OMPT-C mixed paper

Answer: $\frac{13}{10}$

Write both over the common denominator $10$, combine the numerators, then cancel: $\frac{13}{10}$.

Common mistakes, and the answer each one gives:

• Added (or subtracted) numerators and denominators separately → $- \frac{2}{3}$

Answer: $4$

A negative exponent turns the fraction over, and the fraction $\frac{2}{3}$ means the cube root raised to the power $2$: $\left(\dfrac{8}{1}\right)^{2/3} = 4$.

Common mistakes, and the answer each one gives:

• Ignored the minus sign, so the fraction was never turned over → $\frac{1}{4}$
• Took the root but never applied the power → $2$
• Turned the fraction over and stopped there → $8$

Answer: $\frac{9}{5}$

The bars mean the inside is $9$ away from zero in EITHER direction, so there are two equations: $5 x = 9$ and $5 x = -9$. They give $\frac{9}{5}$ and $- \frac{9}{5}$; the larger is $\frac{9}{5}$.

Common mistakes, and the answer each one gives:

• Gave the smaller of the two solutions → $- \frac{9}{5}$
• Solved the positive branch, then negated the answer instead of solving the second branch → $- \frac{9}{5}$

Answer: $\frac{25}{22}$

Multiplication comes before addition, so work out $\dfrac{4}{11} \times \dfrac{9}{8} = \frac{9}{22}$ first, then add $\dfrac{8}{11}$: $\frac{25}{22}$.

Common mistakes, and the answer each one gives:

• Worked strictly left to right, adding before multiplying → $\frac{27}{22}$

Answer: $\frac{9 x + 44}{\left(x + 4\right) \left(x + 6\right)}$

The common denominator is $(x+4)(x+6)$. $\dfrac{4(x+6) + 5(x+4)}{(x+4)(x+6)} = \frac{9 x + 44}{\left(x + 4\right) \left(x + 6\right)}$.

Common mistakes, and the answer each one gives:

• Added numerators and denominators separately → $\frac{9}{2 x + 10}$
• Used $(x+4+6)$ as the common denominator → $\frac{9}{x + 10}$

Answer: $25 x^{2} - 36$

$(p \pm q)^2 = p^2 \pm 2pq + q^2$ and $(p+q)(p-q) = p^2 - q^2$. Here that gives $25 x^{2} - 36$.

Common mistakes, and the answer each one gives:

• Kept a middle term; in this product the two cancel → $25 x^{2} + 60 x - 36$
• Added the squares instead of subtracting → $25 x^{2} + 36$
• Squared only the first term → $25 x^{2} - 6$

Answer: $\left(x - 4\right) \left(x + 1\right)$

Find two numbers whose PRODUCT is $-4$ and whose SUM is $3$: they are $4$ and $-1$. So the factorisation is $\left(x - 4\right) \left(x + 1\right)$.

Common mistakes, and the answer each one gives:

• Used the numbers with their signs unchanged in the brackets → $x^{2} + 3 x - 4$
• Matched the sum to the constant and the product to the middle term → $x^{2} - 4 x + 3$

Answer: $x^{3}$

Multiplying adds the exponents and dividing subtracts them: $7 + 2 - 6 = 3$, so the answer is $x^{3}$.

Common mistakes, and the answer each one gives:

• Multiplied the exponents instead of adding them → $x^{8}$
• Divided the exponents instead of subtracting → $x^{\frac{3}{2}}$

Answer: $\frac{x + 7}{x - 2}$

Dividing by a fraction is multiplying by its reciprocal: $\dfrac{x + 7}{x - 3} \times \dfrac{x - 3}{x - 2}$. The $x - 3$ then cancels, leaving $\frac{x + 7}{x - 2}$.

Common mistakes, and the answer each one gives:

• Multiplied straight across without flipping the second fraction → $\frac{x^{2} + 5 x - 14}{x^{2} - 6 x + 9}$
• Flipped the FIRST fraction instead of the second → $\frac{x^{2} - 6 x + 9}{x^{2} + 5 x - 14}$

Answer: $\frac{5}{4}$

$m = \dfrac{y_2 - y_1}{x_2 - x_1} = \dfrac{-7 - (-2)}{-2 - (2)} = \frac{5}{4}$.

Common mistakes, and the answer each one gives:

• Inverted the fraction (run over rise) → $\frac{4}{5}$
• Subtracted the coordinates in opposite orders → $- \frac{5}{4}$

Answer: $- \frac{71}{2}$

Expand the bracket: $7 x + 63 = 5 x - 8$. Collect the $x$ terms on one side and the numbers on the other, then divide: $x = - \frac{71}{2}$.

Common mistakes, and the answer each one gives:

• Multiplied only the $x$ by the number outside the bracket → $- \frac{17}{2}$
• Moved a term across without changing its sign → $- \frac{47}{2}$

Answer: $\frac{41}{2}$

Cross-multiply to clear both denominators: $3\left(x + 2\right) = 5\left(x - 7\right)$. Expand, collect the $x$ terms and divide: $x = \frac{41}{2}$.

Common mistakes, and the answer each one gives:

• Multiplied only the numerators by the other denominator → $\frac{13}{2}$
• Added the fractions instead of cross-multiplying → $\frac{29}{8}$

Answer: $24$

Rearrange Pythagoras: the missing leg squared is $26^2 - 10^2 = 576$, so the leg is $\sqrt{576} = 24$.

Common mistakes, and the answer each one gives:

• Subtracted the lengths instead of their squares → $16$
• Added the squares, which is the rule for the HYPOTENUSE → $2 \sqrt{194}$

Answer: $85$

The three angles of any triangle add to $180^\circ$, so the third is $180 - 61 - 34 = 85$.

Common mistakes, and the answer each one gives:

• Used $360^\circ$, which is the angles round a POINT → $265$
• Subtracted only one of the two given angles → $119$

Answer: $1080$

A polygon with $8$ sides splits into $6$ triangles from one corner, and each triangle contributes $180^\circ$: $(8 - 2) \times 180 = 1080$.

Common mistakes, and the answer each one gives:

• Multiplied the number of sides by 180 without subtracting 2 → $1440$
• Gave the EXTERIOR angle sum, which is 360 for every polygon → $360$

Answer: $64 \pi$

Area $= \pi r^2 = \pi \times 8^2 = 64\pi$.

Common mistakes, and the answer each one gives:

• Used the circumference formula $2\pi r$ → $16 \pi$
• Doubled the radius instead of squaring it → $16 \pi$

Answer: $26$

The horizontal gap is $10$ and the vertical gap is $24$, so Pythagoras gives $\sqrt{10^2 + 24^2} = \sqrt{676} = 26$.

Common mistakes, and the answer each one gives:

• Added the two gaps instead of using Pythagoras → $34$
• Forgot the square root → $676$

Answer: $\frac{33}{56}$

Write both over the common denominator $56$, combine the numerators, then cancel: $\frac{33}{56}$.

Common mistakes, and the answer each one gives:

• Added (or subtracted) numerators and denominators separately → $5$

Answer: $\frac{27}{343}$

A negative exponent turns the fraction over, and the fraction $\frac{3}{2}$ means the square root raised to the power $3$: $\left(\dfrac{9}{49}\right)^{3/2} = \frac{27}{343}$.

Common mistakes, and the answer each one gives:

• Ignored the minus sign, so the fraction was never turned over → $\frac{343}{27}$
• Took the root but never applied the power → $\frac{3}{7}$
• Turned the fraction over and stopped there → $\frac{9}{49}$

Answer: $1$

The bars mean the inside is $9$ away from zero in EITHER direction, so there are two equations: $6 x + 3 = 9$ and $6 x + 3 = -9$. They give $1$ and $-2$; the larger is $1$.

Common mistakes, and the answer each one gives:

• Gave the smaller of the two solutions → $-2$
• Solved the positive branch, then negated the answer instead of solving the second branch → $-1$

Answer: $\frac{74}{45}$

Multiplication comes before addition, so work out $\dfrac{1}{10} \times \dfrac{4}{9} = \frac{2}{45}$ first, then add $\dfrac{8}{5}$: $\frac{74}{45}$.

Common mistakes, and the answer each one gives:

• Worked strictly left to right, adding before multiplying → $\frac{34}{45}$

Answer: $\frac{14 x + 23}{\left(x + 1\right) \left(x + 2\right)}$

The common denominator is $(x+2)(x+1)$. $\dfrac{5(x+1) + 9(x+2)}{(x+2)(x+1)} = \frac{14 x + 23}{\left(x + 1\right) \left(x + 2\right)}$.

Common mistakes, and the answer each one gives:

• Added numerators and denominators separately → $\frac{14}{2 x + 3}$
• Used $(x+2+1)$ as the common denominator → $\frac{14}{x + 3}$

Answer: $36 x^{2} - 24 x + 4$

$(p \pm q)^2 = p^2 \pm 2pq + q^2$ and $(p+q)(p-q) = p^2 - q^2$. Here that gives $36 x^{2} - 24 x + 4$.

Common mistakes, and the answer each one gives:

• Squared each term and left out the middle term → $36 x^{2} + 4$
• Gave the middle term the wrong sign → $36 x^{2} + 24 x + 4$
• Doubled the whole bracket instead of squaring it → $12 x - 4$

Answer: $\left(x - 3\right) \left(x + 1\right)$

Find two numbers whose PRODUCT is $-3$ and whose SUM is $2$: they are $-1$ and $3$. So the factorisation is $\left(x - 3\right) \left(x + 1\right)$.

Common mistakes, and the answer each one gives:

• Used the numbers with their signs unchanged in the brackets → $x^{2} + 2 x - 3$
• Matched the sum to the constant and the product to the middle term → $x^{2} - 3 x + 2$

Answer: $x^{7}$

Multiplying adds the exponents and dividing subtracts them: $7 + 3 - 3 = 7$, so the answer is $x^{7}$.

Common mistakes, and the answer each one gives:

• Multiplied the exponents instead of adding them → $x^{18}$
• Divided the exponents instead of subtracting → $x^{\frac{10}{3}}$

Answer: $\frac{x + 6}{x + 5}$

Dividing by a fraction is multiplying by its reciprocal: $\dfrac{x + 6}{x + 7} \times \dfrac{x + 7}{x + 5}$. The $x + 7$ then cancels, leaving $\frac{x + 6}{x + 5}$.

Common mistakes, and the answer each one gives:

• Multiplied straight across without flipping the second fraction → $\frac{x^{2} + 11 x + 30}{x^{2} + 14 x + 49}$
• Flipped the FIRST fraction instead of the second → $\frac{x^{2} + 14 x + 49}{x^{2} + 11 x + 30}$

Answer: $- \frac{9}{4}$

$m = \dfrac{y_2 - y_1}{x_2 - x_1} = \dfrac{4 - (-5)}{-5 - (-1)} = - \frac{9}{4}$.

Common mistakes, and the answer each one gives:

• Inverted the fraction (run over rise) → $- \frac{4}{9}$
• Subtracted the coordinates in opposite orders → $\frac{9}{4}$

Answer: $- \frac{13}{3}$

Expand the bracket: $4 x - 4 = 7 x + 9$. Collect the $x$ terms on one side and the numbers on the other, then divide: $x = - \frac{13}{3}$.

Common mistakes, and the answer each one gives:

• Multiplied only the $x$ by the number outside the bracket → $- \frac{10}{3}$
• Moved a term across without changing its sign → $\frac{14}{3}$

Answer: $-31$

Cross-multiply to clear both denominators: $6\left(x + 1\right) = 5\left(x - 5\right)$. Expand, collect the $x$ terms and divide: $x = -31$.

Common mistakes, and the answer each one gives:

• Multiplied only the numerators by the other denominator → $-11$
• Added the fractions instead of cross-multiplying → $\frac{19}{11}$

Answer: $29$

Pythagoras: the square on the hypotenuse equals the sum of the squares on the legs. $20^2 + 21^2 = 841$, and $\sqrt{841} = 29$.

Common mistakes, and the answer each one gives:

• Added the legs instead of their squares → $41$
• Subtracted the squares, which finds a LEG not the hypotenuse → $\sqrt{41}$