MATHEMATICS. Time allowed : 3 hours Maximum Marks : 100

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1 MATHEMATICS Time allowed : hours Maimum Marks : General Instructions:. All questions are compulsory.. The question paper consists of 9 questions divided into three sections, A, B and C. Section A comprises of questions of one mark each, Section B comprises of questions of four marks each and Section C comprises of 7 questions of si marks each.. All questions in Section A are to be answered in one word, one sentence or as per the eact requirement of the question. 4. There is no overall choice. However, internal choice has been provided in 4 questions of four marks each and questions of si marks each. You have to attempt only one of the alternatives in all such questions. 5. Use of calculators is not permitted. QUESTION PAPER CODE 65// SECTION A Question numbers to carry ark each.. State the reason for the relation R in the set {,, } given by R {(l, ), (, )} not to be transitive.. Write the value of sin sin. For a matri, A [a ij ], whose elements are given by a ij j i, write the value of a For what value of, the matri is singular? 4 89

2 5 5. Write A for A. 6. Write the value of sec (sec + tan ). 7. Write the value of For what value of 'a' the vectors î ĵ+ 4kˆ and aî + 6ĵ 8kˆ are collinear? 9. Write the direction cosines of the vector î + ĵ 5kˆ.. Write the intercept cut off by the plane + y z 5 on -ais. SECTION - B Question numbers to carry 4 marks each.. Consider the binary operation * on the set {,,, 4, 5} defined by a * b min. {a, b}. Write the operation table of the operation *.. Prove the following: cot + sin + + sin sin, sin, 4 Find the value of tan y tan y + y. Using properties of determinants, prove that a ba ca ab b cb ac bc c 4a b c 9

3 4. Find the value of 'a' for which the function f defined as a sin ( + ), f( ) tan sin, > is continuous at. 5. Differentiate cos + + w.r.t. d y If a ( θ sin θ ), y a ( + cos θ ), find 6. Sand is pouring from a pipe at the rate of cm /s. The falling sand forms a cone on the grou;nd in such a way that the height of the cone is always one-sith of the radius of the base. How fast is the height of the sand cone increasing when the height is 4cm? Find the points on the curve + y at which the tangents are parallel to -ais. 7. Evaluate: Evaluate: + + ( )( ) 8. Solve the following differential equation: e tan y + ( e ) sec y dy 9. Solve the following differential equation: cos dy + y tan. 9

4 . Find a unit vector perpendicular to each of the vectors a r r r r + b and a b, where r r a î + ĵ+ kˆ and b î + ĵ kˆ.. Find the angle between the following pair of lines: + y z + + y 8 z 5 and and check whether the lines are parallel or perpendicular.. Probabilities of solving a specific problem independently by A and B are and respectively. If both try to solve the problem independently, find the probability that (i) the problem is solved (ii) eactly one of them solves the problem. SECTION - C Question numbers to 9 carry 6 marks each.. Using matri method, solve the following system of equations: , +, + ;, y, z y z y z y z Using elementary transformations, find the inverse of the matri 4. Show that of all the rectangles inscribed in a given fied circle, the square has the maimum area. 5. Using integration find the area of the triangular region whose sides have equations y +, y + and 4. 9

5 6. Evaluate: sin cos tan (sin ) Evaluate: sin cos 4 4 sin + cos 7. Find the equation of the plane which contains the line of intersection of the planes r r ( î + ĵ + kˆ ) 4, ( î + ĵ kˆ ) + 5 and which is perpendicular to the r plane ( 5î + ĵ 6kˆ ) A factory makes tennis rackets and cricket bats. A tennis racket takes.5 hours of machine time and hours of craftman's time in its making while a cricket bat takes hours of machine time and hour of craftman's time. In a day, the factory has the availability of not more than 4 hours of machine time and 4 hours of craftsman's time. If the profit on a racket and on a bat is Rs. and Rs. respectively, find the number of tennis rackets and crickets bats that the factory must manufacture to earn the maimum profit. Make it as an L.P.P. and solve graphically. 9. Suppose 5% of men and.5% of women have grey hair. A grey haired person is selected at random. What is the probability of this person being male? Assume that there are equal number of males and females. Question numbers to carry ark each. QUESTION PAPER CODE 65/ SECTION A. Let A {,, }, B {4, 5, 6, 7} and let f {(, 4), (, 5), (, 6)} be a function from A to B. State whether f is one-one or not.. What is the principal value of cot cos + sin sin? 9

6 . Evaluate : cos5 sin 75 o o sin 5 o cos 75 o 4. If A, write A in terms of A If a matri has 5 elements, write all possible orders it can have. 6. Evaluate: (a + b) 7. Evaluate: 8. Write the direction-cosines of the line joining the points (,, ) and (,, ). 9. Write the projection of the vector î ĵ on the vector î + ĵ.. Write the vector equation of a line given by Question numbers to carry 4 marks each. SECTION B 5 y + 4 z Let f : R R be defined as f() + 7. Find the function g : R R such that gof fog I R. A binary operation * on the set {,,,, 4, 5} is defined as : a + b, a * b a + b 6, if if a + b < 6 a + b 6 94

7 Show that zero is the identity for this operation and each element 'a' of the set is, invertible with 6 a, being the inverse of 'a'.. Prove that: tan cos,. Using properties of determinants, solve the following for : Find the relationship between 'a' and 'b' so that the function 'f' defined by: f() a +, if b +, if > is continuous at. If y e y, show that dy log. { log ( e) } 4 sin θ 5. Prove that y θ is an increasing function in + cos θ,. If the radius of a sphere is measured as 9 cm with an error of. cm, then find the approimate error in calculating its surface area. 6. If tan log y, show that a d y ( + ) dy + ( a) 95

8 7. Evaluate: + sin + cos 8. Solve the following differential equation: dy y + y 9. Solve the following differential equation: (y + ) dy. Using vectors, find the area of the triangle with vertices A(,, ), B(,, 5) and C(, 5, 5).. Find the shortest distance between the following lines whose vector equations are: r ( t) î + (t ) ĵ+ ( t) kˆ r (s + ) î + (s ) ĵ (s + ) kˆ and. A random variable X has the following probability distribution: X P(X) K K K K K K 7K + K Determine: (i) K (ii) P(X < ) (iii) P(X > 6) (iv) P( < X < ) Find the probability of throwing at most sies in 6 throws of a single die. 96

9 Question numbers to 9 carry 6 marks each. SECTION C. Using matrices, solve the following system of equations: 4 + y + z 6 + y + z y + z 7 4. Show that the right-circular cone of least curved surface and given volume has an altitude equal to times the radius of the base. A window has the shape of a rectangle surmounted by an equilateral triangle. If the perimeter of the window is, find the dimensions of the rectangle that will produce the largest area of the window. 5. Evaluate: / Evaluate: /6 + tan ( 5) ( 4 6. Sketch the graph of y + and evaluate the area under the curve y + above -ais and between 6 to. 7. Find the distance of the point (-, - 5, - ), from the point of intersection of the line r r î ĵ + kˆ + λ î + 4 ĵ + kˆ î ĵ + kˆ 5. ( ) ( ) and the plane ( ) 8. Given three identical boes I, II and III each containing two coins. In bo I, both coins are gold coins, in bo II, both are silver coins and in bo III, there is one gold 97

10 and one silver coin. A person chooses a bo at random and takes out a coin. If the coin is of gold, what is the probability that the other coin in the bo is also of gold? 9. A merchant plans to sell two types of personal computers - a desktop model and a portable, model that will cost Rs. 5, and Rs. 4, respectively. He estimates that the total monthly demand of computers will not eceed 5 units. Determine the number of units of each type of computers which the merchant should stock to get maimum profit if he does not want to invest more than Rs. 7 lakhs and his profit on the desktop model is Rs. 4,5 and on the portable model is Rs. 5,. Make an L.P.P. and solve it graphically. 98

11 Marking Scheme ---- Mathematics General Instructions :. The Marking Scheme provides general guidelines to reduce subjectivity in the marking. The answers given in the Marking Scheme are suggested answers. The content is thus indicative. If a student has given any other answer which is different from the one given in the Marking Scheme, but conveys the meaning, such answers should be given full weightage.. Evaluation is to be done as per instructions provided in the marking scheme. It should not be done according to one's own interpretation or any other consideration Marking Scheme should be strictly adhered to and religiously followed.. Alternative methods are accepted. Proportional marks are to be awarded. 4. In question(s) on differential equations, constant of integration has to be written. 5. If a candidate has attempted an etra question, marks obtained in the question attempted first should be retained and the other answer should be scored out. 6. A full scale of marks - to has to be used. Please do not hesitate to award full marks if the answer deserves it. 99

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24 is the identity for *. Also, a {,,,, 4, 5}, a * (6 a) a + (6 a) 6 (which is identity) m Each element a of the set is invertible with (6 a), being the inverse of a. + cos θ cos θ. Putting cos θ to get LHS tan + cos θ + cos θ LHS tan cos θ sin θ cos θ + sin θ tan tan 4 θ + θ cos 4 4. Applying C C and C C C, we get C m Applying R R + R and R R R, We get Epanding along C, we get [ ] or 48 i.e L.H.L. a + f () a + RHL b + since f() is continuous at, a + b + or a b, which is the required relation. y e y y. log ( y) log e y y + log m

25 dy ( + log ) log + ( + log ) ( + log ) log log (loge + log ) (e) [ log ] 5. dy dθ ( + cosθ) 4 cosθ 4 sin θ ( sin θ) ( + cosθ) ( cos θ + sin θ) ( 4 + cos θ + 4cosθ) 8 cosθ + 4 ( + cosθ) 4 cosθ cos θ 4 cosθ cosθ ( + cosθ) ( + cosθ) since, 4 cosθ > for θ and cosθ in +, ( cosθ) dy dθ in,, Hence the function is increasing in, Here r 9 cm and r. cm. da Error in surface area A r where A 4r dr 8r r 8 (9) (.).6 cm 6. log y a log y atan y e a tan tan dy a tan e a +

26 dy ( + ) a y differentiating again w.r.t., we get ( ) d y dy dy + + a d y dy ( ) + ( a) + sin 7. I + + cos + cos sec + tan tan tan + tan 8. Given equation can be written as dy y y + + v + dv v + + v where y v dv + v log v + + v log c v + + v c y + + y c 4

27 9. Given equation can be written as dy y or dy y I.F. log log e e e solution is, y + c y + c. Area ABC AB BC Here, AB î + ĵ + kˆ and BC î + ĵ î ĵ kˆ AB BC 6î ĵ + 4kˆ Area sq. units. Equations of the lines are, r r shortest distance ( î ĵ kˆ ) + t ( î + ĵ kˆ ) and ( î ĵ kˆ ) + s ( î + ĵ kˆ ) ( a a) ( b b ) b b where a î ĵ + kˆ, a î ĵ kˆ, b î + ĵ kˆ, b î + ĵ kˆ 5

28 a a ĵ 4kˆ, b b î 4ĵ kˆ ½+ 4 + S.D Here k + k + k + k + k + k + 7k + k k + 9k (k ) (k + ) k (i) k (ii) P( < ) + k + k k (iii) P( > 6) 7k + k (iv) P( < < ) k + k k Here n 6, probability of success (p) 6 probability of failure (q) 6 5 P (at most sies) P() + P() + P() ( 5 ) + 6 ( 5 ) + 6 ( 5 ) 4 6 C C C ( 5 ) ( ) 5 ( ) ( 5 )

29 SECTION - C. Given system of equations can be written as 4 6 y z or A X B A 4() ( 5) + ( ) 45 5 X A B C Cofactors are C C 5 5 C C C + 5 C C C 5 ark for any 4 correct cofactors m A y z , y 8, z 8 4. Let radius of cone be r and height h v r h (given) h v r 9v C.S.A. A rl r r + h r r + 4 r 9v 9v Let S + r r 4 r r + r 4 7

30 ds dr 8v r 6 4 r or 8v 4 r 8 9 r 4 h 4 r 6 h r d s 54v r + > curved surface area 4 is least when h r dr r Correct fiqure let sides of rectangle be and y and the sides of equilateral triangle be + y y Area y + 4 ( ) + 4 A 4 [4 6 + ] da or 4 ( 6 + ) m y 6 m d A ( + ) < Area is maimum for 4 ( 6 + ) 6 m and y m 8

31 5. I cos + tan cos + sin 6 ( + ) 6 6 cos ( ) ( ) ( ) + sin... (i) sin + sin cos...(ii) cos 6 6 Adding (i) and (ii) to get I. [ ] I I ( 5) ( 4) ( 9) ½ log c log c ½ 9

32 6. For correct graph A ( + ) + ( + 6 ) m A ( + ) ( + ) + m sq. U. 7. Any point on the given line is ( ) ( ) ( ) kˆ If this point lies on plane, it must satisfy its equation + λ î + + 4λ ĵ + + λ [( + λ) î + ( + 4λ) ĵ + ( + λ) kˆ ] ( î ĵ + kˆ ) 5 + λ + 4λ + + λ 5 λ Point of intersection is (,, ) Distance ( ) + ( + 5) + ( + ) + m 8. Let E : selecting bo I, E : selecting bo II and E : selecting bo III P (E ) P (E ) P (E ) let event A : Getting a gold coin P (A/E ) P (A/E ) P (A/E ) P(E /A) P(E) P(A/E) P(E ) P(A/E ) + P(E ) P(A/E ) + P(E ) P(A/E ) + + +

33 9. Let the number of desktop models, he stock be and the number of portable model be y L.P.P. is, Maimise P y subject to + y < y < 7 (or 5 + 8y < 4) m >, y > For correct graph m Vertices of feasible region are A (, 75), B (, 5), C (5, ) P(A) Rs. 875 P(B) Rs Rs. 5 P(C) Rs. 5 For ma. Profit destop model portable model 5

CBSE Board Paper Class-XII. Time allowed : 3 hours Maximum Marks : 100

CBSE Board Paper Class-XII. Time allowed : 3 hours Maximum Marks : 100 L.K.Gupta (Mathematic Classes) www.poineermathematics.com. MOBILE: 98155771, 461771 CBSE Board Paper -011 Class-XII (SET-1) Time allowed : hours Maimum Marks : 100 General Instructions: (i) All questions