MATHS IPE

INTER 1st YEAR Maths 1a june 2005

SECTION – A

VERY SHORT ANSWER TYPE QUESTIONS
Answer all questions. Each question carries 2 marks.

1. Find the domain of the real function f(x) = 1/ log (1-x).

2. Show that the points -2a + 3b + 5dc, a + 2b + 3c, 7a – c are collinear where a, b, c are three non – coplanar vectors.

3. Fine the ratio in which I + 2j + 3k divides the join of -2i + 3j + 5k and 7i – k.

4. If a = 2i + tj –k and b = 4i – 2j + 2k , find the value of t, so that a and b are perpendicular.

5. if (cos α )/a = (sin α)/b , show that a cos 2α + b sin 2α = a.

6. Draw the graph of tan x between 0 and π/2.

7. Show that Tanh^-1 (1/2) =(1/2)log_e3.

8. If tan (A/2) = 5/6 and tan (C/2) = 2/5, determine the relation between a, b, c.

9. Find all the values of (1 + i )^1/2

10. Show that 2⁴ cos⁵θ = cos 5θ + 5 cos 3θ + 10 cos θ.

SECTIN – B

SHORT ANSWER TYPE QUESTIONS
Attempt any 5 questions. Each question carries 4 marks.

11. If Q is the set of all rational numbers, and f : Q → Q is defined by f(x) = 5x + 4, where x є Q, show that f is a bijection.

12. If x = 3 + 3 ^1/3 + 3 ^2/3 , then show that x³ – 9x² + 18x – 12 = 0.

13. If x = log abc, y = log bca and z = log cab, then show that

1/((x+1) + 1/(y + 1) + 1 / ( z + 1) = 1

14. Prove by vector method that x/a + y/b = 1 is the equation of a straight line in intercept from.

15. Determine the value of λ. For which the volume of the parallelepiped having coterminous edges I + j, 3i + λk is 16 cubic units.

16. in the interval 0 ≤ θ ≤ π/2 , solve sin θ + sin 4θ + sin 7 θ = 0.

17. If sin -1 (x) + sin -1 (y) + sin -1 ( z) = π,
prove that x √( 1 – x ²) + y (√1 – y ² ) + z (√1 – z ² ) = 2 xyz.

SECTION C

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

18. if f : A → B is a bijection. Then show that f o f ^-1 = I_B and f ^-1 o f = I _A.

19. Show that 49 ⁿ + 16 n – 1 is divisible by 64 for all positive integral values of n.

20. In a ∆ ABC ,using vector method, prove that ( α – β) = cos α cos β + sin α + sin β

21. If A + B + C = 180^o , then prove that

sin 2 (A/2) + sin ²(B/2) + sin ²(C/2) = 1 – 2sin (A/2) sin (B/2) sin (C/2).

22. In a ∆ ABC, prove that

r₁r + r₂ r_3)/bc =( r₂r + r₂r_{1) /ca =} (r₃r + r₁ r_{2)/ ab}

23. A building with three floors is build vertically on the level ground ( AB, BC, CD are the three floors respectively). From a point P, x units away from the foot of the building on the ground, the angles of elevations of B, C, D are α, β, and γ respectively. If AB = a, AC = b, AD = c, AP = x and α + β + γ = 180 ^o. then show that ( a + b + c) x2 = abc.

24. Show that the points in the Argand diagram represented by the complex numbers -2 + 7 i, -3/2 + 1/2 i , 4 – 3i , 7( 1 + i ) /2 are the vertices of a rhombus.

FIRST YEAR MATHS 1A BLUE PRINT

distribution of marks

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