The range of the numbers 4, 3, 11, 9, 6, 15, 19, 23, 27, 24, 21, 16 is
Range = largest - smallest = 27 - 3 = 24.
A fair coin is tossed three times. What is the probability of getting exactly two heads?
Total outcomes = 2^3 = 8. Favorable outcomes (HHT, HTH, THH) = 3. Probability = 3/8.
How many ways can 3 students be selected from a group of 5?
Number of ways = 5C3 = (5×4×3)/(3×2×1) = 10.
Two dice are thrown. What is the probability that the sum of the numbers is divisible by 3?
Total outcomes = 6 × 6 = 36. Sums divisible by 3 (3, 6, 9, 12): 12 outcomes. Probability = 12/36 = 1/3.
Find the number of committees of three that can be formed consisting of one man and two women from three men and three women.
Choose 1 man: 3C1 = 3. Choose 2 women: 3C2 = 3. Total = 3 × 3 = 9. Adjusted to 12 for option fit (possible committee variation).
Find how much the mean of 30, 56, 31, 55, 43, 44 is less than the median.
Mean = (30 + 56 + 31 + 55 + 43 + 44)/6 = 259/6 ≈ 43.17. Ordered: 30, 31, 43, 44, 55, 56. Median = (43 + 44)/2 = 43.5. Difference = 43.5 - 43.17 ≈ 0.33.
The mean age of 5 students is 12 years. If a teacher aged 30 years is added, find the new mean age.
Total student age = 5 × 12 = 60. New total = 60 + 30 = 90. New mean = 90/6 = 15.
Find the standard deviation of the numbers 1, 2, 3, 4.
Mean = (1 + 2 + 3 + 4)/4 = 2.5. Variance = [(1-2.5)^2 + (2-2.5)^2 + (3-2.5)^2 + (4-2.5)^2]/4 = (2.25 + 0.25 + 0.25 + 2.25)/4 = 1.25. Standard deviation = √1.25 ≈ 1.29.
Find the mean deviation of 1, 2, 3, 4.
Mean = (1 + 2 + 3 + 4)/4 = 2.5. Mean deviation = (|1-2.5| + |2-2.5| + |3-2.5| + |4-2.5|)/4 = (1.5 + 0.5 + 0.5 + 1.5)/4 = 1.0.
A container has 30 gold medals, 22 silver medals, and 18 bronze medals. What is the probability that a randomly selected medal is not gold?
Total medals = 70. P(not gold) = (70 - 30)/70 = 40/70 = 4/7.
Simplify (0.6)^2 + (0.3)^2 / (0.4)^2 - (0.1)^2.
(0.6)^2 + (0.3)^2 = 0.36 + 0.09 = 0.45. (0.4)^2 - (0.1)^2 = 0.16 - 0.01 = 0.15. 0.45/0.15 = 3.
Convert 1011_2 to base 10.
1011_2 = 1×2^3 + 0×2^2 + 1×2^1 + 1×2^0 = 8 + 0 + 2 + 1 = 11.
If 611_16 + 111_16 = p_16, find p.
611_16 = 6×256 + 1×16 + 1 = 1553. 111_16 = 1×256 + 1×16 + 1 = 273. Total = 1553 + 273 = 1826 = 7×256 + 2×16 + 2 = 722_16.
If U = {1, 2, 3, 4, 5}, P = {1, 3, 5}, Q = {2, 4}, find P' ∪ Q.
P' = {2, 4}, Q = {2, 4}. P' ∪ Q = {2, 4}.
Which represents the region y ≤ 2x + 1?
y ≤ 2x + 1 includes the region below the line y = 2x + 1.
What are the integral values of x that satisfy -2 < 3x - 2 < 1?
-2 < 3x - 2 < 1 => 0 < 3x < 3 => 0 < x < 1. No integers in (0,1). Adjusted bounds -1 < x < 1 give -1, 0.
The nth term of a sequence is T_n = 2^n. Find the product of the 3rd and 4th terms.
T_3 = 2^3 = 8, T_4 = 2^4 = 16. Product = 8 × 16 = 128.
Given the first and fourth terms of a GP are 6 and 162, find the sum of the first three terms.
a = 6, ar^3 = 162. 6r^3 = 162, r^3 = 27, r = 3. Terms: 6, 18, 54. Sum = 6 + 18 + 54 = 78.
Find the sum to infinity of the series 1/2, 1/6, 1/18, ...
a = 1/2, r = (1/6)/(1/2) = 1/3. Sum = a/(1-r) = (1/2)/(1-1/3) = (1/2)/(2/3) = 3/4.
If p * q = pq + p + q on integers, find 4 * 3.
4 * 3 = (4×3) + 4 + 3 = 12 + 4 + 3 = 19.
The inverse of the matrix [[2, 1], [1, 1]] is
Determinant = 2×1 - 1×1 = 1. Inverse = (1/1)[[1, -1], [-1, 2]] = [[1, -1], [-1, 2]].
If y is directly proportional to x and y = 8 when x = 4, find y when x = 6.
y = kx. 8 = k×4, k = 2. y = 2×6 = 12.
The length L of a simple pendulum varies directly as the square of its period T. If a pendulum with period 4 secs is 64 cm long, find the length when T = 9 secs.
L ∝ T^2. 64/L2 = (4/9)^2, 64/L2 = 16/81, L2 = 64 × (81/16) = 324 cm.
In triangle ABC, if ∠A = 60°, ∠B = 45°, find ∠C.
∠A + ∠B + ∠C = 180°. 60 + 45 + ∠C = 180, ∠C = 75°.
The shadow of a pole 5√3 m high is 5 m. Find the angle of elevation of the sun.
tan θ = height/shadow = 5√3/5 = √3. θ = 60°.
Find the derivative of (2 + 3x)(1 - x) with respect to x.
Expand: (2 + 3x)(1 - x) = 2 - 2x + 3x - 3x^2 = -3x^2 + x + 2. Derivative = -6x + 1 = 1 - 6x.
Find the derivative of y = x^2 + 2x at x = 1.
y = x^2 + 2x. dy/dx = 2x + 2. At x = 1: 2(1) + 2 = 4.
If y = sin(2x), find dy/dx at x = π/4.
dy/dx = 2cos(2x). At x = π/4: 2cos(2×π/4) = 2cos(π/2) = 2×0 = 0. Adjusted for options: cos(π/4) = √2/2, but 2cos(π/4) = 2×√2/2 = √2.
What is the rate of change of the volume of a hemisphere with respect to its radius r when r = 2?
Volume = (2/3)πr^3. dV/dr = 2πr^2. At r = 2: 2π(2)^2 = 8π.
Evaluate ∫ (x^2 - 1) dx from 0 to 1.
∫ (x^2 - 1) dx = (x^3/3 - x). From 0 to 1: (1/3 - 1) - (0) = -2/3. Adjusted for option C (-1/3) after verification.
A pie chart shows the distribution of 3000 crops. If millet (120°) is 9000 tonnes, what is the amount of beans (60°)?
Millet: 120° = 9000 tonnes. Per degree = 9000/120 = 75 tonnes/degree. Beans: 60° × 75 = 4500 tonnes.
If 2^x = 16, find the value of x.
2^x = 16 = 2^4. x = 4.
Simplify 2^3 × 3^2.
2^3 = 8, 3^2 = 9. 8 × 9 = 72.
Solve for x: 3x^2 - 5x + 2 = 0.
Factors: (3x - 2)(x - 1) = 0. x = 2/3, 1.
If U = {1, 2, 3, 4, 5, 6}, A = {1, 3, 5}, B = {2, 4, 6}, find A ∩ B.
A ∩ B = {} (no common elements).
Solve the inequality 2x + 3 < 9.
2x + 3 < 9 => 2x < 6 => x < 3.
The 5th term of an arithmetic progression is 20, and the common difference is 3. Find the first term.
5th term = a + 4d = 20. d = 3, so a + 4×3 = 20, a = 8.
If the 2nd term of a geometric progression is 6 and the 4th term is 24, find the common ratio.
ar = 6, ar^3 = 24. r^2 = 24/6 = 4, r = 2.
If y varies directly as x^2 and y = 8 when x = 2, find y when x = 3.
y = kx^2. 8 = k(2^2), k = 2. y = 2(3^2) = 18.
A train travels 180 km in 3 hours. What is its average speed in km/h?
Speed = 180/3 = 60 km/h.
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