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MHT-CET 2024 (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
    ABC triangle: A(2,3,5), B(-1,3,2), C(λ\lambda,5,μ\mu). Median through A equally inclined to co-ordinate axes. Value of λ+μ\lambda+\mu is

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  2. 2
    Square plate contracting at 4 cm2^2/sec. Rate of decrease of perimeter when side = 20 cm is

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  3. 3
    Solution of tan⁡−1(1+x)+tan⁡−1(1−x)=π2\tan^{-1}(1+x)+\tan^{-1}(1-x)=\frac{\pi}{2} is

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  4. 4
    p.d.f. f(x)=kx2,  0≤x≤6f(x)=kx^2,\;0\le x\le6. Value of F(4) (c.d.f.) is

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  5. 5
    One side of triangle is double the other; angles opposite differ by 60°. Triangle is

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  6. 6
    A ladder of length 17 m rests with one end against a vertical wall and the other on the level ground. If the lower end slips away at the rate of 1 m/sec, then when it is 8 m away from the wall, its upper end descends at

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  7. 7
    Vertices of feasible region for x+y≤4,  x≤2,  y≤1,  x+y≥1,  x,y≥0x+y\le4,\;x\le2,\;y\le1,\;x+y\ge1,\;x,y\ge0 are

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  8. 8
    Line through A(3,4,1) and B(5,1,6) crosses XZ-plane at

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  9. 9
    a⃗=2i^+3j^+2k^\vec{a}=2\hat{i}+3\hat{j}+2\hat{k}, b⃗=2i^+j^\vec{b}=2\hat{i}+\hat{j}, c⃗=i^+3j^\vec{c}=\hat{i}+3\hat{j}. (a⃗+λb⃗)⊥c⃗(\vec{a}+\lambda\vec{b})\perp\vec{c}. Value of λ\lambda is

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  10. 10
    If I=∫dxx2(x4+1)3/4I=\int\frac{dx}{x^2(x^4+1)^{3/4}}, then I is

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  11. 11
    The 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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  12. 12
    ∫x−7x2−16x+63 dx=Ax2−16x+63+log⁡∣(x−8)+x2−16x+63∣+c\int\frac{x-7}{\sqrt{x^2-16x+63}}\,dx=A\sqrt{x^2-16x+63}+\log|(x-8)+\sqrt{x^2-16x+63}|+c. Value of A is

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  13. 13
    Plane P through (2,1,0),(4,1,1),(5,0,1). Image of R(2,1,6) in plane P is

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  14. 14
    Open metallic tank: square base, vertical sides, volume 500 m3^3. Dimensions for minimum metal area are

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  15. 15
    f(x)=2x−3f(x)=2x-3, g(x)=x3+5g(x)=x^3+5. Value of (f∘g)−1(−9)(f\circ g)^{-1}(-9) is

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  16. 16
    ∫dxsin⁡x+cos⁡x=\int\frac{dx}{\sin x+\cos x}=

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  17. 17
    f(x)=exf(x)=e^x, g(x)=sin⁡−1xg(x)=\sin^{-1}x, h(x)=f(g(x))h(x)=f(g(x)). Value of h′(x)h(x)\frac{h'(x)}{h(x)} is

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  18. 18
    The value of tan⁡π8\tan\frac{\pi}{8} is

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  19. 19
    a⃗=2i^+j^−2k^\vec{a}=2\hat{i}+\hat{j}-2\hat{k}, b⃗=i^+j^\vec{b}=\hat{i}+\hat{j}. c⃗\vec{c} such that a⃗⋅c⃗=∣c⃗∣\vec{a}\cdot\vec{c}=|\vec{c}|, ∣c⃗−a⃗∣=22|\vec{c}-\vec{a}|=2\sqrt{2}, angle between a⃗×b⃗\vec{a}\times\vec{b} and c⃗\vec{c} is π4\frac{\pi}{4}. ∣(a⃗×b⃗)×c⃗∣|(\vec{a}\times\vec{b})\times\vec{c}| is

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  20. 20
    Area of region bounded by y=49−x2y=\sqrt{49-x^2} and X-axis is

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  21. 21
    dPdt=0.5P−450\frac{dP}{dt}=0.5P-450, P(0)=850. Time when P=0 is

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  22. 22
    Number of integral values of p in [-5,5] such that 2x2+4xy−py2+4x+qy+1=02x^2+4xy-py^2+4x+qy+1=0 represents a pair of lines

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  23. 23
    Number of solutions of sin⁡θ+sin⁡4θ+sin⁡7θ=0,  θ∈(0,π)\sin\theta+\sin4\theta+\sin7\theta=0,\;\theta\in(0,\pi)

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  24. 24
    For x>1x>1, (2x)2y=4e2x−2y(2x)^{2y}=4e^{2x-2y}. Value of (1+log⁡2x)2dydx(1+\log2x)^2\frac{dy}{dx} is

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  25. 25
    General solution of cos⁡x(1+cos⁡y) dx−sin⁡y(1+sin⁡x) dy=0\cos x(1+\cos y)\,dx-\sin y(1+\sin x)\,dy=0 is

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  26. 26
    If ∫01/2x2(1−x2)3/2dx=k6\int_0^{1/2}\frac{x^2}{(1-x^2)^{3/2}}dx=\frac{k}{6}, value of k is

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  27. 27
    If αx+by+c=0\alpha x+by+c=0 is normal to curve xy=1xy=1, then

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  28. 28
    Kite at 120 m high, 130 m string out, moving away at 9 m/sec. Rate of string being let out is

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  29. 29
    If w=zz−13iw=\frac{z}{z-\frac{1}{3}i} and ∣w∣=1|w|=1, z lies on

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  30. 30
    B=(1α2122233)\begin{pmatrix}1&\alpha&2\\1&2&2\\2&3&3\end{pmatrix} is adjoint of 3×33\times 3 matrix A, ∣A∣=5|A|=5. Value of α\alpha is

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  31. 31
    Player tosses 2 fair coins: wins ₹5 (2H), ₹2 (1H), ₹1 (0H). Variance of winning amount is

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  32. 32
    A group consists of 8 boys and 5 girls, then the number of committees of 5 persons that can be formed, if the committee consists of at least 2 girls and at most 2 boys, is

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  33. 33
    (1,3) and (5,1) are opposite vertices of a rectangle. Other two vertices lie on y=2x+c. One vertex on other diagonal is

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  34. 34
    Differential equation dydx=1−y2y\frac{dy}{dx}=\frac{1-y^2}{y} determines family of circles with

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  35. 35
    a⃗=i^+2j^+k^\vec{a}=\hat{i}+2\hat{j}+\hat{k}, b⃗=i^−j^+k^\vec{b}=\hat{i}-\hat{j}+\hat{k}, c⃗=i^+j^−k^\vec{c}=\hat{i}+\hat{j}-\hat{k}. Vector in plane of a⃗\vec{a} and b⃗\vec{b}, projection on c⃗\vec{c} is 13\frac{1}{\sqrt{3}}

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  36. 36
    If y=cos⁡−1 ⁣(a2x4+a4)y=\cos^{-1}\!\left(\frac{a^2}{x^4+a^4}\right), then dydx\frac{dy}{dx} is

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  37. 37
    Set A={x≥0∣tan⁡−1x+tan⁡−1(6x)=π4}A=\{x\ge0\mid\tan^{-1}x+\tan^{-1}(6x)=\frac{\pi}{4}\}

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  38. 38
    Number of solutions of tan⁡x+sec⁡x=2cos⁡x\tan x+\sec x=2\cos x in [0,2π][0,2\pi]

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  39. 39
    f(x)f(x) continuous on [0,π][0,\pi] (piecewise: x+a2sin⁡xx+a\sqrt{2}\sin x on [0,π/4)[0,\pi/4), 2xcot⁡x+b2x\cot x+b on [π/4,π/2][\pi/4,\pi/2], acos⁡2x−bsin⁡xa\cos2x-b\sin x on (π/2,π](\pi/2,\pi]). Value of 2a+3b2a+3b is

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  40. 40
    Line x−23=y−1−5=z+22\frac{x-2}{3}=\frac{y-1}{-5}=\frac{z+2}{2} lies in plane x+3y−αz+β=0x+3y-\alpha z+\beta=0. Value of α2+αβ+β2\alpha^2+\alpha\beta+\beta^2 is

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  41. 41
    Variance for first six prime numbers greater than 5 is

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  42. 42
    Vector projection of AB→\overrightarrow{AB} on CD→\overrightarrow{CD}, A(2,-3,0), B(1,-4,-2), C(4,6,8), D(7,0,10)

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  43. 43
    Binomial X, n=6, P(X=4)=135212P(X=4)=\frac{135}{2^{12}}. Variance is

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  44. 44
    The given circuit is equivalent to

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  45. 45
    ∫dx7−6x−x2=\int\frac{dx}{\sqrt{7-6x-x^2}}=

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  46. 46
    Circles x2+y2=9x^2+y^2=9 and x2+y2+2αx+2y+1=0x^2+y^2+2\alpha x+2y+1=0 touch internally. Value of α3\alpha^3 is

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  47. 47
    Three critics: odds in favour 2:5, 3:4, 4:3. Probability majority is in favour

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  48. 48
    (a⃗×b⃗)×c⃗=13∣b⃗∣∣c⃗∣a⃗(\vec{a}\times\vec{b})\times\vec{c}=\frac{1}{3}|\vec{b}||\vec{c}|\vec{a}. Angle θ\theta between b⃗\vec{b} and c⃗\vec{c}: value of sin⁡θ\sin\theta is

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  49. 49
    Scalar projection of segment AB on line with d.r.s 6,-2,3. A(-2,0,3), B(1,4,2).

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  50. 50
    Value of lim⁡x→aa+2x−3x3a+x−2x\lim_{x\to a}\frac{\sqrt{a+2x}-\sqrt{3x}}{\sqrt{3a+x}-2\sqrt{x}} is

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