MCQ 11 Mark
Directions: In the following questions, a statement of assertion $(A)$ is followed by a statement of reason $(R)$. Mark the correct choice as:
Assertion: Roots of quadratic equation $x^2 + 3x + 5 = 0$ is $\text{x}=\frac{-3\pm\text{i}\sqrt{11}}{2}.$
Reason: If $x^2 - x + 2 = 0$ is a quadratic equation, then its roots are $\frac{1\pm\text{i}\sqrt{7}}{2}.$
- A
$A$ is true, $R$ is true; $R$ is a correct explanation of $A.$
- ✓
$A$ is true, $R$ is true; $R$ is not a correct explanation of $A.$
- C
$A$ is true; $R$ is false.
- D
$A$ is false; $R$ is true.
AnswerCorrect option: B. $A$ is true, $R$ is true; $R$ is not a correct explanation of $A.$
Assertion: Given, $x^2 + 3x + 5 = 0$
On comparing the given equation with
$ax^2 + bx + c = 0,$ we get
$a = 1, b = 3, c = 5$
Now, $D = b^2 - 4ac = (3)^2 - 4 . 1 . 5$
$= 9 - 20 = -11 < 0$
$\Rightarrow\text{x}=\frac{-3\pm\sqrt{-11}}{2\cdot1}$
$\therefore\text{x}=\frac{-3\pm\text{i}\sqrt{11}}{2}$
$[\because\sqrt{-1}=\text{i}]$
Reason: Given, $x^2 - x + 2 = 0$
On comparing the given equation with
$ax^2 + bx + c = 0,$ we get
$a = 1, b = -1, c = 2$
Now, $D = b^2 - 4ac = (-1)^2 - 4 . 1 . 5$
$= 1 - 8 = -7 < 0$
$\Rightarrow\text{x}=\frac{-(-1)\pm\sqrt{-7}}{2\cdot1}$
$=\frac{1\pm\text{i}\sqrt{7}}{2\cdot1}$
$[\because\sqrt{-1}=\text{i}]$
View full question & answer→MCQ 21 Mark
Directions: In the following questions, a statement of assertion $(A)$ is followed by a statement of reason $(R).$ Mark the correct choice as:
Assertion: If $\text{z}=\frac{1+2\text{i}}{1-3\text{i}},$ then $\mid\text{z}\mid=\frac{1}{\sqrt{2}}.$
Reason: If $z = a + ib,$ then $\mid\text{z}\mid=\sqrt{\text{a}^{2}+\text{b}^{2}}.$
- ✓
$A$ is true, $R$ is true; $R$ is a correct explanation of $A.$
- B
$A$ is true, $R$ is true; $R$ is not a correct explanation of $A.$
- C
$A$ is true; $R$ is false.
- D
$A$ is false; $R$ is true.
AnswerCorrect option: A. $A$ is true, $R$ is true; $R$ is a correct explanation of $A.$
Assertion: Let $\text{z}=\frac{1+2\text{i}}{1-3\text{i}}$
$\therefore\text{z}=\frac{1+2\text{i}}{1-3\text{i}}\cdot\frac{1+3\text{i}}{1+3\text{i}}=\frac{1+3\text{i}}{1+3\text{i}}=\frac{1+3\text{i}+2\text{i}+6\text{i}^2}{1^{2}-(3\text{i})^2}$
$[\because (a + b)(a - b) = a2 - b2]$
$=\frac{1+5\text{i}+6(-1)}{1-9\text{i}^{2}}$ $[\because\text{i}^{2}=-1]$
$=\frac{1+5\text{i}-6}{1+9}=\frac{-5+5\text{i}}{10}=\frac{-1+\text{i}}{2}$
$\Rightarrow\text{z}=-\frac{1}{2}+\frac{1}{2}\text{i}$
$\therefore\mid\text{z}\mid=\sqrt{\big(-\frac{1}{2}\big)^{2}+\big(\frac{1}{2}\big)^{2}}$ $\big[\because\mid\text{a}+\text{ib}\mid=\sqrt{\text{a}^{2}+\text{b}^{2}}\big]$
$=\sqrt{\frac{1}{4}+\frac{1}{4}}$
$=\sqrt{\frac{2}{4}}$
$=\sqrt{\frac{1}{2}}$
$=\frac{1}{\sqrt{2}}.$
View full question & answer→MCQ 31 Mark
Directions: In the following questions, a statement of assertion $(A)$ is followed by a statement of reason $(R).$ Mark the correct choice as:
Assertion: If $3x^2 + 4x + 2 = 0,$ then equation has imaginary roots.
Reason: In a quadratic equation, In a quadratic equation, $ax^2 + bx + c = 0,$ if $D = b^2 -4ac$ is less than zero, then the equation will have imaginary roots.
- ✓
$A$ is true, $R$ is true; $R$ is a correct explanation of $A.$
- B
$A$ is true, $R$ is true; $R$ is not a correct explanation of $A.$
- C
$A$ is true; $R$ is false.
- D
$A$ is false; $R$ is true.
AnswerCorrect option: A. $A$ is true, $R$ is true; $R$ is a correct explanation of $A.$
Assertion: $3x^2 + 4x + 2 = 0, a = 3, b = 4, c = 2$
$D = b^2 - 4ac$
$= 16 - 4(3)(2)$
$= 16 - 24$
$= -8$
$\Rightarrow D < 0$
$\therefore b^2 - 4ac < 0,$
so above equation has imaginary roots.
View full question & answer→MCQ 41 Mark
Directions: In the following questions, a statement of assertion $(A)$ is followed by a statement of reason $(R)$. Mark the correct choice as:
Assertion: If $Z_1 = 2 + 3i$ and $Z_2 = 3 - 2i,$ then $Z_1 - Z_2 = -1 + 5i.$
Reason: If $Z, = a + ib$ and $Z_2 = c + id,$ then $Z_1 - Z_2 = (a - c) + i(b - d).$
- ✓
$A$ is true, $R$ is true; $R$ is a correct explanation of $A.$
- B
$A$ is true, $R$ is true; $R$ is not a correct explanation of $A.$
- C
$A$ is true; $R$ is false.
- D
$A$ is false; $R$ is true.
AnswerCorrect option: A. $A$ is true, $R$ is true; $R$ is a correct explanation of $A.$
Assertion: Given, $Z, = 2 + 3i, Z_2 = 3 - 2i.$
$\therefore Z, - Zy = (2 + 3i) - (3 - 2i)$
$= (2 - 3) + i(3 - (-2)) = -1 + 5i.$
View full question & answer→MCQ 51 Mark
Directions: In the following questions, a statement of assertion $(A)$ is followed by a statement of reason $(R)$. Mark the correct choice as:
Assertion: If $(1 + i)(x + iy) = 2 - 5i,$ then $\text{x}=\frac{-3}{2}$ and $\text{y}=\frac{-7}{2}.$
Reason: If $a + ib = c + id,$ then $a = c$ and $b = d.$
- ✓
$A$ is true, $R$ is true; $R$ is a correct explanation of $A.$
- B
$A$ is true, $R$ is true; $R$ is not a correct explanation of $A.$
- C
$A$ is true; $R$ is false.
- D
$A$ is false; $R$ is true.
AnswerCorrect option: A. $A$ is true, $R$ is true; $R$ is a correct explanation of $A.$
Assertion: We have,
$(1 + i) (x + iy) = 2 - 5i$
$\Rightarrow x + iy + ix + i^2y = 2 - 5i$
$\Rightarrow x + i(y + x) - y = 2 - 5i [\because i^2 = -1]$
$\Rightarrow (x + y) + i(x + y) = 2 - 5i$
On equating real and imaginary parts from both sides, we get
$x - y = 2 ...(i)$
and $x + y = -5 ...(ii)$
On adding Eqs. $(i)$ and we get
$x - y + x + y = 2 - 5$
$\Rightarrow 2x = -3$
$\Rightarrow\text{x}=\frac{-3}{2}$
On substituting $\text{x}=\frac{-3}{2}$ in Eq. $(ii),$ we get
$\frac{-3}{2}+\text{y}=-5$
$\Rightarrow\text{y}=-5+\frac{3}{2}=\frac{-10+3}{2}=\frac{-7}{2}$
$\therefore\text{x}=\frac{-3}{2}$ and $\text{y}=\frac{-7}{2}.$
View full question & answer→MCQ 61 Mark
Directions: In the following questions, a statement of assertion $(A)$ is followed by a statement of reason $(R).$ Mark the correct choice as:
Assertion: $(2 + 3i)[(3 + 2i) + (2 + i)] = 1 + 21i.$
Reason: $z_1 (z_2 + Z_3) = z_1z_2 + z_1z_3.$
- ✓
$A$ is true, $R$ is true; $R$ is a correct explanation of $A.$
- B
$A$ is true, $R$ is true; $R$ is not a correct explanation of $A$.
- C
$A$ is true; $R$ is false.
- D
$A$ is false; $R$ is true.
AnswerCorrect option: A. $A$ is true, $R$ is true; $R$ is a correct explanation of $A.$
Assertion: For any three complex numbers $z_1, z_2$ and $z_3, $ distributive law is
$z_1(z_2 + z_3) = z_1z_2 + z_1z_2$ and $(z_1 + z_2)Z_3 = z_1z_2 + z_2z_3.$
$\therefore (2 + 3i) [(3 + 2i) + (2 + i)]$
$= (2 + 3i) (3 + 22) + (2 + 3i)(2 + i)$
$= (6 - 6) + 13i + (4 - 3) + 8i$
$= 1 + 21i$
View full question & answer→MCQ 71 Mark
Directions: In the following questions, a statement of assertion $(A)$ is followed by a statement of reason $(R).$ Mark the correct choice as:
Assertion: If $z = i^9 + i^{19},$ then $z$ is equal to $0 + 0i.$
Reason: The value of $1 + i^2 + i^4 + i^6 + .... + i^{20} $ is equal to $-l.$
- A
$A$ is true, $R$ is true; $R$ is a correct explanation of $A.$
- B
$A$ is true, $R$ is true; $R$ is not a correct explanation of $A.$
- ✓
$A$ is true; $R$ is false.
- D
$A$ is false; $R$ is true.
AnswerCorrect option: C. $A$ is true; $R$ is false.
Assertion: $i^9 + i^{19} = i^9(1 + 7^{10}) = i^9 + [1 + (i^2)^5]$
$= i^9 [1 + (-1)^5] = i^9 (1 - 1) = 0 = 0 + 0i$
Reason: $1 + i^2 + i^4 + .... i^{20}$
$=\frac{1[(\text{i}^{2})^{11}-1]}{(\text{i})^{2}-1}=\frac{1(-1-1)}{-1-1}=1$
View full question & answer→MCQ 81 Mark
Directions: In the following questions, a statement of assertion $(A)$ is followed by a statement of reason $(R).$ Mark the correct choice as:
Assertion: If $x + 4iy = ix + y + 3,$ then $x = 1$ and $y = 4$.
Reason: The reciprocal of $3+\sqrt{7}\text{i}$ is equal to $\frac{3}{16}-\frac{\sqrt{7}}{16}\text{i}.$
- A
$A$ is true, $R$ is true; $R$ is a correct explanation of $A.$
- B
$A$ is true, $R$ is true; $R$ is not a correct explanation of $A.$
- C
$A$ is true; $R$ is false.
- ✓
$A$ is false; $R$ is true.
AnswerCorrect option: D. $A$ is false; $R$ is true.
Assertion: $x + 4iy = ix + y + 3$
$\Rightarrow x = y + 3 ...(i)$
$\Rightarrow 4y = x ...(ii)$
From Egg. $(i)$ and $(ii),$ we get
$4y = y + 3$
$\Rightarrow 3y = 3$
$\Rightarrow y = 1$
From Eq. $(i),$ we get
$x = 1 + 3 = 4$
Reason: Let $\text{z}=3+\sqrt{7}\text{i}$
$\therefore\frac{1}{\text{z}}=\frac{1}{3+\sqrt{7}\text{i}}\cdot\frac{3-\sqrt{7}\text{i}}{3-\sqrt{7}\text{i}}$
$=\frac{3-\sqrt{7}\text{i}}{9+7}=\frac{3}{16}-\frac{\sqrt{7}}{16}\text{i}.$
View full question & answer→MCQ 91 Mark
Directions: In the following questions, a statement of assertion $(A)$ is followed by a statement of reason $(R).$ Mark the correct choice as:
Assertion: Simplest form of $\frac{5+\sqrt{2\text{i}}}{1-\sqrt{2\text{i}}}$ is $1-2\sqrt{2\text{i}}.$
Reason: The value of $(1 + i)^5 (1 - i)^5$ is $32.$
- ✓
$A$ is true, $R$ is true; $R$ is a correct explanation of $A.$
- B
$A$ is true, $R$ is true; $R$ is not a correct explanation of $A.$
- C
$A$ is true; $R$ is false.
- D
$A$ is false; $R$ is true.
AnswerCorrect option: A. $A$ is true, $R$ is true; $R$ is a correct explanation of $A.$
Assertion: We have,
$\frac{5+\sqrt{2\text{i}}}{1-\sqrt{2\text{i}}}=\frac{5+\sqrt{2\text{i}}}{1-\sqrt{2\text{i}}}\cdot\frac{1+\sqrt{2\text{i}}}{1-\sqrt{2\text{i}}}$
$=\frac{5+5\sqrt{2\text{i}}+\sqrt{2}\text{i}-2}{1-(\sqrt{2\text{i}})^{2}}$
$=\frac{3+6\sqrt{2}\text{i}}{1+2}=\frac{3(1+2\sqrt{2}\text{i})}{3}$
$=1+2\sqrt{2}\text{i}$
Reason: $(1 + i)^5(1 - i)^5 = (1 - i^2)^5$
$= 2^5= 32.$
View full question & answer→MCQ 101 Mark
Directions: In the following questions, a statement of assertion $(A)$ is followed by a statement of reason $(R)$. Mark the correct choice as:
Assertion: Simplest form of $i^{-35}$ is $-i.$
Reason: Additive inverse of $(1 - i)$ is equal to $-1 + i.$
- A
$A$ is true, $R$ is true; $R$ is a correct explanation of $A.$
- B
$A$ is true, $R$ is true; $R$ is not a correct explanation of $A.$
- C
$A$ is true; $R$ is false.
- ✓
$A$ is false; $R$ is true.
AnswerCorrect option: D. $A$ is false; $R$ is true.
Assertion: $\text{i}^{35}=\frac{1}{\text{i}^{35}}=\frac{1}{(\text{i}^{2})^{17}}=\frac{1}{-\text{i}}\cdot\frac{\text{i}}{\text{i}}=\frac{\text{i}}{-\text{i}^{2}}=\text{i}$
Reason: Additive inverse of $z$ is $-z.$
$\therefore$ Additive inverse of $(1 - i)$ is $-(1 - i) = -1 + i.$
View full question & answer→MCQ 111 Mark
Directions: In the following questions, a statement of assertion $(A)$ is followed by a statement of reason $(R)$. Mark the correct choice as:
Assertion: If $\text{i}=\sqrt{-1},$ then $i^{4k} = 1, i^{4k+1} = i, i^{4k+2} = -1$ and $i^{4k+3} = -i.$
Reason: $i^{4k} + i^{4k+1} i^{4k+2} i^{4k+3} = 1.$
- A
$A$ is true, $R$ is true; $R$ is a correct explanation of $A.$
- B
$A$ is true, $R$ is true; $R$ is not a correct explanation of $A.$
- ✓
$A$ is true; $R$ is false.
- D
$A$ is false; $R$ is true.
AnswerCorrect option: C. $A$ is true; $R$ is false.
Assertion: We know that, $\text{i}=\sqrt{-1},$
$\because i^{4k} = (i^4)^k = 1^k = 1$
$\Rightarrow i^{4k+1} = i^{4k} . i = 1 . i = i$
$\Rightarrow i^{4k+2} = i^{4k} . i^2 = 1 . i = -1$
$\Rightarrow i^{4k+3} = i^{4k} . i^3 = 1 . i = -i$
Reason: $i^{4k} + i^{4k+1} i^{4k+2} i^{4k+3}$
$= i^{4k}(1 + i + i^2 +i^3)$
$= i^{4k}(1 + i - 1 - i) = i^{4k} 0 = 0$
View full question & answer→MCQ 121 Mark
Directions: In the following questions, a statement of assertion $(A)$ is followed by a statement of reason $(R).$ Mark the correct choice as:
Assertion: Multiplicative inverse of $2 - 3i$ is $2 + 3i.$
Reason: If $z = 3 + 4i,$ then $\bar{\text{Z}}=3-4\text{i}.$
- A
$A$ is true, $R$ is true; $R$ is a correct explanation of $A.$
- B
$A$ is true, $R$ is true; $R$ is not a correct explanation of $A.$
- C
$A$ is true; $R$ is false.
- ✓
$A$ is false; $R$ is true.
AnswerCorrect option: D. $A$ is false; $R$ is true.
Assertion: Let $z = 2 - 3i$
Then, $\bar{\text{Z}}=2-3\text{i}$ and $\mid\text{z}\mid^{2}=2^{2}+(-3)^{2}=13$
Therefore, the multiplicative inverse of $2 - 3i$ is
$\text{z}^{_1}=\frac{\bar{\text{z}}}{\mid\text{z}\mid^{2}}=\frac{2+3\text{i}}{13}=\frac{2}{13}+\frac{3}{13}\text{i}$
The above working can be reproduced in the following manner also,
$\text{z}^{-1}=\frac{1}{2-3\text{i}}=\frac{2+3\text{i}}{(2-3\text{i})(2+3\text{i})}$
$=\frac{2+3\text{i}}{2^{2}-(3\text{i})^{2}}=\frac{2+3\text{i}}{13}$
$=\frac{2}{13}+\frac{3}{13}\text{i}$
Reason: If $Z = a + ib,$ then conjugate of $Z$
i.e. $\bar{\text{z}}=\text{a}-\text{ib}$
$\therefore\text{z}=3+4\text{i}$
$\Rightarrow\text{z}=3-4\text{i}$
View full question & answer→MCQ 131 Mark
Directions: In the following questions, a statement of assertion $(A)$ is followed by a statement of reason $(R).$ Mark the correct choice as:
Assertion: If $x^2 + 1 = 0,$ then solution is $\pm\ \text{i}.$
Reason: The value of $i^{-1097}$ is equal to $i.$
- A
$A$ is true, $R$ is true; $R$ is a correct explanation of $A.$
- B
$A$ is true, $R$ is true; $R$ is not a correct explanation of $A.$
- ✓
$A$ is true; $R$ is false.
- D
$A$ is false; $R$ is true.
AnswerCorrect option: C. $A$ is true; $R$ is false.
Assertion: $x^2 + 1 = 0$
$x^2= -1$
$\text{x}=\pm\sqrt{-1}$
$\Rightarrow\text{x}=\pm\text{i}$
Reason: $\text{i}^{-1097}=\frac{1}{4\cdot274+1}$
$\Rightarrow\frac{1}{\text{i}}=\frac{1}{\text{i}}\cdot\frac{\text{i}}{\text{i}^{2}}=\frac{\text{i}}{-1}=-\text{i}$
View full question & answer→MCQ 141 Mark
Directions: In the following questions, a statement of assertion $(A)$ is followed by a statement of reason $(R).$ Mark the correct choice as:
Assertion: If $\text{z}=5\text{i}\big(\frac{-3}{5}\text{i}\big),$ then $z$ is equal to $3 + 0i.$
Reason: If $z_1 = a + ib$ and $z_2 = c + id,$ then $z_1 + Z_2 = (a + c) + i(b + d).$
- A
$A$ is true, $R$ is true; $R$ is a correct explanation of $A.$
- ✓
$A$ is true, $R$ is true; $R$ is not a correct explanation of $A.$
- C
$A$ is true; $R$ is false.
- D
$A$ is false; $R$ is true.
AnswerCorrect option: B. $A$ is true, $R$ is true; $R$ is not a correct explanation of $A.$
Assertion: $5\text{i}\big(\frac{-3}{5}\text{i}\big)=5\cdot\frac{-3}{5}\text{i}^{2}$
$= -3(-1) = 3 = 3 + 0i$
View full question & answer→MCQ 151 Mark
Directions: In the following questions, a statement of assertion $(A)$ is followed by a statement of reason $(R).$ Mark the correct choice as:
Assertion: If $(1 + i)^6 = a + ib,$ then $b = -8.$
Reason: If $(1 - i)^3 = a + ib,$ then $\frac{\text{a}}{\text{b}}=1.$
- A
$A$ is true, $R$ is true; $R$ is a correct explanation of $A.$
- ✓
$A$ is true, $R$ is true; $R$ is not a correct explanation of $A$.
- C
$A$ is true; $R$ is false.
- D
$A$ is false; $R$ is true.
AnswerCorrect option: B. $A$ is true, $R$ is true; $R$ is not a correct explanation of $A$.
Assertion: We have,
$(1 + i)^6 = ((1 + i)^2)^3$
$= (1 + i^2 + 2i)^3 [\because (z_1 + z_2)^2 z_1^2 + z_2^2 + 2z_1z_2]$
$= (1 - 1 + 2i)^3 [\because i^2 = -1]$
$\Rightarrow (1 + i)^6 = (2i)^3 = 8i^3 = -8i [\because i^3 = -1]$
$= a + ib$
$\therefore b = -8$
Reason: $(1 - i)^3 = 1^3 - i^3 - 3(1)^2i + 3(1)(i)^2 [\because (z_1 + z_2)^3 = z_1^3 - 3z_1^2 + 3z_1z_2^2 - z_2^3]$
$= 1 - (-i) - 3i - 3 [\because i^3 = -i$ and $i^2 = -1]$
$\Rightarrow (1 - i)^3 = -2 - 2i$
$= a = ib$
$\Rightarrow a = -2$ and $b = -2$
$\therefore\frac{\text{a}}{\text{b}}=\frac{-2}{-2}=1.$
View full question & answer→