COMPLEX NUMBERS AND QUADRATIC EQUATIONS - Gujarat Board (GSEB) STD 11 Science MATHS Questions

Q 1MCQ1 Mark

If z is a comp lex num ber, then |3z – 1|= 3|z – 2| represents:

A
y-axis
B
a circle
C
x-axis
✓
a line parallel to y-axis

Answer: D.

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Q 2MCQ1 Mark

If $z_1=2+3 i$ and $z_2=5+2 i$, then find sum of two complex numbers:

A
4 + 8i
B
3 - i
✓
7 + 5i
D
7 - 5i

Answer: C.

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Q 3MCQ1 Mark

Choose the correct answer. The real value of $\alpha$ for which the expression $\frac{1-\text{i}\sin\alpha}{1+2\text{i}\sin\alpha}$ is purely real is:

A
$(\text{n}+1)\frac{\pi}{2}$
B
$(2\text{n}+1)\frac{\pi}{2}$
✓
$\text{n}\pi$
D
None of these, where $\text{n}\in\text{N}$

Answer: C.

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Q 4MCQ1 Mark

What will be the sum of b + c if the equations $x^2 + bx + c = 0$ and $x^2 + 3x + 3 = 0$ have one common root:

A
2
B
4
✓
6
D
8

Answer: C.

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Q 5MCQ1 Mark

If the roots of $x^2− bx + c = 0$ are two consecutive integers, then $b^2 − 4 c$ is:

A
0
✓
1
C
2
D
None of these.

Answer: B.

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Q 6(Each question 1 marks)1 Mark

Let $z_1 = 2 - i, z_2 = -2 + i$. Find $\operatorname{Im} \left( {\frac{1}{{{z_1}{z_1}}}} \right)$

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Q 7(Each question 1 marks)1 Mark

Express the complex numbers $(1 - i)^4$ in the form of a + ib.

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Q 8(Each question 1 marks)1 Mark

Express the complex number$\left[ {\left( {\frac{1}{3} + \frac{7}{3}i} \right) + \left( {4 + \frac{1}{3}i} \right)} \right] - \left[ {\frac{{ - 4}}{3} + i} \right]$ in the form of a + ib.

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Q 9(Each question 1 marks)1 Mark

Express the complex number $\left( {\frac{1}{5} + \frac{2}{5}i} \right) - \left( {4 + \frac{5}{2}i} \right)$ in the form of a + ib.

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Q 10(Each question 1 marks)1 Mark

Express the complex number (1 + i) - (- 1 + i6) in form of a + ib.

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Q 11(Each question 2 marks)2 Marks

If $\left(\frac{1+i}{1-i}\right)^{m} = 1$ then find the least positive integral value of m.

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Q 12(Each question 2 marks)2 Marks

$\text { If }(a+i b)(c+i d)(e+i f)(g+i h)=A+i B \text { then show that }$
$\left(a^2+b^2\right)\left(c^2+d^2\right)\left(e^2+f^2\right)\left(g^2+h^2\right)=A^2+B^2$

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Q 13(Each question 2 marks)2 Marks

Find the number of non-zero integral solutions of the equation ${\left| {1 - i} \right|^x} = {2^x}$.

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Q 14(Each question 2 marks)2 Marks

Find the modulus of $\frac{{1 + i}}{{1 - i}} - \frac{{1 - i}}{{1 + i}}$.

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Q 15(Each question 2 marks)2 Marks

Let $z_1 = 2 - i, z_2 = -2 + i$. Find $\operatorname{Re} \left( {\frac{{{z_1}{z_2}}}{{{{\overline z }_1}}}} \right)$

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Q 16(Each question 3 marks)3 Marks

Solve $21x^2- 28x + 10 = 0$

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Q 17(Each question 3 marks)3 Marks

Solve: $27x^2 - 10x+ 1 = 0$

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Q 18(Each question 3 marks)3 Marks

Solve: ${x^2} - 2x + \frac{3}{2} = 0$

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Q 19(Each question 3 marks)3 Marks

Solve $3{x^2} - 4x + \frac{{20}}{3} = 0$

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Q 20(Each question 3 marks)3 Marks

Convert in the polar form: $\frac{1 + 3i}{1 - 2i}$

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Q 21(Each question 4 marks)4 Marks

Find the real numbers x and y if (x – iy) (3 + 5i) is the conjugate of -6 - 24i.

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Q 22(Each question 4 marks)4 Marks

If $\text{a+ib}=\frac{(\text{x+i})^2}{2\text{x}^2+1},$ prove that $\text{a}^2+\text{b}^2=\frac{(\text{x}^2+1)^2}{(2\text{x}^2+1)^2}.$

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Q 23(Each question 4 marks)4 Marks

Reduce $\Big(\frac{1}{1-4\text{i}}-\frac{2}{1+\text{i}}\Big)\Big(\frac{3-4\text{i}}{5+\text{i}}\Big)$ to the standard form.

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Q 24(Each question 4 marks)4 Marks

For any two complex numbers $z_1$ and $z_2$, prove that $Re(z_1z_2) = Rez_1 Rez_2 – Imz_1 Imz_2$.

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Q 25(Each question 4 marks)4 Marks

Solve the following equation: $\sqrt{3}\text{x}^2-\sqrt{2}\text{x}+3\sqrt{3}=0.$

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COMPLEX NUMBERS AND QUADRATIC EQUATIONS question types

COMPLEX NUMBERS AND QUADRATIC EQUATIONS questions

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COMPLEX NUMBERS AND QUADRATIC EQUATIONS MATHS - COMPLEX NUMBERS AND QUADRATIC EQUATIONS

COMPLEX NUMBERS AND QUADRATIC EQUATIONS question types

COMPLEX NUMBERS AND QUADRATIC EQUATIONS questions

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