Grade 10 Maths · Algebra

Reject solutions that do not fit

Check each answer against the situation and throw out any that make no sense (for example a length cannot be negative, so reject x = −4).

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Question 1.3 · November 2024 · Paper 1 · National (DBE) · 3 marks

1.3 Consider the product

(1 + \frac{1}{2}) (1 + \frac{1}{3}) (1 + \frac{1}{4}) ... (1 + \frac{1}{n})

. Determine ALL the values of

n

(

n \geq 2

) for which the product will be an integer value.(3)

Work it out on paper first. Then mark yourself with the official memo, one tick at a time.

More past paper questions: reject solutions that do not fit

Work each one out first, then open the memo and check every tick.

3. Question 3.1 · March 2023 · Provincial · 3 marks
A convergent geometric series is given by: $\frac{5 ( x + 1 )}{3} + \frac{5 ( x + 1 ) ^{2}}{9} + \frac{5 ( x + 1 ) ^{3}}{27} + ...$
3.1 Calculate the values of $x$ .(3)
Show the memo (4 ticks)Memo
1. ✓
  Identify common ratio $r = \frac{x + 1}{3}$
  0 marks · Find the common ratio
2. ✓
  Set convergence condition $- 1 < r < 1$
  $- 1 < \frac{x + 1}{3} < 1$
  1 mark · Use the convergence condition
3. ✓
  Solve inequality $- 3 < x + 1 < 3$
  $- 4 < x < 2$
  1 mark · Solve a linear inequality
4. ✓
  State final range with restriction $- 4 < x < 2; x \neq = - 1$ (since $r \neq = 0$ )
  1 mark · Reject solutions that do not fit
4. Question 1.1.3 · March 2025 · Provincial · 4 marks
1.1 Given: The sum to $n$ terms of an arithmetic sequence is $S_{n} = 3 n^{2} - 5 n$ .
1.1.3 How many terms of this sequence must be added to obtain a sum of 8162?(4)
Show the memo (4 ticks)Memo
1. ✓
  Equate $S_{n}$ to 8162 $3 n^{2} - 5 n = 8162$
  1 mark · Calculate an arithmetic series sum
2. ✓
  Standard form $3 n^{2} - 5 n - 8162 = 0$
  1 mark · Write a quadratic in standard form
3. ✓
  Substitution into the quadratic formula $n = \frac{- ( - 5 ) \pm ( - 5 ) ^{2} - 4 ( 3 ) ( - 8162 )}{2 ( 3 )}$
  1 mark · Use the quadratic formula
4. ✓
  Answer (53 only) $n = 53$ or $n = - \frac{154}{3}$ (not valid: $n$ must be a natural number)
  53 terms have to be added
  1 mark · Reject solutions that do not fit
5. Question 1.2.3 · March 2022 · Provincial · 5 marks
1.2 Given: $\sum_{k = 1}^{\infty} 4 (0, 2)^{k - 1}$
1.2.3 Hence calculate the smallest number of terms of the series whose sum will differ by less than $0, 0001$ from the sum to infinity of the series.(5)
Show the memo (5 ticks)Memo
1. ✓
  Set up inequality $S_{\infty} - S_{n} < 0, 0001$
  $5 - \frac{4 ( 1 - ( 0 , 2 ) ^{n} )}{1 - 0 , 2} < 0, 0001$
  1 mark · Calculate a geometric series sum
2. ✓
  Simplify inequality $5 - 5 (1 - (0, 2)^{n}) < 0, 0001$
  $5 - 5 + 5 (0, 2)^{n} < 0, 0001$
  $5 (0, 2)^{n} < 0, 0001$
  1 mark · Solve a linear inequality
3. ✓
  Solve for n using logs $(0, 2)^{n} < 0, 00002$
  $n > lo g_{0, 2} (0, 00002)$ (Note: sign flips if taking log base < 1, or rearrange)
  Using base 5: $5 (5^{- n}) < 1 0^{- 4} ⟹ 5^{1 - n} < 1 0^{- 4}$
  1 mark · Find an exponent using logs
4. ✓
  Calculate n $n > 6, 7$
  1 mark · Find an exponent using logs
5. ✓
  Final answer $n = 7$
  1 mark · Reject solutions that do not fit
6. Question 3.2.2 · June 2024 · Paper 1 · Provincial · 1 mark
3.2 Given: $2 - \frac{3 x}{4} < 14$
3.2.2 Write down the first natural number that will satisfy the inequality.(1)
Show the memo (1 tick)Memo
1. ✓
  Identify first natural number ( $> 0$ ) $x = 1$
  1 mark · Reject solutions that do not fit

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