Trigonometry
Matrix match — trig equations counting solutions
MJAT_TS6_P1
Grade 12
Question:
Match List-I (number of solutions in given intervals) with List-II (counts):
P) $\cos3x\cos6x=\cos4x\cos7x$, $x\in[0,\pi/2)$
Q) $\sin2x\sin6x=\cos x\cos3x$, $x\in[0,\pi/2]$
R) $\cos3x\cos7x=\cos2x\cos8x$, $x\in[0,\pi/2]$
S) $\sin5x\cos3x=\sin6x\sin2x$, $x\in[0,\pi/2)$
List-II: 1)1, 2)2, 3)3, 4)4, 5)5
A) P-1, Q-4, R-5, S-3
B) P-5, Q-4, R-3, S-2
C) P-4, Q-5, R-2, S-3
D) P-1, Q-5, R-3, S-2
Step-by-Step Solution
Key Concept: Use product-to-sum: $\cos A\cos B=\frac{1}{2}[\cos(A-B)+\cos(A+B)]$. Each equation reduces to $\cos(\text{something})=0$ or $\sin(\text{something})=0$, giving solutions at multiples of $\pi/k$. Count solutions in the given interval.
Answer: **B**.
Correct Answer: B