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MHT-CET 2023 (10 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
    ∫0π/2∣sin⁡x−cos⁡x∣ dx\int_0^{\pi/2}|\sin x - \cos x|\,dx has the value

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  2. 2
    Negation of the statement "The payment will be made if and only if the work is finished in time." is

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  3. 3
    If y=sin⁡x1+cos⁡x⋅1+sin⁡x1+cos⁡x⋯y=\frac{\sin x}{1+\cos x}\cdot\frac{1+\sin x}{1+\cos x}\cdots, then dydx\frac{dy}{dx} is given by

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  4. 4
    If a⃗=i^−k^, b⃗=xi^+j^+(1−x)k^\vec{a}=\hat{i}-\hat{k},\,\vec{b}=x\hat{i}+\hat{j}+(1-x)\hat{k} and c⃗=yi^+xj^+(1+x−y)k^\vec{c}=y\hat{i}+x\hat{j}+(1+x-y)\hat{k}, then a⃗⋅(b⃗×c⃗)\vec{a}\cdot(\vec{b}\times\vec{c}) depends on

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  5. 5
    Let a line intersect the co-ordinate axes in points A and B such that the area of the triangle OAB is 12 sq. units. If the line passes through the point (2,3), then the equation of the line is

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  6. 6
    If for certain x, 3cos⁡x≠2sin⁡x3\cos x\neq 2\sin x, then the general solution of sin⁡2x−cos⁡2x=2−sin⁡2x\sin^2 x - \cos 2x = 2 - \sin 2x is

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  7. 7
    In a game, 3 coins are tossed. A person is paid ₹100, if he gets all heads or all tails; and he is supposed to pay ₹40, if he gets one head or two heads. The amount he can expect to win/lose on an average per game in (₹) is

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  8. 8
    The curve x4−2xy2+y2+3x−3y=0x^4-2xy^2+y^2+3x-3y=0 cuts the X-axis at (0,0)(0,0) at an angle of

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  9. 9
    The abscissae of the 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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  10. 10
    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 cm)

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  11. 11
    S1=∑r=1nr,  S2=∑r=1nr2S_1=\sum_{r=1}^n r,\;S_2=\sum_{r=1}^n r^2 and S3=∑r=1nr3S_3=\sum_{r=1}^n r^3, then the value of lim⁡n→∞S1 ⁣(1+S3)4S22\lim_{n\to\infty}\frac{S_1\!\left(1+S_3\right)}{4S_2^2} is

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  12. 12
    The numerical value of tan⁡ ⁣(2tan⁡−1 ⁣15+π4)\tan\!\left(2\tan^{-1}\!\frac{1}{5}+\frac{\pi}{4}\right)

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  13. 13
    Let a⃗,b⃗\vec{a},\vec{b} and c⃗\vec{c} be three vectors having magnitudes 1, 1 and 2 respectively. If a⃗×(a⃗×c⃗)+b⃗=0⃗\vec{a}\times(\vec{a}\times\vec{c})+\vec{b}=\vec{0}, then the acute angle between a⃗\vec{a} and c⃗\vec{c} is

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  14. 14
    Area (in sq. units) lying in the first quadrant and bounded by the circle x2+y2=4x^2+y^2=4 and the lines x=0x=0 and x=2x=2 is

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  15. 15
    If A+B=(1tan⁡θ2−tan⁡θ21)A+B=\begin{pmatrix}1 & \tan\frac{\theta}{2}\\-\tan\frac{\theta}{2} & 1\end{pmatrix}, where A is symmetric and B is skew-symmetric matrix, then the matrix A−1B+AB−1A^{-1}B+AB^{-1} at θ=π6\theta=\frac{\pi}{6} is given by

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  16. 16
    If ∫dθcos⁡2θ(tan⁡2θ+sec⁡2θ)=λtan⁡θ+2log⁡e∣f(θ)∣+c\int\frac{d\theta}{\cos^2\theta(\tan 2\theta+\sec 2\theta)}=\lambda\tan\theta+2\log_e|f(\theta)|+c (where c is a constant of integration), then the ordered pair (λ,∣f(θ)∣)(\lambda,|f(\theta)|) is equal to

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  17. 17
    The domain of definition of the function y(x)y(x) given by the equation 2x+2y=22^x+2^y=2, is

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  18. 18
    If cos⁡xdydx−ysin⁡x=6x,  0<x<π2\cos x\frac{dy}{dx}-y\sin x=6x,\;0<x<\frac{\pi}{2}, then general solution of the differential equation is

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  19. 19
    Equation of the plane, through the points (−1,2,−2)(-1,2,-2) and (−1,3,2)(-1,3,2) and perpendicular to yzyz-plane, is

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  20. 20
    The values of a and b, so that the function f(x)={x+a2sin⁡x,0≤x≤π42xcot⁡x+b,π4≤x≤π2acos⁡2x−bsin⁡x,π2<x≤πf(x)=\begin{cases}x+a\sqrt{2}\sin x,&0\leq x\leq\frac{\pi}{4}\\2x\cot x+b,&\frac{\pi}{4}\leq x\leq\frac{\pi}{2}\\a\cos2x-b\sin x,&\frac{\pi}{2}<x\leq\pi\end{cases} is continuous for 0≤x≤π0\leq x\leq\pi, are respectively given by

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  21. 21
    If a⃗\vec{a} and c⃗\vec{c} are unit vectors inclined at π3\frac{\pi}{3} with each other and (a⃗×(b⃗×c⃗))⋅(a⃗×c⃗)=5(\vec{a}\times(\vec{b}\times\vec{c}))\cdot(\vec{a}\times\vec{c})=5, then the value of 5[a⃗ b⃗ c⃗]=5[\vec{a}\,\vec{b}\,\vec{c}]=

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  22. 22
    If the lines x−12=y−23=z−14\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-1}{4} and x−3−1=y−k2=z1\frac{x-3}{-1}=\frac{y-k}{2}=\frac{z}{1} intersect, then k is equal to

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  23. 23
    A point moves along the arc of parabola y=2x2y=2x^2. Its abscissa increases uniformly at the rate of 2 units/sec. At the instant, the point is passing through (1,2)(1,2), its distance from origin is increasing at the rate of

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  24. 24
    If ∫(2x+4)x−1 dx=a(x−1)5/2+b(x−1)3/2+c\int(2x+4)\sqrt{x-1}\,dx=a(x-1)^{5/2}+b(x-1)^{3/2}+c (where c is a constant of integration), then the value of a+ba+b is

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  25. 25
    If the angles A, B and C of a triangle ABC are in the ratio 2:3:7 respectively, then the sides a, b and c are respectively in the ratio

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  26. 26
    The value of (1+cos⁡π8) ⁣(1+cos⁡3π8) ⁣(1+cos⁡5π8) ⁣(1+cos⁡7π8)\left(1+\cos\frac{\pi}{8}\right)\!\left(1+\cos\frac{3\pi}{8}\right)\!\left(1+\cos\frac{5\pi}{8}\right)\!\left(1+\cos\frac{7\pi}{8}\right) is

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  27. 27
    The value of tan⁡−1 ⁣(tan⁡7π6)\tan^{-1}\!\left(\tan\frac{7\pi}{6}\right) is

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  28. 28
    In a Binomial distribution consisting of 5 independent trials, probabilities of exactly 1 and 2 successes are 0.4096 and 0.2048 respectively, then the probability of getting exactly 4 successes is

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  29. 29
    The function to be maximized is given by Z=3x+2yZ=3x+2y. The feasible region for this function is the shaded region given below, then the linear constraints for this region are given by

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  30. 30
    If the line x−32=y+21=z+43\frac{x-3}{2}=\frac{y+2}{1}=\frac{z+4}{3} lies in the plane lx+my−z=9lx+my-z=9, then l2+m2l^2+m^2 is equal to

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  31. 31
    ∫xx+1 dx=\int\frac{\sqrt{x}}{x+1}\,dx=

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  32. 32
    The equation of the normal to the curve y=xlog⁡xy=x\log x, which is parallel to the line 2x−2y+3=02x-2y+3=0, is

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  33. 33
    ∫1+sin⁡(log⁡x)1+cos⁡(log⁡x) dx=\int\frac{1+\sin(\log x)}{1+\cos(\log x)}\,dx=

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  34. 34
    If w=−1+i32w=\frac{-1+i\sqrt{3}}{2}, where i=−1i=\sqrt{-1}, then the value of (3+w+3w2)4(3+w+3w^2)^4 is

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  35. 35
    If ∣a⃗∣=2,∣b⃗∣=3|\vec{a}|=2,|\vec{b}|=3 and a⃗,b⃗\vec{a},\vec{b} are mutually perpendicular vectors, then the area of the triangle whose vertices are 0⃗, a⃗+2b⃗, a⃗−2b⃗\vec{0},\,\vec{a}+2\vec{b},\,\vec{a}-2\vec{b} is

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  36. 36
    If y is a function of x and log⁡(x+y)=2xy\log(x+y)=2xy, then the value of y′(0)y'(0) is

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  37. 37
    A random variable X has the following probability distribution: | X | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | | P(X=x) | 0.15 | 0.23 | 0.12 | 0.10 | 0.20 | 0.08 | 0.07 | 0.05 | For the events E={X is prime number}E=\{X \text{ is prime number}\}, F={X<4}F=\{X<4\}. Then P(E∪F)=P(E \cup F)=

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  38. 38
    If y=axn+1+bx−ny=ax^{n+1}+bx^{-n}, then x2d2ydx2=x^2\frac{d^2y}{dx^2}=

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  39. 39
    Let A⃗,B⃗,C⃗\vec{A},\vec{B},\vec{C} be vectors of lengths 3 units, 4 units, 5 units respectively. Let A⃗⊥(B⃗+C⃗)\vec{A}\perp(\vec{B}+\vec{C}), B⃗⊥(C⃗+A⃗)\vec{B}\perp(\vec{C}+\vec{A}) and C⃗⊥(A⃗+B⃗)\vec{C}\perp(\vec{A}+\vec{B}), then the length of vector A⃗+B⃗+C⃗\vec{A}+\vec{B}+\vec{C} is

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  40. 40
    The approximate value of x3−2x2+3x+2x^3-2x^2+3x+2 at x=2.01x=2.01 is

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  41. 41
    The general solution of dydx=x+y+1x+y−1\frac{dy}{dx}=\frac{x+y+1}{x+y-1} is

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  42. 42
    There are three events A, B, C, one of which must and only one can happen. The odds are 8:3 against A, 5:2 against B and the odds against C is 4317k\frac{43}{17k}, then value of k is

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  43. 43
    Let a⃗,b⃗\vec{a},\vec{b} and c⃗\vec{c} be three non-zero vectors such that no two of them are collinear and (a⃗×b⃗)×c⃗=13∣b⃗∣∣c⃗∣a⃗(\vec{a}\times\vec{b})\times\vec{c}=\frac{1}{3}|\vec{b}||\vec{c}|\vec{a}. If θ\theta is the angle between vectors b⃗\vec{b} and c⃗\vec{c}, then the value of cosec θ\text{cosec}\,\theta is

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  44. 44
    If cos⁡−1x+cos⁡−1y+cos⁡−1z=π\cos^{-1}x+\cos^{-1}y+\cos^{-1}z=\pi and x2+y2+z2+kxyz=1x^2+y^2+z^2+kxyz=1, then k is

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  45. 45
    A radio-active substance has a half-life of h days, then its initial decay rate is given by (where radio-active substance has initial mass m0m_0)

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  46. 46
    The inverse of p→(q→r)p\to(q\to r) is logically equivalent to

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  47. 47
    If the line x−23=y−1−5=z+22\frac{x-2}{3}=\frac{y-1}{-5}=\frac{z+2}{2} lies in the plane x+3y−αz+β=0x+3y-\alpha z+\beta=0, then (α,β)=(\alpha,\beta)=

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  48. 48
    Mean and variance of six observations are 6 and 12 respectively. If each observation is multiplied by 3, then new variance of the resulting observations is

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  49. 49
    Words of length 10 are formed by using the letters A, B, C, D, E, F, G, H, I, J. Let x be number of such words where no letter is repeated and y be number of such words where exactly two letters are repeated twice and no other letter is repeated, then the value of yx\frac{y}{x} is

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  50. 50
    △OAB\triangle OAB is formed by the lines x2−4xy+y2=0x^2-4xy+y^2=0 and the line AB. The equation of line AB is 2x+3y−1=02x+3y-1=0. Then the equation of the median of the triangle drawn from the origin is

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