INTERMEDIATE PUBLIC EXAMINATION

INTERMEDIATE PUBLIC EXAMINATION

MARCH 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 inverse of the function f(x) = 5^x.

2. Find the value of cos²45 – sin ²15.

3. Prove that sin (π/5). Sin (2π/5). Sin (3π/5) Sin (4π/5) = 5/16.

4. Prove that (cosh x – sinh x )ⁿ = cosh nx – sinh nx.

5. In a triangle ABC, So that ( b – a cos C ) sin A = a cos A sin C.

6. Simplify

( cos θ – i sin θ)⁷
(Sin 2θ – i cos 2θ)⁴

7. Show that sin 6θ = 6 cos⁵ θ sin θ – 20 cos³ sin³ θ + 6 cos θ sin ⁵ θ.

8. The position vector of the two points A and B are i + j + k
and i + j – 4 k respectively. Find the position vector of the point
which divides AB in the ratio 2: 3 internally.

9. Find the vector equation of the plane passing through the points
(1, -2, 5) (0, -5, -1) (-3. 5. 0)

10. Find the angle made by the straight line through the points
(1, -3, 2) and ( 3, -5, 1) with the x – axis.

SECTION – B

SHORT ANSWER TYPE QUESTIONS.

Attempt any 5 questions, Each question carries 4 marks.

11. A functioned is defined as follows

3x – 2 if x > 3

f ( x) = x² – 2 if -2 ≤ x ≤ 2.

2x + 1 if x < - 3

then find the values of f(4) , f( 2.5) f(-2) , f( 0).

12.Show that

__1____ _ ______1___ _ ____2_____ = 0
√(12- √140) √(8 – √60) √ (10+ √ 84)

13. If (3.7)^x = (0.037)y = 10,000, then find the value of 1/x – 1/y.

14. solve the equation tan θ + tan 2θ + √3 tan θ tan 2θ = √3

15. If sin ^{– 1} x + sin ^–1 y + sin ^–1 z = π, prove that
x √ (1 – x²) + y√ (1 – y²) + z√ (1-z²) = 2 xyz.

16. if A = (2, 4, -1) B = ( 4, 5, 1) and C = ( 3, 6, -3 ) are the vertices
of a triangle ABC, find the lengths of the sides and
show that it is a right angled triangle.

17.Find the area of triangle with the points A ( 1,2,3) B ( 2, 3, 1)
and C ( 3,1,2)

SECTION – C

LONG ANSWER TYPE QUESTIONS
Attempt any 5 questions, each question carries 7 marks.

18. If f: A → B , and g: B → C , are two bijective functions, then
prove that ( g o f ) ^-1 = f ^-1 o g ^-1 .

19, By using mathematical induction show that
1/1.4 + 1/ 4.7 + 1/ 7. 10 + ………… upto n terms. = n / ( 3n + 1 )
for all n є N .

20. If A + B + C = 180 ^o then, prove that
sin ² A/2 + sin ² B/2 – sin ² C/2 = 1 – cos A/2 cos B/2 cos C/2.

21. in ∆ ABC, show that r₁ + r₂ + r₃ = 4 R.

22. from the top of a tree on the bank of a lake, an aeroplane
in the sky makes an angle of elevation ‘α’ and its image in the
river makes an angle of depression ‘ β’ . If the height of the tree
from the water surface is ‘a’ and that of the height of the aeroplane h,
show that h = a sin (α + β) /sin(β-α).

23.a) if (x+iy)=1/1+cosθ+I sin θ,then show that 4x²=1.

b) if the point ‘P’ denotes the complex number z=x+iy
in the argand plane and if z-i/z-1 is a purely imaginary number,
find the locus of ‘P’.

24.By vector method , prove that ∆=√{s(s-a)(s-b)(s-c)}.