IEB-style Grade 12 Mathematics Paper 1: Practice Set 2

Solve for $x$: $x(x-3)=10$

Solve for $x$, correct to two decimal places: $2x^2+3x-7=0$

Solve for $x$: $x(x+4)>12$

Simplify without a calculator: $\frac{\sqrt{48}-\sqrt{12}}{\sqrt{3}}$

Calculate $\sum _{k=1}^{20}(4k-1)$.

Given the quadratic pattern $5;8;13;20;\dots$ Determine $T_n$.

Which term of the pattern is 404?

Prove that the sum of the first $n$ terms of an arithmetic series is $S_n=\frac{n}{2}[2a+(n-1)d]$.

Determine $a$, $p$ and $q$.

Determine the $x$-intercept.

Determine the axis of symmetry with a negative gradient.

For which values of $x$ is $\frac{4}{x+2}+1\ge 3$?

Write down the equation of the reflection of the hyperbola in the $y$-axis.

Determine $f^{\prime }(x)$ from first principles if $f(x)=x^2-4x+1$.

Determine $D_x\left[\frac{(2x-1{)}^2}{x}\right]$.

A ball is kicked straight up. Its height after $t$ seconds is $h(t)=25t-5t^2$ metres. Calculate its velocity after 1 second.

Calculate the velocity of the ball when it hits the ground.

Write down $f^{-1}(x)$.

$g(x)=f(x-2)+1$. Write down the domain of $g$.

Determine the $x$-intercept of $g$.

Solve for $x$: ${\log }_{3}x+{\log }_3(x-2)=1$

For which values of $x$ is $f(x)<2$?

Show that $f(x)=-2x^2-4x+6$.

Determine the coordinates of the other point of intersection of $f$ and $g$.

Write down the range of $f$, and the turning point of $h(x)=f(x-2)-8$. What does the turning point tell you about $h$?

For which values of $k$ will $f(x)=k$ have one positive and one negative root?

150 Grade 12 learners were asked whether they belong to a gym. 70 are boys and 28 of the boys belong to a gym. 60 learners in total belong to a gym. Are being a boy and belonging to a gym independent? Show calculations.

A bag contains 5 red and 3 green sweets. Two are taken without replacement. Calculate the probability that at least one is green.

Seven friends sit in a row at the movies. In how many ways can they sit?

In how many ways can they sit if three particular friends must sit together?

What is the probability that those three friends sit together in a random arrangement?

Calculate the nominal annual rate, compounded quarterly, that is equivalent to an effective rate of 12% p.a.

A bakery buys an oven for R420 000. It will be replaced in 5 years. The price of ovens rises at 6,5% p.a., and the old oven depreciates at 15% p.a. on the reducing-balance method and will be sold. A sinking fund earns 9% p.a. compounded monthly, with monthly payments starting one month from now and ending at the replacement date.

Calculate the monthly payment.

A retiree invests R2 500 000 at 7,2% p.a. compounded monthly and withdraws R25 000 at the end of every month. How many full withdrawals can she make?

A loan of R180 000 is taken at 13,5% p.a. compounded monthly. It is repaid with 48 equal monthly payments, starting one month after the loan is granted. Calculate the balance outstanding immediately after the 20th payment.

Determine the coordinates of the turning points of $f$.

Determine the coordinates of the point of inflection.

Determine the equation of the tangent to $f$ at its point of inflection.

For which values of $x$ is $f$ increasing, and for which values is $f^{\prime }(x)<9$?

The graph of $f$ is shifted vertically to give $y=f(x)+k$. Determine the values of $k$ for which the new graph has exactly two $x$-intercepts.

Solve for $x$: $(x^2-x{)}^2-8(x^2-x)+12=0$

Solve for $x$: $3^{2x}-4\cdot 3^x+3=0$

Prove that the roots of $x^2-(k+3)x+3k=0$ are real for all real values of $k$, and rational when $k$ is rational.

How many terms of $2+7+12+\dots$ must be added to give 1 428?

The second term of a convergent geometric series is 12 and its sum to infinity is 64. Determine the possible values of the first term and the common ratio.

How many terms of $3+6+12+\dots$ must be added for the sum to exceed 10 000?

A closed cylindrical tin must hold 1 000 cm³. Determine the radius that uses the least metal, and calculate the minimum surface area. Give answers to two decimal places.