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CBSE Class 12 Mathematics 2022 question paper (65/1)

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 the sum of the order and the degree of the differential equation : (x+dydx)2=(dydx)2+1\left(x + \frac{dy}{dx}\right)^2 = \left(\frac{dy}{dx}\right)^2 + 1
  2. Q.22 marks
    In a parallelogram PQRS, PQ→=3i^−2j^+2k^\overrightarrow{PQ} = 3\hat{i} - 2\hat{j} + 2\hat{k} and PS→=−i^−2k^\overrightarrow{PS} = -\hat{i} - 2\hat{k}. Find ∣PR→∣|\overrightarrow{PR}| and ∣QS→∣|\overrightarrow{QS}|.
  3. Q.3 (a)2 marks
    If ddx[F(x)]=sec⁡4xcosec⁡4x\frac{d}{dx}[F(x)] = \frac{\sec^4 x}{\operatorname{cosec}^4 x} and F(π4)=π4F\left(\frac{\pi}{4}\right) = \frac{\pi}{4}, then find F(x)F(x).

  4. OR

    Q.3 (b)2 marks
    Find : ∫log⁡x−3(log⁡x)4 dx\int \frac{\log x - 3}{(\log x)^4}\,dx.
  5. Q.42 marks
    Let A and B be two events such that P(A)=58P(A) = \frac{5}{8}, P(B)=12P(B) = \frac{1}{2} and P(A/B)=34P(A/B) = \frac{3}{4}. Find the value of P(B/A)P(B/A).
  6. Q.52 marks
    Two balls are drawn at random from a bag containing 2 red balls and 3 blue balls, without replacement. Let the variable X denotes the number of red balls. Find the probability distribution of X.
  7. Q.62 marks
    Find the values of λ\lambda, for which the distance of point (2,1,λ)(2, 1, \lambda) from plane 3x+5y+4z=113x + 5y + 4z = 11 is 222\sqrt{2} units.

Section B

3 marks each

  1. Q.7 (a)3 marks
    If a⃗\vec{a}, b⃗\vec{b}, c⃗\vec{c} and d⃗\vec{d} are four non-zero vectors such that a⃗×b⃗=c⃗×d⃗\vec{a} \times \vec{b} = \vec{c} \times \vec{d} and a⃗×c⃗=4b⃗×d⃗\vec{a} \times \vec{c} = 4\vec{b} \times \vec{d}, then show that (a⃗−2d⃗)(\vec{a} - 2\vec{d}) is parallel to (2b⃗−c⃗)(2\vec{b} - \vec{c}) where a⃗≠2d⃗\vec{a} \neq 2\vec{d}, c⃗≠2b⃗\vec{c} \neq 2\vec{b}.

  2. OR

    Q.7 (b)3 marks
    The two adjacent sides of a parallelogram are represented by 2i^−4j^−5k^2\hat{i} - 4\hat{j} - 5\hat{k} and 2i^+2j^+3k^2\hat{i} + 2\hat{j} + 3\hat{k}. Find the unit vectors parallel to its diagonals. Using the diagonal vectors, find the area of the parallelogram also.
  3. Q.83 marks
    Find the vector equation of the plane passing through the intersection of the planes r⃗⋅(2i^+2j^−3k^)=7\vec{r} \cdot (2\hat{i} + 2\hat{j} - 3\hat{k}) = 7 and r⃗⋅(2i^+5j^+3k^)=9\vec{r} \cdot (2\hat{i} + 5\hat{j} + 3\hat{k}) = 9 and through the point (2,1,3)(2, 1, 3).
  4. Q.9 (a)3 marks
    Find : ∫dxx+x3\int \frac{dx}{\sqrt{x} + \sqrt[3]{x}}.

  5. OR

    Q.9 (b)3 marks
    Evaluate : ∫0π/2cos⁡x(1+sin⁡x)(4+sin⁡x) dx\int_0^{\pi/2} \frac{\cos x}{(1 + \sin x)(4 + \sin x)}\,dx.
  6. Q.103 marks
    Find the particular solution of the differential equation xdydx+xcos⁡2(yx)=yx\frac{dy}{dx} + x\cos^2\left(\frac{y}{x}\right) = y; given that when x=1x = 1, y=π4y = \frac{\pi}{4}.

Section C

  1. Q.11 (a)4 marks
    Using integration, find the area of the region {(x,y):4x2+9y2≤36,2x+3y≥6}\{(x, y) : 4x^2 + 9y^2 \leq 36, 2x + 3y \geq 6\}.

  2. OR

    Q.11 (b)4 marks
    Using integration, find the area of the region bounded by lines x−y+1=0x - y + 1 = 0, x=−2x = -2, x=3x = 3 and xx-axis.
  3. Q.124 marks
    A card from a pack of 52 playing cards is lost. From the remaining cards, 2 cards are drawn at random without replacement, and are found to be both aces. Find the probability that lost card being an ace.
  4. Q.134 marks
    Evaluate : ∫0πx1+sin⁡x dx\int_0^{\pi} \frac{x}{1 + \sin x}\,dx.
  5. Electrical transmission wires which are laid down in winters are stretched tightly to accommodate expansion in summers. Two such wires lie along the following lines : l1:x+13=y−3−2=z+2−1l_1 : \frac{x + 1}{3} = \frac{y - 3}{-2} = \frac{z + 2}{-1} l2:x−1=y−73=z+7−2l_2 : \frac{x}{-1} = \frac{y - 7}{3} = \frac{z + 7}{-2} Based on the given information, answer the following questions :
    Q.14 (i)2 marks
    Are the lines l1l_1 and l2l_2 coplanar ? Justify your answer.
  6. Q.14 (ii)2 marks
    Find the point of intersection of the lines l1l_1 and l2l_2.