15 questions. Try each one, then open the answer to check yourself and read the explanation.
Simplify √[(8x3y6)/(45z2w7) × (5z2w)/(2x5y4)]
Answer: A (2y/(3xw3))
Multiply out: 40x3y6z2w/(90x5y4z2w7) = 4y2/(9x2w6). Its square root is 2y/(3xw3).
Solve for x if (21−x × 2x−1)/2x+2 = 82−3x
Answer: A (1)
Left side: 2(1−x)+(x−1)−(x+2) = 2−x−2. Right side: 26−9x. So −x − 2 = 6 − 9x, 8x = 8, x = 1.
Simplify √(4 + √(21 + √(11 + √25)))
Answer: B (3)
Work from the inside: √25 = 5, √(11 + 5) = 4, √(21 + 4) = 5, √(4 + 5) = 3.
A = {x : x is odd and x < 10}, B = {2, 3, 5, 7, 11, 13}, find n(A∩B).
Answer: B (3)
A = {1, 3, 5, 7, 9}, so A∩B = {3, 5, 7}. n( ) asks for the number of elements, which is 3.
If f(x + 2) = 6x2 + 5x − 8, find f(5).
Answer: D (61)
f(5) means x + 2 = 5, so x = 3: 6(9) + 5(3) − 8 = 54 + 15 − 8 = 61.
The roots of the equation 2x2 + kx + 5 = 0 are α and β, where k is a constant. If α2 + β2 = −1, find the values of k
Answer: A (±4)
α + β = −k/2 and αβ = 5/2. α² + β² = (α + β)² − 2αβ = k²/4 − 5 = −1, so k² = 16 and k = ±4.
A committee of 5 members is to be formed from three females and four males. In how many ways can these be formed if at least one female is included as a member?
Answer: B (21 ways)
There are 7 people, so 7C5 = 21 committees in all. An all-male committee of 5 is impossible (only 4 males), so all 21 include a female.
Without using calculator, find the value of cos105°
Answer: B ((√2 − √6)/4)
cos(60° + 45°) = cos60 cos45 − sin60 sin45 = (½)(√2/2) − (√3/2)(√2/2) = (√2 − √6)/4.
Differentiate the function y = ln((x + 2)/(x + 3)).
Answer: A (1/((x + 2)(x + 3)))
y = ln(x + 2) − ln(x + 3), so dy/dx = 1/(x + 2) − 1/(x + 3) = 1/((x + 2)(x + 3)).
If y = 2xcos2x − sin2x, find dy/dx when x = π/4.
Answer: B (−π)
dy/dx = 2cos2x − 4x sin2x − 2cos2x = −4x sin2x. At x = π/4: −4(π/4)(sin π/2) = −π.
To what value does this progression converge? 0.4444…
Answer: D (4/9)
0.444… = 4/10 + 4/100 + … is a G.P. with a = 4/10 and r = 1/10. Sum to infinity = (4/10)/(9/10) = 4/9.
The first and the second terms of a geometric progression are 4 and 8 respectively. Find the sum of the first 10 terms of the progression
Answer: C (4092)
a = 4 and r = 2. S10 = 4(210 − 1)/(2 − 1) = 4 × 1023 = 4092.
The internal and external radii of a hemispherical bowl are 3cm and 4cm respectively. What is the volume of material used in making the bowl?
Answer: B ((74/3)π cm3)
Volume = ⅔π(R³ − r³) = ⅔π(64 − 27) = ⅔π × 37 = (74/3)π cm³.
Five digit numbers are formed from digits 4, 5, 6, 7 and 8. How many of the numbers are odd, if repetitions of digits are not allowed?
Answer: C (48)
An odd number must end in 5 or 7 (2 choices). The other 4 digits fill the first four places in 4! = 24 ways. 2 × 24 = 48.
If x − 3 is a factor of 2x2 − 2x + p, find the value of the constant p.
Answer: B (−12)
If x − 3 is a factor, putting x = 3 gives 0: 2(9) − 6 + p = 0, so p = −12.