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MHT-CET 2026 (13 April 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 quadratic polynomial p(x)p(x) has roots 11 and α\alpha, while quadratic polynomial q(x)q(x) has roots 11 and β\beta. Let α\alpha and β\beta be the roots of r(x)=p(x)+q(x)r(x) = p(x) + q(x). Then lim⁡x→∞[p(x)−q(x)]=\lim_{x \to \infty}\left[\sqrt{p(x)} - \sqrt{q(x)}\right] =

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  2. 2
    If x=−1−−1−−1−…∞x = \sqrt{-1 - \sqrt{-1 - \sqrt{-1 - \ldots\infty}}}, where ω\omega is a non-real complex cube root of unity, then the value of xx is...

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  3. 3
    If sin⁡3α=4sin⁡α⋅sin⁡(x+α)⋅sin⁡(x−α)\sin 3\alpha = 4\sin\alpha \cdot \sin(x + \alpha) \cdot \sin(x - \alpha) where α≠nπ,n∈Z\alpha \neq n\pi, n \in Z, then all possible values of xx are given as

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  4. 4
    If cos⁡(θ−α)=a\cos(\theta - \alpha) = a and sin⁡(θ−β)=b\sin(\theta - \beta) = b, then the value of cos⁡2(α−β)+2absin⁡(α−β)+cos⁡2(θ−α)\cos^2(\alpha - \beta) + 2ab\sin(\alpha - \beta) + \cos^2(\theta - \alpha) is...

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  5. 5
    The general solution of cos⁡θ−sin⁡θ=1\cos\theta - \sin\theta = 1 is

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  6. 6
    The equation of the straight line passing through the point (−5,3)(-5, 3) such that the portion of the line intercepted between the axes is divided by the point in the ratio 4:34 : 3 (from the X-axis to the Y-axis) is...

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  7. 7
    If the equation kx2−5xy+6y2+x−3y=0kx^2 - 5xy + 6y^2 + x - 3y = 0 represents a pair of straight lines, then the co-ordinates of their point of intersection are

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  8. 8
    The pair of straight lines represented by the equation 3x2−4xy+3y2=0\sqrt{3}x^2 - 4xy + \sqrt{3}y^2 = 0 are

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  9. 9
    The equation of the circle concentric with circle x2+y2−6x+7=0x^2 + y^2 - 6x + 7 = 0 and which touches the line x+y+3=0x + y + 3 = 0 is ....

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  10. 10
    The distance of the focus of the parabola y2=16xy^2 = 16x from its directrix is...

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  11. 11
    The value of lim⁡n→∞(n+2)!+(n+1)!(n+2)!−(n+1)!\lim_{n \to \infty}\dfrac{(n + 2)! + (n + 1)!}{(n + 2)! - (n + 1)!} is ____

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  12. 12
    If the truth value of the compound statement [(p∨q)∧(q→r)∧(∼r)]→(p∧q)[(p \vee q) \wedge (q \to r) \wedge (\sim r)] \to (p \wedge q) is false, then the truth values of p→qp \to q and q→pq \to p are respectively...

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  13. 13
    Which of the following statements is logically equivalent to ∼(p↔q)\sim(p \leftrightarrow q) ?

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  14. 14
    If A=[sec⁡θ−tan⁡θ−tan⁡θsec⁡θ],θ∈(0,π2)A = \begin{bmatrix} \sec\theta & -\tan\theta \\ -\tan\theta & \sec\theta \end{bmatrix}, \theta \in \left(0, \dfrac{\pi}{2}\right) such that A+adjA=4IA + \mathrm{adj}A = 4I, then θ=\theta =

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  15. 15
    The element in 1st row and 2nd column of the inverse of the matrix [13−2−30−5250]\begin{bmatrix} 1 & 3 & -2 \\ -3 & 0 & -5 \\ 2 & 5 & 0 \end{bmatrix} is ...

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  16. 16
    If y=tan⁡−1(1x2+x+1)+tan⁡−1(1x2+3x+3)+tan⁡−1(1x2+5x+7)+…y = \tan^{-1}\left(\dfrac{1}{x^2 + x + 1}\right) + \tan^{-1}\left(\dfrac{1}{x^2 + 3x + 3}\right) + \tan^{-1}\left(\dfrac{1}{x^2 + 5x + 7}\right) + \ldots upto nn terms, then y′(0)=y'(0) =

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  17. 17
    Let f(x)=(sin⁡−1x)2+(cos⁡−1x)2f(x) = (\sin^{-1}x)^2 + (\cos^{-1}x)^2 be a real-valued function defined on its domain. Then the sum of the greatest and the least values of f(x)f(x) is

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  18. 18
    The value of tan⁡−1(1)+cos⁡−1(12)+sin⁡−1(12)tan⁡−1(3)−sec⁡−1(−2)\dfrac{\tan^{-1}(1) + \cos^{-1}\left(\dfrac{1}{2}\right) + \sin^{-1}\left(\dfrac{1}{2}\right)}{\tan^{-1}(\sqrt{3}) - \sec^{-1}(-2)} is equal to

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  19. 19
    If g(x)g(x) is the inverse function of f(x)f(x), where f(x)=5x+34x−1f(x) = \dfrac{5x + 3}{4x - 1}, then g(1)=g(1) =

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  20. 20
    If f(x)f(x) is continuous at x=0x = 0, where f(x)=8x−2xkx−1f(x) = \dfrac{8^x - 2^x}{k^x - 1}, for x≠0x \neq 0 and f(0)=2f(0) = 2, then the value of kk is ...

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  21. 21
    ddx[sin⁡2{cot⁡−11−x1+x}]=…\dfrac{d}{dx}\left[\sin^2\left\{\cot^{-1}\sqrt{\dfrac{1 - x}{1 + x}}\right\}\right] = \ldots

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  22. 22
    If f(x)=x2+1,g(x)=x+1x2+1,h(x)=2x−3f(x) = \sqrt{x^2 + 1}, g(x) = \dfrac{x + 1}{x^2 + 1}, h(x) = 2x - 3, then f′(h′(g′(x)))=f'\left(h'(g'(x))\right) =

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  23. 23
    If y+x+y−x=c\sqrt{y + x} + \sqrt{y - x} = c such that dydx=f(x)−[f(x)]2−1\dfrac{dy}{dx} = f(x) - \sqrt{[f(x)]^2 - 1}, then f(x)=f(x) = ____

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  24. 24
    If Rolle's theorem is applicable for the function f(x)=log⁡(x2+ax)f(x) = \log\left(\dfrac{x^2 + a}{x}\right) on [3,4][3, 4] with c∈(3,4)c \in (3, 4) such that f′(c)=0f'(c) = 0, then the value of f′′(c)f''(c) is

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  25. 25
    If the tangent to the curve 2y3=x3+ax22y^3 = x^3 + ax^2 at the point (a,a)(a, a) cuts off intercepts α\alpha and β\beta on the coordinate axes such that α2+β2=61\alpha^2 + \beta^2 = 61, then the value of aa is

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  26. 26
    Let f(x)f(x) be the differentiable function for all xx such that f′(x)≤5f'(x) \le 5 and f(1)=4f(1) = 4. The maximum value of f(5)f(5) is...

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  27. 27
    If ∫x2⋅ex dx=exf(x)+c\displaystyle\int x^2 \cdot e^x\,dx = e^x f(x) + c, then the minimum value of f(x)f(x) is ...

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  28. 28
    ∫(x+1)x(1+xex)2 dx=\displaystyle\int\dfrac{(x + 1)}{x(1 + xe^x)^2}\,dx =

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  29. 29
    If ∫x+1x2+1 dx=tan⁡−1x+g(x)+c\displaystyle\int\dfrac{x + 1}{x^2 + 1}\,dx = \tan^{-1}x + g(x) + c, where cc is constant of integration, then the function g(x)g(x) is monotonically increasing in the interval

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  30. 30
    ∫1+sin⁡x dx\displaystyle\int\sqrt{1 + \sin x}\,dx is equal to

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  31. 31
    If ∫0a(x2−4x+1) dx=6\int_0^a(x^2 - 4x + 1)\,dx = 6, then the real value of aa is

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  32. 32
    If ∫π/6π/311+sin⁡x+cos⁡x dx=log⁡2\int_{\pi/6}^{\pi/3}\dfrac{1}{1 + \sin x + \cos x}\,dx = \log 2, then the value of ∫π/6π/3cos⁡x1+sin⁡x+cos⁡x dx=\int_{\pi/6}^{\pi/3}\dfrac{\cos x}{1 + \sin x + \cos x}\,dx = ............

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  33. 33
    The area of the region bounded by the curve y=2kxy = 2^{kx} and x=0,x=2x = 0, x = 2, in first quadrant is 2log⁡2\dfrac{2}{\log 2}, then the value of kk is

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  34. 34
    The area of the region bounded by the curve y=x3y = x^3 and the lines y=8y = 8 and x=0x = 0 is ... square units.

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  35. 35
    The differential equation of 3y=x+c33y = \sqrt[3]{x + c} is

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  36. 36
    The general solution of the differential equation (x+y)dydx=1(x + y)\dfrac{dy}{dx} = 1 is

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  37. 37
    A body cools according to Newton's law of cooling from 100∘100^\circC to 60∘60^\circC in 2020 minutes. The temperature of the surroundings being 20∘20^\circC, then the total time required for the body to cool down to 30∘30^\circC is

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  38. 38
    If aˉ=i^+j^\bar{a} = \hat{i} + \hat{j} and bˉ=i^−k^\bar{b} = \hat{i} - \hat{k}, then the point of intersection of the lines rˉ×aˉ=bˉ×aˉ\bar{r} \times \bar{a} = \bar{b} \times \bar{a} and rˉ×bˉ=aˉ×bˉ\bar{r} \times \bar{b} = \bar{a} \times \bar{b} is

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  39. 39
    If aˉ\bar{a} and bˉ\bar{b} have the same magnitude and angle between them is 60∘60^\circ and their scalar product is 12\dfrac{1}{2}, then ∣aˉ∣|\bar{a}| is ____

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  40. 40
    If aˉ=i^+j^+k^,bˉ=i^,cˉ=c1i^+c2j^+c3k^\bar{a} = \hat{i} + \hat{j} + \hat{k}, \bar{b} = \hat{i}, \bar{c} = c_1\hat{i} + c_2\hat{j} + c_3\hat{k} with c1=1,c2=2c_1 = 1, c_2 = 2, then value of c3c_3 such that aˉ,bˉ,cˉ\bar{a}, \bar{b}, \bar{c} are coplanar is ____

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  41. 41
    If aˉ,bˉ\bar{a}, \bar{b} and cˉ\bar{c} are non-coplanar unit vectors such that the angle between any two of them is 60∘60^\circ, and the vector dˉ=xaˉ+ybˉ+zcˉ\bar{d} = x\bar{a} + y\bar{b} + z\bar{c} is perpendicular to both aˉ\bar{a} and bˉ\bar{b}, then the value of (x+y)z\dfrac{(x + y)}{z} is

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  42. 42
    If aˉ,bˉ,cˉ\bar{a}, \bar{b}, \bar{c} are three non zero and non-coplanar vectors such that aˉ×(bˉ×cˉ)=bˉ2\bar{a} \times (\bar{b} \times \bar{c}) = \dfrac{\bar{b}}{2}, then the angle between aˉ\bar{a} and bˉ\bar{b} is ...

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  43. 43
    The vector equation of the plane which is at a distance of 55 units from the origin and normal to the vector 2i^+j^−2k^2\hat{i} + \hat{j} - 2\hat{k} is

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  44. 44
    The equation of a line passing through a point (4,−2,3)(4, -2, 3) and perpendicular to the XZ-plane is....

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  45. 45
    The equation of the plane passing through the intersection of planes 2x−y+z=32x - y + z = 3, 4x−3y+5z=−94x - 3y + 5z = -9 and parallel to the line x+12=y+34=z−35\dfrac{x + 1}{2} = \dfrac{y + 3}{4} = \dfrac{z - 3}{5} is...

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  46. 46
    If a line makes angles α,β,γ\alpha, \beta, \gamma with the coordinate axes, then the sum of values of sin⁡2α+sin⁡2β+sin⁡2γ\sin^2\alpha + \sin^2\beta + \sin^2\gamma and cos⁡2α+cos⁡2β+cos⁡2γ\cos 2\alpha + \cos 2\beta + \cos 2\gamma is ...

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  47. 47
    The difference between the maximum value and minimum value of the objective function z=3x+5yz = 3x + 5y of a linear programming problem subject to constraints 5x+10y≤50,x+y≥1,y≤45x + 10y \le 50, x + y \ge 1, y \le 4 and x≥0,y≥0x \ge 0, y \ge 0 is 3λ3\lambda. Then the value of λ\lambda is

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  48. 48
    In a multiple-choice examination, there are 1010 questions with one correct option out of 44 options for each question. A student gets 44 marks for each correct answer and 11 mark is deducted for each incorrect answer. The probability that a student, guessing randomly on every question, scores 3030 marks in this exam is...

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  49. 49
    It is known that a box of 88 batteries contains 33 defective pieces and a person randomly selects 22 batteries from this box. Then the probability distribution of the number of defective batteries is

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  50. 50
    A certain disease has a prevalence of 1%1\% in the population. A diagnostic test for the disease has a sensitivity of 98%98\% and a specificity of 95%95\%. If a person from this population tests positive, then the probability that they actually have the disease is...

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