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CENTRE FOR ADVANCEMENT OF STANDARDS IN EXAMINATIONS (GEMS ASIAN SCHOOLS) COMMON REHEARSAL EXAMINATIONS JANUARY 2016 ( ALL INDIA SENIOR SCHOOL CERTIFICATE EXAMINATION) Subject: : : Mathematics Time:3 hours Max.Marks:100 Subject Code: 041 General Instructions (i) (ii) (iii) (iv) (v) (vi) All questions are compulsory Please check that this question paper contains 26 questions. Questions 1-6 in section A are very short-answer type questions carrying 1 mark each Questions 7-19 in Section B are long-answer I type questions carrying 4 marks each. Questions 20-26 in Section C are long-answer II type questions carrying 6 marks each. Please write down the serial number of the question before attempting it. SECTION A 1. 2. If A is a square matrix of order 3 and |3 | =k | |, then write the value of k. If m and n are the order and degree of the differential equation 5 2 + 4 3. 2 3 3 + 3 = 2 1 Then find m-n. Write the integrating factor in the differential equation 1 (1 + ) -( )y = 1+ 4. If | | = 4 , 5. For what value of a the vectors 6. 2 | . | = 2 then find | |2 | | and a +6 -8 are collinear? 2 -3 +4 If a line makes angle , with the axes respectively then Find 2 + 2 + 2 SECTION B 7. If = [ 1 2 ] , show that 1 = [ 1 2 OR Page 1 of 4 2 ] 2 1 1 If A= [ 2 5 0 2 8. If 0 3 ] , find A 1 using elementary row transformations. 1 = 1 1 prove that = 2 2 OR 5 2 Find x if( 1 )2 + ( 1 )2 = 8 For what value of k such that the function 3 9. { ( ) = 2 (1+3 ) 2 1 10. If x13y7 = (x+y) 11. Evaluate 20 then prove that 2 (1+ ) 1+ 2 dy = dx . <0 =0 >0 is continuous at x = 0. 12. Using the properties of determinants, solve the following for x 13. If +2 | +6 1 1 +2|=0 +6 x = + and = , prove that 2 2 2 14. +6 1 +2 Evaluate 1 4 +3 2 +1 OR Evaluate + = 0. 1+2 2 Page 2 of 4 be three vectors. Let = 2 + , = + + and = 4 3 + 7 Find a vector which satisfies = and = 0. 16. Show that the lines = + + (3 ), = (4 + 2 ) + (3 1) intersect. Find the point of intersection? 15. OR Find the equation of the perpendicular drawn from the point (1, - 2, 3) to the plane 2x - 3y +4z +9 =0. Also find the coordinates of the foot of the perpendicular. 17. A letter is known to have come either from LONDON or from CLIFTON .On the envelope just two consecutive letters ON are visible. What is the probability that the letter has come from (1) LONDON (2) CLIFTON 18. 19. 2 ( 2 + 1) as the limit of a sum Find 0 A farmer possess 30 acre cultivated land that must be cultivated into different mode of cultivations organic and inorganic. The yield for organic and inorganic system of cultivation is 11 quintals/ acre and 14 quintals/acre respectively. Using matrix method, determine how to divide 30 acre land among two mode of cultivation to obtain yield 390 quintals. Which mode of cultivations you prefer more and why? SECTION C 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) in AxA. Prove that R is an equivalence relation. Also obtain the equivalence class [(2,5)] 21. From a lot of 15 bulbs which include 5 defectives, a sample of 4 bulbs is drawn one by on with replacement. Find the mean and variance of the distribution. 22. Find the equation of the plane passing through the intersection of the planes ) = 9 such that it makes equal (2 + + 3 ) = 7 and (2 + 5 + 3 intercepts on x and z axis. 23. Show that differential equation 2 + ( 2 ) = 0 is homogenous. Find the particular solution of the differential equation, given that = 0 when = 1. Page 3 of 4 OR Solve the differential equation = ( 3 + 2 ) , (0) = 1. 24. Using integration find the area of the following region 25. Find the shortest distance between the line = 1 and the curve = 2 . OR Show that the height of the cylinder of maximum volume that can be inscribed in a sphere of radius R is 2R/ 3 . Also find the maximum volume. 26. A company produces two types of leather belts say type A and B. Belt A is a superior quality and belt B is of a lower quality. Profits on each type of belt are 2/- and 1.50 per belt, respectively. Each belt of type A requires twice as much time as required by a belt of type B. If all belts were of type B, the company could produce 1000 belts per day. But the supply of leather is sufficient only for 800 belts per day (both A and B combined). Belt A requires a fancy buckle and only 400 fancy buckles are available per day. How should the company manufacture the two type of belts in order to have a maximum overall profit? The use of leather products should be discouraged. Why? Page 4 of 4
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