IEB Grade 11 Mathematics Paper 1: Practice Set 1

Simplify without a calculator: $27^{\frac{2}{3}}\times 9^{-\frac{1}{2}}$

Simplify: $\frac{2^{x+1}+2^{x+3}}{5\cdot 2^x}$

Simplify without a calculator: $\sqrt{50}+\sqrt{8}-\sqrt{18}$

Rationalise the denominator: $\frac{10}{\sqrt{7}-\sqrt{2}}$

Solve for $x$: $3x^2-7x+2=0$

Solve by completing the square, leaving your answer in surd form: $x^2-4x-6=0$

Solve for $x$, correct to two decimal places: $2x^2+5x-1=0$

Solve for $x$: $2^{2x}-3\cdot 2^{x+1}+8=0$

Solve for $x$: $\sqrt{2x+7}=x+2$

Solve for $x$: $x^2-x-12>0$

Solve for $x$ and $y$: $x-2y=1$ and $x^2-xy=6$

For which values of $k$ does $x^2-kx+9=0$ have non-real roots?

Prove that $2x^2+px-3=0$ has real, unequal roots for all real values of $p$.

Solve for $x$: $x^{\frac{3}{2}}=64$

Given the quadratic pattern $6;11;18;27;\dots$

(a) Determine $T_n$. (4)

(b) Calculate $T_{20}$. (1)

(c) Which term is 258? (3)

$2;x;12;y;30$ is a quadratic pattern with a constant second difference of 2. Determine $x$ and $y$.

Blocks are stacked to make staircases. Staircases 1 to 4 use $1;3;6;10$ blocks.

(a) Determine a formula for the number of blocks in staircase $n$. (3)

(b) Which staircase uses 465 blocks? (3)

Given the quadratic pattern $4;11;16;19;\dots$

(a) Determine $T_n$. (3)

(b) Determine the largest term of the pattern. (2)

(c) Which is the first negative term? (2)

Write $f(x)$ in the form $a(x+p{)}^2+q$.

Determine the intercepts of $f$ with the axes.

Sketch the graph of $f$.

Write down the range of $f$.

$g(x)=f(x+1)+2$. Write down the equation of $g$ in turning-point form and its turning point.

For which values of $k$ does $f(x)=k$ have no real roots?

Calculate the average gradient of $f$ between $x=0$ and $x=3$.

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

Determine the intercepts of $h$ with the axes.

Determine the equations of the axes of symmetry of $h$.

Write down the domain and range of $h$.

Write down the asymptote of $k$ and determine its intercepts with the axes.

Write down the range of $k$, and state whether $k$ is increasing or decreasing.

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

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

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

For which values of $x$ is $f(x)\cdot g(x)\le 0$?

Calculate the average gradient of $f$ between $x=-1$ and $x=3$.

R25 000 is invested for 4 years at 9% p.a. Calculate the value if the interest is (a) simple and (b) compounded annually.

Convert 8,4% p.a. compounded monthly to an effective annual rate.

A laptop bought for R16 000 is worth R8 192 after 3 years, on the reducing-balance method. Calculate the annual rate of depreciation.

Zanele invests R40 000 at 7,5% p.a. compounded quarterly. After 18 months she deposits a further R10 000. After 3 years the rate changes to 9% p.a. compounded monthly. Calculate the value of her investment after 5 years.

$P(A)=0,6$, $P(B)=0,25$ and $P(A\text{ and }B)=0,15$.

(a) Are $A$ and $B$ independent? Show calculations. (3)

(b) Calculate $P(A\text{ or }B)$. (2)

In a class of 80 learners, 30 play tennis ($T$), 35 play hockey ($H$) and 28 swim ($S$). 10 play $T$ and $H$, 8 play $T$ and $S$, and 12 play $H$ and $S$. $x$ do all three, and 9 do none.

(a) Calculate $x$. (3)

(b) Calculate the probability that a learner does exactly one of the sports. (3)

A box contains 4 blue and 6 red pens. Two pens are taken without replacement. Calculate the probability that they are different colours.

Learners were asked whether they enjoy Maths:

• Grade 10: enjoy 30, do not enjoy 20
• Grade 11: enjoy $a$, do not enjoy 30

Determine $a$ if being in Grade 10 and enjoying Maths are independent events.