If and then is equal to

13 Explanation: To find the value of , we need to eliminate from the given equations. The most effective method is to square both equations and then add them together. 1. Square the First Equation Given: Squaring both sides: Using the identity : 2. Square the Second Equation Given: Squaring both sides: Using the identity : 3. Add Equation 1 and Equation 2 Adding the left-hand sides and the right-hand sides: The cross-product terms ( and ) cancel each other out: 4. Factor Out and Since we know from trigonometric identities that : Correct Option: (C)

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NIMCET 2021 — Mathematics PYQ

NIMCET | Mathematics | 2021

If $α cos θ + b sin θ = 2$ and $α$ $sin θ - b cos θ = 3$ then $a^{2} + b^{2}$ is equal to

Choose the correct answer:

Explanation

To find the value of $a^2 + b^2$ , we need to eliminate $\theta$ from the given equations. The most effective method is to square both equations and then add them together.

1. Square the First Equation

Given: $a\cos\theta + b\sin\theta = 2$

Squaring both sides:

$(a\cos\theta + b\sin\theta)^2 = 2^2$

Using the identity $(x+y)^2 = x^2 + y^2 + 2xy$ :

$a^2\cos^2\theta + b^2\sin^2\theta + 2ab\sin\theta\cos\theta = 4 \quad \dots \text{(Equation 1)}$

2. Square the Second Equation

Given: $a\sin\theta - b\cos\theta = 3$

Squaring both sides:

$(a\sin\theta - b\cos\theta)^2 = 3^2$

Using the identity $(x-y)^2 = x^2 + y^2 - 2xy$ :

$a^2\sin^2\theta + b^2\cos^2\theta - 2ab\sin\theta\cos\theta = 9 \quad \dots \text{(Equation 2)}$

3. Add Equation 1 and Equation 2

Adding the left-hand sides and the right-hand sides:

$(a^2\cos^2\theta + b^2\sin^2\theta + 2ab\sin\theta\cos\theta) + (a^2\sin^2\theta + b^2\cos^2\theta - 2ab\sin\theta\cos\theta) = 4 + 9$

The cross-product terms ( $2ab\sin\theta\cos\theta$ and $-2ab\sin\theta\cos\theta$ ) cancel each other out:

$a^2\cos^2\theta + a^2\sin^2\theta + b^2\sin^2\theta + b^2\cos^2\theta = 13$

4. Factor Out $a^2$ and $b^2$

$a^2(\cos^2\theta + \sin^2\theta) + b^2(\sin^2\theta + \cos^2\theta) = 13$

Since we know from trigonometric identities that $\sin^2\theta + \cos^2\theta = 1$ :

$a^2(1) + b^2(1) = 13$

$a^2 + b^2 = 13$

Correct Option: (C)

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