(27x3)1/3 = ?

$$(3^3)^{1/3} (x^3)^{1/3} = 3^{3 \times \frac{1}{3}} x^{3 \times \frac{1}{3}} = 3^1 x^1 = 3x$$

The final answer is $\boxed{\text{(d) } 3x}$

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Question 2: Write $\sqrt[5]{x^2}$ in exponential form.

$\sqrt[n]{a^m} = a^{m/n}$۔

$$\sqrt[5]{x^2} = x^{2/5}$$

The final answer is $\boxed{\text{(c) } x^{2/5}}$

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Question 3: Write $x^{3/4}$ with radical sign.

$a^{m/n} = \sqrt[n]{a^m}$۔

$$x^{3/4} = \sqrt[4]{x^3}$$

The final answer is $\boxed{\text{(b) } \sqrt[4]{x^3}}$

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Question 4: In $\sqrt[3]{35}$ the radicand is

$$\text{In } \sqrt[3]{35}, \text{ the radicand is } 35$$

The final answer is $\boxed{\text{(c) } 35}$

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Question 5: $(25/16)^{-1/2} = ?$

$$\left(\frac{16}{25}\right)^{1/2} = \sqrt{\frac{16}{25}} = \frac{\sqrt{16}}{\sqrt{25}} = \frac{4}{5}$$

The final answer is $\boxed{\text{(b) } \frac{4}{5}}$

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Question 6: The conjugate of $5+4i$ is

$$\text{The conjugate of } 5+4i \text{ is } 5-4i$$

The final answer is $\boxed{\text{(a) } 5-4i}$

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Question 7: The value of $i^9$ is

$$i^9 = i^{4 \times 2 + 1} = (i^4)^2 \cdot i^1 = (1)^2 \cdot i = 1 \cdot i = i$$

The final answer is $\boxed{\text{(a) } i}$

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Question 8: Every real number is

$$\text{Every real number } x \text{ can be written as } x+0i, \text{ which is a complex number.}$$

The final answer is $\boxed{\text{(d) A complex number}}$

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Question 9: Real part of $2ab(i+i^2)$ is

$$\text{The real part is } -2ab$$

The final answer is $\boxed{\text{(b) } -2ab}$

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Question 10: Imaginary part of $-i(3i+2)$ is

$$\text{The imaginary part is } -2$$

The final answer is $\boxed{\text{(b) } -2}$

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Question 11: Which of the following sets have the closure property w.r.t. addition?

• (a) $\{0\}$: $0+0=0$۔ 0 سیٹ میں ہے، لہذا یہ بند (closed) ہے۔
• (b) $\{0,1\}$: $1+1=2$۔ 2 سیٹ میں نہیں ہے، لہذا یہ بند نہیں ہے۔
• (c) $\{1, \sqrt{2}\}$: $1+1=2$۔ 2 سیٹ میں نہیں ہے، لہذا یہ بند نہیں ہے۔
• (d) $\{0, -1\}$: $(-1)+(-1)=-2$۔ $-2$ سیٹ میں نہیں ہے، لہذا یہ بند نہیں ہے۔ The final answer is $\boxed{\text{(a) } \{0\}}$

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Question 12: Name the property of real numbers used in $\frac{-5}{-\sqrt{5} \times 1} = \frac{-5}{-\sqrt{5}}$.

The final answer is $\boxed{\text{(b) Multiplicative Identity}}$

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Question 13: If $x, y, z \in \mathbb{R}$ then $x \leq y \implies xz \leq yz$ if $z>0$.

اگر آپ ایک عدم مساوات کے دونوں اطراف کو ایک مثبت عدد سے ضرب دیں تو عدم مساوات کا نشان تبدیل نہیں ہوتا۔
The final answer is $\boxed{\text{(a) } x \leq y \implies xz \leq yz}$

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Question 14: If $a, b \in \mathbb{R}$ then only one of $a=b$ or $a<b$ or $a>b$ holds is called...

The final answer is $\boxed{\text{(b) Trichotomy property}}$

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Additional MCQ

Question 15: A decimal representation which is non-terminating, non-recurring is called

The final answer is $\boxed{\text{(c) An irrational number}}$

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Question 16: The union of the set of rational numbers and irrational numbers is known as set of

The final answer is $\boxed{\text{(c) Real number}}$

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Question 17: $\sqrt{3} \cdot \sqrt{5}$ is a ____ number.

The final answer is $\boxed{\text{(b) Irrational}}$

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Question 18: $\sqrt{a^5 b^6} =$

$$\sqrt{a^4} \cdot \sqrt{b^6} \cdot \sqrt{a^1} = a^{4/2} \cdot b^{6/2} \cdot \sqrt{a} = a^2 b^3 \sqrt{a}$$

The final answer is $\boxed{\text{(b) } a^2 b^3 \sqrt{a}}$

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Question 19: $\sqrt[3]{-8} =$

$$(-2) \times (-2) \times (-2) = 4 \times (-2) = -8$$ $$\text{So, } \sqrt[3]{-8} = -2$$

The final answer is $\boxed{\text{(c) } -2}$

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Question 20: The value of $i^{10}$ is

$$i^{10} = i^{4 \times 2 + 2} = (i^4)^2 \cdot i^2 = (1)^2 \cdot (-1) = 1 \cdot (-1) = -1$$

The final answer is $\boxed{\text{(c) } -1}$

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Question 21: The conjugate of $2+3i$ is

$$\text{The conjugate of } 2+3i \text{ is } 2-3i$$

The final answer is $\boxed{\text{(a) } 2-3i}$

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Question 22: Real part of $(-1+\sqrt{-2})^2$ is

$$(-1+\sqrt{2}i)^2 = (-1)^2 + 2(-1)(\sqrt{2}i) + (\sqrt{2}i)^2$$ $$= 1 - 2\sqrt{2}i + 2i^2$$ $$= 1 - 2\sqrt{2}i + 2(-1)$$ $$= 1 - 2\sqrt{2}i - 2$$ $$= -1 - 2\sqrt{2}i$$

$$\text{The real part is } -1$$

The final answer is $\boxed{\text{(a) } -1}$

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Question 23: Imaginary part of $(-1+\sqrt{-2})^2$ is

$$\text{The imaginary part is } -2\sqrt{2}$$

The final answer is $\boxed{\text{(b) } -2\sqrt{2}}$

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Matrices Section

Question 14: A square matrix A is called singular if

The final answer is $\boxed{\text{(b) } |A|=0}$

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Question 15: A square matrix A is called non-singular if

The final answer is $\boxed{\text{(a) } |A| \neq 0}$

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Question 16: $(AB)^{-1} =$

$$(AB)^{-1} = B^{-1}A^{-1}$$

The final answer is $\boxed{\text{(b) } B^{-1}A^{-1}}$

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Question 17: Additive inverse of $\begin{bmatrix} 1 & -2 \\ 0 & -1 \end{bmatrix}$ is

$$-\begin{bmatrix} 1 & -2 \\ 0 & -1 \end{bmatrix} = \begin{bmatrix} -1 & -(-2) \\ -0 & -(-1) \end{bmatrix} = \begin{bmatrix} -1 & 2 \\ 0 & 1 \end{bmatrix}$$

The final answer is $\boxed{\text{(a) } \begin{bmatrix} -1 & 2 \\ 0 & 1 \end{bmatrix}}$

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Question 18: If A is a matrix then its transpose is denoted by

The final answer is $\boxed{\text{(a) } A^t}$

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Question 19: Which of the following is singular matrix?

• (a) $\begin{bmatrix} 1 & 2 \\ 4 & 7 \end{bmatrix}$: ڈیٹرمیننٹ $= (1)(7) - (2)(4) = 7 - 8 = -1 \neq 0$
• (b) $\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$: ڈیٹرمیننٹ $= (1)(1) - (0)(0) = 1 - 0 = 1 \neq 0$
• (c) $\begin{bmatrix} 1 & 2 \\ 3 & 6 \end{bmatrix}$: ڈیٹرمیننٹ $= (1)(6) - (2)(3) = 6 - 6 = 0$
• (d) $\begin{bmatrix} 2 & 1 \\ 3 & 4 \end{bmatrix}$: ڈیٹرمیننٹ $= (2)(4) - (1)(3) = 8 - 3 = 5 \neq 0$ صرف آپشن (c) کا ڈیٹرمیننٹ صفر ہے۔ The final answer is $\boxed{\text{(c) } \begin{bmatrix} 1 & 2 \\ 3 & 6 \end{bmatrix}}$

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Question 20: If $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$ then the det. A is:

The final answer is $\boxed{\text{(a) } ad - bc}$

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