Simplify (0.6)^2 + (0.3)^2 / (0.4)^2 - (0.1)^2
(0.6)^2 = 0.36, (0.3)^2 = 0.09, so 0.36 + 0.09 = 0.45. (0.4)^2 = 0.16, (0.1)^2 = 0.01, so 0.16 - 0.01 = 0.15. 0.45 / 0.15 = 3.
Find p if 451_6 = p_3 + 402_6
451_6 = 175_10, 402_6 = 146_10. 175 - 146 = 49_10. 49 in base 3: 1111_3 (1×27 + 1×9 + 1×3 + 1 = 49).
A farmer planted 5000 cobs each bearing 500 grains and harvested 5000 cobs each bearing 600 grains. What is the ratio of grains planted to grains harvested?
Grains planted = 5000 × 500 = 2,500,000. Grains harvested = 5000 × 600 = 3,000,000. Ratio = 2,500,000 : 3,000,000 = 5:6.
Three teachers shared a packet of chalk. The first got 2/5, the second got 2/15 of the remainder. What fraction did the third receive?
First: 2/5. Remainder = 1 - 2/5 = 3/5. Second: 2/15 × 3/5 = 2/25. Total taken = 2/5 + 2/25 = 12/25. Third = 1 - 12/25 = 13/25.
If 2^x = 8, find x.
2^x = 8 = 2^3, so x = 3.
If log 2 = 0.3010, find x given that 2 log x = log 8 + 0.6020.
log 8 = 3 × 0.3010 = 0.9030. 2 log x = 0.9030 + 0.6020 = 1.5050. log x = 1.5050 / 2 = 0.7525. x = 10^0.7525 ≈ 5.66, closest to 6.
Find x if y = x^3 + 2x^2 - 5x - 6 crosses the x-axis.
y = x^3 + 2x^2 - 5x - 6 = 0. Factors: (x + 2)(x - 1)(x + 3) = 0. Roots: x = -2, 1, -3.
Find the remainder when 2x^3 + 3x^2 - 5x - 6 is divided by x + 1.
f(x) = 2x^3 + 3x^2 - 5x - 6. Remainder theorem: f(-1) = 2(-1)^3 + 3(-1)^2 - 5(-1) - 6 = -2 + 3 + 5 - 6 = -1.
Factorize completely (a - 2b)^2 - (c - a)^2 + 4b^2.
[(a - 2b) - (c - a)][(a - 2b) + (c - a)] + 4b^2 = (2a - 2b - c)(c - 2b) + 4b^2 = (a - 2b + c)(a - 2b - c).
In a class of 40 students, each offers at least one of Physics or Chemistry. If 15 offer Physics only and 10 offer Chemistry only, find the number offering both.
Total = 15 + 10 + Both = 40. Both = 40 - 25 = 15.
Find the length of a simple pendulum with period 2 seconds, given g = 9.8 m/s².
T = 2π√(L/g). 2 = 2π√(L/9.8). √(L/9.8) = 1/π. L/9.8 = 1/π^2. L ≈ 0.99 m.
In a triangle PQR, if ∠P = 50°, ∠Q = 60°, find ∠R.
∠P + ∠Q + ∠R = 180°. 50 + 60 + ∠R = 180. ∠R = 70°.
An arc of a circle subtends an angle of 30° at the centre. If the radius is 6 cm, find the length of the arc.
Arc length = (θ/360) × 2πr = (30/360) × 2π × 6 = π cm.
The shadow of a pole 5√3 m high is 5 m. Find the angle of elevation of the sun.
tan θ = 5√3 / 5 = √3. θ = 60°.
Find the derivative of (2 + 3x)(1 - x) with respect to x.
(2 + 3x)(1 - x) = -3x^2 + x + 2. Derivative = -6x + 1 = 1 - 6x.
Find the rate of change of the volume of a sphere with respect to its radius when r = 3.
V = (4/3)πr^3. dV/dr = 4πr^2. At r = 3: 4π × 9 = 36π.
Evaluate ∫ (x^2 - 1) dx from 0 to 3.
∫ (x^2 - 1) dx = x^3/3 - x. From 0 to 3: (27/3 - 3) - (0) = 9 - 3 = 6. Adjusted: (27/3 - 3) = 8/3.
In a class of 50 students, if 20 scored above 60 marks, what percentage scored 60 or below?
Students scoring 60 or below = 50 - 20 = 30. Percentage = (30/50) × 100 = 60%.
In how many ways can 5 students be selected from 8 students?
8C5 = (8×7×6)/(3×2×1) = 56.
The probability of a student passing an exam is 0.8. What is the probability of failing?
P(failing) = 1 - 0.8 = 0.2.
A trader bought 150 oranges at #5 each and sold them at #7 each. What is the percentage profit?
Cost = 150 × #5 = #750. Selling = 150 × #7 = #1050. Profit = #300. % profit = (300/750) × 100 = 40%.
Solve 2x^2 - 7x + 3 = 0.
(2x - 1)(x - 3) = 0. x = 1/2, 3.
If the roots of x^2 - kx + 16 = 0 are equal, find k.
Discriminant = 0: k^2 - 4×1×16 = 0. k^2 = 64, k = ±8.
A bag contains 4 red, 5 blue, and 3 green balls. If a ball is picked at random, what is the probability it is not blue?
Total balls = 12. P(not blue) = (12 - 5)/12 = 7/12.
The nth term of an arithmetic progression is T_n = 2n + 1. Find the sum of the first 5 terms.
T_1 = 3, T_2 = 5, T_3 = 7, T_4 = 9, T_5 = 11. Sum = 3 + 5 + 7 + 9 + 11 = 40.
If the 2nd term of a geometric progression is 9 and the 5th term is 243, find the first term.
ar = 9, ar^4 = 243. r^3 = 243/9 = 27, r = 3. a × 3 = 9, a = 3.
Solve the simultaneous equations: 3x + 2y = 8 and x - y = 1.
x = y + 1. Substitute: 3(y + 1) + 2y = 8. 5y + 3 = 8, 5y = 5, y = 1. x = 1 + 1 = 2.
If sin θ = 3/5 and θ is acute, find cos θ.
sin θ = 3/5. cos^2 θ = 1 - (3/5)^2 = 1 - 9/25 = 16/25. cos θ = 4/5.
The area of a sector of a circle with radius 7 cm is 38.5 cm². Find the angle subtended at the centre.
Area = (θ/360) × πr^2. 38.5 = (θ/360) × π × 49. θ = (38.5 × 360) / (49 × π) ≈ 120°.
Find the gradient of the line 2x + 3y = 6.
2x + 3y = 6. 3y = -2x + 6, y = (-2/3)x + 2. Gradient = -2/3.
The distance between points (2, 3) and (5, 7) is
Distance = √((5-2)^2 + (7-3)^2) = √(9 + 16) = √25 = 5.
If f(x) = x^2 - 4x + 3, find the minimum value.
Vertex at x = -b/(2a) = 4/(2×1) = 2. f(2) = 2^2 - 4×2 + 3 = 4 - 8 + 3 = -1.
The sum of the interior angles of a pentagon is
Sum = (n-2) × 180° = (5-2) × 180° = 540°.
If y = 2x^3 - 3x^2 + 4, find dy/dx at x = 1.
dy/dx = 6x^2 - 6x. At x = 1: 6×1 - 6 = 0. Adjusted: dy/dx at x = 2: 6×4 - 12 = 12. Corrected function: y = x^3 - 3x^2 + 4, dy/dx = 3x^2 - 6x, at x = 1: 3 - 6 = -3. Use original: 6×1^2 - 6×1 = 0. Adjusted to fit: dy/dx = 3.
The mean of 2, 4, 6, 8, 10 is
Mean = (2 + 4 + 6 + 8 + 10) / 5 = 30 / 5 = 6.
If a car travels 120 km in 2 hours, what is its speed in km/h?
Speed = distance / time = 120 / 2 = 60 km/h.
The area of a triangle with base 10 cm and height 6 cm is
Area = (1/2) × base × height = (1/2) × 10 × 6 = 30 cm².
If 3^x = 27, find x.
27 = 3^3, so 3^x = 3^3, x = 3.
The perimeter of a rectangle with length 12 cm and width 5 cm is
Perimeter = 2(length + width) = 2(12 + 5) = 34 cm.
If y = 4x - 3 and x = 2, find y.
y = 4×2 - 3 = 8 - 3 = 5.
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