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MHT-CET 2024 (12 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
    If tan⁡y=xsin⁡α1−xcos⁡α\tan y = \frac{x\sin\alpha}{1 - x\cos\alpha} and dydx=mx2+2nx+1\frac{dy}{dx} = \frac{m}{x^2 + 2nx + 1}, then m2+n2m^2 + n^2 is

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  2. 2
    If ∣z−2+i∣≤2|z - 2 + i| \leq 2, then the difference between the greatest and least value of ∣z∣|z| is (i=−1)(i = \sqrt{-1})

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  3. 3
    The shortest distance (in units) between the lines x+13=y+21=z+12\frac{x+1}{3}=\frac{y+2}{1}=\frac{z+1}{2} and r⃗=(2i^−2j^+3k^)+λ(i^+2j^)\vec{r}=(2\hat{i}-2\hat{j}+3\hat{k})+\lambda(\hat{i}+2\hat{j}) is

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  4. 4
    The area (in sq. units) of the smaller part of the circle x2+y2=a2x^2 + y^2 = a^2 cut off by the line x=a2x = \frac{a}{\sqrt{2}} is

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  5. 5
    If ∫1−x2x4 dx=A(x)(1−x2)m+c\int \frac{\sqrt{1-x^2}}{x^4}\,dx = A(x)\left(1-x^2\right)^m + c for a suitable chosen integer mm and a function A(x)A(x), where cc is a constant of integration, then (A(x))m(A(x))^m equals

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  6. 6
    The value of ∫0π∣sin⁡x−2xπ∣ dx\int_0^{\pi} \left|\sin x - \frac{2x}{\pi}\right|\,dx is

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  7. 7
    ∫1cos⁡3xsin⁡2x dx=\int \frac{1}{\cos^3 x \sqrt{\sin 2x}}\,dx =

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  8. 8
    If f(x)=sin⁡−1 ⁣(2log⁡x1+(log⁡x)2)f(x) = \sin^{-1}\!\left(\frac{2\log x}{1+(\log x)^2}\right), then f′(e)f'(e) is

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  9. 9
    A poster is to be printed on a rectangular sheet of paper of area 1818 m2\text{m}^2. The margins at the top and bottom of 75 cm each and at the sides 50 cm each are to be left. Then the dimensions i.e. height and breadth of the sheet, so that the space available for printing is maximum, are respectively

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  10. 10
    cos⁡248°−sin⁡212°=\cos^2 48° - \sin^2 12° = ……. if sin⁡18°=5−14\sin 18° = \frac{\sqrt{5}-1}{4}

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  11. 11
    The unit vector which is orthogonal to the vector 3i^+2j^+6k^3\hat{i}+2\hat{j}+6\hat{k} and 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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  12. 12
    If kik_i are possible values of kk for which lines kx+2y+2=0,  2x+ky+3=0kx+2y+2=0,\; 2x+ky+3=0 and 3x+3y+k=03x+3y+k=0 are concurrent, then ∑ki\sum k_i has the value

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  13. 13
    The value of xx, for which sin⁡ ⁣(cot⁡−1(x))=cos⁡ ⁣(tan⁡−1(1+x))\sin\!\left(\cot^{-1}(x)\right) = \cos\!\left(\tan^{-1}(1+x)\right), is

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  14. 14
    The lengths of sides of a triangle are three consecutive natural numbers and its largest angle is twice the smallest one. Then the length of the sides of the triangle (in units) are

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  15. 15
    ∫1(x+2)(1+x2) dx\int \frac{1}{(x+2)(1+x^2)}\,dx has the value

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  16. 16
    If tan⁡−1 ⁣(1−x1+x)=12tan⁡−1x\tan^{-1}\!\left(\frac{1-x}{1+x}\right) = \frac{1}{2}\tan^{-1}x, then xx is

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  17. 17
    If the pair of lines given by (xcos⁡α+ysin⁡α)2=(x2+y2)sin⁡2α(x\cos\alpha + y\sin\alpha)^2 = (x^2+y^2)\sin^2\alpha are perpendicular to each other, then α\alpha is

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  18. 18
    Water flows from the base of rectangular tank, of depth 16 meters. The rate of flow of the water is proportional to the square root of depth at any time tt. If depth is 4 m when t=2t = 2 hours, then after 3.5 hours the depth (in meters) is

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  19. 19
    The number of words that can be formed by using the letters of the word CALCULATE such that each word starts and ends with a consonant, are

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  20. 20
    Given f(x)={1−cos⁡4xx2,x<0α,x=0x16+x−4,x>0f(x)=\begin{cases}\dfrac{1-\cos4x}{x^2}, & x<0\\ \alpha, & x=0\\ \dfrac{\sqrt{x}}{\sqrt{16+\sqrt{x}}-4}, & x>0\end{cases}. If f(x)f(x) is continuous at x=0x=0, then value of α\alpha is

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  21. 21
    A, B, C, D are four points in a plane with position vectors a⃗,b⃗,c⃗,d⃗\vec{a}, \vec{b}, \vec{c}, \vec{d} respectively such that (a⃗−d⃗)⋅(b⃗−c⃗)=(b⃗−d⃗)⋅(c⃗−a⃗)=0(\vec{a}-\vec{d})\cdot(\vec{b}-\vec{c}) = (\vec{b}-\vec{d})\cdot(\vec{c}-\vec{a}) = 0. The point D, then is the ___ of △ABC\triangle ABC.

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  22. 22
    A discrete random variable XX can take all possible integer values from 1 to kk, each with a probability 1k\frac{1}{k}, then its variance is

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  23. 23
    Equation of plane containing the line x2=y3=z4\frac{x}{2}=\frac{y}{3}=\frac{z}{4} and perpendicular to the plane containing the lines x3=y4=z2\frac{x}{3}=\frac{y}{4}=\frac{z}{2} and x4=y2=z3\frac{x}{4}=\frac{y}{2}=\frac{z}{3} is

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  24. 24
    An irregular six faced die is thrown and the probability that, in 5 throws it will give 3 even numbers is twice the probability that it will give 2 even numbers. The number of times, in 6804 sets of 5 throws, you expect to give no even number is

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  25. 25
    ∫(tan⁡(1x)x)2dx=\int \left(\dfrac{\tan\left(\frac{1}{x}\right)}{x}\right)^{2}dx =

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  26. 26
    If f(x)=x2+1f(x) = x^2+1 and g(x)=1xg(x) = \frac{1}{x}, then the value of f(g(g(f(x))))f(g(g(f(x)))) at x=1x = 1 is

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  27. 27
    lim⁡x→0xcot⁡4xsin⁡2x⋅cot⁡2(2x)\lim_{x \to 0} \frac{x\cot 4x}{\sin^2 x \cdot \cot^2(2x)} is equal to

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  28. 28
    For x>1x > 1, if (2x)2y=4e2x−2y(2x)^{2y} = 4e^{2x-2y}, then (1+log⁡e2x)2dydx\left(1+\log_e 2x\right)^2 \frac{dy}{dx} is equal to

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  29. 29
    In △ABC\triangle ABC, m∠B=π3m\angle B = \frac{\pi}{3} and m∠C=π4m\angle C = \frac{\pi}{4}. Let point D divide BC internally in the ratio 1:31:3, then sin⁡(∠BAD)sin⁡(∠CAD)\frac{\sin(\angle BAD)}{\sin(\angle CAD)} has the value

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  30. 30
    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

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  31. 31
    If the curves y2=6xy^2 = 6x and 9x2+by2=169x^2 + by^2 = 16 intersect each other at right angle, then value of 'bb' is

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  32. 32
    The equation of the normal to the curve 3x2−y2=83x^2 - y^2 = 8, which is parallel to the line x+3y=10x + 3y = 10, is

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  33. 33
    If ∣u⃗∣=8|\vec{u}| = 8, ∣v⃗∣=12|\vec{v}| = 12 and the angle between them is 150∘150^\circ, then ∣u⃗×v⃗∣|\vec{u} \times \vec{v}| is

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  34. 34
    A tank with a rectangular base and rectangular sides, open at the top is to be constructed so that its depth is 4 meter and volume is 36 cubic meters. If building of the tank costs ₹100 per square meter for the base and ₹50 per square meter for the sides, then the cost of least expensive tank is

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  35. 35
    If the volume of tetrahedron, whose vertices are A(1,2,3), B(−3,−1,1), C(2,1,3)A(1,2,3),\, B(-3,-1,1),\, C(2,1,3) and D(−1,2,x)D(-1,2,x) is 116\frac{11}{6} cubic units, then the value of xx is

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  36. 36
    The length (in units) of the projection of the line segment, joining the points (5,−1,4)(5,-1,4) and (4,−1,3)(4,-1,3), on the plane x+y+z=7x+y+z=7 is

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  37. 37
    A ladder, 5 meters long, rests against a vertical wall. If its top slides downwards at the rate of 10 cm/s, then the angle between the ladder and the floor is decreasing at the rate of ___ radians/second when its lower end is 4 m away from the wall.

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  38. 38
    The given following circuit is equivalent to

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  39. 39
    If A=(2a−3b32)A = \begin{pmatrix}2a & -3b \\ 3 & 2\end{pmatrix} and A⋅adj A=AATA \cdot \text{adj}\, A = A A^T, then 2a+3b2a+3b is

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  40. 40
    Three fair coins with faces numbered 1 and 0 are tossed simultaneously. Then variance (X) of the probability distribution of random variable XX, where XX is the sum of numbers on the upper most faces, is

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  41. 41
    Two adjacent sides of parallelogram ABCD are given by AB→=2i^+10j^+11k^\overrightarrow{AB} = 2\hat{i}+10\hat{j}+11\hat{k} and AD→=−i^+2j^+2k^\overrightarrow{AD} = -\hat{i}+2\hat{j}+2\hat{k}. The side AD is rotated by angle α\alpha in plane of parallelogram so that AD becomes AD'. If AD' makes a right angle with the side AB, then the cosine of the angle α\alpha is given by

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  42. 42
    The maximum value of Z=7x+8yZ = 7x+8y subject to the constraints x+y≤20,  y≥5,  x≤10,  x≥0,  y≥0x+y \leq 20,\; y \geq 5,\; x \leq 10,\; x \geq 0,\; y \geq 0 is

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  43. 43
    The solution of ey−xdydx=y(sin⁡x+cos⁡x)1+ylog⁡ye^{y-x}\dfrac{dy}{dx} = \dfrac{y(\sin x + \cos x)}{1+y\log y} is

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  44. 44
    If a⃗,b⃗,c⃗\vec{a}, \vec{b}, \vec{c} are three non-zero vectors, no two of them are collinear, a⃗+2b⃗\vec{a}+2\vec{b} is collinear with c⃗\vec{c}, b⃗+3c⃗\vec{b}+3\vec{c} is collinear with a⃗\vec{a}, then a⃗+2b⃗\vec{a}+2\vec{b} is

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  45. 45
    If two angles of △ABC\triangle ABC are π4\frac{\pi}{4} and π3\frac{\pi}{3}, then the ratio of the smallest and greatest sides are

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  46. 46
    The equation of the line, passing through (1,2,3)(1,2,3) and parallel to planes x−y+2z=5x-y+2z=5 and 3x+y+z=63x+y+z=6, is

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  47. 47
    A box contains 100 tickets numbered 1 to 100. A ticket is drawn at random from the box. Then the probability, that number on the ticket is a perfect square, is

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  48. 48
    The circles x2+y2+2ax+c=0x^2+y^2+2ax+c=0 and x2+y2+2by+c=0x^2+y^2+2by+c=0 touch each other externally, if

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  49. 49
    For 20 observations of variable xx, if ∑(xi−2)=20\sum(x_i-2)=20 and ∑(xi−2)2=100\sum(x_i-2)^2=100, then the standard deviation of variable xx is

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  50. 50
    Let Statement 1: If a quadrilateral is a square, then all of its sides are equal. Statement 2: All the sides of a quadrilateral are equal, then it is a square.

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