CCL Maths Quiz

If Log10(3x-1) + log10(4) = log10(9x + 2), find the value of x.

• 1/3
• 1
• 2
• 3

Answer explanation

Using log(a) + log(b) = log(ab), we get log10(4(3x-1)) = log10(9x+2). This means 4(3x-1) = 9x+2. So, 12x - 4 = 9x + 2. Solving gives 3x = 6, so x = 2.

Simplify: (9 * 3^(n+1) - 3^(n+2)) / (3^(n+1) - 3^n). (Note: OCR was corrected from 9x3n+1)

• 3
• 9
• 27
• 81

Answer explanation

Numerator: 3^2 * 3^(n+1) - 3^(n+2) = 3^(n+3) - 3^(n+2) = 3^(n+2)(3-1) = 2 * 3^(n+2). Denominator: 3^n(3-1) = 2 * 3^n. The fraction simplifies to (2 * 3^(n+2)) / (2 * 3^n) = 3^(n+2-n) = 3^2 = 9.

Consider the statements: X: All wrestlers are strong. Y: Some wrestlers are not weightlifters. Which of the following is a valid conclusion?

• All strong wrestlers are weightlifters
• Some strong wrestlers are not weightlifters
• Some weak wrestlers are weightlifters
• All weightlifters are wrestlers.

Answer explanation

From Y, some wrestlers aren't weightlifters. From X, all wrestlers are strong. Combining these, it means some strong people (wrestlers) are not weightlifters.

Given P = {x: x is a multiple of 5}, Q = {x: x is a multiple of 3} and R = {x: x is an odd number}, are subsets of U = {x: 20 ≤ x ≤ 35}. Find (P U Q) ∩ R.

• {20, 21, 25, 30, 33}
• {21, 25, 27, 33, 35}
• {20, 21, 25, 27, 33, 36}
• {21, 25, 27, 30, 33, 35}

Answer explanation

P={20,25,30,35}, Q={21,24,27,30,33}, R={21,23,25,27,29,31,33,35}. P U Q = {20,21,24,25,27,30,33,35}. The intersection of (P U Q) with R is {21, 25, 27, 33, 35}.

If Un = kn² + pn, U₁ = -1, and U₅ = 15, find the values of k and p. (Assuming variable is n)

• k = -1, p = 2
• k = -1, p = -2
• k = 1, p = -2
• k = 1, p = 2

Answer explanation

U₁ = k(1)²+p(1) = k+p = -1. U₅ = k(5)²+p(5) = 25k+5p = 15. Divide by 5: 5k+p = 3. Solve the system: (5k+p)-(k+p) = 3-(-1) => 4k=4 => k=1. Substitute k=1 into k+p=-1 to get 1+p=-1, so p=-2.

Solve: 3^(2x-2) - 28(3^(x-2)) + 3 = 0.

• x = -2 or x = 1
• x = 0 or x = -3
• x = 2 or x = 1
• x = 0 or x = 3

Answer explanation

Let z = 3^x. The equation is (z²/9) - 28(z/9) + 3 = 0. Multiply by 9: z² - 28z + 27 = 0. Factor: (z-1)(z-27)=0. So, z=1 or z=27. If 3^x = 1, x=0. If 3^x = 27, x=3.

The first, second and third terms of a G.P are (x-4), (x+2) and (3x+1) respectively. Find the values of x.

• -1/2, 8
• 1/2, -8
• -1/2, -8
• 1/2, 8

Answer explanation

The common ratio is constant: (x+2)/(x-4) = (3x+1)/(x+2). Cross-multiply: (x+2)² = (x-4)(3x+1) => x² + 4x + 4 = 3x² - 11x - 4. Rearrange to 2x² - 15x - 8 = 0. Factoring gives (2x+1)(x-8)=0. So x = -1/2 or x = 8.

Find the range of values of x for which 2x² + 7x - 15 ≥ 0.

• x >= -5 or x <= 3/2
• x <= -5 or x >= 3/2
• =-5<=x<=3/2
• =-3/2<=x<=5

Answer explanation

Find roots of 2x²+7x-15=0. Factoring gives (2x-3)(x+5)=0, so roots are x=3/2 and x=-5. Since the parabola opens upwards, the function is ≥ 0 outside the roots. Thus, x ≤ -5 or x ≥ 3/2.

The probability that a student will graduate is 0.4. If 3 students are selected, what is the probability that at least one student will graduate?

• 0.06
• 0.22
• 0.784
• 0.8

Answer explanation

P(at least one) = 1 - P(none graduate). P(not graduating) = 1 - 0.4 = 0.6. P(all 3 not graduating) = 0.6 * 0.6 * 0.6 = 0.216. P(at least one) = 1 - 0.216 = 0.784. Option C is the closest.

If α and β are the roots of x² + mx - n = 0, form the equation whose roots are 1/α and 1/β.

• mnx² - n²x - m = 0
• mx² - nx + 1 = 0
• nx² - mx + 1 = 0
• nx² - mx - 1 = 0

Answer explanation

Original: α+β = -m, αβ = -n. New roots: Sum = 1/α + 1/β = (α+β)/αβ = -m/-n = m/n. Product = 1/αβ = -1/n. New equation: x² - (sum)x + (product) = 0 => x² - (m/n)x - 1/n = 0. Multiply by n: nx² - mx - 1 = 0.