SAMPLE QUESTION PAPER MATHEMATICS CLASS XII :

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2 SAMPLE QUESTION PAPER MATHEMATICS CLASS XII : 05-6 TYPOLOGY VSA ( M) L A I (4M) L A II (6M) MARKS %WEIGHTAGE Remembering 3, 6, 8, 9, 5 0 0% Understanding, 9, 0 4, 6 % Applications 4 3, 5, 6, 7, 0 9 9% Hots 5 7, % Evaluation - 8, 4 4% Total 6 =6 =5 = % Question- wise break up Type of Questions Marks per Question Total number of Questions Total Marks VSA 6 06 L A - I L A - II Total 6 00

3 Blue Print of Sample Paper UNIT VSA LA-I LA-II TOTAL Relations and *4-5() functions Inverse trigonometric functions Matrices 4-8() - 9(3) 5() 0 Determinants - *4-4 3 Continuity and differentiability 4 *4-9(3) Applications of derivative VBQ 3(3) Integrals Application of integrals Differential Equations Vectors - * () - 8() () 44 Three dimensional geometry 4 *6 (3) 7 Linear Programming Probability - 4 *6 0() 0 Total 6(6) 5(3) 4(7) 00(6) 00 Note:- * indicates the questions with internal choice. The number of questions is given in the brackets and the marks are given outside the brackets.

4 Question numbers to 6 carry mark each. SAMPLE PAPER Section A Q. Evaluate: - 3 sin(cos (- )). 5 Q. State the reason for the following Binary Operation *, defined on the set Z of integers, to be not commutative. 3 a *b ab. Q3. Give an example of a skew symmetric matrix of order 3. Q4. Using derivative, find the approximate percentage increase in the area of a circle if its radius is increased by %. Q5. Find the derivative of tanx f(e ) w.r. to x at x = 0. It is given that f 5. Q6. If the lines x y 4 z 3 x and y 5 z are perpendicular to each other, 3p 4 4p 7 then find the value of p. Question numbers 7 to 9 carry 4 marks each. Q7. Let f : W W be defined as Also, find the inverse of f. Section B n, if n is odd f (n). Then show that f is invertible. n, if n is even Show that the relation R in the set N N defined by (a, b)r(c,d) iff a d b c a, b,c,d N, is an equivalence relation. Q8. Prove that: cos x cos x x 3 tan, where x cos x cos x 4 3

5 Q9. Let A. 3 4 Then verify the following: A(adjA) (adja)a A I, where I is the identity matrix of order. Q0. Using properties of determinants, prove that p p q 3 4 3p 4p 3q p 7p 4q Without expanding the determinant at any stage, prove that Q. Let A,B. 4 Then compute AB. Hence, solve the following system of equations: x y 4,3x y. Q. If the following function is differentiable at x =, then find the values of a and b. x, if x f (x). ax b, if x Q3. Let x x cos x dy y (log x) x. Then find. If dy x a sin pt, y bcospt. Then find at t 0. Q4. Evaluate the following indefinite integral:. sin x sin x Evaluate the following indefinite integral: sin d. sin cos 3 4

6 Q5. Evaluate the following definite integral: x( sin x). cos x Q6. Solve the following differential equation: Q7. Solve the following differential equation: dy y x y, x 0. x x ( y ) (tan y x)dy. Q8. Find the shortest distance between the following pair of skew lines: x y z x y 3 z, Q9. A problem in mathematics is given to 4 students A, B, C, D. Their chances of solving the problem, respectively, are /3, /4, /5 and /3. What is the probability that (i) the problem will be solved? (ii) at most one of them will solve the problem? Question numbers 0 to 6 carry 6 marks each. Section C Q0. Find the intervals in which the following function is strictly increasing or strictly decreasing. Also, find the points of local maximum and local minimum, if any. 3 f(x) (x ) (x ) Q. If a,b,c are unit vectors such that a.b a.c 0 that (i)a (b c),(ii) a b b c c a and the angle between b and c is, 6 then prove Q. Using integration, find the area bounded by the tangent to the curve 4y x at the point (, ) and the lines whose equations are x y and x 3y 3. Q3. Find the distance of the point 3i ˆ j ˆ kˆ from the plane 3 x + y z + = 0 measured parallel to the line x y z. Also, find the foot of the perpendicular from the given point upon the 3 5

7 given plane. Find the equation of the plane passing through the line of intersection of the planes ˆ ˆ ˆ r.(i 3j k) and r.(i ˆ ˆj k) ˆ 0 and passing through the point (3, -, -). Also, find the angle between the two given planes. Q4. A Bag I contains 5 red and 4 white balls and a Bag II contains 3 red and 3 white balls. Two balls are transferred from the Bag I to the Bag II and then one ball is drawn from the Bag II. If the ball drawn from the Bag II is red, then find the probability that one red ball and one white ball are transferred from the Bag I to the Bag II. Find the mean, the variance and the standard deviation of the number of doublets in three throws of a pair of dice. Q5. A farmer wants to construct a circular garden and a square garden in his field. He wants to keep the sum of their perimeters 600 m. Prove that the sum their areas is the least, when the side of the square garden is double the radius of the circular garden. Do you think that a good planning can save energy, time and money? Q6. A manufacturing company makes two models A and B of a product. Each piece of model A requires 9 hours of labour for fabricating and hour for finishing. Each piece of model B requires hours of labour for fabricating and 3 hours for finishing. The maximum number of labour hours, available for fabricating and for finishing, are 80 and 30 respectively. The company makes a profit of Rs 8000 and Rs 000 on each piece of model A and model B respectively. How many pieces of each model should be manufactured to get maximum profit? Also, find the maximum profit. 6

8 Marking Scheme (Sample Paper) Section A Q. 4 5 Q. We have, Z such that * = 8 and * =. This implies that * *. Hence, * is not commutative. Q Q4. 4% Q5. 5 Q6. -4 Section B Q7. Here, ff : W W is such that, if n is odd, ff(n) f(f(n)) f(n ) n n (+/) and if n is even, ff(n) f(f(n)) f(n ) n n (+/) Hence, ff I This implies that f is invertible and f f Let Hence, R is reflexive. (a, b) NN. Then a b b a (a, b)r(a, b) 7

9 Let (a,b),(c,d) NN be such that (a,b)r(c,d) a d b c c b d a (c, d)r(a, b) Hence, R is symmetric. Let (a,b),(c,d),(e,f ) N N be such that (a,b)r(c,d),(c,d)r(e,f ). a d b c and c f d e a d c f b c d e a f b e (a,b)r(e,f ) Hence, R is transitive. (+/) Since R is reflexive, symmetric and transitive. Therefore R is an equivalence relation. (/) Q8. tan x x cos sin cos x cos x tan cos x cos x x x cos sin x x cos sin 3 x 3 x x tan ( x cos 0,sin 0) x x cos sin 4 (+/) x tan x tan tan (tan( )) x tan 4 = x x ( ) (/) Q adja 3 8 ()

10 0 0 (adja)a A(adjA) 0 0 (/) (/) A (/) 3 4 Hence, A(adjA) (adja)a A I verified. (/) p p q p p q Q(0) LHS = 3 4 3p 4p 3q 0 p (R R 3R, R3 R3 4R ) 4 7 4p 7p 4q 0 3 3p () p p q 0 p (R R 3R ) = = =RHS. Hence, proved Let (interchanging rows and columns) ( + /) ( )( )( ) ( +/)) (/) 0 0 (/) 9

11 Q. 0 AB I 0 (/) 3 A( B) I A B The given system of equations is equivalent to x 4 AX C, where X,C y (/) X (A ) C (A ) C 4 7 x 7, y Q. Since, f is differentiable at x =, therefore, f is continuous at x =. (/) x x x x lim f (x) lim f (x) f () lim x lim (ax b) 4 4 a b (+/) Since, f is differentiable at x =, f ( h) f () f ( h) f () Lf () Rf () lim lim (h 0) h0 h h0 h ( h) 4 a( h) b 4 lim lim h0 h h0 h 4 ah 4 lim ( h 4) lim 4 a h0 h0 h (+/) b = -4 (/) Q3. x du Let u (log x).then log u x log(log x) log(log x) u log x du x (log x) log(log x) log x 0

12 (+/) xcosx dv x cos x Let v x. Then log v x cos x log x cos x(log x) x sin x logx v x dv xcosx x cos x cos x(log x) x sin x log x (+/) dy du dv x y u v (log x) log(log x) log x xcosx x cos x cos x(log x) xsin x log x dy dy dy dt b ap cos pt, bpsin pt, tan pt (+/) dt dt a dt d y b dt psec pt a (+/) b (/) 3 d y a cos pt d y b ( ) t0 (/) a Q4. Given integral = sin x sin x sin x sin x( cos x) ( cos x)( cos x)( cos x) dt ( t)( t)( t) (cos x t sin x dt)

13 A B C ( t)( t)( t) t t t A( t)( t) B( t)( t) C( t ) (An identity) Putting, t=, ½, -, we get A = /6, B= -/, C = 4/3 (+/) Therefore, the given integral 4 log t log t log t c 6 6 log cos x log cos x log cos x c 6 3 (+/) sin sin d d sin cos 3 cos cos 3 (/) sin d dt (cos t sin d dt) cos cos 4 t t 4 dt (+/) ( 5) (t ) t cos sin c sin c 5 5 Q5. Let x( sin x) x x sin x I cos x cos x cos x (as x sin x 0 cos x 0 x cos x is odd and

14 x sin x cos x is even) x sin x. 0 cos x 4 Let x sin x ( x)sin( x) I cos x cos ( x) 0 0 I ( x)sin x cos x 0 Adding, I sin x cos x 0 dt (cos x t sin x dt) t tan t (/) I. Hence, I (/) 4 Q6. Given differential equation is dy y x y dy y y y, x 0 or, ( ) f ( ), hence, homogeneous. (/) x x x x x Put y = v x dy dv v x. The differential equation becomes dv v x v v 3

15 or, dv v x (/). Integrating, we get log v v log x log k log v v log x k v v x k v y y v kx ( ) cx x x y x y cx, which gives the general solution. Q7. We have the following differential equation: (tan y x) x tan y 0r, dy dy y y y, which is linear in x (/) dy y tan y I.F. e e Multiplying both sides by I. F. and integrating we get tan y tan y tan y (/) xe e dy y tan y t xe e tdt (tan y t dy dt) y tan y t t tan y tan y tan y xe te e c xe tan ye e c which gives the general solution of the differential equation. () Q8. The vector equations of the given lines are ˆ ˆ ˆ ˆ ˆ ˆ r i j k (i 3j 4k), r i ˆ 3j ˆ ( ˆi j ˆ 3k) ˆ a ˆi j ˆ k, ˆ b i ˆ 3j ˆ 4k, ˆ a i ˆ 3j, ˆ b ˆi j ˆ 3kˆ 4

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