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MHT-CET 2023 (15 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
    If y=[(x+1)(2x+1)(3x+1)⋯(nx+1)]2y = [(x+1)(2x+1)(3x+1)\cdots(nx+1)]^{2}, then dydx\frac{dy}{dx} at x=0x=0 is

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  2. 2
    For the probability distribution: x:0,1,2,3,4,5x: 0,1,2,3,4,5; p(x):k,0.3,0.15,0.15,0.1,2kp(x): k, 0.3, 0.15, 0.15, 0.1, 2k. The Expected value of X is

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  3. 3
    The value of ∫dx7+6x−x2\int \frac{dx}{7+6x-x^{2}} is equal to

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  4. 4
    The value of cos⁡(18∘−A)cos⁡(18∘+A)−cos⁡(72∘−A)cos⁡(72∘+A)\cos(18^{\circ}-A)\cos(18^{\circ}+A) - \cos(72^{\circ}-A)\cos(72^{\circ}+A) is equal to

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  5. 5
    If ∫dx1+3sin⁡2x=12tan⁡−1(f(x))+c\int \frac{dx}{1+3\sin^{2}x} = \frac{1}{2}\tan^{-1}(f(x))+c, where c is a constant of integration, then f(x)f(x) is equal to

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  6. 6
    A random variable X has the following probability distribution: X:1,2,3,4,5X: 1,2,3,4,5; p(x):k2,2k,k,2k,5k2p(x): k^{2}, 2k, k, 2k, 5k^{2}. Then p(x≥2)p(x\geq2) is equal to

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  7. 7
    Let f:[−1,3]→Rf:[-1,3]\to\mathbb{R} be defined as f(x)={∣x∣+[x],−1≤x<1x+∣x∣,1≤x<2x+[x],2≤x≤3f(x) = \begin{cases}|x|+[x],&-1\leq x<1\\x+|x|,&1\leq x<2\\x+[x],&2\leq x\leq3\end{cases} where [t][t] denotes the greatest integer function. Then ff is discontinuous at

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  8. 8
    Let a⃗=i^+j^+k^\vec{a}=\hat{i}+\hat{j}+\hat{k}, b⃗=i^−j^+k^\vec{b}=\hat{i}-\hat{j}+\hat{k} and c⃗=i^−j^−k^\vec{c}=\hat{i}-\hat{j}-\hat{k} be three vectors. A vector v⃗\vec{v} in the plane of a⃗\vec{a} and b⃗\vec{b}, whose projection on c⃗\vec{c} is 13\frac{1}{\sqrt{3}}, is given by

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  9. 9
    The differential equation, having general solution as Ax2+By2=1Ax^{2}+By^{2}=1, where A and B are arbitrary constants, is

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  10. 10
    Let zz be a complex number such that ∣z∣+z=2+i|z|+z = 2+i, where i=−1i=\sqrt{-1}, then ∣z∣|z| is equal to

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  11. 11
    If a⃗\vec{a} and b⃗\vec{b} are two unit vectors such that 5a⃗+4b⃗5\vec{a}+4\vec{b} and a⃗−2b⃗\vec{a}-2\vec{b} are perpendicular to each other, then the angle between a⃗\vec{a} and b⃗\vec{b} is

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  12. 12
    The value of ∫sec⁡x⋅tan⁡x9−16tan⁡2x dx\int \frac{\sec x \cdot \tan x}{\sqrt{9-16\tan^{2}x}}\,dx is equal to

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  13. 13
    If n(A)=4,n(B)=2n(A)=4, n(B)=2. Then the number of subsets of the set A×BA\times B each having at least 3 elements are

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  14. 14
    A radio-active substance has half-life of hh days, then its initial decay rate is given by. Note that at t=0,M=m0t=0, M=m_{0}

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  15. 15
    If (a+2bcos⁡x)(a−2bcos⁡y)=a2−b2(a+2b\cos x)(a-2b\cos y) = a^{2}-b^{2}, where a>b>0a>b>0, then dxdy\frac{dx}{dy} at (π4,π4)\left(\frac{\pi}{4},\frac{\pi}{4}\right) is

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  16. 16
    Contrapositive of the statement 'If two numbers are equal, then their squares are equal' is

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  17. 17
    Let A and B be 3×33\times 3 real matrices such that A is symmetric matrix and B is skew-symmetric matrix. Then the system of linear equations (A2B2−B2A2)X=O(A^{2}B^{2}-B^{2}A^{2})X = O, where X is a 3×13\times 1 column matrix and O is a 3×13\times 1 null matrix, has

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  18. 18
    If f(x)=x3−10x2+200x−10f(x) = x^{3}-10x^{2}+200x-10, then

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  19. 19
    The number of common tangents to the circles x2+y2−x=0x^{2}+y^{2}-x=0 and x2+y2+x=0x^{2}+y^{2}+x=0 is/are

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  20. 20
    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 30∘30^{\circ}, then ∣(a⃗×b⃗)×c⃗∣|(\vec{a}\times\vec{b})\times\vec{c}| is equal to

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  21. 21
    The equation esin⁡x−e−sin⁡x=4e^{\sin x}-e^{-\sin x}=4 has

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  22. 22
    The value of ∫dx5+4sin⁡x\int \frac{dx}{5+4\sin x} is equal to

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  23. 23
    If p→(q∨r)p\to(q\vee r) is false, then the truth values of p,q,rp, q, r are respectively

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  24. 24
    The area of the region bounded by curves y=3x+1y=3x+1, y=4x+1y=4x+1 and x=2x=2 is

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  25. 25
    The region represented by the inequations 2x+3y≤18,x+y≥10,x≥0,y≥02x+3y\leq18, x+y\geq10, x\geq0, y\geq0 is

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  26. 26
    Let f(x)=(x+1)2−1,x≥−1f(x)=(x+1)^{2}-1, x\geq-1, then the set {x:f(x)=f−1(x)}\{x: f(x)=f^{-1}(x)\} is

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  27. 27
    The probability, that a year selected at random will have 53 Mondays, is

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  28. 28
    Let L1:x+25=y−32=z−61L_{1}:\frac{x+2}{5}=\frac{y-3}{2}=\frac{z-6}{1} and L2:x−34=y+23=z−35L_{2}:\frac{x-3}{4}=\frac{y+2}{3}=\frac{z-3}{5} be the given lines. Then the unit vector perpendicular to both L1L_{1} and L2L_{2} is

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  29. 29
    A multiple-choice examination has 5 questions. Each question has three alternative answers of which exactly one is correct. The probability that a student will get 4 or more correct answers just by guessing is

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  30. 30
    The perpendicular distance from the origin to the plane containing the two lines x+23=y−25=z+57\frac{x+2}{3}=\frac{y-2}{5}=\frac{z+5}{7} and x−11=y−44=z+47\frac{x-1}{1}=\frac{y-4}{4}=\frac{z+4}{7}, is

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  31. 31
    The number of integral values of kk for which the equation 7cos⁡x+5sin⁡x=2k+17\cos x+5\sin x=2k+1 has a solution, is

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  32. 32
    The value of integral ∫−20[x3+3x2+3x+5+(x+1)cos⁡(x+1)]dx\int_{-2}^{0}\left[x^{3}+3x^{2}+3x+5+(x+1)\cos(x+1)\right]dx is equal to

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  33. 33
    If x=2cos⁡θ−cos⁡2θx=2\cos\theta-\cos 2\theta and y=2sin⁡θ−sin⁡2θy=2\sin\theta-\sin 2\theta, then d2ydx2\frac{d^{2}y}{dx^{2}} is equal to

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  34. 34
    2π−(sin⁡−145+sin⁡−1513+sin⁡−11665)2\pi - \left(\sin^{-1}\frac{4}{5}+\sin^{-1}\frac{5}{13}+\sin^{-1}\frac{16}{65}\right) is equal to

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  35. 35
    If two sides of a square are 4x+3y−20=04x+3y-20=0 and 4x+3y+15=04x+3y+15=0, then the area of the square is

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  36. 36
    Twenty meters of wire is available for fencing off a flower-bed in the form of a circular sector. Then the maximum area (in sq. m) of the flowerbed is

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  37. 37
    Let a,b,ca, b, c be three non-zero real numbers such that the equation 3 acos⁡x+2bsin⁡x=c\sqrt{3}\,a\cos x + 2b\sin x = c, x∈[−π/2,π/2]x \in [-\pi/2, \pi/2], has two distinct real roots α\alpha and β\beta with α+β=π/3\alpha + \beta = \pi/3. Then the value of b/ab/a is

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  38. 38
    Let P(2,1,5) be a point in space and Q be a point on the line r⃗=(i^−j^+2k^)+μ(−3i^+j^+5k^)\vec{r}=(\hat{i}-\hat{j}+2\hat{k})+\mu(-3\hat{i}+\hat{j}+5\hat{k}). Then the value of μ\mu for which the vector PQ is parallel to the plane 3x−y+4z=13x-y+4z=1 is

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  39. 39
    lim⁡x→π/2(1−tan⁡x2)(1−sin⁡x)(1+tan⁡x2)(π−2x)3\lim_{x\to\pi/2}\frac{\left(1-\tan\frac{x}{2}\right)(1-\sin x)}{\left(1+\tan\frac{x}{2}\right)(\pi-2x)^{3}} is

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  40. 40
    A ladder 5 m in length is leaning against a wall. The bottom of the ladder is pulled along the ground away from the wall, at the rate of 22 m/sec. How fast is the height on the wall decreasing when the foot of the ladder is 4 m away from the wall?

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  41. 41
    If the equation cos⁡4θ+sin⁡4θ+λ=0\cos^{4}\theta+\sin^{4}\theta+\lambda=0 has real solutions for θ\theta, then λ\lambda lies in the interval

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  42. 42
    The equation of the tangent to the parabola y2=8xy^{2}=8x, which is parallel to the line 4x−y+3=04x-y+3=0 is

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  43. 43
    The centroid of tetrahedron with vertices P(5,-7,0), Q(a,5,3), R(4,-6,b) and S(6,c,2) is (4,-3,2), then the value of 2a+3b+c2a+3b+c is equal to

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  44. 44
    The approximate value of 32.0013^{2.001}, if log⁡3=1.0986\log 3=1.0986 is

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  45. 45
    If the vectors a⃗=i^−j^+2k^\vec{a}=\hat{i}-\hat{j}+2\hat{k}, b⃗=2i^+4j^+k^\vec{b}=2\hat{i}+4\hat{j}+\hat{k} and c⃗=pi^+j^+qk^\vec{c}=p\hat{i}+\hat{j}+q\hat{k} are mutually orthogonal, then (p,q)(p,q) is equal to

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  46. 46
    If u⃗,v⃗\vec{u}, \vec{v} and w⃗\vec{w} are three non-coplanar vectors, then (u⃗+v⃗−w⃗)⋅[(u⃗−v⃗)×(v⃗−w⃗)](\vec{u}+\vec{v}-\vec{w})\cdot[(\vec{u}-\vec{v})\times(\vec{v}-\vec{w})] is equal to

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  47. 47
    The value of kk, if the slope of one of the lines given by 4x2+kxy+y2=04x^{2}+kxy+y^{2}=0 is four times that of the other, is given by

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  48. 48
    The differential equation of y=ex(a+bx+x2)y=e^{x}(a+bx+x^{2}) is

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  49. 49
    The mean of the numbers a,b,8,5,10a,b,8,5,10 is 6 and the variance is 6.8. Then which of the following gives possible values of aa and bb?

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  50. 50
    Let u⃗=i^+j^,v⃗=i^−j^\vec{u}=\hat{i}+\hat{j}, \vec{v}=\hat{i}-\hat{j} and w⃗=i^+2j^+3k^\vec{w}=\hat{i}+2\hat{j}+3\hat{k}. If n^\hat{n} is a unit vector such that u⃗⋅n^=0\vec{u}\cdot\hat{n}=0 and v⃗⋅n^=0\vec{v}\cdot\hat{n}=0, then ∣w⃗⋅n^∣|\vec{w}\cdot\hat{n}| is equal to

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