Let α be the angle between the lines whose direction cosines satisfy the equations l + m - n = 0 and l² + m² - n² = 0. Then the value of sin⁴α + cos⁴α is :

Solution

Maths sir · 4 steps

Substitute n = l + m into the square equation, and it splits neatly into two lines.

1. Eliminate n. The first equation gives n = l + m. Put this into the second: l² + m² = (l + m)², so 2lm = 0.
2. Two cases. If l = 0, then n = m, giving direction ratios (0, 1, 1). If m = 0, then n = l, giving direction ratios (1, 0, 1). These are the two lines.

   cos α = (0 × 1 + 1 × 0 + 1 × 1)/(√2 × √2) = 1/2

3. Find the value. So α = 60°, with cos² α = 1/4 and sin² α = 3/4. Then sin⁴ α + cos⁴ α = 9/16 + 1/16 = 10/16.
4. So. sin⁴ α + cos⁴ α = 5/8, which is option (d).

The idea to keep. When one condition is linear and one is quadratic, use the linear one to remove a variable. The quadratic then usually factors into the two lines. Direction ratios are enough to find the angle.

Answer from the NTA final answer key. Solution by PrepWiser.