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MHT-CET 2024 (11 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 the angles of a triangle are in the ratio 4:1:14:1:1, then the ratio of the longest side to its perimeter is

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  2. 2
    If f(x)={x−3∣x−3∣+a,x<3a+b,x=3∣x−3∣x−3+b,x>3f(x) = \begin{cases} \frac{x-3}{|x-3|}+a, & x<3 \\ a+b, & x=3 \\ \frac{|x-3|}{x-3}+b, & x>3 \end{cases} is continuous at x=3x=3, then the value of a−ba-b is:

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  3. 3
    A fair die is tossed twice in succession. If X denotes the number of fours in two tosses, then the probability distribution of X is given by

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  4. 4
    If the lines x−12=y+13=z−14\frac{x-1}{2}=\frac{y+1}{3}=\frac{z-1}{4} and x−31=y−k2=z1\frac{x-3}{1}=\frac{y-k}{2}=\frac{z}{1} intersect, then the value of k is:

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  5. 5
    The maximum value of Z=3x+5yZ = 3x+5y subject to the constraints 3x+2y≤183x+2y \leq 18, x≤4x \leq 4, y≤6y \leq 6, x,y≥0x,y \geq 0, is

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  6. 6
    If the line x−12=y+13=z−24\frac{x-1}{2}=\frac{y+1}{3}=\frac{z-2}{4} meets the plane x+2y+3z=15x+2y+3z=15 at the point P, then the distance of P from the origin is:

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  7. 7
    The solution of dxdy+xy=x2\frac{dx}{dy} + xy = x^2 is:

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  8. 8
    If the general solution of the equation tan⁡3x−1tan⁡3x+1=3\frac{\tan 3x - 1}{\tan 3x + 1} = \sqrt{3} is nπp+7πq, n,p,q∈Z\frac{n\pi}{p} + \frac{7\pi}{q},\ n,p,q \in \mathbb{Z}, then pqpq is

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  9. 9
    The value of ∫(1−cos⁡x)csc⁡2x dx\int(1-\cos x)\csc^2 x\,dx is

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  10. 10
    If a⃗=i^+4j^+2k^\vec{a} = \hat{i}+4\hat{j}+2\hat{k}, b⃗=3i^−2j^+7k^\vec{b} = 3\hat{i}-2\hat{j}+7\hat{k}, c⃗=2i^−j^+4k^\vec{c} = 2\hat{i}-\hat{j}+4\hat{k}, then a vector d⃗\vec{d} which is parallel to a⃗×b⃗\vec{a}\times\vec{b} and with c⃗⋅d⃗=15\vec{c}\cdot\vec{d}=15 is

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  11. 11
    Let a random variable X have a Binomial distribution with mean 8 and variance 4. If P(X≤2)=K216P(X\leq 2) = \frac{K}{2^{16}}, then K is

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  12. 12
    The angle between the line x+12=y−21=z−3−2\frac{x+1}{2}=\frac{y-2}{1}=\frac{z-3}{-2} and plane x−2y−λz=3x-2y-\lambda z=3 is cos⁡−1 ⁣(223)\cos^{-1}\!\left(\frac{2\sqrt{2}}{3}\right), then value of λ\lambda is:

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  13. 13
    If a and b are positive numbers such that a>ba>b, then the minimum value of asec⁡θ−btan⁡θa\sec\theta - b\tan\theta (0<θ<π2)\left(0<\theta<\frac{\pi}{2}\right) is

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  14. 14
    If a⃗=i^+j^+k^\vec{a} = \hat{i}+\hat{j}+\hat{k}, b⃗=4i^+3j^+4k^\vec{b} = 4\hat{i}+3\hat{j}+4\hat{k} and c⃗=i^+αj^+βk^\vec{c} = \hat{i}+\alpha\hat{j}+\beta\hat{k} are linearly dependent vectors and ∣c⃗∣=3|\vec{c}|=\sqrt{3}, then the value of α\alpha and β\beta are respectively.

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  15. 15
    The set of all points, where the derivative of the function f(x)=x1+∣x∣f(x) = \frac{x}{1+|x|} exists, is

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  16. 16
    If x=51−2ix = \frac{5}{1-2i}, i=−1i=\sqrt{-1}, then the value of x3+x2−x+22x^3+x^2-x+22 is

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  17. 17
    In triangle ABC with usual notations b=3b=\sqrt{3}, c=1c=1, ∠A=30∘\angle A=30^{\circ}, then the largest angle of the triangle is

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  18. 18
    If the area of the parallelogram with a⃗\vec{a} and b⃗\vec{b} as two adjacent sides is 16 sq.units, then the area of the parallelogram having 3a⃗+2b⃗3\vec{a}+2\vec{b} and a⃗+3b⃗\vec{a}+3\vec{b} as two adjacent sides (in sq.units) is

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  19. 19
    If A and B are two events such that P(A)=13P(A)=\frac{1}{3}, P(B)=15P(B)=\frac{1}{5}, P(A∪B)=13P(A\cup B)=\frac{1}{3}, then the value of P(A′/B′)+P(B′/A′)P(A'/B')+P(B'/A') is:

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  20. 20
    If the function f is given by f(x)=x3−3(a−2)x2+3ax+7f(x)=x^3-3(a-2)x^2+3ax+7, for some a∈Ra\in\mathbb{R}, is increasing in (0,1](0,1] and decreasing in [1,5)[1,5), then a root of the equation f(x)−14(x−1)2=0  (x≠1)\frac{f(x)-14}{(x-1)^2}=0\;(x\neq1) is

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  21. 21
    If ∫cos⁡8x+1cot⁡2x−tan⁡2x dx=Acos⁡8x+c\int\frac{\cos 8x+1}{\cot 2x - \tan 2x}\,dx = A\cos 8x+c, where c is an arbitrary constant, then the value of A is

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  22. 22
    At present a firm is manufacturing 1000 items. It is estimated that the rate of change of production P w.r.t. additional number of workers x is given by dPdx=100−12x\frac{dP}{dx}=100-12\sqrt{x}. If the firm employs 9 more workers, then the new level of production of items is

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  23. 23
    The slope of the tangent to a curve y=f(x)y=f(x) at (x,f(x))(x,f(x)) is 2x+12x+1. If the curve passes through the point (1,2)(1,2), then the area (in sq.units), bounded by the curve, the X-axis and the line x=1x=1, is

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  24. 24
    If cos⁡−1p+cos⁡−11−p+cos⁡−11−q=3π4\cos^{-1}\sqrt{p}+\cos^{-1}\sqrt{1-p}+\cos^{-1}\sqrt{1-q}=\frac{3\pi}{4}, then q is:

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  25. 25
    A rod AB, 13 feet long moves with its ends A and B on two perpendicular lines OX and OY respectively. When A is 5 feet from O, it is moving away at the rate of 3 feet/sec. At this instant, B is moving at the rate

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  26. 26
    If both mean and variance of 50 observations x1,x2,…,x50x_1, x_2,\ldots,x_{50} are equal to 16 and 256 respectively, then mean of (x1−5)2,(x2−5)2,…,(x50−5)2(x_1-5)^2,(x_2-5)^2,\ldots,(x_{50}-5)^2 is

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  27. 27
    If truth value of logical statement (p↔∼q)→(∼p∧q)(p\leftrightarrow\sim q)\to(\sim p\wedge q) is false, then the truth values of p and q are respectively

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  28. 28
    If f(x)=3x+45x−7f(x)=\frac{3x+4}{5x-7} and g(x)=7x+45x−3g(x)=\frac{7x+4}{5x-3}, then f(g(x))=f(g(x))=

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  29. 29
    If cos⁡x+cos⁡y−cos⁡(x+y)=32\cos x+\cos y-\cos(x+y)=\frac{3}{2}, then

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  30. 30
    If f(x)=3xf(x)=3^x, g(x)=4xg(x)=4^x, then f′(0)−g′(0)1+f′(0)g′(0)\frac{f'(0)-g'(0)}{1+f'(0)g'(0)} is

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  31. 31
    The equation of line passing through the point (1,2,3)(1,2,3) and perpendicular to the lines x−23=y−12=z+1−2\frac{x-2}{3}=\frac{y-1}{2}=\frac{z+1}{-2} and x2=y−3=z1\frac{x}{2}=\frac{y}{-3}=\frac{z}{1} is:

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  32. 32
    If the vertices of a triangle are (−2,3)(-2,3), (6,−1)(6,-1) and (4,3)(4,3), the coordinates of the circumcentre of the triangle are

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  33. 33
    If y=[(x+1)(2x+1)(3x+1)⋯(nx+1)]3/2y=[(x+1)(2x+1)(3x+1)\cdots(nx+1)]^{3/2}, then dydx\frac{dy}{dx} at x=0x=0 is

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  34. 34
    π+sin⁡−1 ⁣45+sin⁡−1 ⁣513+sin⁡−1 ⁣1665\pi+\sin^{-1}\!\frac{4}{5}+\sin^{-1}\!\frac{5}{13}+\sin^{-1}\!\frac{16}{65} is equal to

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  35. 35
    The solution of the differential equation dydx+yx=sin⁡x\frac{dy}{dx}+\frac{y}{x}=\sin x is

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  36. 36
    Let f:[−1,2]→[0,∞)f:[-1,2]\to[0,\infty) be a continuous function such that f(x)=f(1−x),  ∀x∈[−1,2]f(x)=f(1-x),\;\forall x\in[-1,2]. Let R1=∫−12xf(x) dxR_1=\int_{-1}^{2}xf(x)\,dx and R2R_2 be the area of the region bounded by y=f(x)y=f(x), x=−1x=-1, x=2x=2 and the X-axis, then R2R_2 is

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  37. 37
    If I=∫sin⁡(log⁡x) dxI=\int\sin(\log x)\,dx, then I is given by

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  38. 38
    The teacher wants to arrange 5 students on the platform such that the boy B1B_1 occupies the second position and the girls G1G_1 and G2G_2 are always adjacent to each other, then the number of such arrangements is

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  39. 39
    Let a⃗=2i^+j^−2k^\vec{a}=2\hat{i}+\hat{j}-2\hat{k}, b⃗=i^+j^\vec{b}=\hat{i}+\hat{j} and c⃗\vec{c} be a vector such that ∣c⃗−a⃗∣=4|\vec{c}-\vec{a}|=4, ∣(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 π6\frac{\pi}{6}, then a⃗⋅c⃗\vec{a}\cdot\vec{c} is equal to

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  40. 40
    The logical statement (∼(∼p∨q)∨(p∧r))∧(∼q∧r)(\sim(\sim p\vee q)\vee(p\wedge r))\wedge(\sim q\wedge r) is equivalent to

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  41. 41
    ∫x2sin⁡(x2+1)−sin⁡2(x2+1)2sin⁡(x2+1)+sin⁡2(x2+1) dx=\int\frac{x^2\sin(x^2+1)-\sin 2(x^2+1)}{2\sin(x^2+1)+\sin 2(x^2+1)}\,dx =

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  42. 42
    Two cards are drawn successively with replacement from a well shuffled pack of 52 cards. Then mean of number of tens is

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  43. 43
    For all real x, the minimum value of 1−x+x21+x+x2\frac{1-x+x^2}{1+x+x^2} is

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  44. 44
    If A=(i110)A=\begin{pmatrix}i&1\\1&0\end{pmatrix} where i=−1i=\sqrt{-1} and B=A2029B=A^{2029}, then B−1=B^{-1}=

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  45. 45
    If a⃗=2i^+3j^+2k^\vec{a}=2\hat{i}+3\hat{j}+2\hat{k}, b⃗=2i^+j^−k^\vec{b}=2\hat{i}+\hat{j}-\hat{k} and c⃗=3i^−j^\vec{c}=3\hat{i}-\hat{j} are such that a⃗+λb⃗\vec{a}+\lambda\vec{b} is perpendicular to c⃗\vec{c}, then the value of λ\lambda is

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  46. 46
    If a circle passes through points (4,0)(4,0) and (0,2)(0,2) and its centre lies on Y-axis. If the radius of the circle is r, then the value of r2−r+1r^2-r+1 is

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  47. 47
    The joint equation of the pair of lines passing through the point (3,−2)(3,-2) and perpendicular to the lines 5x2+2xy−3y2=05x^2+2xy-3y^2=0, is

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  48. 48
    The equation of the tangent to the curve y=9−2x2y=\sqrt{9-2x^2}, at the point where the ordinate and abscissa are equal, is

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  49. 49
    If l=lim⁡x→0x∣x∣+x2l=\lim_{x\to0}\frac{x}{|x|+x^2}, then the value of l is:

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  50. 50
    The unit vector perpendicular to each of the vectors a⃗+b⃗\vec{a}+\vec{b} and a⃗−b⃗\vec{a}-\vec{b}, where a⃗=i^+j^+k^\vec{a}=\hat{i}+\hat{j}+\hat{k} and b⃗=3i^−2j^+5k^\vec{b}=3\hat{i}-2\hat{j}+5\hat{k} is:

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