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CBSE Class 12 Mathematics 2022 question paper (65/3)

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 : ∫dxx2−6x+13\int \dfrac{dx}{x^2 - 6x + 13}
  2. Q.22 marks
    Find the general solution of the differential equation : edy/dx=x2e^{dy/dx} = x^2.
  3. Q.32 marks
    Write the projection of the vector (b⃗+c⃗)(\vec{b} + \vec{c}) on the vector a⃗\vec{a}, where a⃗=2i^−2j^+k^\vec{a} = 2\hat{i} - 2\hat{j} + \hat{k}, b⃗=i^+2j^−2k^\vec{b} = \hat{i} + 2\hat{j} - 2\hat{k} and c⃗=2i^−j^+4k^\vec{c} = 2\hat{i} - \hat{j} + 4\hat{k}.
  4. Q.42 marks
    If the distance of the point (1, 1, 1) from the plane x−y+z+λ=0x - y + z + \lambda = 0 is 53\dfrac{5}{\sqrt{3}}, find the value(s) of λ\lambda.
  5. Q.52 marks
    Two cards are drawn successively with replacement from a well shuffled pack of 52 cards. Find the probability distribution of the number of spade cards.
  6. Q.6 (a)2 marks
    A pair of dice is thrown and the sum of the numbers appearing on the dice is observed to be 7. Find the probability that the number 5 has appeared on atleast one die.

  7. OR

    Q.6 (b)2 marks
    The probability that A hits the target is 13\dfrac{1}{3} and the probability that B hits it, is 25\dfrac{2}{5}. If both try to hit the target independently, find the probability that the target is hit.

Section B

3 marks each

  1. Q.73 marks
    Evaluate : ∫02πdx1+esin⁡x\int_{0}^{2\pi} \dfrac{dx}{1 + e^{\sin x}}
  2. Q.8 (a)3 marks
    Find the particular solution of the differential equation xdydx−y=x2⋅exx \dfrac{dy}{dx} - y = x^2 \cdot e^x, given y(1)=0y(1) = 0.

  3. OR

    Q.8 (b)3 marks
    Find the general solution of the differential equation xdydx=y(log⁡y−log⁡x+1)x \dfrac{dy}{dx} = y(\log y - \log x + 1).
  4. Q.9 (a)3 marks
    The two adjacent sides of a parallelogram are represented by vectors 2i^−4j^+5k^2\hat{i} - 4\hat{j} + 5\hat{k} and i^−2j^−3k^\hat{i} - 2\hat{j} - 3\hat{k}. Find the unit vector parallel to one of its diagonals. Also, find the area of the parallelogram.

  5. OR

    Q.9 (b)3 marks
    If a⃗=2i^+2j^+3k^\vec{a} = 2\hat{i} + 2\hat{j} + 3\hat{k}, b⃗=−i^+2j^+k^\vec{b} = -\hat{i} + 2\hat{j} + \hat{k} and c⃗=3i^+j^\vec{c} = 3\hat{i} + \hat{j} are such that the vector (a⃗+λb⃗)(\vec{a} + \lambda \vec{b}) is perpendicular to vector c⃗\vec{c}, then find the value of λ\lambda.
  6. Q.103 marks
    Show that the lines : 1−x2=y−34=z−1\dfrac{1 - x}{2} = \dfrac{y - 3}{4} = \dfrac{z}{-1} and x−43=2y−2−4=z−1\dfrac{x - 4}{3} = \dfrac{2y - 2}{-4} = z - 1 are coplanar.

Section C

  1. Q.114 marks
    Find the area of the region bounded by curve 4x2=y4x^2 = y and the line y=8x+12y = 8x + 12, using integration.
  2. Q.12 (a)4 marks
    Find : ∫x2(x2+1)(3x2+4) dx\int \dfrac{x^2}{(x^2 + 1)(3x^2 + 4)} \, dx

  3. OR

    Q.12 (b)4 marks
    Evaluate : ∫−215−4x−x2 dx\int_{-2}^{1} \sqrt{5 - 4x - x^2} \, dx
  4. Q.134 marks
    Find the distance of the point (1, -2, 9) from the point of intersection of the line r⃗=4i^+2j^+7k^+λ(3i^+4j^+2k^)\vec{r} = 4\hat{i} + 2\hat{j} + 7\hat{k} + \lambda(3\hat{i} + 4\hat{j} + 2\hat{k}) and the plane r⃗⋅(i^−j^+k^)=10\vec{r} \cdot (\hat{i} - \hat{j} + \hat{k}) = 10.
  5. A shopkeeper sells three types of flower seeds A1, A2, A3. They are sold in the form of a mixture, where the proportions of these seeds are 4 : 4 : 2, respectively. The germination rates of the three types of seeds are 45%, 60% and 35% respectively. Based on the above information :
    Q.14 (a)2 marks
    Calculate the probability that a randomly chosen seed will germinate;
  6. Q.14 (b)2 marks
    Calculate the probability that the seed is of type A2, given that a randomly chosen seed germinates.