Question 1: Prove that \( \frac{\sin A – \cos A}{\sin A + \cos A} = \frac{1 – \tan A}{1 + \tan A} \).
To prove this, divide both the numerator and denominator by \( \cos A \), then simplify using trigonometric identities to get the desired result.
Question 2: If \( \sec \theta + \tan \theta = x \), prove that \( \sec \theta = \frac{x^2 + 1}{2x} \).
Square both sides of the given equation and simplify using the identity \( \sec^2 \theta – \tan^2 \theta = 1 \) to derive the result.
Question 3: Solve the system of equations by substitution: \( 3x – 4y = -5 \) and \( 2x + y = 7 \).
From the second equation, express \( y \) in terms of \( x \), substitute into the first equation, and solve for \( x \). Then use this value to find \( y \).
Question 4: Solve the pair of equations: \( 2x + 3y = 9 \) and \( 4x – y = 3 \) using the elimination method.
Multiply the second equation by 3 and eliminate \( y \). Solve for \( x \), and then substitute back to find \( y \).
Question 5: A box contains 6 white balls and 4 black balls. Two balls are drawn randomly without replacement. What is the probability that both balls are white?
The probability is \( \frac{6}{10} \times \frac{5}{9} = \frac{1}{3} \).