FSc Pre-Engineering Mathematics: Trigonometric Equations MCQs

FSc Pre-Engineering Mathematics: Trigonometric Equations MCQs — sample questions

1. Question 1

   Q1. The general solution of sin(x) = 0 is

   • A) nπ
   • B) 2nπ
   • C) (2n + 1)π/2
   • D) (n + 1/2)π

   Answer: nπ

   Explanation: The sine function equals zero at integer multiples of π, hence the general solution is nπ.

2. Question 2

   Q2. The general solution of cos(x) = 0 is

   • A) nπ
   • B) 2nπ
   • C) (2n + 1)π/2
   • D) (n + 1/2)π

   Answer: (2n + 1)π/2

   Explanation: The cosine function equals zero at odd multiples of π/2, hence the general solution is (2n + 1)π/2.

3. Question 3

   Q3. If sin(x) = 1/2, then the general solution is

   • A) nπ + (-1)^n π/6
   • B) 2nπ + π/6
   • C) nπ + π/3
   • D) 2nπ + π/3

   Answer: nπ + (-1)^n π/6

   Explanation: Using the formula for sin(x) = sin(α), the general solution is nπ + (-1)^n α, where α = π/6.

4. Question 4

   Q4. The number of solutions of tan(x) = 1 in the interval [0, 2π] is

   • A) 1
   • B) 2
   • C) 3
   • D) 4

   Answer: 2

   Explanation: tan(x) = 1 has solutions x = π/4 and 5π/4 in the given interval.

5. Question 5

   Q5. The general solution of 2cos²(x) - 1 = 0 is

   • A) nπ ± π/4
   • B) 2nπ ± π/3
   • C) nπ ± π/3
   • D) 2nπ ± π/4

   Answer: 2nπ ± π/4

   Explanation: Using the identity cos(2x) = 2cos²(x) - 1, we get cos(2x) = 0, hence 2x = (2n + 1)π/2, giving x = nπ ± π/4, but in the form 2nπ ± π/4 for a specific n.

6. Question 6

   Q6. The solution of sin(2x) = sin(x) is

   • A) 2nπ/3
   • B) nπ
   • C) 2nπ or nπ + (-1)^n π/3
   • D) nπ or 2nπ ± π/3

   Answer: 2nπ or nπ + (-1)^n π/3

   Explanation: Using the identity sin(a) = sin(b), we have 2x = nπ + (-1)^n x, giving two sets of solutions.

7. Question 7

   Q7. For the equation tan(3x) = cot(x), the general solution is

   • A) nπ/4 + π/8
   • B) nπ/2 + π/4
   • C) (2n + 1)π/8
   • D) nπ + π/4

   Answer: nπ/4 + π/8

   Explanation: tan(3x) = cot(x) gives tan(3x)tan(x) = 1, which simplifies to a form that leads to the general solution nπ/4 + π/8.

8. Question 8

   Q8. The general solution of the equation cosec(x) = -√2 is

   • A) nπ + (-1)^n (5π/4)
   • B) nπ + (-1)^n (π/4)
   • C) 2nπ + (5π/4)
   • D) 2nπ - (π/4)

   Answer: nπ + (-1)^n (5π/4)

   Explanation: cosec(x) = -√2 implies sin(x) = -1/√2, so x = nπ + (-1)^n (5π/4).

9. Question 9

   Q9. The number of solutions of the equation sin(x) + cos(x) = √2 in the interval [0, 2π] is

   • A) 0
   • B) 1
   • C) 2
   • D) 4

   Answer: 1

   Explanation: Dividing by √2, we get sin(x + π/4) = 1, which has one solution in the given interval.

10. Question 10

    Q10. The general solution of 3tan²(x) = 1 is

    • A) nπ ± π/6
    • B) nπ ± π/3
    • C) 2nπ ± π/6
    • D) nπ + (-1)^n π/6

    Answer: nπ ± π/6

    Explanation: 3tan²(x) = 1 gives tan(x) = ±1/√3, hence x = nπ ± π/6.

11. Question 11

    Q11. The general solution of sin(3x) + sin(x) = 0 is

    • A) nπ/2
    • B) nπ
    • C) 2nπ
    • D) nπ/4

    Answer: nπ

    Explanation: Using sum-to-product identity, we simplify to 2sin(2x)cos(x) = 0, which gives solutions nπ and nπ/2 ± π/4, but the correct general form simplifies to nπ.

12. Question 12

    Q12. For the equation 2sin(x)cos(x) = sin(x), the general solution is

    • A) nπ or nπ + (-1)^n π/6
    • B) nπ
    • C) nπ or 2nπ ± π/3
    • D) nπ (2)

    Answer: nπ (2)

    Explanation: Factoring out sin(x), we get sin(x)(2cos(x) - 1) = 0, giving sin(x) = 0 or cos(x) = 1/2, hence the general solution is nπ.

13. Question 13

    Q13. The general solution of the equation cos(2x) = cos(x) is

    • A) 2nπ
    • B) 2nπ/3
    • C) 2nπ or 2nπ/3
    • D) nπ/3

    Answer: 2nπ or 2nπ/3

    Explanation: Using the identity cos(a) = cos(b), we have 2x = 2nπ ± x, giving two sets of solutions: 2nπ and 2nπ/3.

14. Question 14

    Q14. The number of solutions of sin(x) = sin(π/3) in the interval [0, 2π] is

    • A) 1
    • B) 2
    • C) 3
    • D) 4

    Answer: 2

    Explanation: sin(x) = sin(π/3) has solutions x = π/3 and 2π - π/3 = 2π/3 in the given interval.

15. Question 15

    Q15. For the equation tan(x) + tan(2x) + tan(3x) = 0, the general solution is

    • A) nπ/3
    • B) nπ/2
    • C) nπ/4
    • D) nπ

    Answer: nπ

    Explanation: Using the tangent sum identity, the equation simplifies to a form that leads to the general solution nπ.

16. Question 16

    Q16. The general solution of cos(x) + cos(3x) + cos(5x) = 0 is

    • A) nπ/3 + π/6
    • B) (2n + 1)π/6
    • C) nπ/3 or (2n + 1)π/2
    • D) nπ

    Answer: nπ/3 or (2n + 1)π/2

    Explanation: Using sum-to-product identities, we simplify the equation, which gives solutions nπ/3 and (2n + 1)π/2.

17. Question 17

    Q17. The general solution of sin(x) + sin(3x) + sin(5x) = 0 is

    • A) nπ/3
    • B) nπ/5
    • C) nπ/3 or (2n + 1)π/6
    • D) nπ

    Answer: nπ/3 or (2n + 1)π/6

    Explanation: Using sum-to-product identity, we simplify the equation to a form that gives the general solution.

18. Question 18

    Q18. The number of solutions of cos(x) = 1/2 in the interval [0, 2π] is

    • A) 1
    • B) 2
    • C) 3
    • D) 4

    Answer: 2

    Explanation: cos(x) = 1/2 has solutions x = π/3 and 5π/3 in the given interval.

19. Question 19

    Q19. For the equation sin(2x) + sin(4x) + sin(6x) = 0, the general solution is

    • A) nπ/4
    • B) nπ/3
    • C) (2n + 1)π/4
    • D) nπ/2 or (2n + 1)π/4

    Answer: nπ/2 or (2n + 1)π/4

    Explanation: Using sum-to-product identity, the equation simplifies to a form that gives the general solution nπ/2 and (2n + 1)π/4.

20. Question 20

    Q20. The general solution of 2sin²(x) + sin²(2x) = 2 is

    • A) nπ ± π/4
    • B) nπ/2
    • C) nπ + (-1)^n π/4
    • D) nπ

    Answer: nπ

    Explanation: Simplifying the equation using trigonometric identities leads to the general solution nπ.