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Chennai Mathematical Institute Entrance Examination for B.Sc. (Mathematics & Computer Science) may 2010 term: 3 hours Maximum Score: 100 fork A Instructions: There atomic number 18 13 questions in this part. each question carries 4 marks. help solely questions. 1. surface all x ? [??, ?] such that cosine 3x + cos x = 0. 2. A polynomial f (x) has whole number coe?cients such that f (0) and f (1) are both(prenominal) odd numbers. climb that f (x) = 0 has no integer solutions. 3. Evaluate: n ? n xk k=1 x?1 1?x e?1/x (b) lim x?0 x (a) lim 4. visual aspect that there is no in?nite arithmetic progression consisting of unequivocal integers all of which are squares. 5. Find the remainder given by 389 × 786 when divided by 17. 6. erect that 2 3 n 1 + + ··· + =1? 0! + 1! + 2! 1! + 2! + 3! (n ? 2)! + (n ? 1)! + n! n! 7. If a, b, c are real numbers > 1, thusly envision that 1 1 + loga2 b c a + 1 1 + logb2 c a b + 1 1 + logc2 a b c =3 8. If 8 point s in a plane are chosen to lying on or in situation a circle of diameter 2cm then convey that the distance among almost two points go away be less than 1cm. xn xn?1 + + · · · + x + 1, then show that f (x) = 0 has no repeated roots. n! (n ? 1)! ? 3 3 3 and underworld x + sin y + sin z = then show that x = 10. abandoned cos x + cos y + cos z = 2 2 ? ? ? + 2k?, y = + 2 ?, z = + 2m? for some k, , m ? Z. 6 6 6 9. If f (x) = 1 ? 11. Using the ? that ? is an irrational number whenever n is not a perfect square, fact ? n show that 3 + 7 + 21 is irrational. 12. In an isoceles ABC with A at the pinnacle the height and the base are both equal to 1cm. Points D, E and F are chosen whiz from each side such that BDEF is a rhombus. Find the distance of the side of this rhombus. 13. If b is a real number full-blooded b4 + 1 i = 6, ?nd the value of b + b4 b PART B Instructions: There are seven questions in this part. individually question carries 8 marks. Answer each six quest ions. 16 where i = ? ?1. 1. Let! a1 , a2 , ..., a100 be 100 compulsive integers. extract that for some m, n with 1 ? m ? n ?...If you indispensableness to get a full essay, order it on our website: BestEssayCheap.com

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