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

Question 1. \( x^2 + 3 = 0 \)
Answer: Given the equation: \( x^2 + 3 = 0 \)
Subtracting 3 from both sides gives: \( x^2 = -3 \)
Taking the square root of both sides, we get: \( x = \pm\sqrt{-3} \)
Since \( \sqrt{-1} = i \), this simplifies to: \( x = \pm\sqrt{3}i \)
In simple words: To solve this, move the number 3 to the other side, making it negative. Then, take the square root of both sides. Since you're taking the square root of a negative number, the answer will include 'i', which represents the imaginary unit.

Exam Tip: Remember to express the square root of a negative number using the imaginary unit \( i \) in your final answer.

Question 2. \( 2x^2 + x + 1 = 0 \)
Answer: We have the quadratic equation \( 2x^2 + x + 1 = 0 \).
Comparing this with the general quadratic form \( ax^2 + bx + c = 0 \), we find: \( a = 2 \), \( b = 1 \), \( c = 1 \).
First, calculate the discriminant \( \Delta = b^2 - 4ac \):
\( \Delta = (1)^2 - 4(2)(1) \)
\( \Delta = 1 - 8 \)
\( \Delta = -7 \)
Now, use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\( x = \frac{-1 \pm \sqrt{-7}}{2(2)} \)
\( x = \frac{-1 \pm \sqrt{7}i}{4} \)
In simple words: This equation is a quadratic one. We identify the values for a, b, and c, then calculate the discriminant. Because the discriminant is negative, our solutions will involve imaginary numbers. We then use the quadratic formula to find the two complex roots.

Exam Tip: Always correctly identify \( a \), \( b \), and \( c \) before calculating the discriminant. A negative discriminant means the roots are complex conjugates.

Question 3. \( x^2 + 3x + 9 = 0 \)
Answer: We have the quadratic equation \( x^2 + 3x + 9 = 0 \).
Comparing with \( ax^2 + bx + c = 0 \), we get: \( a = 1 \), \( b = 3 \), \( c = 9 \).
Calculate the discriminant \( \Delta = b^2 - 4ac \):
\( \Delta = (3)^2 - 4(1)(9) \)
\( \Delta = 9 - 36 \)
\( \Delta = -27 \)
Now, use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\( x = \frac{-3 \pm \sqrt{-27}}{2(1)} \)
\( x = \frac{-3 \pm \sqrt{9 \times -3}}{2} \)
\( x = \frac{-3 \pm 3\sqrt{-3}}{2} \)
\( x = \frac{-3 \pm 3\sqrt{3}i}{2} \)
In simple words: For this equation, first find the a, b, c values. Then, calculate the discriminant. If it's negative, the roots are complex. Use the quadratic formula to find the values of x, simplifying the square root of the negative number using 'i'.

Exam Tip: Simplify the square root of a negative number by factoring out \( \sqrt{-1} = i \) and simplifying any perfect squares from the positive part.

Question 4. \( -x^2 + x - 2 = 0 \)
Answer: We have the equation \( -x^2 + x - 2 = 0 \).
Multiplying by -1, we get the standard form: \( x^2 - x + 2 = 0 \).
Comparing with \( ax^2 + bx + c = 0 \), we have: \( a = 1 \), \( b = -1 \), \( c = 2 \).
Calculate the discriminant \( \Delta = b^2 - 4ac \):
\( \Delta = (-1)^2 - 4(1)(2) \)
\( \Delta = 1 - 8 \)
\( \Delta = -7 \)
Now, apply the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\( x = \frac{-(-1) \pm \sqrt{-7}}{2(1)} \)
\( x = \frac{1 \pm \sqrt{7}i}{2} \)
In simple words: First, change the equation so that the \( x^2 \) term is positive. Then, determine a, b, and c. Calculate the discriminant, which will be negative here. Finally, use the quadratic formula to find the complex solutions.

Exam Tip: It is usually easier to solve quadratic equations when the \( x^2 \) coefficient is positive. Always adjust the equation into this form before finding \( a, b, c \).

Question 5. \( x^2 + 3x + 5 = 0 \)
Answer: We have the quadratic equation \( x^2 + 3x + 5 = 0 \).
Comparing this with \( ax^2 + bx + c = 0 \), we find: \( a = 1 \), \( b = 3 \), \( c = 5 \).
Calculate the discriminant \( \Delta = b^2 - 4ac \):
\( \Delta = (3)^2 - 4(1)(5) \)
\( \Delta = 9 - 20 \)
\( \Delta = -11 \)
Now, use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\( x = \frac{-3 \pm \sqrt{-11}}{2(1)} \)
\( x = \frac{-3 \pm \sqrt{11}i}{2} \)
In simple words: Start by identifying the 'a', 'b', and 'c' values from the equation. Then, calculate the discriminant to see if the roots are real or complex. Because it's negative, we use the quadratic formula and the imaginary unit 'i' to find the final complex solutions.

Exam Tip: Always show the steps for calculating the discriminant clearly. This helps in understanding the nature of the roots and avoids calculation errors.

Question 6. \( x^2 - x + 2 = 0 \)
Answer: We have the quadratic equation \( x^2 - x + 2 = 0 \).
Comparing with \( ax^2 + bx + c = 0 \), we get: \( a = 1 \), \( b = -1 \), \( c = 2 \).
Calculate the discriminant \( \Delta = b^2 - 4ac \):
\( \Delta = (-1)^2 - 4(1)(2) \)
\( \Delta = 1 - 8 \)
\( \Delta = -7 \)
Now, use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\( x = \frac{-(-1) \pm \sqrt{-7}}{2(1)} \)
\( x = \frac{1 \pm \sqrt{7}i}{2} \)
In simple words: First, determine the coefficients 'a', 'b', and 'c' from the given equation. Then, compute the discriminant. Since the discriminant is a negative value, the solutions will be complex numbers. Finally, use the quadratic formula to solve for x, expressing the imaginary part with 'i'.

Exam Tip: Be careful with the sign of 'b' when substituting into the quadratic formula, especially with \( -b \).

Question 7. \( \sqrt{2}x^2 + x + \sqrt{2} = 0 \)
Answer: We have the quadratic equation \( \sqrt{2}x^2 + x + \sqrt{2} = 0 \).
Comparing with \( ax^2 + bx + c = 0 \), we get: \( a = \sqrt{2} \), \( b = 1 \), \( c = \sqrt{2} \).
Calculate the discriminant \( \Delta = b^2 - 4ac \):
\( \Delta = (1)^2 - 4(\sqrt{2})(\sqrt{2}) \)
\( \Delta = 1 - 4(2) \)
\( \Delta = 1 - 8 \)
\( \Delta = -7 \)
Now, use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\( x = \frac{-1 \pm \sqrt{-7}}{2(\sqrt{2})} \)
\( x = \frac{-1 \pm \sqrt{7}i}{2\sqrt{2}} \)
In simple words: Identify the values of a, b, and c, noting that some are square roots. Calculate the discriminant, which turns out to be negative. Then, apply the quadratic formula and simplify the result, making sure to include 'i' for the imaginary part.

Exam Tip: Be careful when multiplying terms with square roots, such as \( \sqrt{2} \times \sqrt{2} = 2 \), during discriminant calculation.

Question 8. \( \sqrt{3}x^2 - \sqrt{2}x + 3\sqrt{3} = 0 \)
Answer: We have the quadratic equation \( \sqrt{3}x^2 - \sqrt{2}x + 3\sqrt{3} = 0 \).
Comparing with \( ax^2 + bx + c = 0 \), we get: \( a = \sqrt{3} \), \( b = -\sqrt{2} \), \( c = 3\sqrt{3} \).
Calculate the discriminant \( \Delta = b^2 - 4ac \):
\( \Delta = (-\sqrt{2})^2 - 4(\sqrt{3})(3\sqrt{3}) \)
\( \Delta = 2 - 4(3 \times 3) \)
\( \Delta = 2 - 4(9) \)
\( \Delta = 2 - 36 \)
\( \Delta = -34 \)
Now, use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\( x = \frac{-(-\sqrt{2}) \pm \sqrt{-34}}{2(\sqrt{3})} \)
\( x = \frac{\sqrt{2} \pm \sqrt{34}i}{2\sqrt{3}} \)
In simple words: First, find the 'a', 'b', and 'c' values, which include square roots. Calculate the discriminant carefully, remembering to square negative square roots correctly. Since the discriminant is negative, use the quadratic formula to get the complex solutions, showing the imaginary part with 'i'.

Exam Tip: Pay close attention to squaring negative square root terms, for example, \( (-\sqrt{2})^2 = 2 \), and ensure correct multiplication of square root terms in the \( 4ac \) part.

Question 9. \( x^2 + x + \frac{1}{\sqrt{2}} = 0 \)
Answer: We have the equation \( x^2 + x + \frac{1}{\sqrt{2}} = 0 \).
To remove the fraction, multiply the entire equation by \( \sqrt{2} \):
\( \sqrt{2}x^2 + \sqrt{2}x + 1 = 0 \)
Comparing this with \( ax^2 + bx + c = 0 \), we get: \( a = \sqrt{2} \), \( b = \sqrt{2} \), \( c = 1 \).
Calculate the discriminant \( \Delta = b^2 - 4ac \):
\( \Delta = (\sqrt{2})^2 - 4(\sqrt{2})(1) \)
\( \Delta = 2 - 4\sqrt{2} \)
Now, use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\( x = \frac{-\sqrt{2} \pm \sqrt{2 - 4\sqrt{2}}}{2\sqrt{2}} \)
We need to simplify \( \sqrt{2 - 4\sqrt{2}} \). Since \( 2 - 4\sqrt{2} \) is negative (as \( 4\sqrt{2} \approx 5.65 \)), we will have an imaginary component.
\( x = \frac{-\sqrt{2} \pm \sqrt{-(4\sqrt{2} - 2)}}{2\sqrt{2}} \)
\( x = \frac{-\sqrt{2} \pm \sqrt{4\sqrt{2} - 2}i}{2\sqrt{2}} \)
This can also be written as:
\( x = \frac{-\sqrt{2} \pm \sqrt{2(2\sqrt{2}-1)}i}{2\sqrt{2}} \)
\( x = \frac{-1 \pm \sqrt{2\sqrt{2}-1}i}{2} \) (by dividing numerator and denominator by \( \sqrt{2} \))
In simple words: First, clear the fraction by multiplying the whole equation by \( \sqrt{2} \). Then, identify a, b, and c. Calculate the discriminant, which will be negative, leading to complex roots. Use the quadratic formula to solve for x, simplifying the expression and ensuring 'i' is included for the imaginary part.

Exam Tip: When dealing with fractions involving square roots in the denominator, multiply the entire equation by that square root to clear the fraction first. Be very careful with the signs under the square root in the quadratic formula.

Question 10. \( x^2 + \frac{x}{\sqrt{2}} + 1 = 0 \)
Answer: We have the equation \( x^2 + \frac{x}{\sqrt{2}} + 1 = 0 \).
To remove the fraction, multiply the entire equation by \( \sqrt{2} \):
\( \sqrt{2}x^2 + x + \sqrt{2} = 0 \)
Comparing this with \( ax^2 + bx + c = 0 \), we get: \( a = \sqrt{2} \), \( b = 1 \), \( c = \sqrt{2} \).
Calculate the discriminant \( \Delta = b^2 - 4ac \):
\( \Delta = (1)^2 - 4(\sqrt{2})(\sqrt{2}) \)
\( \Delta = 1 - 4(2) \)
\( \Delta = 1 - 8 \)
\( \Delta = -7 \)
Now, use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\( x = \frac{-1 \pm \sqrt{-7}}{2(\sqrt{2})} \)
\( x = \frac{-1 \pm \sqrt{7}i}{2\sqrt{2}} \)
In simple words: Start by multiplying the whole equation by \( \sqrt{2} \) to get rid of the fraction. Then, find the values for a, b, and c. Calculate the discriminant, which will be negative, meaning the roots are complex. Finally, use the quadratic formula to find the values of x, including the imaginary unit 'i'.

Exam Tip: Rationalizing the denominator of complex numbers is often expected. Multiply the numerator and denominator by \( \sqrt{2} \) if required to get \( \frac{-\sqrt{2} \pm \sqrt{14}i}{4} \).