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Federal (FBISE) · Chapter test · Free

2nd Year Mathematics Chapter 9 Test09: Solution of Trigonometric Equations

29 marks · 40 minutes · 20 questions. Covers: 09: Solution of Trigonometric Equations.

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2nd Year Mathematics Chapter 9 Test

2nd Year · Mathematics · Federal (FBISE)

Name: ____________Roll No: ________Class/Section: ________Date: ________Total marks: 29Time: 40 minutes

Q1. Multiple choice questions (Attempt all · 9 × 1 = 9)

  1. For what value of x∈(0,2π), sin⁡x=−12? \text{For what value of } x \in (0, 2\pi) \text{, } \sin x = -\frac{1}{2}?
    1. 7π6,π6 \frac{7\pi}{6}, \frac{\pi}{6}
    2. 7π6,11π6 \frac{7\pi}{6}, \frac{11\pi}{6}
    3. 5π6,11π6 \frac{5\pi}{6}, \frac{11\pi}{6}
    4. π6,5π6 \frac{\pi}{6}, \frac{5\pi}{6}
  2. If cos⁡2x=12 with x∈[0,π] then value of x is : \text{If } \cos^2 x = \frac{1}{2} \text{ with } x \in [0, \pi] \text{ then value of } x \text{ is :}
    1. π4,7π4 \frac{\pi}{4}, \frac{7\pi}{4}
    2. π4,3π4 \frac{\pi}{4}, \frac{3\pi}{4}
    3. 3π4,5π4 \frac{3\pi}{4}, \frac{5\pi}{4}
    4. 5π4,7π4 \frac{5\pi}{4}, \frac{7\pi}{4}
  3. What is the solution of sec⁡x=2, if x∈[0,π]? \text{What is the solution of } \sec x = 2 \text{, if } x \in [0, \pi]?
    1. {−π6} \left\{ -\frac{\pi}{6} \right\}
    2. {−π3} \left\{ -\frac{\pi}{3} \right\}
    3. {π3} \left\{ \frac{\pi}{3} \right\}
    4. {π6} \left\{ \frac{\pi}{6} \right\}
  4. One solution of a trigonometric equation cos⁡x+sin⁡x=2, if x∈[0,2π] ? \text{One solution of a trigonometric equation } \cos x + \sin x = \sqrt{2} \text{, if } x \in [0, 2\pi] \text{ ?}
    1. {π4} \left\{ \frac{\pi}{4} \right\}
    2. {−π4} \left\{ -\frac{\pi}{4} \right\}
    3. {2π4} \left\{ \frac{2\pi}{4} \right\}
    4. {3π2} \left\{ \frac{3\pi}{2} \right\}
  5. If sin⁡(2x)=32 with x∈[0,2π], then value of x is: \text{If } \sin(2x) = \frac{\sqrt{3}}{2} \text{ with } x \in [0, 2\pi] \text{, then value of } x \text{ is:}
    1. π6,π3 \frac{\pi}{6}, \frac{\pi}{3}
    2. π6,5π12 \frac{\pi}{6}, \frac{5\pi}{12}
    3. π6,5π6 \frac{\pi}{6}, \frac{5\pi}{6}
    4. π6,2π3 \frac{\pi}{6}, \frac{2\pi}{3}
  6. Solution set of a trigonometric equation 2sin⁡x+3=0 in the 4th quadratic is : \text{Solution set of a trigonometric equation } 2\sin x + \sqrt{3} = 0 \text{ in the } 4^{\text{th}} \text{ quadratic is :}
    1. -π6 \frac{\pi}{6}
    2. -π3 \frac{\pi}{3}
    3. -5π6 \frac{5\pi}{6}
    4. 11π6 \frac{11\pi}{6}
  7. What is the solution of sin⁡x+cos⁡x=0, if x∈[0,π]? \text{What is the solution of } \sin x + \cos x = 0 \text{, if } x \in [0, \pi]?
    1. {π4} \left\{ \frac{\pi}{4} \right\}
    2. {π2} \left\{ \frac{\pi}{2} \right\}
    3. {2π3} \left\{ \frac{2\pi}{3} \right\}
    4. {3π4} \left\{ \frac{3\pi}{4} \right\}
  8. What is the general solution of a trigonometric equation 1 + cos x = 0 ? Where n is an integer.
    1. (π+nπ)\left({\pi} + n\pi \right)
    2. (π2+nπ)\left( \frac{\pi}{2} + n\pi \right)
    3. (π+2nπ)\left({\pi} + 2n\pi \right)
    4. (2nπ)\left(2 n\pi \right)
  9. For what value of x, a trigonometric equation sin x = cos x remains true ?
    1. π4,3π4 \frac{\pi}{4}, \frac{3\pi}{4}
    2. π4,5π4 \frac{\pi}{4}, \frac{5\pi}{4}
    3. 3π4,5π4 \frac{3\pi}{4}, \frac{5\pi}{4}
    4. π4 \frac{\pi}{4}

Q2. Short questions (Attempt any 6 of 8 · 6 × 2 = 12)

  1. Solve graphically the trigonometric equation : sin⁡θ=θ2\sin \theta = \frac{\theta}{2} where θ∈[−π2,π2]\theta \in \left[ -\frac{\pi}{2}, \frac{\pi}{2} \right]
  2. Solve the following trigonometric equations graphically. sin⁡(θ2)=θ;θ∈[−π2,π2]\sin\left(\frac{\theta}{2}\right) = \theta; \quad \theta \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right]
  3. Solve tan⁡(θ2)=1\tan\left( \frac{\theta}{2}\right) = 1 for θ\theta
  4. Solve the following trigonometric equations for the values of θ in the range 0∘≤ θ ≤ 360∘: cos5θ + cosθ = cos3θ
  5. Solve : sinxcosx=14sin x cos x =\frac{1}{4} where x∈[0,π] x \in [0, \pi] .
  6. Solve the following trigonometric equations when Ø ∈ [0 , 2π]: cosØ + cos3Ø = 0
  7. A mechanical arm in a factory is required to reach an object 5 meters away and 2 meters above its base. Find the angle at which the arm should be positioned using the suitable inverse function.
  8. Solve the trigonometric equations within the domain [0 , 2π]. cosx = 0.71

Q3. Long questions (Attempt any 2 of 3 · 2 × 4 = 8)

  1. Find the general solution, in radians, of the following equations: cos4x + cos6x = cos5x
  2. Solve the following trigonometric equations when Ø ∈ [0 , 2π]: cos² Ø - 2 cosØ + 1 = 0
  3. A lighthouse is 120 meters tall, and a ship is spotted at a distance of 250 meters from the base of the lighthouse. Find the angle of depression from the lighthouse to the ship using the inverse tangent function.