**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) = x^{2}+x-2.

2. Find a unit vector parallel to the resultant of the vectors, r_{1}= 2i + 4j – 5k and r_{2 }= 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 x^{2} + xy + y^{2} = 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 = 180^{0}, 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 + r_{1}+ r_{2} - r_{3 }= 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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