Date: 2026-09-23 - Number of Questions: 17
1.
Solve for $x$:
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Hint
2.
Solve simultaneously for $x$ and $y$:
$x + y = 3$ and $2x^2 + 4xy - y = 15$ (6)
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Hint
3.
If $n$ is the largest integer for which $n^{200} < 5^{300}$, determine the value of $n$. (3)
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4.
$7 ; x ; y ; -11 ; \ldots$ is an arithmetic sequence. Determine the values of $x$ and $y$. (4)
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5.
Given the quadratic number pattern: $-3 ; 6 ; 27 ; 60 ; \ldots$
Determine the general term of the pattern in the form $T_n = an^2 + bn + c$. (4)
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Hint
6.
Prove that $\sum_{k=1}^{\infty} 4.3^{2-k}$ is a convergent geometric series. Show ALL your calculations. (3)
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Hint
7.
If $\sum_{k=p}^{\infty} 4.3^{2-k} = \frac{2}{9}$, determine the value of $p$. (5)
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8.
Given: $h(x) = \frac{-3}{x-1} + 2$
Write down the equations of the asymptotes of $h$. (2)
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Hint
9.
The graphs of $f(x) = \frac{1}{2}(x + 5)^2 - 8$ and $g(x) = \frac{1}{2}x + \frac{9}{2}$ are sketched below.
- A is the turning point of $f$.
- The axis of symmetry of $f$ intersects the x-axis at E and the line $g$ at $D(m; n)$.
- C is the y-intercept of $f$ and $g$.
Write down the coordinates of A. (2)
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Hint
10.
The graph of $f(x)=3^{-x}$ is sketched below. A is the the y-intercept of $f$. B is the point of intersection of $f$ and the line $y=9$.
Write down the coordinates of A. (1)
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Hint
11.
On 31 January 2020, Tshepo made the first of his monthly deposits of R1 000 into a savings account. He continues to make monthly deposits of R1 000 at the end of each month up until 31 January 2032. The interest rate was fixed at 7,5% p.a., compounded monthly.
What will the investment be worth immediately after the last deposit? (4)
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12.
Jim bought a new car for R250 000. The value of the car depreciated at a rate of 22% p.a. annually according to the reducing-balance method. After how many years will its book value be R92 537,64? (3)
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13.
Mpho is granted a loan under the following conditions:
- The interest rate is 11,3% p.a., compounded monthly.
- The period of the loan is 6 years.
- The monthly repayment on the loan is R1 500.
- Her first repayment is made one month after the loan is granted.
Calculate the value of the loan. (3)
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14.
Determine $f'(x)$ from first principles if $f(x) = 2x^2 - 1$. (5)
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15.
The graph of $g(x) = ax^3 + bx^2 + cx$, a cubic function having a $y$-intercept of 0, is drawn below. The $x$-coordinates of the turning points of $g$ are -1 and 2.
For which values of $x$ will $g$ increase? (2)
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16.
A closed rectangular box has to be constructed as follows:
- Dimensions: length ($l$), width ($w$) and height ($h$).
- The length ($l$) of the base has to be 3 times its width ($w$).
- The volume has to be 5 m$^3$.
The material for the top and the bottom parts costs R15 per square metre and the material for the sides costs R6 per square metre.
Show that the cost to construct the box can be calculated by: Cost = $90w^2 + 48wh$ (4)
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17.
In a certain country, 10-digit telephone numbers with the following format were introduced:
Digits may be repeated.
How many possible 10-digit telephone numbers could be formed? (2)
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Hint
