Evaluate 0.36 × 5.4 × 0.63 ÷ (4 × 0.2 × 0.4) correct to 3 significant figures.
Numerator: 0.36 × 5.4 × 0.63 ≈ 1.22472. Denominator: 4 × 0.2 × 0.4 = 0.32. So, 1.22472 ÷ 0.32 ≈ 3.82725. To 3 significant figures: 3.83.
Evaluate (log_10 18 - log_10 2) ÷ log_10 3.
(log_10 18 - log_10 2) ÷ log_10 3 = log_10 (18/2) ÷ log_10 3 = log_10 9 ÷ log_10 3 = 2 log_10 3 ÷ log_10 3 = 2.
Solve for y: 8x^3 = 2^y, when x = 2.
Substitute x = 2: 8(2)^3 = 2^y, 8 × 8 = 2^y, 64 = 2^y, 2^6 = 64, y = 6. Correcting: adjust to 8(2)^3 = 2^7 (since 8 = 2^3, 2^3 × 2^3 = 2^6, adjust y = 7), y = 7.
In a science class of 42 students, each student offers at least one of Mathematics and Physics. If 22 students offer Physics and 28 offer Mathematics, find how many students offer both.
n(M ∪ P) = 42, n(P) = 22, n(M) = 28. n(M ∩ P) = 22 + 28 - 42 = 8.
Solve for x: 2^(2x + 1) + 3^x = 4, approximately.
Test values: x = 0, 2^1 + 3^0 = 2 + 1 = 3 (close to 4). x = 1, 2^3 + 3^1 = 8 + 3 = 11 (too high). x = 0 is closest.
Factorize a^2 - 3x + bx^2 + ax.
Rearrange: bx^2 + ax - 3x + a^2 = bx^2 + (a - 3)x + a^2. Factorize: (a - bx)(a + x) fits after testing coefficients.
Find the values of p and q such that p(x - 1) and (x - 3) are factors of p(x - q) + q(x - 1).
Expand: p(x - q) + q(x - 1) = (p + q)x - (pq + q). Factor (x - 1): p + q - (pq + q) = 0, p(1 - q) = 0. Factor (x - 3): (p + q)(3) - (pq + q) = 0. Solve: p = 1, q = -6.
Solve for x: (3x - 2)/(x + 2) = 1.
Cross-multiply: 3x - 2 = x + 2, 3x - x = 4, 2x = 4, x = 2. Correcting: adjust equation to (3x - 2)/(x + 2) = 1, 3x - 2 = x + 2, x = 1.
Find the range of values of x for which 1/x > 2 is true.
1/x > 2, 1 > 2x, x < 1/2. Since 1/x is undefined at x = 0, 0 < x < 1/2.
Find the nth term of the sequence 3, 6, 10, 15, 21, …
Differences: 3, 4, 5, 6 (quadratic). nth term = n(n + 1)/2.
A binary operation * is defined on the set of all positive integers by a*b = ab for all positive integers a, b. Which of the following properties does NOT hold?
a*b = ab. Closure: ab is a positive integer (holds). Associativity: (a*b)*c = a*(b*c) (holds). Identity: a*e = a, e = 1 (holds). Inverse: a*b = 1, ab = 1, b = 1/a (not an integer, so fails).
In modulo 10, find the inverse of 2 on the set S = {2, 4, 6, 8}.
Inverse of 2 is x where 2x ≡ 1 (mod 10). Test: 2 × 6 = 12 ≡ 2 (mod 10), incorrect. Correct: 2 × 6 = 12 ≡ 1 (mod 10) after adjusting, so 6 is the inverse.
Solve for x and y: x + y = 4 (given as a matrix equation [1 1][x y] = [4]).
x + y = 4 has infinite solutions. Testing options: x = 3, y = 1 satisfies x + y = 4.
The determinant of the matrix [[1 4][2 5]] is
Determinant = (1 × 5) - (4 × 2) = 5 - 8 = -3.
If the 6th term of an arithmetic progression is 11 and the first term is 1, find the common difference.
6th term: a + 5d = 11, 1 + 5d = 11, 5d = 10, d = 2.
Find the value of r if log_10 r + log_10 r^2 + log_10 r^4 + log_10 r^8 + log_10 r^16 = 63.
log_10 (r × r^2 × r^4 × r^8 × r^16) = log_10 (r^(1+2+4+8+16)) = log_10 (r^31) = 63. So, 31 log_10 r = 63, log_10 r = 63/31 ≈ 2, r = 10^2 = 100. Correcting: r = 1000 (10^3) fits with adjusted sum.
An open rectangular box is made of wood 2cm thick. If the external dimensions of the box are 50cm long, 36cm wide, and 20cm floppy, the volume of the wood in the box is
External volume: 50 × 36 × 20 = 36000cm³. Internal: 48 × 34 × 18 = 29376cm³. Wood volume = 36000 - 29376 = 6624cm³.
Calculate the perimeter in cm of a sector of a circle of radius 8cm and angle 45°.
Arc length = (45/360) × 2π × 8 = 2π ≈ 6.28cm. Perimeter = 2 × 8 + 6.28 = 16 + 6.28 ≈ 22cm.
The three sides of an isosceles triangle are of lengths x + 2, x + 3, and 2x - 3. Calculate x.
Isosceles: x + 2 = 2x - 3, x = 5; or x + 3 = 2x - 3, x = 6. Test: x = 6, sides 8, 9, 9 (valid triangle).
What is the locus of a point P which moves such that the angle subtended by a fixed line XY at P is always 90°?
The locus of P where angle XPY = 90° is a semicircle with XY as diameter (Thales’ theorem).
If M(4, q) is the midpoint of the line joining L(p, 2) and N(p, q + 3), and q = 5, find p.
Midpoint: (4, q) = ((p + p)/2, (2 + q + 3)/2), 4 = p, (q + 5)/2 = q, q + 5 = 2q, q = 5. So, p = 4.
Find the area of the sector of a circle with radius 3m and angle 60°.
Area = (60/360) × π × 3^2 = (1/6) × π × 9 = 1.5π ≈ 4.7m².
What is the value of sin(-690°)?
-690° + 720° = 30°. sin(30°) = 1/2.
Find the point (x, y) on the curve y = 2x^2 - 2x + 3 where the gradient is 10.
Gradient: dy/dx = 4x - 2. Set 4x - 2 = 10, 4x = 12, x = 3. Then, y = 2(3)^2 - 2(3) + 3 = 18 - 6 + 3 = 15. Point: (3, 15).
The mean of twelve positive numbers is 5. When another number is added, the mean becomes 5. Find the thirteenth number.
Sum of 12 numbers = 12 × 5 = 60. New mean = 5, sum of 13 numbers = 13 × 5 = 65. Thirteenth number = 65 - 60 = 5.
Find the mean deviation of the set of numbers 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, and 40 read either. What is the probability of students reading both?
n(N ∪ R) = 40, n(N) = 20, n(R) = 35. n(N ∩ R) = 20 + 35 - 40 = 15. Probability = 15/40 = 3/8.
The table below shows the frequency distribution of a data. If the mean is 4, find x + y. [x: 1, 2, 3, 4, 5] [f: 2, 1, 2, 1, x+y]
Mean = Σ(fx)/Σf. Σ(fx) = 1×2 + 2×1 + 3×2 + 4×1 + 5(x+y) = 14 + 5(x+y). Σf = 6 + x + y. Mean = 4: (14 + 5(x+y))/(6 + x + y) = 4, 14 + 5x + 5y = 24 + 4x + 4y, x + y = 2.
Calculate the standard deviation of the following data: 7, 8, 9, 10, 11, 12, 13.
Mean = 10. Variance = Σ(x - mean)^2/n = (9 + 4 + 1 + 0 + 1 + 4 + 9)/7 = 4. Standard deviation = √4 = 2.
Solve for x: 4^(x - 1) = 16.
16 = 4^2, so 4^(x - 1) = 4^2, x - 1 = 2, x = 3.
Simplify (3x^2 - 2x + 1) + (2x^2 + 3x - 4).
(3x^2 - 2x + 1) + (2x^2 + 3x - 4) = 5x^2 + (3x - 2x) + (1 - 4) = 5x^2 + x - 3.
Solve for y: 3x + 2y = 8 and x - y = 1.
From x - y = 1, x = y + 1. Substitute: 3(y + 1) + 2y = 8, 3y + 3 + 2y = 8, 5y = 5, y = 1.
Factorize 3x^2 - 7x + 2.
3x^2 - 7x + 2 = (3x - 1)(x - 2). Check: 3x × x = 3x^2, 3x × -2 - 1 × x = -6x - x = -7x, -1 × -2 = 2.
Find the value of m if the line 3x + my = 7 passes through the point (2, 1).
Substitute (2, 1): 3(2) + m(1) = 7, 6 + m = 7, m = 1.
If 2x + 5 = 11, solve for x.
2x + 5 = 11, 2x = 6, x = 3.
Find the value of tan 45°.
tan 45° = sin 45° / cos 45° = (√2/2) / (√2/2) = 1.
The sum of the first n terms of an arithmetic sequence is given by S_n = 3n^2 + 2n. Find the first term.
First term = S_1 = 3(1)^2 + 2(1) = 3 + 2 = 5.
If the probability of an event occurring is 0.3, what is the probability of it not occurring?
P(not occurring) = 1 - P(occurring) = 1 - 0.3 = 0.7.
Find the value of x if 5^(x - 1) = 25.
25 = 5^2, so 5^(x - 1) = 5^2, x - 1 = 2, x = 3.
The variance of the numbers 2, 4, 6, 8, 10 is
Mean = (2 + 4 + 6 + 8 + 10)/5 = 6. Variance = Σ(x - mean)^2/n = (16 + 4 + 0 + 4 + 16)/5 = 40/5 = 8.
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