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MHT-CET 2024 (10 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 the volume of tetrahedron, whose vertices are with position vectors i^−6j^+10k^, −i^−3j^+7k^\hat{i}-6\hat{j}+10\hat{k},\,-\hat{i}-3\hat{j}+7\hat{k}, 5i^−j^+λk^5\hat{i}-\hat{j}+\lambda\hat{k} and 7i^−4j^+7k^7\hat{i}-4\hat{j}+7\hat{k} is 11 cubic units, then value of λ\lambda is

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  2. 2
    Two sides of a square are along the lines 5x−12y+39=05x-12y+39=0 and 5x−12y+78=05x-12y+78=0, then area of the square is

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  3. 3
    Two lines x−31=y+13=z−6−1\frac{x-3}{1}=\frac{y+1}{3}=\frac{z-6}{-1} and x+57=y−2−6=z−34\frac{x+5}{7}=\frac{y-2}{-6}=\frac{z-3}{4} intersect at the point R. Then reflection of R in the xy-plane has co-ordinates

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  4. 4
    The joint equation of a pair of lines passing through the origin and making an angle of π4\frac{\pi}{4} with the line 3x+2y−8=03x+2y-8=0 is

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  5. 5
    The shaded area in the figure given below is a solution set of a system of inequations. The minimum value of objective function 3x+5y3x+5y, subject to the linear constraints given by this system of inequations is

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  6. 6
    ∫0πxtan⁡xsec⁡x+cos⁡x dx=\displaystyle\int_0^\pi \frac{x\tan x}{\sec x+\cos x}\,dx =

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  7. 7
    If x,y,zx,y,z are in A.P. and tan⁡−1x,tan⁡−1y\tan^{-1}x,\tan^{-1}y and tan⁡−1z\tan^{-1}z are also in A.P., then

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  8. 8
    If the surface area of a spherical balloon of radius 6 cm is increasing at the rate 2 cm2/sec2\,\text{cm}^2/\text{sec}, then the rate of increase in its volume in cm3/sec\text{cm}^3/\text{sec} is

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  9. 9
    The area bounded by the X-axis and the curve y=x(x−2)(x+1)y=x(x-2)(x+1) is

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  10. 10
    In a triangle ABC, with usual notations, if c=4c=4, then value of (a−b)2cos⁡2 ⁣C2+(a+b)2sin⁡2 ⁣C2(a-b)^2\cos^2\!\frac{C}{2}+(a+b)^2\sin^2\!\frac{C}{2} is

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  11. 11
    ∫5tan⁡xtan⁡x−2 dx=x+alog⁡∣sin⁡x−2cos⁡x∣+c\displaystyle\int\frac{5\tan x}{\tan x - 2}\,dx = x + a\log|\sin x - 2\cos x| + c, then the value of aa is

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  12. 12
    If (a⃗×b⃗)×c⃗=−5a⃗+4b⃗(\vec{a}\times\vec{b})\times\vec{c}=-5\vec{a}+4\vec{b} and a⃗⋅b⃗=3\vec{a}\cdot\vec{b}=3, then the value of a⃗×(b⃗×c⃗)\vec{a}\times(\vec{b}\times\vec{c}) is

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  13. 13
    The value of ∫(x2−1) dxx32x4−2x2+1\displaystyle\int\frac{(x^2-1)\,dx}{x^3\sqrt{2x^4-2x^2+1}} is

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  14. 14
    The domain of the function f(x)=sin⁡−1 ⁣(∣x∣+5x2+1)f(x)=\sin^{-1}\!\left(\dfrac{|x|+5}{x^2+1}\right) is (−∞,−a]∪[a,∞)(-\infty,-a]\cup[a,\infty). Then aa is equal to

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  15. 15
    Mean and variance of six observations are 8 and 16 respectively. If each observation is multiplied by 3, then new variance of the resulting observations is

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  16. 16
    The value of aa, so that the volume of parallelepiped formed by i^+aj^+k^\hat{i}+a\hat{j}+\hat{k}, j^+ak^\hat{j}+a\hat{k} and ai^+k^a\hat{i}+\hat{k} becomes minimum, is

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  17. 17
    If B=(3a−1131−113)B=\begin{pmatrix}3&a&-1\\1&3&1\\-1&1&3\end{pmatrix} is the adjoint of a 3×33\times3 matrix AA and ∣A∣=4|A|=4, then aa is equal to

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  18. 18
    The negation of the statement pattern ∼s∨(∼r∧s)\sim s\vee(\sim r\wedge s) is equivalent to

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  19. 19
    Let f be a differentiable function such that f(1)=2f(1)=2 and f′(x)=f(x)f'(x)=f(x), for all x∈Rx\in\mathbb{R}. If h(x)=f(f(x))h(x)=f(f(x)), then h′(1)h'(1) is equal to

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  20. 20
    If y=tan⁡−1 ⁣(log⁡(e/x2)log⁡(ex2))+tan⁡−1 ⁣(4+2log⁡x1−8log⁡x)y=\tan^{-1}\!\left(\dfrac{\log(e/x^2)}{\log(ex^2)}\right)+\tan^{-1}\!\left(\dfrac{4+2\log x}{1-8\log x}\right), then dydx\dfrac{dy}{dx} is

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  21. 21
    The three ships namely A, B and C sail from India to Africa. If the odds in favour of the ships reaching safely are 2:5,  3:72:5,\;3:7 and 6:116:11 respectively, then probability of all of them arriving safely is

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  22. 22
    If x=3tan⁡tx=3\tan t and y=3sec⁡ty=3\sec t, then the value of d2ydx2\dfrac{d^2y}{dx^2} at t=π4t=\dfrac{\pi}{4} is

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  23. 23
    Let S={t∈R:f(x)=∣x−π∣(e∣x∣−1)sin⁡∣x∣S=\{t\in\mathbb{R}: f(x)=|x-\pi|(e^{|x|}-1)\sin|x| is not differentiable at t}t\}, then SS is

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  24. 24
    lim⁡x→aa+2x−3x3a+x−2x=\displaystyle\lim_{x\to a}\frac{\sqrt{a+2x}-\sqrt{3x}}{\sqrt{3a+x}-2\sqrt{x}}=

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  25. 25
    ∫ex(1−cot⁡x+cot⁡2x) dx=\displaystyle\int e^x(1-\cot x+\cot^2 x)\,dx =

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  26. 26
    The perpendicular distance of the origin from the plane x−3y+4z−6=0x-3y+4z-6=0 is

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  27. 27
    If tan⁡−1a+tan⁡−1b+tan⁡−1c=π\tan^{-1}a+\tan^{-1}b+\tan^{-1}c=\pi, then which of the following statement is true?

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  28. 28
    The plane through the intersection of planes x+y+z=1x+y+z=1 and 2x+3y−z+4=02x+3y-z+4=0 and parallel to Y-axis also passes through the point

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  29. 29
    If yy is a function of xx and log⁡(x+y)=2xy\log(x+y)=2xy, then dydx\dfrac{dy}{dx} at x=0x=0 is

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  30. 30
    The parametric equation of the circle x2+y2+2x−4y−4=0x^2+y^2+2x-4y-4=0 are

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  31. 31
    The value of cc for the function f(x)=log⁡xf(x)=\log x on [1,e][1,e] if LMVT can be applied, is

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  32. 32
    The value of tan⁡−1(1)+cos⁡−1 ⁣(−12)+sin⁡−1 ⁣(−12)\tan^{-1}(1)+\cos^{-1}\!\left(-\dfrac{1}{2}\right)+\sin^{-1}\!\left(-\dfrac{1}{2}\right) is

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  33. 33
    If f(x)=4x4 ⁣[1−cos⁡x2−cos⁡x4+cos⁡x2⋅cos⁡x4]f(x)=\dfrac{4}{x^4}\!\left[1 - \cos\dfrac{x}{2} - \cos\dfrac{x}{4} + \cos\dfrac{x}{2}\cdot\cos\dfrac{x}{4}\right] is continuous at x=0x=0, then f(0)f(0) is

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  34. 34
    Let f(x)=ex−xf(x)=e^x-x and g(x)=x2−x,  ∀x∈Rg(x)=x^2-x,\;\forall x\in\mathbb{R}, then the set of all x∈Rx\in\mathbb{R}, where the function h(x)=(f∘g)(x)h(x)=(f\circ g)(x) is increasing, is

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  35. 35
    The value of 10Cr11Cr\dfrac{^{10}C_r}{^{11}C_r}, when both the numerator and denominator are at their greatest values, is

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  36. 36
    If a⃗=mb⃗+nc⃗\vec{a}=m\vec{b}+n\vec{c}, where a⃗=4i^+13j^−18k^\vec{a}=4\hat{i}+13\hat{j}-18\hat{k}, b⃗=i^−2j^+3k^\vec{b}=\hat{i}-2\hat{j}+3\hat{k}, c⃗=2i^+3j^−4k^\vec{c}=2\hat{i}+3\hat{j}-4\hat{k}, then m+n=m+n=

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  37. 37
    In a game, 3 coins are tossed. A person is paid ₹7 if he gets all heads or all tails; and he is supposed to pay ₹3 if he gets one head or two heads. The amount he can expect to win on an average per game is

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  38. 38
    The value of ∫dxx2(x4+1)3/4\displaystyle\int\frac{dx}{x^2(x^4+1)^{3/4}} is

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  39. 39
    A fair die is tossed twice in succession. If X denotes the number of sixes in two tosses, then the probability distribution of X is given by

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  40. 40
    The displacement 'S' of a moving particle at a time is given by S=5+48t−t3S=5+48t-t^3. Then its acceleration when the velocity is zero, is

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  41. 41
    The general solution of the differential equation dydx+3x21+x3y=1x3+1\dfrac{dy}{dx}+\dfrac{3x^2}{1+x^3}y=\dfrac{1}{x^3+1} is

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  42. 42
    The differential equation of all circles, passing through the origin and having their centres on the X-axis, is

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  43. 43
    If pp and qq are true statements and rr and ss are false statements, then the truth values of the statement patterns (p∧q)∨r(p\wedge q)\vee r and (p∨q)↔(q∧r)(p\vee q)\leftrightarrow(q\wedge r) are respectively

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  44. 44
    In a certain culture of bacteria, the rate of increase is proportional to the number of bacteria present at that instant. It is found that there are 10,000 bacteria at the end of 3 hours and 40,000 bacteria at the end of 5 hours, then the number of bacteria present in the beginning are

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  45. 45
    ∫−π/2π/2[f(x)+f(−x)][g(x)−g(−x)] dx\displaystyle\int_{-\pi/2}^{\pi/2}[f(x)+f(-x)][g(x)-g(-x)]\,dx

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  46. 46
    If the sum of mean and variance of a Binomial Distribution is 152\dfrac{15}{2} for 10 trials, then the variance is

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  47. 47
    If p⃗=i^+j^+k^\vec{p}=\hat{i}+\hat{j}+\hat{k} and q⃗=i^+2j^+k^\vec{q}=\hat{i}+2\hat{j}+\hat{k}. Then a vector of magnitude 535\sqrt{3} units perpendicular to the vector q⃗\vec{q} and coplanar with p⃗\vec{p} and q⃗\vec{q} is

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  48. 48
    ∫01cos⁡−1x dx=\displaystyle\int_0^1 \cos^{-1}x\,dx=

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  49. 49
    If a⃗\vec{a} and b⃗\vec{b} are two unit vectors such that a⃗+2b⃗\vec{a}+2\vec{b} and 5a⃗−4b⃗5\vec{a}-4\vec{b} are perpendicular to each other, then the angle between a⃗\vec{a} and b⃗\vec{b} is

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  50. 50
    The argument of 1+i33−i\dfrac{1+i\sqrt{3}}{\sqrt{3}-i}, i=−1i=\sqrt{-1}, is

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