OMPT-E mixed paper

Answer: $- \frac{8}{63}$

Write both over the common denominator $63$, combine the numerators, then cancel: $- \frac{8}{63}$.

Common mistakes, and the answer each one gives:

• Added (or subtracted) numerators and denominators separately → $1$

Answer: $\frac{8}{125}$

A negative exponent turns the fraction over, and the fraction $\frac{3}{2}$ means the square root raised to the power $3$: $\left(\dfrac{4}{25}\right)^{3/2} = \frac{8}{125}$.

Common mistakes, and the answer each one gives:

• Ignored the minus sign, so the fraction was never turned over → $\frac{125}{8}$
• Took the root but never applied the power → $\frac{2}{5}$
• Turned the fraction over and stopped there → $\frac{4}{25}$

Answer: $\frac{8}{3}$

The bars mean the inside is $8$ away from zero in EITHER direction, so there are two equations: $6 x - 8 = 8$ and $6 x - 8 = -8$. They give $\frac{8}{3}$ and $0$; the larger is $\frac{8}{3}$.

Common mistakes, and the answer each one gives:

• Gave the smaller of the two solutions → $0$
• Solved the positive branch, then negated the answer instead of solving the second branch → $- \frac{8}{3}$

Answer: $\frac{13}{5}$

Multiplication comes before addition, so work out $\dfrac{1}{12} \times \dfrac{6}{5} = \frac{1}{10}$ first, then add $\dfrac{5}{2}$: $\frac{13}{5}$.

Common mistakes, and the answer each one gives:

• Worked strictly left to right, adding before multiplying → $\frac{31}{10}$

Answer: $\frac{4 \left(2 x + 7\right)}{\left(x + 3\right) \left(x + 5\right)}$

The common denominator is $(x+3)(x+5)$. $\dfrac{2(x+5) + 6(x+3)}{(x+3)(x+5)} = \frac{4 \left(2 x + 7\right)}{\left(x + 3\right) \left(x + 5\right)}$.

Common mistakes, and the answer each one gives:

• Added numerators and denominators separately → $\frac{8}{2 x + 8}$
• Used $(x+3+5)$ as the common denominator → $\frac{8}{x + 8}$

Answer: $9 x^{2} + 54 x + 81$

$(p \pm q)^2 = p^2 \pm 2pq + q^2$ and $(p+q)(p-q) = p^2 - q^2$. Here that gives $9 x^{2} + 54 x + 81$.

Common mistakes, and the answer each one gives:

• Squared each term and left out the middle term → $9 x^{2} + 81$
• Gave the middle term the wrong sign → $9 x^{2} - 54 x + 81$
• Doubled the whole bracket instead of squaring it → $6 x + 18$

Answer: $\left(x - 2\right) \left(x + 3\right)$

Find two numbers whose PRODUCT is $-6$ and whose SUM is $-1$: they are $2$ and $-3$. So the factorisation is $\left(x - 2\right) \left(x + 3\right)$.

Common mistakes, and the answer each one gives:

• Used the numbers with their signs unchanged in the brackets → $x^{2} - x - 6$
• Matched the sum to the constant and the product to the middle term → $x^{2} - 1$

Answer: $x^{5}$

Multiplying adds the exponents and dividing subtracts them: $4 + 5 - 4 = 5$, so the answer is $x^{5}$.

Common mistakes, and the answer each one gives:

• Multiplied the exponents instead of adding them → $x^{16}$
• Divided the exponents instead of subtracting → $x^{\frac{9}{4}}$

Answer: $\frac{x - 2}{x + 6}$

Dividing by a fraction is multiplying by its reciprocal: $\dfrac{x - 2}{x + 3} \times \dfrac{x + 3}{x + 6}$. The $x + 3$ then cancels, leaving $\frac{x - 2}{x + 6}$.

Common mistakes, and the answer each one gives:

• Multiplied straight across without flipping the second fraction → $\frac{x^{2} + 4 x - 12}{x^{2} + 6 x + 9}$
• Flipped the FIRST fraction instead of the second → $\frac{x^{2} + 6 x + 9}{x^{2} + 4 x - 12}$

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

$m = \dfrac{y_2 - y_1}{x_2 - x_1} = \dfrac{15 - (6)}{7 - (3)} = \frac{9}{4}$.

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: $-16$

Expand the bracket: $3 x - 15 = 4 x + 1$. Collect the $x$ terms on one side and the numbers on the other, then divide: $x = -16$.

Common mistakes, and the answer each one gives:

• Multiplied only the $x$ by the number outside the bracket → $-6$
• Moved a term across without changing its sign → $-13$

Answer: $- \frac{33}{2}$

Cross-multiply to clear both denominators: $7\left(x + 6\right) = 3\left(x - 8\right)$. Expand, collect the $x$ terms and divide: $x = - \frac{33}{2}$.

Common mistakes, and the answer each one gives:

• Multiplied only the numerators by the other denominator → $- \frac{25}{2}$
• Added the fractions instead of cross-multiplying → $- \frac{9}{5}$

Answer: $0$

Eliminate $y$ by scaling and adding, or substitute. The solution is $x = 0$, $y = -4$.

Common mistakes, and the answer each one gives:

• Solved for y instead of x → $-4$

Answer: $5$

The first equation already gives $y$, so put it straight into the second: $2x + 3\left(3 x - 19\right) = -2$. That leaves one unknown, and $x = 5$.

Common mistakes, and the answer each one gives:

• Gave the value of $y$ rather than of $x$ → $-4$
• Substituted into the equation it came from, which says nothing → $8$

Answer: $8$

$\left(x - 8\right) \left(x - 7\right) = 0$, so $x = 8$ or $x = 7$. The larger is $8$.

Common mistakes, and the answer each one gives:

• Gave the smaller root → $7$
• Read the roots off with the signs unchanged → $-7$

Answer: $- \frac{1}{2} + \frac{\sqrt{19}}{2}$

The discriminant is $b^2 - 4ac = 76$, which is positive but not a perfect square, so the roots are irrational and the formula is needed: $x = \dfrac{-b \pm \sqrt{b^2-4ac}}{2a}$. The larger root is $- \frac{1}{2} + \frac{\sqrt{19}}{2}$.

Common mistakes, and the answer each one gives:

• Used $+b$ in the numerator instead of $-b$ → $\frac{1}{2} + \frac{\sqrt{19}}{2}$
• Divided by $a$ instead of $2a$ → $-1 + \sqrt{19}$
• Gave the smaller root → $- \frac{\sqrt{19}}{2} - \frac{1}{2}$

Answer: $-5$

The turning point of $ax^2+bx+c$ sits at $x = -\dfrac{b}{2a}$. Here $a = -3$ and $b = -30$, so $x = -\dfrac{-30}{2(-3)} = -5$. (Completing the square gives $-3\left(x + 5\right)^2 - 7$, which shows the same thing.)

Common mistakes, and the answer each one gives:

• Used $\dfrac{b}{2a}$ without the minus sign → $5$
• Gave the MAXIMUM VALUE instead of where it happens → $-7$

Answer: $4$

Where the graphs meet, the two expressions are equal. Bring everything to one side: $x^{2} + 2 x - 24 = 0$, which factorises as $\left(x - 4\right) \left(x + 6\right)$. The roots are $-6$ and $4$.

Common mistakes, and the answer each one gives:

• Gave the smaller root → $-6$
• Read the roots straight off the brackets with the signs unchanged → $6$

Answer: $-6$

The vertical asymptote sits where the denominator is zero: $x + 6 = 0$, so $x = -6$. (The horizontal one is $y = 1$, the ratio of the leading coefficients.)

Common mistakes, and the answer each one gives:

• Set the NUMERATOR to zero (that gives the x-intercept) → $-5$
• Gave the horizontal asymptote instead → $1$
• Forgot the sign when solving → $6$

Answer: $\frac{x - 2}{x + 6}$

Factorise both: $\dfrac{\left(x - 2\right) \left(x + 1\right)}{\left(x + 1\right) \left(x + 6\right)}$. The factor $x + 1$ is common to both and cancels, leaving $\frac{x - 2}{x + 6}$.

Common mistakes, and the answer each one gives:

• Cancelled the $x^2$ terms, which are terms and not factors → $\frac{- x - 2}{7 x + 6}$
• Cancelled the wrong bracket → $\frac{x + 1}{x + 6}$

Answer: $50 x^{2} - 100 x + 44$

Square the bracket FIRST, then multiply by $2$, then add the constant — the order matters. $\left(5 x - 5\right)^2 = 25 x^{2} - 50 x + 25$, and the whole thing comes to $50 x^{2} - 100 x + 44$.

Common mistakes, and the answer each one gives:

• Multiplied the bracket by the outside number before squaring → $100 x^{2} - 200 x + 94$
• Squared each term inside the bracket separately → $50 x^{2} + 44$

Answer: $18$

Multiply both sides by $x - 5$: $4 x - 7 = 5\left(x - 5\right)$. Expand, collect and divide: $x = 18$. (Check it is not $x = 5$, which the original forbids.)

Common mistakes, and the answer each one gives:

• Multiplied only the numerator by the right-hand side → $3$
• Set the denominator to zero, which is the EXCLUDED value → $5$

Answer: $-5$

A cube is undone by a cube root, and unlike a square root it keeps the sign: $\sqrt[3]{-1} = -1$. So $x + 4 = -1$ and $x = -5$.

Common mistakes, and the answer each one gives:

• Divided by 3 instead of taking the cube root → $- \frac{13}{3}$
• Took the cube root but lost the sign → $-3$

Answer: $5 x^{4}$

The power rule turns $4\log_{7}(x)$ into $\log_{7}(x^{4})$, and the product rule combines the two into $\log_{7}(5 x^{4})$.

Common mistakes, and the answer each one gives:

• Added the arguments instead of multiplying them → $x^{4} + 5$
• Multiplied by the power instead of raising to it → $20 x$

Answer: $5$

Equal logarithms with the same base have equal arguments, so $x + 1 = 6$ and $x = 5$. Check the domain: $x + 1 > 0$ holds.

Common mistakes, and the answer each one gives:

• Subtracted the shift from the wrong side → $7$
• Treated the log as a multiplier and divided → $6$

Answer: $0$

Write the right-hand side as a power of $5$: $625 = 5^{4}$. With the same base on both sides the exponents must be equal, so $2 x + 4 = 4$ and $x = 0$.

Common mistakes, and the answer each one gives:

• Set the exponent equal to 625 instead of to 4 → $\frac{621}{2}$
• Forgot to move the constant in the exponent across → $2$

Answer: $\frac{2}{3}$

Write both numbers as powers of $2$: $8 = 2^{3}$ and $4 = 2^{2}$. Then $\log_{8}4 = \dfrac{2}{3} = \frac{2}{3}$, because the base has to be raised to that power to reach the argument.

Common mistakes, and the answer each one gives:

• Divided the two numbers → $\frac{1}{2}$
• Turned the fraction of exponents upside down → $\frac{3}{2}$

Answer: $24$

Put the values in order first — $6, 18, 20, 24, 26, 29, 30$ — then take the middle one: $24$.

Common mistakes, and the answer each one gives:

• Gave the mean instead of the median → $\frac{153}{7}$
• Took the middle of the list AS WRITTEN, without sorting it first → $6$

Answer: $27$

The range is the largest value minus the smallest: $29 - 2 = 27$.

Common mistakes, and the answer each one gives:

• Gave the variance instead of the range → $\frac{2546}{25}$
• Added the distances from the mean without squaring them → $\frac{228}{25}$

Answer: $210$

The first place has $7$ candidates, the next $6$, and so on for $3$ places: $\dfrac{7!}{(7 - 3)!} = 210$.

Common mistakes, and the answer each one gives:

• Used combinations, which ignore the order → $35$
• Arranged all of them rather than only the chosen ones → $5040$