OMPT-D mixed paper

Answer: $\frac{93}{88}$

Write both over the common denominator $88$, combine the numerators, then cancel: $\frac{93}{88}$.

Common mistakes, and the answer each one gives:

• Added (or subtracted) numerators and denominators separately → $\frac{9}{19}$

Answer: $\frac{125}{64}$

A negative exponent turns the fraction over, and the fraction $\frac{3}{2}$ means the square root raised to the power $3$: $\left(\dfrac{25}{16}\right)^{3/2} = \frac{125}{64}$.

Common mistakes, and the answer each one gives:

• Ignored the minus sign, so the fraction was never turned over → $\frac{64}{125}$
• Took the root but never applied the power → $\frac{5}{4}$
• Turned the fraction over and stopped there → $\frac{25}{16}$

Answer: $\frac{7}{3}$

The bars mean the inside is $9$ away from zero in EITHER direction, so there are two equations: $3 x + 2 = 9$ and $3 x + 2 = -9$. They give $\frac{7}{3}$ and $- \frac{11}{3}$; the larger is $\frac{7}{3}$.

Common mistakes, and the answer each one gives:

• Gave the smaller of the two solutions → $- \frac{11}{3}$
• Solved the positive branch, then negated the answer instead of solving the second branch → $- \frac{7}{3}$

Answer: $\frac{4}{7}$

Multiplication comes before addition, so work out $\dfrac{8}{7} \times \dfrac{2}{11} = \frac{16}{77}$ first, then add $\dfrac{4}{11}$: $\frac{4}{7}$.

Common mistakes, and the answer each one gives:

• Worked strictly left to right, adding before multiplying → $\frac{232}{847}$

Answer: $\frac{10 x + 43}{\left(x + 4\right) \left(x + 5\right)}$

The common denominator is $(x+5)(x+4)$. $\dfrac{7(x+4) + 3(x+5)}{(x+5)(x+4)} = \frac{10 x + 43}{\left(x + 4\right) \left(x + 5\right)}$.

Common mistakes, and the answer each one gives:

• Added numerators and denominators separately → $\frac{10}{2 x + 9}$
• Used $(x+5+4)$ as the common denominator → $\frac{10}{x + 9}$

Answer: $9 x^{2} - 48 x + 64$

$(p \pm q)^2 = p^2 \pm 2pq + q^2$ and $(p+q)(p-q) = p^2 - q^2$. Here that gives $9 x^{2} - 48 x + 64$.

Common mistakes, and the answer each one gives:

• Squared each term and left out the middle term → $9 x^{2} + 64$
• Gave the middle term the wrong sign → $9 x^{2} + 48 x + 64$
• Doubled the whole bracket instead of squaring it → $6 x - 16$

Answer: $\left(x - 9\right) \left(x + 3\right)$

Find two numbers whose PRODUCT is $-27$ and whose SUM is $6$: they are $9$ and $-3$. So the factorisation is $\left(x - 9\right) \left(x + 3\right)$.

Common mistakes, and the answer each one gives:

• Used the numbers with their signs unchanged in the brackets → $x^{2} + 6 x - 27$
• Matched the sum to the constant and the product to the middle term → $x^{2} - 7 x + 6$

Answer: $x^{10}$

Multiplying adds the exponents and dividing subtracts them: $8 + 4 - 2 = 10$, so the answer is $x^{10}$.

Common mistakes, and the answer each one gives:

• Multiplied the exponents instead of adding them → $x^{30}$
• Divided the exponents instead of subtracting → $x^{6}$

Answer: $\frac{x + 4}{x}$

Dividing by a fraction is multiplying by its reciprocal: $\dfrac{x + 4}{x + 7} \times \dfrac{x + 7}{x}$. The $x + 7$ then cancels, leaving $\frac{x + 4}{x}$.

Common mistakes, and the answer each one gives:

• Multiplied straight across without flipping the second fraction → $\frac{x^{2} + 4 x}{x^{2} + 14 x + 49}$
• Flipped the FIRST fraction instead of the second → $\frac{x^{2} + 14 x + 49}{x^{2} + 4 x}$

Answer: $4$

$m = \dfrac{y_2 - y_1}{x_2 - x_1} = \dfrac{16 - (8)}{3 - (1)} = 4$.

Common mistakes, and the answer each one gives:

• Inverted the fraction (run over rise) → $\frac{1}{4}$
• Subtracted the coordinates in opposite orders → $-4$

Answer: $-18$

Expand the bracket: $2 x - 12 = 3 x + 6$. Collect the $x$ terms on one side and the numbers on the other, then divide: $x = -18$.

Common mistakes, and the answer each one gives:

• Multiplied only the $x$ by the number outside the bracket → $-12$
• Moved a term across without changing its sign → $0$

Answer: $-25$

Cross-multiply to clear both denominators: $4\left(x + 4\right) = 3\left(x - 3\right)$. Expand, collect the $x$ terms and divide: $x = -25$.

Common mistakes, and the answer each one gives:

• Multiplied only the numerators by the other denominator → $-19$
• Added the fractions instead of cross-multiplying → $-1$

Answer: $4$

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

Common mistakes, and the answer each one gives:

• Solved for y instead of x → $-4$

Answer: $-3$

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

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 → $2$

Answer: $2$

$\left(x - 2\right) \left(x + 4\right) = 0$, so $x = 2$ or $x = -4$. The larger is $2$.

Common mistakes, and the answer each one gives:

• Gave the smaller root → $-4$
• Read the roots off with the signs unchanged → $4$

Answer: $-1 + \frac{\sqrt{33}}{3}$

The discriminant is $b^2 - 4ac = 132$, 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 $-1 + \frac{\sqrt{33}}{3}$.

Common mistakes, and the answer each one gives:

• Used $+b$ in the numerator instead of $-b$ → $1 + \frac{\sqrt{33}}{3}$
• Divided by $a$ instead of $2a$ → $-2 + \frac{2 \sqrt{33}}{3}$
• Gave the smaller root → $- \frac{\sqrt{33}}{3} - 1$

Answer: $-6$

The turning point of $ax^2+bx+c$ sits at $x = -\dfrac{b}{2a}$. Here $a = -3$ and $b = -36$, so $x = -\dfrac{-36}{2(-3)} = -6$. (Completing the square gives $-3\left(x + 6\right)^2 + 2$, which shows the same thing.)

Common mistakes, and the answer each one gives:

• Used $\dfrac{b}{2a}$ without the minus sign → $6$
• Gave the MAXIMUM VALUE instead of where it happens → $2$

Answer: $5$

Where the graphs meet, the two expressions are equal. Bring everything to one side: $x^{2} - 3 x - 10 = 0$, which factorises as $\left(x - 5\right) \left(x + 2\right)$. The roots are $-2$ and $5$.

Common mistakes, and the answer each one gives:

• Gave the smaller root → $-2$
• Read the roots straight off the brackets with the signs unchanged → $2$

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

The vertical asymptote sits where the denominator is zero: $2 x + 7 = 0$, so $x = - \frac{7}{2}$. (The horizontal one is $y = \frac{3}{2}$, the ratio of the leading coefficients.)

Common mistakes, and the answer each one gives:

• Set the NUMERATOR to zero (that gives the x-intercept) → $- \frac{2}{3}$
• Gave the horizontal asymptote instead → $\frac{3}{2}$
• Forgot the sign when solving → $\frac{7}{2}$

Answer: $\frac{x - 6}{x - 7}$

Factorise both: $\dfrac{\left(x - 6\right) \left(x + 1\right)}{\left(x - 7\right) \left(x + 1\right)}$. The factor $x + 1$ is common to both and cancels, leaving $\frac{x - 6}{x - 7}$.

Common mistakes, and the answer each one gives:

• Cancelled the $x^2$ terms, which are terms and not factors → $\frac{5 x + 6}{6 x + 7}$
• Cancelled the wrong bracket → $\frac{x + 1}{x - 7}$

Answer: $48 x^{2} - 96 x + 45$

Square the bracket FIRST, then multiply by $3$, then add the constant — the order matters. $\left(4 x - 4\right)^2 = 16 x^{2} - 32 x + 16$, and the whole thing comes to $48 x^{2} - 96 x + 45$.

Common mistakes, and the answer each one gives:

• Multiplied the bracket by the outside number before squaring → $144 x^{2} - 288 x + 141$
• Squared each term inside the bracket separately → $48 x^{2} + 45$

Answer: $-12$

Multiply both sides by $x + 3$: $6 x + 9 = 7\left(x + 3\right)$. Expand, collect and divide: $x = -12$. (Check it is not $x = -3$, which the original forbids.)

Common mistakes, and the answer each one gives:

• Multiplied only the numerator by the right-hand side → $- \frac{1}{3}$
• Set the denominator to zero, which is the EXCLUDED value → $-3$

Answer: $1$

A cube is undone by a cube root, and unlike a square root it keeps the sign: $\sqrt[3]{-8} = -2$. So $x - 3 = -2$ and $x = 1$.

Common mistakes, and the answer each one gives:

• Divided by 3 instead of taking the cube root → $\frac{1}{3}$
• Took the cube root but lost the sign → $5$

Answer: $11 x^{5}$

The power rule turns $5\log_{5}(x)$ into $\log_{5}(x^{5})$, and the product rule combines the two into $\log_{5}(11 x^{5})$.

Common mistakes, and the answer each one gives:

• Added the arguments instead of multiplying them → $x^{5} + 11$
• Multiplied by the power instead of raising to it → $55 x$

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: $\frac{5}{2}$

Write the right-hand side as a power of $2$: $4 = 2^{2}$. With the same base on both sides the exponents must be equal, so $2 x - 3 = 2$ and $x = \frac{5}{2}$.

Common mistakes, and the answer each one gives:

• Set the exponent equal to 4 instead of to 2 → $\frac{7}{2}$
• Forgot to move the constant in the exponent across → $1$

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

Write both numbers as powers of $2$: $4 = 2^{2}$ and $8 = 2^{3}$. Then $\log_{4}8 = \dfrac{3}{2} = \frac{3}{2}$, 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 → $2$
• Turned the fraction of exponents upside down → $\frac{2}{3}$

Answer: $x^{3} \left(8 x + 4\right) e^{8 x}$

$u = x^{4}$, $v = e^{8x}$, so $f' = u'v + uv'$. This gives $f'(x) = x^{3} \left(8 x + 4\right) e^{8 x}$.

Common mistakes, and the answer each one gives:

• Differentiated each part separately and combined them → $32 x^{3} e^{8 x}$

Answer: $-3$

$f'(x) = 6 x^{2} - 24 x - 126 = 0$ at $x = -3$ and $x = 7$. $f''(x) = 12 x - 24$, and $f''(-3) = -60 < 0$, so $x = -3$ is the maximum.

Common mistakes, and the answer each one gives:

• Gave the local MINIMUM instead → $7$
• Gave the y-value rather than the x-coordinate → $216$
• Solved f(x) = 0 instead of f'(x) = 0 → $0$

Answer: $\frac{- x^{2} - 8 x + 6}{x^{4} + 12 x^{2} + 36}$

The quotient rule is $\dfrac{u'v - uv'}{v^2}$ — and the ORDER of the two products matters, unlike the product rule. With $u = x + 4$ and $v = x^{2} + 6$: $u' = 1$ and $v' = 2x$, giving $\frac{- x^{2} - 8 x + 6}{x^{4} + 12 x^{2} + 36}$.

Common mistakes, and the answer each one gives:

• Wrote $uv' - u'v$ in the numerator, the two products swapped → $\frac{x^{2} + 8 x - 6}{x^{4} + 12 x^{2} + 36}$
• Differentiated top and bottom separately → $\frac{1}{2 x}$