1) In any Δ𝐴𝐵𝐶 , if the angles are in the ratio 1:2:3, prove that the corresponding sides are in the ratio 1: √3 : 2.
2) If a cosA = b cosB, then prove that the triangle is either isosceles or right angled. 𝑠𝑖𝑛𝐶
3) In a Δ𝐴𝐵𝐶 , if cosC =
2 𝑠𝑖𝑛𝐵
, prove that the triangle is isosceles.
4) Solve √3 cos 𝜃 + sin 𝜃 = √2 .
5) Prove that : 𝑠𝑒𝑐8𝜃−1 𝑠𝑒𝑐4𝜃−1
= tan 8𝜃 tan 4𝜃
6) Show that :
√2 + √2 + √2 + 2 cos 8𝜃
= 2 cos 𝜃
7) Prove that : sin 20° sin 40° sin 60° sin 80° =
3
16
8) A horse is tied to a post by a rope. If the horse moves along a circular path, always keeping the rope tight and describes 88 metres when it traces 72 ° at the centre, find the length of the rope.
9) If the angles of a triangle are in the ratio 3:4:5, find the smallest angle in degrees
10) and the greatest angle in radians.
Prove that tan 50 ° = tan 40° + 2 tan 10°
11)
12)
13)
Draw the graph of cos x in [0,2 𝜋]
Draw the graph of the function y = 2 cos 𝑥
2
, -2 𝜋 ≤ 𝑥 ≤ 2𝜋
If 2 sin
𝐴
2 sin
𝐶
2
= sin
𝐵
2
, then prove that a,b,c are in A.P.
In a triangle ABC, the angles A,B,C are in A.P., show that 14)
2 cos
𝐴−𝐶
2
=
√𝑎 2 𝑎+𝑐
−𝑎𝑐+ 𝑐 2

1) Find the real values of x and y if, (x + iy) (2-3i) = 4 + i
2) Prove that 𝑥 4 + 4 = (x + 1 + i) (x + 1- i) (x - 1 + i) (x - 1 - i)
3) If i 𝑧 3 + 𝑧 2 – z + I = 0, then show that |𝑧| = 1.
4) Evaluate √−16 + 3 √−25 + √−36 - √−625
5) If a = cos 𝜃 + 𝑖 sin 𝜃 , then show that
1+𝑎
1−𝑎
= i cot 𝜃
2
6) Show that
√7+ √3𝑖
√7− √3𝑖
+
√7− √3𝑖
√7+ √3𝑖
is purely real.
7) For what values of x and y are the numbers 3 + i 𝑥 2 y and 𝑥 2 + y + 4i conjugate complexes? (x,y are reals)
8) Where does z lie, if | 𝑧−5𝑖 𝑧+5𝑖
| = 1?
9) Find the smallest positive integer n for which (
1+𝑖
√2
) 𝑛
= 1
10) Find the complex number which when multiplied by 2 + 5i gives 3 – 7i.
11)
13)
Solve the quadratic equation (𝑥 2 − 5𝑥 + 7) 2 – (x-2) (x-3) = 7
12) Find x ∈ 𝑅 that satisfies the equation
1−𝑖𝑥
1+𝑖𝑥
= a – ib, where a,b ∈ R and 𝑎 2 +
𝑏 2 = 1.
Find the value of tan 𝜃 , if 𝜃 is the amplitude of 𝑎+𝑖𝑏 𝑎−𝑖𝑏
.