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MHT-CET 2023 (2 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
    The vector equation of a line whose Cartesian equations are y=2,4x−3z+5=0y = 2, 4x - 3z + 5 = 0 is

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  2. 2
    If 3 books on Physics, 2 books on Chemistry and 4 books on Mathematics are to be arranged on a shelf so that all the Physics books are together and all the Mathematics books are together, then the number of such arrangements is

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  3. 3
    The value of lim⁡x→0x∣x∣+x2\lim_{x \to 0} \dfrac{x}{|x| + x^2} is

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  4. 4
    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}. Let c⃗\vec{c} be a vector such that ∣c⃗−a⃗∣=3|\vec{c} - \vec{a}| = 3 and ∣(a⃗×b⃗)×c⃗∣=3|(\vec{a} \times \vec{b}) \times \vec{c}| = 3 and the angle between c⃗\vec{c} and a⃗×b⃗\vec{a} \times \vec{b} is 30∘30^\circ, then a⃗⋅c⃗\vec{a} \cdot \vec{c} is equal to

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  5. 5
    The value of cos⁡(2cos⁡−1x+sin⁡−1x)\cos\left(2\cos^{-1}x + \sin^{-1}x\right) at x=15x = \frac{1}{5}, where 0≤cos⁡−1x≤π0 \le \cos^{-1}x \le \pi and −π2≤sin⁡−1x≤π2-\frac{\pi}{2} \le \sin^{-1}x \le \frac{\pi}{2}, is

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  6. 6
    Considering only the principal values of inverse function, the set A={x≥0 | tan⁡−1(2x)+tan⁡−1(3x)=π4}A = \left\{x \ge 0\,\middle|\, \tan^{-1}(2x) + \tan^{-1}(3x) = \frac{\pi}{4}\right\}

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  7. 7
    The slopes of the lines given by x2+2hxy+2y2=0x^2 + 2hxy + 2y^2 = 0 are in the ratio 1:21:2, then hh is

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  8. 8
    If y=e4x⋅(x−4)3/4(x+3)3/4y = e^{4x} \cdot \dfrac{(x-4)^{3/4}}{(x+3)^{3/4}} then dydx=\dfrac{dy}{dx} =

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  9. 9
    ∫33x⋅3x dx=\displaystyle\int 3^{3^x} \cdot 3^x\,dx =

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  10. 10
    If f(x)=1+cos⁡(πx)π(1−x)2f(x) = \dfrac{1 + \cos(\pi x)}{\pi(1-x)^2} for x≠1x \ne 1 is continuous at x=1x = 1, then f(1)f(1) is equal to

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  11. 11
    The length of the longest interval, in which the function 3sin⁡x−4sin⁡3x3\sin x - 4\sin^3 x is increasing, is

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  12. 12
    The scalar a⃗⋅[(b⃗+c⃗)×(a⃗+b⃗+c⃗)]\vec{a} \cdot [(\vec{b} + \vec{c}) \times (\vec{a} + \vec{b} + \vec{c})] equals

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  13. 13
    The volume of parallelopiped formed by vectors i^+mj^+k^\hat{i} + m\hat{j} + \hat{k}, j^+mk^\hat{j} + m\hat{k} and mi^+k^m\hat{i} + \hat{k} becomes minimum when mm is

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  14. 14
    If the vectors a⃗=i^−j^+2k^\vec{a} = \hat{i} - \hat{j} + 2\hat{k}, b⃗=2i^+4j^+k^\vec{b} = 2\hat{i} + 4\hat{j} + \hat{k} and c⃗=mi^+j^+nk^\vec{c} = m\hat{i} + \hat{j} + n\hat{k} are mutually perpendicular, then (m,n)(m, n) is

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  15. 15
    ∫log⁡((1+x)1+x) dx=\displaystyle\int \log\left((1+x)^{1+x}\right)\,dx =

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  16. 16
    ∫x+2(x+4)2 ex dx=\displaystyle\int \dfrac{x+2}{(x+4)^2}\,e^x\,dx =

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  17. 17
    In △ABC\triangle ABC, with usual notations, if 1b+c+1c+a=3a+b+c\dfrac{1}{b+c} + \dfrac{1}{c+a} = \dfrac{3}{a+b+c}, then m∠Cm\angle C is equal to

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  18. 18
    If a>0a>0 and z=(1+i)2a−iz=\dfrac{(1+i)^2}{a-i}, i=−1i=\sqrt{-1}, has magnitude 25\sqrt{\dfrac{2}{5}}, then zˉ\bar{z} is equal to

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  19. 19
    A bag contains 4 Red and 6 Black balls. A ball is drawn at random from the bag, its colour is observed and this ball along with 3 additional balls of the same colour are returned to the bag. If now a ball is drawn at random from the bag, then the probability that this drawn ball is red is

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  20. 20
    Let KK be the set of all real values of xx, where the function f(x)=sin⁡∣x∣−∣x∣+2(x−π)cos⁡∣x∣f(x) = \sin|x| - |x| + 2(x-\pi)\cos|x| is not differentiable. Then the set KK is

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  21. 21
    Let ff and gg be continuous functions on [0,a][0,a] such that f(x)=f(a−x)f(x) = f(a-x) and g(x)+g(a−x)=4g(x) + g(a-x) = 4, then ∫0af(x)g(x) dx\displaystyle\int_0^a f(x)g(x)\,dx is equal to

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  22. 22
    The principal solutions of the equation 3sec⁡x+2=0\sqrt{3}\sec x + 2 = 0 are

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  23. 23
    The number of real solutions of tan⁡−1x(x+1)+sin⁡−1x2+x+1=π2\tan^{-1}\sqrt{x(x+1)} + \sin^{-1}\sqrt{x^2+x+1} = \dfrac{\pi}{2} is

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  24. 24
    If a⃗=(2i^+2j^+3k^)\vec{a} = (2\hat{i} + 2\hat{j} + 3\hat{k}), b⃗=(−i^+2j^+k^)\vec{b} = (-\hat{i} + 2\hat{j} + \hat{k}) and c⃗=(3i^+j^)\vec{c} = (3\hat{i} + \hat{j}) such that (a⃗+λb⃗)(\vec{a} + \lambda\vec{b}) is perpendicular to c⃗\vec{c}, then the value of λ\lambda is

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  25. 25
    The solution of the differential equation dydx=(x−y)2\dfrac{dy}{dx} = (x-y)^2 when y(1)=1y(1) = 1 is

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  26. 26
    If x0x_0 is the point of local minima of f(x)=a⃗⋅(b⃗×c⃗)f(x) = \vec{a}\cdot(\vec{b}\times\vec{c}) where a⃗=xi^−2j^+3k^\vec{a} = x\hat{i} - 2\hat{j} + 3\hat{k}, b⃗=−2i^+xj^−k^\vec{b} = -2\hat{i} + x\hat{j} - \hat{k}, c⃗=7i^−2j^+xk^\vec{c} = 7\hat{i} - 2\hat{j} + x\hat{k}, then value of a⃗⋅b⃗\vec{a}\cdot\vec{b} at x=x0x = x_0 is

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  27. 27
    a⃗\vec{a}, b⃗\vec{b}, and c⃗\vec{c} are three unit vectors such that a⃗×(b⃗×c⃗)=32(b⃗+c⃗)\vec{a} \times (\vec{b} \times \vec{c}) = \dfrac{\sqrt{3}}{2}(\vec{b} + \vec{c}). If b⃗\vec{b} is not parallel to c⃗\vec{c}, then the angle between a⃗\vec{a} and b⃗\vec{b} is

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  28. 28
    For a suitable chosen real constant aa, let a function f:R∖{−a}→Rf: \mathbb{R} \setminus \{-a\} \to \mathbb{R} be defined by f(x)=a−xa+xf(x) = \dfrac{a-x}{a+x}. Further suppose that for any real number x≠−ax \ne -a and f(x)≠−af(x) \ne -a, (f∘f)(x)=x(f \circ f)(x) = x. Then f(−15)f\left(-\dfrac{1}{5}\right) is equal to

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  29. 29
    If the statement p∨∼(q∧r)p \vee \sim(q \wedge r) is false, then the truth values of pp, qq and rr are respectively

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  30. 30
    If (mi,1mi)\left(m_i, \dfrac{1}{m_i}\right), mi>0m_i > 0, i=1,2,3,4i = 1, 2, 3, 4 are four distinct points on a circle, then the product m1m2m3m4m_1 m_2 m_3 m_4 is equal to

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  31. 31
    If two lines x+(a−1)y=1x + (a-1)y = 1 and 2x+a2y=12x + a^2y = 1 (a∈R∖{0,1}a \in \mathbb{R} \setminus \{0,1\}) are perpendicular, then the distance of their point of intersection from the origin is

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  32. 32
    If xdydx=y(log⁡y−log⁡x+1)x\dfrac{dy}{dx} = y(\log y - \log x + 1), then general solution of this equation is

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  33. 33
    A spherical metal ball at 80∘80^\circC cools in 5 minutes to 60∘60^\circC, in surrounding temperature of 20∘20^\circC, then the temperature of the ball after 20 minutes is approximately

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  34. 34
    If a discrete random variable X takes values 0,1,2,3,…0, 1, 2, 3, \ldots with probability P(X=x)=k(x+1)5−xP(X=x) = k(x+1)5^{-x}, where kk is a constant, then P(X=0)P(X=0) is

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  35. 35
    The Cartesian equation of the plane, passing through the points (3,1,1)(3,1,1), (1,2,3)(1,2,3) and (−1,4,2)(-1,4,2), is

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  36. 36
    The equation of the line passing through the point (−1,3,−2)(-1, 3, -2) and perpendicular to each of the lines x1=y2=z3\dfrac{x}{1} = \dfrac{y}{2} = \dfrac{z}{3} and x+2−3=y−12=z+15\dfrac{x+2}{-3} = \dfrac{y-1}{2} = \dfrac{z+1}{5}, is

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  37. 37
    If y=asin⁡x+bcos⁡xy = a\sin x + b\cos x (where aa and bb are constants), then y2+(dydx)2y^2 + \left(\dfrac{dy}{dx}\right)^2 is

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  38. 38
    If Rolle's theorem holds for the function f(x)=x3+bx2+ax+5f(x) = x^3 + bx^2 + ax + 5 on [1,3][1,3] with c=2+13c = 2 + \dfrac{1}{\sqrt{3}}, then the values of aa and bb respectively are

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  39. 39
    Ten bulbs are drawn successively, with replacement, from a lot containing 10% defective bulbs, then the probability that there is at least one defective bulb, is

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  40. 40
    If statement I: If the work is not finished on time, the contractor is in trouble. Statement II: Either the work is finished on time or the contractor is in trouble. Then

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  41. 41
    The value of sin⁡(2cos⁡−1(−35))\sin\left(2\cos^{-1}\left(-\dfrac{3}{5}\right)\right) is

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  42. 42
    If y=1−sin⁡−1x1+sin⁡−1xy = \sqrt{\dfrac{1 - \sin^{-1}x}{1 + \sin^{-1}x}}, then dydx\dfrac{dy}{dx} at x=0x = 0 is

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  43. 43
    The point, at which the maximum value of 10x+6y10x + 6y subject to the constraints x+y≤12x+y \le 12, 2x+y≤202x+y \le 20, x≥0x \ge 0, y≥0y \ge 0 occurs, is

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  44. 44
    If the line x−32=y+2−1=z+43\dfrac{x-3}{2} = \dfrac{y+2}{-1} = \dfrac{z+4}{3} lies in the plane lx+my−z=9lx + my - z = 9, then l2+m2l^2 + m^2 is

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  45. 45
    The mean of 100 observations is 50 and their standard deviation is 5, then the sum of all squares of all the observations is

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  46. 46
    The area of the region bounded by hyperbola x2−y2=9x^2 - y^2 = 9 and its latus rectum is

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  47. 47
    ∫dx3−2cos⁡2x=tan⁡−1(f(x))5+c\displaystyle\int \dfrac{dx}{3 - 2\cos 2x} = \dfrac{\tan^{-1}(f(x))}{\sqrt{5}} + c, (where cc is constant of integration), then f(π/4)f(\pi/4) has the value

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  48. 48
    The normal to the curve y(x−2)(x−3)=x+6y(x-2)(x-3) = x+6 at the point, where the curve intersects the Y-axis, passes through the point

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  49. 49
    For all real xx, the vectors Cxi^−6j^−3k^Cx\hat{i} - 6\hat{j} - 3\hat{k} and xi^+2j^+2Cxk^x\hat{i} + 2\hat{j} + 2Cx\hat{k} make an obtuse angle with each other, then the value of CC can be in

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  50. 50
    If A=(3−1−42)A = \begin{pmatrix} 3 & -1 \\ -4 & 2 \end{pmatrix}, then A−1A^{-1} is

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