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MHT-CET 2023 (16 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 g(x)=[f(2f(x)+2)]2g(x) = [f(2f(x)+2)]^2 and f(0)=−1f(0) = -1, f′(0)=1f'(0) = 1, then g′(0)g'(0) is

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  2. 2
    If tan⁡−114+tan⁡−129=12cos⁡−1x\tan^{-1}\frac{1}{4} + \tan^{-1}\frac{2}{9} = \frac{1}{2}\cos^{-1} x, then xx is

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  3. 3
    The shaded area in the given figure is a solution set for some system of inequalities. The maximum value of the function z=4x+3yz = 4x + 3y subject to linear constraints given by the system is

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  4. 4
    If z1=5−2iz_1 = 5-2i and z2=3+iz_2 = 3+i, where i=−1i = \sqrt{-1}, then arg⁡(z1+z2z1−z2)\arg\left(\frac{z_1+z_2}{z_1-z_2}\right) is

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  5. 5
    The co-ordinates of the foot of the perpendicular from the point (0,2,3)(0,2,3) on the line x+35=y+12=z+43\frac{x+3}{5} = \frac{y+1}{2} = \frac{z+4}{3} is

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  6. 6
    The slope of tangent at (x,y)(x,y) to a curve passing through (1,π4)\left(1, \frac{\pi}{4}\right) is yx−cos⁡2yx\frac{y}{x} - \cos^2\frac{y}{x}, then the equation of curve is

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  7. 7
    If cos⁡−11213+sin⁡−135=sin⁡−1P\cos^{-1}\frac{12}{13} + \sin^{-1}\frac{3}{5} = \sin^{-1} P, then the value of PP is

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  8. 8
    The rate of change of the volume of a sphere with respect to its surface area, when its radius is 2 cm, is ___ cm3/cm2\text{cm}^3/\text{cm}^2.

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  9. 9
    In a triangle ABC, l(AB)=23l(AB) = \sqrt{23} units, l(BC)=3l(BC) = 3 units, l(CA)=4l(CA) = 4 units, then cot⁡A+cot⁡Ccot⁡B\frac{\cot A + \cot C}{\cot B} is

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  10. 10
    Truth values of p→rp \to r is F and p↔qp \leftrightarrow q is F. Then the truth values of (∼p∨q)→(p∨∼q)(\sim p \vee q) \to (p \vee \sim q) and (p∧∼q)→(∼p∧q)(p \wedge \sim q) \to (\sim p \wedge q) are respectively

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  11. 11
    Water is being poured at the rate of 36 m3/min\text{m}^3/\text{min} into a cylindrical vessel, whose circular base is of radius 3 meters. Then the water level in the cylinder is rising at the rate of

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  12. 12
    A line having direction ratios 1,−4,21, -4, 2 intersects the lines x−73=y−1−1=z+21\frac{x-7}{3} = \frac{y-1}{-1} = \frac{z+2}{1} and x2=y−73=z1\frac{x}{2} = \frac{y-7}{3} = \frac{z}{1} at points A and B respectively. Then co-ordinates of A and B are

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  13. 13
    ∫x2−4x4+9x2+16 dx=tan⁡−1(f(x))+c\int \frac{x^2-4}{x^4+9x^2+16}\,dx = \tan^{-1}(f(x)) + c (where cc is a constant of integration), then value of f(2)f(2) is

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  14. 14
    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} are two vectors, then the angle between the vectors 3a⃗+5b⃗3\vec{a} + 5\vec{b} and 5a⃗+3b⃗5\vec{a} + 3\vec{b} is

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  15. 15
    Let A=(12−14)A = \begin{pmatrix}1 & 2 \\ -1 & 4\end{pmatrix} and A−1=αI+βAA^{-1} = \alpha I + \beta A, α,β∈R\alpha, \beta \in \mathbb{R}, II is the identity matrix of order 2, then 4(α−β)4(\alpha - \beta) is

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  16. 16
    lim⁡y→01+1+y4−2y4=\lim_{y \to 0} \frac{\sqrt{1 + \sqrt{1+y^4}} - \sqrt{2}}{y^4} =

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  17. 17
    If a⃗\vec{a} is perpendicular to b⃗\vec{b} and c⃗\vec{c}, ∣a⃗∣=2|\vec{a}|=2, ∣b⃗∣=3|\vec{b}|=3, ∣c⃗∣=4|\vec{c}|=4 and the angle between b⃗\vec{b} and c⃗\vec{c} is π3\frac{\pi}{3}, then [a⃗ b⃗ c⃗]=[\vec{a}\,\vec{b}\,\vec{c}] =

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  18. 18
    The mean and variance of seven observations are 8 and 16 respectively. If five of the observations are 2, 4, 10, 12, 14, then the product of remaining two observations is

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  19. 19
    If y=tan⁡−12+3x3−2x+tan⁡−14x1+5x2y = \tan^{-1}\frac{2+3x}{3-2x} + \tan^{-1}\frac{4x}{1+5x^2}, then dydx=\frac{dy}{dx} =

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  20. 20
    If f(x)=asin⁡x+bcos⁡xcsin⁡x+dcos⁡xf(x) = \frac{a\sin x + b\cos x}{c\sin x + d\cos x} is decreasing for all xx, then

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  21. 21
    If a discrete random variable X is defined as follows: P[X=x]=k(x+1)5xP[X=x] = k\frac{(x+1)}{5^x}, if x=0,1,2,…x = 0,1,2,\ldots, 0 otherwise, then k=k =

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  22. 22
    If a⃗=2i^−j^+k^\vec{a} = 2\hat{i}-\hat{j}+\hat{k}, b⃗=i^+j^−2k^\vec{b} = \hat{i}+\hat{j}-2\hat{k} and c⃗=4i^−2j^+k^\vec{c} = 4\hat{i}-2\hat{j}+\hat{k}, then the unit vector in the direction of 3a⃗+b⃗−2c⃗3\vec{a}+\vec{b}-2\vec{c} is

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  23. 23
    Numbers are selected at random, one at a time from two-digit numbers 10, 11, 12, \ldots, 99 with replacement. An event E occurs if and only if the product of the two digits of a selected number is 18. If four numbers are selected, then probability that the event E occurs at least 3 times is

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  24. 24
    The function y(x)y(x) represented by x=sin⁡tx = \sin t, y=aet2+be−t2y = ae^{t\sqrt{2}} + be^{-t\sqrt{2}}, t∈(−π2,π2)t \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) satisfies the equation (1−x2)y′′−xy′=ky(1-x^2)y'' - xy' = ky, then the value of kk is

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  25. 25
    The value of cos⁡20∘+2sin⁡255∘−2sin⁡65∘\cos 20^\circ + 2\sin^2 55^\circ - \sqrt{2}\sin 65^\circ is

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  26. 26
    If [x][x] denotes the greatest integer function, then ∫05x2[x] dx=\int_0^5 x^2[x]\,dx =

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  27. 27
    If the function ff defined on (π6,π3)\left(\frac{\pi}{6}, \frac{\pi}{3}\right) by f(x)=2cos⁡x−1cot⁡x−1,x≠π4f(x) = \frac{\sqrt{2}\cos x - 1}{\cot x - 1}, x \neq \frac{\pi}{4}; f(x)=k,x=π4f(x) = k, x = \frac{\pi}{4} is continuous, then kk is equal to

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  28. 28
    ∫cos⁡−3/7x⋅sin⁡−11/7x dx=\int \cos^{-3/7}x \cdot \sin^{-11/7}x\,dx =

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  29. 29
    If a⃗=a1i^+a2j^+a3k^\vec{a} = a_1\hat{i}+a_2\hat{j}+a_3\hat{k}, b⃗=b1i^+b2j^+b3k^\vec{b} = b_1\hat{i}+b_2\hat{j}+b_3\hat{k}, c⃗=c1i^+c2j^+c3k^\vec{c} = c_1\hat{i}+c_2\hat{j}+c_3\hat{k} and [3a⃗+b⃗,3b⃗+c⃗,3c⃗+a⃗]=λ[a⃗ b⃗ c⃗][3\vec{a}+\vec{b}, 3\vec{b}+\vec{c}, 3\vec{c}+\vec{a}] = \lambda[\vec{a}\,\vec{b}\,\vec{c}], then the value of λ\lambda is

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  30. 30
    The domain of the function f(x)=sin⁡−1(x−3)9−x2f(x) = \frac{\sin^{-1}(x-3)}{\sqrt{9-x^2}} is

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  31. 31
    If a⃗=2i^+2j^+3k^\vec{a} = 2\hat{i}+2\hat{j}+3\hat{k}, b⃗=−i^+2j^+k^\vec{b} = -\hat{i}+2\hat{j}+\hat{k} and c⃗=3i^+j^\vec{c} = 3\hat{i}+\hat{j} are vectors such that a⃗+λb⃗\vec{a}+\lambda\vec{b} is perpendicular to c⃗\vec{c}, then value of λ\lambda is

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  32. 32
    The area (in sq. units) bounded between the parabolas x2=y4x^2 = \frac{y}{4} and x2=9yx^2 = 9y and the line y=2y = 2 is

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  33. 33
    A plane makes positive intercepts of unit length on each of X and Y axis. If it passes through the point (−1,1,2)(-1,1,2) and makes angle θ\theta with the X-axis, then θ\theta is

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  34. 34
    The rate of growth of bacteria in a culture is proportional to the number of bacteria present and the bacteria count is 1000 at t=0t=0. The number of bacteria is increased by 20% in 2 hours. If the population of bacteria is 2000 after klog⁡65\dfrac{k}{\log\frac{6}{5}} hours, then (klog⁡2)2\left(\dfrac{k}{\log 2}\right)^2 is

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  35. 35
    One end of the diameter of the circle x2+y2−6x−5y−1=0x^2+y^2-6x-5y-1=0 is (−1,3)(-1,3), then the equation of the tangent at the other end of the diameter is

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  36. 36
    If tan⁡−1x+1x−1+tan⁡−1x−1x=tan⁡−1(−7)\tan^{-1}\frac{x+1}{x-1} + \tan^{-1}\frac{x-1}{x} = \tan^{-1}(-7), then xx is equal to

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  37. 37
    The equation of the normal to the curve y=xlog⁡xy = x\log x parallel to 2x−2y+3=02x-2y+3=0 is

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  38. 38
    ∫etan⁡−1x1+x2[(sec⁡−11+x2)2+cos⁡−11−x21+x2]dx\int \dfrac{e^{\tan^{-1}x}}{1+x^2}\left[\left(\sec^{-1}\sqrt{1+x^2}\right)^2 + \cos^{-1}\dfrac{1-x^2}{1+x^2}\right]dx where x>0x > 0 is

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  39. 39
    The equation of plane through the point (2,−1,−3)(2,-1,-3) and parallel to lines x−13=y+22=z−4\frac{x-1}{3} = \frac{y+2}{2} = \frac{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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  40. 40
    The co-ordinates of the foot of perpendicular, drawn from the point (−2,3)(-2,3) on the line 3x−y−1=03x-y-1=0 are

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  41. 41
    The value of tan⁡−1(−3)−sin⁡−112+cos⁡−1(−12)\tan^{-1}(-\sqrt{3}) - \sin^{-1}\dfrac{1}{\sqrt{2}} + \cos^{-1}\left(-\dfrac{1}{2}\right) is

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  42. 42
    The general solution of the differential equation dydx=ytan⁡x−y2sec⁡x\frac{dy}{dx} = y\tan x - y^2\sec x is

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  43. 43
    If a⃗,b⃗,c⃗\vec{a}, \vec{b}, \vec{c} are three vectors such that a⃗≠0\vec{a} \neq 0 and a⃗×b⃗=2a⃗×c⃗\vec{a} \times \vec{b} = 2\vec{a} \times \vec{c}, ∣a⃗∣=∣c⃗∣=1|\vec{a}|=|\vec{c}|=1, ∣b⃗∣=4|\vec{b}|=4 and ∣b⃗×c⃗∣=15|\vec{b}\times\vec{c}|=\sqrt{15}. If b⃗−2c⃗=λa⃗\vec{b}-2\vec{c} = \lambda\vec{a}, then λ\lambda is

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  44. 44
    Two friends A and B apply for a job in the same company. The probabilities of A getting selected is 25\frac{2}{5} and that of B is 47\frac{4}{7}. Then the probability, that one of them is selected, is

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  45. 45
    If P1P_1 and P2P_2 are perpendicular distances (in units) from point (2,−1)(2,-1) to the pair of lines 2x2−5xy+2y2=02x^2-5xy+2y^2=0, then the value of P1P2P_1 P_2 is

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  46. 46
    ∫x3−7x+6x2+3x dx=\int \frac{x^3-7x+6}{x^2+3x}\,dx =

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  47. 47
    The cumulative distribution function of a discrete random variable X is given by: F(-1)=0.3, F(0)=0.7, F(1)=0.8, F(2)=1. Then E(X2)=E(X^2) =

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  48. 48
    A five-digit number divisible by 3 is to be formed using the digits 0, 1, 2, 3, 4, 5 without repetition, then the total number of ways this can be done is

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  49. 49
    The statement ∼(p↔∼q)\sim(p \leftrightarrow \sim q) is

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  50. 50
    If x=−1x=-1 and x=2x=2 are extreme points of f(x)=αlog⁡∣x∣+βx2+xf(x) = \alpha\log|x| + \beta x^2 + x, then

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