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MHT-CET 2023 (11 May Shift 1), Paper I, Mathematics question paper

50 questions · 100 marks · 90 minutes · +2 per correct answer, no negative marking

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  1. 1
    ∫x dx(x−1)2(x+2)=\int \frac{x\,dx}{(x-1)^2(x+2)} =

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  2. 2
    Let Z be a complex number such that ∣Z∣+Z=2+i|Z| + Z = 2 + i (where i=−1i = \sqrt{-1}), then ∣Z∣|Z| is equal to

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  3. 3
    How many numbers greater than a million can be formed with the digits 2, 3, 0, 3, 4, 2, 3?

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  4. 4
    For the following shaded region, the linear constraints are

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  5. 5
    lim⁡x→25x+53−x−3053−x−5x/2=\lim_{x \to 2} \frac{5^x + 5^{3-x} - 30}{5^{3-x} - 5^{x/2}} =

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  6. 6
    If ∫(7x−2)3x+2 dx=A(3x+2)5/2+B(3x+2)3/2+c\int(7x-2)\sqrt{3x+2}\,dx = A(3x+2)^{5/2} + B(3x+2)^{3/2} + c (where c is a constant of integration), then the values of A and B are respectively

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  7. 7
    If y=sin⁡−1(log⁡x21+(log⁡x)2)y = \sin^{-1}\left(\frac{\log x^2}{1+(\log x)^2}\right), then dydx∣x=1=\left.\frac{dy}{dx}\right|_{x=1} =

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  8. 8
    If f(x)={a2(x−∣x∣),x<00,x=0bx2sin⁡1x,x>0f(x) = \begin{cases} a^2(x-|x|), & x < 0 \\ 0, & x = 0 \\ bx^2\sin\frac{1}{x}, & x > 0 \end{cases} is continuous at x=0x = 0, then

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  9. 9
    The p.m.f. of a random variable X is given by P[X=x]=(5x)125P[X=x] = \binom{5}{x}\frac{1}{2^5} if x=0,1,2,3,4,5x = 0,1,2,3,4,5; 0 otherwise. Then which of the following is not correct?

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  10. 10
    If f(x)=(1+x)(1+x2)(1+x4)(1+x8)f(x) = (1+x)(1+x^2)(1+x^4)(1+x^8), then f′(1)=f'(1) =

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  11. 11
    ∫0.23.5[x] dx=\int_{0.2}^{3.5} [x]\,dx = (where [x]=[x] = greatest integer not greater than x)

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  12. 12
    Let p, q and r be the statements: p: X is an equilateral triangle; q: X is isosceles triangle; r:q∨∼pr: q \vee \sim p, then the equivalent statement of r is

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  13. 13
    The bacteria increase at the rate proportional to the number of bacteria present. If the original number N doubles in 8 hours, then the number of bacteria in 24 hours will be

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  14. 14
    If f(x)=x3−6x2+9x+3f(x) = x^3 - 6x^2 + 9x + 3 is monotonically decreasing function, then x lies in

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  15. 15
    If [x]2−5[x]+6=0[x]^2 - 5[x] + 6 = 0, where [x][x] denotes the greatest integer function, then

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  16. 16
    The perpendicular distance of the origin from the plane 2x+y−2z−18=02x + y - 2z - 18 = 0 is

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  17. 17
    Let p: A man is judge. q: He is honest. The inverse of p→qp \to q is

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  18. 18
    If equation of normal to the curve x=tx = \sqrt{t}, y=t−1ty = t - \frac{1}{t} at t=4t = 4 is

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  19. 19
    If three fair coins are tossed, then variance of number of heads obtained, is

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  20. 20
    The plane 2x+3y+4z=12x + 3y + 4z = 1 meets X-axis in A, Y-axis in B and Z-axis in C. Then the centroid of △ABC\triangle ABC is

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  21. 21
    If x2+y2=t+1tx^2 + y^2 = t + \frac{1}{t}, x4+y4=t2+1t2x^4 + y^4 = t^2 + \frac{1}{t^2}, then dydx=\frac{dy}{dx} =

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  22. 22
    The general solution of dydx+sin⁡x+y2=sin⁡x−y2\frac{dy}{dx} + \sin\frac{x+y}{2} = \sin\frac{x-y}{2} is

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  23. 23
    If A and B are two independent events such that P(A′)=0.75P(A') = 0.75, P(A∪B)=0.65P(A \cup B) = 0.65 and P(B)=pP(B) = p, then value of p is

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  24. 24
    If the lines x+1−10=y+k−1=z−41\frac{x+1}{-10} = \frac{y+k}{-1} = \frac{z-4}{1} and x+10−1=y+1−3=z−14\frac{x+10}{-1} = \frac{y+1}{-3} = \frac{z-1}{4} intersect each other, then the value of k is

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  25. 25
    The approximate value of (3.978)3/2(3.978)^{3/2} is

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  26. 26
    The value of tan⁡−1(1+x−1−x1+x+1−x)+12cos⁡−1x\tan^{-1}\left(\frac{\sqrt{1+x}-\sqrt{1-x}}{\sqrt{1+x}+\sqrt{1-x}}\right) + \frac{1}{2}\cos^{-1}x is

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  27. 27
    Let a, b and c be three non-zero vectors such that no two of them are collinear and (a⃗×b⃗)×c⃗=13∣b⃗∣∣c⃗∣a⃗(\vec{a}\times\vec{b})\times\vec{c} = \frac{1}{3}|\vec{b}||\vec{c}|\vec{a}. If 'θ\theta' is the angle between the vectors b and c, then value of sin⁡θ\sin\theta is

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  28. 28
    In an experiment with 15 observations for x, the following results were available ∑x2=2830\sum x^2 = 2830, ∑x=170\sum x = 170. One observation 20 was found to be wrong and was replaced by the correct value 30. Then the corrected variance is

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  29. 29
    The area of the region lying in the first quadrant by y=4x2y = 4x^2, x=0x = 0, y=2y = 2, y=4y = 4 is

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  30. 30
    The equation of the line passing through the point (3,1,2) and perpendicular to the lines x−11=y−22=z−33\frac{x-1}{1}=\frac{y-2}{2}=\frac{z-3}{3} and x−3=y2=z5\frac{x}{-3}=\frac{y}{2}=\frac{z}{5} is

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  31. 31
    Let A=(1−1121−3111)A = \begin{pmatrix}1&-1&1\\2&1&-3\\1&1&1\end{pmatrix} and B=(402)B = \begin{pmatrix}4\\0\\2\end{pmatrix} such that AX=BAX = B, then X=X =

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  32. 32
    If x⃗=b⃗×c⃗[a⃗b⃗c⃗]\vec{x} = \frac{\vec{b}\times\vec{c}}{[\vec{a}\vec{b}\vec{c}]}, y⃗=c⃗×a⃗[a⃗b⃗c⃗]\vec{y} = \frac{\vec{c}\times\vec{a}}{[\vec{a}\vec{b}\vec{c}]} and z⃗=a⃗×b⃗[a⃗b⃗c⃗]\vec{z} = \frac{\vec{a}\times\vec{b}}{[\vec{a}\vec{b}\vec{c}]} where a, b, c are non-coplanar vectors, then value of x⃗⋅(a⃗+b⃗)+y⃗⋅(b⃗+c⃗)+z⃗⋅(c⃗+a⃗)\vec{x}\cdot(\vec{a}+\vec{b})+\vec{y}\cdot(\vec{b}+\vec{c})+\vec{z}\cdot(\vec{c}+\vec{a}) is

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  33. 33
    If in a triangle ABC, with usual notations, the angles are in A.P. and b:c=3:2b:c = \sqrt{3}:\sqrt{2}, then angle A =

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  34. 34
    The rate of change of the volume of a sphere with respect to its surface area, when its radius is 2 cm, is

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  35. 35
    If angle θ\theta in [0,2π][0, 2\pi] satisfies both the equations cot⁡θ=3\cot\theta = \sqrt{3} and 3sec⁡θ+2=0\sqrt{3}\sec\theta + 2 = 0, then θ\theta is equal to

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  36. 36
    The area of the triangle with vertices (1,2,0), (1,0,2) and (0,3,1) is

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  37. 37
    The particular solution of the differential equation xydydx=x2+2y2xy\frac{dy}{dx} = x^2 + 2y^2 when y(1)=0y(1) = 0 is

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  38. 38
    The equation of the tangent to the circle, given by x=5cos⁡θx = 5\cos\theta, y=5sin⁡θy = 5\sin\theta at the point θ=π/3\theta = \pi/3 on it, is

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  39. 39
    The joint equation of two lines through the origin, each making an angle with measure of 30∘30^\circ with the positive Y-axis, is

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  40. 40
    With usual notations, if the lengths of the sides of a triangle are 7 cm, 434\sqrt{3} cm and 13\sqrt{13} cm, then the measures of the smallest angle is

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  41. 41
    If the volume of tetrahedron whose vertices are A≡(1,−6,10)A\equiv(1,-6,10), B≡(−1,−3,7)B\equiv(-1,-3,7), C≡(5,−1,k)C\equiv(5,-1,k) and D≡(7,−4,7)D\equiv(7,-4,7) is 11 cu. units, then the value of k is

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  42. 42
    If the slope of one of the lines given by Kx2+6xy+y2=0Kx^2 + 6xy + y^2 = 0 is three times the other, then the value of K is

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  43. 43
    cos⁡3π8cos⁡3π8+sin⁡3π8sin⁡3π8=\cos^3\frac{\pi}{8}\cos\frac{3\pi}{8} + \sin^3\frac{\pi}{8}\sin\frac{3\pi}{8} =

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  44. 44
    If a and b are two unit vectors such that 5a⃗+4b⃗5\vec{a} + 4\vec{b} and a⃗−2b⃗\vec{a} - 2\vec{b} are perpendicular to each other, then the angle between a and b is

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  45. 45
    The value of ∫cos⁡3xsin⁡2x+sin⁡x dx\int\frac{\cos^3 x}{\sin^2 x + \sin x}\,dx is

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  46. 46
    Let P≡(−5,0)P \equiv (-5,0), Q≡(0,0)Q \equiv (0,0) and R≡(2,23)R \equiv (2,2\sqrt{3}) be three points. Then the equation of the bisector of the angle PQR is

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  47. 47
    cos⁡(sin⁡−135+cos⁡−11213)=\cos\left(\sin^{-1}\frac{3}{5} + \cos^{-1}\frac{12}{13}\right) =

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  48. 48
    If x∈[−1,1]x \in [-1,1], then the value of ∫esin⁡−1x(x+1−x2)1−x2 dx\int\frac{e^{\sin^{-1}x}(x + \sqrt{1-x^2})}{\sqrt{1-x^2}}\,dx is

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  49. 49
    The probability that a person who undergoes a bypass surgery will recover is 0.6. The probability that of the six patients who undergo similar operations, half of them will recover is

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  50. 50
    The distance 's' in meters covered by a body in t seconds is given by s=3t2−8t+5s = 3t^2 - 8t + 5. The body will stop after

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