HEC USAT-E (Pre-Engineering) Mathematics Trigonometric Functions — Set 3

HEC USAT-E (Pre-Engineering) Mathematics Trigonometric Functions — Set 3

1. Question 1

   Q1. If cos(x) = -1/2, what is x in 0 to 2π?

   • A) 2π/3, 4π/3
   • B) π/3, 5π/3
   • C) π/2, 3π/2
   • D) π/4, 7π/4

   Answer: 2π/3, 4π/3

   Explanation: cos(2π/3) = -1/2 and cos(2π - 2π/3) = cos(4π/3) = -1/2.

2. Question 2

   Q2. The period of y = sin(x) + cos(x) is?

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

   Answer: 2π

   Explanation: Both sin(x) and cos(x) have a period of 2π, so their sum also has a period of 2π.

3. Question 3

   Q3. If tan(x) = 1, what is x in 0 to π?

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

   Answer: π/4

   Explanation: tan(π/4) = 1.

4. Question 4

   Q4. What is the amplitude of y = 3cos(2x)?

   • A) 3
   • B) 2
   • C) 1
   • D) 6

   Answer: 3

   Explanation: The amplitude of acos(bx) is |a|. Here, a = 3, so amplitude = 3.

5. Question 5

   Q5. If sin(x) = -1, what is x in -π to π?

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

   Answer: -π/2

   Explanation: sin(-π/2) = -1.

6. Question 6

   Q6. The graph of y = cos(x) is reflected about the x-axis. What is the new equation?

   • A) y = -cos(x)
   • B) y = cos(x)
   • C) y = sin(x)
   • D) y = -sin(x)

   Answer: y = -cos(x)

   Explanation: Reflecting cos(x) about the x-axis gives -cos(x).

7. Question 7

   Q7. If tan(x) = -1, what is x in 0 to 2π?

   • A) 3π/4, 7π/4
   • B) π/4, 5π/4
   • C) π/2, 3π/2
   • D) 0, π

   Answer: 3π/4, 7π/4

   Explanation: tan(3π/4) = -1 and tan(2π - π/4) = tan(7π/4) = -1.

8. Question 8

   Q8. The period of y = sin(2x) + cos(3x) is?

   • A) 2π
   • B) π
   • C) 2π/lcm(2, 3)
   • D) lcm(2π/2, 2π/3)

   Answer: lcm(2π/2, 2π/3)

   Explanation: The period is the lcm of the periods of sin(2x) and cos(3x), which is lcm(π, 2π/3) = 2π.

9. Question 9

   Q9. What is the amplitude of y = 2sin(x + π/4)?

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

   Answer: 2

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

10. Question 10

    Q10. The period of sin(2x) + cos(3x) is

    • A) 2π
    • B) π
    • C) 2π / gcd(2, 3)
    • D) 2π (2)

    Answer: 2π / gcd(2, 3)

    Explanation: The period of sin(ax) + cos(bx) is 2π / gcd(a, b). Here, gcd(2, 3) = 1, so period = 2π.

11. Question 11

    Q11. If sin(x) = 1/2, then x =

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

    Answer: π / 6

    Explanation: sin(π / 6) = 1/2. sin(x) = 1/2 implies x = π / 6 in the first quadrant.

12. Question 12

    Q12. The graph of y = tan(x) has a period of

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

    Answer: π

    Explanation: The period of tan(x) is π, as it repeats every π radians.

13. Question 13

    Q13. If cos(x) = -1/2, then x =

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

    Answer: 2π / 3

    Explanation: cos(2π / 3) = -1/2. cos(x) = -1/2 implies x = 2π / 3 in the second quadrant.

14. Question 14

    Q14. The amplitude of y = 2sin(x) is

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

    Answer: 2

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

15. Question 15

    Q15. The range of y = cosec(x) is

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

    Answer: (-∞, -1] ∪ [1, ∞)

    Explanation: cosec(x) = 1 / sin(x). Since sin(x) lies between -1 and 1, cosec(x) lies outside this range.

16. Question 16

    Q16. If sin(A) = 3/5 and cos(B) = 4/5, then sin(A+B) =

    • A) 24/25
    • B) 7/25
    • C) 1
    • D) 0

    Answer: 24/25

    Explanation: Using sin(A+B) = sin(A)cos(B) + cos(A)sin(B). We need to find cos(A) and sin(B) using Pythagorean identity.

17. Question 17

    Q17. The general solution of tan(x) = 1 is

    • A) x = nπ + π / 4
    • B) x = 2nπ + π / 4
    • C) x = nπ - π / 4
    • D) x = nπ

    Answer: x = nπ + π / 4

    Explanation: The general solution of tan(x) = tan(α) is x = nπ + α. Here, α = π / 4.

18. Question 18

    Q18. If f(x) = sin(x) + cos(x), then f(π / 4) =

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

    Answer: √2

    Explanation: f(π / 4) = sin(π / 4) + cos(π / 4) = 1/√2 + 1/√2 = √2.

19. Question 19

    Q19. The number of solutions of sin(x) = x/2 is

    • A) 2
    • B) 3
    • C) 4
    • D) 5

    Answer: 3

    Explanation: By plotting the graphs of y = sin(x) and y = x/2, we can see they intersect 3 times.

20. Question 20

    Q20. The period of y = sin(3x) + cos(2x) is

    • A) 2π
    • B) π
    • C) 2π / gcd(3, 2)
    • D) lcm(2π/3, 2π/2)

    Answer: 2π / gcd(3, 2)

    Explanation: The period of sin(ax) + cos(bx) is 2π / gcd(a, b). Here, gcd(3, 2) = 1, so period = 2π.