Question 1: If \( \sin \theta + \cos \theta = \frac{17}{15} \), find \( \tan \theta \).
We know that \( \sin^2 \theta + \cos^2 \theta = 1 \). Solve using the identity to find \( \tan \theta = 8 \).
Question 2: Evaluate \( \sin 60^\circ \cdot \cos 30^\circ + \sin 30^\circ \cdot \cos 60^\circ \).
The value is \( \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{1}{2} \cdot \frac{1}{2} = 1 \).
Question 3: Find the distance between the points (7, -5) and (1, 3).
The distance is \( \sqrt{(7 – 1)^2 + (-5 – 3)^2} = \sqrt{6^2 + (-8)^2} = 10 \) units.
Question 4: Find the coordinates of the centroid of a triangle with vertices (2, 3), (4, -5), and (6, 7).
The centroid is given by \( \left( \frac{2+4+6}{3}, \frac{3-5+7}{3} \right) = (4, \frac{5}{3}) \).
Question 5: Solve the system of equations: \( 3x + 2y = 11 \) and \( 2x – y = 3 \).
The solution is \( x = 3 \) and \( y = 1 \).
Question 6: Solve the equation: \( 7x – 3 = 5x + 9 \).
Rearranging and solving, \( 2x = 12 \), hence \( x = 6 \).
Question 7: A bag contains 3 red balls and 5 black balls. One ball is drawn randomly. What is the probability that the ball drawn is red?
The probability is \( \frac{3}{8} \).