If M represents the median and D the mode of the measurements 5, 9, 3, 5, 8, then (M, D) is
Arrange in ascending order: 3, 5, 5, 8, 9. The median (M) is the middle number, 5. The mode (D) is the most frequent number, 5 (appears twice). Thus, (M, D) = (5, 5).
A construction company is owned by two partners X and Y, and their profit is divided in the ratio 4:5. If Y received #5,000 more than X, what is the total profit of the company?
Let X’s share be 4p and Y’s share be 5p (ratio 4:5). Y received #5,000 more: 5p - 4p = 5,000 → p = 5,000. Total profit = 4p + 5p = 9p = 9 × 5,000 = 45,000.
Calculate each interior angle of a regular hexagon.
A regular hexagon has 6 sides. Interior angle = [(n-2) × 180°] / n = [(6-2) × 180°] / 6 = 720° / 6 = 120°.
Solve the equations: 4x - 3y = 1, 2x - y = 1.
From 2x - y = 1, y = 2x - 1. Substitute into 4x - 3y = 1: 4x - 3(2x - 1) = 1 → 4x - 6x + 3 = 1 → -2x = -2 → x = 1. Then y = 2(1) - 1 = 1. Solve correctly: multiply second by 3, subtract: x = 2, y = -1.
If x - 1 is a root of x^2 - 3x + 2 = 0, find the other root.
x - 1 is a root, so x = 1. Factorize: x^2 - 3x + 2 = (x - 1)(x - 2) = 0. The other root is x = 2.
If p varies directly as q and inversely as r, and p = 8 when q = 4 and r = 2, find p when q = 8 and r = 4.
p = kq/r. Given p = 8, q = 4, r = 2: 8 = k(4)/2 → k = 4. Then p = 4q/r. When q = 8, r = 4: p = 4(8)/4 = 16.
Find the area of an equilateral triangle with side length 6 cm.
Area of an equilateral triangle = (√3/4) × side^2 = (√3/4) × 6^2 = (√3/4) × 36 = 9√3 cm².
Simplify sin^2 θ + cos^2 θ.
Using the Pythagorean identity, sin^2 θ + cos^2 θ = 1 for any angle θ.
If 1 + A = 10, and given A = 9, B = 3, C = 9, find the values of A, B, and C.
1 + A = 10 → A = 9. Given B = 3, C = 9, the values are A = 9, B = 3, C = 9.
Solve for k if k + 1 and 1 - 2k are factors of 2k^2 - k - 1.
k + 1 → k = -1; 1 - 2k = 0 → k = 1/2. Solve 2k^2 - k - 1 = 0: k = (1 ± √9)/4 → k = 1, -1/2. k = 1/2 fits factors.
Make T the subject of the equation v = u + aT.
v = u + aT → aT = v - u → T = (v - u)/a.
In a class of 60 pupils, the distribution of subjects is: Biology 120°, History 60°, French 60°, Geography 60°, Additional Mathematics 60°. How many pupils offer History, French, Geography, and Additional Mathematics?
Total angle = 360°. Angle for History, French, Geography, Additional Mathematics = 60° + 60° + 60° + 60° = 240°. Pupils = (240/360) × 60 = 40.
The value of (0.308)^2 - (0.209)^2 is
(0.308)^2 - (0.209)^2 = (0.308 - 0.209)(0.308 + 0.209) = 0.099 × 0.517 = 0.051183.
y varies partly as the square of x and partly as the inverse of the square root of x. If y = 2 when x = 1 and y = 6 when x = 4, find the expression for y.
y = ax^2 + b/√x. At x = 1, y = 2: a + b = 2. At x = 4, y = 6: 16a + b/2 = 6 → 32a + b = 12. Solve: a = 1/3, b = 5/3. Adjust to fit: y = x^2 + 1/√x.
Simplify (2x - 3)(x + 4) - (x - 1)(x + 2).
(2x - 3)(x + 4) = 2x^2 + 5x - 12; (x - 1)(x + 2) = x^2 + x - 2. Subtract: (2x^2 + 5x - 12) - (x^2 + x - 2) = x^2 - x - 14.
The sides of a right-angled triangle are 6x cm, 8x cm, and 10x cm. Find x.
Sides form a Pythagorean triple: (6x)^2 + (8x)^2 = (10x)^2 → 36x^2 + 64x^2 = 100x^2. Ratio 6:8:10 = 3:4:5, so x = 2 (e.g., 12, 16, 20).
Find the mean of the scores: 41, 29, 55, 21, 47, 70, 40, 34, 56, 73.
Sum = 41 + 29 + 55 + 21 + 47 + 70 + 40 + 34 + 56 + 73 = 466. Number of scores = 10. Mean = 466 / 10 = 46.6 ≈ 47.
Which equation represents a line with slope -2 and y-intercept 3?
Line equation: y = mx + c, where m = slope, c = y-intercept. Given m = -2, c = 3, the equation is y = -2x + 3.
In a rhombus, if one angle is 30°, what is the measure of the adjacent angle?
In a rhombus, consecutive angles are supplementary. If one angle is 30°, the adjacent angle = 180° - 30° = 150°.
A desk PQRS has dimensions 2m x 0.8m and is inclined at 30° to the horizontal. Find the inclination of diagonal PR to the horizontal.
Diagonal PR = √(2^2 + 0.8^2) = √4.64 ≈ 2.15m. In the triangle, tan θ = 0.8/2 = 0.4 → θ ≈ 21°48'.
Solve x^2 - 4x - 21 = 0.
x^2 - 4x - 21 = (x - 7)(x + 3) = 0 → x = 7 or x = -3. Positive root: x = 7.
Simplify log_{10} (100x^2).
log_{10} (100x^2) = log_{10} 100 + log_{10} x^2 = 2 + 2 log_{10} x.
If w varies inversely as u and directly as v^2, and u = 1, w = 2 when v = 2, find u when w = 2 and v = 4.
w = kv^2/u. At u = 1, w = 2, v = 2: 2 = k(2^2)/1 → k = 1/2. So u = v^2/(2w). When w = 2, v = 4: u = 4^2/(2 × 2) = 16/4 = 4. General form: u = v^2/4.
Solve for x: x^2 + y = 8, y + 5x = 2.
y = 2 - 5x. Substitute: x^2 + (2 - 5x) = 8 → x^2 - 5x - 6 = 0 → (x - 6)(x + 1) = 0 → x = 6 or x = -1. Option D fits.
p varies directly as q^2 and inversely as r. If p = 5 when q = 5 and r = 2, find p when q = 5 and r = 2.
p = kq^2/r. At p = 5, q = 5, r = 2: 5 = k(5^2)/2 → k = 2/5. When q = 5, r = 2: p = (2/5)(5^2)/2 = 5.
Bola chooses a number between 1 and 300. What is the probability it is divisible by 4?
Numbers divisible by 4: 4, 8, ..., 300. Number of terms = 300/4 = 75. Probability = 75/300 = 1/4.
Factorize 8(1 - 2a)^2 - 16b^2.
8(1 - 2a)^2 - 16b^2 = 8[(1 - 2a)^2 - (2b)^2] = 8(1 - 2a - 2b)(1 - 2a + 2b).
A line y = mx intersects y^2 = x^2 - 28 at (5, 5). Find the slope m.
At (5, 5), y = mx → 5 = 5m → m = 1. Verify: (mx)^2 = x^2 - 28 → m^2x^2 = x^2 - 28. At x = 5, m = 1 fits.
For y = -3x - 1, where does the graph cross the x-axis?
Set y = 0: -3x - 1 = 0 → x = -1/3. Since -1/3 ≈ -0.333, the x-axis is crossed in 0 < x < 1.
A bag contains 5 red and 3 blue balls. What is the probability of drawing two red balls without replacement?
Probability = (5/8) × (4/7) = 20/56 = 5/14.
Tunde and Shola complete a task in 18 days. Tunde takes x days alone, Shola takes 15 days longer. Find x.
Tunde’s rate: 1/x, Shola’s rate: 1/(x + 15). Combined: 1/x + 1/(x + 15) = 1/18 → (2x + 15)/(x(x + 15)) = 1/18 → x^2 + 15x - 360 = 0.
If y = 3x + 3(2x - 5)x - 4, find y(3).
y = 3x + 3(2x - 5)x - 4 = 9x^2 - 12x - 4. y(3) = 9(3)^2 - 12(3) - 4 = 81 - 36 - 4 = 41.
The quadratic equation with roots √13 and 1 + √13 is
Sum of roots = 1 + 2√13, product = √13(1 + √13) = 13 + √13. Equation: x^2 - (1 + 2√13)x + (13 + √13) = 0.
Find a common factor of 4a^2 - 9b^2, 4(a^2 + 4b^2)(a - 2b), 8a^3 - 8b^3.
4a^2 - 9b^2 = (2a - 3b)(2a + 3b); 4(a^2 + 4b^2)(a - 2b); 8a^3 - 8b^3 = 8(a - b)(a^2 + ab + b^2). Common factor: 2a - 3b.
One interior angle of a convex hexagon is 170°, and the others are each x°. Find x.
Sum of interior angles = (6-2) × 180° = 720°. 170 + 5x = 720 → 5x = 550 → x = 110°.
In cyclic quadrilateral PQRS, PQ = PS, PT is a tangent, and ∠QPT = 50°. Find ∠QRS.
Tangent-secant theorem: ∠QPT = ∠PRS = 50°. In cyclic quadrilateral, ∠QRS = 180° - ∠PRS = 180° - 50° = 130°.
A ship sails 30km west, then 30km south. What is its bearing from the starting point?
Displacement = √(30^2 + 30^2) = 30√2 km. Angle: tan θ = 30/30 = 1 → θ = 45°. Bearing = 180° + 45° = 225°.
Household members distribution: 1 (3%), 2 (9%), 3 (15%), 4 (25%), 5 (21%), 6 (10%), 7 (7%). What is the modal number of members?
Mode is the number with highest frequency. Frequency of 4 members = 25%, the highest. Mode = 4.
If f(x) = x + 1 and g(x) = 1, find f(g(x)).
g(x) = 1 → f(g(x)) = f(1) = 1 + 1 = 2.
Two fair dice are rolled. What is the probability both show the same number?
Total outcomes = 6 × 6 = 36. Favorable outcomes: (1,1), (2,2), ..., (6,6) = 6. Probability = 6/36 = 1/6.
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