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MHT-CET 2024 (11 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
    ∫0πdx4+3cos⁡x=\int_0^{\pi} \frac{dx}{4 + 3\cos x} =

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  2. 2
    If y=log⁡sin⁡x(tan⁡x)y = \log_{\sin x}(\tan x), then dydx∣x=π/4\left.\frac{dy}{dx}\right|_{x=\pi/4} has the value

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  3. 3
    lim⁡x→2(1x−2−2xx3−3x2+2x)\lim_{x \to 2} \left(\frac{1}{x-2} - \frac{2x}{x^3-3x^2+2x}\right) is equal to

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  4. 4
    From a lot of 20 baskets, which includes 6 defective baskets, a sample of 2 baskets is drawn at random one by one without replacement. The expected value of number of defective baskets is

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  5. 5
    If the angle between the lines represented by the equation x2+λxy−y2tan⁡2θ=0x^2 + \lambda xy - y^2\tan^2\theta = 0 is 2θ2\theta, then the value of λ\lambda is

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  6. 6
    Number of common tangents to the circles x2+y2−6x−14y+48=0x^2+y^2-6x-14y+48=0 and x2+y2−6x=0x^2+y^2-6x=0 are

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  7. 7
    The left-hand derivative of f(x)=[x]sin⁡(πx)f(x) = [x]\sin(\pi x), at x=kx = k, k is an integer and [.] is the greatest integer function, is

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  8. 8
    Let f(x)=∫x(1+x)2 dx,  x≥0f(x) = \int \frac{\sqrt{x}}{(1+x)^2}\,dx,\; x \geq 0, then f(3)−f(1)f(3) - f(1) is equal to

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  9. 9
    Value of c satisfying the conditions and conclusions of Rolle's theorem for the function f(x)=xx+6,  x∈[−6,0]f(x) = x\sqrt{x+6},\; x \in [-6,0] is

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  10. 10
    Three of six vertices of a regular hexagon are chosen at random. The probability that the triangle with these three vertices is equilateral, equals

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  11. 11
    If the direction cosines l,m,nl, m, n of two lines are connected by relations l−5m+3n=0l - 5m + 3n = 0 and 7l2+5m2−3n2=07l^2 + 5m^2 - 3n^2 = 0, then value of l+m+nl+m+n is

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  12. 12
    Let f(x)=log⁡(sin⁡x),  0<x<πf(x) = \log(\sin x),\; 0 < x < \pi and g(x)=sin⁡−1(e−x),  x≥0g(x) = \sin^{-1}(e^{-x}),\; x \geq 0. If α\alpha is a positive real number such that a=(f∘g)′(α)a = (f \circ g)'(\alpha) and b=(f∘g)(α)b = (f\circ g)(\alpha), then

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  13. 13
    Five students are to be arranged on a platform such that the boy B1 occupies the second position and such that the girl G1 is always adjacent to the girl G2. Then, the number of such possible arrangements is

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  14. 14
    If the volume of the parallelepiped is 158 cu. units whose coterminous edges are given by the vectors a⃗=(i^+j^+nk^)\vec{a} = (\hat{i}+\hat{j}+n\hat{k}), b⃗=2i^+4j^−nk^\vec{b} = 2\hat{i}+4\hat{j}-n\hat{k} and c⃗=i^+nj^+3k^\vec{c} = \hat{i}+n\hat{j}+3\hat{k}, where n≥0n \geq 0, then the value of n is

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  15. 15
    Let f(x)f(x) be positive for all real xx. If I1=∫1−hhxf(x(1−x)) dxI_1 = \int_{1-h}^{h} xf(x(1-x))\,dx and I2=∫1−hhf(x(1−x)) dxI_2 = \int_{1-h}^{h} f(x(1-x))\,dx, where (2h−1)>0(2h-1) > 0, then I1I2\frac{I_1}{I_2} is

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  16. 16
    The mirror image of the point (1,2,3)(1,2,3) in a plane is (−73,−43,−13)\left(-\frac{7}{3},-\frac{4}{3},-\frac{1}{3}\right). Thus, the point lies on this plane.

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  17. 17
    If α=3sin⁡−1611\alpha = 3\sin^{-1}\frac{6}{11} and β=3cos⁡−149\beta = 3\cos^{-1}\frac{4}{9}, where the inverse trigonometric functions take only the principal values, then the correct option is

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  18. 18
    The value of sec⁡2(tan⁡−12)+csc⁡2(cot⁡−13)\sec^2(\tan^{-1} 2) + \csc^2(\cot^{-1} 3) is

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  19. 19
    The value of tan⁡π8\tan\frac{\pi}{8} is

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  20. 20
    A plane is parallel to two lines, whose direction ratios are (1,0,−1)(1,0,-1) and (−1,1,0)(-1,1,0) and it contains the point (1,1,1)(1,1,1). If it cuts coordinate axes (X, Y, Z axes resp.) at A, B, C, then the volume of the tetrahedron OABC is cu. Units.

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  21. 21
    The domain of the function given by 2x+2y=22^x + 2^y = 2 is

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  22. 22
    If f(x)=xex(1−x),  x∈Rf(x) = xe^{x(1-x)},\; x \in \mathbb{R}, then f(x)f(x) is

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  23. 23
    The solution of the differential equation dydx=1+y21+x2\frac{dy}{dx} = \frac{1+y^2}{1+x^2} is

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  24. 24
    If a⃗,b⃗,c⃗\vec{a}, \vec{b}, \vec{c} are three vectors such that a⃗⋅(b⃗+c⃗)+b⃗⋅(c⃗+a⃗)+c⃗⋅(a⃗+b⃗)=0\vec{a}\cdot(\vec{b}+\vec{c}) + \vec{b}\cdot(\vec{c}+\vec{a}) + \vec{c}\cdot(\vec{a}+\vec{b}) = 0 and ∣a⃗∣=1|\vec{a}|=1, ∣b⃗∣=8|\vec{b}|=8 and ∣c⃗∣=4|\vec{c}|=4, then ∣a⃗+b⃗+c⃗∣|\vec{a}+\vec{b}+\vec{c}| has the value

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  25. 25
    Let a⃗=2i^+j^−2k^\vec{a}=2\hat{i}+\hat{j}-2\hat{k} and b⃗=i^+j^\vec{b}=\hat{i}+\hat{j}. If c⃗\vec{c} is a vector such that a⃗⋅c⃗=∣c⃗∣\vec{a}\cdot\vec{c}=|\vec{c}|, ∣c⃗−a⃗∣=22|\vec{c}-\vec{a}|=2\sqrt{2} and the angle between (a⃗×b⃗)(\vec{a}\times\vec{b}) and c⃗\vec{c} is π6\frac{\pi}{6}, then ∣(a⃗×b⃗)×c⃗∣|(\vec{a}\times\vec{b})\times\vec{c}| is

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  26. 26
    ∫−13[cot⁡−1xx2+1+cot⁡−1x2+1x]dx=\int_{-1}^{3} \left[\cot^{-1}\frac{x}{x^2+1} + \cot^{-1}\frac{x^2+1}{x}\right]dx =

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  27. 27
    If P=(1α3133244)P = \begin{pmatrix}1&\alpha&3\\1&3&3\\2&4&4\end{pmatrix} is the adjoint of a 3×33\times3 matrix A and ∣A∣=4|A|=4, then value of α\alpha is

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  28. 28
    If f(x)={1+mx−1−mxx,−1≤x<02x+1x−2,0≤x≤1f(x)=\begin{cases}\dfrac{\sqrt{1+mx}-\sqrt{1-mx}}{x}, & -1\leq x<0\\ \dfrac{2x+1}{x-2}, & 0\leq x\leq1\end{cases} is continuous in the interval [−1,1][-1,1], then m is equal to

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  29. 29
    The area of the region bounded by the parabola y=x2y = x^2 and the curve y=∣x∣y = |x| is

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  30. 30
    Derivative of tan⁡−1 ⁣(1+x2−1−x21+x2+1−x2)\tan^{-1}\!\left(\frac{\sqrt{1+x^2}-\sqrt{1-x^2}}{\sqrt{1+x^2}+\sqrt{1-x^2}}\right) w.r.t. cos⁡−1(x2)\cos^{-1}(x^2) is

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  31. 31
    A binomial random variable X satisfies 9P(X=4)=P(X=2)9P(X=4) = P(X=2) when n=6n=6. Then p is equal to

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  32. 32
    In △ABC\triangle ABC, with usual notations, if acos⁡2C2+ccos⁡2A2=3b2a\cos^2\frac{C}{2} + c\cos^2\frac{A}{2} = \frac{3b}{2}, then

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  33. 33
    The lines x−13=y+12=z−15\frac{x-1}{3}=\frac{y+1}{2}=\frac{z-1}{5} and x+24=y−13=z+12\frac{x+2}{4}=\frac{y-1}{3}=\frac{z+1}{2}

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  34. 34
    A curve passes through the point (1,π6)\left(1,\frac{\pi}{6}\right). Let the slope of the curve at each point (x,y)(x,y) be yx+sec⁡(yx),  x>0\frac{y}{x}+\sec\left(\frac{y}{x}\right),\; x>0, then, the equation of the curve is

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  35. 35
    For the following shaded area, the linear constraints except x,y≥0x, y \geq 0 are

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  36. 36
    Let ω≠1\omega \neq 1 be a cube root of unity and S be the set of all non-singular matrices of the form (1abω1cω2ω1)\begin{pmatrix}1&a&b\\\omega&1&c\\\omega^2&\omega&1\end{pmatrix} where each of a, b and c is either ω\omega or ω2\omega^2, then the number of distinct matrices in the set S is

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  37. 37
    The value of 2tan⁡−1 ⁣12+tan⁡−1 ⁣172\tan^{-1}\!\frac{1}{2}+\tan^{-1}\!\frac{1}{7}

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  38. 38
    ∫ex(1+x)cos⁡2(ex⋅x) dx=\int \frac{e^x(1+x)}{\cos^2(e^x \cdot x)}\,dx =

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  39. 39
    In a triangle ABC, with usual notations, if m∠A=60∘m\angle A = 60^\circ, b=8b=8, a=6a=6 and B=sin⁡−1(x)B = \sin^{-1}(x), then x has the value

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  40. 40
    If variance of x1,x2,…,xnx_1, x_2, \ldots, x_n is σx2\sigma_x^2, then the variance of λx1,λx2,…,λxn\lambda x_1, \lambda x_2, \ldots, \lambda x_n (λ≠0)(\lambda \neq 0) is

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  41. 41
    If a⃗=i^+j^\vec{a}=\hat{i}+\hat{j}, b⃗=2j^−k^\vec{b}=2\hat{j}-\hat{k} and r⃗×a⃗=b⃗×a⃗\vec{r}\times\vec{a}=\vec{b}\times\vec{a}, r⃗×b⃗=a⃗×b⃗\vec{r}\times\vec{b}=\vec{a}\times\vec{b}, then the value r⃗∣r⃗∣\frac{\vec{r}}{|\vec{r}|} is

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  42. 42
    If z=x+iyz=x+iy and z1/3=p+iqz^{1/3}=p+iq, where x,y,p,q∈Rx,y,p,q\in\mathbb{R} and i=−1i=\sqrt{-1}, then value of xp+yq\frac{x}{p}+\frac{y}{q} is

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  43. 43
    If ∫dxx1−x3=klog⁡∣1−x3−11−x3+1∣+c\int\frac{dx}{x\sqrt{1-x^3}} = k\log\left|\frac{\sqrt{1-x^3}-1}{\sqrt{1-x^3}+1}\right| + c, (where c is a constant of integration), then value of k is

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  44. 44
    The statement pattern p→∼(p∧∼q)p \to \sim(p \wedge \sim q) is equivalent to

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  45. 45
    a and b are the intercepts made by a line on the coordinate axes. If 3a=b3a = b and the line passes through (1,3)(1,3), then the equation of the line is

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  46. 46
    Let a⃗,b⃗\vec{a},\vec{b} and c⃗\vec{c} be three unit vectors such that a⃗×(b⃗×c⃗)=32(b⃗+c⃗)\vec{a}\times(\vec{b}\times\vec{c})=\frac{\sqrt{3}}{2}(\vec{b}+\vec{c}). If b⃗\vec{b} is not parallel to c⃗\vec{c}, then the angle between a⃗\vec{a} and b⃗\vec{b} is

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  47. 47
    The vector equation of the line 2x+4=3y+1=6z−32x+4=3y+1=6z-3 is

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  48. 48
    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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  49. 49
    ∫log⁡(cot⁡x)sin⁡2x dx=\int \frac{\log(\cot x)}{\sin 2x}\,dx =

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  50. 50
    If y=1−sin⁡−1(x)1+sin⁡−1(x)y = \sqrt{\frac{1-\sin^{-1}(x)}{1+\sin^{-1}(x)}}, then dydx\frac{dy}{dx} at x=0x=0 and y=1y=1 is

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