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

Question 1

1.1.1 [3]

$(x-7)(x+2)=0$

$x=7$

or $x=-2$

Answer:

$x=7$ or $x=-2$

1.1.2 [4]

$x=\frac{1\pm \sqrt{1+60}}{6}$

$x\approx 1,47$

or $x\approx -1,14$

Answer:

$x\approx 1,47$ or $x\approx -1,14$

1.1.3 [3]

$(x-5)(x+3)\le 0$

Critical values $-3$ and $5$

$-3\le x\le 5$

Answer:

$-3\le x\le 5$

1.2 [2]

$\Delta =16-4k=0$

$k=4$

Answer:

$k=4$

Question 2

2.1.1 [2]

$a=4$, $d=7$

$T_n=4+7(n-1)=7n-3$

Answer:

$T_n=7n-3$

2.1.2 [3]

$S_{30}=\frac{30}{2}[2(4)+29(7)]$

$=3165$

Answer:

3 165

2.2.1 [3]

$\frac{a{r}^5}{a{r}^2}=r^3=27$

$r=3$

$a=\frac{18}{9}=2$

Answer:

$a=2$, $r=3$

2.2.2 [2]

$S_8=\frac{2(3^8-1)}{3-1}$

$=6560$

Answer:

6 560

2.3 [3]

$a=12$, $r=\frac{1}{3}$

$S_{\infty }=\frac{12}{1-\frac{1}{3}}$

$=18$

Answer:

18

2.4 [3]

Second difference 4, so $a=2$

$3(2)+b=7\Rightarrow b=1$

$c=0$, so $T_n=2n^2+n$

Answer:

$T_n=2n^2+n$

Question 3

3.1 [5]

$f^{\prime }(x)=\underset{h\to 0}{\lim }\frac{f(x+h)-f(x)}{h}$

$f(x+h)=3x^2+6xh+3h^2-x-h$

$f(x+h)-f(x)=6xh+3h^2-h$

$\underset{h\to 0}{\lim }(6x+3h-1)$

$f^{\prime }(x)=6x-1$

Answer:

$f^{\prime }(x)=6x-1$

3.2 [4]

$y=4x^3-2x^{-1}+x^{\frac{1}{2}}$

$12x^2$

$+2x^{-2}$

$+\frac{1}{2}{x}^{-\frac{1}{2}}$

Answer:

$\frac{dy}{dx}=12x^2+\frac{2}{x^2}+\frac{1}{2\sqrt{x}}$

3.3 [4]

$\frac{dy}{dx}=3x^2-6x=9$

$(x-3)(x+1)=0$, so $x=3$ or $x=-1$

At $(3;0)$: $y=9x-27$

At $(-1;-4)$: $y+4=9(x+1)$, so $y=9x+5$

Answer:

$y=9x-27$ and $y=9x+5$

Question 4

4.1 [4]

$P(A\text{ and }B)=0,135$

$P(A\text{ or }B)=0,45+0,3-0,135$

$=0,615$

$P(\text{neither})=0,385$

Answer:

0,135; 0,615; 0,385

Question 5

5.1 [2]

$x^2-2x-8=0\Rightarrow (x-4)(x+2)=0$

$x=4$ or $x=-2$

Answer:

$(-2;0)$ and $(4;0)$

5.2 [2]

$x=-\frac{2}{2(-1)}=1$

$f(1)=9$, so $(1;9)$

Answer:

$(1;9)$

5.3 [3]

$-x^2+2x+8=x+2\Rightarrow x^2-x-6=0$

$x=3$ or $x=-2$

$(3;5)$ and $(-2;0)$

Answer:

$(3;5)$ and $(-2;0)$

5.4 [2]

$g(x)\left[f(x)-g(x)\right]=(x+2)(-x^2+x+6)=-(x-3)(x+2{)}^2\ge 0$

$(x+2{)}^2\ge 0$ for all $x$, so $x\le 3$ (this includes $x=-2$)

Answer:

$x\le 3$

5.5 [4]

$d(x)=-x^2+x+6$

$d^{\prime }(x)=-2x+1=0$

$x=\frac{1}{2}$

$d=6,25$ units

Answer:

6,25 units

Question 6

6.1 [2]

${\left(1+\frac{0,105}{12}\right)}^{12}-1$

$\approx 11,02\%$

Answer:

11,02%

6.2 [2]

$280000(0,85{)}^4$

$\approx$ R146 161,75

Answer:

R146 161,75

6.3 [4]

$F=\frac{2500\left[{\left(1+\frac{0,085}{12}\right)}^{72}-1\right]}{\frac{0,085}{12}}\approx$ R233 752,97, which is less than R250 000

Shortfall $\approx$ R16 247,03

$P=\frac{16247,03}{{\left(1+\frac{0,085}{12}\right)}^{72}}$

$\approx$ R9 773,83

Answer:

R9 773,83

6.4.1 [3]

$950000=\frac{x\left[1-{\left(1+\frac{0,11}{12}\right)}^{-240}\right]}{\frac{0,11}{12}}$

$x\approx$ R9 805,79

Answer:

R9 805,79

6.4.2 [3]

Remaining payments: 144

Balance $=\frac{9805,79\left[1-{\left(1+\frac{0,11}{12}\right)}^{-144}\right]}{\frac{0,11}{12}}$

$\approx$ R782 238,33

Answer:

R782 238,33

6.5 [2]

$(1,01{)}^{-n}=1-\frac{60000\times 0,01}{2000}=0,7\Rightarrow n=-\frac{\log 0,7}{\log 1,01}\approx 35,85$

36 payments (35 of R2 000 and a smaller final payment)

Answer:

36 payments ($n\approx 35,85$)

Question 7

7.1 [3]

$f(1)=1-6+9-4=0$

$f(x)=(x-1)(x^2-5x+4)$

$=(x-1{)}^2(x-4)$

Answer:

$f(x)=(x-1{)}^2(x-4)$

7.2 [4]

$f^{\prime }(x)=3x^2-12x+9=0$

$3(x-1)(x-3)=0$

$(1;0)$

$(3;-4)$

Answer:

$(1;0)$ and $(3;-4)$

7.3 [3]

Shape: rises to the right, touching the $x$-axis at $(1;0)$

Intercepts $(4;0)$ and $(0;-4)$

Minimum $(3;-4)$ labelled

Answer:

Sketch

7.4 [2]

$y=k$ must lie between the turning point $y$-values

$-4<k<0$

Answer:

$-4<k<0$

Question 8

8.1 [3]

$x=2y^2\Rightarrow y^2=\frac{x}{2}$

$f^{-1}(x)=-\sqrt{\frac{x}{2}}$ (negative root, since $y\le 0$)

Domain: $x\ge 0$

Answer:

$f^{-1}(x)=-\sqrt{\frac{x}{2}}$, $x\ge 0$

8.2 [3]

$f$: left half of a parabola through $(0;0)$ and $(-1;2)$

$f^{-1}$: through $(0;0)$ and $(2;-1)$

Line $y=x$ shown as the mirror line

Answer:

Sketch

8.3 [4]

$q^{-1}(x)={\log }_{\frac{1}{3}}x$

${\log }_{\frac{1}{3}}x=-2$ when $x=9$

The graph decreases, so $q^{-1}(x)\ge -2$ when $x\le 9$

and $x>0$: $0<x\le 9$

Answer:

$q^{-1}(x)={\log }_{\frac{1}{3}}x$; $0<x\le 9$

Question 9

9.1 [2]

$x=1$

$y=2$

Answer:

$x=1$ and $y=2$

9.2 [3]

$2^x=4$

$x=2$

Range: $y>-4$

Answer:

$(2;0)$; $y>-4$

9.3 [2]

$x=2^y-4\Rightarrow 2^y=x+4$

$y={\log }_2(x+4)$

Answer:

$k^{-1}(x)={\log }_2(x+4)$

9.4 [4]

$-x+c=\frac{3}{x-1}+2\Rightarrow (-x+c)(x-1)=2x+1$

$x^2+(1-c)x+(c+1)=0$

No intersection when $\Delta <0$: $(1-c{)}^2-4(c+1)=c^2-6c-3<0$

$3-2\sqrt{3}<c<3+2\sqrt{3}$

Answer:

$3-2\sqrt{3}<c<3+2\sqrt{3}$ (about $-0,46<c<6,46$)

Question 10

10.1 [3]

$S_n=\frac{n}{2}[8+7(n-1)]=\frac{n(7n+1)}{2}>1000$

$7n^2+n-2000>0\Rightarrow n>16,83$

$n=17$ ($S_{16}=904$ and $S_{17}=1020$)

Answer:

17 terms ($n>16,83$)

10.2 [3]

Geometric with $r=2x-1$; it converges when $-1<r<1$

$-1<2x-1<1$

$0<x<1$

Answer:

$0<x<1$

10.3 [4]

First differences $7;11;15;\dots$ are arithmetic

$7+4(n-1)=4n+3$

$4n+3=203\Rightarrow n=50$

Between $T_{50}$ and $T_{51}$

Answer:

Between $T_{50}$ and $T_{51}$

Question 11

11.1 [5]

$\sqrt{x+10}=x+4$

$x+10=x^2+8x+16$

$(x+1)(x+6)=0$

Check $x=-6$: $\sqrt{4}=2\ne -2$, so reject

$x=-1$

Answer:

$x=-1$

11.2 [5]

$y=5-x$

$\frac{1}{x}+\frac{1}{5-x}=\frac{5}{6}\Rightarrow 6(5-x)+6x=5x(5-x)$

$30=25x-5x^2\Rightarrow x^2-5x+6=0$

$x=2$ or $x=3$

$(2;3)$ and $(3;2)$

Answer:

$x=2$, $y=3$ or $x=3$, $y=2$

11.3 [4]

$\Delta =(p+4{)}^2-16p=p^2-8p+16$

$=(p-4{)}^2\ge 0$, so the roots are always real

$(x-4)(x-p)=0$, so the roots are $4$ and $p$

Positive and unequal: $p>0$ and $p\ne 4$

Answer:

Real for all $p$ since $\Delta =(p-4{)}^2\ge 0$; positive and unequal when $p>0$, $p\ne 4$

Question 12

12.1 [7]

(a) 9 letters, F twice and E twice: $\frac{9!}{2!2!}$

$=90720$

(b) Fix D and N; 7 letters remain with F twice and E twice

$\frac{7!}{2!2!}=1260$

$P=\frac{1260}{90720}$

$=\frac{1}{72}$

Answer:

(a) 90 720 (b) $\frac{1}{72}\approx 0,014$

12.2 [4]

Total codes: $9\times 9\times 8\times 7=4536$

First digit 6 or 8: $2\times 8\times 7\times 4=448$ (4 even digits left for the last place)

First digit 5, 7 or 9: $3\times 8\times 7\times 5=840$

$P=\frac{448+840}{4536}=\frac{1288}{4536}=\frac{23}{81}\approx 0,28$

Answer:

$\frac{23}{81}\approx 0,28$

Question 13

13.1 [8]

$V=x(30-2x{)}^2$

$V^{\prime }(x)=(30-2x{)}^2-4x(30-2x)$

$=(30-2x)(30-6x)=0$

$x=5$ ($x=15$ gives no box)

$V=5\times 20^2$

$=2000$ cm³

Answer:

$x=5$ cm, volume 2 000 cm³