INTERMEDIATE PUBLIC EXAMINATION

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
2 2

4sin ( A + π ) = ±1/√2 .
4

17. Show that Tan ^‑1 1/8 + Tan ^‑1 1/2 + Tan ^‑1 1/5 = π/4.

SECTION – C

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.

FIRST YEAR MATHEMATICS – I B
(Blue print)

TYPE OF QUESTION	VSA	SA	LA	TOTAL
No. of questions to be given	10	7	7	24
No. of questions to be answered	10	5	5	20
Marks Allotted	20/20	20/28	35/49	75/97
Estimated time (In minutes)	10 x 4 = 40	5 x 9 = 45	5 x 18 = 90	175

S. No	Weightage to content: units / Sub-units:	marks
1	Locus	04
2	Change of Axes	04
3	Straight Lines	15
4	Pair of Straight Lines	14
5	3D- Coordinates	02
6	Direction cosines & Direction ratios – 3D Geometry	07
7	The Plane – 3D Geometry	02
8	Functions, Limits, Continuity	06
9	Differentiations	17
10	Errors – Approximations	02
11	Rate measure	04
12	Tangent – Normal	09
13	Maxima – Minima	07
14	Partial Differentiation	04
	Total	97

Scheme of Options:

Section – A: No Option

Section – B: 5 out of 7

Section – C: 5 out of 7