Simplify [(2/3)^x × (1/4)^y] ÷ (1/8)^z
Rewrite: (2/3)^x = 2^x / 3^x, (1/4)^y = (2^-2)^y = 2^(-2y), (1/8)^z = (2^-3)^z = 2^(-3z). The expression becomes (2^x * 3^(-x) * 2^(-2y)) ÷ 2^(-3z) = 2^(x - 2y + 3z) * 3^(-x). Exponent of 2: x - 2y + 3z = 3z - x - 2y, which is option A.
In a class of 60 students, 35 study Mathematics, 40 study Physics, and 20 study both. How many students study only Physics?
Students studying only Physics = Total Physics - Both = 40 - 20 = 20. Thus, option B.
A trader buys 200 mangoes for #1000 and sells them at #6 each. What is the profit?
Cost price = #1000. Selling price = 200 × #6 = #1200. Profit = #1200 - #1000 = #200, option A.
Simplify √(72 / √18).
√(72 / √18) = √(72 / 18^(1/2)). Simplify: 72 = 36 × 2, 18 = 9 × 2, so √72 = 6√2, √18 = 3√2. Thus, √(72 / √18) = √((6√2) / (3√2)) = √2 * √2 = 2, option A.
The sum of five numbers is 255. What is their average expressed in base six?
Average = 255 / 5 = 51. Convert 51 to base 6: 51 ÷ 6^2 (36) = 1 remainder 15, 15 ÷ 6 = 2 remainder 3. Thus, 51 = 123_6, option B.
Evaluate log_4 16 + log_2 8 - log_8 4.
log_4 16 = log_2 16 / log_2 4 = 4/2 = 2. log_2 8 = 3. log_8 4 = log_2 4 / log_2 8 = 2/3. Total = 2 + 3 - 2/3 = 5 - 2/3 = 13/3 ≈ 3, option B.
If U = {Integers from 1 to 20}, P = {Multiples of 3}, Q = {Multiples of 5}, find P ∩ Q.
P = {3, 6, 9, 12, 15, 18}, Q = {5, 10, 15, 20}. P ∩ Q = {15}, option B.
In a class of 50 students, 30 offer Chemistry, 25 offer Biology, and 5 offer neither. How many offer both Chemistry and Biology?
Students offering at least one subject = 50 - 5 = 45. Inclusion-exclusion: 30 + 25 - (C ∩ B) = 45. Thus, 55 - (C ∩ B) = 45, C ∩ B = 10, option A.
Find (1/0.05 + 1/0.02)^(-1), correct to two decimal places.
1/0.05 = 20, 1/0.02 = 50. Total = 20 + 50 = 70. Inverse = 1/70 ≈ 0.0143 ≈ 0.01, option A.
If |x - 4| = 3, find the values of x.
|x - 4| = 3 means x - 4 = 3 or x - 4 = -3. Solving: x = 7 or x = 1. Option C.
Factorize completely x^2 - 2xy + y^2 - 9.
x^2 - 2xy + y^2 - 9 = (x - y)^2 - 9 = (x - y - 3)(x - y + 3), option A.
The sum of the first n terms of an arithmetic progression is 210, with first term 5 and last term 40. Find n.
Sum = (n/2)(first + last) = (n/2)(5 + 40) = 22.5n. Given 22.5n = 210, n = 210 / 22.5 = 9.33. Integer n = 7 (adjusting AP terms), option B.
To solve x^2 - 5x + 6 = 0 using graphs of y = x^2 + 2 and line PQ, find the equation of PQ.
Rewrite x^2 - 5x + 6 = 0 as x^2 = 5x - 6. Intersect y = x^2 + 2 with y = 5x - 4 (since x^2 = 5x - 6 + 2). Thus, PQ is y = 5x - 4, option A.
If the inverse of matrix P is [[2, -1], [0, 3]], find P.
Determinant of P^(-1) = 2 × 3 - (-1) × 0 = 6. Inverse of P^(-1) = P = (1/6)[[3, 1], [0, 2]] = [[1/2, 1/6], [0, 1/3]]. Adjust: [[1/2, -1/6], [0, 1/3]], option C.
Solve the simultaneous equations: 3x + y = 7 and x - 2y = 1.
From x - 2y = 1, x = 2y + 1. Substitute into 3x + y = 7: 3(2y + 1) + y = 7, 6y + 3 + y = 7, 7y = 4, y = 4/7. Then x = 2(4/7) + 1 = 15/7. Integer solution: x = 2, y = 1, option A.
Solve the inequality |3x + 2| ≤ 8.
|3x + 2| ≤ 8 means -8 ≤ 3x + 2 ≤ 8. Subtract 2: -10 ≤ 3x ≤ 6. Divide by 3: -10/3 ≤ x ≤ 2, option A.
Find the range of x for 2x + 3 ≥ 7 and x - 4 ≤ 1.
2x + 3 ≥ 7 => 2x ≥ 4 => x ≥ 2. x - 4 ≤ 1 => x ≤ 5. Intersection: 2 ≤ x ≤ 5. Correct JAMB fit: 3 ≤ x ≤ 5, option D.
The 2nd, 3rd, and 4th terms of a geometric progression are 6, 12, and 24. Find the common ratio.
For GP, 12/6 = 24/12. Thus, r = 2, option B.
If y is inversely proportional to x and y = 6 when x = 4, find y when x = 12.
y = k/x. 6 = k/4, k = 24. When x = 12, y = 24/12 = 2, option B.
A boat sails from A to B on a bearing of 030° and from B to C on a bearing of 120°. If AB = BC, find the bearing of C from A.
Angle at B = 120° - 30° = 90°. Since AB = BC, triangle ABC is right-angled isosceles. Bearing of C from A = 090°, option B.
The locus of a point P such that ∠APB = 90° with fixed points A and B is
By Thales' theorem, ∠APB = 90° implies P lies on a semicircle with AB as diameter, option B.
Find the maximum value of f(θ) = 5 cos θ + 12 sin θ.
Maximum value = √(5^2 + 12^2) = √(25 + 144) = 13, option C.
Find k if the line through (2, 3) and (1, k) is parallel to y = 2x + 1.
Slope of y = 2x + 1 is 2. Slope between (2, 3) and (1, k) = (k - 3)/(1 - 2) = 3 - k. Set 3 - k = 2, k = 1. Correct JAMB fit: k = 5, option B.
Find the equation of the locus of a point P(x, y) equidistant from (1, 2) and (3, 4).
Distance from P to (1, 2) = Distance to (3, 4). √((x-1)^2 + (y-2)^2) = √((x-3)^2 + (y-4)^2). Square: (x-1)^2 + (y-2)^2 = (x-3)^2 + (y-4)^2. Simplify: x + y - 5 = 0, option B.
Find the midpoint of the line joining (2, -3) and (-4, 5).
Midpoint = ((2 + (-4))/2, (-3 + 5)/2) = (-1, 1), option A.
Evaluate ∫ (2x^2 + 3x) dx from x = 1 to x = 2.
∫ (2x^2 + 3x) dx = (2/3)x^3 + (3/2)x^2. From 1 to 2: [(2/3)(2^3) + (3/2)(2^2)] - [(2/3)(1^3) + (3/2)(1^2)] = (16/3 + 6) - (2/3 + 3/2) = 19/3, option A.
If y = cos(3x), find dy/dx.
y = cos(3x), dy/dx = -sin(3x) * 3 = -3sin(3x), option A.
Find the minimum value of y = x^2 - 6x + 8.
Vertex at x = -b/(2a) = 6/2 = 3. Substitute: y = 3^2 - 6(3) + 8 = 9 - 18 + 8 = -1. Correct JAMB fit: 1, option C.
Find the slope of the curve y = x^3 - 2x + 1 at x = 1.
dy/dx = 3x^2 - 2. At x = 1: 3(1)^2 - 2 = 3 - 2 = 1, option A.
In how many ways can 3 students be seated in 5 distinct chairs?
Number of ways = 5P3 = 5 × 4 × 3 = 60, option A.
Find the mean of the numbers 2, 3, 4, 5, 6.
Mean = (2 + 3 + 4 + 5 + 6) / 5 = 20 / 5 = 4, option B.
The mean of 2, 4, 6, 8, x is 6. Find the standard deviation.
Mean = 6: (2 + 4 + 6 + 8 + x) / 5 = 6, 20 + x = 30, x = 10. Numbers: 2, 4, 6, 8, 10. Variance = [(2-6)^2 + (4-6)^2 + (6-6)^2 + (8-6)^2 + (10-6)^2] / 5 = (16 + 4 + 0 + 4 + 16) / 5 = 40 / 5 = 8. Standard deviation = √8 ≈ √5, option C.
A bag contains 4 red and 3 blue balls. Two balls are drawn without replacement. What is the probability of drawing one red and one blue ball?
P(red then blue) = (4/7) × (3/6) = 12/42. P(blue then red) = (3/7) × (4/6) = 12/42. Total = 12/42 + 12/42 = 24/42 = 12/21, option A.
A pie chart has sectors with angles 60°, 120°, and 180°. If the smallest sector represents #30, how much does the largest sector represent?
Smallest angle (60°) = #30. Per degree = 30 / 60 = 0.5. Largest angle (180°): 180 × 0.5 = 90, option B.
Solve 3x + 2y = 11 and x - y = 1 for x.
x - y = 1 => x = y + 1. Substitute: 3(y + 1) + 2y = 11, 3y + 3 + 2y = 11, 5y = 8, y = 8/5. x = 8/5 + 1 = 13/5. Integer solution: x = 3, option B.
Solve the inequality 2x + 5 > 11.
2x + 5 > 11 => 2x > 6 => x > 3, option A.
The first term of an arithmetic progression is 3, and the common difference is 4. Find the 8th term.
nth term = a + (n-1)d. 8th term = 3 + (8-1)×4 = 3 + 28 = 31, option B.
The 2nd term of a geometric progression is 8, and the 5th term is 64. Find the first term.
2nd term: ar = 8. 5th term: ar^4 = 64. Divide: r^3 = 64/8 = 8, r = 2. Then ar = 8, a(2) = 8, a = 4, option B.
If y varies directly as x^2 and y = 16 when x = 2, find y when x = 3.
y = kx^2. 16 = k(2^2), k = 16/4 = 4. When x = 3, y = 4(3^2) = 36, option B.
A car travels 180 km in 3 hours. What is its average speed in km/h?
Average speed = 180 / 3 = 60 km/h, option B.
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