Test paper -2(J++)

1) In the expansion of (√x - √k/x²)¹⁰, the terms independent of x be 405: find the value of k.

2) If the equation x²+ bx+ ca=0 and x²+ cx+ ab=0 have a common root, prove that their other roots will satisfy the equation x²+ ax+ bc=0.

3) If x - 1/x = 2i sink, show that x⁴- 1/x⁴ = 2i sin 4k.

4) Prove tan6 tan42 tan66 tan78=1.

Or

Show that tan20 tan40 tan80= tan60

5) If a cosk + b sink= a cosx + b sinx, show that sin(k+ x)= 2ab/(a²+ b²)

6) Solve: sin⁸x + cos⁸x =17/32

7) If a²(1- sinx) + b²(1+ sinx) 2ab cosx, show that a/b - b/a = 2tanx.

8) y= mx is the equation of a chord of the circle x²+ y²-2ax=0 . Show that the equation of the circle on this chord as diameter is (1+ m²)(x²+ y²) - 2a(x+ my)=0

9) The points (1,3) and (5,1) are two opposite vertices of a rectangle. The other two vertices lie on the line y= 2x + c. Find c and the remaining vertices.

10) P is a variable point on the hyperbola x² - y²=a and A is the fixed point (2a,0). Show that the locus of the midpoint of the line segment AP is another hyperbola.

11) Given the ellipse 4x²+ 9y²=36, find the equation of the chord which is bisected at (2,-1).

12) A function f(x) is defined as follows:

f(x)= 2 - |x|/x, when -2 ≤ x ≤2