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Pre-engineering mathematics
NED · Mathematics · 24 questions
Name: ____________________ Date: ____________
cosec 2x − cot 2x is equal to:
- A cos x
- B sec x
- C tan x
- D cot x
The limit of (x² − 4) ÷ (x² − 2x) as x tends to 2 is:
- A 0
- B 2
- C 4
- D −2
The equation of the normal to the curve y = 5 − x² at the point where x = 2 is:
- A 4y − x = 2
- B y + 4x = 7
- C y + 4x = 9
- D 4y − x = 8
The value of the integral of 3x² from 0 to 2 is:
- A 8
- B 12
- C 4
- D 6
The dot product of the vectors (2, 3, 1) and (1, −1, 2) is:
- A 3
- B 7
- C 1
- D −1
The cross product i × k is equal to:
- A −j
- B k
- C j
- D −k
The sum of the first 10 terms of the arithmetic progression 5, 8, 11, … is:
- A 155
- B 185
- C 175
- D 195
The coefficient of x⁴ in the expansion of (1 + x)⁸ is:
- A 56
- B 70
- C 35
- D 28
The determinant of the matrix with rows (1, 2, 3), (0, 1, 4) and (5, 6, 0) is:
- A 24
- B 0
- C 1
- D −1
The value of i¹⁰, where i = √−1, is:
- A 1
- B −1
- C i
- D −i
The equation of the tangent to the parabola y = x² at the point (1, 1) is:
- A y = x + 1
- B y = 2x + 1
- C y = 2x − 1
- D y = x
The eccentricity of the hyperbola x²/9 − y²/16 = 1 is:
- A 5/3
- B 5/4
- C 4/3
- D 3/5
The derivative of x eˣ with respect to x is:
- A eˣ (x − 1)
- B eˣ
- C eˣ (1 + x)
- D x eˣ
The integral of x eˣ with respect to x is:
- A eˣ ÷ x + c
- B eˣ (x + 1) + c
- C x eˣ + c
- D eˣ (x − 1) + c
The number of different arrangements of 5 people in a row is:
- A 24
- B 60
- C 120
- D 25
Two fair dice are thrown. The probability that the sum is 8 is:
- A 1/9
- B 7/36
- C 5/36
- D 1/6
The distance of the point (1, 2) from the line 3x + 4y − 10 = 0 is:
- A 1/5
- B 1
- C 3/5
- D 5
The inverse of the function f(x) = eˣ is:
- A ln x
- B e⁻ˣ
- C 1 ÷ eˣ
- D x ÷ e
The solution of |2x − 3| < 5 is:
- A −1 < x < 4
- B x > −1
- C x < −1 or x > 4
- D x < 4
If cos x = 3/5, then cos 2x is:
- A 6/5
- B −9/25
- C −7/25
- D 7/25
The sum to infinity of the geometric series 9 + 3 + 1 + … is:
- A 13.5
- B 27
- C 10
- D 12
The limit of (sin 3x) ÷ x as x tends to 0 is:
- A 1/3
- B 3
- C 0
- D 1
The area between the curve y = x², the x-axis and the line x = 3 is:
- A 6
- B 27
- C 3
- D 9
The magnitude of the vector 2i − 2j + k is:
- A 3
- B 9
- C 1
- D 5
Answer key
1. C, 2. B, 3. A, 4. A, 5. C, 6. A, 7. B, 8. B, 9. C, 10. B, 11. C, 12. A, 13. C, 14. D, 15. C, 16. C, 17. A, 18. A, 19. A, 20. C, 21. A, 22. B, 23. D, 24. A
cosec 2x − cot 2x = (1 − cos 2x) ÷ sin 2x = 2 sin²x ÷ (2 sin x cos x) = tan x.
The expression simplifies to (x + 2) ÷ x, which is 2 at x = 2.
At x = 2 the point is (2, 1) and the tangent slope is −4, so the normal slope is 1/4 and the line is 4y − x = 2.
The antiderivative is x³, which gives 8 − 0.
The product is 2 − 3 + 2 = 1.
Using the right hand rule, k × i = j, so i × k = −j.
The sum is (10 ÷ 2)(2 × 5 + 9 × 3) = 5 × 37 = 185.
The coefficient is 8C4 = 70.
Expanding along the first row gives −24 + 40 − 15 = 1.
i⁴ = 1, so i¹⁰ = i² = −1.
The slope is 2x = 2 at x = 1, so y − 1 = 2(x − 1).
Here c² = 9 + 16 = 25, so e = c ÷ a = 5/3.
The product rule gives eˣ + x eˣ.
Integration by parts gives x eˣ − eˣ.
The number is 5! = 120.
Five outcomes give a sum of 8: (2, 6), (3, 5), (4, 4), (5, 3) and (6, 2).
The distance is |3 + 8 − 10| ÷ 5 = 1/5.
The natural logarithm undoes the exponential.
The inequality means −5 < 2x − 3 < 5.
cos 2x = 2 cos²x − 1 = 18/25 − 1 = −7/25.
The sum is 9 ÷ (1 − 1/3) = 13.5.
The limit is 3 × (sin 3x ÷ 3x), which tends to 3.
The area is the integral of x² from 0 to 3, which is 27 ÷ 3 = 9.
The magnitude is √(4 + 4 + 1) = 3.