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CBSE XII Sample / Mock 2015 : MATHEMATICS with Answers

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SAMPLE PAPER: MATHEMATICS CLASS XII: 2014 15 TYPOLOGY VSA (1 M) LA I (4 M) LA II (6 M) 100 Remembering 2, 5 11, 15, 19 24 20 Understanding 1, 4 8, 12 23 16 Applications 6 14, 18, 13 21, 26 25 HOTS 3 10, 17 20, 22 21 Evaluation & MD 7, 9, 16 25 18 1 SECTION A Question number 1 to 6 carry 1 mark each. 1. The position vectors of points A and B are and respectively. P divides AB in the ratio 3 : 1 and Q is mid-point of AP. Find the position vector of Q. 1 2. Find the area of the parallelogram, whose diagonals are =5 and =2 1 3. If P(2, 3, 4) is the foot of perpendicular from origin to a plane, then write the vector equation of this plane. 4. 1 If = 4 3 3 5 5 1 2 6 , Write the cofactor of a32 (the element of third row and 2nd 2 column). 5. If m and n are the order and degree, respectively of the differential equation y 6. 1 + x3 -xy = sin x, then write the value of m+n. 1 Write the differential equation representing the curve y2 = 4ax, where a is an arbitrary constant. 1 SECTION-B Question numbers 7 to 19 carry 4 marks each. 7. To raise money for an orphanage, students of three schools A, B and C organized an exhibition in their locality, where they sold paper bags, scrap-books and pastel sheets made by them using recycled paper, at the rate of Rs. 20, Rs.15 and Rs. 5 per unit respectively. School A sold 25 paper-bags 12 scrap-books and 34 pastel sheets. School B sold 22 paper-bags, 15 scrapbooks and 28 pastel-sheets while school C sold 26 paper-bags, 18 scrap-books and 36 pastel sheets. Using matrices, find the total amount raised by each school. By such exhibition, which values are inculcated in the students? 8. Let A = 2 1 3 , then show that A2 4A + 7I = O. 2 2 4 Using this result calculate A3 also. OR 1 1 0 If A = 2 5 3 , find A-1 , using elementary row operations. 0 2 1 9. 4 If x, y, z are in GP, then using properties of determinants, show that + + 0 = o, where x y z and p is any real number. + 10. Evaluate : | 11. 4 + |dx. Evaluate : 4 . e2x dx. 4 OR Evaluate : ( 12. )( ) dx Consider the experiment of tossing a coin. If the coin shows tail, toss it again but if it shows head, then throw a die. Find the conditional probability of the event that the die shows a number greater than 3 given that there is at least one head . 4 OR How many times must a man toss a fair coin so that the probability of having at least one head is more than 90%? 13. For three vectors , and if = and = , then prove that , and are mutually perpendicular vectors, | |= | |and| | = 1 14. 4 Find the equation of the line through the point (1,-1,1) and perpendicular to the lines joining the points (4,3,2), (1,-1,0) and (1,2,-1), (2,1,1) OR 3 4 Find the position vector of the foot of perpendicular drawn from the point P(1,8,4) to the line joining A(O,-1,3) and B(5,4,4). Also find the length of this perpendicular. 15. Solve for x: sin-1 6x + sin-1 6 3 = OR Prove that: 2 sin-1 16. If x = sin t, (1-x2) -x - tan-1 = 4 y = sin kt, show that + k2 y = 0 4 17. If yx + xy + xx = ab, find 18. It is given that for the function f(x) = x3 + bx2 + ax + 5 on [1, 3], Rolle s theorem holds with c = 2 + 4 . Find values of a and b. 19. 4 Evaluate : 4 dx SECTION-C Question numbers 20 to 26 carry 6 marks each. 20. Let A = {1, 2, 3, , 9} and R be the relation in A x A defined by (a, b) R (c, d) if a+d = b+c for a, b, c, d A. Prove that R is an equivalence relation. Also obtain the equivalence class [(2, 5)]. 6 OR Let f : N R be a function defined as f(x) = 4x2 + 12x + 15. Show that f : N S is invertible, where S is the range of f. Hence find inverse of f. 21. Compute, using integration, the area bounded by the lines x+2y = 2, y-x=1 and 2x+y= 7 4 6 22. Find the particular solution of the differential equation sin + 6 = o, given that y = 0, when x = 1 OR Obtain the differential equation of all circles of radius r. 23. Show that the lines = ( 3 + + 5 ) + (-3 + + 5 ) and = + 2 + 5 +2 +5 are coplanar. Also, find the equation of the plane containing these lines. 24. + 6 40% students of a college reside in hostel and the remaining reside outside. At the end of year, 50% of the hosteliers got A grade while from outside students, only 30% got A grade in the examination. At the end of year, a student of the college was chosen at random and was found to get A grade. What is the probability that the selected student was a hostelier? 25. 6 A man rides his motorcycle at the speed of 50km/h. He has to spend Rs. 2 per km on petrol. If he rides it at a faster speed of 80km/h, the petrol cost increases to Rs. 3 per km. He has atmost Rs. 120 to spend on petrol and one hour s time. Using LPP find the maximum distance he can travel. 26. 6 A jet of enemy is flying along the curve y = x2+2 and a soldier is placed at the point (3, 2). Find the minimum distance between the soldier and the jet. 5 6 MARKING SCHEME SAMPLE PAPER SECTION-A 1. 5 +3 1 2. 5 sq. units 1 3. . 2 + 3 + 4 4. 14 1 5. m+n=4 1 6. 2x 1 = 29 1 y=0 SECTION-B Sale matrix for A, B and C is Price matrix is 7. 25 22 26 20 15 5 25 12 22 15 26 18 34 28 36 12 34 15 28 18 36 20 500 + = 15 440 + 5 520 + 180 + 170 225 + 140 270 + 180 850 = 805 970 Amount raised by School A = Rs 850, school B = Rs 805, school C = Rs 970 Values 1 8. Helping the orphans Use of recycled paper 1 A2 = 2 1 3 2 2 1 3 1 12 = 2 4 1 1 6 1 4 A2 4A + 7I = 12 8 12 7 + + 1 4 8 0 0 0 = 7 0 0 0 2 A2 = 4A-7I A3 = 4A2 7A = 4(4A-7I) -7A = 9A 28I = = 10 9 18 9 27 28 0 + 18 0 28 27 10 1 OR Write A = IA we get 1 2 0 1 5 2 0 1 0 3 = 0 1 1 0 0 R2 R2-2R1 1 0 0 1 7 2 R2 R2-3R3 1 0 0 1 1 2 1 0 0 + 2 0 0 .A 1 0 1 = 3 2 1 0 0 0 1 0 A 0 1 1 0 1 0 = 2 1 0 0 0 1 3 A 0 1 1 0 0 1 1 3 1 0 = 2 1 3 A 0 1 4 2 7 1 1 1 3 A-1 = 2 1 3 4 2 7 9. = + + 0 + + 0 0 C1 C1- pC2 C3, = 1 + + Expanding by R3 = (-p2x-2py-z) (xz-y2) 1 7 Since x, y, z are in GP, y2 = xz or y2 xz = 0 1 =0 10. | . | dx=2 | cos = 2 ( cos =2 1 1 = . e2x dx = = 1 . et dt (where 2x=t) et dt + 1 t 2 + tan 2 e dt = tan )dx + 2 2 11. I = 1 ) dx+2 ( cos + =2 | 2 2 = f(t) then f (t) = sec 1 2 Using ( ( ) + ( )) et dt = f(t) et + C, we get t 2x 2. e + C = tan x. e + C 1 I = tan OR We have ( )( ) = ( + 1)+ = ( + 1) + ( Now express ( )( ) =( )( ) .. (1) ) +( 8 ) .. (2) 1 So, + 1)+ ( 1= ( =( + ) + ) ( 1) +( ) + Equating coefficients, A + B = 0, C B = 0 and A C = 1, Which give A = , B = C = - . Substituting values of A, B, and C in (2), we get ( )( ) = ( ) - ( ) - ( .. (3) ) 1 Again, substituting (3) in (1), we have ( )( ) =( + 1) + ( ) - ( ) - ( ) Therefore ( )( = ) + + log | 1| log( + 1) + 1+1 12. Let E : Die shows a number > 3 E : {H4, H5, H6] and F : there is atleast one head. F : {HT, H1, H2, H3, H4, H5, H6} P(F) = 1 = 1 P(E F) = = 1 P(E/F) = ( ) ( ) = = 1 9 OR p = , q = , let the coin be tossed n times P(r 1) > or 1-P(r=0) > P(r=0) < 1- = nC 0 < < 1 2n > 10, n = 4 13. 1 = and = and 1 = | | and | | = | | sin = | | and | | | | sin 1 2= | | = | | | | | | = | | = 1 | | = 1 1 1. = | | = | | 14. DR s of line (L1) joining (4, 3, 2) and (1, -1, 0) are <3, 4, 2> DR s of line (L2) joining (1, 2, -1) and (2, 1, 1) are <1, -1, 2> A vector to L1 and L2 is 3 4 1 1 2 = 10 -4 -7 2 1 Equation of the line passing through (1, -1, 1) and to L1 and L2 is = ( - + ) + (10 -4 -7 ) 1 10 OR Equation of line AB is = (- +3 ) + (5 +5 + ) 1 Point Q is (5 , -1+5 , 3+ ) = (5 -1) +(5 -9) + ( -1) PQ AB 5(5 -1) + 5 (5 -9) + 1 ( -1) = 0 51 = 51 =1 foot of perpendicular (Q) is (5, 4, 4) Length of perpendicular PQ = 4 + ( 4) + 0 = 4 2 units 1 15. sin-1 6x + sin-1 6 3 x = sin-1 6x = sin 6 3 6x = sin sin 6 3 = - sin + sin = - cos [sin-1 6 3 ] = - 1 108 6 3 1 36x2 = 1-108 x2 144 x2 = 1 x= since x = does not satisfy the given equation x= 1 OR LHS = 2 sin-1 - tan-1 11 = 2 tan-1 - tan-1 = tan-1 = tan-1 = tan-1 . 1 - tan-1 1 - tan-1 1 = tan-1 (1) = . 1 4 16. x = sin t and y = sin kt = cost and = k cost kt =k or cost. 1 = k. coskt cos2t = k2 cos2 kt cos2t = k2 cos2 kt (1-x2) = k2 (1-y2) 1 Differentiating w.r.t.x (1-x2) 2 (1-x2) 17. let u = yx, (i) ( 2 ) = -2k2y + -x 1 + k2y = 0 v = xy, logu = x logy (ii) log v = y log x w = xx = yx log y + = xy 1 + 12 (iii) log w = x logx log y + = = xx , (1+logx) + log + xy ( + xx (1+logx) = 0 ) 1 1 . 18. f(x) = x3 + bx2 + ax + 5 on [1, 3] f (x) = 3x2 + 2bx+a f (c) = 0 3 2 + + 2b 2 + + a = 0 - - - - - - (i) 1 f(1) = f(3) b+a+6 = 32 + 9b +3a or a + 4b = -13 - - - - - - - - - - (ii) Solving (i) and (ii) to get a=11, b= -6 19. 1 1 Let 3x + 1 = A ( 2x 2) + B 1 I= ( ) = 3 5 2 A = -3/2, B = 2 2 1+1 2. sin ( ) + 1 SECTION C 20. (i) for all a, b A, (a, b) R (a, b), as a + b = b + a R is reflexive 1 (ii) for a, b, c, d A, let (a, b) R (c, d) a + d = b + c c + b = d + a (c, d) R (a, b) R is symmetric 1 (iii) for a, b, c, d, e, f, A, (a, b) R (c, d) and (c, d) R (e, f) 13 a + d = b + c and c + f = d + e a + d + c + f = b + c + d + e or a + f = b + e (a, b) R (e, f) R is Transitive 2 Hence R is an equivalence relation and equivalence class [(2, 5)] is {(1, 4), (2, 5), (3, 6), (4, 7), (5, 8), (6, 9)} 1 OR Let y S, then y=4x2+12x+15, for some x N y = (2x + 3)2 + 6 x = , as y > 6 1 Let g : S N is defined by g(y) = 1 gof (x) = g (4x2+12x+15) = g ((2x+3)2+6) = and fog (y) = f ( ) + 3 +6 = y = =x 1 1 Hence fog (y) = IS and gof(x) = IN f is invertible and f-1 = g 1 21. Let the lines be, AB: x+2y = 2, BC: 2x+y = 7, AC = y-x = 1 1 Points of intersection are A(0,1), B(4,-1) and C(2, 3) 1 A = (7 ) dy - (2 2 ) dy ( 1) dy 1 = 7 (2 ) 1 = 12 4 2 = 6sq.Unit. 22. Given differential equation is homogenous. 14 Putting y = vx to get =v+x v+ v+ = or = = =v = 1 = + - - - - - - - - (i) 1 I1 = sinv.e-v + cos = -sinv.e-v cosv e sin v. e dv I1 = (sin v + cosv) 1 Putting (i), (sinv + cosv) sin + cos = logx +C2 = log +C 1 x = 1, y = 0 c = 1 1 Hence, Solution is sin + cos = log +1 OR (x a)2 + (y b)2 = r2 ..........(i) 2(x a) + 2(y b) 1+(y b) + =0 .........(ii) = 0 .........(iii) (y b) = 1 From (ii), (x a) = 1 15 Putting these values in (i) + = = 1+ or 1 1 23. Here = 3 + + 5k , = 3 + + 5k = + 2 + 5k , = + 2 + 5k . x 2 = 3 1 1 0 1 5 = 2 5 1( 15 + 5) 2 5 1 = 10 + 10 = 0 lines are co planer. Perpendicular vector (n ) to the plane = b x b j 3 1 1 2 k 5 = 5 + 10 5k 5 2 or 2 + k 2 Eqn. of plane is r . 2 + k = 2 + . 3 + +5 =0 1 or x 2y + z = 0 24. Let E1: Student resides in the hostel E2: Student resides outside the hostel P(E ) = = , P (E ) = + A: Getting A grade in the examination P = = P = 16 = 1+1 P ( = = ( ) ( ) . . . ) ( ) ( 1 ) = 1+1 25. Let the distance travelled @ 50 km/h be x km. and that @ 80 km/h be y km. LPP is Maximize D = x + y St. 2x + 3y 120 + 1 or 8 + 5 400 2 x 0,y 0 2 Vertices are. (0, 40), , , (50,0) 17 , Max. D is at Max. D = = 54 km. 1 26. Let P(x, y) be the position of the jet and the soldier is placed at A(3, 2) AP = 3 + (y 2) As y = x2 + 2 y 2 = x2 = 2(x 3) + 4x3 and = 0 x = 1 and .......(i) .......(ii) AP2 = (x 3)2 + x4 = z (say) = 12x2 + 2 2 (at x = 1) > 0 1+1 z is minimum when x = 1, when x = 1, y = 1+2 = 3 minimum distance = (3 1) + 1 = 5 18 1

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