NTS NAT-IE (Engineering Track) Mathematics Trigonometric Functions — Set 2

NTS NAT-IE (Engineering Track) Mathematics Trigonometric Functions — Set 2

1. Question 1

   Q1. What is the amplitude of y = 2sin(x)?

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

   Answer: 2

   Explanation: The amplitude of y = asin(x) is |a|, so for y = 2sin(x), it is 2.

2. Question 2

   Q2. For y = sin(x) + cos(x), what is the maximum value?

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

   Answer: √2

   Explanation: Using the identity sin(A+B), the max value is √(1²+1²) = √2.

3. Question 3

   Q3. What is the range of y = 2 + sin(x)?

   • A) [1, 3]
   • B) [0, 2]
   • C) [-1, 1]
   • D) (-∞, ∞)

   Answer: [1, 3]

   Explanation: The range of sin(x) is [-1, 1], so for 2 + sin(x), it is [1, 3].

4. Question 4

   Q4. The graph of y = sin(-x) is?

   • A) Same as y = sin(x)
   • B) Reflection of y = sin(x) about x-axis
   • C) Reflection of y = sin(x) about y-axis
   • D) None of these

   Answer: Reflection of y = sin(x) about x-axis

   Explanation: sin(-x) = -sin(x), which is a reflection of y = sin(x) about the x-axis.

5. Question 5

   Q5. What is the value of sin(π/4)?

   • A) 1/√2
   • B) 1/2
   • C) √3/2
   • D) 1

   Answer: 1/√2

   Explanation: sin(π/4) = sin(45°), which is 1/√2.

6. Question 6

   Q6. For y = cos(2x), what is the period?

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

   Answer: π

   Explanation: The period of cos(ax) is 2π/a, so for cos(2x), it is 2π/2 = π.

7. Question 7

   Q7. The minimum value of y = cos(x) is?

   • A) -1
   • B) 0
   • C) 1
   • D) 1/2

   Answer: -1

   Explanation: The minimum value of cos(x) is -1, as it oscillates between -1 and 1.

8. Question 8

   Q8. What is the domain of y = tan(x)?

   • A) (-∞, ∞)
   • B) (-∞, ∞) - π/2 + kπ
   • C) (-∞, ∞) - kπ
   • D) [0, π]

   Answer: (-∞, ∞) - π/2 + kπ

   Explanation: y = tan(x) is undefined when x = π/2 + kπ, so the domain excludes these points.

9. Question 9

   Q9. For y = 3cos(x), what is the amplitude?

   • A) 1
   • B) 3
   • C) 1/3
   • D) 0

   Answer: 3

   Explanation: The amplitude of y = acos(x) is |a|, so for y = 3cos(x), it is 3.

10. Question 10

    Q10. The function y = sin(x) is increasing in which interval?

    • A) [0, π/2]
    • B) [π/2, π]
    • C) [π, 3π/2]
    • D) [3π/2, 2π]

    Answer: [0, π/2]

    Explanation: y = sin(x) is increasing in the first quadrant, [0, π/2].

11. Question 11

    Q11. What is the value of cos(0)?

    • A) 0
    • B) 1
    • C) -1
    • D) 1/2

    Answer: 1

    Explanation: cos(0) = 1, as it is the maximum value of the cosine function.

12. Question 12

    Q12. For y = sin(x)cos(x), what is the period?

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

    Answer: π

    Explanation: Using the identity sin(2x) = 2sin(x)cos(x), the period is π.

13. Question 13

    Q13. The graph of y = cos(x) + 1 is shifted?

    • A) 1 unit up
    • B) 1 unit down
    • C) 1 unit left
    • D) 1 unit right

    Answer: 1 unit up

    Explanation: Adding 1 to cos(x) shifts the graph 1 unit up.

14. Question 14

    Q14. For y = tan(2x), what is the period?

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

    Answer: π/2

    Explanation: The period of tan(ax) is π/a, so for tan(2x), it is π/2.

15. Question 15

    Q15. The maximum value of y = sin(x) + 2 is?

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

    Answer: 3

    Explanation: The max value of sin(x) is 1, so for sin(x) + 2, it is 3.

16. Question 16

    Q16. What is the range of y = sin(x) - 1?

    • A) [-2, 0]
    • B) [-1, 1]
    • C) [0, 2]
    • D) (-∞, ∞)

    Answer: [-2, 0]

    Explanation: The range of sin(x) is [-1, 1], so for sin(x) - 1, it is [-2, 0].

17. Question 17

    Q17. For y = 2sin(3x), what is the period?

    • A) 2π/3
    • B) 2π
    • C) π/3
    • D) 3π/2

    Answer: 2π/3

    Explanation: The period of sin(ax) is 2π/a, so for sin(3x), it is 2π/3.

18. Question 18

    Q18. If sin(x) = 1/2, what is the value of x in 0 to 2π?

    • A) π/6, 5π/6
    • B) π/3, 2π/3
    • C) π/4, 3π/4
    • D) π/2, 3π/2

    Answer: π/6, 5π/6

    Explanation: sin(π/6) = 1/2 and sin(π - π/6) = sin(5π/6) = 1/2.

19. Question 19

    Q19. What is the amplitude of 2sin(x) + 3?

    • A) 2
    • B) 3
    • C) 5
    • D) 1

    Answer: 2

    Explanation: The amplitude of asin(x) + b is |a|. Here, a = 2, so amplitude = 2.

20. Question 20

    Q20. The range of y = tan(x) is?

    • A) (-∞, ∞)
    • B) [-1, 1]
    • C) [0, ∞)
    • D) (-∞, 0]

    Answer: (-∞, ∞)

    Explanation: tan(x) can take any real value, so the range is all real numbers.