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MHT-CET 2026 (14 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
    The domain of function f(x)=1∣x∣−xf(x) = \dfrac{1}{\sqrt{|x| - x}} is :

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  2. 2
    In ΔABC\Delta ABC, ∠C=2π3\angle C = \dfrac{2\pi}{3}, then the value of cos⁡2A+cos⁡2B−cos⁡Acos⁡B\cos^2 A + \cos^2 B - \cos A \cos B is

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  3. 3
    The number of solutions of cos⁡2x=cos⁡x\cos 2x = \cos x in the interval [0,2π][0, 2\pi] is

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  4. 4
    The value of ∑n=14(in+in+1)\displaystyle\sum_{n=1}^{4}\left(i^n + i^{n+1}\right) is

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  5. 5
    A man has seven relatives, 4 of them are ladies and 3 gentlemen, his wife also has 7 relatives out of which 3 are ladies and 4 gentlemen. The number of ways in which they can invite a dinner party of 3 ladies and 3 gentlemen, so that there are 3 of the man's relatives and 3 of the wife's relatives, are

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  6. 6
    Four Rupee 5 coins, three rupee 2 coins and two rupee 1 coin are stacked together in a column at random. The probability that the coins of the same denomination are consecutive, is

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  7. 7
    Point C and D divide segment AB respectively internally and externally, in the ratio m:n then B divides CD internally in the ratio --------

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  8. 8
    If the polar coordinates of the point P are (1,7π4)\left(1, \dfrac{7\pi}{4}\right), then the sum of the squares of abscissa and ordinate of Cartesian coordinates of point P is

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  9. 9
    If P(2,7)P(2, 7) is the mid-point of a line segment intercepted between the co-ordinate axes, then the equation of line is

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  10. 10
    If the sum of the cubes of slopes of two lines represented by the equation ax2+2hxy+by2=0ax^2 + 2hxy + by^2 = 0 is 11, then

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  11. 11
    If the tangent to the curve y2=4xy^2 = 4x at (4,4)(4, 4) touches the circle x2+y2−c=0x^2 + y^2 - c = 0 then the value of 5c5c is

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  12. 12
    Tangents are drawn from the point P(3,4)P(3, 4) to the ellipse x29+y24=1\dfrac{x^2}{9} + \dfrac{y^2}{4} = 1 touching the ellipse at points A and B. Then length of segment AB is

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  13. 13
    If lim⁡x→∞[x3+1x2+1−(ax+b)]=2\displaystyle\lim_{x \to \infty}\left[\dfrac{x^3 + 1}{x^2 + 1} - (ax + b)\right] = 2 then

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  14. 14
    [(p∧q)∨(∼r∨∼s∨∼t)]∧[(p∧q)∨(r∧s∧t)]≡[(p \wedge q) \vee (\sim r \vee \sim s \vee \sim t)] \wedge [(p \wedge q) \vee (r \wedge s \wedge t)] \equiv

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  15. 15
    Let p, q, r be three logical statements. Consider the compound statements S1:[(p→q)∧∼q]→rS_1 : [(p \rightarrow q) \wedge \sim q] \rightarrow r and S2:(p∨q)∨rS_2 : (p \vee q) \vee r Then which of the following is not true?

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  16. 16
    The negation of converse of "I can travel by train if I get my ticket reserved" is

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  17. 17
    If u=tan⁡−1(1+x2−1x)u = \tan^{-1}\left(\dfrac{\sqrt{1 + x^2} - 1}{x}\right) and v=tan⁡−1(2x1−x21−2x2)v = \tan^{-1}\left(\dfrac{2x\sqrt{1 - x^2}}{1 - 2x^2}\right) then dudv=…\dfrac{du}{dv} = \ldots

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  18. 18
    If −1≤x≤−12-1 \le x \le -\dfrac{1}{\sqrt{2}} then sin⁡−1(2x1−x2)=\sin^{-1}\left(2x\sqrt{1 - x^2}\right) =

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  19. 19
    If sin⁡−1x+cos⁡−1(15)=π4\sin^{-1}x + \cos^{-1}\left(\dfrac{1}{\sqrt{5}}\right) = \dfrac{\pi}{4}, then x=…x = \ldots

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  20. 20
    The value of cot⁡−1(ab+1a−b)+tan⁡−1(b−cbc+1)+cot⁡−1(ac+1c−a)\cot^{-1}\left(\dfrac{ab + 1}{a - b}\right) + \tan^{-1}\left(\dfrac{b - c}{bc + 1}\right) + \cot^{-1}\left(\dfrac{ac + 1}{c - a}\right) is

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  21. 21
    If the matrix A=[123115247]A = \begin{bmatrix} 1 & 2 & 3 \\ 1 & 1 & 5 \\ 2 & 4 & 7 \end{bmatrix} and its inverse is A−1=[ac−7b−1c−201]A^{-1} = \begin{bmatrix} a & c & -7 \\ b & -1 & c \\ -2 & 0 & 1 \end{bmatrix}, then ....

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  22. 22
    If f(x)=1−cos⁡(pπx2)x2f(x) = \dfrac{1 - \cos\left(\dfrac{p\pi x}{2}\right)}{x^2} is continuous at x=0x = 0 and f(0)=9π232f(0) = \dfrac{9\pi^2}{32}, then the value of pp is

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  23. 23
    If x=1+log⁡tt2x = \dfrac{1 + \log t}{t^2} and y=3+2log⁡tty = \dfrac{3 + 2\log t}{t}, then the value of dydx\dfrac{dy}{dx} is

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  24. 24
    If xyyx=1x^y y^x = 1, then dydx\dfrac{dy}{dx} is equal to

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  25. 25
    If 3x2+4xy+5y2=03x^2 + 4xy + 5y^2 = 0, then dydx=…\dfrac{dy}{dx} = \ldots

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  26. 26
    A rectangular sheet of paper has its area 3232 meters2\text{meters}^2. The margins at the top and bottom are 5050 cm each and at the sides 2525 cm each. If the area of the printed space is maximum, then the dimensions of the paper are

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  27. 27
    A stone is dropped into a quiet lake and waves move in circles at the speed of 7 cm s−17\ \mathrm{cm\,s^{-1}}. At the instant when the radius of circular wave is 55 cm, the enclosed area is increasing at the rate -------- cm2 s−1\mathrm{cm^2\,s^{-1}}

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  28. 28
    If Rolle's theorem holds for the function f(x)=x2−px+10f(x) = x^2 - px + 10, x∈[0,4]x \in [0, 4] with c=2c = 2, then p=p =

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  29. 29
    The value of the integral ∫dxx4+x3\displaystyle\int\dfrac{dx}{x^4 + x^3} is

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  30. 30
    Evaluate ∫ex(sin⁡x+2cos⁡x)sin⁡x dx\displaystyle\int e^x(\sin x + 2\cos x)\sin x\,dx

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  31. 31
    If ∫1+sec⁡x dx=2sin⁡−1(f(x))+c\displaystyle\int\sqrt{1 + \sec x}\,dx = 2\sin^{-1}(f(x)) + c, then

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  32. 32
    ∫x3sin⁡(tan⁡−1(x4))1+x8 dx=\displaystyle\int\dfrac{x^3\sin(\tan^{-1}(x^4))}{1 + x^8}\,dx =

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  33. 33
    ∫01sin⁡−1x(2−x)cos⁡−1x+sin⁡−1x(2−x) dx=…\displaystyle\int_0^1\dfrac{\sin^{-1}\sqrt{x(2 - x)}}{\cos^{-1}x + \sin^{-1}\sqrt{x(2 - x)}}\,dx = \ldots

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  34. 34
    ∫0π/4log⁡(1+tan⁡x) dx=…\displaystyle\int_0^{\pi/4}\log(1 + \tan x)\,dx = \ldots

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  35. 35
    The value of the integral ∫0π/4cos⁡x1−sin⁡2x dx\displaystyle\int_0^{\pi/4}\dfrac{\cos x}{\sqrt{1 - \sin^2 x}}\,dx is

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  36. 36
    The area of the region bounded by the parabola y2=16xy^2 = 16x and the line x=4x = 4 is

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  37. 37
    If the function f(x)f(x) satisfying the equation [f′(x)]2+4f(x)f′(x)+f2(x)=0[f'(x)]^2 + 4f(x)f'(x) + f^2(x) = 0, then f(x)=f(x) = (where cc is constant of integration)

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  38. 38
    A person's assets start reducing in such a way that the rate of reduction of assets is proportional to the square root of the assets existing at that moment. If the assets at the beginning are Rs. 40\text{Rs. }40 lakhs and they dwindle down to Rs. 40000\text{Rs. }40000 after 22 years. Then the person will be bankrupt in k years from the start. Then k=k =

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  39. 39
    The differential equation of the equation x2+(y−b)2=1x^2 + (y - b)^2 = 1 is

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  40. 40
    If the volume of tetrahedron with its coterminous edges aˉ,bˉ,cˉ\bar{a}, \bar{b}, \bar{c} is 66 cu. units then the volume of a tetrahedron with coterminous edges (aˉ×bˉ),(bˉ×cˉ),(cˉ×aˉ)(\bar{a} \times \bar{b}), (\bar{b} \times \bar{c}), (\bar{c} \times \bar{a}) is

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  41. 41
    If pˉ,qˉ,rˉ\bar{p}, \bar{q}, \bar{r} are unit vectors such that pˉ×(qˉ×rˉ)=17qˉ\bar{p} \times (\bar{q} \times \bar{r}) = \dfrac{1}{7}\bar{q}, then the angle between pˉ\bar{p} and rˉ\bar{r} is

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  42. 42
    The equation of the line passing through the point of intersection of the lines x−31=y−52=z−1−1\dfrac{x - 3}{1} = \dfrac{y - 5}{2} = \dfrac{z - 1}{-1} & x−42=y−2−1=z−42\dfrac{x - 4}{2} = \dfrac{y - 2}{-1} = \dfrac{z - 4}{2} and in the direction of i^−2j^+k^\hat{i} - 2\hat{j} + \hat{k} is

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  43. 43
    The distance of the point (4,5,3)(4, 5, 3) from the plane x+y+z=2x + y + z = 2 measured parallel to the line x=y=2zx = y = 2z is ..... units

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  44. 44
    The Cartesian equation of the line passing through A(3,2,1)A(3, 2, 1) and B(1,3,1)B(1, 3, 1) is:

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  45. 45
    If θ\theta is the angle between the line rˉ=(i^+j^−k^)+λ(i^−j^+k^)\bar{r} = (\hat{i} + \hat{j} - \hat{k}) + \lambda(\hat{i} - \hat{j} + \hat{k}) and the plane rˉ⋅(i^−2j^+3k^)=5\bar{r} \cdot (\hat{i} - 2\hat{j} + 3\hat{k}) = 5 then the value of cos⁡2θ\cos^2\theta is

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  46. 46
    In △ABC\triangle ABC, mid-points of the sides AB, BC and CA are respectively (l,0,0)(l, 0, 0), (0,m,0)(0, m, 0) and (0,0,n)(0, 0, n). Then (AB)2+(BC)2+(CA)2l2+m2+n2=\dfrac{(AB)^2 + (BC)^2 + (CA)^2}{l^2 + m^2 + n^2} =

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  47. 47
    If the points A(1,p,1),B(2,5,−1),C(q,11,r)A(1, p, 1), B(2, 5, -1), C(q, 11, r) are collinear. If the line passing through these points is perpendicular to the vector 4i^−2j^+k^4\hat{i} - 2\hat{j} + \hat{k}, then the value of p+q+rp + q + r is

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  48. 48
    The point at which the maximum value of 3x+2y3x + 2y subject to the constraints x+y≤2,x≥0,y≥0x + y \le 2, x \ge 0, y \ge 0 is obtained is

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  49. 49
    A coin is tossed n times. The probability of getting a head at least once is greater than 0.80.8, then the least value of n is

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  50. 50
    Two numbers are selected at random without replacement from the first nn natural numbers. Let X denote the larger of the two numbers obtained. If E(X)=143E(X) = \dfrac{14}{3}, then the value of nn is

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