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MHT-CET 2021, Paper I, Mathematics question paper

50 questions · 100 marks · 90 minutes · +2 per correct answer, no negative marking

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  1. 1
    With usual notations, in any △ABC\triangle ABC, if acos⁡B=bcos⁡Aa\cos B = b\cos A, then the triangle is

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  2. 2
    If θ\theta lies in the first quadrant and 5tan⁡θ=45\tan\theta = 4, then 5sin⁡θ−3cos⁡θsin⁡θ+2cos⁡θ\frac{5\sin\theta - 3\cos\theta}{\sin\theta + 2\cos\theta} is equal to

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  3. 3
    If cos⁡−1x>sin⁡−1x\cos^{-1} x > \sin^{-1} x, then

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  4. 4
    The value of tan⁡3A−tan⁡2A−tan⁡A\tan 3A - \tan 2A - \tan A is

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  5. 5
    The three straight lines ax+by=cax+by=c, bx+cy=abx+cy=a and cx+ay=bcx+ay=b are collinear if

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  6. 6
    A line through the point A(2,0)A(2,0) which makes an angle of 30∘30^\circ with the positive direction of X-axis is rotated about AA in clockwise direction through an angle of 15∘15^\circ. Then, the equation of the straight line in the new position is

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  7. 7
    Find the value of cos⁡x2\cos\frac{x}{2}, if tan⁡x=512\tan x = \frac{5}{12} and xx lies in the third quadrant.

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  8. 8
    The locus of a point which moves such that the ratio of the length of the tangents to the circles x2+y2+4x+3=0x^2+y^2+4x+3=0 and x2+y2−6x+5=0x^2+y^2-6x+5=0 is 2:32:3, is

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  9. 9
    The circles x2+y2+6x+6y=0x^2+y^2+6x+6y=0 and x2+y2−12x−12y=0x^2+y^2-12x-12y=0

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  10. 10
    The mean and variance of nn observations x1,x2,…,xnx_1, x_2, \ldots, x_n are 5 and 0, respectively. If ∑i=1nxi2=400\sum_{i=1}^n x_i^2 = 400, then the value of nn is equal to

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  11. 11
    The variance of first nn natural numbers is

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  12. 12
    Out of 50 tickets numbered 00, 01, 02, \ldots, 49, one ticket is drawn randomly. The probability of the ticket having the product of its digits 7, given that the sum of the digits is 8, is

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  13. 13
    The probability of India winning a test match against South Africa is 1/2, assuming independence from match to match. The probability that in a match series India's second win occurs at the third match is

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  14. 14
    Consider the following statements: r: If a number is a multiple of 9, then it is a multiple of 3. Let pp: A number is a multiple of 9, and qq: A number is a multiple of 3. Then, 'if p then q' is the same as

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  15. 15
    The negation of the statement '7 is greater than 4 or 6 is less than 7'.

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  16. 16
    For an invertible matrix AA, if A(adj A)=(200020)A(\text{adj}\,A) = \begin{pmatrix}20 & 0\\0 & 20\end{pmatrix}, then ∣A∣=|A| =

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  17. 17
    If matrix A=(1−123)A = \begin{pmatrix}1&-1\\2&3\end{pmatrix}, then A2−4A+5IA^2 - 4A + 5I is where II is a unit matrix.

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  18. 18
    If f(x)={3sin⁡(πx)5xx≠02kx=0f(x) = \begin{cases}\frac{3\sin(\pi x)}{5x} & x \neq 0 \\ 2k & x = 0\end{cases} is continuous at x=0x = 0, then the value of kk is

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  19. 19
    If f(x)={sin⁡3(x)⋅log⁡(1+3x)(tan⁡−1x)2(e5x−1)xx≠0ax=0f(x) = \begin{cases} \dfrac{\sin^{3}(\sqrt{x}) \cdot \log(1 + 3x)}{\left(\tan^{-1}\sqrt{x}\right)^{2}\left(e^{5\sqrt{x}} - 1\right)x} & x \neq 0 \\ a & x = 0 \end{cases} is continuous in [0,1][0,1], then aa equals

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  20. 20
    The differential equation of all parabolas having vertex at the origin and axis along positive Y-axis is

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  21. 21
    The order and degree of the differential equation dydx−4dydx−7x=0\sqrt{\frac{dy}{dx}} - 4\frac{dy}{dx} - 7x = 0 are

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  22. 22
    The solution of the differential equation e−x(y+1)dy+(cos⁡2x−sin⁡2x)y dx=0e^{-x}(y+1)dy + (\cos^2 x - \sin 2x)y\,dx = 0 subjected to the condition that y=1y = 1 when x=0x = 0 is

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  23. 23
    The value of ∫−1117x5−x4+29x3−31x+1x2+1 dx\int_{-1}^{1}\frac{17x^5 - x^4 + 29x^3 - 31x + 1}{x^2+1}\,dx

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  24. 24
    The number of ways of arranging letters of the word 'HAVANA', so that V and N do not appear together, is

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  25. 25
    Out of 7 consonants and 4 vowels, the number of words (not necessarily meaningful) that can be made, each consisting of 3 consonants and 2 vowels, is

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  26. 26
    If f(x)=sin⁡2x+sin⁡2 ⁣(x+π3)+cos⁡xcos⁡ ⁣(x+π3)f(x) = \sin^2 x + \sin^2\!\left(x+\frac{\pi}{3}\right) + \cos x\cos\!\left(x+\frac{\pi}{3}\right) and g ⁣(54)=1g\!\left(\frac{5}{4}\right) = 1, then g(f(x))g(f(x)) is equal to

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  27. 27
    Range of the function f(x)=x1+x2f(x) = \frac{x}{1+x^2} is

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  28. 28
    The domain of f(x)=sin⁡−1 ⁣(log⁡2x2)f(x) = \sin^{-1}\!\left(\log_2\frac{x}{2}\right) is

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  29. 29
    If f(x)=k⋅2xf(x) = k\cdot 2^x is a probability distribution of a random variable XX that can take on the values x=0,1,2,3,4x = 0, 1, 2, 3, 4, then kk is equal to

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  30. 30
    lim⁡x→0ex2−cos⁡xx2\lim_{x\to0}\frac{e^{x^2}-\cos x}{x^2} is equal to

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  31. 31
    Evaluate lim⁡x→2x4−4x2+32 x−8\lim_{x \to \sqrt{2}} \dfrac{x^{4} - 4}{x^{2} + 3\sqrt{2}\,x - 8}.

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  32. 32
    What will be the projection of the vector 4i^−3j^+k^4\hat{i} - 3\hat{j} + \hat{k} on the line joining the points (2,3,−1)(2,3,-1) and (−2,−4,3)(-2,-4,3)?

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  33. 33
    If a⃗=3i^−2j^+k^\vec{a} = 3\hat{i} - 2\hat{j} + \hat{k} and b⃗=2i^−4j^−3k^\vec{b} = 2\hat{i} - 4\hat{j} - 3\hat{k}, then ∣a⃗−2b⃗∣|\vec{a} - 2\vec{b}| will be

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  34. 34
    What will be the equation of the plane passing through a point (1,4,−2)(1,4,-2) and parallel to the given plane −2x+y−3z=9-2x + y - 3z = 9?

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  35. 35
    If the line joining two points A(2,0)A(2,0) and B(3,1)B(3,1) is rotated about AA in anticlockwise direction through an angle of 15∘15^\circ, then the equation of the line in the new position is

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  36. 36
    ∫(log⁡x)2x dx\int\frac{(\log x)^2}{x}\,dx

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  37. 37
    ∫tan⁡4x⋅sec⁡2xx dx\int\frac{\tan^4 x\cdot\sec^2 x}{\sqrt{x}}\,dx

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  38. 38
    The slant height of a right circular cone is 3 cm. The height of the cone for maximum volume is

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  39. 39
    If y=p[x(x−2)]2y = p[x(x-2)]^2 is an increasing function, then the value of xx is

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  40. 40
    If y=tan⁡−1 ⁣1+cos⁡x1−cos⁡xy = \tan^{-1}\!\sqrt{\frac{1+\cos x}{1-\cos x}}, then dydx=\frac{dy}{dx} =

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  41. 41
    Derivative of tan⁡−1 ⁣1+x2−1x\tan^{-1}\!\frac{\sqrt{1+x^2}-1}{x} w.r.t. sin⁡−1 ⁣2x1+x2\sin^{-1}\!\frac{2x}{1+x^2} is

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  42. 42
    The region represented by the inequation system x,y≥0x, y \geq 0, y≤6y \leq 6, x+y≤3x + y \leq 3 is

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  43. 43
    The constraints −x1+x2≤1-x_1 + x_2 \leq 1, −x1+3x2≤9-x_1 + 3x_2 \leq 9 and x1,x2≥0x_1, x_2 \geq 0 define

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  44. 44
    If z=(3+i)3(3i+4)2(8+6i)2z = \dfrac{(\sqrt{3}+i)^3(3i+4)^2}{(8+6i)^2}, then ∣z∣|z| is equal to

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  45. 45
    If 32+cos⁡θ+isin⁡θ=a+ib\frac{3}{2+\cos\theta+i\sin\theta} = a+ib, then (a−2)2+b2(a-2)^2+b^2 is equal to

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  46. 46
    A box contains 100 bulbs, out of which 10 are defective. A sample of 5 bulbs is drawn. The probability that none is defective is

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  47. 47
    A random variable XX has the probability distribution: X:1,2,3,4,5X: 1, 2, 3, 4, 5; P(X=x):K,2K,3K,2K,KP(X=x): K, 2K, 3K, 2K, K. Its variance is

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  48. 48
    The area (in sq. units) of the region bounded by the curves y2=4axy^2 = 4ax and x2=4ayx^2 = 4ay, a>0a>0 is

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  49. 49
    The area in the positive quadrant enclosed by the circle x2+y2=4x^2+y^2=4, the line x=y3x=y\sqrt{3} and x-axis is

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  50. 50
    ∫π/6π/2csc⁡x⋅cot⁡x1+csc⁡2x dx\int_{\pi/6}^{\pi/2}\frac{\csc x\cdot\cot x}{1+\csc^2 x}\,dx

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