20 questions. Try each one, then open the answer to check yourself and read the explanation.
Find the angle in degree which the line x − √3y = 0 makes with the positive y-axis
Answer: C (60)
y = x/√3 has gradient 1/√3 = tan 30°, so the line makes 30° with the x-axis and 90° − 30° = 60° with the y-axis.
If equation 6 − kx + 2x2 = 0 has equal roots, find k2 + 4.
Answer: B (52)
Equal roots need b² = 4ac: k² = 4 × 2 × 6 = 48. So k² + 4 = 52.
Simplify log100√(10−1).
Answer: B (−1/4)
√(10−1) = 10−1/2 and 100 = 10². log10010−1/2 = (−½)/2 = −¼.
Evaluate (1/25)−1/2 + (1/8)−2/3
Answer: C (9)
(1/25)−1/2 = 251/2 = 5 and (1/8)−2/3 = 82/3 = 4. 5 + 4 = 9.
Find the remainder when x4 − 11x + 2 is divided by x
Answer: A (2)
By the remainder theorem, put x = 0: 0 − 0 + 2 = 2.
if α and β are the roots of equation cx2 + ax + b = 0, find αβ
Answer: C (b/c)
Product of roots = constant term ÷ coefficient of x² = b/c.
The binary operation ⊗ is defined by a ⊗ b = 2a − 1. Find 3 ⊗ (2 ⊗ 1)
Answer: C (5)
2 ⊗ 1 = 2(2) − 1 = 3. Then 3 ⊗ 3 = 2(3) − 1 = 5.
Two coins tossed; find the probability of having at least two heads
Answer: C (¼)
With two coins, at least two heads means HH, 1 of the 4 outcomes: ¼.
If x is a real number and x + 11 < 0, evaluate |x|/x
Answer: B (−1)
x < −11, so x is negative and |x| = −x. |x|/x = −x/x = −1.
Simplify: 30/√2 + √50
Answer: B (20√2)
30/√2 = 15√2 and √50 = 5√2. Together 20√2.
If m is the gradient of the line pq − px − qy = 0 and q ≠ 0, find 1/m
Answer: C (−q/p)
qy = −px + pq, so y = (−p/q)x + p. m = −p/q and 1/m = −q/p.
If P is directly proportional to √Q; P = 20 when Q = 4. Find Q when P = 100
Answer: C (100)
P = k√Q: 20 = 2k, so k = 10. 100 = 10√Q gives √Q = 10, Q = 100.
Obtain the centre of the circle 7(y2 + 10y) + 7x2 = 1
Answer: C ((0, −5))
Divide by 7 and complete the square: x² + (y + 5)² = 1/7 + 25. The centre is (0, −5).
Given ∫−aa15x2dx = 3430, find the value of the constant a.
Answer: C (7)
[5x³] from −a to a = 5a³ + 5a³ = 10a³ = 3430, so a³ = 343 and a = 7.
Evaluate d/dx (ln sin 3x)
Answer: A (3 cot 3x)
d/dx ln u = u'/u: (3 cos 3x)/(sin 3x) = 3 cot 3x.
Find the equation of a line which passes through a point (−2, 3) and makes an angle of 45° with positive x-axis
Answer: A (y − x − 5 = 0)
Gradient = tan 45° = 1. y − 3 = x + 2, so y − x − 5 = 0.
If cos A = 12/13 and A is an acute angle, find (1 + tan2A)
Answer: D (169/144)
1 + tan²A = sec²A = 1/cos²A = (13/12)² = 169/144.
Integrate the function 1 − 2x.
Answer: A (x − x2 + k)
∫1 dx = x and ∫2x dx = x², so the answer is x − x² + k.
Differentiate 2 − sin(2 − ax) with respect to x
Answer: A (acos(2 − ax))
The derivative of −sin(2 − ax) is −cos(2 − ax) × (−a) = a cos(2 − ax); the constant 2 gives 0.
Simplify (8√n/m3/2)2(4−1m2/(2n−2))
Answer: B (8n3m−1)
The first bracket squared is 64n/m³. The second is m²n²/8. Multiplying gives 8n³/m = 8n³m−1.