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MHT-CET 2024 (14 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
    lim⁡x→∞x3{x2+1+x4−x2}=\lim_{x\to\infty}x^3\left\{\sqrt{x^2+\sqrt{1+x^4}}-x\sqrt2\right\}=

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  2. 2
    The range of the function f(x)=x2x2+1f(x)=\dfrac{x^2}{x^2+1} is

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  3. 3
    Let a⃗,b⃗,c⃗\vec{a},\vec{b},\vec{c} be the vectors such that ∣a⃗∣=3,∣b⃗∣=5,b⃗⋅c⃗=10|\vec{a}|=3,|\vec{b}|=5,\vec{b}\cdot\vec{c}=10 and the angle between b⃗\vec{b} and c⃗\vec{c} is π3\dfrac{\pi}{3}. If a⃗\vec{a} is perpendicular to the vector b⃗×c⃗\vec{b}\times\vec{c}, then ∣a⃗×(b⃗×c⃗)∣|\vec{a}\times(\vec{b}\times\vec{c})| is equal to

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  4. 4
    The money invested in a company is compounded continuously. If Rs. 200 invested today becomes Rs. 400 in 6 years, then at the end of 33 years it will become Rs.

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  5. 5
    The area bounded by the curves y=(x−1)2y=(x-1)^2, y=(x+1)2y=(x+1)^2 and y=14y=\dfrac{1}{4} is

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  6. 6
    If ∫x3 dx1+x2=a(1+x2)1+x2+b1+x2+c\int\dfrac{x^3\,dx}{\sqrt{1+x^2}}=a(1+x^2)\sqrt{1+x^2}+b\sqrt{1+x^2}+c (Where c is a constant of integration), then the value of 3ab3ab is:

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  7. 7
    The solution set of the inequalities 4x+3y≤604x+3y\leq60, y≥2xy\geq2x, x≥3x\geq3, x,y≥0x,y\geq0 is represented by region

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  8. 8
    If the line x−2y=mx-2y=m (m∈Zm\in\mathbb{Z}) intersects the circle x2+y2=2x+4yx^2+y^2=2x+4y at two distinct points, then the number of possible values of mm are

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  9. 9
    If f(x)={3(1−2x2)4;0<x<10;otherwisef(x)=\begin{cases}\dfrac{3(1-2x^2)}{4}; & 0<x<1\\0; & \text{otherwise}\end{cases} is a probability density function of XX, then P ⁣(14<x<13)P\!\left(\dfrac{1}{4}<x<\dfrac{1}{3}\right) is:

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  10. 10
    The angle between the lines, whose direction cosines l,m,nl,m,n satisfy the equations l+m+n=0l+m+n=0 and 2l2+2m2−n2=02l^2+2m^2-n^2=0, is:

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  11. 11
    The maximum value of the function f(x)=3x3−18x2+27x−40f(x)=3x^3-18x^2+27x-40 on the set S={x∈R ∣ x2+30≤11x}S=\{x\in\mathbb{R}\,|\,x^2+30\leq11x\} is

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  12. 12
    pp is the length of perpendicular from the origin to the line whose intercepts on the axes are aa and bb respectively, then 1a2+1b2\dfrac{1}{a^2}+\dfrac{1}{b^2} equals.

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  13. 13
    The negation of the statement "The number is an odd number if and only if it is divisible by 3."

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  14. 14
    The integral ∫π/6π/3sec⁡2/3x⋅csc⁡4/3x dx\displaystyle\int_{\pi/6}^{\pi/3}\sec^{2/3}x\cdot\csc^{4/3}x\,dx is equal to

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  15. 15
    If a⃗,b⃗,c⃗\vec{a},\vec{b},\vec{c} are unit vectors and θ\theta is angle between a⃗\vec{a} and c⃗\vec{c} and a⃗+2b⃗+2c⃗=0⃗\vec{a}+2\vec{b}+2\vec{c}=\vec{0}, then ∣a⃗×c⃗∣=|\vec{a}\times\vec{c}|=

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  16. 16
    Consider the lines L1:x+13=y+21=z+12L_1:\dfrac{x+1}{3}=\dfrac{y+2}{1}=\dfrac{z+1}{2} and L2:x−21=y+22=z−33L_2:\dfrac{x-2}{1}=\dfrac{y+2}{2}=\dfrac{z-3}{3}, then the unit vector perpendicular to both L1L_1 and L2L_2 is

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  17. 17
    If y=sin⁡−1x2+cos⁡−1x2y=\sin^{-1}x^2+\cos^{-1}x^2, then (1−x2)y2−xy1=(1-x^2)y_2-xy_1=

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  18. 18
    A linguistic club consists of 6 girls and 4 boys. A team of 4 members is to be selected from this group including the selection of a leader (from among these 4 members) for the team. If the team has to include at most one boy, the number of ways of selecting the team is

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  19. 19
    The function f(x)=[x]⋅cos⁡(2x−12π)f(x)=[x]\cdot\cos\left(\dfrac{2x-1}{2}\pi\right), where [⋅][\cdot] denotes the greatest integer function, is discontinuous at

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  20. 20
    The perpendiculars are drawn to lines L1L_1 and L2L_2 from the origin making an angle π4\dfrac{\pi}{4} and 3π4\dfrac{3\pi}{4} respectively with positive direction of X-axis. If both the lines are at unit distance from the origin, then their joint equation is:

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  21. 21
    Equation of the plane passing through (1,−1,2)(1,-1,2) and perpendicular to the planes x+2y−2z=4x+2y-2z=4 and 3x+2y+z=63x+2y+z=6 is:

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  22. 22
    ∫sin⁡2x(1−32cos⁡x)esin⁡2x+cos⁡3x dx=\int\dfrac{\sin2x\left(1-\dfrac{3}{2}\cos x\right)}{e^{\sin^2 x+\cos^3 x}}\,dx=

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  23. 23
    If the angles A,BA,B and CC of a triangle are in an Arithmetic Progression and if a,ba,b and cc denote the lengths of the sides opposite to A,BA,B and CC respectively, then the value of the expression acsin⁡2C+casin⁡2A\dfrac{a}{c}\sin2C+\dfrac{c}{a}\sin2A is:

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  24. 24
    Let f(x)=∫x2−3x+2x4+1 dxf(x)=\int\dfrac{x^2-3x+2}{x^4+1}\,dx, then function decreases in the interval.

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  25. 25
    If a⃗,b⃗,c⃗\vec{a},\vec{b},\vec{c} are three vectors such that ∣a⃗+b⃗+c⃗∣=1|\vec{a}+\vec{b}+\vec{c}|=1, c⃗=λ(a⃗×b⃗)\vec{c}=\lambda(\vec{a}\times\vec{b}) and ∣a⃗∣=13|\vec{a}|=\dfrac{1}{\sqrt{3}}, ∣b⃗∣=12|\vec{b}|=\dfrac{1}{\sqrt{2}}, ∣c⃗∣=16|\vec{c}|=\dfrac{1}{\sqrt{6}}, then the angle between a⃗\vec{a} and b⃗\vec{b} is:

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  26. 26
    Let the curve be represented by x=2(cos⁡t+tsin⁡t)x=2(\cos t+t\sin t), y=2(sin⁡t−tcos⁡t)y=2(\sin t-t\cos t). Then normal at any point 'tt' of the curve is at a distance of units from the origin.

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  27. 27
    If ∫cos⁡θ5+7sin⁡θ−2cos⁡2θ dθ=Alog⁡e∣f(θ)∣+c\int\dfrac{\cos\theta}{5+7\sin\theta-2\cos^2\theta}\,d\theta=A\log_e|f(\theta)|+c (where cc is a constant of integration), then f(θ)A\dfrac{f(\theta)}{A} can be

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  28. 28
    If the shortest distance between the lines x−12=y−23=z−3λ\dfrac{x-1}{2}=\dfrac{y-2}{3}=\dfrac{z-3}{\lambda} and x−21=y−44=z−55\dfrac{x-2}{1}=\dfrac{y-4}{4}=\dfrac{z-5}{5} is 13\dfrac{1}{\sqrt{3}}, then sum of possible values of λ\lambda is

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  29. 29
    Let A=(2−102)A=\begin{pmatrix}2&-1\\0&2\end{pmatrix}. If B=I−3C1(adj A)+3C2(adj A)2−3C3(adj A)3B=I-{}^3C_1(\text{adj}\,A)+{}^3C_2(\text{adj}\,A)^2-{}^3C_3(\text{adj}\,A)^3, then the sum of all elements of the matrix BB is

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  30. 30
    A line with positive direction cosines passes through the point P(2,−1,2)P(2,-1,2) and makes equal angles with the co-ordinate axes. The line meets the plane 2x+y+z=92x+y+z=9 at point QQ. The length of the line segment PQPQ equals

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  31. 31
    Three critics review a book. For the three critics the odds in favour of the book are 2:5, 3:4 and 4:3 respectively. The probability that the majority is in favour of the book, is given by

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  32. 32
    The equation x3+x−1=0x^3+x-1=0 has

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  33. 33
    If f′(x)=tan⁡−1(sec⁡x+tan⁡x)f'(x)=\tan^{-1}(\sec x+\tan x), −π2<x<π2-\dfrac{\pi}{2}<x<\dfrac{\pi}{2} and f(0)=0f(0)=0, then f(1)f(1) is:

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  34. 34
    If y=[(x+1)(2x+1)⋯(nx+1)]ny=[(x+1)(2x+1)\cdots(nx+1)]^n, then dydx\dfrac{dy}{dx} at x=0x=0 is

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  35. 35
    General solution of the differential equation 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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  36. 36
    If a>0a>0 and z=(1+i)2a−iz=\dfrac{(1+i)^2}{a-i}, i=−1i=\sqrt{-1}, has magnitude 25\dfrac{2}{\sqrt{5}}, then zz is

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  37. 37
    Let f:R→Rf:R\rightarrow R be a function such that f(x)=x3+x2f′(1)+xf′′(2)+6f(x)=x^3+x^2f'(1)+xf''(2)+6, x∈Rx\in R, then f(2)f(2) equals

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  38. 38
    For an initial screening of an entrance exam, a candidate is given fifty problems to solve. If the probability that the candidate can solve any problem is 45\dfrac{4}{5}, then the probability that he is unable to solve less than two problems, is

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  39. 39
    Two cards are drawn successively with replacement from well shuffled pack of 52 cards, then the probability distribution of number of queens is

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  40. 40
    If △ABC\triangle ABC is right angled at AA, where A=(4,2,x)A=(4,2,x), B=(3,1,8)B=(3,1,8) and C≡(2,−1,2)C\equiv(2,-1,2), then the value of xx is:

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  41. 41
    If a⃗,b⃗,c⃗\vec{a},\vec{b},\vec{c} are three vectors with magnitudes 3, 1, 2 respectively, such that a⃗×(a⃗×c⃗)+3b⃗=0⃗\vec{a}\times(\vec{a}\times\vec{c})+3\vec{b}=\vec{0}, if θ\theta is the angle between a⃗\vec{a} and c⃗\vec{c}, then sec⁡2θ\sec^2\theta is

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  42. 42
    The differential equation of y=ex(acos⁡x+bsin⁡x)y=e^x(a\cos x+b\sin x) is

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  43. 43
    ∫sin⁡x+sin⁡3xcos⁡2x dx=Acos⁡x+Blog⁡∣f(x)∣+c\int\dfrac{\sin x+\sin^3 x}{\cos2x}\,dx=A\cos x+B\log|f(x)|+c (where c is a constant of integration). Then values of A, B and f(x)f(x) are

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  44. 44
    If sin⁡(θ−α)\sin(\theta-\alpha), sin⁡θ\sin\theta and sin⁡(θ+α)\sin(\theta+\alpha) are in H.P., then the value of cos⁡2θ\cos2\theta is

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  45. 45
    If cos⁡−1x+cos⁡−1y+cos⁡−1z=3π\cos^{-1}x+\cos^{-1}y+\cos^{-1}z=3\pi, then the value of x2025+x2026+x2027x^{2025}+x^{2026}+x^{2027} is

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  46. 46
    The principal solutions of the equation sec⁡x+tan⁡x=2cos⁡x\sec x+\tan x=2\cos x are

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  47. 47
    In △ABC\triangle ABC, with usual notations, 2acsin⁡12(A−B+C)2ac\sin\dfrac{1}{2}(A-B+C) is equal to

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  48. 48
    Let B≡(0,3)B\equiv(0,3) and C≡(4,0)C\equiv(4,0). The point AA is moving on the line y=2xy=2x at the rate of 2 units/second. The area of △ABC\triangle ABC is increasing at the rate of

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  49. 49
    The statement [(p→q)∧¬q]→r[(p\to q)\wedge\neg q]\to r is a tautology, when rr is equivalent to

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  50. 50
    The variance of 20 observations is 5. If each observation is multiplied by 2, then variance of resulting observations is

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