OMPT-E mixed paper

Answer: $\dfrac{21}{22}$

Dividing by a fraction is multiplying by it turned over: $\dfrac{7}{11} \div \dfrac{2}{3} = \dfrac{7}{11} \times \dfrac{3}{2} = \dfrac{21}{22}\mathclose{.}$

Common mistakes, and the answer each one gives:

• Multiplied without turning the second fraction over → $\frac{14}{33}$
• Turned the FIRST fraction over instead of the second → $\frac{22}{21}$

Answer: $0.014$

Multiply as whole numbers, $7 \times 2 = 14\mathclose{,}$ then count the decimal places of BOTH factors — $1 + 2 = 3$ — so the answer is $0.014\mathclose{.}$

Common mistakes, and the answer each one gives:

• Counted the decimal places of only one factor → $0.14$

Answer: $3$

Work out each inside first, then make it non-negative: $-2 - 7 = -9\mathclose{,}$ so the first bar gives $9\mathclose{;}$ $-9 + 3 = -6\mathclose{,}$ so the second gives $6\mathclose{.}$ Then $9 - 6 = 3\mathclose{.}$

Common mistakes, and the answer each one gives:

• Dropped the bars without making the insides positive → $-3$
• Applied the bars to each number on its own, as if $|a - b| = |a| - |b|$ → $-17$

Answer: $x^{9}$

Multiplying adds the exponents and dividing subtracts them: $5 + 6 - 2 = 9\mathclose{,}$ so the answer is $x^{9}\mathclose{.}$

Common mistakes, and the answer each one gives:

• Multiplied the exponents instead of adding them → $x^{28}$
• Divided the exponents instead of subtracting → $x^{\frac{11}{2}}$

Answer: $36 x^{2} - 64$

$(p + q)(p - q) = p^2 - q^2\mathclose{,}$ and the two middle terms cancel: $\left(6 x + 8\right)\left(6 x - 8\right) = \left(6x\right)^{2} - 8^{2} = 36 x^{2} - 64\mathclose{.}$

Common mistakes, and the answer each one gives:

• Treated it as a square, as if both brackets were the same → $36 x^{2} + 96 x + 64$
• Added the squares instead of subtracting → $36 x^{2} + 64$
• Squared the $x$ but not the $6$ in front of it → $6 x^{2} - 64$

Answer: $\dfrac{5 x + 9}{x - 6}$

The denominators are the same, so keep the denominator and add the numerators: $\dfrac{3 x + 9 + \left(2 x\right)}{x - 6} = \dfrac{5 x + 9}{x - 6}\mathclose{.}$

Common mistakes, and the answer each one gives:

• Added the denominators as well → $\dfrac{5 x + 9}{2 x - 12}$

Answer: $y = - x + 7$

Parallel lines have the SAME slope, $-1\mathclose{.}$ Through $(2, 5)\mathclose{:}$ $y = -\left(x - 2\right) + 5\mathclose{,}$ which is $y = - x + 7\mathclose{.}$

Common mistakes, and the answer each one gives:

• Used the perpendicular slope → $y = x + 3$
• Kept the intercept of the given line → $y = - x + 3$

Answer: $\dfrac{1}{2}$

Expand the bracket: $2 x - 2 = 4 x - 3\mathclose{.}$ Put the $x$ terms on the left and the numbers on the right, changing the sign of each term that crosses over: $- 2 x = -1\mathclose{,}$ so $x = \dfrac{1}{2}\mathclose{.}$

Common mistakes, and the answer each one gives:

• Multiplied only the $x$ by the number outside the bracket → $1$
• Moved the number across without changing its sign → $\frac{5}{2}$

Answer: $8$

Multiply both sides by $18$ to clear the denominators: $3x = 6\left(x - 4\right)\mathclose{.}$ Expand: $3 x = 6 x - 24\mathclose{.}$ Collect: $- 3 x = -24\mathclose{,}$ so $x = 8\mathclose{.}$

Common mistakes, and the answer each one gives:

• Multiplied out $6\left(x - 4\right)$ as $6 x - 4\mathclose{,}$ multiplying only the $x$ → $\frac{4}{3}$
• Cross-multiplied the wrong way, writing $6\left(x\right) = 3\left(x - 4\right)$ → $-4$

Answer: $6$

The first equation already gives $y\mathclose{,}$ so put it straight into the second: $x + 4\left(x - 12\right) = -18\mathclose{.}$ Expand: $5 x - 48 = -18\mathclose{,}$ so $5 x = 30$ and $x = 6\mathclose{.}$ (Then the first equation gives $y = -6\mathclose{.})$

Common mistakes, and the answer each one gives:

• Gave the value of $y$ instead of $x$ → $-6$
• Multiplied only the first term of the bracket by $4$ → $- \frac{6}{5}$

Answer: $-4$

The first equation already gives $y\mathclose{,}$ so put it straight into the second: $3 x + \left(- x - 6\right) = -14\mathclose{.}$ Expand: $2 x - 6 = -14\mathclose{,}$ so $2 x = -8$ and $x = -4\mathclose{.}$ (Then the first equation gives $y = -2\mathclose{.})$

Common mistakes, and the answer each one gives:

• Gave the value of $y$ instead of $x$ → $-2$

Answer: $y = 3 x^{2} - 30 x + 73$

In that order: the stretch gives $y = 3x^{2}\mathclose{,}$ the shifts then give $y = 3\left(x - 5\right)^{2} - 2\mathclose{,}$ which is $y = 3 x^{2} - 30 x + 73\mathclose{.}$

Common mistakes, and the answer each one gives:

• Stretched AFTER shifting up or down, so the vertical shift was stretched too → $y = 3 x^{2} - 30 x + 69$

Answer: $9$

Factorise: $2 x^{2} - 15 x - 27 = \left(2 x + 3\right)\left(x - 9\right) = 0\mathclose{,}$ so $2 x + 3 = 0$ or $x - 9 = 0\mathclose{:}$ $x = -\dfrac{3}{2}$ or $x = 9\mathclose{.}$ The larger is $9\mathclose{.}$

Common mistakes, and the answer each one gives:

• Gave the smaller solution → $- \frac{3}{2}$
• Read the solutions off the brackets with the signs unchanged → $\frac{3}{2}$

Answer: $\dfrac{-7 + \sqrt{61}}{6}$

Here $a = 3\mathclose{,}$ $b = 7$ and $c = -1\mathclose{,}$ so the discriminant is $b^2 - 4ac = 49 - (-12) = 61$ — positive but not a perfect square, so the formula is needed: $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \dfrac{-7 \pm \sqrt{61}}{6}\mathclose{.}$ The larger solution is $\dfrac{-7 + \sqrt{61}}{6}\mathclose{.}$

Common mistakes, and the answer each one gives:

• Used $+b$ in the numerator instead of $-b$ → $\dfrac{7 + \sqrt{61}}{6}$
• Divided by $a$ instead of $2a$ → $\dfrac{-7 + \sqrt{61}}{3}$
• Gave the smaller solution → $\dfrac{-7 - \sqrt{61}}{6}$

Answer: $(-\infty, -8)$

$2^{x}$ takes every POSITIVE value and never reaches $0\mathclose{.}$ Times $-2$ it is every NEGATIVE value, and taking $8$ off moves the asymptote to $y = -8\mathclose{,}$ which is approached and never reached: the range is $(-\infty, -8)\mathclose{.}$

Common mistakes, and the answer each one gives:

• Included the asymptote itself → $(-\infty, -8]$
• Ignored the minus sign in front → $(-8, \infty)$

Answer: $[-1, 7) \cup (7, \infty)$

The root needs $x + 1 \ge 0\mathclose{,}$ so $x \ge -1\mathclose{,}$ and the denominator is zero at $x = 7\mathclose{,}$ which is left out: the domain is $[-1, 7) \cup (7, \infty)\mathclose{.}$

Common mistakes, and the answer each one gives:

• Forgot that $x = 7$ makes the denominator zero → $[-1, \infty)$
• Required the inside of the root to be strictly positive → $(-1, 7) \cup (7, \infty)$

Answer: $\dfrac{x + 7}{x - 5}$

Factorise both: $\dfrac{\left(x + 7\right) \left(x + 8\right)}{\left(x - 5\right) \left(x + 8\right)}\mathclose{.}$ The factor $x + 8$ is common to both and cancels, leaving $\dfrac{x + 7}{x - 5}\mathclose{.}$

Common mistakes, and the answer each one gives:

• Cancelled the wrong bracket, keeping the common one → $\frac{x + 8}{x - 5}$
• Cancelled the $x^2$ terms, which are terms and not factors → $\frac{15 x + 56}{3 x - 40}$

Answer: $x = 3$

Square both sides: $x + 22 = \left(x + 2\right)^{2}\mathclose{,}$ so $x^{2} + 3 x - 18 = 0$ and $x = 3$ or $x = -6\mathclose{.}$ SQUARING CAN ADD SOLUTIONS, so check both in the original: at $x = 3$ both sides are $5\mathclose{;}$ at $x = -6$ the right-hand side is $-4\mathclose{,}$ negative, while a square root never is. Only $x = 3$ survives.

Common mistakes, and the answer each one gives:

• Kept both solutions of the squared equation without checking them → $x = -6 \text{ or } x = 3$
• Gave the solution that fails the check → $x = -6$

Answer: $\dfrac{- 9 x - 1}{x + 1}$

Write $y = \dfrac{- x - 1}{x + 9}$ and solve for $x\mathclose{:}$ $y\left(x + 9\right) = - x - 1\mathclose{,}$ so $x y + 9 y = - x - 1\mathclose{.}$ Gather the $x$ terms: $x\left(y + 1\right) = - 9 y - 1\mathclose{,}$ so $x = \dfrac{- 9 y - 1}{y + 1}\mathclose{.}$ Swap the letters: $f^{-1}(x) = \dfrac{- 9 x - 1}{x + 1}\mathclose{.}$

Common mistakes, and the answer each one gives:

• Gave the reciprocal $\frac{1}{f(x)}$ instead of the inverse → $\frac{x + 9}{- x - 1}$
• Swapped $a$ and $d$ but did not change the signs of $b$ and $c$ → $\frac{9 x - 1}{x - 1}$

Answer: $\frac{x^{4}}{9}$

The power rule turns $4\log_{3}(x)$ into $\log_{3}(x^{4})\mathclose{,}$ and the quotient rule combines the two into $\log_{3}\left(\frac{x^{4}}{9}\right)\mathclose{.}$

Common mistakes, and the answer each one gives:

• Subtracted the arguments instead of dividing them → $x^{4} - 9$
• Divided the wrong way round → $\frac{9}{x^{4}}$
• Multiplied by the power instead of raising to it → $\frac{4 x}{9}$

Answer: $\log_{5}(13)$

$13$ is not a whole power of $5\mathclose{,}$ so take the logarithm with base $5$ of both sides: $x = \log_{5}(13)\mathclose{.}$ (As a quotient of natural logarithms, $x = \dfrac{\ln 13}{\ln 5}\mathclose{.})$

Common mistakes, and the answer each one gives:

• Divided $13$ by $5$ → $\dfrac{13}{5}$
• Took $\ln 13$ and forgot to divide by $\ln 5$ → $\ln 13$

Answer: $\text{mean}$

Interval values can be added and divided, so the mean is meaningful — a true zero is only needed for ratios.

Common mistakes, and the answer each one gives:

• Thought a mean needs a true zero → $\text{median}$

Answer: $1$

The mode is the value with the tallest bar: $1\mathclose{,}$ which occurs $8$ times.

Common mistakes, and the answer each one gives:

• Gave the height of the tallest bar instead of its value → $8$
• Gave the largest value → $4$

Answer: $123.44$

The mean is $16.4\mathclose{.}$ Square each value's distance from it, add those up — that gives $617.2$ — and divide by $5\mathclose{:}$ $123.44\mathclose{.}$

Common mistakes, and the answer each one gives:

• Divided by $n - 1\mathclose{,}$ the formula for a sample → $154.3$
• Added the distances from the mean without squaring them → $10.72$

Answer: $84$

The mean is $\bar{x} = 15\mathclose{.}$ The deviations from it are $-1, \allowbreak 5, \allowbreak 1, \allowbreak -5, \allowbreak 4, \allowbreak -4\mathclose{;}$ squared and added: $1 + 25 + 1 + 25 + 16 + 16 = 84\mathclose{.}$

Common mistakes, and the answer each one gives:

• Divided by $n\mathclose{,}$ which gives the variance instead → $14$
• Added the deviations without squaring them → $0$

Answer: $0.65$

$P(A \cup B) = P(A) + P(B) - P(A \cap B)\mathclose{:}$ the overlap is in both $A$ and $B\mathclose{,}$ so adding them counts it twice. $0.2 + 0.6 - 0.15 = 0.65\mathclose{.}$

Common mistakes, and the answer each one gives:

• Added $P(A)$ and $P(B)$ without taking away the overlap → $0.8$
• Took the overlap away twice → $0.5$

Answer: $\dfrac{5}{12}$

Given that they smoke, only the smokers count: $20$ of the $48$ smokers exercise, so the probability is $\dfrac{20}{48} = \dfrac{5}{12}\mathclose{.}$

Common mistakes, and the answer each one gives:

• Divided by everyone in the survey instead of the smokers → $\dfrac{4}{23}$
• Divided by everyone who exercises instead of the smokers → $\dfrac{20}{53}$

Answer: $4.1$

Multiply each value by its probability and add: $E(X) = (-1) \cdot 0.3 + 2 \cdot 0.2 + 8 \cdot 0.5 = 4.1\mathclose{.}$

Common mistakes, and the answer each one gives:

• Averaged the values, ignoring the probabilities → $3$

Answer: $0.15\%$

$67$ is $3$ standard deviations above the mean. $99.7\%$ lie within $3$ of the mean, so $0.3\%$ lie outside, split equally between the two tails: $0.15\%\mathclose{.}$

Common mistakes, and the answer each one gives:

• Gave the percentage inside the band instead of beyond it → $99.7\%$
• Did not halve: counted both tails → $0.3\%$

Answer: $985$

$x = \mu + z\sigma = 1000 + (-1.5) \cdot 10 = 985$ g.

Common mistakes, and the answer each one gives:

• Subtracted $z\sigma$ instead of adding it → $1015$
• Added $z$ itself, without multiplying by $\sigma$ → $998.5$