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CBSE XII Pre Boards 2015 : MATHEMATICS with Answers (KV Chennai)

8 pages, 31 questions, 10 questions with responses, 26 total responses,

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KENDRIYA VIDYALAYA SANGATHAN, CHENNAI REGION PRE BOARD EXAMINATION (2014-15) Class: XII Max.Marks: 100 Subject: Mathematics Time: 3Hours General Instructions: (i) Total number of questions in the paper is 26. (ii) The question paper consists of 6 questions (Q.No.1 to 6) of 1 mark each, 13 questions (Q.No.7 to 19) of 4 marks each and 7 questions (Q.No.20 to 26) of 6 marks each. (iii) There is no overall choice. However internal choice has been provided in 4 questions four marks each and two questions of six marks each. You have to attempt only one of the alternatives in all such questions. . (iv) If you wish to answer any question, already answered for any reason, cancel the previous answer. (v) Use of calculator is not permitted. 1. What is the range of cos 1 x ? d2y dy 2. Find the order and degree of the differential equation 1 . dx 2 dx 3 3. If e 4. If p e 4 x 3x 2 , find the values of a and b. 4 2 is a unit vector and x p . x p 80 , then find x ax bx dx . 5. Write a unit vector of magnitude 9 units in the direction of vector 2i j 2k . 6. Write the direction ratios and direction cosines of the vector 2i j 3k SECTION- B 7. Let A=NXN and * be a binary operation on A defined by (a , b) * (c , d ) = ( a + c , b + d ). Show that *is commutative and associative. Also, find the identity element for * on A, if any. a a 2b 1 1 cos 1 tan cos 1 b b a 4 2 4 2 8. Prove that tan (OR) 12 1 3 1 56 sin sin 13 5 65 1 Prove the following cos 1 9. Using properties of determinant, prove that b c c a a b c a a b b c 2(3abc a 3 b 3 c 3 ) a b b c c a 3 2 5 10. Express 4 1 3 as sum of two matrices such that one is symmetric and the other is skew 0 6 7 symmetric. sin 1 x 11. If y 1 x2 d2y dy , show that 1 x 3x y 0 2 dx dx 2 12. If x a (cos t t sin t ), y a (sin t t cos t ), find d 2 y d 2x , dt 2 dt 2 4 sin x 2 x x cos x 13. Find the intervals in which the function f given by f ( x) 2 cos x is V i) increasing ii) decreasing (OR) R x tan x dx (OR) sec x tan x 0 14. Evaluate 15. Evaluate: x tan 1 x dx x 16. Evaluate: x f ( x) x2 2 x 8, x 4,2 9 16 sin 2 x dx 4 sin x cos x 0 2 1 x2 2 (OR) Evaluate: x sin x dx dx 2 3 x2 4 1 17. If 3i j and 2i j 3k , then in the form 1 2 , where 1 is parallel to 2 3 and 2 is perpendicular to 18. Find the equation of the plane passing through the intersection of the planes 3 x y + 2 z 4 = 0 and x + y + z 2 = 0 and the point (2,2,1). 19. How many time must a man toss a fair coin so that the probability of having at least one head is more than 90%?. 2 SECTION C 2 4 3 20. If A= 2 1 3 , find A-1. Using this solve 3 1 2 3x 2 y 3z 8 ; 2 x y z 1 ; 4 x 3 y 2 z 4 . (OR) 4 4 4 Using elementary transformation find the inverse of 7 1 3 5 3 1 21. Show that the height of the cylinder of maximum volume that can be inscribed in a sphere of radius R is 2R . Also find the maximum volume. 3 (OR) Show that the semi vertical angle of the cone of maximum volume and of given slant height is tan 1 2 . 22. Using integration, find the area of triangle whose are (1 , 4) , (2 , -3 ) and (4 , 2) 23. Suppose that the reliability of a HIV test is specified as follows: Of people having HIV, 90% of the test detect the disease but 10% go undetected . Of people free of HIV, 99% of the test are judged HIV-ive but 1% are diagnosed as showing HIV+ive. From a large population of which only 0.1% have HIV, one person is selected at random, given the HIV test, and the pathologist reports him/her as HIV+ive. What is the probability that the person actually has HIV? Can we keep away the people affected by HIV from the society ? 24. Find the distance of the point (-2, 3, -4) from the line parallel to the plane 4 x 12 y 3z 1 0 . x 2 2 y 3 3z 4 measured 3 4 5 25. Find the particular solution of the differential equation: given that y = 0 when x 2 dy y cot x 2 x x2 cot x, ( x 0) , dx . 26. An aero plane can carry a maximum of 200 passengers. A profit of Rs.500 is made on each executive class ticket out of which 20% will go to the welfare fund of the employees. Similarly a profit of Rs.400 is made on each economy ticket out of which 25% will go for the improvement of facilities provided to the economy class passengers. In both cases the remaining profit goes to the air lines fund. The air line reserves at least 20 seats for executive class. However at least four times as many passengers prefer to travel by economy class than executive class. Determine how many tickets of each type must be sold in order to maximize the net profit of the air line .Make the as an LPP and solve graphically. Do you think, more passengers would prefer to travel by such an air line than by others? 3 KENDRIYA VIDYALAYA SANAGATHAN CHENNAI REGION Common pre-board examination:2014-15 SCORING KEY Q.No. 1 2 3 4 5 6 Value points/suggestive points Range is [0, ] Order 2, degree 2 a = 4, b=3 9 . ) 3( 2i j 2k -2,1,3 and 2 14 7 Marks 1 1 1 1 1 , 1 14 , 1 3 14 1.5 For applying tan-1 x+ tan-1 y and simplifying to prove the result 10 2 (OR) for changing to tan-1 each 0.5 mark 9 1+2 1 1+1 For simplification 8 For commutative and associative For saying no identity element For each expansion 2.5 For each transformation 0.5 mark For simplification to get the result 2 2 2 1 1 ( A A ) P and ( A A ) Q 2 2 For proving P P and Q Q and conclusion For 1+1 11 For finding first derivative and second derivative 1+1 12 For simplifying For 2 4x1=4 13 dx dy d 2x d2y at cos t , at sin t , at cos t a sin t , at sin t a cos t dt dt dt 2 dt 2 4 cos x cos 2 x f ( x) (2 cos x) 2 3 cos x(4 cos x) 0 if cos x 0. we knowcos x 0 in (0, ) ( ,2 ) . 2 2 2 (2 cos x) 3 Hence increasing in the above interval and decreasing in ( , ) ( , ) 2 2 (OR) For saying f(x) is continuous, differentiable and f(a),f(b) F -1 ( 4,2) 1 2 1 1 2 14 For applying property and getting I tan x dx 2 0 sec x tan x ( 2) 2 1 = (sec x- tan x + x)(with in the limit) 25 16t 0 (OR) = dt 1 2 2 dx where t sin x cos x For integrating and getting the answer 15 log 9 40 2 For using integration by parts For integrating second part and to get the answer 16 For dividing For getting A=C=0,B=2,D=-6 x 2 3 tan 1 x 3 1 3 tan 1 x2 x 1 tan 1 x tan 1 x c 2 2 2 2 1 3 1 2 x c 2 1 (OR) x sin x dx x sin x dx 1 3 2 1 1 1 For integration Substituting the limits and getting the answer 17 18 , (1 ) =1/2 2 = (2 3 )i j 3k 1 1 3 Finding the required vectors. 1 = (3i j 3k j) , 2 = i 2 2 2 The equation of the plane passing through the intersection of planes (3 x y + 2 z 4 ) + ( x + y + z 2 = 0) 1 j) , 1 = (3i 2 3 1+1 1.5 1.5 7x-5y+4z-8=0 P(x 1) >90%, 1-p(x=0)>90% 2 1 1 n 10 2 1 1 n=4 20 1 1+1 1+1 19 1 1 2 2 1 8 10 1 A-1= 5 6 1 17 7 1 9 3 2 AX B X=(A-1 B X=1,y=2,z=3 1 OR For A=IA for row transformation and substituting A and I correctly 1 1 1 1 Answer 1 2 2 (for each correct entry 0.5 mark) 8 3 2 1 21 For correct figure Vol.Cy= ( R 2 h volume = 4.5 1 h3 ) 4 2 For I derivative and second derivative 2R h= 3 2 1.5 1 1 4 R3 1 3 3 OR V= 3 l 2 h h 3 2 1+1 For I derivative and second derivative l= 3h For proving the result. 22 1 1 11 y Eqn.of AB x 7 2 y 16 Eqn. of BC x 5 14 3 y Eqn. of AC x 2 0.5 0.5 0.5 Dig-1 Required area= 3 14 3 y 16 2 y 11 y 10 30 21 19 dy dy dy 2 5 7 2 2 2 2 2 3 3 4 23 2 4 E=the person selected having HIV PE PE A:HIV test is diagonised as positive PAE PAE P(E/A)=0.083 Value based question 3+0.5 1 2 2 1 24 4 3 5 4 , . 3 2 4 9 5 8 AB 3 , , 3 2 Let Abe the point (-2,3,-4). Let B be the point (x1,y1,z1) Since B lies on the line . B 3 2, T D 2 1 The point B(4, AB= 25 2 5 ,2) 2 1 17 2 1 1 1 1 3 y sin x sin x(2 x x2 cot x) dx c IF= sin x x 2 sin x c y sin x x sin x 2 2 1 4 4 26 Let no. of tickets of EXECU . be x, No. of tickets of Eco. Class be y constra int 2, graph 2 and corner points1 and VBQ 1 x y 200 x 20 y 4x Z=400x+300y Corner points (20,80) (`40,160) (20,180) The profit is maximum at (40,160) Z=400x+300y 32000 64000 62000 5

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