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

Solve for $x$: $x^2-5x-14=0$

Solve for $x$, correct to two decimal places: $3x^2-x-5=0$

Solve for $x$: $x^2-2x-15\le 0$

For which value of $k$ does $x^2+4x+k=0$ have equal roots?

Given the arithmetic sequence $4;11;18;\dots$ Determine $T_n$.

Calculate the sum of the first 30 terms of this sequence.

In a geometric sequence, $T_3=18$ and $T_6=486$. Determine $a$ and $r$.

Calculate $S_8$ for this sequence.

Calculate $\sum _{k=1}^{\infty }12{\left(\frac{1}{3}\right)}^{k-1}$.

Given the quadratic pattern $3;10;21;36;\dots$ Determine $T_n$.

Determine $f^{\prime }(x)$ from first principles if $f(x)=3x^2-x$.

Determine $\frac{dy}{dx}$ if $y=4x^3-\frac{2}{x}+\sqrt{x}$.

Determine the equations of the two tangents to $y=x^3-3x^2$ that are parallel to the line $y=9x+1$.

$A$ and $B$ are independent events with $P(A)=0,45$ and $P(B)=0,3$. Calculate $P(A\text{ and }B)$, $P(A\text{ or }B)$ and $P(\text{neither})$.

Determine the $x$-intercepts of $f$.

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

Determine the coordinates of the points of intersection of $f$ and $g$.

For which values of $x$ is $g(x)\cdot \left[f(x)-g(x)\right]\ge 0$?

Calculate the maximum vertical distance between $f$ and $g$ for $-2\le x\le 3$.

Convert 10,5% p.a. compounded monthly to an effective annual rate.

A bakkie costs R280 000 and depreciates at 15% p.a. on the reducing-balance method. Calculate its value after 4 years.

Thabo needs R250 000 in 6 years' time. He saves R2 500 at the end of every month for 6 years at 8,5% p.a. compounded monthly. Show that he will not reach his target, and calculate the single amount he must deposit into the same account today to make up the shortfall.

A home loan of R950 000 is repaid over 20 years at 11% p.a. compounded monthly. Calculate the monthly repayment.

Calculate the outstanding balance immediately after the 96th payment.

A loan of R60 000 at 12% p.a. compounded monthly is repaid at R2 000 per month. How many payments are needed?

Show that $(x-1)$ is a factor of $f(x)$ and factorise $f(x)$ fully.

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

Sketch the graph of $f$, showing all intercepts and turning points.

For which values of $k$ will $f(x)=k$ have three distinct real roots?

Given $f(x)=2x^2$, $x\le 0$. Determine $f^{-1}(x)$ and its domain.

Sketch $f$ and $f^{-1}$ on the same set of axes. Show the line of symmetry.

Given $q(x)={\left(\frac{1}{3}\right)}^x$. Write down $q^{-1}(x)$ and solve $q^{-1}(x)\ge -2$.

Write down the equations of the asymptotes of $h$.

Determine the $x$-intercept and the range of $k$.

Determine $k^{-1}(x)$ in the form $y=\dots$

Determine the values of $c$ for which the line $y=-x+c$ does not meet the graph of $h$.

Determine the smallest number of terms of the arithmetic sequence $4;11;18;\dots$ whose sum is greater than 1 000.

For which values of $x$ does $\sum _{k=1}^{\infty }(2x-1{)}^k$ converge?

The quadratic pattern $3;10;21;36;\dots$ has $T_n=2n^2+n$. Between which two consecutive terms of the pattern is the difference 203?

Solve for $x$: $\sqrt{x+10}-x=4$

Solve for $x$ and $y$: $x+y=5$ and $\frac{1}{x}+\frac{1}{y}=\frac{5}{6}$

Show that the roots of $x^2-(p+4)x+4p=0$ are real for all real values of $p$, and determine the values of $p$ for which the roots are positive and unequal.

The letters of DIFFERENT are arranged in a row (the two F's are identical, and so are the two E's).

(a) How many different arrangements are possible? (3)

(b) Calculate the probability that an arrangement starts with D and ends with N. (4)

A four-digit code is made from the digits 0 to 9 without repetition, and the first digit may not be 0. Calculate the probability that a randomly chosen code is even and greater than 5 000.

Determine the value of $x$ that gives the largest volume, and calculate this volume.