19 questions. Try each one, then open the answer to check yourself and read the explanation.
What is the value of (x + y) in the equations log3x = y = log9(2x − 1)?
Answer: B (1)
x = 3y and 2x − 1 = 9y = (3y)2 = x2. So x2 − 2x + 1 = 0, x = 1, and y = log31 = 0. x + y = 1.
There are 6 blue and 4 red identical balls in a bag. Two balls are drawn at random one after the other without replacement. What is the probability that the two balls are of the same colour?
Answer: B (7/15)
P(both red) = 4/10 × 3/9 = 12/90 and P(both blue) = 6/10 × 5/9 = 30/90. Add: 42/90 = 7/15.
A circle has its centre on the x-axis and passes through (1, 2) and (3, 2), find its equation.
Answer: A (x2 + y2 − 4x − 1 = 0)
The centre (a, 0) is equally far from both points, so a = 2. r² = (1 − 2)² + 2² = 5. (x − 2)² + y² = 5 expands to x² + y² − 4x − 1 = 0.
Calculate the mean deviation of the following scores: 1, 5, 7, 8, 9.
Answer: B (2.4)
Mean = 30/5 = 6. Distances from 6: 5, 1, 1, 2, 3, total 12. Mean deviation = 12/5 = 2.4.
If α and β are the roots of 7x2 − x(x + 13) = 15, find the value of 1/(α + 1) + 1/(β + 1).
Answer: A (25/4)
The equation is 6x² − 13x − 15 = 0, so α + β = 13/6 and αβ = −15/6. The sum = (α + β + 2)/(αβ + α + β + 1) = (25/6)/(4/6) = 25/4.
Find the nth term of the sequence 3, 7, 11, 15 …?
Answer: B (4n − 1)
It is an A.P. with a = 3 and d = 4: 3 + (n − 1)4 = 4n − 1.
What must be subtracted from 6x3 + 4x2 − x + 5 so that it will be divisible by x − 2?
Answer: B (67)
The remainder on dividing by x − 2 is f(2) = 48 + 16 − 2 + 5 = 67; subtracting it leaves no remainder.
Find the equation of the line which is parallel to the line 5x + 4y = 9 and makes an intercept of 4 units on the x-axis.
Answer: D (4y + 5x = 20)
A parallel line has the form 5x + 4y = c. It passes through (4, 0), so c = 20: 4y + 5x = 20.
Given the matrix M = [2 −4 −4; 1 8 2; 1 1 −2], find |M|.
Answer: A (−24)
Expand along the top row: 2(−16 − 2) + 4(−2 − 2) − 4(1 − 8) = −36 − 16 + 28 = −24.
The common ratio of a Geometric Progression is 2. If the 5th term is greater than the 1st term by 45, find the 5th term
Answer: D (48)
ar4 − a = 16a − a = 15a = 45, so a = 3 and the 5th term is 3 × 16 = 48.
Solve the equation 8x−2 = 2/25.
Answer: D (10)
8/x² = 2/25 gives x² = 100, so x = 10 (the positive root is the one offered).
Solve sin2x + 2 sin x + 1 = 0 for 0° < x < 360°.
Answer: D (270°)
(sin x + 1)² = 0, so sin x = −1, which happens at x = 270°.
Evaluate (16y2 × 22y1/2)/(641/2 × y5/2).
Answer: A (8)
Top: 64y5/2. Bottom: 8y5/2. The y terms cancel, leaving 64/8 = 8.
Find the equation of the circle which has its centre at (−2, 3) and passes through the point (4, 5).
Answer: B (x2 + y2 + 4x − 6y − 27 = 0)
r² = 6² + 2² = 40. (x + 2)² + (y − 3)² = 40 expands to x² + y² + 4x − 6y − 27 = 0.
Find the differential coefficient of (x2 − 5x + 7)/(2x) at the point (2, ¼).
Answer: C (−3/8)
Write y = x/2 − 5/2 + 7/(2x). Then dy/dx = ½ − 7/(2x²). At x = 2: ½ − 7/8 = −3/8.
Evaluate ∫12(2x2 − 3x + 5)dx.
Answer: A (31/6)
Integral is 2x³/3 − 3x²/2 + 5x. At 2: 16/3 + 4 = 28/3. At 1: 2/3 + 7/2 = 25/6. Difference = 56/6 − 25/6 = 31/6.
An operation * is defined on the set of real numbers R such at if x, y ∈ R, then x * y = (x + y)/3. Find ½ * ½.
Answer: A (1/3)
(½ + ½)/3 = 1/3.
Evaluate 645/3 + log2128 + log5125
Answer: B (1034)
645/3 = 45 = 1024, log2128 = 7 and log5125 = 3. Total 1034.
In a single throw of two fair coins, find the probability that one head and one tail appear.
Answer: C (½)
The outcomes are HH, HT, TH, TT. Two of the four (HT, TH) have one head and one tail, so the probability is 2/4 = ½.