GSEB Solutions Class 11 Maths Chapter 5 Complex Numbers and Quadratic Equations Ex 5.3

Gujarat Board GSEB Textbook Solutions Class 11 Maths Chapter 5 Complex Numbers and Quadratic Equations Ex 5.3 Textbook Questions and Answers.

Gujarat Board Textbook Solutions Class 11 Maths Chapter 5 Complex Numbers and Quadratic Equations Ex 5.3

Solve each of the following equations:
1. x² + 3 = 0
2. 2x² + x + 1 = 0
3. x² + 3x + 9 = 0
4. – x² + x – 2 = 0
5. x² + 3x + 5 = 0
6. x² – x + 2 = 0
7. $\sqrt{2}$x² + x + $\sqrt{2}$ = 0
8. $\sqrt{3}$x² – $\sqrt{2}$x + 3$\sqrt{3}$ = 0
9. x² + x + $\frac{1}{\sqrt{2}}$ = 0
10. x² + $\frac{x}{\sqrt{2}}$ + 1 = 0.
Solutions to questions 1 to 10:
1. x² + 3 = 0 ⇒ x² = – 3
∴ x = ±$\sqrt{- 3}$ = ±$\sqrt{3i}$.

2. 2x² + x + 1 = 0.
Comparing with ax² + bx + c = 0,
a = 2, b = 1, c = 1.
∴ b² – 4ac = 1² – 4.2.1 = 1 – 8 = – 7
∴ x = $\frac{-b \pm \sqrt{b^{2}-4 a c}}{2 a}$ = $\frac{-1 \pm \sqrt{-7}}{2.2}$ = $\frac{-1 \pm \sqrt{7} i}{4}$.

3. x² + 3x + 9 = 0
∴ a = 1, b = 3, c = 9
∴ b² – 4ac = 32 – 4.1.9
= 9 – 36
= – 27.

4. – x² + x – 2 = 0 or x² – x + 2 = 0.
Here, a = 1, b = – 1, c = 2.
∴ b² – 4ac = (- 1)² – 4.1.2
= 1 – 8 = – 7.
∴ x = $\frac{-b \pm \sqrt{b^{2}-4 a c}}{2 a}$ = $\frac{-1 \pm \sqrt{-7}}{2.1}$ = $\frac{-1 \pm \sqrt{7} i}{2}$.

5. x² + 3x + 5 = 0
∴ a = 1, b = 3, c = 5.
∴ b² – 4ac = 9 – 4.1.5 = 9 – 20 = – 11.

6. x² – x + 2 = 0
∴ a = 1, b = – 1, c = 2.
∴ b² – 4ac = (- 1)² – 4.1.2 = 1 – 8 = – 7.
∴ x = $\frac{-b \pm \sqrt{b^{2}-4 a c}}{2 a}$ = $\frac{-1 \pm \sqrt{-7}}{2.1}$ = $\frac{-1 \pm \sqrt{7} i}{2}$.

7. $\sqrt{2}$x² + x + $\sqrt{2}$ = 0
∴ a = $\sqrt{2}$, b = 1, c = $\sqrt{2}$.
∴ b² – 4ac = 1² – 4($\sqrt{2}$.$\sqrt{2}$) = 1 – 8 = – 7.
∴ $\frac{-b \pm \sqrt{b^{2}-4 a c}}{2 a}$ = $\frac{-1 \pm \sqrt{-7}}{2 \cdot \sqrt{2}}$ = $\frac{-1 \pm \sqrt{-7} i}{2 \sqrt{2}}$.

8. $\sqrt{3}$x² – $\sqrt{2}$x + 3$\sqrt{3}$ = 0
∴ a = $\sqrt{3}$, b = – $\sqrt{2}$, c = 3$\sqrt{3}$.
∴ b² – 4ac = (- $\sqrt{2}$)² – 4.$\sqrt{3}$.3$\sqrt{3}$ = 2 – 36 = – 34.

9. x² + x + $\frac{1}{\sqrt{2}}$ = 0, Multiplying by $\sqrt{2}$, we get
$\sqrt{2x}$ + $\sqrt{2x}$ + 1 = 0.
∴ a = $\sqrt{2}$, b = $\sqrt{2}$, c = 1.
∴ b² – 4ac = ($\sqrt{2}$)² – 4.$\sqrt{2}$.1 = 2 – 4$\sqrt{2}$.

10. x² + $\frac{x}{\sqrt{2}}$ + 1 = 0
Multiplying by $\sqrt{2}$, we get
$\sqrt{2}$x² + x + $\sqrt{2}$ = 0.
∴ a = $\sqrt{2}$, b = 1, c = $\sqrt{2}$
∴ b² – 4ac = 1² – 4$\sqrt{2}$.$\sqrt{2}$ = 1 – 8 = – 7.
∴ x = $\frac{-b \pm \sqrt{b^{2}-4 a c}}{2 a}$ = $\frac{-1 \pm \sqrt{-7}}{2 \sqrt{2}}$ = $\frac{-1 \pm \sqrt{7} i}{2 \sqrt{2}}$.