2005 Andhra Pradesh State Maths 1A Question paper
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 (1x).
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 a )/a = (sin a)/b , show that a cos 2a + b sin 2a = a.
6. Draw the graph of tan x between 0 and p/2.
7. Show that Tanh1 (1/2) =(1/2)loge3.
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 cos5? = cos 5? + 5 cos 3? + 10 cos ?.
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 ? 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 ?. For which the volume of the parallelepiped having coterminous edges I + j, 3i + ?k is 16 cubic units.
16. in the interval 0 = ? = p/2 , solve sin ? + sin 4? + sin 7 ? = 0.
17. If sin 1 (x) + sin 1 (y) + sin 1 ( z) = p, prove that x v( 1 – x 2) + y (v1 – y 2 ) + z (v1 – 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 = IB 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 ( a – ß) = cos a cos ß + sin a + sin ß
21. If A + B + C = 180o , 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 r3)/bc =( r2r + r2r1) /ca = (r3r + r1 r2)/ 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 a, ß, and ? respectively. If AB = a, AC = b, AD = c, AP = x and a + ß + ? = 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.
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