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MHT-CET 2023 (9 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 A=(1111a3322)A = \begin{pmatrix}1&1&1\\1&a&3\\3&2&2\end{pmatrix} and B=(−20b7−1−2c11)B = \begin{pmatrix}-2&0&b\\7&-1&-2\\c&1&1\end{pmatrix} and if matrix B is the inverse of matrix A, then value of 4a+2b−c4a+2b-c is

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  2. 2
    If ∫f(x) dx=ψ(x)\int f(x)\,dx = \psi(x), then ∫x5f(x3) dx\int x^5 f(x^3)\,dx is equal to

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  3. 3
    The population of a town increases at a rate proportional to the population at that time. If the population increases from 40 thousand to 80 thousand in 40 years, then the population in another 40 years will be

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  4. 4
    If X is a random variable with distribution given below: X=xi: 0, 1, 2, 3; P(X=xi): k, 3k, 3k, k Then the value of k and its variance are respectively given by

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  5. 5
    An urn contains nine balls of which three are red, four are blue and two are green. Three balls are drawn at random without replacement from the urn. The probability that the three balls have different colours is

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  6. 6
    The area (in sq. units) bounded by the curves y=(x+1)2y=(x+1)^2, y=(x−1)2y=(x-1)^2 and the line y=14y=\frac{1}{4} is

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  7. 7
    If ∫dx3sin⁡11xcos⁡x=−38f(x)+32g(x)+c\int\frac{dx}{3\sin^{11}x\cos x} = -\frac{3}{8}f(x) + \frac{3}{2}g(x) + c then

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  8. 8
    A random variable X takes values −1,0,1,2-1,0,1,2 with probabilities 1+3p4,1−p4,1+2p4,1−4p4\frac{1+3p}{4},\frac{1-p}{4},\frac{1+2p}{4},\frac{1-4p}{4} respectively, where pp varies over R\mathbb{R}. Then the minimum and maximum values of the mean of X are respectively.

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  9. 9
    Let C be a curve given by y(x)=1+4x−3y(x) = 1 + \sqrt{4x-3}, x>34x > \frac{3}{4}. If P is a point on C, such that the tangent at P has slope 23\frac{2}{3}, then a point through which the normal at P passes, is

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  10. 10
    The equation of the circle, the end points of whose diameter are the centres of the circles x2+y2+6x−14y+5=0x^2+y^2+6x-14y+5=0 and x2+y2−4x+10y−4=0x^2+y^2-4x+10y-4=0 is

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  11. 11
    The equation of the plane, passing through the point (1,1,1)(1,1,1) and perpendicular to the planes 2x+y−2z=52x+y-2z=5 and 3x−6y−2z=73x-6y-2z=7, is

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  12. 12
    If for x∈(0,14)x\in\left(0,\frac{1}{4}\right), the derivative of tan⁡−1 ⁣(6xx1−9x3)\tan^{-1}\!\left(\frac{6x\sqrt{x}}{1-9x^3}\right) is x⋅g(x)x\cdot g(x), then g(x)g(x) equals

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  13. 13
    If the vectors ai^+j^+k^a\hat{i}+\hat{j}+\hat{k}, i^+bj^+k^\hat{i}+b\hat{j}+\hat{k}, i^+j^+ck^\hat{i}+\hat{j}+c\hat{k} (a≠b,c≠1a\neq b, c\neq1) are coplanar, then 11−a+11−b+11−c\frac{1}{1-a}+\frac{1}{1-b}+\frac{1}{1-c} has the value

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  14. 14
    The sides of a triangle are sin⁡θ\sin\theta, cos⁡θ\cos\theta and 1+sin⁡θcos⁡θ\sqrt{1+\sin\theta\cos\theta} for some 0<θ<π20<\theta<\frac{\pi}{2}, then the greatest angle of a triangle is

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  15. 15
    If lim⁡x→∞(x2+x+1x+1−ax−b)=4\lim_{x\to\infty}\left(\frac{x^2+x+1}{x+1} - ax - b\right) = 4 then

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  16. 16
    If y=y(x)y=y(x) is the solution of the differential equation xdydx+2y=x2x\frac{dy}{dx}+2y=x^2 satisfying y(1)=1y(1)=1, then the value of y ⁣(12)y\!\left(\frac{1}{2}\right) is

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  17. 17
    The value of cos⁡−1 ⁣(12(cos⁡9π10−sin⁡9π10))\cos^{-1}\!\left(\frac{1}{\sqrt{2}}\left(\cos\frac{9\pi}{10}-\sin\frac{9\pi}{10}\right)\right) is

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  18. 18
    Consider the statements: (A) If 3+3=73+3=7, then 4+3=84+3=8. (B) If 5+3=85+3=8, then earth is flat. (C) If both (A) and (B) are true, then 5+6=175+6=17. Then which of the following statements is correct?

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  19. 19
    If α+β=π2\alpha+\beta=\frac{\pi}{2} and β+γ=α\beta+\gamma=\alpha, then tan⁡α\tan\alpha equals

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  20. 20
    If f(x)=log⁡xxf(x) = \frac{\log x}{x} (x>0)(x>0), then it is increasing in

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  21. 21
    If a⃗\vec{a}, b⃗\vec{b} and c⃗\vec{c} are three non-coplanar vectors, then (a⃗+b⃗+c⃗)⋅[(a⃗+b⃗)×(a⃗+c⃗)](\vec{a}+\vec{b}+\vec{c})\cdot[(\vec{a}+\vec{b})\times(\vec{a}+\vec{c})] equals

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  22. 22
    In a class of 300 students, every student reads 5 newspapers and every newspaper is read by 60 students. Then the number of newspapers is

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  23. 23
    The maximum value of log⁡xx\frac{\log x}{x} is

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  24. 24
    Derivative of exe^{x} w.r.t. x\sqrt{x} is

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  25. 25
    The curve satisfying the differential equation y dx−(x+3y2) dy=0y\,dx - (x+3y^2)\,dy = 0 and passing through the point (1,1)(1,1) also passes through the point

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  26. 26
    The maximum value of z=x+yz=x+y, subjected to x+y≤10x+y\leq10, 5x+3y≥155x+3y\geq15, x≤6x\leq6, x,y≥0x,y\geq0

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  27. 27
    The distance of the point (1,3,−7)(1,3,-7) from the plane passing through the point (1,−1,−1)(1,-1,-1) having normal perpendicular to both the lines x−11=y+2−2=z−43\frac{x-1}{1}=\frac{y+2}{-2}=\frac{z-4}{3} and x−22=y+1−1=z+7−1\frac{x-2}{2}=\frac{y+1}{-1}=\frac{z+7}{-1} is

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  28. 28
    Domain of definition of the real valued function f(x)=sin⁡−1(2x)+π6f(x) = \sqrt{\sin^{-1}(2x)+\frac{\pi}{6}} is

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  29. 29
    The value of m, such that x−41=y−21=z−m2\frac{x-4}{1}=\frac{y-2}{1}=\frac{z-m}{2} lies in the plane 2x−4y+z=72x-4y+z=7, is

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  30. 30
    Suppose that p⃗\vec{p}, q⃗\vec{q} and r⃗\vec{r} are three non-coplanar vectors in R3\mathbb{R}^3. Let the components of a vector s⃗\vec{s} along p⃗\vec{p}, q⃗\vec{q} and r⃗\vec{r} be 4, 3 and 5 respectively. If the components of this vector s⃗\vec{s} along (−p⃗+q⃗+r⃗)(-\vec{p}+\vec{q}+\vec{r}), (p⃗−q⃗+r⃗)(\vec{p}-\vec{q}+\vec{r}) and (−p⃗−q⃗+r⃗)(-\vec{p}-\vec{q}+\vec{r}) are xx, yy and zz respectively, then the value of 2x+y+z2x+y+z is

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  31. 31
    ∫x4+x2+1x2−x+1 dx\int\frac{x^4+x^2+1}{x^2-x+1}\,dx is equal to

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  32. 32
    If the curves y2=6xy^2=6x, 9x2+by2=169x^2+by^2=16 intersect each other at right angles, then the value of bb is

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  33. 33
    Let a⃗=2i^+j^−2k^\vec{a}=2\hat{i}+\hat{j}-2\hat{k} and b⃗=i^+j^\vec{b}=\hat{i}+\hat{j}. If c⃗\vec{c} is a vector such that a⃗⋅c⃗=∣c⃗∣\vec{a}\cdot\vec{c}=|\vec{c}|, ∣c⃗−a⃗∣=22|\vec{c}-\vec{a}|=2\sqrt{2} and the angle between (a⃗×b⃗)(\vec{a}\times\vec{b}) and c⃗\vec{c} is 60∘60^\circ, then the value of ∣(a⃗×b⃗)×c⃗∣|(\vec{a}\times\vec{b})\times\vec{c}| is

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  34. 34
    The mean and the standard deviation of 10 observations are 20 and 2 respectively. Each of these 10 observations is multiplied by pp and then reduced by qq, where p≠0p\neq0 and q≠0q\neq0. If the new mean and new standard deviation (s.d.) become half of the original values, then qq is equal to

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  35. 35
    If Mean Value Theorem holds for the function f(x)=(x−1)(x−2)(x−3)f(x)=(x-1)(x-2)(x-3), x∈[0,4]x\in[0,4] then the values of c as per the theorem are

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  36. 36
    The value of I=∫−π/2π/2x2cos⁡x1+e−x dxI = \int_{-\pi/2}^{\pi/2}\frac{x^2\cos x}{1+e^{-x}}\,dx is equal to

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  37. 37
    If a⃗=i^+j^+k^\vec{a}=\hat{i}+\hat{j}+\hat{k}, a⃗⋅b⃗=1\vec{a}\cdot\vec{b}=1 and a⃗×b⃗=j^−k^\vec{a}\times\vec{b}=\hat{j}-\hat{k}, then b⃗\vec{b} is

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  38. 38
    If the mean and the variance of Binomial variate X are 2 and 1 respectively, then the probability that X takes a value greater than or equal to one is

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  39. 39
    The value of I=∫(x−1)ex(x+1)3 dxI=\int\frac{(x-1)e^x}{(x+1)^3}\,dx is

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  40. 40
    If Z=−21+3iZ=\frac{-2}{1+\sqrt{3}i}, i=−1i=\sqrt{-1}, then the value of arg⁡(Z)\arg(Z) is

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  41. 41
    In (0,2π)(0,2\pi), the number of solutions of tan⁡θ+sec⁡θ=2cos⁡θ\tan\theta+\sec\theta=2\cos\theta are

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  42. 42
    If the area of the parallelogram with a⃗\vec{a} and b⃗\vec{b} as two adjacent sides is 15 square units, then the area (in square units) of the parallelogram, having 3a⃗+2b⃗3\vec{a}+2\vec{b} and a⃗+3b⃗\vec{a}+3\vec{b} as two adjacent sides, is

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  43. 43
    If an equation hxy+gx+fy+c=0hxy+gx+fy+c=0 represents a pair of lines, then

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  44. 44
    The general solution of sin⁡x−3sin⁡2x+sin⁡3x=cos⁡x−3cos⁡2x+cos⁡3x\sin x-3\sin 2x+\sin 3x=\cos x-3\cos 2x+\cos 3x is

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  45. 45
    The length of the perpendicular from the point A(1,−2,−3)A(1,-2,-3) on the line x−12=y+3−1=z+1−2\frac{x-1}{2}=\frac{y+3}{-1}=\frac{z+1}{-2} is

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  46. 46
    The straight line 2x−3y+17=02x-3y+17=0 is perpendicular to the line passing through the points (7,17)(7,17) and (15,β)(15,\beta), then β\beta equals

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  47. 47
    If f(x)=x2−xx2+2xf(x)=\frac{x^2-x}{x^2+2x} then ddxf−1(x)\frac{d}{dx}f^{-1}(x) at x=2x=2 is

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  48. 48
    For the triangle ABC, with usual notations, if the angles A, B, C are in A.P. and m∠A=30∘m\angle A=30^\circ, c=3c=3, then the values of a and b are respectively

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  49. 49
    Let pp, qq, rr be three statements such that the truth value of (p∧q)→(∼q∨r)(p\wedge q)\to(\sim q\vee r) is F. Then the truth values of pp, qq, rr are respectively

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  50. 50
    Let k be a non-zero real number. If f(x)={(ex−1)2sin⁡(xk)log⁡(1+x4),x≠012,x=0f(x)=\begin{cases}\dfrac{(e^x-1)^2}{\sin\left(\frac{x}{k}\right)\log\left(1+\frac{x}{4}\right)}, & x\neq0 \\ 12, & x=0\end{cases} is a continuous function, then the value of k is

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