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MHT-CET 2023 (4 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
    Suppose A is any 3×33\times3 non-singular matrix and (A−3I)(A−5I)=O(A-3I)(A-5I)=O where I=I3I=I_3 and O=O3O=O_3. Here O3O_3 represent zero matrix of order 3 and I3I_3 is an identity matrix of order 3. If αA+βA−1=4I\alpha A + \beta A^{-1} = 4I, then α+β\alpha + \beta is equal to

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  2. 2
    The lines x−21=y−31=z−4−k\frac{x-2}{1}=\frac{y-3}{1}=\frac{z-4}{-k} and x−1k=y−42=z−51\frac{x-1}{k}=\frac{y-4}{2}=\frac{z-5}{1} are coplanar if

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  3. 3
    The variance of first 50 even natural numbers is

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  4. 4
    The statement pattern [p∧(q∨r)]∨[∼r∧∼q∧p][p\wedge(q\vee r)]\vee[\sim r\wedge\sim q\wedge p] is equivalent to

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  5. 5
    If 8f(x)+6f ⁣(1x)=x+58f(x)+6f\!\left(\frac{1}{x}\right)=x+5 and y=x2f(x)y=x^2 f(x), then dydx\frac{dy}{dx} at x=−1x=-1 is

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  6. 6
    The number of ways in which 5 boys and 3 girls can be seated on a round table, if a particular boy B1B_1 and a particular girl G1G_1 never sit adjacent to each other, is

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  7. 7
    The value of cot⁡ ⁣(csc⁡−1 ⁣53+tan⁡−1 ⁣23)\cot\!\left(\csc^{-1}\!\frac{5}{3}+\tan^{-1}\!\frac{2}{3}\right) is

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  8. 8
    ∫ ⁣(1+x−1x)ex+1x dx\int\!\left(1+x-\frac{1}{x}\right)e^{x+\frac{1}{x}}\,dx equals

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  9. 9
    The function f(x)=log⁡e(π+x)log⁡e(e+x)f(x)=\frac{\log_e(\pi+x)}{\log_e(e+x)} is

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  10. 10
    Let y=y(x)y=y(x) be the solution of the differential equation sin⁡x dydx+ycos⁡x=4x, x∈(0,π)\sin x\,\frac{dy}{dx}+y\cos x=4x,\,x\in(0,\pi). If y ⁣(π2)=0y\!\left(\frac{\pi}{2}\right)=0, then y ⁣(π6)y\!\left(\frac{\pi}{6}\right) is equal to

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  11. 11
    The value of I=∫x2(a+bx)2 dxI=\int\frac{x^2}{(a+bx)^2}\,dx is

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  12. 12
    Let a⃗=αi^+3j^−k^\vec{a}=\alpha\hat{i}+3\hat{j}-\hat{k}, b⃗=3i^−βj^+4k^\vec{b}=3\hat{i}-\beta\hat{j}+4\hat{k} and c⃗=i^+2j^−2k^\vec{c}=\hat{i}+2\hat{j}-2\hat{k}, where α,β∈R\alpha,\beta\in\mathbb{R}, be three vectors. If the projection of a⃗\vec{a} on c⃗\vec{c} is 103\frac{10}{3} and b⃗×c⃗=−6i^+10j^+7k^\vec{b}\times\vec{c}=-6\hat{i}+10\hat{j}+7\hat{k}, then the value of 2α+β2\alpha+\beta is

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  13. 13
    If 4ab=3h24ab = 3h^2, then the ratio of the slopes of lines represented by ax2+2hxy+by2=0ax^2+2hxy+by^2=0 is

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  14. 14
    If sin⁡(cot⁡−1(x+1))=cos⁡(tan⁡−1x)\sin(\cot^{-1}(x+1))=\cos(\tan^{-1}x), then considering positive square roots, xx has the value

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  15. 15
    A random variable has the following probability distribution XX: 0, 1, 2, 3, 4, 5, 6, 7 P(x)P(x): 0,2p,2p,3p,p2,2p2,7p2,2p0, 2p, 2p, 3p, p^2, 2p^2, 7p^2, 2p The then value of pp is

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  16. 16
    Let A and B be two events such that the probability that exactly one of them occurs is 25\frac{2}{5} and the probability that A or B occurs is 12\frac{1}{2}, then the probability of both of them occur together is

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  17. 17
    Let (−2−13i)3=x+iy27\left(-2-\dfrac{1}{3}i\right)^{3}=\dfrac{x+iy}{27}, i=−1i=\sqrt{-1}, where xx and yy are real numbers, then (y−x)(y-x) has the value

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  18. 18
    The shaded region in the following figure is the solution set of the inequations

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  19. 19
    Let p⃗\vec{p} and q⃗\vec{q} be the position vectors of P and Q respectively, with respect to O and ∣p⃗∣=p|\vec{p}|=p, ∣q⃗∣=q|\vec{q}|=q. The points R and S divide PQ internally and externally in the ratio 2:3 respectively. If OR and OS are perpendiculars, then

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  20. 20
    The value of aa for which the volume of parallelepiped formed by i^+aj^+k^\hat{i}+a\hat{j}+\hat{k}, j^+ak^\hat{j}+a\hat{k} and ai^+k^a\hat{i}+\hat{k} becomes minimum is

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  21. 21
    The approximate value of log⁡101002\log_{10}1002 is (Given log⁡10e=0.4343\log_{10}e=0.4343)

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  22. 22
    The sum of intercepts on coordinate axes made by tangent to the curve x+y=a\sqrt{x}+\sqrt{y}=\sqrt{a} is

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  23. 23
    If the angles of a triangle are in the ratio 4:1:14:1:1, then the ratio of the longest side to the perimeter is

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  24. 24
    Considering only the Principal values of inverse functions, the set A={x≥0∣tan⁡−1(2x)+tan⁡−1(3x)=π4}A = \{x\geq0 \mid \tan^{-1}(2x)+\tan^{-1}(3x)=\frac{\pi}{4}\}

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  25. 25
    The line L given by x5+yb=1\frac{x}{5}+\frac{y}{b}=1 passes through the point (13,32)(13,32). The line K is parallel to line L and has the equation xc+y3=1\frac{x}{c}+\frac{y}{3}=1. Then the distance between L and K is ___ units.

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  26. 26
    The integral ∫−1/21/2([x]+log⁡e1+x1−x)dx\int_{-1/2}^{1/2}\left([x]+\log_e\frac{1+x}{1-x}\right)dx, where [x][x] represents greatest integer function, equals

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  27. 27
    If f(x)=x3+ex/2f(x)=x^3+e^{x/2} and g(x)=f−1(x)g(x)=f^{-1}(x), then the value of g′(1)g'(1) is

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  28. 28
    A wire of length 2 units is cut into two parts, which are bent respectively to form a square of side xx units and a circle of radius rr units. If the sum of the areas of square and the circle so formed is minimum, then

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  29. 29
    Let L1:x+13=y+21=z+12L_1:\frac{x+1}{3}=\frac{y+2}{1}=\frac{z+1}{2} and L2:x−21=y+22=z−33L_2:\frac{x-2}{1}=\frac{y+2}{2}=\frac{z-3}{3} be two given lines. Then the unit vector perpendicular to L1L_1 and L2L_2 is

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  30. 30
    Let a,b∈Ra,b\in\mathbb{R}. If the mirror image of the point P(a,6,9)P(a,6,9) w.r.t. line x−37=y−25=z−1−9\frac{x-3}{7}=\frac{y-2}{5}=\frac{z-1}{-9} is (20,b,−a−9)(20,b,-a-9), then ∣a+b∣|a+b| is equal to

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  31. 31
    A random variable X has the following probability distribution XX: 1, 2, 3, 4, 5 P(X)P(X): k2k^2, 2k2k, kk, 2k2k, 5k25k^2 Then P(X>2)P(X>2) is equal to

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  32. 32
    The number of distinct real values of λ\lambda, for which the vectors −λ2i^+j^+k^-\lambda^2\hat{i}+\hat{j}+\hat{k}, i^−λ2j^+k^\hat{i}-\lambda^2\hat{j}+\hat{k} and i^+j^−λ2k^\hat{i}+\hat{j}-\lambda^2\hat{k} are coplanar, is

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  33. 33
    If f(x)=log⁡e ⁣1−x1+x, ∣x∣<1f(x)=\log_e\!\frac{1-x}{1+x},\,|x|<1, then f ⁣(2x1+x2)f\!\left(\frac{2x}{1+x^2}\right) is equal to

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  34. 34
    Let the vectors a⃗,b⃗,c⃗\vec{a},\vec{b},\vec{c} and d⃗\vec{d} be such that (a⃗×b⃗)×(c⃗×d⃗)=0⃗(\vec{a}\times\vec{b})\times(\vec{c}\times\vec{d})=\vec{0}. Let P1P_1 and P2P_2 be the planes determined by the pair of vectors a⃗,b⃗\vec{a},\vec{b} and c⃗,d⃗\vec{c},\vec{d} respectively, then the angle between P1P_1 and P2P_2 is

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  35. 35
    If I=∫esin⁡θ(log⁡sin⁡θ+csc⁡2θ)cos⁡θ dθI=\int e^{\sin\theta}(\log\sin\theta+\csc^2\theta)\cos\theta\,d\theta, then I is equal to

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  36. 36
    The equation of the circle which has its centre at the point (3,4)(3,4) and touches the line 5x+12y−11=05x+12y-11=0 is

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  37. 37
    A plane which is perpendicular to two planes 2x−2y+z=02x-2y+z=0 and x−y+2z=4x-y+2z=4, passes through (1,−2,1)(1,-2,1). The distance of the plane from the point (1,2,2)(1,2,2) is

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  38. 38
    The value of the expression 3csc⁡20∘−sec⁡20∘\sqrt{3}\csc20^\circ-\sec20^\circ is equal to

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  39. 39
    Let a⃗,b⃗\vec{a},\vec{b} and c⃗\vec{c} be three vectors having magnitude 1, 1 and 2 respectively. If a⃗×(a⃗×c⃗)+b⃗=0⃗\vec{a}\times(\vec{a}\times\vec{c})+\vec{b}=\vec{0}, then the acute angle between a⃗\vec{a} and c⃗\vec{c} is

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  40. 40
    The area bounded between the curves y=ax2y=ax^2 and x=ay2 (a>0)x=ay^2\,(a>0) is 1 sq. units, then the value of aa is

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  41. 41
    The integral ∫sec⁡2/3x⋅csc⁡4/3x dx\int\sec^{2/3}x\cdot\csc^{4/3}x\,dx is equal to

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  42. 42
    If sum of two numbers is 3, then the maximum value of the product of first number and square of the second number is

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  43. 43
    Given that the slope of the tangent to a curve y=y(x)y=y(x) at any point (x,y)(x,y) is 2yx2\frac{2y}{x^2}. If the curve passes through the centre of the circle x2+y2−2x−2y=0x^2+y^2-2x-2y=0, then its equation is

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  44. 44
    If y=(x+1)(4x+1)(9x+1)⋯(n2x+1)2y=(x+1)(4x+1)(9x+1)\cdots(n^2x+1)^2, then dydx\frac{dy}{dx} at x=0x=0 is

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  45. 45
    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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  46. 46
    A wet substance in the open air loses its moisture at a rate proportional to the moisture content. If a sheet hung in the open air loses half its moisture during the first hour, then the time tt, in which 99% of the moisture will be lost, is

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  47. 47
    lim⁡x→0(1−cos⁡2x)(3+cos⁡x)xtan⁡4x\lim_{x\to0}\frac{(1-\cos2x)(3+\cos x)}{x\tan4x} has the value

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  48. 48
    Let a,b∈R (a≠0)a,b\in\mathbb{R}\,(a\neq0). If the function ff is defined as f(x)={2x2a,0≤x<1a,1≤x<22b2−4bx,2≤x<∞f(x)=\begin{cases}\frac{2x^2}{a},&0\leq x<1\\a,&1\leq x<\sqrt{2}\\\frac{2b^2-4b}{x},&\sqrt{2}\leq x<\infty\end{cases} is continuous in [0,∞)[0,\infty), then an ordered pair (a,b)(a,b) is

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  49. 49
    If (p∧∼q)∧(p∧r)→∼p∨q(p\wedge\sim q)\wedge(p\wedge r)\to\sim p\vee q is false, then the truth values of pp, qq and rr are respectively

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  50. 50
    In △ABC\triangle ABC, with usual notations, if b=3b=3, c=8c=8, m∠A=60∘m\angle A=60^\circ, then the circumradius of the triangle is ___ units.

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