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CBSE Class 12 Mathematics 2022 question paper (65/5)

Maximum marks 40 · Time 2 hours · 3 sets

Try each question first, then open its model answer. Where the paper offers a choice, both questions are shown with OR between them.

Section A

2 marks each

  1. Q.12 marks
    Find : ∫dx4x−x2\int \dfrac{dx}{\sqrt{4x - x^2}}
  2. Q.22 marks
    Find the general solution of the following differential equation : dydx=ex−y+x2e−y\dfrac{dy}{dx} = e^{x-y} + x^2 e^{-y}
  3. Q.32 marks
    Let X be a random variable which assumes values x1x_1, x2x_2, x3x_3, x4x_4 such that 2P(X=x1)=3P(X=x2)=P(X=x3)=5P(X=x4)2P(X = x_1) = 3P(X = x_2) = P(X = x_3) = 5P(X = x_4). Find the probability distribution of X.
  4. Q.42 marks
    If a⃗=i^+j^+k^\vec{a} = \hat{i} + \hat{j} + \hat{k}, a⃗⋅b⃗=1\vec{a} \cdot \vec{b} = 1 and a⃗×b⃗=j^−k^\vec{a} \times \vec{b} = \hat{j} - \hat{k}, then find ∣b⃗∣\left|\vec{b}\right|.
  5. Q.52 marks
    If a line makes an angle α\alpha, β\beta, γ\gamma with the coordinate axes, then find the value of cos⁡2α+cos⁡2β+cos⁡2γ\cos 2\alpha + \cos 2\beta + \cos 2\gamma.
  6. Q.6 (a)2 marks
    Events A and B are such that P(A)=12P(A) = \dfrac{1}{2}, P(B)=712P(B) = \dfrac{7}{12} and P(A‾∪B‾)=14P\left(\overline{A} \cup \overline{B}\right) = \dfrac{1}{4} Find whether the events A and B are independent or not.

  7. OR

    Q.6 (b)2 marks
    A box B1B_1 contains 1 white ball and 3 red balls. Another box B2B_2 contains 2 white balls and 3 red balls. If one ball is drawn at random from each of the boxes B1B_1 and B2B_2, then find the probability that the two balls drawn are of the same colour.

Section B

3 marks each

  1. Q.73 marks
    Evaluate : ∫0π/4dx1+tan⁡x\int_0^{\pi/4} \dfrac{dx}{1 + \tan x}
  2. Q.8 (a)3 marks
    If a⃗\vec{a} and b⃗\vec{b} are two vectors such that ∣a⃗+b⃗∣=∣b⃗∣\left|\vec{a} + \vec{b}\right| = \left|\vec{b}\right|, then prove that (a⃗+2b⃗)\left(\vec{a} + 2\vec{b}\right) is perpendicular to a⃗\vec{a}.

  3. OR

    Q.8 (b)3 marks
    If a⃗\vec{a} and b⃗\vec{b} are unit vectors and θ\theta is the angle between them, then prove that sin⁡θ2=12∣a⃗−b⃗∣\sin \dfrac{\theta}{2} = \dfrac{1}{2} \left|\vec{a} - \vec{b}\right|.
  4. Q.93 marks
    Find the equation of the plane passing through the line of intersection of the planes r⃗⋅(i^+j^+k^)=10\vec{r} \cdot \left(\hat{i} + \hat{j} + \hat{k}\right) = 10 and r⃗⋅(2i^+3j^−k^)+4=0\vec{r} \cdot \left(2\hat{i} + 3\hat{j} - \hat{k}\right) + 4 = 0 and passing through the point (−2,3,1)(-2, 3, 1).
  5. Q.10 (a)3 marks
    Find : ∫ex⋅sin⁡2x dx\int e^x \cdot \sin 2x\, dx

  6. OR

    Q.10 (b)3 marks
    Find : ∫2x(x2+1)(x2+2) dx\int \dfrac{2x}{\left(x^2 + 1\right)\left(x^2 + 2\right)}\, dx

Section C

  1. Q.114 marks
    Three persons A, B and C apply for a job of manager in a private company. Chances of their selection are in the ratio 1 : 2 : 4. The probability that A, B and C can introduce changes to increase the profits of a company are 0.8, 0.5 and 0.3 respectively. If increase in the profit does not take place, find the probability that it is due to the appointment of A.
  2. Q.124 marks
    Find the area bounded by the curves y=∣x−1∣y = |x - 1| and y=1y = 1, using integration.
  3. Q.13 (a)4 marks
    Solve the following differential equation : (y−sin⁡2x)dx+tan⁡x dy=0\left(y - \sin^2 x\right) dx + \tan x\, dy = 0

  4. OR

    Q.13 (b)4 marks
    Find the general solution of the differential equation : (x3+y3)dy=x2y dx\left(x^3 + y^3\right) dy = x^2 y\, dx
  5. Two motorcycles A and B are running at the speed more than the allowed speed on the roads represented by the lines r⃗=λ(i^+2j^−k^)\vec{r} = \lambda\left(\hat{i} + 2\hat{j} - \hat{k}\right) and r⃗=(3i^+3j^)+μ(2i^+j^+k^)\vec{r} = \left(3\hat{i} + 3\hat{j}\right) + \mu\left(2\hat{i} + \hat{j} + \hat{k}\right) respectively. Based on the above information, answer the following questions :
    Q.14 (a)2 marks
    Find the shortest distance between the given lines.
  6. Q.14 (b)2 marks
    Find the point at which the motorcycles may collide.