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MHT-CET 2024 (12 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 a⃗,b⃗,c⃗\vec{a}, \vec{b}, \vec{c} are three vectors, ∣a⃗∣=2|\vec{a}|=2, ∣b⃗∣=4|\vec{b}|=4, ∣c⃗∣=1|\vec{c}|=1, ∣b⃗×c⃗∣=15|\vec{b}\times\vec{c}|=\sqrt{15} and b⃗=2c⃗+λa⃗\vec{b}=2\vec{c}+\lambda\vec{a}, then the value of λ\lambda is

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  2. 2
    The centroid of tetrahedron with vertices at A(−1,2,3)A(-1,2,3), B(3,−2,1)B(3,-2,1), C(2,1,3)C(2,1,3) and D(−1,−2,4)D(-1,-2,4) is

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  3. 3
    Two adjacent sides of a 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 an acute angle α\alpha in the plane of the parallelogram so that AD becomes AD'. If AD' makes a right angle with side AB, then the cosine of the angle α\alpha is given by

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  4. 4
    The values of aa and bb, so that the function f(x)={x+a2sin⁡x,0≤x≤π42xcot⁡x+b,π4≤x≤π2acos⁡2x−bsin⁡x,π2<x≤πf(x)=\begin{cases}x+a\sqrt{2}\sin x, & 0\le x\le\frac{\pi}{4}\\ 2x\cot x+b, & \frac{\pi}{4}\le x\le\frac{\pi}{2}\\ a\cos2x-b\sin x, & \frac{\pi}{2}<x\le\pi\end{cases} is continuous, are respectively given by

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  5. 5
    For a feasible region OCDBO given below, the maximum value of the objective function z=3x+4yz=3x+4y is

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  6. 6
    If g(x)=1+xg(x)=1+\sqrt{x} and f(g(x))=3+2x+xf(g(x))=3+2\sqrt{x}+x, then f(f(x))f(f(x)) is

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  7. 7
    The approximate value of sin⁡60∘0′10′′\sin 60^{\circ}0'10'' is (given that 3=1.732\sqrt{3}=1.732, 1∘=0.0175∘1^{\circ}=0.0175^{\circ})

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  8. 8
    The decay rate of radioactive material at any time tt is proportional to its mass at that time. The mass is 27 grams when t=0t=0. After three hours it was found that 8 grams are left. Then the substance left after one more hour is

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  9. 9
    The derivative of f(tan⁡x)f(\tan x) w.r.t. g(sec⁡x)g(\sec x) at x=π4x=\frac{\pi}{4}, where f′(1)=2f'(1)=2 and g′(2)=4g'(\sqrt{2})=4, is

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  10. 10
    The p.m. of random variate XX is P(X)=2xn(n+1),  x=1,2,3,…,nP(X)=\frac{2x}{n(n+1)},\;x=1,2,3,\ldots,n. Then E(X)=E(X)=

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  11. 11
    The angle between the tangents to the curves y=2x2y=2x^{2} and x=2y2x=2y^{2} at (1,1)(1,1) is

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  12. 12
    If the area of the triangle with vertices (1,2,0)(1,2,0), (1,0,2)(1,0,2) and (0,x,1)(0,x,1) is 6\sqrt{6} square units, then the value of xx is

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  13. 13
    An experiment succeeds twice as often as it fails. Then the probability, that in the next 6 trials there will be at least 4 successes, is

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  14. 14
    The co-ordinates of the points on the line 2x−y=52x-y=5 which are the distance of 1 unit from the line 3x+4y=53x+4y=5 are

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  15. 15
    If x=cosec(tan⁡−1(cos⁡(cot⁡−1(sec⁡(sin⁡−1a)))))x=\text{cosec}\left(\tan^{-1}\left(\cos\left(\cot^{-1}\left(\sec\left(\sin^{-1}a\right)\right)\right)\right)\right), a∈[0,1]a\in[0,1]

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  16. 16
    Let A⃗\vec{A} be a vector parallel to line of intersection of planes P1P_1 and P2P_2 through origin. P1P_1 is parallel to the vectors 2j^+3k^2\hat{j}+3\hat{k} and 4j^−3k^4\hat{j}-3\hat{k} and P2P_2 is parallel to j^−k^\hat{j}-\hat{k} and 3i^+3j^3\hat{i}+3\hat{j}, then the angle between A⃗\vec{A} and 2i^+j^−2k^2\hat{i}+\hat{j}-2\hat{k} is

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  17. 17
    ∫cosec x dxcos⁡2(1+log⁡tan⁡x2)=\displaystyle\int\frac{\text{cosec}\,x\,dx}{\cos^{2}\left(1+\log\tan\frac{x}{2}\right)}=

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  18. 18
    If the variance of the numbers −1,0,1,k-1,0,1,k is 5, where k>0k>0, then kk is equal to

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  19. 19
    The differential equation cos⁡(x+y) dy=dx\cos(x+y)\,dy=dx has the general solution given by

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  20. 20
    The area of the region bounded by the curves y=exy=e^{x}, y=log⁡xy=\log x and lines x=1x=1, x=2x=2 is

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  21. 21
    If x=−1x=-1 and x=2x=2 are extreme points of f(x)=αlog⁡∣x∣+βx2+xf(x)=\alpha\log|x|+\beta x^{2}+x, α\alpha and β\beta are constants, then the value of α2+2β\alpha^{2}+2\beta is

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  22. 22
    A plane is parallel to two lines whose direction ratios are 1,0,−11,0,-1 and −1,1,0-1,1,0 and it contains the point (1,1,1)(1,1,1). If it cuts the co-ordinate axes at A,B,CA,B,C, then the volume of the tetrahedron OABC (in cubic units) is

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  23. 23
    The function f(x)=sin⁡4x+cos⁡4xf(x)=\sin^{4}x+\cos^{4}x is increasing in

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  24. 24
    If a>0a>0 and z=(1+i)2a+iz=\frac{(1+i)^{2}}{a+i} (i=−1i=\sqrt{-1}) has magnitude 25\frac{2}{\sqrt{5}}, then zˉ\bar{z} is equal to

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  25. 25
    The integral ∫sin⁡2xcos⁡2x(sin⁡5x+cos⁡3xsin⁡2x+sin⁡3xcos⁡2x+cos⁡5x)2dx\displaystyle\int\frac{\sin^{2}x\cos^{2}x}{(\sin^{5}x+\cos^{3}x\sin^{2}x+\sin^{3}x\cos^{2}x+\cos^{5}x)^{2}}dx is equal to

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  26. 26
    The equation of the plane through (−1,1,2)(-1,1,2) whose normal makes equal acute angles with coordinate axes is

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  27. 27
    If TnT_n denotes the number of triangles which can be formed using the vertices of regular polygon of nn sides and Tn+1−Tn=21T_{n+1}-T_n=21, then n=n=

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  28. 28
    Two cards are drawn successively with replacement from a well shuffled pack of 52 cards. Then the probability distribution of number of jacks is

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  29. 29
    If tan⁡θ=sin⁡α−cos⁡αsin⁡α+cos⁡α\tan\theta=\frac{\sin\alpha-\cos\alpha}{\sin\alpha+\cos\alpha}, 0≤α≤π20\leq\alpha\leq\frac{\pi}{2}, then the value of cos⁡2θ\cos2\theta is

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  30. 30
    The solution set of 8cos⁡2θ+14cos⁡θ+5=08\cos^{2}\theta+14\cos\theta+5=0, in the interval [0,2π][0,2\pi] is

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  31. 31
    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 rad./s when its lower end is 4 m away from the wall.

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  32. 32
    If dydx=y+3\frac{dy}{dx}=y+3 and y(0)=2y(0)=2, then y(log⁡2)=y(\log2)=

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  33. 33
    If log⁡(x+y)=2xy\log(x+y)=2xy, then dydx\frac{dy}{dx} at x=0x=0 is

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  34. 34
    If general solution of cos⁡2θ−2sin⁡θ+14=0\cos^{2}\theta-2\sin\theta+\frac{1}{4}=0 is θ=nπA+(−1)nπB\theta=\frac{n\pi}{A}+(-1)^{n}\frac{\pi}{B}, n∈Zn\in\mathbb{Z}, then A+BA+B has the value

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  35. 35
    u⃗\vec{u}, v⃗\vec{v}, w⃗\vec{w} are three vectors such that ∣u⃗∣=1|\vec{u}|=1, ∣v⃗∣=2|\vec{v}|=2, ∣w⃗∣=3|\vec{w}|=3. If the projection of v⃗\vec{v} along u⃗\vec{u} is equal to projection of w⃗\vec{w} along u⃗\vec{u} and v⃗\vec{v}, w⃗\vec{w} are perpendicular to each other, then ∣u⃗−v⃗+w⃗∣=|\vec{u}-\vec{v}+\vec{w}|=

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  36. 36
    The distance of the point P(−2,4,−5)P(-2,4,-5) from the line x+33=y−45=z+86\frac{x+3}{3}=\frac{y-4}{5}=\frac{z+8}{6} is

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  37. 37
    A and B are independent events with P(A)=14P(A)=\frac{1}{4} and P(A∪B)=2P(B)−P(A)P(A\cup B)=2P(B)-P(A), then P(B)P(B) is

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  38. 38
    ∫x2+1x(x2−1)dx=\displaystyle\int\frac{x^{2}+1}{x(x^{2}-1)}dx=

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  39. 39
    If the matrix A=(12−51)A=\begin{pmatrix}1&2\\-5&1\end{pmatrix} and A−1=xA+yIA^{-1}=xA+yI when II is a unit matrix of order 2, then the value of 2x+3y2x+3y is

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  40. 40
    The inverse of the statement "If the surface area increases, then the pressure decreases" is

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  41. 41
    In a triangle, the sum of lengths of two sides is xx and the product of the lengths of the same two sides is yy. If x2−c2=yx^{2}-c^{2}=y where cc is the length of the third side of the triangle, then the circumradius of the triangle is

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  42. 42
    The contrapositive of "If xx and yy are integers such that xyxy is odd, then both xx and yy are odd" is

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  43. 43
    If y=(1+x)(1+x2)(1+x4)⋯(1+x2n)y=(1+x)(1+x^{2})(1+x^{4})\cdots(1+x^{2^{n}}) then the value of dydx\frac{dy}{dx} at x=0x=0 is

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  44. 44
    lim⁡x→0cos⁡7x∘−cos⁡2x∘x2\displaystyle\lim_{x\to0}\frac{\cos7x^{\circ}-\cos2x^{\circ}}{x^{2}} is

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  45. 45
    ∫04∣2x−5∣ dx=\displaystyle\int_{0}^{4}|2x-5|\,dx=

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  46. 46
    If λ\lambda is the perpendicular distance of a point PP on the circle x2+y2+2x+2y−3=0x^{2}+y^{2}+2x+2y-3=0 from the line 2x+y+13=02x+y+13=0 then maximum possible value of λ\lambda is

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  47. 47
    If the lines 1−x3=7y−142p=z−32\frac{1-x}{3}=\frac{7y-14}{2p}=\frac{z-3}{2} and 7−7x3p=y−51=6−z5\frac{7-7x}{3p}=\frac{y-5}{1}=\frac{6-z}{5} are at right angles, then P=P=

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  48. 48
    The value of sin⁡(cot⁡−1x)\sin(\cot^{-1}x) is

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  49. 49
    If ∫cos⁡3/5x⋅sin⁡3x dx=−1mcos⁡mx+1ncos⁡nx+c\displaystyle\int\cos^{3/5}x\cdot\sin^{3}x\,dx=-\frac{1}{m}\cos^{m}x+\frac{1}{n}\cos^{n}x+c (where cc is the constant of integration), then (m,n)=(m,n)=

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  50. 50
    Let PQR be a right angled isosceles triangle, right angled at P(2,1)P(2,1). If the equation of the line QR is 2x+y=32x+y=3 then the equation representing the pair of lines PQ and PR is

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