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MHT-CET 2024 (14 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
    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∣\vec{a} \cdot \vec{c} = |\vec{c}|, ∣c⃗−a⃗∣=22|\vec{c} - \vec{a}| = 2\sqrt{2} and the angle between a⃗×b⃗\vec{a} \times \vec{b} and c⃗\vec{c} is 2π3\frac{2\pi}{3}, then ∣(a⃗×b⃗)×c⃗∣|(\vec{a} \times \vec{b}) \times \vec{c}| =

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  2. 2
    The value of tan⁡−118+tan⁡−112+tan⁡−115\tan^{-1}\frac{1}{8} + \tan^{-1}\frac{1}{2} + \tan^{-1}\frac{1}{5} is

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  3. 3
    The differential equation representing the family of curves y2=2c(x+c)y^2 = 2c(x + \sqrt{c}), where cc is a positive parameter, is of

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  4. 4
    ∣a⃗∣=2,∣b⃗∣=3,∣c⃗∣=5|\vec{a}| = 2, |\vec{b}| = 3, |\vec{c}| = 5 and each angle between pairs is 60°60°, then the value of ∣a⃗+b⃗+c⃗∣|\vec{a} + \vec{b} + \vec{c}| is

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  5. 5
    The area bounded by the curve y=∣x−2∣y = |x-2|, x=1x = 1, x=3x = 3 and X-axis is

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  6. 6
    There are 6 positive and 8 negative numbers. Four numbers are chosen at random and multiplied. The probability that the product is a negative number is

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  7. 7
    If f(x)=2x−33x−4,x≠43f(x) = \frac{2x-3}{3x-4}, x \neq \frac{4}{3}, then f−1(x)f^{-1}(x) is

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  8. 8
    Let z∈Cz \in \mathbb{C} with Im(z)=10\text{Im}(z) = 10 and it satisfies 2z−n2z+n=2i−1\frac{2z-n}{2z+n} = 2i-1 for some natural number n, then

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  9. 9
    If f′(x)=sin⁡(log⁡x)f'(x) = \sin(\log x) and y=f ⁣(2x+33−2x)y = f\!\left(\frac{2x+3}{3-2x}\right), then dydx\frac{dy}{dx} at x=1x=1 is

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  10. 10
    Let u⃗,v⃗,w⃗\vec{u}, \vec{v}, \vec{w} be vectors with ∣u⃗∣=1,∣v⃗∣=2,∣w⃗∣=3|\vec{u}|=1, |\vec{v}|=2, |\vec{w}|=3. Projection of v⃗\vec{v} along u⃗\vec{u} equals projection of w⃗\vec{w} along u⃗\vec{u}, and v⃗⊥w⃗\vec{v} \perp \vec{w}. Then ∣u⃗−v⃗+w⃗∣|\vec{u} - \vec{v} + \vec{w}| equals

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  11. 11
    If I=∫dxsin⁡(x−a)sin⁡(x−b)I = \int \frac{dx}{\sin(x-a)\sin(x-b)}, then II is given by

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  12. 12
    If y=(x−sin⁡x)+(x−sin⁡x)+x−sin⁡x+⋯y = \sqrt{(x-\sin x) + \sqrt{(x-\sin x) + \sqrt{x - \sin x + \cdots}}}, then dydx=\frac{dy}{dx} =

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  13. 13
    The length of the perpendicular drawn from the point (1,2,3)(1,2,3) to the line x−63=y−72=z−7−2\frac{x-6}{3} = \frac{y-7}{2} = \frac{z-7}{-2} is

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  14. 14
    In a regular polygon, the number of diagonals are 54, then the number of sides of the polygon are

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  15. 15
    In triangle ABC, ∠A,∠B,∠C\angle A, \angle B, \angle C are in A.P. and lengths of two larger sides are 10 and 9 units, then the length of the third side is

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  16. 16
    If A=(1−12−1)A = \begin{pmatrix}1 & -1\\2 & -1\end{pmatrix}, B=(114−1)B = \begin{pmatrix}1 & 1\\4 & -1\end{pmatrix}, then (A+B)−1(A+B)^{-1} is

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  17. 17
    If f(a)=2,f′(a)=1,g(a)=−1,g′(a)=2f(a)=2, f'(a)=1, g(a)=-1, g'(a)=2, then as xx approaches aa, g(x)f(a)−g(a)f(x)x−a\frac{g(x)f(a) - g(a)f(x)}{x-a} approaches

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  18. 18
    A vector parallel to the line of intersection of the planes r⃗⋅(3i^−j^+k^)=1\vec{r}\cdot(3\hat{i}-\hat{j}+\hat{k})=1 and r⃗⋅(i^+4j^−2k^)=2\vec{r}\cdot(\hat{i}+4\hat{j}-2\hat{k})=2 is

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  19. 19
    If cos⁡−1x−cos⁡−1y3=α\cos^{-1}x - \cos^{-1}\frac{y}{3} = \alpha, where −1≤x≤1-1\leq x\leq1, −3≤y≤3-3\leq y\leq3, x≤y3x\leq\frac{y}{3}, then for all x,yx,y: 9x2−6xycos⁡α+y29x^2 - 6xy\cos\alpha + y^2 is equal to

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  20. 20
    Let a⃗=i^+2j^−k^\vec{a} = \hat{i}+2\hat{j}-\hat{k} and b⃗=i^+j^−k^\vec{b} = \hat{i}+\hat{j}-\hat{k}. If c⃗\vec{c} satisfies b⃗×c⃗=b⃗×a⃗\vec{b}\times\vec{c} = \vec{b}\times\vec{a} and c⃗⋅a⃗=0\vec{c}\cdot\vec{a} = 0, then c⃗⋅b⃗\vec{c}\cdot\vec{b} is

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  21. 21
    If x=esin⁡−1tx = e^{\sin^{-1}t} and y=ecos⁡−1ty = e^{\cos^{-1}t}, then d2ydx2\frac{d^2y}{dx^2} is

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  22. 22
    Let f(x)=5−∣x−2∣f(x) = 5 - |x-2| and g(x)=∣x+1∣g(x) = |x+1|, x∈Rx\in\mathbb{R}. If f(x)f(x) attains maximum at α\alpha and g(x)g(x) attains minimum at β\beta, then lim⁡x→−αβ(x−1)(x2−5x+6)x2−6x+8\lim_{x\to -\alpha\beta}\frac{(x-1)(x^2-5x+6)}{x^2-6x+8} is equal to

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  23. 23
    If the slope of the tangent of the curve at any point is equal to −y+e−x-y + e^{-x}, then the equation of the curve passing through origin is

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  24. 24
    The function f(x)=x3−6x2+9x+2f(x) = x^3 - 6x^2 + 9x + 2 has maximum value when xx is

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  25. 25
    If the statement p↔(q→p)p \leftrightarrow (q \to p) is false, then true statement/statement pattern is

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  26. 26
    The value of ∫ex(x2+4x+4)(x+4)2 dx\int \frac{e^x(x^2+4x+4)}{(x+4)^2}\,dx is

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  27. 27
    Let x0x_0 be 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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  28. 28
    The p.m.f. of a random variable X is P(x)=2xn(n+1),  x=1,2,…,nP(x) = \frac{2x}{n(n+1)},\; x=1,2,\ldots,n, then E(x)E(x) is

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  29. 29
    The centroid of the triangle formed by the lines x+3y=10x+3y=10 and 6x2+xy−y2=06x^2+xy-y^2=0 is

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  30. 30
    If y=4x−5y = 4x - 5 is a tangent to the curve y2=px3+qy^2 = px^3 + q at (2,3)(2,3), then p−qp - q is

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  31. 31
    A lot of 100 bulbs contains 10 defective bulbs. Five bulbs are selected at random and sent to retail store. Then the probability that the store will receive at most one defective bulb is

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  32. 32
    The shaded region in the following figure represents the solution set for a certain linear programming problem. Then linear constraints for this region are given by

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  33. 33
    The diagonal of a square is changing at the rate of 0.5 cm/sec. Then the rate of change of area when the area is 400 cm2^2 is equal to

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  34. 34
    If ∑r=150tan⁡−112r2=p\sum_{r=1}^{50} \tan^{-1}\frac{1}{2r^2} = p, then tan⁡p\tan p is

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  35. 35
    If a body cools from 80°C80°C to 50°C50°C in a room temperature of 25°C25°C in 30 minutes, then the temperature of the body after 1 hour is

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  36. 36
    If ∫dxcot⁡2x−1=1Alog⁡∣sec⁡2x+tan⁡2x∣−xB+c\int \frac{dx}{\cot^2 x - 1} = \frac{1}{A}\log|\sec 2x + \tan 2x| - \frac{x}{B} + c, then A+BA+B =

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  37. 37
    If the lines x−k2=y+13=z−14\frac{x-k}{2}=\frac{y+1}{3}=\frac{z-1}{4} and x−31=y−9/22=z1\frac{x-3}{1}=\frac{y-9/2}{2}=\frac{z}{1} intersect, then the value of kk is

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  38. 38
    The statement [p∧(q∨r)]∨[∼r∧∼q∧p][p\wedge(q\vee r)]\vee[\sim r\wedge\sim q\wedge p] is equivalent to

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  39. 39
    Given 0≤x≤120\leq x\leq\frac{1}{\sqrt{2}}, then the value of tan⁡[sin⁡−1(x2+1−x22)−sin⁡−1x]\tan\left[\sin^{-1}\left(\frac{x}{\sqrt{2}}+\frac{\sqrt{1-x^2}}{\sqrt{2}}\right)-\sin^{-1}x\right] is

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  40. 40
    Let P(x)P(x) be a polynomial of degree 2 with P(2)=−1,P′(2)=0,P′′(2)=2P(2)=-1, P'(2)=0, P''(2)=2, then P(1.001)P(1.001) is

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  41. 41
    If ∣a⃗∣=3,∣b⃗∣=5,b⃗⋅c⃗=10|\vec{a}|=3, |\vec{b}|=5, \vec{b}\cdot\vec{c}=10, angle between b⃗\vec{b} and c⃗\vec{c} is π3\frac{\pi}{3}, a⃗\vec{a} is perpendicular to b⃗×c⃗\vec{b}\times\vec{c}. Then the value of ∣a⃗×(b⃗×c⃗)∣|\vec{a}\times(\vec{b}\times\vec{c})| is

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  42. 42
    If In=∫0π/4tan⁡nθ dθI_n = \int_0^{\pi/4} \tan^n\theta\,d\theta, then I12+I10I_{12} + I_{10} =

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  43. 43
    Let P(−3,0),Q(0,0)P(-3,0), Q(0,0) and R(3,33)R(3,3\sqrt{3}) be three points. Then the equation of the bisector of the angle PQR is

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  44. 44
    A fair die with numbers 1 to 6 on their faces is thrown. Let X denote the number of factors of the number on the uppermost face, then the probability distribution of X is

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  45. 45
    If ∫x21−x dx=p1−x(3x2+4x+8)+c\int x^2\sqrt{1-x}\,dx = p\sqrt{1-x}(3x^2+4x+8) + c, then the value of pp is

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  46. 46
    The centre of the circle whose radius is 3 units and touching internally the circle x2+y2−4x−6y−12=0x^2+y^2-4x-6y-12=0 at the point (−1,−1)(-1,-1) is

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  47. 47
    If acos⁡2θ+bsin⁡2θ=ca\cos 2\theta + b\sin 2\theta = c has α\alpha and β\beta as its roots, then tan⁡α+tan⁡β\tan\alpha + \tan\beta is

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  48. 48
    The mirror image of P(2,4,−1)P(2,4,-1) in the plane x−y+2z−2=0x-y+2z-2=0 is (a,b,c)(a,b,c), then the value of a+b+ca+b+c is

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  49. 49
    If both mean and variance of 50 observations x1,x2,…,x50x_1, x_2, \ldots, x_{50} are equal to 16 and 256 respectively, then mean of (x1−5)2,(x2−5)2,…,(x50−5)2(x_1-5)^2, (x_2-5)^2, \ldots, (x_{50}-5)^2 is

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  50. 50
    The function ff defined on [−13,13]\left[-\frac{1}{3},\frac{1}{3}\right] by f(x)={1xlog⁡1+3x1−2x,x≠0k,x=0f(x) = \begin{cases}\frac{1}{x}\log\frac{1+3x}{1-2x}, & x\neq0\\k, & x=0\end{cases} is continuous at x=0x=0, then kk is

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