−1(2x1+x2)+12cos⁡−1(1−x21+x2)}\tan\left\{\dfrac{1}{2}\sin^{-1}\left(\dfrac{2x}{1 + x^2}\right) + \dfrac{1}{2}\cos^{-1}\left(\dfrac{1 - x^2}{1 + x^2}\right)\right\} is:

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  21. 21
    If A−1=[10−40−12−121]=adj(A)kA^{-1} = \begin{bmatrix} 1 & 0 & -4 \\ 0 & -1 & 2 \\ -1 & 2 & 1 \end{bmatrix} = \dfrac{\mathrm{adj}(A)}{k}, then kk is a root of the equation...

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  22. 22
    If A=[0121233a1]A = \begin{bmatrix} 0 & 1 & 2 \\ 1 & 2 & 3 \\ 3 & a & 1 \end{bmatrix} and A−1=[12−1212−43c52−3212]A^{-1} = \begin{bmatrix} \dfrac{1}{2} & -\dfrac{1}{2} & \dfrac{1}{2} \\ -4 & 3 & c \\ \dfrac{5}{2} & -\dfrac{3}{2} & \dfrac{1}{2} \end{bmatrix}, then ...

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  23. 23
    If the planes ax=y+zax = y + z, by=z+xby = z + x, cz=x+ycz = x + y pass through a common line, then the value of (a+b+c−abc)=⋅s(a + b + c - abc) = \cdot s

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  24. 24
    If f′(12)=3f'\left(\dfrac{1}{2}\right) = \sqrt{3} and g′(32)=−2g'\left(\dfrac{\sqrt{3}}{2}\right) = -2, then the derivative of f(sin⁡x)f(\sin x) with respect to g(cos⁡x)g(\cos x) at x=π6x = \dfrac{\pi}{6} is:

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  25. 25
    If the derivative of tan⁡−1(a+bx)\tan^{-1}(a + bx) at x=0x = 0 is 11, then 1−b3+a61 - b^3 + a^6 is:

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  26. 26
    If f(x)=x−14+(x−1)312+(x−1)520+(x−1)728+…f(x) = \dfrac{x - 1}{4} + \dfrac{(x - 1)^3}{12} + \dfrac{(x - 1)^5}{20} + \dfrac{(x - 1)^7}{28} + \ldots where 0<x<20 < x < 2, then f′(x)=f'(x) =

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  27. 27
    The coordinates of the point on the curve y=sin⁡(πx2)y = \sin\left(\dfrac{\pi x}{2}\right) at which the normal is parallel to the Y- axis are:

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  28. 28
    A right circular cone is melting such that its volume is decreasing at the rate of 12π cm3/sec12\pi\ \mathrm{cm^3/sec}. The radius of the base is always equal to half of its height. Then the rate of change of the radius when the radius is 22 cm is:

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  29. 29
    If ∫(tan⁡x+cot⁡x) dx=atan⁡−1(tan⁡x−1btan⁡x)+c\int(\sqrt{\tan x} + \sqrt{\cot x})\, dx = a\tan^{-1}\left(\dfrac{\tan x - 1}{\sqrt{b\tan x}}\right) + c, for x∈(0,π2)x \in \left(0, \dfrac{\pi}{2}\right), then a10+b5=⋅sa^{10} + b^5 = \cdot s

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  30. 30
    The value of ∫tan⁡−1(6x1−9x2)dx\int \tan^{-1}\left(\dfrac{6x}{1 - 9x^2}\right) dx, where ∣x∣<13|x| < \dfrac{1}{3} is:

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  31. 31
    If ∫x(a−x)9 dx=[f(x)]1111−a[f(x)]1010+c\int x(a - x)^9\, dx = \dfrac{[f(x)]^{11}}{11} - a\dfrac{[f(x)]^{10}}{10} + c, where cc is a constant of integration, then f′(x)=⋅sf'(x) = \cdot s

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  32. 32
    ∫π/53π/1011+tan⁡x dx=⋅s\displaystyle\int_{\pi/5}^{3\pi/10}\dfrac{1}{1 + \sqrt{\tan x}}\, dx = \cdot s

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  33. 33
    The value of the integral ∫1/221xsin⁡101(x−1x)dx\int_{1/2}^{2}\dfrac{1}{x}\mathrm{\sin}^{101}\left(x - \dfrac{1}{x}\right) dx is equal to:

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  34. 34
    The value of I=∫13ex(x+2)2(x+4)2 dxI = \int_1^3 e^x\dfrac{(x + 2)^2}{(x + 4)^2}\, dx is:

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  35. 35
    The area bounded by the parabola y=4x−x2y = 4x - x^2 and the x-axis (in sq. units) is:

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  36. 36
    If the general solution of the differential equation ysec⁡2(xy)dx−(xsec⁡2(xy)−y)dy=0y\sec^2\left(\dfrac{x}{y}\right)dx - \left(x\sec^2\left(\dfrac{x}{y}\right) - y\right)dy = 0 is tan⁡(xy)+g(y)=c\tan\left(\dfrac{x}{y}\right) + g(y) = c, then g(y)=…g(y) = \ldots

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  37. 37
    The general solution of the differential equation (x+1)dydx−y=ex(x+1)2(x + 1)\dfrac{dy}{dx} - y = e^x(x + 1)^2 is given by:

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  38. 38
    For a certain substance, the rate of change of vapor pressure PP with respect to temperature TT is directly proportional to the vapor pressure PP, and inversely proportional to the square of the temperature TT. The representation of this statement in the form of a differential equation is: (where kk is a constant of proportionality)

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  39. 39
    If aˉ=2i^−j^+k^\bar{a} = 2\hat{i} - \hat{j} + \hat{k}, bˉ=i^+3j^−k^\bar{b} = \hat{i} + 3\hat{j} - \hat{k}, cˉ=−2i^+j^−3k^\bar{c} = -2\hat{i} + \hat{j} - 3\hat{k} and dˉ=3i^+2j^+5k^\bar{d} = 3\hat{i} + 2\hat{j} + 5\hat{k}, then the value of x2+z2−yx^2 + z^2 - y is ⋅s\cdot s, where dˉ=xaˉ+ybˉ+zcˉ\bar{d} = x\bar{a} + y\bar{b} + z\bar{c}

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  40. 40
    Let aˉ=2i^−j^+k^\bar{a} = 2\hat{i} - \hat{j} + \hat{k} and bˉ=i^+2j^−2k^\bar{b} = \hat{i} + 2\hat{j} - 2\hat{k}. If a third vector cˉ=xi^+yj^+zk^\bar{c} = x\hat{i} + y\hat{j} + z\hat{k} is coplanar with aˉ\bar{a} and bˉ\bar{b}, is perpendicular to aˉ\bar{a}, and satisfies ∣cˉ∣=45|\bar{c}| = \sqrt{45}, then the sum of all possible values of yy is:

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  41. 41
    A unit vector uˉ\bar{u} makes an angle of 45∘45^\circ with the X- axis. If uˉ+j^+k^\bar{u} + \hat{j} + \hat{k} is also a unit vector, then the sum of the projections of uˉ\bar{u} on the coordinate axes is:

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  42. 42
    The position vectors of vertices AA, BB and CC of triangle ABCABC are i^−j^−3k^\hat{i} - \hat{j} - 3\hat{k}, 2i^+j^−2k^2\hat{i} + \hat{j} - 2\hat{k}, and −5i^+2j^−6k^-5\hat{i} + 2\hat{j} - 6\hat{k} respectively. The length of the bisector ADAD of angle BACBAC, where DD is on the line segment BCBC, is

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  43. 43
    The equation of the plane containing the line rˉ=(2i^−k^)+λ(i^+j^+2k^)\bar{r} = (2\hat{i} - \hat{k}) + \lambda(\hat{i} + \hat{j} + 2\hat{k}) and perpendicular to the plane rˉ⋅(3i^−j^+k^)=20\bar{r} \cdot (3\hat{i} - \hat{j} + \hat{k}) = 20 is…\ldots

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  44. 44
    If the direction cosines of the line 2x+53=2−3z−1; 3y=−2\dfrac{2x + 5}{3} = \dfrac{2 - 3z}{-1};\ 3y = -2 are 985, m, 285\dfrac{9}{\sqrt{85}},\ m,\ \dfrac{2}{\sqrt{85}} then the value of mm is:

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  45. 45
    If the line x+4a=y−2b=z+1c\dfrac{x + 4}{a} = \dfrac{y - 2}{b} = \dfrac{z + 1}{c} passes through the point (−3,1,−2)(-3, 1, -2) and its direction is chosen such that it makes an acute angle γ\gamma with the positive direction of Z- axis, then value of cos⁡γ\cos\gamma is:

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  46. 46
    If l,m,nl, m, n are direction cosines of a vector rˉ\bar{r} such that l+m+n=1l + m + n = 1 and l2+m2+n2=1l^2 + m^2 + n^2 = 1, then: (where l, m, n≥0l,\ m,\ n \ge 0)

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  47. 47
    The linear programming problem: Maximize : Z=x+yZ = x + y Subject to : x+2y≤200, x+y≤150, 0≤y≤60, x≥0x + 2y \le 200,\ x + y \le 150,\ 0 \le y \le 60,\ x \ge 0 has:

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  48. 48
    It is known that 10%10\% of certain articles manufactured are defective. What is the probability that in a random sample of 1212 such articles, 99 are defective?

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  49. 49
    The probability distribution of a discrete random variable (r.v.) XX is given below:
    X=xX = x0011223344
    P(X=x)P(X = x)kk2k2k4k4k2k2kkk
    Then the value of P[X≤E(X)]P[X \le E(X)] is:

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  50. 50
    A husband and wife appeared for an interview for two vacancies in an office. The probability of the husband's selection is 15\dfrac{1}{5} and that of the wife's selection is 17\dfrac{1}{7}, then the probability that exactly one of them is selected is...

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