C.B.S.E Class XII Delhi & Outside Delhi Sets

SOLVED PAPER With CBSE Marking Scheme C.B.S.E. 8 Class XII Delhi & Outside Delhi Sets Mathematics Time : Hours Ma. Marks : General Instructions : (i) All questions are compulsory. (ii) The question paper consists of 9 questions divided into four sections A, B, C and D. Section A comprises of questions of one mark each, Section B comprises of 8 questions of two marks each, Section C comprises of questions of four marks each and Section D comprises of 6 questions of si marks each. (iii) All questions in Section A are to be answered in one word, one sentence or as per the eact requirement of the question. (iv) There is no overall choice. However, internal choice has been provided in questions of four marks each and questions of si marks each. You have to attempt only one of the alternatives in all such questions. (v) Use of calculators in not permitted. You may ask for logarithmic tables, if required. SECTION - A. Find the value of tan cot ( ). a. If the matri A is skew symmetric, b find the values of a and b.. Find the magnitude of each of the vectors a and b, having the same magnitude such that the angle between them is 6 and their scalar product is 9.. If a * b denotes the large of a and b if a o b (a * b) +, then write the value of () o (), where * and o are binary operations. SECTION - B. Prove that : sin sin ( ),, 6. Given A 7, compute A and show that A 9I A. + cos 7. Differentiate tan with respect to. sin 8. The total cost C() associated with the production units of an item is given by C().. + +. Find the marginal cost when units are produced, where by marginal cost we mean the instantaneous rate of change of total cost at any level of output. 9. Evaluate : cos+ sin d cos. Find the differential equation representing the family of curves y a e b+, where a and b are arbitrary constants.. If q is the angle between two vectors i j+ k and i j+ k, find sin q.. A black and a red die are rolled together. Find the conditional probability of obtaining the sum 8, given that the red die resulted in a number less than. SECTION - C. Using properties of determinants, prove that + + y + z. If ( + y ) y, find d. 9 ( yz+ y+ yz+ z) If a (q sin q) and y a ( cos q), find d when q π. To know about more useful books for class- click here. If y sin (sin ), prove that + tan + ycos. d d

] Oswaal CBSE Solved Paper - 8, MATHEMATICS, Class-XII 6. Find the equations of the tangent and the normal, to the curve 6 + 9y at the point (, y ), where and y >. Find the intervals in which the function f () + + is (a) strictly increasing, (b) strictly decreasing. 7. An open tank with a square base and vertical sides is to be constructed from a metal sheet so as to hold a given quantity of water. Show that the cost of material will be least when depth of the tank is half of its width. If the cost is to be borne by nearby settled lower income families, for whom water will be provided, what kind of value is hidden in this question? 8. Find : cos ( sin )( + sin ) d 9. Find the particular solution of the differential equation e tan y d + ( e ) sec y, given that y π when. Find the particular solution of the differential equation + y tan sin, given that y d when π.. Let a i+ j k, b i j+ k and c i+ j k. Find a vector d which is perpendicular to both c and b and d. a.. Find the shortest distance between the lines r ( i j) + λ( i+ j k) and r ( i j+ k) + µ ( i+ jk).. Suppose a girl throws a die. If she gets or, she tosses a coin three times and notes the number of tails. If she gets,, or 6, she tosses a coin once and notes whether a head or tail is obtained. If she obtained eactly one tail, what is the probability that she threw,, or 6 with the die?. Two numbers are selected at random (without replacement) from the first five positive integers. Let X denote the larger of the two numbers obtained. Find the mean and variance of X. SECTION - D. Let A { Z : < < }. Show that R {(a, b) : a, b A,} a b is divisible by } is an equivalence relation. Find the set of all elements To know about more useful books for class- click here related to. Also write the equivalence class []. Show that the function f : R R defined by f() +, " R is neither one-one nor onto. Also, if g : R R is defined as g(), find fog().. If A, find A. Use it to solve the system of equations y + z + y z + y z. Using elementary row transformations, find the inverse of the matri A 7. 6. Using integration, find the area of the region in the first quadrant enclosed by the -ais, the line y and the circle + y. 7. Evaluate : π / sin+ cos + sin d 6 9 Evaluate : ( + + e ) d, as the limit of the sum. 8. Find the distance of the point (,, ) from the point of intersection of the line r i j+ k+ λ ( i+ j+ k) and the plane r.( i j+ k). 9. A factory manufactures two types of screws A and B, each type requiring the use of two machines, an automatic and a hand-operated. It takes minutes on the automatic and 6 minutes on the hand-operated machines to manufacture a packet of screws A while it takes 6 minutes on the automatic and minutes on the hand-operated machine to manufacture a packet of screws B. Each machine is available for at most hours on any day. The manufacturer can sell a packet of screws A at a profit of 7 paise and screws B at a profit of `. Assuming that he can sell all the screws he manufactures, how many packets of each type should the factory owner produce in a day in order to maimize his profit? Formulate the above LPP and solve it graphically and find the maimum profit.

Oswaal CBSE Solved Paper - 8, MATHEMATICS, Class-XII [ CBSE Marking Scheme (Issued by Board). π π π π 6 SECTION - A + Note : m. for any one of the two correct values and m. for final answer. [CBSE Marking Scheme, 8] tan ( ) cot ( ) π tan tan cot cot π 6 π tan tan cot cot π π 6 π π π 6 π π π 6. a, b + [CBSE Marking Scheme, 8] Since, the given matri is skew symmetric. So, we have A T A. (A T is transpose of A) b a a b a + and b. Hence, a and b.. a b + Given, [CBSE Marking Scheme, 8] 9 angle q 6 and a. b Q cos q a. b a. b cos 6 9 / a. a ( a b ) a a 9 a b... o ( * ) + + For * For Final Answer SECTION - B. In RHS, put sin q RHS sin ( sin q sin q) sin (sin q) q sin LHS 6. A, A 7 LHS A 7 LHS RHS RHS 9 7 7 + cos 7. f() tan sin tan cos sin cos π tan cot f'() [CBSE Marking Scheme, 8] Let + cos y tan sin tan tan + cos.sin.cos cos sin tan cot To know about more useful books for class- click here

] Oswaal CBSE Solved Paper - 8, MATHEMATICS, Class-XII π tan tan y π Differentiating with respect to. d. 8. Marginal cost C'() + At, C'() 9. I. sin + sin d cos To know about more useful books for class- click here sec d tan + C d baeb+ d by d b d The differential equation is : y d d Differentiate w. r. to Again, D.w.r. to. [CBSE Marking Scheme, 8] y a e b+...() d a.b eb+...() d a.b e b+...() put b. from () and () in equation (). we y d get y.. d y d Required differential equation is d. y d ( i j+ k ) ( i j+ k ). sin q i j+ j i j+ k ( i j+ k ) ( i j+ k ) i + 8 j+ k 6 sin q 6 6 7 [CBSE Marking Scheme, 8] Let a i j+ k a + ( ) + b i j+ k b + ( ) + Q cos q a. b a. b ( i j+ k ).( i j+ k ). + + sin q cos θ sin q 9. 7 7.. A : Getting a sum of 8, B : Red die resulted in no. <.. LHS ( ) P(A/B) P A B PB ( ) / 6 8 / 6 9 SECTION - C + + y + z + y y y z (Using C C C & C C C (9yz) + (z + 9yz + y) (Epanding along R ) 9(yz + y + yz + z) RHS. Differentiating with respect to '' ( + y ) + y d + y d d y y y+ y

Oswaal CBSE Solved Paper - 8, MATHEMATICS, Class-XII [ d d a( cos θ) a sin θ a sin θ a sinθ cosθ d asinθcosθ cotθ asin θ θ π [CBSE Marking Scheme, 8] ( + y ) y Differentiating with respect to..( + y ) + y. d. + y d ( + y ) + y( + y ) y d d [y( + y ) ] d y ( + y ) d [ ( y + y )] [ y ( + y ) ] Given a(q sin q) Differentiate with respect to q. d a( cos q) d a( cos q) and y a.( cos q) a sin q d d d θ π θ π asin θ d a( cos θ) π π sin. sin π π cos + cos / +.. y sin (sin) d. cos (sin) cos To know about more useful books for class- click here and sin (sin) cos sin cos (sin) d + LHS sin (sin) cos sin cos (sin) + sin cos (sin) cos + sin (sin) cos cos RHS 6. y ( y > ) Differentiating the given equation, we get, 6 d 9y Slope of tangent at (, ) d (, ) 7 Slope of Normal at (, ) 7 Equation of tangent: + 7y Equation of Normal: 7 y f '() + ( ) ( )( + ) f '(),,. sign of f '(): + + f() is strictly increasing on (, ) È (, ) and f() is strictly decreasing on (, ) È (, ) [CBSE Marking Scheme, 8] Given, 6 + 9y...() It passes through (, y ) 6 + 9y put 6 + 9y 9y 8 y (Q y > ) Diff. equation () with respect to. + 8y d d Slope of the tangent at (, ) d (, ) 8y The equation of tangent at (, ) is y.( ) 7 7y 8 + 6 + 7y 8 7 7 Slope of Normal at (, ) d (, )

6 ] Oswaal CBSE Solved Paper - 8, MATHEMATICS, Class-XII Now, equation of Normal is given by y 7 ( ) y 96 7 7 y + Given f() + + Differentiating with respect to. put f () + f () + ( ) ( ) ( ) ( ) ( ) ( ) ( + ) ( ), and The intervals are (, ), (, ) (, ) and (, ). + + - - (a) f () > " (, ) (, ) f() is strictly increasing " (, ) (, ) (b) f () < " (, ) (, ) f() is strictly decreasing " (, ) (, ) 7. Let side of base and depth of tank y V y y V, (V Quantity of water constant) Cost of material is least when area of sheet used is minimum. A (Surface area of tank) + y + V da d V, y V, da d + + d A + 8 V d >, + Area is minimum, thus cost is minimum when y Value: Any relevant value. 8. Put sin cos d dt Let I d t t cos ( sin) + sin ( ) ( ) + dt ( ) Let To know about more useful books for class- click here ( t) + t ( ) A t + Bt + C, solving we get + t A, B, C I + + t dt dt + dt tt + t log t + log + t + tan t + C log( sin) + log ( + sin ) + tan (sin) +C [CBSE Marking Scheme, 8] cos ( sin ).( + sin ). d Let sin y cos. d cos ( sin ).( + sin ). d ( y).( + y ). Let ( y)( + y ) A By C y + + + y A ( + y ) + (By + C) ( y) A + Ay + By + C By Cy (A B)y + (B C).y + A + C Comparing the Coefficients of y, y and Constant terms. A B, B C and A + C. Solving all these equation we get A B C cos ( sin ).( + sin ). d ( y)( + y ). - y + y +.. + y dt y +. +. - y + y + y log y + log + y + tan y + C + y log + tan y+ C y log + sin + tan (sin ) + C sin 9. Separating the variables, we get: y sec tan y e d e log tan y log e + log C

Oswaal CBSE Solved Paper - 8, MATHEMATICS, Class-XII [ 7 tan y C(e ), for, y p/, C Particular solution is: tan y e. Integrating factor tan d e sec Solution is: y sec sin sec d sec tan d y sec sec + C, for π, y, C + Particular solution is: y sec sec or y cos cos Given differential equation is e.tan y.d + ( e ).sec y. y put tan y t e z [CBSE Marking Scheme, 8] sec tan y. e d ( e ). sec y. dt e d dz y π when t. dt z. dz log t log z + log C log t log C.z t C.z tan y C.(e ) tan π C.(e ) C y when π y. sec sec.sin d. + c y sec sec.tan d. + c y sec sec + c sec π + c + c c required particular solution is. d λ c b y sec sec i j k ( ) l d λi 6λ jλk da l 8l + l l d 6 i + j+ k [CBSE Marking Scheme, 8] Since, vector d is perpendicular to both c and b. d. b and d. c Let d i+ yj+ zk Now, Q d. b ( i+ yj+ zk).( i j+ k) C Required particular solution is Given d (it is linear in y) y tan ( e ) + tan y. sin Comparing with + Py. Q. d Here, P tan, Q sin Integrating factor (I.F.) d e tan. The solution is log sec e sec y (I.F.) (. IF.) Qd. + c To know about more useful books for class- click here y + z...(i) d. c ( i+ yj+ zk).( i+ jk) + y z...(ii) and d. a ( i+ yj+ zk).( i+ jk) + y z...(iii) Solving equation (i), (ii) and (iii) we get, y 6 and z. So, d + 6 + i j k.. Here a i j, a i j + k a a i + k

8 ] Oswaal CBSE Solved Paper - 8, MATHEMATICS, Class-XII i j k b b i j ( a a b b ) ( ) Shortest distance b b 6 6. E : She gets or on die. E : She gets,, or 6 on die. A : She obtained eactly tail P(E ) or 6, P (E ) P(A/E ) 8, P (A/E ) P(E /A) PE ( ) P( A/ E) PE P A/ E P E P A/ E ( ) ( ) + ( ) ( ) + 8 8. Let X denote the larger of two numbers X P(X) / / / / X P(X) / 6/ / / X P(X) / 8/ 8/ / Mean SX P(X) Variance SX P(X) [SX P (X)] 7 [CBSE Marking Scheme, 8] We have, the first five positive integers are,,, and. We can selected two numbers from numbers is P 6. Here, given X denote the larger of two numbers we observe that X can be take values,,,. P (X ) Probability that larger number is. P(X ) Probability of getting in first selection and in second selection or getting in first selection and in second selection P(X ) + To know about more useful books for class- click here P (X ) Probability that the larger of two number is. P(X ) + P(X ) Probability that the larger of two number is P(X ) + 6 P(X ) + 8. Thus, the probability distribution of X is X : P(X) : 6 8 Mean [E(X)] 6 8 + + + E(X ) + + + 8. 6 8 + + + 8 6 + 96 + + 7 Variance of X E(X ) [E(X)] 7. Hence, E(X) and Var (X). SECTION - D. Refleive : a a, which is divisible by, a A (a, a) Î R, a A R is refleive Symmetric : let (a, b) R a b is divisible by ( ) b a is divisible by a b b a (b, a) R R is symmetric Transitive: let (a, b), (b, c) R a b & b c are divisible by a b ±m, b c ±n, n Î Z Adding we get, a c (±m ± n) (a c) is divisible by a c is divisible by (a, c) Î R R is transitive Hence R is an equivalence relation in A set of elements related to is {,, 9} and [] {, 6, }.

Oswaal CBSE Solved Paper - 8, MATHEMATICS, Class-XII [ 9 Here f() f but ¹ f is not for y let f() + As D ( ) ()() <, No real solution f() ¹, for any R (D f ) f is not onto fog () f( ) + ( ) + [CBSE Marking Scheme, 8] Given R {(a, b} : a, b A, a b is divisible by } Where A { Z : < < } {,,,,,, 6, 7, 8, 9,,,,}. Refleivity : For any a A, we have a a is divisible by. Thus R is Refleive. (a, a) R. Symmetry : Let (a, b) R then (a, b) R a b is divisible by. b a is divisible by. (b, a) R. Thus, (a, b) R (b, a) R R is Symmetric.. Transitivity : Let (a, b) R and (b, c) R then (a, b) R and (b, c) R a b is divisible by and b c is divisible by. a b l and b c m for some l, m N a b ± l and b c ± m. a c ± l + m a c is divisible by. (a, c) R Thus, (a, b) R and (b, c) R (a, c) R. R is transitive. Hence, R is an equivalence Relation Now, Let be an element of A such that R or (, ) R then, is divisible by. To know about more useful books for class- click here,, 8,,, 8,,, 9 (Q A) Hence, the set of all element which is related to is {,, 9}. Also, equivalence class []. {, 6, }. Given, f : R R; f() + g : R R; g(). To show that f is neither one-one nor onto (i) f is one-one : Let, R (domain) and f( ) f( ) + + ( + ) ( + ) +. ( ).(. ) Taking, R. f is not one-one. f () f ( ) 7 f ( ) f 7 (ii) f is onto : Let y R (co-domain) f () y + y + y y y.( + ) ± y y since, R, y > ( + y) ( y) > < y < So Range (f ), Range (f ) R (Co-domain) f is not onto. Hence f is neither one-one nor not. Now, fog() f(g())

] Oswaal CBSE Solved Paper - 8, MATHEMATICS, Class-XII + ( ) + + +. A : ¹ A eists Co-factors of A are : A ; A ; A m for A ; A 9; A correct A ; A ; A cofactors adj(a) 9 A A. adj( A) 9 For, X y and B, the system of equation z is A X B X A B 9, y, z Using elementary Row operations : Let : A IA 7 A A {Using, R R R ; R R + R } A {Using, R R R } A {Using, R R R ; R R R } A 6. Correct figure : y O +y Point of intersection, Area of shaded region d+ d 6 + + sin X π 8+ 6 8 π π Sq. units. 7. Put sin cos t, (cos + sin ) d dt, sin t when, t and p/, t π / sin+ cos I d 6 + 9sin dt 6 + 9 ( t ) 9t dt I log + t t log log or log Here, f() + + e, a, b, nh ( + + e ) d lim f() + f( + h) +... + f( + nh) h To know about more useful books for class- click here

Oswaal CBSE Solved Paper - 8, MATHEMATICS, Class-XII [ ( nh h)( nh)( nhh) ( nh) + 6 lim h ( nh h)( nh) h nh + + e ( e h e ) 8 8 + + + ee ( ) 6 + e e [CBSE Marking Scheme, 8] Here, the line r i j+ k+ λ ( i+ j+ k)...() meet the plane r( i j+ k)...() Put the value of vector r from () in eq n () [ ( i j+ k) + λ ( i+ j+ k)].( i j+ k). or [( i j+ k).( i j+ k) + λ( i+ j+ k)].( i j+ k) Given ( + + e ). d Let I ( + + e ) d and f() + + e a, b h b a n n I ( + + e ). d lim h [f() + f( + h) + f( + h) h +... f( + (n )h] lim h [ + e + ( + h) + ( + h) + e +h h + ( + h) +.( + h) + e +h +... + ( + (n ) h) ( + (n ).h) + ( + (n ) h) + e lim h [n + e + h ( + + +... + (n ) ] h + h(+ ++...+(n ) + h.( + + +... + (n )) + e.(e h + e h + e h +... e (n ).h ] ( n)( n)( n ) lim h. n+ e+ h h 6 + h nn + h nn + e e h e h.( n ) ( ) ( ) [ ] h e 8 8 + + + 6+ e e 6 + e e 8. General point on the line is : ( + l, + l, + l) As the point lies on the plane + l + l + + l l Point is (,, ) Distance ( ) + + ( ) ( ( )) ( ( )) [CBSE Marking Scheme, 8] or ( + + ) + l.( + ) or + l l putting the value of l in equation (i) To know about more useful books for class- click here 9. r ( i j+ k) The points are (,, )_ and (,, ) The distance between the points (,, ) and (,, ) in ( + ) + ( + ) + ( + ) 9+ 6 + 69. X' 8 6 Y A(, ) B(, ) C(, ) 6 8 X Let number of packets of type A and number of packets of type B y L.P.P. is: Maimize, Z.7 + y Subject to constraints: + 6y or + y 6 + y or + y 8, y Correct graph Z(, ), Z(, ) Z(, ) 8, Z(, ) (Ma.) Ma. profit is ` at, y.