Match the following. List-I List-II (a) If sin θ = 3 sin ( θ + 2 α ) , then the value of tan ( θ + α ) + 2 tan α is (p) 0 (b) If p sin θ + q cos θ = a and p cos θ − q sin θ = b , then q + b p + a +

Question

Match the following. List-I List-II (a) If sin θ = 3 sin ( θ + 2 α ) , then the value of tan ( θ + α ) + 2 tan α is (p) 0 (b) If p sin θ + q cos θ = a and p cos θ − q sin θ = b , then q + b p + a ​ + p − a q − b ​ + 1 is equal to (q) 1 (c) The value of the expression cos 7 π ​ cos 7 2 π ​ cos 7 10 π ​ − sin 14 π ​ sin 14 3 π ​ sin 14 5 π ​ is (r) sec θ (d) If sec θ + tan θ = 1 , then one root of the equation ( a − 2 b + c ) x 2 + ( b − 2 c + a ) x + ( c − 2 a + b ) = 0 is (s) − 4 1 ​ (t) − 2 1 ​ (One or more entries can match more than one option.)

TestHub · Accepted Answer

(a) sin ( θ + α − α ) = 3 sin ( θ + α + α ) ⇒ − 2 sin ( θ + α ) cos α = 4 cos ( θ + α ) sin α ⇒ − tan ( θ + α ) = 2 tan α , so tan ( θ + α ) + 2 tan α = 0 ( p ) . (b) Squaring and adding the two equations: p 2 + q 2 = a 2 + b 2 ⇒ ( p + a ) ( p − a ) + ( q + b ) ( q − b ) = 0 ⇒ q + b p + a ​ + p − a q − b ​ = 0 , so the expression = 1 ( q ) . (c) cos 7 10 π ​ = cos 7 4 π ​ = − cos 7 3 π ​ , so cos 7 π ​ cos 7 2 π ​ cos 7 10 π ​ = − 8 1 ​ ; also sin 14 π ​ sin 14 3 π ​ sin 14 5 π ​ = cos 7 3 π ​ cos 7 2 π ​ cos 7 π ​ = 8 1 ​ . Value = − 8 1 ​ − 8 1 ​ = − 4 1 ​ ( s ) . (d) Sum of coefficients = 0 , so x = 1 is a root (q); sec θ + tan θ = 1 ⇒ sec θ − tan θ = 1 ⇒ sec θ = 1 , so x = sec θ is the same root (r).

List-I	List-II
(a) If $sin θ = 3 sin (θ + 2 α)$ , then the value of $tan (θ + α) + 2 tan α$ is	(p) 0
(b) If $p sin θ + q cos θ = a$ and $p cos θ - q sin θ = b$ , then $\frac{p + a}{q + b} + \frac{q - b}{p - a} + 1$ is equal to	(q) 1
(c) The value of the expression $cos \frac{π}{7} cos \frac{2 π}{7} cos \frac{10 π}{7} - sin \frac{π}{14} sin \frac{3 π}{14} sin \frac{5 π}{14}$ is	(r) $sec θ$
(d) If $sec θ + tan θ = 1$ , then one root of the equation $(a - 2 b + c) x^{2} + (b - 2 c + a) x + (c - 2 a + b) = 0$ is	(s) $- \frac{1}{4}$
	(t) $- \frac{1}{2}$

Mathematics - Trigonometric Ratios & Identities Question with Solution | TestHub

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