Grade 11 Maths · Algebra · Paper 1

Nature of Roots Grade 11

Nature of Roots practice questions and answers, with each memo split into ticks. Try one, mark yourself, and see exactly which skill cost you marks.

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Practice · Siyavula Grade 11 Maths · Ex 2.7 · 2 marks

Determine the nature of the roots for each of the following equations:

(a)

x^{2} + 3 x = - 2

(2)

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

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More Nature of Roots Grade 11 questions and answers

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4. Practice · Siyavula Grade 11 Maths · Ex 2.7 · 3 marks
Show that $k^{2} x^{2} + 2 = k x - x^{2}$ has non-real roots for all real values for $k$ .
[IEB, Nov. 2002, HG](3)
Show the memo (3 ticks)Memo
1. ✓
  Standard form $(k^{2} + 1) x^{2} - k x + 2 = 0$
  1 mark · Write a quadratic in standard form
2. ✓
  Discriminant $Δ = (- k)^{2} - 4 (k^{2} + 1) (2) = k^{2} - 8 k^{2} - 8 = - 7 k^{2} - 8$
  1 mark · Calculate the discriminant
3. ✓
  Conclusion $Δ = - (7 k^{2} + 8)$ and $7 k^{2} + 8 > 0$ for every real $k$ , so $Δ < 0$ : the roots are non-real.
  1 mark · Determine the nature of roots
Question adapted from Siyavula Education (CC BY 3.0) · Solution by NavyBlue
5. Practice · Siyavula Grade 11 Maths · Ex 2.7 · 3 marks
The equation $x^{2} + 12 x = 3 k x^{2} + 2$ has real roots.
(a) Find the greatest value of value $k$ such that $k \in Z$ .(3)
Show the memo (3 ticks)Memo
1. ✓
  Standard form $(1 - 3 k) x^{2} + 12 x - 2 = 0$
  1 mark · Write a quadratic in standard form
2. ✓
  Discriminant $Δ = 144 - 4 (1 - 3 k) (- 2) = 144 + 8 - 24 k = 152 - 24 k$
  1 mark · Calculate the discriminant
3. ✓
  Real roots $152 - 24 k \geq 0$ , so $k \leq \frac{19}{3} = 6 \frac{1}{3}$
  The greatest integer is $k = 6$ .
  1 mark · Use the discriminant to find k
Question adapted from Siyavula Education (CC BY 3.0) · Solution by NavyBlue
6. Practice · Siyavula Grade 11 Maths · Ex 2.7 · 3 marks
Consider the equation: $k = \frac{x ^{2} - 4}{2 x - 5}$ where $x \neq = \frac{5}{2}$ .
(a) Find a value of $k$ for which the roots are equal.(3)
Show the memo (3 ticks)Memo
1. ✓
  Standard form $k (2 x - 5) = x^{2} - 4$ , so $x^{2} - 2 k x + 5 k - 4 = 0$
  1 mark · Write a quadratic in standard form
2. ✓
  Discriminant $Δ = 4 k^{2} - 4 (5 k - 4) = 4 (k^{2} - 5 k + 4) = 4 (k - 1) (k - 4)$
  1 mark · Calculate the discriminant
3. ✓
  Equal roots $Δ = 0$ when $k = 1$ or $k = 4$
  1 mark · Use the discriminant to find k
Question adapted from Siyavula Education (CC BY 3.0) · Solution by NavyBlue
7. Practice · Siyavula Grade 11 Maths · Ex 2.7 · 3 marks
(a) Prove that the roots of the equation $x^{2} - (a + b) x + ab - p^{2} = 0$ are real for all real values of $a$ , $b$ and $p$ .(3)
Show the memo (3 ticks)Memo
1. ✓
  Discriminant $Δ = (a + b)^{2} - 4 (ab - p^{2}) = a^{2} + 2 ab + b^{2} - 4 ab + 4 p^{2}$
  1 mark · Calculate the discriminant
2. ✓
  Simplify $Δ = a^{2} - 2 ab + b^{2} + 4 p^{2} = (a - b)^{2} + 4 p^{2}$
  1 mark · Calculate the discriminant
3. ✓
  Conclusion $(a - b)^{2} \geq 0$ and $4 p^{2} \geq 0$ , so $Δ \geq 0$ : the roots are real.
  1 mark · Determine the nature of roots
Question adapted from Siyavula Education (CC BY 3.0) · Solution by NavyBlue
8. Practice · Siyavula Grade 11 Maths · Ex 2.7 · 3 marks
If $b$ and $c$ can take on only the values $1$ , $2$ or $3$ , determine all pairs $(b; c)$ such that $x^{2} + b x + c = 0$ has real roots.
[IEB, Nov. 2005, HG](3)
Show the memo (3 ticks)Memo
1. ✓
  ConditionReal roots need $Δ = b^{2} - 4 c \geq 0$ , so $b^{2} \geq 4 c$ .
  1 mark · Calculate the discriminant
2. ✓
  Test the values $b = 1$ : $1 < 4 c$ for all $c$ . $b = 2$ : $4 \geq 4 c$ only for $c = 1$ . $b = 3$ : $9 \geq 4 c$ for $c = 1$ and $c = 2$ .
  1 mark · Calculate the discriminant
3. ✓
  Pairs $(b; c) = (2; 1)$ , $(3; 1)$ or $(3; 2)$
  1 mark · Determine the nature of roots
Question adapted from Siyavula Education (CC BY 3.0) · Solution by NavyBlue
9. Practice · NavyBlue · 2 marks
The circle with centre $C$ has equation $(x - 1)^{2} + (y - 1)^{2} = 8$ . The line $y = - x + 6$ is drawn.
(b) What does your answer to (a) tell you about the line? Explain.(2)
Show the memo (2 ticks)Memo
1. ✓
  InterpretThere is only one (repeated) solution, $(3; 3)$ : the line meets the circle at exactly one point, so it is a tangent to the circle at $(3; 3)$ .
  2 marks · Determine the nature of roots
10. Practice · NavyBlue · 2 marks
Consider the circle $x^{2} + y^{2} = 16$ and the line $y = x + 6$ .
(b) Calculate the discriminant and interpret it.(2)
Show the memo (2 ticks)Memo
1. ✓
  Discriminant $Δ = 6^{2} - 4 (1) (10) = - 4$
  1 mark · Calculate the discriminant
2. ✓
  Interpret $Δ < 0$ : there are no real roots, so the line and the circle do not meet.
  1 mark · Determine the nature of roots

Nature of Roots Grade 11 videos

Short lessons from South African teachers first. Videos load only when you tap play, to save your data.

The Nature of Roots of a Quadratic Equation Grade 11 MathsMiss Martins Maths and Science

How to find the discriminant and label the solutions of a quadraticBrian McLogan

Nature of the roots Grade11: L2 Find a value for kOnline Maths by Miss Pythagoras

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