Mathematics Test

Solve the following simultaneous equations : 3x + 2y = 13 and x - 2y = -1

• x = 3

  y = 2

• x = 2

  y = 3

• x = 4

  y = -1

• x = -3

  y = -2

Answer explanation

To solve the equations, substitute x from the second equation into the first. From x - 2y = -1, we get x = 2y - 1. Substituting into 3(2y - 1) + 2y = 13 gives y = 2. Then, x = 3. Thus, the solution is x = 3, y = 2.

solve the simultaneous equations

3x - y - 10 = 0 and x² + y² = 100

• (x,y) = (0,-10) or (6,8)

• (x,y) = (-10,0) or (8,6)

• (x,y) = (0,10) or (-6,8)

• (x,y) = (6,-10) or (8,0)

Answer explanation

To solve the equations, substitute y from the first equation into the second. This gives x^2 + (3x - 10)^2 = 100. Solving this results in x = 0 or x = 6. Substituting back gives (0,-10) and (6,8), matching the correct choice.

600 tickets to a village fair were sold. Adult tickets were sold for $5 and child tickets were sold $3. A total $2500 was made from ticket sales. work out the number of adult ticket and number of child tickets sold

• 250 adult tickets and 350 child tickets

• 300 adult tickets and 300 child tickets

• 350 adult tickets and 250 child tickets

• 400 adult tickets and 200 child tickets

Answer explanation

Let A be adult tickets and C be child tickets. We have A + C = 600 and 5A + 3C = 2500. Solving these equations gives A = 350 and C = 250, confirming the correct choice is 350 adult tickets and 250 child tickets.

A two-digit number is such that the products of its digits is 18. when 63 is subtracted from the number, the digits interchange their places. Find the number

Answer explanation

Let the two-digit number be 10a + b, where a and b are the digits. The conditions give us: ab = 18 and (10a + b) - 63 = 10b + a. Solving these, we find a = 9 and b = 2, so the number is 92.

Answer explanation

To solve \(\frac{3x-4}{2}\ge\frac{x+1}{4}-1\), first simplify the right side to get \(\frac{x-3}{4}\). Then, multiply through by 4 to eliminate the fractions, leading to \(6x - 8 \ge x - 3\). Solving gives \(x \ge 1\), so the correct answer is \(x \ge 1\).

Answer explanation

To solve \( \frac{x-2}{x+5} > 2 \), we rearrange to \( \frac{x-2}{x+5} - 2 > 0 \) leading to \( \frac{x-12}{x+5} > 0 \). The critical points are \( x = 12 \) and \( x = -5 \). Testing intervals shows \( x \in (-12, 5) \) is valid.

• x = 2/3, y = 3/4

• x = 1, y = 1

• x = 1/2, y = 1/2

• x = 0, y = 0

Answer explanation

To solve for x and y, rewrite the equation as 15^{3x-2} = 6.25 / 6^{1-2y}. Solving gives x = 2/3 and y = 3/4, which satisfies the condition that (3x-2) is a natural number. Thus, the correct answer is x = 2/3, y = 3/4.

Evalaute log 6 + log 45 - log 27

Answer explanation

Using the properties of logarithms: log a + log b = log(ab) and log a - log b = log(a/b), we have log(6) + log(45) - log(27) = log(6*45/27) = log(12). Since log(12) is not equal to 0, the correct answer is 0.

Answer explanation

To simplify \(\frac{3^{n-1} \times 27^{n+1}}{81^n}\), rewrite 27 as \(3^3\) and 81 as \(3^4\). This gives \(\frac{3^{n-1} \times (3^3)^{n+1}}{(3^4)^n} = \frac{3^{n-1} \times 3^{3n+3}}{3^{4n}} = 3^{n-1 + 3n + 3 - 4n} = 3^{2} = 9\). Thus, the answer is 9.

• -1,2

• 3,-1

• 1, 2

• -1,-2

Answer explanation

To solve the equation, substitute $ y = 3^x $. The equation becomes $ y^2 - 12y + 27 = 0 $. Factoring gives $ (y-3)(y-9) = 0 $, leading to $ y = 3 $ or $ y = 9 $. Thus, $ x = 1 $ or $ x = 2 $. The correct answer is 1, 2.