INTERMEDIATE PUBLIC EXAMINATION, MAY 2006
MATHEMATICS PAPER 1(A)
ALGEBRA, VECTOR ALGEBRA AND TRIGONOMETRY
TIME: 3 Hrs. Max. Marks: 75.
SECTION – A
VERY SHORT ANSWER TYPE QUESTIONS
Answer all questions. Each question carries 2 marks.
1. Find the range of the function f: A à R where A = {1, 2, 3, 4} and f(x) = x2+x-2.
2. Find a unit vector parallel to the resultant of the vectors, r1= 2i + 4j – 5k and r2 = I + 2j + 3k.
3. If the position vectors of the vertices A, B, C of ∆ ABC, are 7j + 10k, -i + 6j + 6k and -4i +9j +6k respectively. Prove that the triangle is right angled and isosceles.
4. If a = I + j + k and b = 2i + 3j + k , find the length of the projection of b on a and the length of the projection of a on b.
5. Prove that tan (A + 135) tan (A-135) = -1.
6. If tan A = 8/25, find the values of sin 2A and cos 2A.
7. If cosh x = 5/2, find the value of cosh 2x.
8. In ∆ ABC, express ∑ r1 cot ( A/2) in terms of ‘s’.
9. Find the values of (√3/2 – i/2)12.
10. Expand cos 4A in powers of cos A.
SECTION – B
SHORT ANSWER TYPE QUESTIONS.
Attempt any 5 questions, Each question carries 4 marks.
11. f: R à R are defined by f(x) = 3x – 2 and g(x) = x2 + 1, then find the following
i) (g o f-1) (2) ii) (g o f ) ( x-1)
12. If x = (√3 – √2)/ (√3 +√2), y =(√3 + √2)/ (√3 +√2) then show that x2 + xy + y2 = 99.
13. If x = log 2a a, y = log 3a 2a and z = log 4a 3a, then show that xyz + 1 = 2yz.
14. If, a, b, c are non coplanar vectors, show that a + a2b + c, -a + 3b – 4c, a – b + 2c are non coplanar.
15. If a = 2i + 3j + 4k , b = I + j – k , compute a X (b X c) and verify that is perpendicular to a.
16. If tan ( π sin A ) = cot (π cot A), then show that sin ( A + π ) = ±1/√2 .
2 2 4
17. Show that Tan ‑1 1/8 + Tan ‑1 1/2 + Tan ‑1 1/5 = π/4.
SECTION – C π
LONG ANSWER TYPE QUESTIONS
Attempt any 5 questions, each question carries 7 marks.
18. Let f : A à B and g: B à C be bisections, Prove that g o f : A à C is also bijection.
19. using the principles of Mathematical Induction, prove that 2.3+ 3.4 + 4. 5 + ………..upto π
Terms = n (n2 + 6n + 11)/3, for all n € N.
20. For any vectors,a, b , c prove that a X ( b X c) = ( a. c ) b – ( a . b ) c.
21. If A + B + C = 1800, prove that cos A + cos B + - cos C = - 1 + 4 cos A/2 cos B/2 cos C/2 .
22. In ∆ ABC, prove that r + r1+ r2 - r3 = 4 R cos C.
23. From the top of a tree on the bank of a lake, an aeroplane in the sky makes an angle of elevation A and the of the height of the aeroplane is ‘h’. Show that h = a Sin (A + B)
Sin (A - B)
24. If the amplitude of ( z – 2 )/ ( 2 – 6i ) is π/2 , find the equation of locus of z.
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