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MHT-CET 2023 (4 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 solution set of the equation sin⁡2θ−cos⁡θ=14\sin^2\theta - \cos\theta = \frac{1}{4} in the interval [0,2π][0, 2\pi] is

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  2. 2
    If the points (1,−1,λ)(1,-1,\lambda) and (−3,0,1)(-3,0,1) are equidistant from the plane 3x−4y−12z+13=03x-4y-12z+13=0, then the sum of all possible values of λ\lambda is

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  3. 3
    If a⃗=i^−2j^+3k^\vec{a} = \hat{i}-2\hat{j}+3\hat{k} and b⃗=2i^+3j^−k^\vec{b} = 2\hat{i}+3\hat{j}-\hat{k}, then the angle between the vectors (2a⃗+b⃗)(2\vec{a}+\vec{b}) and (a⃗+2b⃗)(\vec{a}+2\vec{b}) is

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  4. 4
    The maximum value of the objective function z=4x+6yz=4x+6y subject to 3x+2y≤12,x+y≥4,x,y≥03x+2y\leq12, x+y\geq4, x, y\geq0 is

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  5. 5
    ddxcos⁡−1 ⁣(x−1xx+1x)=\frac{d}{dx}\cos^{-1}\!\left(\frac{x-\frac{1}{x}}{x+\frac{1}{x}}\right)=

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  6. 6
    If a⃗,b⃗,c⃗\vec{a}, \vec{b}, \vec{c} are non-coplanar vectors and p⃗=b⃗×c⃗[a⃗b⃗c⃗],q⃗=c⃗×a⃗[a⃗b⃗c⃗],r⃗=a⃗×b⃗[a⃗b⃗c⃗]\vec{p}=\frac{\vec{b}\times\vec{c}}{[\vec{a}\vec{b}\vec{c}]}, \vec{q}=\frac{\vec{c}\times\vec{a}}{[\vec{a}\vec{b}\vec{c}]}, \vec{r}=\frac{\vec{a}\times\vec{b}}{[\vec{a}\vec{b}\vec{c}]}, then 2a⃗⋅p⃗+b⃗⋅q⃗+c⃗⋅r⃗=2\vec{a}\cdot\vec{p}+\vec{b}\cdot\vec{q}+\vec{c}\cdot\vec{r}=

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  7. 7
    The approximate value of tan⁡−1(0.999)\tan^{-1}(0.999) is (use π=3.1415\pi=3.1415)

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  8. 8
    Let P be a plane passing through the points (2,1,0),(4,1,1)(2,1,0),(4,1,1) and (5,0,1)(5,0,1) and R be the point (2,1,6)(2,1,6). Then image of R in the plane P is

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  9. 9
    If O(0,0),A(1,2)O(0,0), A(1,2) and B(3,4)B(3,4) are the vertices of triangle OAB, then the joint equation of the altitude and median drawn from O is

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  10. 10
    The equation of the plane, passing through the point (−1,2,−3)(-1,2,-3) and parallel to the lines x−13=y−22=z−4\frac{x-1}{3}=\frac{y-2}{2}=z-4 and x2=y−1−3=z−22\frac{x}{2}=\frac{y-1}{-3}=\frac{z-2}{2}, is

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  11. 11
    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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  12. 12
    The combined equation of two lines through the origin and making an angle of 45° with the line 3x+y=03x+y=0, is

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  13. 13
    The general solution of the differential equation 1xdydx=tan⁡−1x\frac{1}{x}\frac{dy}{dx}=\tan^{-1}x is

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  14. 14
    The area (in square units) in the first quadrant bounded by the curve y=x2+2y=x^2+2 and the lines y=x+1,x=0y=x+1, x=0 and x=3x=3, is

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  15. 15
    The co-ordinates of the point where the line through A(3,4,1)A(3,4,1) and B(5,1,6)B(5,1,6) crosses the xy-plane are

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  16. 16
    The value of tan⁡−1(3)−sec⁡−1(−2)cosec−1(−2)+cos⁡−1(−12)\frac{\tan^{-1}(\sqrt{3})-\sec^{-1}(-2)}{\text{cosec}^{-1}(-\sqrt{2})+\cos^{-1}\left(-\frac{1}{2}\right)} is

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  17. 17
    The value of ∫dxx2(x4+1)3/4\int\frac{dx}{x^2(x^4+1)^{3/4}} is

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  18. 18
    If (a+b)cos⁡C+(b+c)cos⁡A+(c+a)cos⁡B=72(a+b)\cos C+(b+c)\cos A+(c+a)\cos B=72 and if a=18,b=24a=18, b=24, then area of the triangle ABC is

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  19. 19
    If cot⁡−1(7)+cot⁡−1(8)+cot⁡−1(18)=cot⁡−1x\cot^{-1}(7)+\cot^{-1}(8)+\cot^{-1}(18)=\cot^{-1}x, then the value of x is

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  20. 20
    If P(X=2)=0.3,P(X=3)=0.4,P(X=4)=0.3P(X=2)=0.3, P(X=3)=0.4, P(X=4)=0.3, then the variance of random variable X is

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  21. 21
    The Cartesian equation of a line is 2x−2=3y+1=6z−22x-2=3y+1=6z-2, then the vector equation of the line is

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  22. 22
    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}. Let c⃗\vec{c} be a vector such that ∣c⃗−a⃗∣=3|\vec{c}-\vec{a}|=3 and ∣(a⃗×b⃗)×c⃗∣=3|(\vec{a}\times\vec{b})\times\vec{c}|=3 and the angle between c⃗\vec{c} and a⃗×b⃗\vec{a}\times\vec{b} is 30°, then a⃗⋅c⃗\vec{a}\cdot\vec{c} is equal to

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  23. 23
    If ∫4ex−252ex−5dx=Ax+Blog⁡∣2ex−5∣+c\int\frac{4e^x-25}{2e^x-5}dx = Ax+B\log|2e^x-5|+c (where c is a constant of integration) then

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  24. 24
    The converse of [p∧(∼q)]→r[p\wedge(\sim q)]\rightarrow r is

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  25. 25
    The differential equation obtained by eliminating arbitrary constant from the equation y2=(x+c)3y^2=(x+c)^3 is

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  26. 26
    If the statements p, q and r have the truth values F, T, F respectively, then the truth values of the statement patterns (p∧∼q)→r(p\wedge\sim q)\rightarrow r and (p∨q)→r(p\vee q)\rightarrow r are respectively

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  27. 27
    The decay rate of radium is proportional to the amount present at any time t. If initially 60 gms was present and half life period of radium is 1600 years, then the amount of radium present after 3200 years is

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  28. 28
    If cos⁡−1x+cos⁡−1y+cos⁡−1z=3π\cos^{-1}x+\cos^{-1}y+\cos^{-1}z=3\pi, then the value of x2+y2+z2−2xyzx^2+y^2+z^2-2xyz is

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  29. 29
    Let A=(12−51)A=\begin{pmatrix}1&2\\-5&1\end{pmatrix} and A−1=xA+yI2A^{-1}=xA+yI_2, (where I2I_2 is unit matrix of order 2), then

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  30. 30
    If f(x)=x+x2+x3+…+xn−nx−1f(x)=\frac{x+x^2+x^3+\ldots+x^n-n}{x-1}, for x≠1x\neq1 is continuous at x=1x=1, then f(1)=f(1)=

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  31. 31
    The particular solution of differential equation (1+y2)(1+log⁡x)dx+x dy=0(1+y^2)(1+\log x)dx+x\,dy=0 at x=1,y=1x=1, y=1 is

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  32. 32
    If lim⁡x→1x2−ax+bx−1=7\lim_{x\to1}\frac{x^2-ax+b}{x-1}=7, then a+ba+b is equal to

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  33. 33
    A ladder 5 m long rests against a vertical wall. If its top slides downwards at the rate of 10 cm/sec, then the foot of the ladder is sliding at the rate of ___ m/sec, when it is 4 m away from the wall.

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  34. 34
    If f(x)=x2−xf(x)=\dfrac{x}{2-x}, g(x)=x+1x+2g(x)=\dfrac{x+1}{x+2}, then (g∘g∘f)(x)=(g\circ g\circ f)(x)=

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  35. 35
    If f(x)=cos⁡−1x, g(x)=exf(x)=\cos^{-1}x,\, g(x)=e^x and h(x)=g(f(x))h(x)=g(f(x)), then h′(x)h(x)=\frac{h'(x)}{h(x)}=

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  36. 36
    Five persons A, B, C, D and E are seated in a circular arrangement, if each of them is given a hat of one of the three colours red, blue and green, then the number of ways of distributing the hats such that the person seated in adjacent seats get different coloured hats, is

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  37. 37
    The value of ∫π/6π/41sin⁡2x(tan⁡5x+cot⁡5x)dx\int_{\pi/6}^{\pi/4}\frac{1}{\sin 2x(\tan^5 x+\cot^5 x)}dx is

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  38. 38
    If ∣z1+i∣=2\left|\frac{z}{1+i}\right|=2, where z=x+iy,i=−1z=x+iy, i=\sqrt{-1} represents a circle, then centre 'C' and radius 'r' of the circle are

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  39. 39
    If y=Acos⁡(nx)+Bsin⁡(nx)y=A\cos(nx)+B\sin(nx), then d2ydx2=\frac{d^2y}{dx^2}=

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  40. 40
    A man and his wife appear for an interview for two posts. The probability of the husband's selection is 17\frac{1}{7} and that of the wife's selection is 15\frac{1}{5}. If they appear for the interview independently, then the probability that only one of them is selected, is

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  41. 41
    The expected value of the sum of the two numbers obtained on the uppermost faces, when two fair dice are rolled, is

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  42. 42
    ∫tan⁡−1 ⁣1−sin⁡x1+sin⁡x dx=\int\tan^{-1}\!\sqrt{\frac{1-\sin x}{1+\sin x}}\,dx=

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  43. 43
    The mean and variance of seven observations are 8 and 16 respectively. If 5 of the observations are 2, 4, 10, 12, 14, then the square root of product of remaining two observations is

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  44. 44
    The equation of the tangent to the curve y=1−ex/3y=1-e^{x/3} at the point of intersection with Y-axis is

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  45. 45
    ∫x2+1(x+1)2 dx=\int\frac{x^2+1}{(x+1)^2}\,dx=

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  46. 46
    The equation of the circle which passes through the centre of the circle x2+y2+8x+10y−7=0x^2+y^2+8x+10y-7=0 and concentric with the circle 2x2+2y2−8x−12y−9=02x^2+2y^2-8x-12y-9=0 is

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  47. 47
    The acute angle between the lines xcos⁡30°+ysin⁡30°=3x\cos30°+y\sin30°=3 and xcos⁡60°+ysin⁡60°=5x\cos60°+y\sin60°=5 is

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  48. 48
    If f(x)=sin⁡4x+cos⁡4x, 0<x<π2f(x)=\sin^4x+\cos^4x,\, 0<x<\frac{\pi}{2}, then the function has minimum value ___ at x=x=

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  49. 49
    For an entry to a certain course, a candidate is given twenty problems to solve. If the probability that the candidate can solve any problem is 37\frac{3}{7}, then the probability that he is unable to solve at most two problems is

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  50. 50
    If A>B and tan⁡A−tan⁡B=x\tan A - \tan B = x and cot⁡B−cot⁡A=y\cot B - \cot A = y, then cot⁡(A−B)=\cot(A-B)=

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