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MHT-CET 2023 (16 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
    Let A=(x110)A=\begin{pmatrix} x & 1 \\ 1 & 0 \end{pmatrix}, x∈R+x\in\mathbb{R}^+, and A4=[aij]A^4=[a_{ij}]. If a11=109a_{11}=109, then (A4)−1=(A^4)^{-1}=

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  2. 2
    The proposition (∼p)∨(p∧∼q)(\sim p) \vee (p \wedge \sim q) is equivalent to

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  3. 3
    In a triangle ABC, with usual notations, 2acsin⁡ ⁣(A−B+C2)2ac\sin\!\left(\frac{A-B+C}{2}\right) is equal to

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  4. 4
    If x2a+2xyh+y2b=0x^2a + 2xyh + y^2b = 0 represents a pair of straight lines and slope of one of the lines is twice that of the other, then ab:h2ab : h^2 is

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  5. 5
    If a⃗,b⃗\vec{a}, \vec{b} and c⃗\vec{c} are unit coplanar vectors, then the scalar triple product [2a⃗−b⃗,  2b⃗−c⃗,  2c⃗−a⃗][2\vec{a}-\vec{b},\; 2\vec{b}-\vec{c},\; 2\vec{c}-\vec{a}] has the value

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  6. 6
    If a random variable X has the following probability distribution values: X: 0,1,2,3,4,5,6,7 and P(X): 0,k,2k,2k,3k,k2,2k2,7k2+k0,k,2k,2k,3k,k^2,2k^2,7k^2+k. Then P(X≥6)P(X \geq 6) has the value

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  7. 7
    Let the vectors a⃗,b⃗,c⃗\vec{a},\vec{b},\vec{c} be such that ∣a⃗∣=2,∣b⃗∣=4|\vec{a}|=2, |\vec{b}|=4 and ∣c⃗∣=4|\vec{c}|=4. If the projection of b⃗\vec{b} on a⃗\vec{a} is equal to the projection of c⃗\vec{c} on a⃗\vec{a} and b⃗\vec{b} is perpendicular to c⃗\vec{c}, then the value of ∣a⃗+b⃗−c⃗∣|\vec{a}+\vec{b}-\vec{c}| is

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  8. 8
    If the sum of the deviations of 50 observations from 30 is 50, then the mean of these observations is

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  9. 9
    The general solution of the equation 3cos⁡θ+sin⁡θ=2\sqrt{3}\cos\theta + \sin\theta = \sqrt{2} is

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  10. 10
    If sin⁡−1 ⁣(x5)+csc⁡−1 ⁣(54)=π2\sin^{-1}\!\left(\frac{x}{5}\right) + \csc^{-1}\!\left(\frac{5}{4}\right) = \frac{\pi}{2}, then the value of x is

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  11. 11
    A random variable X takes the values 0, 1, 2, 3 and its mean is 1.3. If P(X=3)=2P(X=1)P(X=3) = 2P(X=1) and P(X=2)=0.3P(X=2) = 0.3, then P(X=0)P(X=0) is

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  12. 12
    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 sin⁡θ\sin\theta is

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  13. 13
    The equation of the plane, passing through the intersection of the planes x+y+z=1x+y+z=1 and 2x+3y−z+4=02x+3y-z+4=0 and parallel to Y-axis is

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  14. 14
    If f(1)=1,f′(1)=3f(1)=1, f'(1)=3, then the derivative of f(f(f(x)))+(f(x))2f(f(f(x))) + (f(x))^2 at x=1x=1 is

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  15. 15
    If [a⃗×b⃗    b⃗×c⃗    c⃗×a⃗]=λ[a⃗  b⃗  c⃗]2[\vec{a}\times\vec{b}\;\; \vec{b}\times\vec{c}\;\; \vec{c}\times\vec{a}] = \lambda[\vec{a}\;\vec{b}\;\vec{c}]^2, then λ\lambda is equal to

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  16. 16
    The value of lim⁡x→0+[(sin⁡x)1/x+1xsin⁡x]\lim_{x \to 0^+} \left[(\sin x)^{1/x} + \frac{1}{x^{\sin x}}\right] where x>0x > 0 is

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  17. 17
    If θ\theta denotes the acute angle between the curves y=10−x2y = 10 - x^2 and y=2+x2y = 2 + x^2, at a point of the intersection, then ∣tan⁡θ∣|\tan\theta| is equal to

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  18. 18
    If y=alog⁡x+bx2+xy = a\log x + bx^2 + x has its extremum values at x=−1x = -1 and x=2x = 2, then

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  19. 19
    Let P={θ:sin⁡θ−cos⁡θ=2cos⁡θ}P = \{\theta : \sin\theta - \cos\theta = \sqrt{2}\cos\theta\} and Q={θ:sin⁡θ+cos⁡θ=2sin⁡θ}Q = \{\theta : \sin\theta + \cos\theta = \sqrt{2}\sin\theta\} be two sets, then

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  20. 20
    A line with positive direction cosines passes through the point P(2,−1,2)P(2,-1,2) and makes equal angles with coordinate axes. The line meets the plane 2x+y+z=92x+y+z=9 at point Q. Then the length of the line segment PQ equals

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  21. 21
    Let f(x)=1−tan⁡x4x−π,  x≠π4,  x∈[0,π2)f(x) = \frac{1-\tan x}{4x-\pi},\; x \neq \frac{\pi}{4},\; x \in \left[0, \frac{\pi}{2}\right). f(x)f(x) is continuous in [0,π2)\left[0, \frac{\pi}{2}\right), then f ⁣(π4)f\!\left(\frac{\pi}{4}\right) is

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  22. 22
    The value of ∫−π/2π/2sin⁡2x1+2x dx\int_{-\pi/2}^{\pi/2} \frac{\sin^2 x}{1+2^x}\,dx is

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  23. 23
    The area (in sq. units) of the region described by {(x,y):y2≤2x}\{(x,y) : y^2 \leq 2x\} and y≥(4x−1)y \geq (4x-1) is

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  24. 24
    Let y=y(x)y = y(x) be the solution of the differential equation xlog⁡x dydx+y=2xlog⁡x  (x≥1)x\log x\,\frac{dy}{dx} + y = 2x\log x\; (x \geq 1), then y(e)y(e) is equal to

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  25. 25
    The set of all points, for which f(x)=x2e−xf(x) = x^2 e^{-x} strictly increases, is

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  26. 26
    Range of the function f(x)=x2+x+2x2+x+1,  x∈Rf(x) = \frac{x^2+x+2}{x^2+x+1},\; x \in \mathbb{R} is

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  27. 27
    Let S be a non-empty subset of R\mathbb{R}. Consider the following statement: p: There is a rational number x∈Sx \in S such that x>0x > 0. Which of the following statements is the negation of the statement p?

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  28. 28
    The number of values of x in the interval (0,5π)(0, 5\pi) satisfying the equation 3sin⁡2x−7sin⁡x+2=03\sin^2 x - 7\sin x + 2 = 0

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  29. 29
    If the vectors AB→=3i^+4k^\overrightarrow{AB} = 3\hat{i} + 4\hat{k} and AC→=5i^−2j^+4k^\overrightarrow{AC} = 5\hat{i} - 2\hat{j} + 4\hat{k} are the sides of the triangle ABC, then the length of the median, through A, is

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  30. 30
    The equation of the circle, concentric with the circle 2x2+2y2−6x+8y+1=02x^2+2y^2-6x+8y+1=0 and double of its area is

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  31. 31
    If y=sec⁡(tan⁡−1x)y = \sec(\tan^{-1}x), then dydx\frac{dy}{dx} at x=1x=1 is equal to

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  32. 32
    Eight chairs are numbered 1 to 8. Two women and three men wish to occupy one chair each. First the women choose chairs from amongst the chairs marked 1 to 4, and then the men select the chairs from amongst the remaining. The number of possible arrangements is

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  33. 33
    If the distance between the plane Ax−2y+z=dAx-2y+z=d and the plane containing the lines x−12=y−23=z−34\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4} and x−23=y−34=z−45\frac{x-2}{3}=\frac{y-3}{4}=\frac{z-4}{5} is 6\sqrt{6} units, then ∣d∣|d| is

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  34. 34
    Maximum value of Z=100x+70yZ = 100x + 70y subject to 2x≥4,y≤3,x+y≤8,x,y≥02x \geq 4, y \leq 3, x+y \leq 8, x,y \geq 0 is

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  35. 35
    Three persons P, Q and R independently try to hit a target. If the probabilities of their hitting the target are 34,12\frac{3}{4}, \frac{1}{2} and 58\frac{5}{8} respectively, then the probability that the target is hit by P or Q but not by R, is

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  36. 36
    If a⃗\vec{a} and b⃗\vec{b} are two unit vectors such that a⃗+2b⃗\vec{a} + 2\vec{b} and 5a⃗−4b⃗5\vec{a} - 4\vec{b} are perpendicular to each other, then the angle between a⃗\vec{a} and b⃗\vec{b} is

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  37. 37
    Let the function g:(−∞,∞)→(−π2,π2)g:(-\infty,\infty) \to \left(-\frac{\pi}{2},\frac{\pi}{2}\right) be given by g(u)=2tan⁡−1(eu)−π2g(u) = 2\tan^{-1}(e^u) - \frac{\pi}{2}. Then g is

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  38. 38
    The length of the projection of the line segment joining the points (5,−1,4)(5,-1,4) and (4,−1,3)(4,-1,3) on the plane x+y+z=7x+y+z=7 is

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  39. 39
    If f(x)=sin⁡2(πx)1+πxf(x) = \frac{\sin^2(\pi x)}{1+\pi^x}, then ∫(f(x)+f(−x)) dx\int(f(x)+f(-x))\,dx is equal to

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  40. 40
    Suppose that the points (h,k)(h,k), (1,2)(1,2) and (−3,4)(-3,4) lie on the line l1l_1. If a line l2l_2 passing through the points (h,k)(h,k) and (4,3)(4,3) is perpendicular to l1l_1, then kh\frac{k}{h} equals

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  41. 41
    If ∫dxcos⁡3x2sin⁡2x=(tan⁡x)A+C(tan⁡x)B+k\int\frac{dx}{\cos^3 x\sqrt{2\sin 2x}} = (\tan x)^A + C(\tan x)^B + k where k is a constant of integration, then A+B+CA+B+C equals

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  42. 42
    The maximum value of cos⁡α1⋅cos⁡α2⋯cos⁡αn\cos\alpha_1 \cdot \cos\alpha_2 \cdots \cos\alpha_n under the constraints 0≤α1,…,αn≤π20 \leq \alpha_1,\ldots,\alpha_n \leq \frac{\pi}{2} and cot⁡α1⋅cot⁡α2⋯cot⁡αn=1\cot\alpha_1 \cdot \cot\alpha_2 \cdots \cot\alpha_n = 1 is

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  43. 43
    If dydx=y+3,  y+3>0\frac{dy}{dx} = y+3,\; y+3 > 0 and y(0)=2y(0)=2, then y(log⁡2)y(\log 2) is equal to

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  44. 44
    The assets of a person are reduced in his business such that the rate of reduction is proportional to the square root of the existing assets. If the assets were initially ₹10,00,000 and due to loss they reduce to ₹10,000 after 3 years, then the number of years required for the person to go bankrupt will be

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  45. 45
    If f(1)=1,f′(1)=5f(1)=1, f'(1)=5, then the derivative of f(f(f(x)))+(f(x))2f(f(f(x))) + (f(x))^2 at x=1x=1 is

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  46. 46
    The abscissa of the point on the curve y=a ⁣(ex/a+e−x/a)y = a\!\left(e^{x/a} + e^{-x/a}\right) where the tangent is parallel to the X-axis is

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  47. 47
    The integral ∫2x3−1x4+x dx\int\frac{2x^3-1}{x^4+x}\,dx is equal to

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  48. 48
    Let z=x+iyz = x+iy be a complex number, where x and y are integers and i=−1i = \sqrt{-1}. Then the area of the rectangle whose vertices are the roots of the equation zzˉ3+zˉz3=350z\bar{z}^3 + \bar{z}z^3 = 350 is

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  49. 49
    If ∫log⁡ ⁣(t+1+t2)1+t2 dt=12(g(t))2+c\int\frac{\log\!\left(t+\sqrt{1+t^2}\right)}{\sqrt{1+t^2}}\,dt = \frac{1}{2}(g(t))^2 + c where c is a constant of integration, then g(2)g(2) is equal to

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  50. 50
    A box contains 15 green and 10 yellow balls. If 10 balls are randomly drawn, one by one, with replacement, then the variance of the number of green balls drawn is

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