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MHT-CET 2023 (15 May Shift 2), 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 x=sin⁡θx = \sin\theta, y=sin⁡3θy = \sin^3\theta, then d2ydx2\frac{d^2y}{dx^2} at θ=π2\theta = \frac{\pi}{2} is

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  2. 2
    Let P,Q,RP,Q,R and SS be the points on the plane with position vectors −2i^−j^, 4i^, 3i^+3j^-2\hat{i}-\hat{j},\, 4\hat{i},\, 3\hat{i}+3\hat{j} and −3i^+2j^-3\hat{i}+2\hat{j} respectively. Then the quadrilateral PQRS must be a

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  3. 3
    If y=x2/3−x−1/3x2/3+x−1/3, x≠0y = \frac{x^{2/3} - x^{-1/3}}{x^{2/3} + x^{-1/3}},\, x \neq 0, then (x+1)2y1=(x+1)^2 y_1 =

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  4. 4
    The number of values of CC that satisfy the conclusion of Rolle's theorem in case of following function f(x)=sin⁡2πx, x∈[−1,1]f(x) = \sin 2\pi x,\, x \in [-1,1] is

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  5. 5
    An open tank with a square bottom, to contain 4000 cubic cm. of liquid, is to be constructed. The dimensions of the tank, so that the surface area of the tank is minimum, are

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  6. 6
    ∫cosec(x−a)⋅cosec x dx=\int \text{cosec}(x-a) \cdot \text{cosec}\,x\, dx =

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  7. 7
    The contrapositive of the inverse of p→(p→q)p \to (p \to q) is

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  8. 8
    The area enclosed between the parabola y2=4xy^2 = 4x and the line y=2x−4y = 2x - 4 is

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  9. 9
    There are 3 sections in a question paper and each section contains 5 questions. A candidate has to answer a total of 5 questions, choosing at least one question from each section. Then the number of ways, in which the candidate can choose the questions, is

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  10. 10
    The order of the differential equation, whose solution is y=C1+C2ex+C3ex+C4y = C_1 + C_2 e^x + C_3 e^{x+C_4}, is

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  11. 11
    If the sides of a rectangle are given by the equations x=−2,x=6,y=−2,y=5x=-2, x=6, y=-2, y=5, then the equation of the circle, drawn on the diagonal of this rectangle as its diameter, is

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  12. 12
    The solution set of the equation tan⁡x+sec⁡x=2cos⁡x\tan x + \sec x = 2\cos x, in the interval [0,2π][0, 2\pi] is

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  13. 13
    If one of the lines represented by ax2+2hxy+by2=0ax^2+2hxy+by^2=0 is perpendicular to mx+ny=18mx+ny=18, then

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  14. 14
    If sin⁡−1x13+cosec−11312=π2\sin^{-1}\frac{x}{13} + \text{cosec}^{-1}\frac{13}{12} = \frac{\pi}{2}, then the value of xx is

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  15. 15
    The graphical solution set of the system of inequations 2x+3y≤6, x+4y≥4, x≥0, y≥02x+3y \leq 6,\, x+4y \geq 4,\, x \geq 0,\, y \geq 0 is given by

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  16. 16
    The derivative of sin⁡−1(2x1−x2)\sin^{-1}(2x\sqrt{1-x^2}) w.r.t. sin⁡−1(3x−4x3)\sin^{-1}(3x-4x^3) is

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  17. 17
    If A=(5a−b32)A = \begin{pmatrix} 5a & -b \\ 3 & 2 \end{pmatrix} and A⋅adj A=AATA \cdot \text{adj}\,A = AA^T, then 5a+b5a+b is equal to

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  18. 18
    The function f(x)=2x3−9x2+12x+2f(x) = 2x^3-9x^2+12x+2 is decreasing in

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  19. 19
    ∫(1+x−1x)ex+1x dx\int\left(1+x-\frac{1}{x}\right)e^{x+\frac{1}{x}}\,dx is equal to

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  20. 20
    If the function f(x)={−2sin⁡xx≤−π/2Asin⁡x+B−π/2<x<π/2cos⁡xx≥π/2f(x) = \begin{cases} -2\sin x & x \leq -\pi/2 \\ A\sin x+B & -\pi/2 < x < \pi/2 \\ \cos x & x \geq \pi/2 \end{cases} is continuous everywhere, then the values of AA and BB are respectively

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  21. 21
    ∫π/3π/21+cos⁡x(1−cos⁡x)5/2 dx=\int_{\pi/3}^{\pi/2} \frac{\sqrt{1+\cos x}}{(1-\cos x)^{5/2}}\,dx =

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  22. 22
    If the complex number z=x+iyz = x+iy, where i=−1i = \sqrt{-1}, satisfies the condition ∣z+1∣=1|z+1| = 1, then zz lies on

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  23. 23
    The general solution of the differential equation dydx=3e2x+3e4xex+e−x\frac{dy}{dx} = \frac{3e^{2x}+3e^{4x}}{e^x+e^{-x}} is

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  24. 24
    If cos⁡−1x=α (0<x<1)\cos^{-1}x = \alpha\,(0 < x < 1) and sin⁡−1(2x1−x2)+sec⁡−1(12x2−1)=2π3\sin^{-1}(2x\sqrt{1-x^2}) + \sec^{-1}\left(\frac{1}{2x^2-1}\right) = \frac{2\pi}{3}, then α\alpha is

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  25. 25
    If the points P,QP, Q and RR are with the position vectors i^−2j^+3k^, −2i^+3j^+2k^\hat{i}-2\hat{j}+3\hat{k},\,-2\hat{i}+3\hat{j}+2\hat{k} and −8i^+13j^-8\hat{i}+13\hat{j} respectively, then these points are

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  26. 26
    A line makes 45∘45^\circ angle with positive X-axis and makes equal angles with positive Y-axis and Z-axis respectively, then the sum of the three angles which the line makes with positive X-axis, Y-axis and Z-axis is

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  27. 27
    If the lines x−12=y+23=z−14\frac{x-1}{2}=\frac{y+2}{3}=\frac{z-1}{4} and x−31=y−k2=z1\frac{x-3}{1}=\frac{y-k}{2}=\frac{z}{1} intersect, then kk has the value

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  28. 28
    The vector equation of the plane through the line of intersection of the planes x+y+z=1x+y+z=1 and 2x+3y+4z=52x+3y+4z=5, which is perpendicular to the plane x−y+z=0x-y+z=0, is

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  29. 29
    The equation of motion of a particle is s=at2+bt+cs = at^2+bt+c. If the displacement after 1 second is 20 m, velocity after 2 seconds is 30 m/sec and the acceleration is 10 m/sec2^2, then

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  30. 30
    One side and one diagonal of a parallelogram are represented by 3i^+j^−k^3\hat{i}+\hat{j}-\hat{k} and 2i^+j^−2k^2\hat{i}+\hat{j}-2\hat{k} respectively, then the area of parallelogram in square units is

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  31. 31
    If ∫ex2⋅x3 dx=ex2f(x)+c\int e^{x^2}\cdot x^3\,dx = e^{x^2}f(x)+c and f(1)=0f(1)=0 (where cc is a constant of integration), then the value of f(x)f(x) is

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  32. 32
    If cot⁡−1(cos⁡α)−tan⁡−1(cos⁡α)=x\cot^{-1}(\cos\alpha)-\tan^{-1}(\cos\alpha)=x, then the value of sin⁡x\sin x is

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  33. 33
    If [x]2−5[x]+6=0[x]^2-5[x]+6=0, where [⋅][\cdot] denotes the greatest integer function, then

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  34. 34
    The value of the integral ∫0π/2cot⁡xcot⁡x+tan⁡x dx\int_0^{\pi/2} \frac{\cot x}{\cot x+\tan x}\,dx is

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  35. 35
    Variance of first nn natural numbers is

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  36. 36
    Suppose three coins are tossed simultaneously. If xx denotes the number of heads, then probability distribution of xx is

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  37. 37
    Four fair dice are thrown independently 27 times. Then the expected number of times, at least two dice show up a three or a five is

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  38. 38
    If 0<x<10 < x < 1, then 1+x2[{xcos⁡(cot⁡−1x)+sin⁡(cot⁡−1x)}2−1]1/2\sqrt{1+x^{2}}\left[\left\{x\cos\left(\cot^{-1}x\right)+\sin\left(\cot^{-1}x\right)\right\}^{2}-1\right]^{1/2} is equal to

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  39. 39
    If the vector c⃗\vec{c} lies in the plane of a⃗\vec{a} and b⃗\vec{b}, where a⃗=i^−j^+2k^, b⃗=i^+j^+k^\vec{a}=\hat{i}-\hat{j}+2\hat{k},\, \vec{b}=\hat{i}+\hat{j}+\hat{k} and c⃗=xi^−(2−x)j^−k^\vec{c}=x\hat{i}-(2-x)\hat{j}-\hat{k}, then the value of xx is

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  40. 40
    The equation of a line passing through the point (2,−1,1)(2,-1,1) and parallel to the line joining the points i^+2j^+2k^\hat{i}+2\hat{j}+2\hat{k} and −i^+4j^+k^-\hat{i}+4\hat{j}+\hat{k} is

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  41. 41
    The foot of the perpendicular drawn from origin to a plane is M(2,1,−2)M(2,1,-2), then vector equation of the plane is

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  42. 42
    If f(1)=1, f′(1)=3f(1)=1,\, f'(1)=3, then the derivative of f(f(f(x)))+(f(x))2f(f(f(x)))+(f(x))^2 at x=1x=1 is

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  43. 43
    In a triangle ABC, with usual notations, cos⁡B+cos⁡Cb+c+cos⁡Aa\frac{\cos B+\cos C}{b+c}+\frac{\cos A}{a} has the value

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  44. 44
    If f(x)=xx+1, x≠−1f(x)=\frac{x}{x+1},\, x\neq -1 and (f∘f)(x)=F(x)(f\circ f)(x) = F(x), then ∫F(x) dx\int F(x)\,dx is

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  45. 45
    If p: The total prime numbers between 2 to 100 are 26. q: Zero is a complex number. r: Least common multiple (L.C.M.) of 6 and 7 is 6. Then which of the following is correct?

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  46. 46
    lim⁡x→23x+33−x−1233−x−3x/2=\lim_{x\to 2}\frac{3^x+3^{3-x}-12}{3^{3-x}-3^{x/2}} =

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  47. 47
    ∫0π/4sec⁡2x(1+tan⁡x)(2+tan⁡x) dx=\int_0^{\pi/4}\frac{\sec^2 x}{(1+\tan x)(2+\tan x)}\,dx =

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  48. 48
    If two fair dice are rolled, then the probability that the sum of the numbers on the upper faces is at least 9, is

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  49. 49
    After tt seconds, the acceleration of a particle, which starts from rest and moves in a straight line is (8−t5) cm/s2\left(8-\frac{t}{5}\right)\text{ cm/s}^2, then velocity of the particle at the instant, when the acceleration is zero, is

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  50. 50
    If A+B=225∘A+B=225^\circ, then cot⁡A1+cot⁡A⋅cot⁡B1+cot⁡B\frac{\cot A}{1+\cot A}\cdot\frac{\cot B}{1+\cot B}, if it exists, is equal to

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