20 questions. Try each one, then open the answer to check yourself and read the explanation.
If N is the set of natural numbers and A = {x : |x − 1| < 2, x ∈ N}, then
Answer: B (A = {1, 2})
|x − 1| < 2 means −2 < x − 1 < 2, so −1 < x < 3. The natural numbers (counting numbers 1, 2, 3, …) in this range are 1 and 2, so A = {1, 2}.
The sum of the digits of a 2-digt number is 11. With the digits interchanged, the new number exceeds the original by 45. Find the original number
Answer: C (38)
A + B = 11 and 9(B − A) = 45, so B − A = 5: A = 3, B = 8, giving 38 (83 − 38 = 45).
Find the constant q such that f(x) = 2x3 + qx + 2 and f(−1) = 2
Answer: B (−2)
f(−1) = 2(−1) − q + 2 = −q. Setting −q = 2 gives q = −2.
Given that α and β are the roots of x2 + 5 = 7x, evaluate α + αβ + β
Answer: A (12)
x² − 7x + 5 = 0: α + β = 7 and αβ = 5, total 12.
A binary operation * is defined on the set of integers by p * q = √(pq). Evaluate 4 * (8 * 32)
Answer: C (8)
8 * 32 = √256 = 16, then 4 * 16 = √64 = 8.
For what range of values of p does the equation x2 − px + p + 3 = 0 fail to have real roots?
Answer: B (−2 < p < 6)
No real roots when b² < 4ac: p² < 4(p + 3), i.e. (p − 6)(p + 2) < 0, so −2 < p < 6.
Find the remainder when the expression x5 + 4x3 − x2 + 1 is divided by x − 1
Answer: D (5)
f(1) = 1 + 4 − 1 + 1 = 5.
Which of the following is a facor of p(x) = ax3 − bx2 + bx − a
Answer: B (x − 1)
p(1) = a − b + b − a = 0, so x − 1 is a factor.
Solve the inequality (y − 3)/2 < (2y − 1)/3
Answer: C (y > −7)
Multiply by 6: 3y − 9 < 4y − 2, so −y < 7 and y > −7.
Find the pair (A, B) such that (x + 3)/(4x2 − 9) = A/(2x + 3) + B/(2x − 3)
Answer: A ((−1/4, 3/4))
A(2x − 3) + B(2x + 3) = x + 3. At x = 3/2, 6B = 9/2 so B = 3/4; at x = −3/2, −6A = 3/2 so A = −1/4.
The first and twenty eight terms of an Arithmetic progression are −14 and −5 respectively. find its common difference
Answer: C (1/3)
−14 + 27d = −5, so d = 9/27 = 1/3.
The sum to infinity of a Geometric series is 250. Find the common ratio of the series if its first term is 350
Answer: C (−2/5)
350/(1 − r) = 250, so 1 − r = 1.4 and r = −2/5.
The interior angles of a pentagon are x°, (x + 30)°, (2x − 65)°, (x + 110)° and 45°. Find x
Answer: D (84)
The angles add to 540°: 5x + 120 = 540, so x = 84.
What is the value of p if p − 3, p − 1, p are three consecutive terms of a Geometric progression?
Answer: C (−1)
(p − 1)² = p(p − 3) gives −2p + 1 = −3p, so p = −1 (terms −4, −2, −1).
Which of the following polygons has each of its interior angles twice the exterior angle?
Answer: C (Regular Hexagon)
Interior = 2 × exterior and they add to 180°, so each exterior angle is 60° and n = 6. Equal angles throughout means a regular hexagon.
How many sides has a polygon whose sum of interior angles is 1800°?
Answer: D (12)
(n − 2) × 180 = 1800 gives n − 2 = 10, n = 12.
Find the value of x for which the determinant |x −1 −2; 2 1 0; −x 0 4| = 0 (rows separated by semicolons)
Answer: C (−4)
Expanding along the first row: x(4) + 1(8) − 2(0 + x) = 2x + 8 = 0, so x = −4.
Matrices P, Q and R are 5 × 2, 2 × 3 and 3 × 5 respectively. Which of the following multiplications is NOT possible
Answer: C (PRQ)
A product needs the columns of the first to equal the rows of the next; P has 2 columns but R has 3 rows, so PRQ is impossible.
Find x in the system of linear equations: ax + by = p; rx + ty = q
Answer: A ((pt − bq)/(at − br))
Eliminate y: multiply the first by t and the second by b and subtract: x(at − br) = pt − bq.
Simplify ((1.25 × 104) × (2.0 × 10−1))/(6.25 × 105)
Answer: A (4.0 × 10−3)
(2.5 × 10³)/(6.25 × 10⁵) = 0.4 × 10⁻² = 4.0 × 10⁻³.