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MHT-CET 2023 (3 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
    If ∫dxcos⁡3x2sin⁡2x=(tan⁡x)A+C(tan⁡x)B+K\int\frac{dx}{\cos^3 x\sqrt{2\sin2x}}=(\tan x)^A+\frac{C}{(\tan x)^B}+K, where K is a constant of integration, then the value of 5(A+B+C)5(A+B+C) is equal to

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  2. 2
    ∫2x2−1(x2+4)(x2−3) dx=\int\frac{2x^2-1}{(x^2+4)(x^2-3)}\,dx=

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  3. 3
    Let f(x)=1−tan⁡x4x−π, x≠π4, x∈ ⁣[0,π2]f(x)=\frac{1-\tan x}{4x-\pi},\ x\neq\frac{\pi}{4},\ x\in\!\left[0,\frac{\pi}{2}\right]. If f(x)f(x) is continuous in [0,π2]\left[0,\frac{\pi}{2}\right], then f ⁣(π4)f\!\left(\frac{\pi}{4}\right) is

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  4. 4
    The number of possible distinct straight lines passing through (2,3)(2,3) and forming a triangle with coordinate axes whose area is 12 sq. units are

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  5. 5
    The differential equation [1+(dydx)2]⋅(d2ydx2)1/2=kx\left[1+\left(\frac{dy}{dx}\right)^2\right]\cdot\left(\frac{d^2y}{dx^2}\right)^{1/2}=kx is of

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  6. 6
    A triangular park is enclosed on two sides by a fence and on the third side by a straight river bank. The two sides having fence are of same length xx. The maximum area (in sq. units) enclosed by the park is

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  7. 7
    If pp and qq are statements, then which of the following is a contingency.

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  8. 8
    If ∫0π/3tan⁡θ2ksec⁡θ dθ=1−12, (k>0)\int_0^{\pi/3}\frac{\tan\theta}{\sqrt{2k\sec\theta}}\,d\theta=1-\frac{1}{\sqrt{2}},\ (k>0), then the value of kk is

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  9. 9
    A random variable X has the following probability distribution. Then value of k is___ and P(3<x≤6)P(3<x\leq6) has the value. \begin{array}{|c|c|c|c|c|c|c|c|c|c|}\hline X=x & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8\\ \hline P(x) & k & 2k & 3k & 4k & 4k & 3k & 2k & k & k\\ \hline\end{array}

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  10. 10
    If y=log⁡e ⁣(x5⋅(3x−4)4/3(x+5)4/3)y=\log_e\!\left(\dfrac{x^{5}\cdot(3x-4)^{4/3}}{(x+5)^{4/3}}\right), then dydx\dfrac{dy}{dx} is equal to

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  11. 11
    Let f be a twice differentiable function such that f′′(x)=−f(x)f''(x)=-f(x), f′(x)=g(x)f'(x)=g(x) and h(x)=[f(x)]2+[g(x)]2h(x)=[f(x)]^2+[g(x)]^2. If h(5)=1h(5)=1, then h(10)h(10) is

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  12. 12
    The area (in sq. units) of the parallelogram whose diagonals are along the vectors 8i^−6j^8\hat{i}-6\hat{j} and 3i^+4j^−12k^3\hat{i}+4\hat{j}-12\hat{k} is

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  13. 13
    The line L given by x5+yb=1\frac{x}{5}+\frac{y}{b}=1 passes through the point (13,32)(13,32). The line K is parallel to L and has the equation xc+y3=1\frac{x}{c}+\frac{y}{3}=1. Then the distance between L and K is

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  14. 14
    If ∣a⃗∣=27|\vec{a}|=\sqrt{27}, ∣b⃗∣=7|\vec{b}|=7 and ∣a⃗×b⃗∣=35|\vec{a}\times\vec{b}|=35, then a⃗⋅b⃗\vec{a}\cdot\vec{b} is equal to

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  15. 15
    The number of real solutions of tan⁡−1 ⁣x(x+1)+sin⁡−1 ⁣x2+x+1=π2\tan^{-1}\!\sqrt{x(x+1)}+\sin^{-1}\!\sqrt{x^2+x+1}=\frac{\pi}{2} is

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  16. 16
    Let S={x∈(−π,π)∣x≠0,±π2}S=\{x\in(-\pi,\pi)\mid x\neq0,\pm\frac{\pi}{2}\}. The sum of all distinct solutions of 3sec⁡x+cosec x+2(tan⁡x−cot⁡x)=0\sqrt{3}\sec x+\text{cosec}\,x+2(\tan x-\cot x)=0 in S is equal to

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  17. 17
    The number of four-letter words that can be formed using letters of the word BARRACK is

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  18. 18
    A stone is dropped into a quiet lake and waves move in circles at speed of 8 cm/sec. At the instant when the radius of the circular wave is 12 cm, how fast is the enclosed area increasing?

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  19. 19
    The area (in sq. units) of the region bounded by the curve x2=4yx^2=4y and the straight line x=4y−2x=4y-2 is

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  20. 20
    If the half-life of substance is 5 years, then the total amount of the substance left after 15 years, when initial amount is 64 gms is

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  21. 21
    Equation of the plane containing the straight line x3=y2=z4\frac{x}{3}=\frac{y}{2}=\frac{z}{4} and perpendicular to the plane containing the straight lines x4=y3=z2\frac{x}{4}=\frac{y}{3}=\frac{z}{2} and x2=y−4=z3\frac{x}{2}=\frac{y}{-4}=\frac{z}{3} is

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  22. 22
    The number of roots of the equation (81)sin⁡2x+(81)cos⁡2x=30(81)^{\sin^2 x}+(81)^{\cos^2 x}=30 in the interval [0,π][0,\pi] is equal to

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  23. 23
    If ∫(2x+4)x−1 dx=a(x−1)5/2+b(x−1)3/2+c\int(2x+4)\sqrt{x-1}\,dx=a(x-1)^{5/2}+b(x-1)^{3/2}+c where cc is a constant of integration, then the value of (2a+b)(2a+b) is

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  24. 24
    Let X∼B ⁣(6,12)X\sim B\!\left(6,\frac{1}{2}\right), then P[∣x−4∣≤2]P[|x-4|\leq2] is

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  25. 25
    The mean of nn observations is xˉ\bar{x}. If three observations n+1,n−1,2n−1n+1, n-1, 2n-1 are added such that mean remains same, then value of nn is

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  26. 26
    If y=sec⁡(tan⁡−1x)y=\sec(\tan^{-1}x), then dydx\frac{dy}{dx} at x=1x=1 is equal to

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  27. 27
    The equation of the concentric circle, with circle C1C_1 having equation x2+y2−6x−4y−12=0x^2+y^2-6x-4y-12=0 and having double the area of C1C_1, is

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  28. 28
    Let α(a)\alpha(a) and β(a)\beta(a) be the roots of the equation (1+a3−1)x2+(1+a−1)x+(1+a6−1)=0(\sqrt[3]{1+a}-1)x^2+(\sqrt{1+a}-1)x+(\sqrt[6]{1+a}-1)=0 where a>−1a>-1. Then lim⁡a→0+α(a)\lim_{a\to0^+}\alpha(a) and lim⁡a→0+β(a)\lim_{a\to0^+}\beta(a) respectively are

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  29. 29
    Let 2sin⁡2x+3sin⁡x−2>02\sin^2 x+3\sin x-2>0 and x2−x−2<0x^2-x-2<0. (x is measured in radians). Then x lies in the interval

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  30. 30
    If A≡(1,−1,0), B≡(0,1,−1)A\equiv(1,-1,0),\ B\equiv(0,1,-1) and C≡(−1,0,1)C\equiv(-1,0,1), then the unit vector d^\hat{d} such that a⃗\vec{a} and d^\hat{d} are perpendicular and b⃗,c⃗,d^\vec{b},\vec{c},\hat{d} are coplanar is

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  31. 31
    A bullet is shot horizontally and its distance S cm at time t second is given by S=1200t−15t2S=1200t-15t^2, then the distance covered by the bullet when it comes to rest is

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  32. 32
    If ∣z∣=1|z|=1 and w=z−1z+1w=\frac{z-1}{z+1} (where z≠−1z\neq-1), then Re(w)\text{Re}(w) is

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  33. 33
    If g(x)=x2+x−1g(x)=x^2+x-1 and (g∘f)(x)=4x2−10x+5(g\circ f)(x)=4x^2-10x+5, then f(2)f(2) is equal to

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  34. 34
    The shaded area in the figure below is the solution set for a certain linear programming problem, then the linear constraints are given by

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  35. 35
    The equation of the 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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  36. 36
    If tan⁡x=34\tan x=\frac{3}{4} and π<x<3π2\pi<x<\frac{3\pi}{2}, then cos⁡x2=\cos\frac{x}{2}=

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  37. 37
    The value of m, such that x−41=y−21=2z−m3\frac{x-4}{1}=\frac{y-2}{1}=\frac{2z-m}{3} lies in the plane 2x−5y+2z=72x-5y+2z=7, is

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  38. 38
    The image of the line x−13=y−31=z−4−5\frac{x-1}{3}=\frac{y-3}{1}=\frac{z-4}{-5} in the plane 2x−y+z+3=02x-y+z+3=0 is the line

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  39. 39
    Let the vectors a⃗,b⃗,c⃗\vec{a},\vec{b},\vec{c} be such that ∣a⃗∣=2,∣b⃗∣=4,∣c⃗∣=4|\vec{a}|=2,|\vec{b}|=4,|\vec{c}|=4. If the projection of b⃗\vec{b} on a⃗\vec{a} equals the projection of c⃗\vec{c} on a⃗\vec{a} and b⃗\vec{b} is perpendicular to c⃗\vec{c}, then the value of ∣a⃗+b⃗−c⃗∣|\vec{a}+\vec{b}-\vec{c}| is equal to

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  40. 40
    A person throws an unbiased die. If the number shown is even, he gains an amount equal to the number shown. If the number is odd, he loses an amount equal to the number shown. Then his expectation is Rs.

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  41. 41
    Let a⃗=2i^+j^−2k^\vec{a}=2\hat{i}+\hat{j}-2\hat{k} and b⃗=i^+j^\vec{b}=\hat{i}+\hat{j}. If c⃗\vec{c} is a vector such that a⃗⋅c⃗=∣c⃗∣, ∣c⃗−a⃗∣=22\vec{a}\cdot\vec{c}=|\vec{c}|,\ |\vec{c}-\vec{a}|=2\sqrt{2} and the angle between (a⃗×b⃗)(\vec{a}\times\vec{b}) and c⃗\vec{c} is 30∘30^\circ, then ∣(a⃗×b⃗)×c⃗∣|(\vec{a}\times\vec{b})\times\vec{c}| is equal to

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  42. 42
    If θ\theta and α\alpha are not odd multiples of π2\frac{\pi}{2}, then tan⁡θ=tan⁡α\tan\theta=\tan\alpha implies principal solution is

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  43. 43
    The approximate value of cos⁡30∘30′\cos30^\circ 30' is, given that 1∘=0.01751^\circ=0.0175 rad and cos⁡30∘=0.8660\cos30^\circ=0.8660

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  44. 44
    Two cards are drawn successively with replacement from a pack of 52 cards. Let X denote the number of jacks obtained. Then P(X=1)+P(X=2)P(X=1)+P(X=2) equals

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  45. 45
    The value of ∫(x−1)ex(x+1)3 dx\int\frac{(x-1)e^x}{(x+1)^3}\,dx is equal to

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  46. 46
    Let P(2,3,6)P(2,3,6) be a point in space and Q be a point on the line r⃗=(i^−j^+2k^)+μ(−3i^+j^+5k^)\vec{r}=(\hat{i}-\hat{j}+2\hat{k})+\mu(-3\hat{i}+\hat{j}+5\hat{k}). Then the value of μ\mu for which vector PQ is parallel to the plane x−4y+4z=1x-4y+4z=1 is

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  47. 47
    Consider the following statements: p: the switch S1 is closed. q: the switch S2 is closed. r: the switch S3 is closed. Then the switching circuit represented by the statement (p∧q)∨(∼p∧(∼q∨p∨r))(p\wedge q)\vee(\sim p\wedge(\sim q\vee p\vee r)) is

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  48. 48
    Let a⃗,b⃗,c⃗\vec{a},\vec{b},\vec{c} be three non-coplanar vectors and p⃗,q⃗,r⃗\vec{p},\vec{q},\vec{r} defined by p⃗=b⃗×c⃗[a⃗ b⃗ c⃗], q⃗=c⃗×a⃗[a⃗ b⃗ c⃗], r⃗=a⃗×b⃗[a⃗ b⃗ c⃗]\vec{p}=\frac{\vec{b}\times\vec{c}}{[\vec{a}\ \vec{b}\ \vec{c}]},\ \vec{q}=\frac{\vec{c}\times\vec{a}}{[\vec{a}\ \vec{b}\ \vec{c}]},\ \vec{r}=\frac{\vec{a}\times\vec{b}}{[\vec{a}\ \vec{b}\ \vec{c}]}, then the value of (a⃗+b⃗)⋅p⃗+(b⃗+c⃗)⋅q⃗+(c⃗+a⃗)⋅r⃗(\vec{a}+\vec{b})\cdot\vec{p}+(\vec{b}+\vec{c})\cdot\vec{q}+(\vec{c}+\vec{a})\cdot\vec{r} is equal to

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  49. 49
    If A=(2−243)A=\begin{pmatrix}2&-2\\4&3\end{pmatrix}, then A−1=A^{-1}=

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  50. 50
    The differential equation of family of circles, whose centres are on the X-axis and also touch the Y-axis is

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