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HSSC mathematics

Written by Hassan Raza (Mathematics)

PIEAS · Mathematics · 34 questions

Name: ____________________ Date: ____________

The derivative of ln x with respect to x is:
- A ln x ÷ x
- B 1/x
- C eˣ
- D x
The integral of eˣ with respect to x is:
- A eˣ⁺¹ + c
- B x eˣ + c
- C ln x + c
- D eˣ + c
The solutions of x² − 9 = 0 are:
- A 3 only
- B 3 and −3
- C 0 and 9
- D 9 and −9
The number of ways to choose 2 people from 5 is:
- A 25
- B 5
- C 20
- D 10
The eccentricity of a circle is:
- A 0
- B 1
- C Greater than 1
- D Between 0 and 1
The domain of f(x) = √(x − 2) is:
- A All real numbers
- B x > 0
- C x ≤ 2
- D x ≥ 2
The derivative of x² sin x is:
- A 2x cos x
- B 2x sin x − x² cos x
- C 2x sin x + x² cos x
- D x² cos x
The sum of the integers from 1 to 100 is:
- A 10100
- B 5050
- C 5100
- D 5000
The magnitude of the vector 3i + 4j is:
- A 25
- B 5
- C 7
- D 12
The value of cos 60° is:
- A 1/2
- B 1
- C 0
- D √3/2
The derivative of sin 2x with respect to x is:
- A −2 cos 2x
- B 2 cos 2x
- C 2 sin 2x
- D cos 2x
The integral of sin x with respect to x is:
- A cos x + c
- B −sin x + c
- C −cos x + c
- D sin x + c
The limit of (1 − cos x) ÷ x² as x approaches 0 is:
- A 2
- B 0
- C 1/2
- D 1
The sum to infinity of the geometric series 4 + 2 + 1 + … is:
- A 6
- B 8
- C 4
- D 7
The period of the function sin x is:
- A 4π
- B 2π
- C π
- D π/2
The value of 10C2 is:
- A 100
- B 90
- C 45
- D 20
The number of terms in the expansion of (a + b)⁸ is:
- A 7
- B 16
- C 8
- D 9
The eccentricity of the ellipse x²/25 + y²/9 = 1 is:
- A 3/5
- B 5/3
- C 16/25
- D 4/5
The slope of the tangent to y = x² at x = 3 is:
- A 2
- B 6
- C 9
- D 3
The area of the triangle with vertices (0, 0), (4, 0) and (0, 6) is:
- A 12
- B 6
- C 10
- D 24
The cross product i × j is equal to:
- A 1
- B k
- C 0
- D −k
The modulus of the complex number 5 − 12i is:
- A 7
- B 13
- C 119
- D 17
The determinant of the 3 × 3 identity matrix is:
- A 1
- B 3
- C −1
- D 0
If 2ˣ⁺¹ = 16, then x is:
- A 3
- B 2
- C 7
- D 4
The domain of the function ln x is:
- A x > 0
- B All real numbers
- C x < 0
- D x ≥ 0
The derivative of ln (x² + 1) is:
- A 1 ÷ (x² + 1)
- B 2x ÷ (x² + 1)
- C 2x
- D (x² + 1) ÷ 2x
The value of the integral of sin x from 0 to π is:
- A 2
- B 1
- C 0
- D π
The inverse of the function f(x) = 2x + 3 is:
- A 2x − 3
- B x ÷ 2 − 3
- C (x + 3) ÷ 2
- D (x − 3) ÷ 2
The sum of the squares of the roots of x² − 5x + 6 = 0 is:
- A 5
- B 13
- C 11
- D 25
The remainder when x³ − 2x + 1 is divided by x − 1 is:
- A 1
- B 0
- C 2
- D −1
The number of different arrangements of the letters of the word BOOK is:
- A 4
- B 12
- C 6
- D 24
A card is drawn from a standard deck of 52 cards. The probability that it is a king is:
- A 4/13
- B 1/4
- C 1/52
- D 1/13
The slope of a line perpendicular to a line of slope 2 is:
- A −2
- B −1/2
- C 2
- D 1/2
The integral of 1 ÷ (1 + x²) with respect to x is:
- A sin⁻¹ x + c
- B 1 ÷ x + c
- C tan⁻¹ x + c
- D ln (1 + x²) + c

Answer key

1. B, 2. D, 3. B, 4. D, 5. A, 6. D, 7. C, 8. B, 9. B, 10. A, 11. B, 12. C, 13. C, 14. B, 15. B, 16. C, 17. D, 18. D, 19. B, 20. A, 21. B, 22. B, 23. A, 24. A, 25. A, 26. B, 27. A, 28. D, 29. B, 30. B, 31. B, 32. D, 33. B, 34. C

This is a standard derivative.
The exponential function is its own integral.
The equation gives x² = 9, so x = 3 or x = −3.
5C2 = (5 × 4) ÷ 2 = 10.
A circle is a conic with eccentricity zero.
The expression under the root must not be negative.
The product rule gives 2x sin x + x² cos x.
The sum is 100 × 101 ÷ 2 = 5050.
The magnitude is √(9 + 16) = 5.
This is a standard value.
The chain rule gives a factor of 2.
The derivative of −cos x is sin x.
This is a standard limit.
The sum is a ÷ (1 − r) = 4 ÷ (1/2) = 8.
The sine curve repeats after a full turn.
10C2 = (10 × 9) ÷ 2 = 45.
The expansion of power n has n + 1 terms.
Here c² = 25 − 9 = 16, so e = c ÷ a = 4/5.
The derivative is 2x, which is 6 at x = 3.
Area = ½ × 4 × 6 = 12.
This follows the right hand rule.
The modulus is √(25 + 144) = 13.
The determinant is the product of the diagonal entries, which is 1.
2ˣ⁺¹ = 2⁴, so x + 1 = 4.
The logarithm is defined only for positive numbers.
The chain rule gives the derivative of the inside divided by the inside.
The value is −cos π + cos 0 = 1 + 1 = 2.
Solve y = 2x + 3 for x.
The roots are 2 and 3, and 4 + 9 = 13.
By the remainder theorem the remainder is f(1) = 1 − 2 + 1 = 0.
The number is 4! ÷ 2! = 12.
There are 4 kings in 52 cards.
The product of the slopes of perpendicular lines is −1.
This is a standard integral.