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MHT-CET 2024 (13 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
    The value of tan⁡ ⁣(sin⁡−1 ⁣35+tan⁡−1 ⁣23)\tan\!\left(\sin^{-1}\!\frac{3}{5}+\tan^{-1}\!\frac{2}{3}\right) is

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  2. 2
    The integral ∫π/6π/3sec⁡2/3x⋅csc⁡4/3x dx\int_{\pi/6}^{\pi/3}\sec^{2/3}x\cdot\csc^{4/3}x\,dx is equal to

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  3. 3
    ∫2+cos⁡x(2x+sin⁡x)2 dx=\int\frac{2+\cos x}{(2x+\sin x)^2}\,dx=

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  4. 4
    The acute angle between the line joining the points (2,1,−3)(2,1,-3), (−3,1,7)(-3,1,7) and a line parallel to x−13=y4=z+35\frac{x-1}{3}=\frac{y}{4}=\frac{z+3}{5} through the point (−1,0,4)(-1,0,4) is

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  5. 5
    Variance of first 2n2n natural numbers is

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  6. 6
    If qq is false and (p∧q)↔r(p\wedge q)\leftrightarrow r is true, then ………. is a tautology.

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  7. 7
    Slope of the tangent to the curve y=2exsin⁡ ⁣(π4−x2)cos⁡ ⁣(π4−x2)y=2e^x\sin\!\left(\frac{\pi}{4}-\frac{x}{2}\right)\cos\!\left(\frac{\pi}{4}-\frac{x}{2}\right) where 0≤x≤2π0\leq x\leq2\pi is minimum at x=x=

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  8. 8
    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+\frac{1}{\sqrt{3}}, then the values of aa and bb respectively are

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  9. 9
    Five students are selected from nn students such that the ratio of number of ways in which 2 particular students are selected to the number of ways 2 particular students not selected is 2:3. Then the value of nn is

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  10. 10
    In △ABC\triangle ABC, with notations, if b+c11=c+a12=a+b13\frac{b+c}{11}=\frac{c+a}{12}=\frac{a+b}{13}, then the value of cos⁡A+cos⁡B+cos⁡C\cos A+\cos B+\cos C is:

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  11. 11
    If (tan⁡−1x)2+(cot⁡−1x)2=5π28\left(\tan^{-1}x\right)^{2}+\left(\cot^{-1}x\right)^{2}=\frac{5\pi^{2}}{8}, then the value of xx is:

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  12. 12
    If a⃗,b⃗,c⃗\vec{a},\vec{b},\vec{c}, are non-coplanar unit vectors such that a⃗×(b⃗×c⃗)=b⃗+c⃗2\vec{a}\times(\vec{b}\times\vec{c})=\frac{\vec{b}+\vec{c}}{2} then the angle between a⃗\vec{a} and b⃗\vec{b} is:

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  13. 13
    If I=∫exe4x+e2x+1dxI=\int\frac{e^x}{e^{4x}+e^{2x}+1}dx and J=∫e−xe−4x+e−2x+1dxJ=\int\frac{e^{-x}}{e^{-4x}+e^{-2x}+1}dx, then for any arbitrary constant cc, that value of J−IJ-I equals

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  14. 14
    If x dy=y(dx+y dy), y(1)=1, y(x)>0x\,dy=y(dx+y\,dy),\ y(1)=1,\ y(x)>0, then y(−3)y(-3) is:

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  15. 15
    The solutions of sin⁡x+sin⁡5x=sin⁡3x\sin x+\sin 5x=\sin 3x in (0,π2)\left(0,\frac{\pi}{2}\right) are

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  16. 16
    lim⁡x→01+xsin⁡x−cos⁡xtan⁡2(x/2)=\lim_{x\to0}\dfrac{\sqrt{1+x\sin x}-\sqrt{\cos x}}{\tan^2(x/2)}=

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  17. 17
    The distance of the point (1,6,2)(1,6,2) from the point of intersection of the line x−23=y+14=z−212\frac{x-2}{3}=\frac{y+1}{4}=\frac{z-2}{12} and the plane x−y+z=16x-y+z=16 is:

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  18. 18
    A random variable X has the probability distribution.
    X=xP(X=x)
    10.15
    20.23
    30.12
    40.20
    50.08
    60.10
    70.05
    80.07
    for the events E={X is a prime number}E=\{X\text{ is a prime number}\} and F={x<5}F=\{x<5\}, P(E∪F)P(E\cup F) is

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  19. 19
    If the vectors pi^+j^+k^,i^+qj^+k^p\hat{i}+\hat{j}+\hat{k},\hat{i}+q\hat{j}+\hat{k} and i^+j^+rk^ (p≠q≠r≠1)\hat{i}+\hat{j}+r\hat{k}\ (p\neq q\neq r\neq1) are coplanar, then the value of pqr−(p+q+r)pqr-(p+q+r) is

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  20. 20
    Let PQR be a right angled isosceles triangle, right angled at Q(2,1)Q(2,1). If the equation of the line PR is 2x+y=32x+y=3, then the combined equation representing the pair of lines PQ and QR is

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  21. 21
    If f(x)f(x) is continuous on its domain [−2,2][-2,2], where f(x)={sin⁡axx+3,−2≤x<02x+7,0≤x≤1x2+8−b,1<x≤2f(x)=\begin{cases}\frac{\sin ax}{x}+3, & -2\leq x<0\\ 2x+7, & 0\leq x\leq1\\ \sqrt{x^2+8}-b, & 1<x\leq2\end{cases} then the value of 2a+3b2a+3b is:

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  22. 22
    If x=log⁡e ⁣cos⁡y2−sin⁡y2cos⁡y2+sin⁡y2, tan⁡y2=1−t1+tx=\log_e\!\frac{\cos\frac{y}{2}-\sin\frac{y}{2}}{\cos\frac{y}{2}+\sin\frac{y}{2}},\ \tan\frac{y}{2}=\frac{1-t}{1+t}. Then y′ ⁣ ⁣∣t=12y'\!\!\left|_{t=\frac{1}{2}}\right. has the value

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  23. 23
    A(1,-3), B(4,3) are two points on the curve y=x−4xy=x-\frac{4}{x}. The points on the curve, the tangents at which are parallel to the chord AB, are

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  24. 24
    If the sum of the mean and the variance of a Binomial distribution for 5 trials is 1.8, then the value of pp is

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  25. 25
    The shaded area in the given figure is a solution set for some system of inequations. The maximum value of the function z=10x+25yz=10x+25y subject to the linear constraints given by the system is

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  26. 26
    If A=(1tan⁡x−tan⁡x1)A=\begin{pmatrix}1 & \tan x\\ -\tan x & 1\end{pmatrix}, then ATA−1=A^T A^{-1}=

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  27. 27
    If slope of the tangent to the curve xy+ax+by=0xy+ax+by=0 at the point (1,1)(1,1) on it is 2, then the value of 3a+b3a+b is

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  28. 28
    The c.d.f. F(x)F(x) associated with p.d.f. f(x)={12x2(1−x),0<x<10,otherwisef(x)=\begin{cases}12x^2(1-x), & 0<x<1\\ 0, & \text{otherwise}\end{cases} is

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  29. 29
    If ∫log⁡(t+1+t2)1+t2 dt=12[g(t)]2+c\int\frac{\log(t+\sqrt{1+t^2})}{\sqrt{1+t^2}}\,dt=\frac{1}{2}[g(t)]^2+c, (Where cc is a constant of integration) then g(2)g(2) is:

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  30. 30
    The foot of the perpendicular from the point (1,2,3)(1,2,3) on the line r⃗=(6i^+7j^+7k^)+λ(3i^+2j^−2k^)\vec{r}=(6\hat{i}+7\hat{j}+7\hat{k})+\lambda(3\hat{i}+2\hat{j}-2\hat{k}) has the co-ordinates.

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  31. 31
    A tetrahedron has vertices at P(2,1,3),Q(−1,1,2)P(2,1,3), Q(-1,1,2), R(1,2,1)R(1,2,1) and O(0,0,0)O(0,0,0), then angle between the faces OPQ and PQR is

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  32. 32
    Let f:R→Rf:R\to R be a function such that f(x)=x3+x2f′(1)+xf′′(2)+6, x∈Rf(x)=x^3+x^2f'(1)+xf''(2)+6,\ x\in R, then f(2)f(2) equals

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  33. 33
    Water is running in a hemispherical bowl of radius 180 cm at the rate of 108 cubic decimeters per minute. How fast the water level is rising when depth of the water level in the bowl is 120 cm? (1 decimeter =10=10 cm)

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  34. 34
    ∫x−3(x−1)3ex dx=\int\frac{x-3}{(x-1)^3}e^x\,dx=

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  35. 35
    The domain of the definition of the function y(x)y(x) is given by the equation 2x+2y=22^x+2^y=2 is

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  36. 36
    The incentre of the triangle ABC, whose vertices are A(0,2,1),B(−2,0,0)A(0,2,1), B(-2,0,0) and C(−2,0,2)C(-2,0,2), is

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  37. 37
    Two dice are rolled. If both dice have six faces numbered 1,2,3,5,7,111,2,3,5,7,11, then the probability that the sum of the numbers on upper most face is prime is

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  38. 38
    A plane is parallel to two lines whose direction ratios are 2,0,−22,0,-2 and −2,2,0-2,2,0 and it contains the point (2,2,2)(2,2,2). If it cuts co-ordinate axes at A, B, C, then the volume of the tetrahedron OABC (in cubic units) is

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  39. 39
    If a curve y=ax+bxy=a\sqrt{x}+bx passes through the point (1,2)(1,2) and the area bounded by the curve, line x=4x=4 and X-axis is 8 sq. units, then

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  40. 40
    A radioactive substance, with initial mass m0m_0, has a half-life of hh days. Then its initial decay rate is given by

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  41. 41
    If (1+1+x)tan⁡x=1+1−x(1+\sqrt{1+x})\tan x=1+\sqrt{1-x}, then sin⁡4x\sin4x is:

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  42. 42
    If a⃗=110(3i^+k^),b⃗=17(2i^+3j^−6k^)\vec{a}=\frac{1}{\sqrt{10}}(3\hat{i}+\hat{k}), \vec{b}=\frac{1}{7}(2\hat{i}+3\hat{j}-6\hat{k}), then the value of (2a⃗−b⃗)⋅[(a⃗×b⃗)×(a⃗+2b⃗)](2\vec{a}-\vec{b})\cdot[(\vec{a}\times\vec{b})\times(\vec{a}+2\vec{b})] is:

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  43. 43
    Negation of contrapositive of statement pattern (p∨∼q)→(p∧∼q)(p\vee\sim q)\to(p\wedge\sim q) is

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  44. 44
    y=(1+3x)1/3(1+4x)1/4(1+5x)1/5(1+7x)1/7(1+8x)1/8y=\frac{(1+3x)^{1/3}(1+4x)^{1/4}(1+5x)^{1/5}}{(1+7x)^{1/7}(1+8x)^{1/8}}. Then dydx\frac{dy}{dx} at x=0x=0 is:

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  45. 45
    If (3x+2)−(5y−3)i(3x+2)-(5y-3)i and (6x+3)+(2y−4)i(6x+3)+(2y-4)i are conjugates of each other, then the value of x−yx+y\frac{x-y}{x+y} is

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  46. 46
    The solution of (1+xy)y dx+(1−xy)x dy=0(1+xy)y\,dx+(1-xy)x\,dy=0 is

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  47. 47
    a⃗=2i^+3j^+4k^,b⃗=i^−2j^+k^,c⃗=i^+j^−k^\vec{a}=2\hat{i}+3\hat{j}+4\hat{k},\vec{b}=\hat{i}-2\hat{j}+\hat{k},\vec{c}=\hat{i}+\hat{j}-\hat{k} are three vectors. For a vector r⃗\vec{r} with r⃗×a⃗=b⃗\vec{r}\times\vec{a}=\vec{b} and r⃗⋅c⃗=3\vec{r}\cdot\vec{c}=3, ∣r⃗∣|\vec{r}| is

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  48. 48
    The abscissae of two points A and B are the roots of the equation x2+2ax−b2=0x^2+2ax-b^2=0 and their ordinates are roots of the equation y2+2py−q2=0y^2+2py-q^2=0. Then the equation of the circle with AB as diameter is given by

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  49. 49
    PS is the median of the triangle with vertices at P(2,2),Q(6,−1)P(2,2), Q(6,-1) and R(7,3)R(7,3), then the intercepts on the co-ordinate axes of the line passing through point (1,−1)(1,-1) and parallel to PS are respectively

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  50. 50
    The principal value of sin⁡−1 ⁣(sin⁡3π4)\sin^{-1}\!\left(\sin\frac{3\pi}{4}\right) is:

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