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MHT-CET 2023 (11 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 number of arrangements, of the letters of the word MANAMA in which two M's do not appear adjacent, is

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  2. 2
    A random variable X has the following probability distribution: X=x: 1,2,3,4,5,6,7,8 with P(X=x): 0.15,0.23,0.10,0.12,0.20,0.08,0.07,0.05. For the event E={X is a prime number}E=\{X \text{ is a prime number}\}, F={X<4}F=\{X<4\}, then P(E∪F)P(E \cup F) is

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  3. 3
    ∫x+1x(1+xex)2 dx=\int\frac{x+1}{x(1+xe^x)^2}\,dx =

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  4. 4
    An ice ball melts at the rate which is proportional to the amount of ice at that instant. Half of the quantity of ice melts in 15 minutes. x0x_0 is the initial quantity of ice. If after 30 minutes the amount of ice left is kx0kx_0, then the value of k is

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  5. 5
    Contrapositive of the statement 'If two numbers are not equal, then their squares are not equal', is

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  6. 6
    If f:[1,∞)→[2,∞)f:[1,\infty)\to[2,\infty) is given by f(x)=x+1xf(x) = x+\frac{1}{x} then f−1(x)f^{-1}(x) equals

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  7. 7
    The angles of a triangle are in the ratio 5:1:6, then ratio of the smallest side to the greatest side is

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  8. 8
    Let y=y(x)y=y(x) be the solution of the differential equation xdydx+y=xlog⁡x, (x>1)x\frac{dy}{dx}+y = x\log x,\ (x>1). If 2(y(2))=log⁡4−12(y(2)) = \log 4-1 then the value of y(e)y(e) is

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  9. 9
    The area (in sq. units) of the region bounded by y−x=2y-x=2 and x2=yx^2=y is equal to

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  10. 10
    Let a⃗,b⃗\vec{a},\vec{b} and c⃗\vec{c} be three unit vectors such that a⃗×(b⃗×c⃗)=32(b⃗+c⃗)\vec{a}\times(\vec{b}\times\vec{c}) = \frac{\sqrt{3}}{2}(\vec{b}+\vec{c}). If b⃗\vec{b} is not parallel to c⃗\vec{c}, then the angle between a⃗\vec{a} and b⃗\vec{b} is

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  11. 11
    lim⁡x→0xtan⁡2x−2xtan⁡x(1−cos⁡2x)2\lim_{x\to 0}\frac{x\tan 2x-2x\tan x}{(1-\cos 2x)^2} is

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  12. 12
    If y=alog⁡x+bx2+xy = a\log x+bx^2+x has its extreme values at x=−1x=-1 and x=2x=2, then the value of ab+ba\frac{a}{b}+\frac{b}{a} is

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  13. 13
    If a⃗=110(3i^+k^)\vec{a}=\frac{1}{\sqrt{10}}(3\hat{i}+\hat{k}) and b⃗=17(2i^+3j^−6k^)\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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  14. 14
    If f(x)=log⁡x2(log⁡x)f(x) = \log_{x^2}(\log x), then at x=ex=e, f′(x)f'(x) has the value

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  15. 15
    If I=∫0π/4log⁡(1+tan⁡x) dxI = \int_0^{\pi/4}\log(1+\tan x)\,dx, then value of I is

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  16. 16
    The parametric equations of the circle x2+y2−ax−by=0x^2+y^2-ax-by=0 are

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  17. 17
    The curve y=ax3+bx2+cx+5y=ax^3+bx^2+cx+5 touches the x-axis at (−2,0)(-2,0) and cuts the y-axis at a point Q where its gradient is 3, then values of a,b,ca,b,c respectively, are

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  18. 18
    A variable plane passes through the fixed point (3,2,1)(3,2,1) and meets X, Y and Z axes at points A, B and C respectively. A plane is drawn parallel to YZ-plane through A, a second plane parallel to ZX-plane through B, a third plane parallel to XY-plane through C. Then locus of the point of intersection of these three planes, is

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  19. 19
    The value of sin⁡(cos⁡−1(−13)−sin⁡−113)\sin\left(\cos^{-1}\left(-\frac{1}{\sqrt{3}}\right)-\sin^{-1}\frac{1}{\sqrt{3}}\right) is

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  20. 20
    Let a⃗,b⃗\vec{a},\vec{b}, and c⃗\vec{c} be three non-zero vectors such that no two of these are collinear. If the vector a⃗+2b⃗\vec{a}+2\vec{b} is collinear with c⃗\vec{c} and b⃗+3c⃗\vec{b}+3\vec{c} is collinear with a⃗\vec{a}, then a⃗+2b⃗+6c⃗\vec{a}+2\vec{b}+6\vec{c} equals

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  21. 21
    Let A, B and C be three events, which are pairwise independent and E denote the complement of an event E. If P(A∩B∩C)=0P(A\cap B\cap C)=0 and P(C)>0P(C)>0, then P(A‾∩B‾/C)P\left(\overline{A}\cap\overline{B}/C\right) is equal to

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  22. 22
    A production unit makes special type of metal chips by combining copper and brass. The standard weight of the chip must be at least 5 gms. The basic ingredients i.e. copper and brass cost ₹8 and ₹5 per gm. The durability considerations dictate that the metal chip must not contain more than 4 gms of brass and should contain minimum 2 gms of copper. Then the minimum cost of the metal chip satisfying the above conditions is

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  23. 23
    The distance of the point (1,−5,9)(1,-5,9) from the plane x−y+z=5x-y+z=5 measured along the line x=y=zx=y=z is

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  24. 24
    If f(x)=1+x; g(x)=log⁡xf(x)=1+x;\ g(x)=\log x, then ∫g(f(x)) dx\int g(f(x))\,dx is equal to

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  25. 25
    The general solution of sin⁡x+cos⁡x=1\sin x+\cos x=1 is

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  26. 26
    In a △PQR\triangle PQR, m∠R=π2m\angle R = \frac{\pi}{2}. If tan⁡P2\tan\frac{P}{2} and tan⁡Q2\tan\frac{Q}{2} are the roots of the equation ax2+bx+c=0 (a≠0)ax^2+bx+c=0\ (a\neq0), then

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  27. 27
    If for some α∈R\alpha\in\mathbb{R}, the lines L1:x+12=y−2−1=z−11L_1:\frac{x+1}{2}=\frac{y-2}{-1}=\frac{z-1}{1} and L2:x+2α=y+15−α=z+11L_2:\frac{x+2}{\alpha}=\frac{y+1}{5-\alpha}=\frac{z+1}{1} are coplanar, then the line L2L_2 passes through the point

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  28. 28
    ∫cos⁡(log⁡x) dx=\int\cos(\log x)\,dx =

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  29. 29
    If a⃗,b⃗,c⃗\vec{a},\vec{b},\vec{c} are non-coplanar unit vectors such that a⃗×(b⃗×c⃗)=(b⃗+c⃗)2\vec{a}\times(\vec{b}\times\vec{c})=\frac{(\vec{b}+\vec{c})}{\sqrt{2}} then the angle between a⃗\vec{a} and b⃗\vec{b} is

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  30. 30
    The joint equation of pair of lines through the origin, each of which makes an angle of 30°30° with Y-axis, is

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  31. 31
    Let f(x)={1−cos⁡4xx2,x<0a,x=0x16+x−4,x>0f(x) = \begin{cases}\frac{1-\cos 4x}{x^2}, & x<0\\ a, & x=0\\ \frac{\sqrt{x}}{\sqrt{16+\sqrt{x}}-4}, & x>0\end{cases}. If f(x)f(x) is continuous at x=0x=0, then the value of a is

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  32. 32
    The number of solutions of tan⁡x+sec⁡x=2cos⁡x\tan x+\sec x=2\cos x in [0,2π][0,2\pi] is

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  33. 33
    The maximum value of the function f(x)=2x3−15x2+36x−48f(x) = 2x^3-15x^2+36x-48 on the set A={x∣x2+20≤9x}A = \{x\mid x^2+20\leq9x\} is

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  34. 34
    The number of unit vectors perpendicular to a⃗=(1,1,0)\vec{a}=(1,1,0) and b⃗=(0,1,1)\vec{b}=(0,1,1) is

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  35. 35
    If the normal to the curve y=f(x)y=f(x) at the point (3,4)(3,4) makes an angle of 3π4\frac{3\pi}{4} with the positive X-axis, then the value of f′(3)f'(3) is

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  36. 36
    ∫2x+57−6x−x2 dx=A7−6x−x2+Bsin⁡−1(x+34)+c\int\frac{2x+5}{\sqrt{7-6x-x^2}}\,dx = A\sqrt{7-6x-x^2}+B\sin^{-1}\left(\frac{x+3}{4}\right)+c (where c is a constant of integration) then the value of A+B is

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  37. 37
    A straight line L through the point (3,−2)(3,-2) is inclined at an angle of 60°60° to the line 3x+y=1\sqrt{3}x+y=1. If L also intersects the X-axis, then the equation of L is

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  38. 38
    If y(x)y(x) is the solution of the differential equation (x+2)dydx=x2+4x−9, x≠−2(x+2)\frac{dy}{dx} = x^2+4x-9,\ x\neq-2 and y(0)=0y(0)=0, then y(−4)y(-4) is equal to

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  39. 39
    If the sides of a triangle a,b,ca,b,c are in A.P., then with usual notations, acos⁡2C2+ccos⁡2A2a\cos^2\frac{C}{2}+c\cos^2\frac{A}{2} is

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  40. 40
    A random variable X takes the values 0,1,2,3,…0,1,2,3,\ldots with probability P(X=x)=k(x+1)(15)xP(X=x) = k(x+1)\left(\frac{1}{5}\right)^x, where k is a constant, then P(X=0)P(X=0) is

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  41. 41
    If A=(2174)A = \begin{pmatrix}2 & 1\\7 & 4\end{pmatrix} then (A2−5A)−1(A^2-5A)^{-1} is

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  42. 42
    The following statement (p→q)→((∼p→q)→q)(p\to q)\to((\sim p\to q)\to q) is

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  43. 43
    Let ω=−12+i32, i=−1\omega = -\frac{1}{2}+i\frac{\sqrt{3}}{2},\ i=\sqrt{-1}, then the value of ∣1111−1−ω2ω21ω2ω4∣\begin{vmatrix}1&1&1\\1&-1-\omega^2&\omega^2\\1&\omega^2&\omega^4\end{vmatrix} is

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  44. 44
    Let f be twice differentiable function such that f′′(x)=−f(x)f''(x)=-f(x), f′(x)=g(x)f'(x)=g(x) and h(x)=(f(x))2+(g(x))2h(x)=(f(x))^2+(g(x))^2. If h(5)=1h(5)=1, then the value of h(10)h(10) is

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  45. 45
    Let P(3,2,6)P(3,2,6) be a point in space and Q be a point on the line r⃗=i^−j^+2k^+μ(−3i^+j^+5k^)\vec{r}=\hat{i}-\hat{j}+2\hat{k}+\mu(-3\hat{i}+\hat{j}+5\hat{k}). Then the value of μ\mu for which the vector PQ→\overrightarrow{PQ} is parallel to the plane x−4y+3z=1x-4y+3z=1 is

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  46. 46
    Let f(x)=xa2+x2−d−xb2+(d−x)2, x∈Rf(x)=\frac{x}{\sqrt{a^2+x^2}}-\frac{d-x}{\sqrt{b^2+(d-x)^2}},\ x\in\mathbb{R} where a,b,da,b,d are non-zero real constants. Then

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  47. 47
    If the vectors a⃗=i^−j^+2k^\vec{a}=\hat{i}-\hat{j}+2\hat{k}, b⃗=2i^+4j^+k^\vec{b}=2\hat{i}+4\hat{j}+\hat{k} and c⃗=λi^+j^+μk^\vec{c}=\lambda\hat{i}+\hat{j}+\mu\hat{k} are mutually orthogonal, then (λ,μ)≡(\lambda,\mu)\equiv

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  48. 48
    If y=(sin⁡x)tan⁡xy=(\sin x)^{\tan x}, then dydx\frac{dy}{dx} is equal to

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  49. 49
    If for some x∈R+∪{0}x\in\mathbb{R}^+\cup\{0\}, the frequency distribution of the marks obtained by 20 students in a test is: Marks: 2,3,5,7; Frequency: (x+1)2, 2x−5, x2−3x, x(x+1)^2,\ 2x-5,\ x^2-3x,\ x. Then the mean of the marks is

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  50. 50
    One hundred identical coins, each with probability p, of showing up heads are tossed once. If 0<p<10<p<1 and the probability of heads showing on 50 coins is equal to that of heads showing on 51 coins, then the value of p is

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