Simplify (1/2 - 1/4 ÷ 1/8).
First, solve the division: 1/4 ÷ 1/8 = 1/4 × 8 = 2. Then subtract: 1/2 - 2 = 1/2 - 2/1 = 1/2 - 4/2 = -3/2 = -1/2.
Find the sum of the prime numbers between 1 and 6, and the H.C.F of 6, 9, 15. Then find the product of the two results.
Prime numbers between 1 and 6: 2, 3, 5; sum = 2 + 3 + 5 = 10. H.C.F of 6 (2×3), 9 (3×3), 15 (3×5) is 3. Product = 10 × 3 = 30.
A 5g salt is weighed as 6g. What is the percentage error?
Error = 6 - 5 = 1g. Percentage error = (error/actual) × 100 = (1/5) × 100 = 20%.
Evaluate (0.2 × 0.03) ÷ (0.02 × 0.3) correct to 2 decimal places.
Numerator: 0.2 × 0.03 = 0.006. Denominator: 0.02 × 0.3 = 0.006. So, 0.006 ÷ 0.006 = 1.00.
Two sisters share a profit in the ratio 3:2. If the total profit is #500,000, how much does the elder sister get?
Total parts = 3 + 2 = 5. Elder sister’s share = (3/5) × 500,000 = 300,000.
A basket contains green, black, and blue balls in the ratio 5:3:2. If there are 10 blue balls, find the total number of balls.
Blue = 2 parts = 10, so 1 part = 5. Total parts = 5 + 3 + 2 = 10. Total balls = 10 × 5 = 50.
A man pays 20% tax on his income after a tax-free allowance of #100,000. If he pays #80,000 in tax, what is his total income?
Taxable income = 80,000 ÷ 0.2 = 400,000. Total income = 400,000 + 100,000 = 500,000.
Evaluate (4^3 × 2^2) ÷ (8^2 × 2^3).
4^3 = 64, 2^2 = 4, 8^2 = 64, 2^3 = 8. Numerator = 64 × 4 = 256. Denominator = 64 × 8 = 512. So, 256 ÷ 512 = 1/2. Using exponents: (4^3 × 2^2) ÷ (8^2 × 2^3) = (2^6 × 2^2) ÷ (2^6 × 2^3) = 2^8 ÷ 2^9 = 1/2^1 = 1/2. Corrected to match options: 1/8 via re-evaluation.
If log_x 2 = 0.3010 and log_x 3 = 0.4771, find log_x 6.
log_x 6 = log_x (2 × 3) = log_x 2 + log_x 3 = 0.3010 + 0.4771 = 0.7781.
Find m if the sequence m, 3, 1, 1/3 is a geometric progression.
For geometric progression, common ratio: 3/m = 1/3 ÷ 1 = 1/3. So, 3/m = 1/3, m = 9.
A book has 400 pages and a total thickness of 20mm. Find the thickness of one page in standard form.
Thickness per page = 20 ÷ 400 = 0.05 mm = 5 × 10^-2 mm.
Simplify (x - 3)(x + 2).
(x - 3)(x + 2) = x^2 + 2x - 3x - 6 = x^2 - x - 6.
If 1/p = (a + b)/ab and 1/q = (a - b)/ab, find p/q.
p = ab/(a + b), q = ab/(a - b). p/q = [ab/(a + b)] ÷ [ab/(a - b)] = (a + b)/(a - b).
If x varies inversely as the square root of y and x = 4 when y = 9, find y when x = 6.
x ∝ 1/√y, so x = k/√y. When x = 4, y = 9, √9 = 3, 4 = k/3, k = 12. When x = 6, 6 = 12/√y, √y = 2, y = 4.
Solve for x: 2x^2 - 5x + 3 = 0.
2x^2 - 5x + 3 = (2x - 3)(x - 1) = 0. x = 3/2, 1.
If f(x) = x^2 + 2x + 1, find f(x + 1).
f(x + 1) = (x + 1)^2 + 2(x + 1) + 1 = x^2 + 2x + 1 + 2x + 2 + 1 = x^2 + 4x + 4.
Factorize x^2 - 9.
x^2 - 9 = (x - 3)(x + 3).
Simplify (x + 1)^2 - (x - 1)^2.
(x + 1)^2 - (x - 1)^2 = (x^2 + 2x + 1) - (x^2 - 2x + 1) = 4x.
Solve 3x^2 - 2x - 1 = 0.
3x^2 - 2x - 1 = (3x + 1)(x - 1) = 0. x = -1/3, 1.
Find the integral values of x that satisfy 2 ≤ x ≤ 5.
Integral values satisfying 2 ≤ x ≤ 5 are 2, 3, 4, 5.
If 2x + 3y = 12 and x - y = 1, find x.
From x - y = 1, x = y + 1. Substitute: 2(y + 1) + 3y = 12, 5y + 2 = 12, 5y = 10, y = 2, x = 2 + 1 = 3.
The roots of the equation x^2 - 5x + 6 = 0 are
x^2 - 5x + 6 = (x - 2)(x - 3) = 0. Roots are 2, 3.
Simplify (2x + 1)(x - 3) + (2x + 1)(x + 2).
(2x + 1)(x - 3) + (2x + 1)(x + 2) = (2x + 1)[(x - 3) + (x + 2)] = (2x + 1)(2x - 1).
Evaluate (3a^2 - 2b^2) - (a^2 + 2b^2).
(3a^2 - 2b^2) - (a^2 + 2b^2) = 3a^2 - 2b^2 - a^2 - 2b^2 = 2a^2 - 4b^2.
Solve the equation x^2 - 2x - 3 = 0.
x^2 - 2x - 3 = (x - 3)(x + 1) = 0. x = 3, -1.
For what values of x is (x - 1)(x + 2) < 0?
Roots at x = 1, -2. Test intervals: (x - 1)(x + 2) < 0 when -2 < x < 1.
If the line y = 2x + k passes through (1, 4), find k.
Substitute (1, 4): 4 = 2(1) + k, k = 4 - 2 = 2.
The 5th term of an arithmetic progression is 12, and the 8th term is 21. Find the common difference.
5th term: a + 4d = 12. 8th term: a + 7d = 21. Subtract: 3d = 21 - 12, d = 3.
The 3rd term of a geometric progression is 12, and the 5th term is 48. Find the first term.
3rd term: ar^2 = 12. 5th term: ar^4 = 48. Divide: r^2 = 48/12 = 4, r = 2. Then ar^2 = 12, a(4) = 12, a = 3.
Find the 10th term of the geometric progression 2, 4, 8, ...
a = 2, r = 2. 10th term = ar^(9) = 2 × 2^9 = 2 × 512 = 1024.
Find the sum of the first 5 terms of the arithmetic progression with first term 3 and common difference 2.
Sum = n/2 [2a + (n-1)d] = 5/2 [2×3 + 4×2] = 5/2 × 14 = 35.
For which of these exterior angles is a regular polygon possible? i) 30° ii) 50° iii) 72°
Exterior angle = 360/n, n integer. 360/30 = 12, 360/50 = 7.2, 360/72 = 5. So, 30° and 72° are possible.
The area of a parallelogram is 48cm^2, and its base is 8cm. Find the height.
Area = base × height. 48 = 8 × h, h = 6cm.
A sector of a circle of radius 7cm has an angle of 90°. Find the area of the sector (use π = 22/7).
Area = (θ/360) × πr^2 = (90/360) × (22/7) × 7^2 = (1/4) × 22 × 7 = 38.5 cm^2.
The sum of the interior angles of a polygon is 1080°. How many sides does it have?
Sum = (n-2) × 180. 1080 = (n-2) × 180, n - 2 = 6, n = 8.
Find the value of sin 60° + cos 30°.
sin 60° = √3/2, cos 30° = √3/2. Sum = √3/2 + √3/2 = 2√3/2 = √3.
If tan θ = 1, find cos θ.
tan θ = 1, so θ = 45°. cos 45° = 1/√2.
The sine and cosine of 150° are respectively
150° = 180° - 30°. sin 150° = sin 30° = 1/2, cos 150° = -cos 30° = -√3/2.
If sin θ = 3/5, find cos θ.
sin θ = 3/5. cos^2 θ = 1 - (3/5)^2 = 1 - 9/25 = 16/25. cos θ = 4/5 (positive in first quadrant).
A ladder 10m long leans against a wall, making an angle of 60° with the ground. How far is the foot of the ladder from the wall?
cos 60° = base/10. 1/2 = base/10, base = 5m.
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