1.         sinø - 2 sin³ø
            --------------------  =  tanø
            2 cos³ø - cosø

 

2. Solve by completing the squares: a²x² - 3abx + 2b² = 0

     Divide whole equation by a² 

      x² - 3(b/a)x + 2b²/a² = 0

      Adding and subtracting (-3b/2a)² 

     (a²x² - 3(b/a)x + 9b²/4a²) - 9b²/4a² + 2b²/a² = 0

     [x - 3b/2a)]² - b²/4a² = 0
    (x - 3b/2a + b/2a)(x - 3b/2a - b/2a) = 0      [Using a² - b² = (a + b)(a - b)]

     (x - b/a)(x - 2b/a) = 0
      x = b/a  OR x = 2b/a  Ans.

3. What is the sum of all integers between 100 & 1000 which are neither divisible by 2 nor 5?

HINT 

Let S₁ = 101 + 102 + 103 + .... + 999

       S₂ = 102 + 104 + 106 + .... + 998

       S₃ = 105 + 110 + 115 + .... + 995

       S₄ = 110 + 120 + 130 + .... + 990 (Terms divisible by lcm of 2 and 5 i.e. 10)

 

Now, sum of numbers divisible by 2 or 5, S_d = S₂ + S₃ - S₄

To find S₂, Put 998 as nth term

a_n = a + (n - 1)d = 998           =>  102 +(n - 1)2 = 998    => n = 449

and obtain the values as 494450, 246950, 98450 and 48950 respectively.

 

Now, sum of numbers divisible by 2 or 5, S_d = S₂ + S₃ - S₄