21 questions. Try each one, then open the answer to check yourself and read the explanation.
If the probability of success in an event is y/x. What is the probability of failure?
Answer: A ((x − y)/x)
P(failure) = 1 − y/x = (x − y)/x.
What is the circumference of the circle? x2 + y2 = (7/π)2?
Answer: B (14 units)
The radius is 7/π, so the circumference is 2π × 7/π = 14 units.
Find the diameter of the circle 2x2 + 2y2 − 50 = 0
Answer: B (10units)
Divide by 2: x² + y² = 25, so r = 5 and the diameter is 10 units.
Find point of intersection of the lines 3x − 2y = 5, 2x + 5y = −7,
Answer: A (x = 11/19, y = −31/19)
Multiply to eliminate y: 15x − 10y = 25 and 4x + 10y = −14 add to 19x = 11, x = 11/19. Then y = (3x − 5)/2 = −31/19.
Solve 4x2 + 20x − 24 = 0
Answer: D (−6, 1)
Divide by 4: x² + 5x − 6 = (x + 6)(x − 1) = 0, so x = −6 or 1.
What is the 15th term of the sequence ……. −3, 2, 7,..?
Answer: D (67)
It is an A.P. with a = −3 and d = 5. T15 = −3 + 14 × 5 = 67.
What is the distance between the points (−1,5) and (−7, −3)?
Answer: B (10)
Differences are 6 and 8, so the distance is √(36 + 64) = 10.
Evaluate (log√27 − log√8)/(log3 − log 2)
Answer: C (3/2)
log√27 = (3/2)log 3 and log√8 = (3/2)log 2, so the top is (3/2)(log 3 − log 2). Dividing gives 3/2.
What is the remainder when x3 + 5x2 − 6x + 1 is divided by x − 1?
Answer: D (1)
Remainder = f(1) = 1 + 5 − 6 + 1 = 1.
If α and β are the roots of the equation 6 + 5x − x2 = 0, find αβ + α + β.
Answer: D (−1)
Write it as x² − 5x − 6 = 0: α + β = 5 and αβ = −6. So αβ + α + β = −1.
What is the value of y for which the function (y − 1)/(y + 1) is undefined?
Answer: A (−1)
A fraction is undefined when its denominator is 0: y + 1 = 0 gives y = −1.
Solve the equation 5x² = 25x+4
Answer: C (4, −2)
25x+4 = 52x+8, so x² = 2x + 8, x² − 2x − 8 = 0, (x − 4)(x + 2) = 0: x = 4 or −2.
Evaluate Σn=2n=4(2n + 1).
Answer: B (31)
(2² + 1) + (2³ + 1) + (2⁴ + 1) = 5 + 9 + 17 = 31.
Integrate 4x3 + 1/x with respect to x
Answer: A (ln x + x4 + k)
∫4x³ dx = x⁴ and ∫1/x dx = ln x, so the answer is ln x + x⁴ + k.
If X = {2,3,6,7,8} and Y = {6,7,10,3,17}, find X∩Y
Answer: D ({6,3,7})
The common elements are 3, 6 and 7. Written as a set (in curly brackets) that is {6, 3, 7}; order does not matter.
What is the coordinate of centre of the circle x2 + y2 + 2x − 4y = 10?
Answer: C ((−1, 2))
Complete the squares: (x + 1)² + (y − 2)² = 15, so the centre is (−1, 2).
Simplify logxx4 + log44x
Answer: C (4 + x)
logxx⁴ = 4 and log44x = x, so the sum is 4 + x.
Solve the equation 3x+1 = 271−x
Answer: A (½)
271−x = 33−3x, so x + 1 = 3 − 3x, 4x = 2, x = ½.
Given f(x) = 3 + x and g(x) = 3 − x, find g[f(x)].
Answer: C (−x)
g(3 + x) = 3 − (3 + x) = −x.
Differentiate sin(2x − 5) with respect to x.
Answer: C (2cos (2x − 5))
Chain rule: cos(2x − 5) × 2 = 2cos(2x − 5).
if ρ, λ are the roots of equation x2 − 5x + 7 = 0, find the value of ρ2 + λ2
Answer: D (11)
ρ + λ = 5 and ρλ = 7. ρ² + λ² = (ρ + λ)² − 2ρλ = 25 − 14 = 11.