# 1 are perpendicular to each other then, find. Q06. If the lines x 1 z 3 and x 2 y 5 z

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1 Useful for CBSE Board Examination of Math (XII) for 6 For more stuffs on Maths, please visit : Time Allowed : 8 Minutes Max. Marks : SECTION A 3 Q. Evaluate : sin cos 5. Q. State the reason for the following Binar Operation *, defined on the set Z of integers, to be not commutative : a*b = ab 3. Q3. Give an example of a skew smmetric matrix of order 3. Q4. Using derivatives, find the approximate percentage increase in the area of a circle if its radius is increased b %. tan x Q5. Find the derivative of f (e ) w.r.t. x at x. It is given that f () 5. Q6. If the lines x 4 z 3 and x 5 z are perpendicular to each other then, find 3p 4 4p 7 the value of p. SECTION B n, if n is odd Q7. Let f : W W be defined as f (n). Then show that f is invertible. Also find n, if n is even the inverse of f. OR Show that the relation R in the set N N defined b (a, b)r(c,d) iff a d b c a, b,c,d N, is an equivalence relation. x 3 Q8. Prove that tan, where x. 4 Q9. Let A 3 4. Then verif A(adjA) (adja)a A I, where I is the identit matrix of order. p p q Q. Using properties of determinants, prove that OR 3 4 3p 4p 3q p 7p 4q Without expanding the determinant at an stage, prove that Q. Let A and B 4. Then compute AB. Hence, solve the following sstem of equations : x 4, 3x. Q. If the following function is differentiable at x =, then find the value of a and b : x, if x f (x). ax b, if x x x Q3. Let (log x) x. Then find d. OR If x a sin pt and, b cos pt. Then find Q4. Evaluate : d at t.. OR Evaluate : sin x sin x sin d sin cos 3. 5//8

2 x( sin x) Q5. Evaluate the definite integral :. d x Q6. Solve :, x. Q7. Solve : ( ) (tan x)d. x x Q8. Find the shortest distance between the following pair of skew lines : x z and x 3 z Q9. A problem in Mathematics is given to 4 students A, B, C and D. Their chances of solving the problems respectivel are /3, /4, /5 and /3. What is the probabilit that (i) the problem will be solved? (ii) at most one of them will solve the problem? SECTION C Q. Find the intervals in which the following function is strictl increasing or strictl decreasing. Also 3 find the points of local maximum and local minimum, if an : f (x) (x ) (x ). Q. If a, b and c are unit vectors such that a.b a.c and the angle between b and c is, then 6 prove that (i) a (b c) (ii) a b b c c a. Q. Using integration, find the area bounded b the tangent to the curve 4 x at the point (, ) and the lines whose equations are x and x 3 3. Q3. Find the distance of the point 3i ˆ j ˆ kˆ from the plane 3x z measured parallel to the x z. Also find the foot of perpendicular from the given point upon the given plane. 3 OR Find the equation of the plane passing through the line of intersection of the planes r.(i ˆ 3j ˆ k) ˆ and r.(i ˆ ˆj k) ˆ 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 probabilit that one red ball and one white ball are transferred from Bag I to the Bag II. OR 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 6m. Prove that the sum of their areas is the least, when the side of the square garden is double the radius of the circular garden. Do ou think that a good planning can save energ, time and mone? Q6. A manufacturing compan 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 8 and 3 respectivel. The compan makes a profit of `8 and ` on each piece of model A and B respectivel. How man pieces of each model should be manufactured to get maximum profit? Also, find the maximum profit. A Compilation B : O. P. Gupta [Call or ] 5//8

3 Marking Scheme Of Sample Question Paper SECTION A 4 Q Q. Note that * 8 but * * where, Z. So * isn t commutative. Q Q4. 4%. Q5. 5. Q SECTION B Q7. Here fof : W W fof (n) f f (n) f (n ) n n and if n is is such that if n is odd, even, fof (n) f f (n) f (n ) n n. + Hence fof = I. This implies that f is invertible and f = f. OR Let (a, b) N N. Then as a b b a (a, b)r(a, b). So, R is reflexive. Let (a, b), (c,d) N N be s.t. (a, b)r(c,d). Then as a d b c c b d a (c, d)r(a, b). So R is smmetric. Let (a, b), (c,d), (e,f ) N N be s.t. (a,b)r(c,d) and (c,d)r(e,f ). Then as 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 ). So R is transitive. Since R is reflexive, smmetric as well as transitive so, R is an equivalence relation. x x cos sin Q8. LHS : tan tan x x cos sin x x cos sin x x 3 x 3 cos sin tan x 4 tan x x cos sin x x cos sin x II quad. x tan x x tan tan tan RHS x. tan Q9. We ve A 3 4 adj.a 3 4 A(adjA) (i) 4 (adja)a (ii) Also, A (iii) 3 4 B (i), (ii) & (iii), A(adjA) (adja)a A I is verified. T 5//8 3

4 Q. LHS : p p q 3 4 3p 4p 3q 4 7 4p 7p 4q p p q p 3 3p p p q B R R 3R, R3 R3 4R B R3 R3 3R p Expanding along C, ( ) RHS. OR LHS: Let Interchanging Rows & Columns, 3 ( )( )( ) RHS Q. Here A and B 4 AB I 4 A B I A B...(i) The given sstem of equations x 4, 3x can be written as x 4 PX C where, P, X and C 3 P PX P C I X P C T T X P C [Note that P A P (A ) x 7 Therefore, X 6 4 X B equalit of matrices, we get : x 7,. Q. As the function f is differentiable at x =, so it is continuous at x = as well. lim f (x) lim f (x) f () lim x lim ax b () 4 a b (i) + x x x x Also, f is differentiable at x = Lf () Rf () f (x) f () f (x) f () i.e., lim lim x x x x x 4 (ax b) 4 (ax b) 4 lim lim lim (x ) lim x x x x x x x b (i), b 4 a (ax 4 a) 4 (x )a 4 lim 4 lim lim a x x x x x a 4 + Replacing value of a in (i), we get : b 4. x x Q3. We have (log x) x. Let Now d du dv...(i) x log u log(log x) log u x log(log x) x xcosx u v where u (log x), v x x u (log x) du du x x log(log x) (log x) log(log x) u log x x log x x x And, v x log v log(x) log v x log x 5//8 4

5 dv dv x log x x sin x x x log x x sin x v x d x x B (i), (log x) log(log x) x ( log x) x sin x log x log x d OR We have x a sin pt and, b cos pt ap cos pt and, bp sin pt dt dt d d dt d bsin pt b tan pt + dt a cos pt a d bp dt bp b 3 sec pt sec pt sec pt + a a ap cos pt a Therefore, d b 3 b sec a a at t. + sin x Q4. Let I I sin x sin x I sin x( ) ( )( )( ) dt Put t sin x dt I ( t)( t)( t) A B C Consider ( t)( t)( t) t t t A( t)( t) B( t)( t) C( t)( t) 4 On equating the coefficients of like terms, we get : A, B, C dt dt 4 dt So, I 6 ( t) ( t) 3 ( t) I log t log t log t C 6 3 I log log log C sin sin OR Let I d I d sin cos 3 cos cos 3 Q5. sin I d Put cos t sin dt 4 cos cos I dt 4 t t t I sin C 5 x sin x Let I dt I + ( 5) (t ) cos I sin C. 5 x x sin x I x x Consider f (x) f ( x) f (x) x sin x ( x)sin( x) x sin x and, g(x) g( x) g(x) cos ( x) Clearl, f (x) is an odd function whereas g (x) is an even function. x sin x x sin x So, I I 4...(i) 5//8 5

6 ( x) sin( x) cos ( x) I 4 ( x)sin x I 4 (ii) x sin x ( x)sin x On adding (i) & (ii), we get : I 4 4 sin x sin x I 4 I sin x sin( x) sin x Consier f (x) f ( x) f (x) cos ( x) a a f x, if f a x f x B using f x, if f a x f x / sin x We have I Put t sin x dt dt When x t & when x t I t I 4 dt t I 4 tan t I 4 tan tan 4 4. Q6. We have d x, x...(i) x x x Consider f (x, ). Put x x, x x x x f ( x, ) f (x, ). So, f is homogeneous. x x x x d dv Let vx v x in (i) dv vx x v x v x x x dv v x x log log Cx x Q7. Given that ( ) (tan x)d The differential equation is in the form So, dv x v log v v log x log C x C x is the required solution. d tan tan x d x tan d tan P()x Q() where P() and Q() d Integration factor, (I.F.) e e + tan solution is given as : x e e d C tan t Put tan t d dt. xe e tdt C tan tan 5//8 6

7 tan t d t xe t e dt (t) e dt C dt tan tan xe e (tan ) C or, tan t t xe te e C x tan Ce Q8. The lines in vector form can be written as : ˆ ˆ ˆ ˆ ˆ ˆ r i j k (i 3j 4k), r iˆ 3j ˆ ( ˆi ˆj 3k) ˆ ˆ ˆ ˆ ˆ ˆ ˆ a i j k, b i 3j 4k, a iˆ 3j, ˆ b ˆi ˆj 3kˆ ˆi ˆj kˆ a a 3i ˆ ˆj k, ˆ b b 3 4 7i ˆ ˆj kˆ tan is the required solution. 3 (a a ).b b Shortest distance between skew lines b b ( 3iˆ ˆj k).( ˆ 7i ˆ ˆj k) ˆ 4 units Q9. Let A, B, C and D denote the events that A, B, C and D solves the problem in Mathematics respectivel. 3 4 P(A), P(B), P(C) and P(D) P(A), P(B), P(C) and P(D) (i) P(the problem will be solved) P(A B C D) P(A B C D) P(A) P(B) P(C) P(D) (ii) P(at most one of them will solve the problem) P(A B C D) P(A B C D) P(A B C D) P(A B C D) P(A B C D) SECTION C 3 Q. Given f (x) (x ) (x ) f (x) (x ) (x )(5x 4) For critical points, f (x) (x ) (x )(5x 4) x,, 5 Interval Sign of f (x) f (x) is (, ) Positive Strictl increasing 4 Negative Strictl decreasing, 5 4 Positive Strictl increasing, 5 (, ) Positive Strictl increasing 4 Hence f (x) is strictl increasing in (, ) and, 5 & strictl decreasing in 4, 5. Also in the left neighbourhood of, f (x) is positive and in right neighbourhood of, f (x) is negative and f ( ). Therefore b the first derivative test, x = is a point of local maximum. Also f (x) changes its sign from negative to positive as x passes through 4/5 so, x = 4/5 is a point of local minimum. + Q. Given that a.b a.c which implies, a is perpendicular to both b and c (since these three vectors are unit vectors). That is, a b c a (b c) + 4 5//8 7

8 a b c b c sin 6 Therefore, a (b c). Hence (i) is proved. Also, a b b c c a (a b) (b c).(c a) a b a c bb bc.(c a) a b a c b c.(c a) a b.c a c.c b c.c a b.a a c.a b c.a [a b c] [b c a] [a b c] a.(b c) [a b c] [b c a] [a b c] + a. a a. Hence (ii) is proved. d d x d Q. The given curve is 4 x 4 x mt at (,) Equation of tangent at (, ) is : (x ) x...(i) Also given lines are x...(ii) and x (iii). Required area 3 6 x x 3 x (x ) x x x 6x x x Mark for 9 9 Labeled-graph sketching 3 sq.unit. 4 4 Q3. The equation of line through 3i ˆ j ˆ kˆ i.e., P(3,, ) and parallel to the given line having d.r. s as (, 3, ), is x 3 z. 3 The coordinates of an random point on this line is : A( 3, 3, ) If A lies on the plane 3x z, then 3( 3) ( 3 ) ( ) 4 Therefore, A( 5,, 3). + Hence, the required distance is, AP (3 5) ( ) ( 3) 4 4 units. Now, the equation of line passing through P(3,, ) and perpendicular to the given equation of plane 3x z is, x 3 z. 3 The coordinates of an random point on this line is : B(3 3,, ) If B lies on the plane 3x z, then 3(3 3) ( ) ( ) Therefore, the required foot of perpendicular is,,. + OR An plane passing through the line of intersection of given planes is r.(i ˆ 3j ˆ k) ˆ r.(i ˆ ˆj k) ˆ i.e., r. (i ˆ 3j ˆ k) ˆ (i ˆ ˆj k) ˆ...(i) If plane (i) contains (3,, ) then, (3iˆ j ˆ k). ˆ (i ˆ 3j ˆ k) ˆ (i ˆ ˆj k) ˆ 6 6 (3 ) 3 5//8 8

9 Replacing value of in (i), we get r. (i ˆ 3j ˆ k) ˆ (i ˆ ˆj k) ˆ 3 3 i.e., r.(4i ˆ 7j ˆ k) ˆ 3. Let be the angle between normals of the given planes (as angle between the planes and their normals remains the same). (i ˆ 3j ˆ k).(i ˆ ˆ ˆj k) ˆ 7 7 So, cos cos Q4. Let E : white balls are transferred from Bag I to the Bag II, E : red balls are transferred from Bag I to the Bag II, E 3 : red and white balls are transferred from Bag I to the Bag II, E : the ball drawn from Bag II is red. / C C C C 4 P(E ), P(E 9 ),P(E 9 3) 9 C 7 C 7 C and, P(E E ), P(E E ), P(E E 3) P(E 3)P(E E 3) B Baes Theorem, P(E 3 E) P(E )P(E E ) P(E )P(E E ) P(E 3)P(E E 3) OR Let X : number of doublets in three throws of a pair of dice. X,,, Here n = 3. Let p be the probabilit of success p, q p r n r n r Therefore b using, P(X r) Crp q 3 5 we have : P(X r) Cr 6 X P( X ) X P(X) X P(X) Mean X P(X) X P(X) 5// X P(X) 3 5 3, Var (X) X P(X) X P(X) 5 5 Also, standard deviation Var X. 6 Q5. Let the radius of circular garden be r (in metres) and the side of the square garden be x (in metres). 6 r Then, 6 r 4x x (i) 4 The sum of the areas, S r x ds 4 r 6 r 4r 3 r dr 6 6 r S r 4 +

10 ds For local points of maxima and/or minima, 4r 3 r dr r 3 4r 3 r metres. 4 Also d S d S 4 4 S is least. dr dr 3 at r 4 B (i), 6 r 4x 6 6 8r 4x x r. Therefore, when the side of the square garden is double the radius of the circular garden, the sum of their areas is least. To achieve an goal, there is ever possibilit that energ, time and mone are required to be invested. One must plan in such a manner that least energ, time and mone are spent. A good planning and execution, therefore, is essentiall needed. Q6. Let the number of pieces of model A and model B respectivel be x and. To maximize : Z ` (8x ) Subject to the constraints : 9x 8, x 3 3, x, That is, 3x 4 6, x 3 3, x, Mark for Graph work on the actual graph paper Corner points Value of Z (in `) O (, ) C (, ) 6 B (, 6) 68 A (, ) Mark for table The number of pieces of model A and model B are and 6 respectivel. Also maximum profit is of `68. # This paper has been issued b CBSE, New Delhi for 6 Board Exams of XII. We ve re-tped the same and done the necessar corrections (as the original paper had some errors). On other hand, if ou still find an error which could have gone un-noticed, please do inform us via or us : 5//8

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