PYQ Vault

MHT-CET 2024 (9 May Shift 2), Paper I, Mathematics question paper

50 questions · 100 marks · 90 minutes · +2 per correct answer, no negative marking

Every question in the order it was set, free to read. Tap an option to check it, or open the answer and worked solution. Answers open as you go; sign in free to keep opening them.

Sit it as a timed mock
  1. 1
    If the slope of one of the lines represented by ax2+(2a+1)xy+2y2=0ax^2 + (2a+1)xy + 2y^2 = 0 is reciprocal of the slope of the other, then the sum of squares of slopes is

    Tap an option to check your answer.

  2. 2
    In △PQR\triangle PQR, sin⁡P\sin P, sin⁡Q\sin Q and sin⁡R\sin R are in A.P., then

    Tap an option to check your answer.

  3. 3
    The value of tan⁡−1 ⁣ ⁣(1+x2+1−x21+x2−1−x2),  ∣x∣<12,  x≠0\tan^{-1}\!\!\left(\dfrac{\sqrt{1+x^2}+\sqrt{1-x^2}}{\sqrt{1+x^2}-\sqrt{1-x^2}}\right),\;|x| < \frac{1}{\sqrt{2}},\; x \neq 0

    Tap an option to check your answer.

  4. 4
    If slope of a tangent to the curve xy+ax+by=0xy + ax + by = 0 at the point (1,1)(1,1) on it is 2, then a−ba - b is

    Tap an option to check your answer.

  5. 5
    The equation of a line, whose perpendicular distance from the origin is 7 units and the angle which the perpendicular to the line from the origin makes is 120∘120^\circ with positive X-axis, is

    Tap an option to check your answer.

  6. 6
    Let a,b,ca, b, c be the lengths of sides of triangle ABC such that a+b7=b+c8=c+a9=k\frac{a+b}{7} = \frac{b+c}{8} = \frac{c+a}{9} = k. Then (A(△ABC))2k4=\dfrac{(A(\triangle ABC))^2}{k^4} =

    Tap an option to check your answer.

  7. 7
    If the mean and S.D. of the data 3,5,7,a,b3, 5, 7, a, b are 5 and 2 respectively, then aa and bb are the roots of the equation

    Tap an option to check your answer.

  8. 8
    A man takes a step forward with probability 0.4 and backwards with probability 0.6. The probability that at the end of eleven steps, he is one step away from the starting point is

    Tap an option to check your answer.

  9. 9
    If f′(x)=x−5x5f'(x) = x - \dfrac{5}{x^5} and f(1)=4f(1) = 4, then f(x)f(x) is

    Tap an option to check your answer.

  10. 10
    A linguistic club consists of 6 girls and 4 boys. A team of 4 members to be selected including selection of a Captain (from among these 4 members). If the team has to include at most one boy, the number of ways of selecting the team is

    Tap an option to check your answer.

  11. 11
    Negation of the statement: 'The payment will be made if and only if the work is finished in time.' is

    Tap an option to check your answer.

  12. 12
    The equation x3+x−1=0x^3 + x - 1 = 0 has

    Tap an option to check your answer.

  13. 13
    If a line LL is the line of intersection of the planes 2x+3y+z=12x + 3y + z = 1 and x+3y+2z=2x + 3y + 2z = 2. If line LL makes an angle α\alpha with the positive X-axis, then the value of sec⁡α\sec\alpha is

    Tap an option to check your answer.

  14. 14
    The range of values of xx for which f(x)=x3+6x2−36x+7f(x) = x^3 + 6x^2 - 36x + 7 is increasing in

    Tap an option to check your answer.

  15. 15
    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 \leq 11x\} is

    Tap an option to check your answer.

  16. 16
    Let p,q,rp, q, r be three statements, then [p→(q→r)]↔[(p∧q)→r][p \to (q \to r)] \leftrightarrow [(p \wedge q) \to r] is

    Tap an option to check your answer.

  17. 17
    ∫12dx(x2−2x+4)3/2=kk+5\int_1^2 \dfrac{dx}{\left(x^2 - 2x + 4\right)^{3/2}} = \dfrac{k}{k+5}, then kk has the value

    Tap an option to check your answer.

  18. 18
    If the sides of a rectangle are given by the equations x=−2, x=6, y=−2, y=5x = -2,\ x = 6,\ y = -2,\ y = 5, then the equation of the circle, drawn on the diagonal of this rectangle as its diameter, is

    Tap an option to check your answer.

  19. 19
    a⃗=i^+j^+k^,  b⃗=j^−k^\vec{a} = \hat{i}+\hat{j}+\hat{k},\;\vec{b} = \hat{j}-\hat{k}, then vector r⃗\vec{r} satisfying a⃗×r⃗=b⃗\vec{a}\times\vec{r} = \vec{b} and a⃗⋅r⃗=3\vec{a}\cdot\vec{r} = 3 is

    Tap an option to check your answer.

  20. 20
    The magnitude of the projection of the vector 2i^+j^+k^2\hat{i}+\hat{j}+\hat{k} on the vector perpendicular to the plane containing the vectors i^+j^+k^\hat{i}+\hat{j}+\hat{k} and i^+2j^+3k^\hat{i}+2\hat{j}+3\hat{k} is

    Tap an option to check your answer.

  21. 21
    The shortest distance between the lines x−12=y−23=z−34\dfrac{x-1}{2}=\dfrac{y-2}{3}=\dfrac{z-3}{4} and x−23=y−44=z−55\dfrac{x-2}{3}=\dfrac{y-4}{4}=\dfrac{z-5}{5} is

    Tap an option to check your answer.

  22. 22
    If a⃗\vec{a}, b⃗\vec{b} and c⃗\vec{c} are any three non-zero vectors, then (a⃗+2b⃗+c⃗)⋅[(a⃗−b⃗)×(a⃗−b⃗−c⃗)]=(\vec{a}+2\vec{b}+\vec{c})\cdot[(\vec{a}-\vec{b})\times(\vec{a}-\vec{b}-\vec{c})]=

    Tap an option to check your answer.

  23. 23
    In △ABC\triangle ABC, with usual notations, ∠C=π2\angle C = \frac{\pi}{2}, if tan⁡A2\tan\frac{A}{2} and tan⁡B2\tan\frac{B}{2} are the roots of the equation a1x2+b1x+c1=0  (a1≠0)a_1x^2 + b_1x + c_1 = 0\;(a_1 \neq 0), then

    Tap an option to check your answer.

  24. 24
    If ∫sin⁡x3+4cos⁡2x dx=Atan⁡−1(Bcos⁡x)+c\int\dfrac{\sin x}{3 + 4\cos^2 x}\,dx = A\tan^{-1}(B\cos x) + c, (where cc is a constant of integration), then the value of A+BA + B is

    Tap an option to check your answer.

  25. 25
    Vectors a⃗\vec{a} and b⃗\vec{b} are such that ∣a⃗∣=1|\vec{a}|=1, ∣b⃗∣=4|\vec{b}|=4 and a⃗⋅b⃗=2\vec{a}\cdot\vec{b}=2. If c⃗=2a⃗×b⃗−3b⃗\vec{c} = 2\vec{a}\times\vec{b} - 3\vec{b}, then the angle between b⃗\vec{b} and c⃗\vec{c} is

    Tap an option to check your answer.

  26. 26
    If yy is a function of xx and log⁡(x+y)=2xy\log(x+y) = 2xy, then the value of y′(0)y'(0) is

    Tap an option to check your answer.

  27. 27
    If cos⁡2B=cos⁡(A+C)cos⁡(A−C)\cos 2B = \dfrac{\cos(A+C)}{\cos(A-C)}, then tan⁡A,tan⁡B,tan⁡C\tan A, \tan B, \tan C are in

    Tap an option to check your answer.

  28. 28
    If log⁡2x+log⁡4x+log⁡8x+log⁡16x=2536\log_2 x + \log_4 x + \log_8 x + \log_{16} x = \dfrac{25}{36} and x=2kx = 2^k, then kk is

    Tap an option to check your answer.

  29. 29
    If ∣cos⁡(A+B)−sin⁡(A+B)cos⁡2Bsin⁡Acos⁡Asin⁡B−cos⁡Asin⁡Acos⁡B∣=0\begin{vmatrix}\cos(A+B) & -\sin(A+B) & \cos 2B \\ \sin A & \cos A & \sin B \\ -\cos A & \sin A & \cos B\end{vmatrix} = 0, then the value of BB is

    Tap an option to check your answer.

  30. 30
    General solution of the differential equation log⁡ ⁣(dydx)=ax+by\log\!\left(\frac{dy}{dx}\right) = ax + by is

    Tap an option to check your answer.

  31. 31
    ∫(tan⁡x+cot⁡x) dx=\int(\tan x + \cot x)\,dx =

    Tap an option to check your answer.

  32. 32
    If Z1=2+iZ_1 = 2+i and Z2=3−4iZ_2 = 3-4i and Z1‾Z2‾=a+bi\dfrac{\overline{Z_1}}{\overline{Z_2}} = a+bi, then the value of −7a+b-7a + b is (where i=−1i = \sqrt{-1} and a,b∈Ra,b \in \mathbb{R})

    Tap an option to check your answer.

  33. 33
    Let α∈(0,π/2)\alpha \in (0,\pi/2) be fixed. If ∫tan⁡x+tan⁡αtan⁡x−tan⁡α dx=A(x)cos⁡2α+B(x)sin⁡2α+c\int\dfrac{\tan x + \tan\alpha}{\tan x - \tan\alpha}\,dx = A(x)\cos 2\alpha + B(x)\sin 2\alpha + c, then functions A(x)A(x) and B(x)B(x) are respectively

    Tap an option to check your answer.

  34. 34
    Two adjacent sides of a parallelogram are 2i^−4j^+5k^2\hat{i}-4\hat{j}+5\hat{k} and i^−2j^−3k^\hat{i}-2\hat{j}-3\hat{k}, then the unit vector parallel to its diagonal is

    Tap an option to check your answer.

  35. 35
    A water tank has a shape of inverted right circular cone whose semi-vertical angle is tan⁡−1 ⁣12\tan^{-1}\!\frac{1}{2}. Water is poured into it at constant rate of 5 cubic meter/minute. The rate in meter/minute at which level of water is rising, at the instant when depth of water in the tank is 10 m is

    Tap an option to check your answer.

  36. 36
    The differential equation of all circles which pass through the origin and whose centres lie on Y-axis is

    Tap an option to check your answer.

  37. 37
    If xk+yk=ak    (a,k>0)x^k + y^k = a^k\;\;(a,k > 0) and dydx+(yx)1/3=0\dfrac{dy}{dx} + \left(\dfrac{y}{x}\right)^{1/3} = 0, then kk has the value

    Tap an option to check your answer.

  38. 38
    The area (in sq. units) bounded by the curve y=x∣x∣y = x|x|, X-axis and the lines x=−1x = -1 and x=1x = 1 is

    Tap an option to check your answer.

  39. 39
    The co-ordinates of the point, where the line x−12=y−2−3=z+54\dfrac{x-1}{2}=\dfrac{y-2}{-3}=\dfrac{z+5}{4} meets the plane 2x+4y−z=32x+4y-z=3, are

    Tap an option to check your answer.

  40. 40
    lim⁡x→0(1−cos⁡2x)⋅sin⁡5xx2sin⁡3x\lim_{x\to 0}\dfrac{(1-\cos 2x)\cdot\sin 5x}{x^2\sin 3x} is

    Tap an option to check your answer.

  41. 41
    If gg is the inverse of ff and f′(x)=11+x3f'(x) = \dfrac{1}{1+x^3}, then g′(x)g'(x) is

    Tap an option to check your answer.

  42. 42
    A problem in statistics is given to three students A, B and C. Their probabilities of solving the problem are 12,13\frac{1}{2},\frac{1}{3} and 14\frac{1}{4} respectively. If all of them try independently, then the probability that the problem is solved is

    Tap an option to check your answer.

  43. 43
    Let A=[1110131−21], B=[6110]A=\begin{bmatrix}1&1&1\\0&1&3\\1&-2&1\end{bmatrix},\ B=\begin{bmatrix}6\\11\\0\end{bmatrix} and X=[abc]X=\begin{bmatrix}a\\b\\c\end{bmatrix}, if AX=BAX=B, then the value of 2a+b+2c2a+b+2c is

    Tap an option to check your answer.

  44. 44
    f(x)={1−cos⁡kxx2,x≤0x16+x−4,x>0f(x) = \begin{cases}\dfrac{1-\cos kx}{x^2}, & x \leq 0 \\ \dfrac{\sqrt{x}}{\sqrt{16+\sqrt{x}}-4}, & x > 0\end{cases} is continuous at x=0x=0, then the value of kk is

    Tap an option to check your answer.

  45. 45
    If D, E and F are the mid-points of the sides BC, CA and AB of triangle ABC respectively, then AD→+23BE→+13CF→=\overrightarrow{AD}+\frac{2}{3}\overrightarrow{BE}+\frac{1}{3}\overrightarrow{CF}=

    Tap an option to check your answer.

  46. 46
    Two cards are drawn successively with replacement from a well shuffled pack of 52 cards, then mean of number of queens is

    Tap an option to check your answer.

  47. 47
    The equation of a plane, containing the line of intersection of the planes 2x−y−4=02x-y-4=0 and y+2z−4=0y+2z-4=0 and passing through the point (2,1,0)(2,1,0), is

    Tap an option to check your answer.

  48. 48
    A random variable X assumes values 1,2,3,…,n1,2,3,\ldots,n with equal probabilities, if Var(X)=E(X)\text{Var}(X) = E(X), then nn is

    Tap an option to check your answer.

  49. 49
    The graphical solution set for the system of inequations x−2y≤2,  5x+2y≥10,  4x+5y≤20,  x≥0,  y≥0x-2y \leq 2,\;5x+2y \geq 10,\;4x+5y \leq 20,\;x \geq 0,\;y \geq 0 is given by

    Tap an option to check your answer.

  50. 50
    Let f(0)=−3f(0) = -3 and f′(x)≤5f'(x) \leq 5 for all real values of xx. The f(2)f(2) can have possible maximum value as

    Tap an option to check your answer.

Want the real exam feel? Sit this paper with a 90-minute timer and get your score straight away.

Sit it as a timed mock