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MHT-CET 2023 (2 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
    Two tangents drawn from P(1,7)P(1,7) to the circle x2+y2=25x^2 + y^2 = 25, touch the circle at Q and R respectively. The area of the quadrilateral PQOR is

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  2. 2
    Let X=(abc)X = \begin{pmatrix}a\\b\\c\end{pmatrix}, A=(1−12201321)A = \begin{pmatrix}1&-1&2\\2&0&1\\3&2&1\end{pmatrix} and B=(314)B = \begin{pmatrix}3\\1\\4\end{pmatrix}. If AX=BAX = B, then the value of 2a−3b+4c2a - 3b + 4c will be

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  3. 3
    Let in a Binomial distribution, consisting of 5 independent trials, probabilities of exactly 1 and 2 successes be 0.4096 and 0.2048 respectively. Then the probability of getting exactly 3 successes is equal to

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  4. 4
    The unit vector which is orthogonal to the vector 5i^+2j^+6k^5\hat{i}+2\hat{j}+6\hat{k} and is coplanar with the vectors 2i^+j^+k^2\hat{i}+\hat{j}+\hat{k} and i^−j^+k^\hat{i}-\hat{j}+\hat{k} is

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  5. 5
    The probability distribution of a random variable X is given by: X = 0,1,2,3,4 with P(X) = 0.4, 0.3, 0.1, 0.1, 0.1. Then the variance of X is

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  6. 6
    If ∫x+12x−1 dx=f(x)2x−1+c\int \frac{x+1}{\sqrt{2x-1}}\,dx = f(x)\sqrt{2x-1} + c, then f(x)f(x) is equal to

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  7. 7
    If α+β+γ=π\alpha+\beta+\gamma = \pi, then the expression sin⁡2α+sin⁡2β−sin⁡2γ\sin^2\alpha + \sin^2\beta - \sin^2\gamma has the value

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  8. 8
    Let A⃗=2i^+k^\vec{A} = 2\hat{i}+\hat{k}, B⃗=i^+j^+k^\vec{B} = \hat{i}+\hat{j}+\hat{k} and C⃗=4i^−3j^+7k^\vec{C} = 4\hat{i}-3\hat{j}+7\hat{k}. If a vector R⃗\vec{R} satisfies R⃗×B⃗=C⃗×B⃗\vec{R}\times\vec{B} = \vec{C}\times\vec{B} and R⃗⋅A⃗=0\vec{R}\cdot\vec{A} = 0, then R⃗\vec{R} is given by

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  9. 9
    Distance between the parallel lines x3=y−1−2=z1\frac{x}{3} = \frac{y-1}{-2} = \frac{z}{1} and x+43=y−3−2=z+21\frac{x+4}{3} = \frac{y-3}{-2} = \frac{z+2}{1} is

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  10. 10
    If f(x)=log⁡x2(log⁡ex)f(x) = \log_{x^2}(\log_e x), then f′(x)f'(x) at x=ex = e is

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  11. 11
    A body cools from 100∘C100^\circ\text{C} to 60∘C60^\circ\text{C} in 15 minutes. If the surrounding temperature is 20∘C20^\circ\text{C}, the temperature of the body after cooling for one hour is

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  12. 12
    The value of k, for which the function f(x)={(45)tan⁡4xtan⁡5x,0<x<π2k+25,x=π2f(x)=\begin{cases}\left(\dfrac{4}{5}\right)^{\frac{\tan4x}{\tan5x}}, & 0<x<\frac{\pi}{2}\\ k+\dfrac{2}{5}, & x=\frac{\pi}{2}\end{cases} is continuous at x=π2x=\frac{\pi}{2}, is

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  13. 13
    The co-ordinates of a point on the curve y=xlog⁡xy = x\log x at which the normal is parallel to the line 2x−2y=32x - 2y = 3 are

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  14. 14
    The value of I=∫log⁡e2log⁡e3xsin⁡(x2)sin⁡(x2)+sin⁡(log⁡e6−x2) dxI = \displaystyle\int_{\sqrt{\log_e 2}}^{\sqrt{\log_e 3}} \frac{x\sin(x^2)}{\sin(x^2)+\sin(\log_e 6 - x^2)}\,dx is

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  15. 15
    The value of I=∫dxx2(x4+1)3/4I = \int \frac{dx}{x^2(x^4+1)^{3/4}} is

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  16. 16
    Three houses are available in a locality. Three persons apply for the houses. Each applies for one house without consulting others. The probability that all three persons apply for the same house is

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  17. 17
    The value of C for which the Mean Value Theorem holds for f(x)=log⁡exf(x) = \log_e x on [1,3][1,3] is

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  18. 18
    The shaded region in the following figure is the solution set of the inequations

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  19. 19
    The diagonals of a parallelogram ABCD are along the lines x+3y=4x+3y=4 and 6x−2y=76x-2y=7. Then ABCD must be a

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  20. 20
    If x=sec⁡θ−cos⁡θ,  y=sec⁡10θ−cos⁡10θx = \sec\theta - \cos\theta,\; y = \sec^{10}\theta - \cos^{10}\theta and (x2+4)(dydx)2=k(y2+4)\left(x^2+4\right)\left(\frac{dy}{dx}\right)^2 = k(y^2+4), then the value of k is

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  21. 21
    If (5+ex)2+ydydx+ex=0(5+e^x)^{2+y}\frac{dy}{dx} + e^x = 0 satisfying y(0)=1y(0)=1, then a value of y(log⁡13)y(\log 13) is

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  22. 22
    The equation of the plane passing through the midpoint of the line segment joining P(1,2,5)P(1,2,5) and Q(3,4,3)Q(3,4,3) and perpendicular to it, is

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  23. 23
    If C is a non-zero scalar and A, B are non-zero vectors with A⃗⊥B⃗\vec{A}\perp\vec{B}, A⃗⋅X⃗=C\vec{A}\cdot\vec{X}=C and A⃗×X⃗=B⃗\vec{A}\times\vec{X}=\vec{B}, then X⃗\vec{X} is given by

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  24. 24
    If the equation 7x2−14xy+py2−12x+qy−4=07x^2-14xy+py^2-12x+qy-4=0 represents a pair of parallel lines, then the value of p2+q2−pqp^2+q^2-pq is

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  25. 25
    If P(x,y)P(x,y) denotes z=x+iyz=x+iy and ∣z−1z+2i∣=1\left|\frac{z-1}{z+2i}\right|=1, then the locus of P is

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  26. 26
    The general solution of 23cos⁡2θ=sin⁡θ2\sqrt{3}\cos^2\theta = \sin\theta is

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  27. 27
    If (2+sin⁡x)dydx+(y+1)cos⁡x=0(2+\sin x)\frac{dy}{dx} + (y+1)\cos x = 0 and y(0)=1y(0)=1, then y ⁣(π2)y\!\left(\frac{\pi}{2}\right) is equal to

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  28. 28
    The area of the triangle whose vertices are A(1,−1,2)A(1,-1,2), B(2,1,−1)B(2,1,-1) and C(3,−1,2)C(3,-1,2) is

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  29. 29
    The maximum value of the function f(x)=3x3−18x2+27x−40f(x) = 3x^3 - 18x^2 + 27x - 40 on the set S={x∈R:x2+30≤11x}S = \{x\in\mathbb{R} : x^2+30 \leq 11x\} is

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  30. 30
    tan⁡ ⁣(π4+12cos⁡−1ab)+tan⁡ ⁣(π4−12cos⁡−1ab)\tan\!\left(\frac{\pi}{4}+\frac{1}{2}\cos^{-1}\frac{a}{b}\right)+\tan\!\left(\frac{\pi}{4}-\frac{1}{2}\cos^{-1}\frac{a}{b}\right) is

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  31. 31
    Consider three observations a,b,ca, b, c such that b=a+cb = a+c. If the standard deviation of a+2,b+2,c+2a+2, b+2, c+2 is dd, then

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  32. 32
    The area (in sq. units) bounded by the curves y=xy = \sqrt{x}, 2y−x+3=02y - x + 3 = 0, X-axis and lying in the first quadrant is

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  33. 33
    If the lengths of the sides of a triangle are 3,5,73, 5, 7, then the largest angle of the triangle is

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  34. 34
    If f(x)=sin⁡−1 ⁣(2⋅3x1+9x)f(x) = \sin^{-1}\!\left(\frac{2\cdot3^x}{1+9^x}\right), then f′ ⁣(12)f'\!\left(\frac{1}{2}\right) equals

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  35. 35
    ∫[f(x)g′′(x)−f′′(x)g(x)] dx\int [f(x)g''(x) - f''(x)g(x)]\,dx is equal to

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  36. 36
    The value of tan⁡ ⁣(2tan⁡−1 ⁣(5−1)/2)\tan\!\left(2\tan^{-1}\!\left(\sqrt{5}-1\right)/2\right) is

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  37. 37
    Number of different nine-digit numbers that can be formed from the digits in 223355888 by rearranging, so that the odd digits occupy even positions, is

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  38. 38
    If a⃗=i^+2j^+3k^\vec{a} = \hat{i}+2\hat{j}+3\hat{k} and b⃗=i^×(a⃗×i^)+j^×(a⃗×j^)+k^×(a⃗×k^)\vec{b} = \hat{i}\times(\vec{a}\times\hat{i})+\hat{j}\times(\vec{a}\times\hat{j})+\hat{k}\times(\vec{a}\times\hat{k}), then ∣b⃗∣|\vec{b}| is

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  39. 39
    The equation of normal to the curve x=θ+sin⁡θ,  y=1+cos⁡θx=\theta+\sin\theta,\; y=1+\cos\theta at θ=π2\theta=\frac{\pi}{2} is

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  40. 40
    The equation of the line through A(1,2,3)A(1,2,3) and perpendicular to the vectors 2i^+j^−k^2\hat{i}+\hat{j}-\hat{k} and i^+3j^+2k^\hat{i}+3\hat{j}+2\hat{k} is

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  41. 41
    ∫log⁡x3x dx\displaystyle\int \dfrac{\log\sqrt{x}}{3x}\,dx is equal to

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  42. 42
    The negation of contrapositive of the statement p→(¬q∧r)p\to(\neg q \wedge r) is

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  43. 43
    If F(x)=f ⁣(x22)+g ⁣(x22)F(x) = f\!\left(\frac{x^2}{2}\right)+g\!\left(\frac{x^2}{2}\right), where f′′(x)=−f(x)f''(x) = -f(x) and g(x)=f′(x)g(x) = f'(x), and F(5)=5F(5) = 5, then F(10)F(10) is equal to

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  44. 44
    Let P be the image of the point (3,1,7)(3,1,7) with respect to the plane x−y+z=3x-y+z=3. Then the equation of the plane passing through P and containing the straight line x1=y2=z1\frac{x}{1}=\frac{y}{2}=\frac{z}{1} is

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  45. 45
    Let f(θ)=sin⁡ ⁣(tan⁡−1 ⁣sin⁡θcos⁡2θ)f(\theta) = \sin\!\left(\tan^{-1}\!\frac{\sin\theta}{\sqrt{\cos2\theta}}\right) where −π4<θ<π4-\frac{\pi}{4}<\theta<\frac{\pi}{4}, then dd(tan⁡θ)(f(θ))\frac{d}{d(\tan\theta)}(f(\theta)) is

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  46. 46
    The approximate value of 0.0263\sqrt[3]{0.026} is

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  47. 47
    The incentre of the triangle whose vertices are P(0,3,0)P(0,3,0), Q(0,0,4)Q(0,0,4) and R(0,3,4)R(0,3,4) is

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  48. 48
    A random variable X has probability distribution: X=1,2,3,4 with P = 0.2, 0.4, 0.3, 0.1. The mean and variance of X are respectively

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  49. 49
    lim⁡x→0sin⁡(πcos⁡2x)x2\lim_{x\to0}\frac{\sin(\pi\cos^2 x)}{x^2} is equal to

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  50. 50
    Which one of the following is the pair of equivalent circuits? (i) through (v) as given

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