CLASS 12 SUBJECT : MATHEMATICS

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1 CLASS 2 SUBJECT : MATHEMATICS CBSE QUESTION PAPER 27(FOREIGN) General Instructions: (i) All questions are compulsory. (ii) Questions 4 in Section A carrying mark each (iii) Questions 5 2 in Section B carrying 2 marks each (iv) Questions 3 23 in Section C carrying 4 marks each (v) Questions in Section D carrying 6 marks each SECTION A ) If A = 3 and A = [ 3 2 ], then write adja sin5x + cosx, if x 2) For what value of k is the function f(x) = { 3x k, if x = continuous at x= 2π 3) Evaluate cos 5 x dx 4) Write the distance of the point (3, -5, 2) from x-axis SECTION B 5) If A and B are square matrices of order 3 such that A =, B = 3, then find the value of 2AB 6) The radius r of right circular cylinder is decreasing at the rate of 3cm/min. and its height h is increasing at the rate of 2cm/min. When r = 7cm and h = 2cm, find the rate of change of volume of cylinder. (use π = 22 7 ) 7) Find dy at t = 2π when x = (t sint) and y = 2( cost). dx 3 8) Show that the function f given by f(x) = tan (sinx + cosx) is decreasing for all x ( π 4, π 2 ). 9) A line passes through the point with position vector 2i 3j+ 4k and is perpendicular to the plane r. (3i + 4j 5k) = 7. Find the equation of the line in cartesian and vector forms. D.SREENIVASULU, M.Sc, M.Phil, B.Ed, PGT(MATHEMATICS), KENDRIYA VIDYALAYA

2 ) If P (A) =.4, P (B) = p, P (A U B) =.6 and A and B are given to be independent events, find the value of p. ) A company produces two types of goods A and B, that require gold and silver. Each unit of type A requires 3g of silver and g of gold while that of type B requires g of silver and 2g of gold. The company can procure a maximum of 9g of silver and 8g of gold. If each unit of type A bring a profit of Rs 4 and that of type B Rs 5, formulate LPP to maximize profit. 2) Find : dx 3 2x x 2 SECTION-C 3) Find the value of cot 2 and xy <. [cos 2x +x 2 + sin y2 +y 4) Using properties of determinants show that 2], x <, y > + x + y = xyz + yz + zx + xy. + z Find matrix X so that X ( ) = ( ) 5) If xy = e (x y), then show that dy dx = y(x ) x(y+). If log y = tan x, then show that ( + x 2 ) d2 y dy dx2 + (2x ) =. dx 6) Find: e x (2+e x )(4+e 2x ) dx 2 7) Evaluate: x 3 x dx Find: e 2x sin(3x + )dx D.SREENIVASULU, M.Sc, M.Phil, B.Ed, PGT(MATHEMATICS), KENDRIYA VIDYALAYA 2

3 8) Find the particular solution of the differential equation x y 2ye x y dx + (y 2x e )dy =, given thatx = when y =. 9) Find the area of a parallelogram ABCD whose side AB and the diagonal AC are given by the vectors 3i + j + 4k and 4i + 5k respectively 2) If a = 2i + j k, b = 4i 7j + k, find a vector c such that a c = b and a. c = 6. 2) There are 4 cards numbered to 4, one number on one card. Two cards are drawn at random without replacement. Let X denote the sum of the numbers on the two drawn cards. Find the mean and variance of X. 22) In a shop X, 3 tins of pure ghee and 4 tins of adulterated ghee which look alike, are kept for sale while in shop Y, similar 5 tins of pure ghee and 6 tins of adulterated ghee are there. One tin of ghee is purchased from one of the randomly selected shops and is found to be adulterated. Find the probability that it is purchased from shop Y. What measures should be taken to stop adulteration? 23) Solve the following LPP graphically: Maximise Z = x + 6y subject to the constraints x + y 2 x 2 y 4x x, y. SECTION-D ) If A = [ 2 3 4] and B = [ 4 2 4] are two square matrices, Find AB and hence solve the system of linear equations x y = 3, 2x + 3y + 4z = 7 and y + 2z = 7. 25) Let A = R {3}, R {}. Let f: A B be defined by f(x) = x 2, X A. Show that f is bijective. Also, find x 3 (i) x, if f (x) = 4 (ii) f (7) D.SREENIVASULU, M.Sc, M.Phil, B.Ed, PGT(MATHEMATICS), KENDRIYA VIDYALAYA 3

4 Let A = R R and let * be a binary operation on A defined by (a, b) (c, d) = (ad + bc, bd) for all (a, b), (c, d) R R. (i) Show that * is commutative on A. (ii) Show that * is associative on A. (iii) Find the identity element of * in A. 26) A wire of length 34 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into a rectangle whose length is twice its breadth. What should be the lengths of the two pieces, so that the combined area of the square and the rectangle is minimum? 27) Using the method of integration, find the area of the triangle ABC, coordinates of whose vertices are A (, 2), B (2, ) and C (4, 3). Using integration, find the area of the region {(x, y): x 2 + y 2 x + y}. 28) Find the particular solution of the differential equation dy dx 3y cot x = sin 2x, given that y = 2 when x = π 2. 29) Find the vector equation of the plane through the line of intersection of the planes x + y + z = and 2x + 3y + 4z = 5 which is perpendicular to the planex y + z =. Hence find whether the plane thus obtained contains the line x+2 = y 3 = z or not Find the image P of the point P having position vector i + 3j + 4k in the plane r. (2i j + k ) + 3 =. Hence find the length of PP. D.SREENIVASULU, M.Sc, M.Phil, B.Ed, PGT(MATHEMATICS), KENDRIYA VIDYALAYA 4

5 ) adja = [ ] 2) k = 8 3 3) Zero 4) ( 5) = 3 5) 2 3. A B = 24 6) 44cm 3 /min 7) dy dx = 6 5 ( sint cost ) = when t = 2π 3 8) f (x) < for all x ( π 4, π 2 ) ANSWERS 27 FORIGN QUESTION PAPER 9) Vector form : r = 2i 3j + 4k + λ(3i + 4j 5k) Cartesian form: x 2 = y + 3 = z ) p = 3 ) Maximise z = 4x + 5y subject to constraints 3x + y 9, x + 2y 8, x, y 2) sin ( x+ 2 ) + c 3) x+y xy 4) X = [ 2 2 ] 5) 6) log ex log e2x + 8 tan ( ex ) + c 2 7) (x x 3 ) dx + 2 (x 3 x) dx (x 3 x) dx = 4 e2x {2 sin(3x + ) 3cos (3x + )} + c 3 8) log y = 2 ( ey) 9) BC = AC AB = i j + k Area = AB BC = 42 2) c = 3i + 2j + 2k x X P(X) Mean = 5, variance = = 5 3 D.SREENIVASULU, M.Sc, M.Phil, B.Ed, PGT(MATHEMATICS), KENDRIYA VIDYALAYA 5

6 2) E = Gheepurchased from shop X E 2 = Gheepurchased from shop Y A = Getting adulterated ghee P ( E 2 A ) = = ) Corner points z = x + 6y (2, 8) z = 68, (4,6) z =,36, (2, 8) z =,28, Maximum of z =,36, at x = 4, y = 6 23) AB = 6I AX = C X = A C 6 BC x = 2, y =, z = 4 24) Bijective means show f(x) is one-one and Onto. f (x) = 3x 2 x f (x) = 4 x = 2 f (7) = ) Identity element is (,) 26) Equation of AB is y = (x + 5) 3 Equation of BC is y = 3 (x 2) 2 Equation of AC is y = 4 2x 4 Area = (x + 5)dx (4 2x) (x 2) = = (y of circle y of line)dx = ( x 2 ( x))dx = π 4 2 sq. units 27) y = 2sin 2 x + 4sin 3 x 28) Equation of plane in vector form r. (i k) + 2 = Yes, the plane contains the given line because (5) + (4) + ( )(5) = D.SREENIVASULU, M.Sc, M.Phil, B.Ed, PGT(MATHEMATICS), KENDRIYA VIDYALAYA 6