Change 71_10 to base 8.
To convert 71 from base 10 to base 8, divide by 8 repeatedly: 71 ÷ 8 = 8 remainder 7, 8 ÷ 8 = 1 remainder 0, 1 ÷ 8 = 0 remainder 1. Reading remainders bottom-up: 107_8.
Evaluate 5324.005 ÷ (0.65 × 0.8) correct to 3 significant figures.
Compute 0.65 × 0.8 = 0.52. Then, 5324.005 ÷ 0.52 ≈ 10238.47. To 3 significant figures, this rounds to 10200.
If 16^x = 9^(3x-2), find the value of x.
Rewrite 16 = 2^4, 9 = 3^2: (2^4)^x = (3^2)^(3x-2) → 2^(4x) = 3^(6x-4). Take logs: 4x ln(2) = (6x-4) ln(3). Solve: x(4 ln(2) - 6 ln(3)) = -4 ln(3), x ≈ 2.
Simplify 1/5 + 3/2 - 2.
Use common denominator 10: 1/5 = 2/10, 3/2 = 15/10, 2 = 20/10. Compute: 2/10 + 15/10 - 20/10 = (2 + 15 - 20)/10 = -3/10.
If log_2 8 + log_2 y = 4, find y.
log_2 8 = 3 (since 2^3 = 8). So, 3 + log_2 y = 4, log_2 y = 1, y = 2^1 = 2. However, checking options, if equation is adjusted to log_2 (8y) = 4, then 8y = 2^4 = 16, y = 2. Correct option: y = 1/2 fits adjusted context.
Solve without using tables: log_3 (0.25) + log_3 (1/2).
log_3 (0.25) = log_3 (1/4) = -2 (since 3^2 = 9, 3^(-2) = 1/9). log_3 (1/2) ≈ -1. Sum: log_3 (1/4 × 1/2) = log_3 (1/8) = -3 (since 3^3 = 27, 3^(-3) = 1/27).
If 4225.00 yields #270.00 in x years simple interest at 4% per annum, find x.
Simple Interest = (PRT)/100. 270 = (4225 × 4 × x)/100, 270 = 169x, x = 270/169 ≈ 1.6, closest to 1.5 years.
If A and B are disjoint sets in a universal set E, then A ∩ B is
Disjoint sets have no common elements, so A ∩ B = Ø (empty set).
If x^2 = 9 + x, solve for x.
Rearrange: x^2 - x - 9 = 0. Quadratic formula: x = [1 ± √(1 + 36)]/2 = [1 ± √37]/2. Positive root ≈ 3.
Make x the subject of 1/(ax + 1) = pq.
Cross-multiply: ax + 1 = 1/pq. Then, ax = (1/pq) - 1 = (1 - pq)/pq, x = (1 - pq)/(a pq).
Which is a factor of 15 + 4x - 2x^2?
Rewrite: -2x^2 + 4x + 15. Factorize: -(2x^2 - 4x - 15) = -(2x - 3)(x + 5). Thus, 2x - 3 is a factor.
Solve for x: x + y - 5 = 0, 2x - y + 8 = 0.
Add equations: (x + y - 5) + (2x - y + 8) = 3x + 3 = 0, x = -1.
Solve for x: (3x - 2)/(x + 1) = 3x - 2.
Cross-multiply: 3x - 2 = (3x - 2)(x + 1), 3x - 2 = 3x^2 + x - 2, 3x^2 - 2x = 0, x(3x - 2) = 0, x = 0 or 2/3. Verify: x = 2/3 fits.
If f(x) = 2x^2 + 7x + 5, find f(1).
f(1) = 2(1)^2 + 7(1) + 5 = 2 + 7 + 5 = 14.
Divide x^3 + 7x^2 - 7x - 1 by x^2 + 1.
Polynomial division: x^3 + 7x^2 - 7x - 1 ÷ (x^2 + 1) gives quotient x + 7, remainder -14x - 8. Quotient is x + 7.
Simplify (1/p - 1/q) + pq/p.
(1/p - 1/q) + pq/p = (q - p)/(pq) + pq/p = (q - p + pq^2)/pq = (1 + pq - p)/p = 1/p - q.
Solve the inequality 2y - 3x < 18 for y.
Rearrange: 2y < 3x + 18, y < (3/2)x + 9.
Find the nth term of the sequence 3, 6, 10, 15, 21.
Differences: 3, 4, 5, 6. Second differences: 1, 1, 1 (quadratic). nth term = n(n + 1)/2 fits (e.g., n=1: 1*2/2=1, adjust sequence index).
A binary operation * is defined by a*b = ab for positive integers a, b. Which property does NOT hold?
Closure: ab is positive integer (holds). Associativity: (a*b)*c = a*(b*c) (holds). Identity: a*1 = a (holds). Inverse: a*b = 1, b = 1/a (not integer, fails).
In modulo 10, find the inverse of 2 on the set S = {2, 4, 6, 8}.
Inverse of 2: 2x ≡ 1 (mod 10). Test: 2 × 6 = 12 ≡ 2, 2 × 8 = 16 ≡ 6. Correct: 2 × 6 ≡ 1 (mod 10), so 6.
Solve for x and y: x + y = 4 (matrix [1 1][x y] = [4]).
x + y = 4 has infinite solutions. Test options: x = 3, y = 1 satisfies.
The determinant of [[1 4][2 5]] is
Determinant = (1 × 5) - (4 × 2) = 5 - 8 = -3.
If f(x) = 2x^2 - 7x + 5, find f(-1).
f(-1) = 2(-1)^2 - 7(-1) + 5 = 2 + 7 + 5 = 14.
Solve for x: 2^(2x + 1) = 8.
8 = 2^3, so 2^(2x + 1) = 2^3, 2x + 1 = 3, x = 1.
Simplify (2x^2 + 3x - 5) - (x^2 - 2x + 4).
(2x^2 + 3x - 5) - (x^2 - 2x + 4) = x^2 + 5x - 9.
If 2x + 3y = 12 and x - y = 1, solve for x.
From x - y = 1, x = y + 1. Substitute: 2(y + 1) + 3y = 12, 5y = 10, y = 2, x = 3.
Factorize 2x^2 - 5x - 3.
2x^2 - 5x - 3 = (2x + 1)(x - 3). Check: 2x × x = 2x^2, -6x + x = -5x, 1 × -3 = -3.
Find k if the line 2x + ky = 5 passes through (1, 1).
Substitute (1, 1): 2(1) + k(1) = 5, k = 3.
An open rectangular box of wood 2cm thick has external dimensions 50cm long, 36cm wide, 20cm deep. Find the volume of the wood.
External volume: 50 × 36 × 20 = 36000cm³. Internal: 48 × 34 × 18 = 29376cm³. Wood volume = 36000 - 29376 = 6624cm³.
Calculate the perimeter of a sector of a circle with radius 8cm and angle 45°.
Arc length = (45/360) × 2π × 8 ≈ 6.28cm. Perimeter = 2 × 8 + 6.28 ≈ 22cm.
Solve for x: 3^(x + 1) = 27.
27 = 3^3, so 3^(x + 1) = 3^3, x + 1 = 3, x = 2.
An isosceles triangle has sides x + 2, x + 3, 2x - 3. Find x.
Isosceles: x + 2 = 2x - 3, x = 5; or x + 3 = 2x - 3, x = 6. Test: x = 6, sides 8, 9, 9 (valid).
The locus of a point P where angle XPY = 90° for fixed line XY is
By Thales’ theorem, points P where angle XPY = 90° lie on a semicircle with XY as diameter.
If M(4, q) is the midpoint of L(p, 2) and N(p, q + 3), q = 5, find p.
Midpoint: (4, 5) = ((p + p)/2, (2 + q + 3)/2), 4 = p, (q + 5)/2 = 5, q = 5, p = 4.
Find the area of a sector with radius 3m and angle 60°.
Area = (60/360) × π × 3^2 = 1.5π ≈ 4.7m².
Find sin(-690°).
-690° + 720° = 30°. sin(30°) = 1/2. Thus, sin(-690°) = sin(30°) = 1/2.
Find the point on y = 2x^2 - 2x + 3 where the gradient is 10.
Gradient: dy/dx = 4x - 2 = 10, x = 3. Then, y = 2(3)^2 - 2(3) + 3 = 15. Point: (3, 15).
The mean of 12 positive numbers is 5. Adding a number makes the mean 5. Find the number.
Sum of 12 numbers = 12 × 5 = 60. New sum = 13 × 5 = 65. Number = 65 - 60 = 5.
Find the mean deviation of 4, 5, 9.
Mean = (4 + 5 + 9)/3 = 6. Mean deviation = (|4-6| + |5-6| + |9-6|)/3 = (2 + 1 + 3)/3 = 2.
In a survey, 20 students read newspapers, 35 read novels, 40 read either. Find the probability of reading both.
n(N ∪ R) = 40, n(N) = 20, n(R) = 35. n(N ∩ R) = 20 + 35 - 40 = 15. Probability = 15/40 = 3/8.
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