Question numbers to carry mark each. CBSE MATHEMATICS SECTION A. If R = {(, y) : + y = 8} is a relation of N, write the range of R. R = {(, y)! + y = 8} a relation of N. y = 8 y must be Integer So Can be,,, 6 y y 6 6y y Range is {,,}. If tan tan, y y <, then write the value of + y + y. tan tan y y y y y y y tan, y y y. If A is a square matri such that A = A, then write the value of 7A (I + A), where I is an identity matri. A is a square matri A = A 7A (I + A) = 7A (I + A)(I + A)(I + A) = 7A (II + IA + AI + A )(I + A) = 7A (I + A + A + A)(I + A) All Rights Reserved 8. Wiley India Pvt. Ltd.
= 7A (I + A)(I + A) = 7A (II + IA + AI + AA) = 7A (I + A + A) = 7A (I + 7A) = I. If y z, y w 5 find the value of + y. y z y 5 y & z y & 5 y y + y = 5. If 7 8 7, 6 find the value of. 7 8 7 6 6. If f ( ) t sin t dt, then write the value of f ( ). f t sin t dt cos cos d By Parts f t t t t cos sin t f cos sin differentiate w.r.t f () = ( sin) cos + cos = sin All Rights Reserved 8. Wiley India Pvt. Ltd.
f () = sin 7. Evaluate: d I d d 7 I ln ln7 ln 5 ln 5 8. Find the value of p for which the vectors iˆ ˆj 9k ˆ and iˆ pj ˆ k ˆ are parallel. let a iˆ ˆj 9kˆ b iˆ pj ˆ kˆ a & b are two parallel vector then a b (i + j + 9k) = λ(i pj + k) equate Coefficient, P, 9 P P P 9. Find a.( b c ), if a iˆ ˆj k ˆ, b iˆ ˆj k ˆ and c iˆ ˆj k ˆ. a iˆ ˆj kˆ, b iˆ ˆj kˆ, c iˆ ˆj kˆ 6 a b c 6 5. If the cartesian equations of a line are y z 6, 5 7 write the vector equation for the line. All Rights Reserved 8. Wiley India Pvt. Ltd.
y z 5 7 y z 5 7 Constant Any point on this line is 5, 7, vector equation of line is ˆ ˆ ˆ 5ˆ 7 ˆ ˆ r i j k i j k SECTION B Question numbers to carry marks each.. If the function f : R R be given by f() = + and g : R R be given by g ( ), find fog and gof and hence find fog () and gof ( )., f R R f! g! R R g fog f g g f gof g f f fog 6 gof. Prove that tan cos, All Rights Reserved 8. Wiley India Pvt. Ltd.
let y tan let cos,, y tan tan cos cos cos cos use cos sin cos sin cos sin cos, tan tan y y cos OR If tan tan, find the value of. tan tan tan tan 8 8 6 8 6. Using properties of determinants, prove that y 5 y 8y 8 All Rights Reserved 8. Wiley India Pvt. Ltd. 5
y y 5 y 5 y 8y 8 8 8y 8 5 C C C 5 y 8 8 8 5 8 7 7 6 C C C. Find the value of d at, if = aeθ (sin θ cos θ) and y = ae θ (sin θ + cos θ). sin cos, sin cos ae y ae d ae sin cos ae cos sin ae sin d ae sin cos ae cos sin ae sin d d ae cos cot d d d ae sin d 5. If y = Pe a + Qe b, show that d y a b y Pe Qe ( a b) aby d d a b ape bqe d differentiate w.r.t to d y a b a Pe b Qe d d y d d a b a b a b a b aby a Pe b Qe a bape bqe abpe Qe a b a b a b a b a Pe b Qe a Pe abqe bape b Qe abpe Qabe 6. Find the value(s) of for which y = [( )] is an increasing function. All Rights Reserved 8. Wiley India Pvt. Ltd. 6
y = ( ) differentiate w.r.t d Sign schemes of d y d f y is increasing,, OR Find the equations of the tangent and normal to the curve Equation of Hyperbola is a y b Equation of tangent to Hyperbola at point P a, b a y b at the point ( ab, ). a y b y a b a b b Slope of tangent a a Slope of normal b Equation of normal at point P a, b a y b a b b y b a a a by a b 7. Evaluate: Let sin d cos All Rights Reserved 8. Wiley India Pvt. Ltd. 7
I sin d cos cos sin I d u sing porperty sin I d cos sin sin I d d cos cos OR tan cos tan cos tan cos Evaluate: d 5 6 I d 56 5 d d 56 5 6 5d d 56 5 5 5 56 5 5 ln C 5 5 6 ln 5 6 C 8. Find the particular solution of the differential equation d y y, =. given that y = when All Rights Reserved 8. Wiley India Pvt. Ltd. 8
y y d ( ) y( ) d ( )( y) d ( )d c (Variable Separable method) y ln( y ) c at, y ln c c / ln( y ) ln( y ) Solution is y e 9. Solve the differential equation tan ( ) y e. d y e d tan tan tan tan e y d linear differential equation. Integrating factor = e d d tan tan tan tan e e ye d C tan tan tan e e ye d C e e t tan tan tan tan d dt tan tan e e e t d t dt C ye e y e C e C All Rights Reserved 8. Wiley India Pvt. Ltd. 9
. Show that the four points A, B, C and D with position vectors iˆ 5 ˆj k ˆ, ˆ j k ˆ, iˆ 9 ˆj k ˆ and ( iˆ ˆj k ˆ) respectively are coplanar. Position vector of Points are OA i 5 j k, OB j k, OC i 9 j k, OD i j k AB OB OA j k i 5 j k i 6 j k AC OC OA i 9 j k i 5 j k i j k AD OD OA i j k i 5 j k 8i j k AD, AC & AD vector are coplanar than scalar triple product these vector is. AB AC AD AB AC AD 6 5 6 8 OR The scalar product of the vector a iˆ ˆj k ˆ with a unit vector along the sum of vectors b iˆ ˆj 5k ˆ and c iˆ ˆj k ˆ is equal to one. Find the value of λ and hence find the unit vector along b c. a i j k u d b c i j 5k i j k i 6 j k Unit vector dˆ iˆ 6ˆj kˆ 6 i j k i 6j k a dˆ 6 6 8 8 8 8 ˆ i 6 j k i 6 j k d 9 7 All Rights Reserved 8. Wiley India Pvt. Ltd.
. A line passes through (,, ) and is perpendicular to the lines r ( iˆ ˆj kˆ) (iˆ ˆj k ˆ) and r (iˆ ˆj kˆ) ( iˆ ˆj k ˆ). Obtain its equation in vector and cartesian form. Line passing through (,,) is ˆ ˆ r i j k li mj nk l, m, n are direction of line which is perpendicular to given lines & r i j k i j k r i j k i j k Dot product of direction is zero. l m n l m n l n l n l m l l m Direction ratio of required line is l, l, l,,,, vector equation of line is r i j k i j k Cartesian equation of line is y z. An eperiment succeeds thrice as often as it fails. Find the probability that in the net five trials, there will be at least successes. Let the chances of success of an eperiment is = p & the chances of an eperiment is = q p + q = p = q q q p All Rights Reserved 8. Wiley India Pvt. Ltd.
Probability of at least success out of 5 trial is C p q C p q C p 5 5 5 5 5 5 7 5 8 5 5 C C C5 5 5 5 59 57 Question numbers to 9 carry 6 marks each. SECTION C. Two schools A and B want to award their selected students on the values of sincerity, truthfulness and helpfulness. The school A wants to award ` each, ` y each and ` z each for the three respective values to, and students respectively with a total award money of `,6. School B wants to spend `, to award its, and students on the respective values (by giving the same award money to the three values as before). If the total amount of award for one prize on each value is < 9, using matrices, find the award money for each value. Apart from these three values, suggest one more value which should be considered for award. Name of schoolof their no of Student Award for Sincenty( ) Award for Truthful ness( y) Award for Helpless( z) A B Student for each value School A award the money + y + z = 6 () School B award the money + y + z = () Total amount of award for one Prize +y + z = 9 () On each value is let P, X y z 6 Q 9 PX Q Det ( P) ( ) ( ) ( ) 6 5 Matri P is non Singular matri All Rights Reserved 8. Wiley India Pvt. Ltd.
X = P t Q Cofactor is of P is C, C, C 5 C ( ), C, C 5 C, C, C 5 5 adjp 5 5 5 t (adjp) P 5 (Det P) 5 5 / 5 / 5 t P / 5 / 5 / 5 / 5 / 5 / 5 6 y / 5 / 5 z / 5 / 5 9, y, z. Show that the altitude of the right circular cone of maimum volume that can be inscribed in a sphere of radius r is r. Also show that the maimum volume of the cone is 8 of the volume of the 7 sphere. OBD OD OD BD cos =, sin OB r OB OD rcos BD rsin Volume of right Circular Cone is All Rights Reserved 8. Wiley India Pvt. Ltd.
V R h R Radiusof Cone h height of Cone h r r cos R rsin V r sin ( r r cos ) V r (sin sin cos ) dv d dv d dv cos cos cos d cos cos r (sin cos sin cos sin ) (differentiatic w.r.t ) r sin ( cos cos sin ) (cos )(cos ) cos 7 cos / / sin r r hieght = r Volumeof Cone r ( r sin ) r sin 9 8 r 9 9 r 8 8 r 7 8 volumeof sphase 7 5. Evaluate: Rh d cos sin All Rights Reserved 8. Wiley India Pvt. Ltd.
Let I d cos sin sec I d tan Put tan t sec d dt sec t I dt dt t t ( / t ) I dt t t I ( / t ) ( t / t) ( ) t z dt dz t t I dz z tan ( ) I z t / t tan C tan cot tan C C 6. Using integration, find the area of the region bounded by the triangle whose vertices are (, ), (, 5) and (, ). Coordinate of Vertices of ΔABC is A(, ) B(,5) & C(, ) Equation of line AB is 5 y z ( ) y y 7 7 y Equation of line AC is All Rights Reserved 8. Wiley India Pvt. Ltd. 5
y z ( ) y 8 y 5 y Equation of line BC is 5 y ( ) y 8 y 7 5 Area of ABCis = d d d 6 sq. unit t t Area of ABC ( 7)d ( )d ( 5)d 7 ) 5 9 7 7 9 5 5 6 8 sq.unit. 7. Find the equation of the plane through the line of intersection of the planes + y + z = and + y + z = 5 which is perpendicular to the plane y + z =. Also find the distance of the plane obtained above, from the origin. Equation of plane passing through the line of Intersection of two given plane is given by ( y z) ( y z 5) ( ) y( ) z( ) ( 5 ) This plane is perpendicular to y + z = ( ) Hence plane is y z z y z 5 z All Rights Reserved 8. Wiley India Pvt. Ltd. 6
Distance of Plane from origin OR Find the distance of the point (,, 5) from the point of intersection of the line r iˆ ˆj kˆ (iˆ ˆj k ˆ) and the plane r.( iˆ ˆj k ˆ). n Eq of plane is r ( i j k) ( i yj zk) ( i j k) y z & Eq n of line is r i j k ( si j k) r i( ) j( ) k( ) any point on this line is ( ) lies in the Plane 8 8 Point of Intersection of plane & line is (,, ) (,,5) Distance b/w given point & this point is ( ) ( ) ( 5) 5 69 8. A manufacturing company makes two types of teaching aids A and B of Mathematics for class XII. Each type of A requires 9 labour hours of fabricating and labour hour for finishing. Each type of B requires labour hours for fabricating and labour hours for finishing. For fabricating and finishing, the maimum labour hours available per week are 8 and respectively. The company makes a profit of ` 8 on each piece of type A and ` on each piece of type B. How many pieces of type A and type B should be manufactured per week to get a maimum profit? Make it as an LPP and solve graphically. What is the maimum profit per week? Let no. of pieces of type A manufactured per week = Let no. of pieces of type B manufactured per week =y Type of pieces hour for fabrication hour for finishing A B 9 All Rights Reserved 8. Wiley India Pvt. Ltd. 7
for A Maimum no. of fabrication 9 hours is 8. 9 y 8 y 6 Ma m no. of finishing hour for B is. y Profit 8 y P Where y y y 6 5 y y Graphically Plotting of this y gy 9 y6 5y y 6 8 P 8 y Q(, b) Pr ofit P 8 6 96 7 68 S(,) P 8 R(,) P 8 6 T(,5) P 8 5 8 Maimum Profit occur at T(, 5) Where Teaching aid from A= All Rights Reserved 8. Wiley India Pvt. Ltd. 8
Teaching aid from B=5 9. There are three coins. One is a two-headed coin (having head on both faces), another is a biased coin that comes up heads 75% of the times and third is also a biased coin that comes up tails % of the times. One of the three coins is chosen at random and tossed, and it shows heads. What is the probability that it was the two-headed coin? Number of Coin= For coin P Probability of P Heads is P(H/P) = For coin Q Probability of P Heads is P(H/Q) = 75 / For coin R Probability of P Heads is in 6 P(H R) 5 Prob of choosing one Coin out of coins is =/ P P P Q P R P(P/H) Prob. Of Head from coin P, same other Prob of Head P(H)=P(P) P(H/P)+P(Q)P(H/Q)+P(R) P(H/R) 7 5 5 7 7 6 P(P) P(H/P) P(P/H)= P(H) 7 7 OR Two numbers are selected at random (without replacement) from the first si positive integers. Let X denote the larger of the two numbers obtained. Find the probability distribution of the random variable X, and hence find the mean of the distribution. First si positive Integer is {,,,,5,6} X is greatest no out of chosen of no. X = 5 6 If = favourable Case is (,) P ( ) 6 C 5 If = favourable Case is All Rights Reserved 8. Wiley India Pvt. Ltd. 9
(,) & (,) P ( ) 6 C 5 If = favourable case is (,), (,), (,) P ( ) 6 C 5 If = s fav. Case is (,5), (,5), (,5), (,5) P ( 5) 6 C 5 If = 6 fav Case is (,6),(,6,),(,6),(,6),(5,6) 5 5 P ( 6) 6 C 5 Variable X 5 6 Corresponding 5 Probability 5 5 5 5 5 5 Mean of distribution 5 6 5 5 5 5 5 ( 6 ) 7 5 5 All Rights Reserved 8. Wiley India Pvt. Ltd.