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CBSE Board, Class XII Mathematics, Sample Paper - 1, Term 2 – 2021-22, Time: 2 hours, , Total Marks: 40, , General Instructions:, 1. This question paper contains three sections – A, B and C. Each part is, compulsory., 2. Section - A has 6 short answer type (SA1) questions of 2 marks each, 3. Section – B has 4 short answer type (SA2) questions of 3 marks each., 4. Section - C has 4 long answer type questions (LA) of 4 marks each., 5. There is an internal choice in some of the questions., 6. Q14 is a case-based problem having 2 sub parts of 2 marks each., Section A, Q1 – Q6 are of 2 marks each., 1. Integrate, , cos2x, ,  sin2 x cos2 xdx, , OR, 8, , Integrate, , 3, , x 1, ,  3 x  1  3 11  x dx, , 2, , 2. Check whether the differential equation, , , , , , 2yex /y dx  y  2x ex /y dy  0, , is, , homogeneous., , 3. Write the direction cosines of the vectors 2i  j  5k., 4. Find the angle between the line, , x 1 y z 3,  , and the plane 10x + 2y – 11z, 2, 3, 6, , =3, , 5. Bag A contains 3 white and 4 red balls, and bag A contains 6 white and 3 red, balls. An unbiased coin, twice as likely to come up heads as tails, is tossed once., If it shows head, a ball drawn from bag A, otherwise, from bag B. Given that a, white ball was drawn, what is the probability that the coin came up tail?

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Section C, Q11 – Q14 are of 4 marks each, , 11. Integrate, ,  3x  2, , x2  x  1dx, , 12. Using integration, find the area between the circle x2 + y2 = 16 and the parabola, y2 = 6x., OR, Find the area of the smaller region bounded by the ellipse, , x2, , y2, , 2, , 2, , a, , b, , 1 and the line, , x, a, , y, b, , 1, , 13. Find the vector equation of the plane passing through three points with position, vector i, , j 2k,2i j k and i 2j k. Also, find the coordinates of the point of, , intersection of this plane and the line r, , 3i, , j k, , 2i 2j k ., , 14. Case Study, Four cards are drawn successively with replacement from a well shuffled deck of, 52 cards. What is the probability that, i. All the four cards are spades?, ii. Only 3 cards are spades?

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Solution, Section A, cos2x, , 1. I  , , sin2 x cos2 x, , dx, , cos2 x  sin2 x, , I, , dx, sin2 x cos2 x,  1, 1 , I  , , dx,  sin2 x cos2 x , , , , , , I   cosec2x  sec2 x dx, , I   cot x  tan x  c, OR, 8, , I, , 3, , x 1, , 3, , 3, 2 x  1  11  x, , 8, , I, , 3, , dx …(i), , 10  x  1, , 3, 2 10  x  1  11  10  x , 3, , 8, , I, , 2, , 3, 3, , 11  x, , 11  x  3 1  x, , dx …(ii), , Adding (i) and (ii), 83, , 2I  , , x  1  3 11  x, , 3, 3, 2 x  1  11  x, 8, , I, , 1, dx, 2, 2, , I, , 1 8, x, 2 2, , I=3, , dx, , dx

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2. Given DE is 2yex/y dx, , dx, dy, , 2xe, , x, y, , y, , 2ye, , y – 2x ex/y dy, , 0, , ... 1, , x, y, , Let, , F x,y, , 2xe, , x, y, , y, , 2ye, , x, y, , Then,, , 2xe, F, , x, y, , y, o, , x, y, 2ye, , x, y, , F x,y, , Thus, F(x, y) is a homogeneous function of degree zero., , 3. Let a  2i  j  5k, then, a  (2)2  (1)2  (5)2,  a , , 4  1  25, ,  a  30, Now,a , , a, a, , , , 2i  j  5k, 30, , , , 2, 30, , i, , 1, 30, , j, , 5, 30, , k, , Thus, the direction cosines of the vector 2i  j  5k are , , 2, 30, , ,, , 1, 30, , and , , 5, 30, , 4. Let  be the angle between the line and plane and θ be the angle between the, line and normal to plane,  = (90 – θ), Let b be vector parallel to line, ˆ, ˆ  6k,  b  2iˆ  3j, Let n̂ be normal to plane, ,  n̂  10iˆ  2jˆ  11kˆ, , .

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cos  , , 2iˆ  3jˆ  6kˆ .10iˆ  2jˆ  11kˆ , 22  32  62, ,  cos  90    ,  sin  , , 102  22  112, , 40, 8, , 7  15 21, , 8,  8 ,    sin 1  , 21,  21 , , 5., Let W be the white ball was drawn and T be the tail come up., ,  T  PT  W , P  , P W , W, 1 6, , 2 9, , 1 6 1 3,   , 2 9 2 7, 1,  3, 1 3, , 3 14, 14, , 17, , 6. Let X be the number of coins showing heads., Let X be a binomial variate with parameter n = 150 and p., P(X = 75) = P(X = 76), 75, 75, 76, 74, ⇒ C150, = C150, 76 p (1 − p), 75 p (1 − p), , ⇒, , 150!, 150!, (1 − p) =, p, 75! 75!, 76! 74!, , ⇒, , 1, 1, (1 − p) =, p, 75, 76, , ⇒, , p, 76, =, 1 − p 75

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12. x2 + y2 = 16 … (i) and y2 = 6x … (ii), , Points of intersection of curve (i) and (ii) is, , , , , , , , , , , , A 2, 2 3 and B 2,-2 3, , Also, , C  4,0  ., , Area, , OBCAO = 2 (Area ODA + Area DCA), , 4, 2, ,  2   y 2dx   y 1dx , 2, 0, , 4, 2, ,  2   6x dx   16  x2 dx , 2, 0, , 4, , 2 , 2, , 2, x, 16, , x, 16, x, , ,  ,  2  6.  x3 2   ,  sin 1  , , 2, 2, 4 , 3, 0 , 2 , 

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2 6,  2.2 3, 1 ,  2, .2 2  0  8sin 1 1  ,  8sin 1 , 2 ,  2,  3, , , , , , , , 16 3,  , ,  16.   4 3  16. , 3, 2 , 6, ,  4 3 16 , , ,   sq. units.,  3, 3 , , OR, , x2 y2, , 1, a2 b2, x y,  1, a b, , ...(1), ...(2), , Equation (i) gives y = b 1 , , , , , x2, a2, , Equation (ii) gives y = b  1 (Area of the smaller region), , a,   y  y dx, 2, 0 1, , , , , , x, a 

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a, a  x, x2, =  b 1- dx -  b  1- dx, a2, 0, 0  a, a x , ba, =  a2 - x2 dx - b   1-  dx, a0, 0 a , a, a, , ,  x2 , b  x a2 - x2 a2, x, =, +, sin-1  - b x - , a, 2, 2, a,  2a , 0, , 0,   a2 , b  a2, =  sin-11 - b a - , a 2,   2a , , , =, , , , , , , , ab π ab 1, . - = ab  π -2 sq. units., 2 2 2 4, , 13. Let the position vectors of the three points be,, , a, , i, , j 2k,b, , 2i, , j, , k and c, , i, , k., , 2j, , So, the equation of the plane passing through the points a,b and c is, , r a . b c, , c a, , 0, , r, , i, , j 2k . i 3j, , r, , i, , j 2k . k 3j 9i, , r., , 9i 3j k, , r. 9i 3j k, , 14, , j 3k, , 0, , 0, , 0, , 14, , ... 1, , So, the vector equation of the required plane is r. 9i, The equation of the given line is r, , 3i, , j k, , 2i 2j, , Position vector of any point on the give line is, , r, , 3, , 2, , i, , 1 2, , j, , 1, , k, , ... 2, , The point (2) lies on plane (1) if,, , 3 2, , i, , 93 2, 11, , 1 2, 3, , j, , 1 2, , 1, , k . 9i 3j k, 1, , 25 14, 1, , Putting, , 1 in (2), we have, , 14, , 3j k, , 14, , k ., , 14.

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r, , 3 2, 3 2, i, , i, 1 i, , 1 2, , j, , 1, , k, , 1 2, , 1 j, , 1, , 1 k, , j 2k, , Thus, the position vector of the point of intersection of the given line and plane, (1) is i, , j 2k and its co-ordinates are 1, 1,, , 2 ., , 14., , This is a case of bernoulli trials., p = P(Success) = P(getting a spade in a single draw) =, q  P(Failure)  1  p  1 , , 13 1, , 52 4, , 1 3, , 4 4, 4, , 1, 1, i. All the four cards are spades = P(X = 4) = C4p q    , 256, 4, 12, 3, ii. Only 3 cards are spades= P(X = 3) = 4 C3p3q1 , , 256 64, 4, , 4 0