Give an example of two complex numbers
Question: Give an example of two complex numbers $z_{1}$ and $z_{2}$ such that $z_{1} \neq z_{2}$ and $\left|z_{1}\right|=$ $\left|\mathbf{z}_{2}\right| .$ Solution: Let $z_{1}=3-4 i$ and $z_{2}=4-3 i$ Here, $z_{1} \neq z_{2}$ Now, calculating the modulus, we get, $\left|z_{1}\right|=\sqrt{3^{2}+(4)^{2}}=\sqrt{25}=5$ $\left|z_{2}\right|=\sqrt{4^{2}+(3)^{2}}=\sqrt{25}=5$...
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Question: Tick (✓) the correct answer: In a regular polygon, each interior angle is thrice the exterior angle. The number os sides of the polygon is (a) 6 (b) 8 (c) 10 (d) 12 Solution: (b) 8 For a regular polygon with $\mathrm{n}$ sides: Each exterior angle $=\frac{360}{n}$ Each interior angle $=180-\frac{360}{n}$ $\therefore 180-\frac{360}{n}=3\left(\frac{360}{n}\right)$ $\Rightarrow 180=4\left(\frac{360}{n}\right)$ $\Rightarrow n=\frac{4 \times 360}{180}=8$...
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Question: Tick (✓) the correct answer: Each interior angle of a polygon is 135. How many sides does it have? (a) 8 (b) 7 (c) 6 (d) 10 Solution: (a) 8 Each interior angle of a regular polygon with $\mathrm{n}$ sides $=180-\left(\frac{360}{n}\right)$ $\Rightarrow 180-\left(\frac{360}{n}\right)=135$ $\Rightarrow \frac{360}{n}=45$ $\Rightarrow n=8$...
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Question: If $|z+i|=|z-i|$, prove that $z$ is real. Solution: Let z = x + iy Consider, $|z+i|=|z-i|$ $\Rightarrow|x+i y+i|=|x+i y-i|$ $\Rightarrow|x+i(y+1)|=|x+i(y-1)|$ $\Rightarrow \sqrt{(x)^{2}+(y+1)^{2}}=\sqrt{(x)^{2}+(y-1)^{2}}$ $\left[\because|\mathrm{z}|=\right.$ modulus $\left.=\sqrt{a^{2}+b^{2}}\right]$ $\Rightarrow \sqrt{x^{2}+y^{2}+1+2 y}=\sqrt{x^{2}+y^{2}+1-2 y}$ Squaring both the sides, we get $\Rightarrow x^{2}+y^{2}+1+2 y=x^{2}+y^{2}+1-2 y$ $\Rightarrow x^{2}+y^{2}+1+2 y-x^{2}-y^{2...
Read More →For all rational numbers
Question: For all rational numbers $x$ and $y, x-y=y-x$. Solution: False. Let $x=2, y=3$ Then. LHS $=x-y$ $=2-3$ $=-1$ $\mathrm{RHS}=\mathrm{y}-\mathrm{x}$ $=3-2$ $=1$ By comparing LHS and RHS $-1 \neq 1$ $\mathrm{LHS} \neq \mathrm{RHS}$...
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Question: Tick (✓) the correct answer: The measure of each exterior angle of a regular polygon is 40. How many sides does it have? (a) 8 (b) 9 (c) 6 (d) 10 Solution: (b) 9 Each exterior angle of a regular $\mathrm{n}-$ sided polygon $=\frac{360}{n}=40$ $\Rightarrow n=\frac{360}{40}=9$...
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Question: Find $\frac{d y}{d x}$, when Solution: as $x=$ cost Differentiating it with respect to $t$, $\frac{\mathrm{dx}}{\mathrm{dt}}=\frac{\mathrm{d}}{\mathrm{dt}}(\cos \mathrm{t})$ $\frac{d x}{d t}=-\sin t$ .....(1) And, $y=\sin t$ Differentiating it with respect to $t$, $\frac{\mathrm{dy}}{\mathrm{dt}}=\frac{\mathrm{d}}{\mathrm{dt}}(\sin t)$ $\frac{d y}{d t}=\cos t \ldots \ldots(2)$ Dividing equation (2) by (1), $\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\frac{\mathrm{dy}}{\mathrm{dt}}}{\frac{\m...
Read More →The negative of 1 is 1 itself.
Question: The negative of 1 is 1 itself. Solution: False. The negative of $1=-1$...
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Question: Tick (✓) the correct answer: Each interior angle of a polygon is 108. How many sides does it have? (a) 8 (b) 6 (c) 5 (d) 7 Solution: (c) 5 Each interior angle for a regular $n$-sided polygon $=180-\left(\frac{360}{n}\right)$ $180-\left(\frac{360}{n}\right)=108$ $\Rightarrow\left(\frac{360}{n}\right)=72$ $\Rightarrow n=\frac{360}{72}=5$...
Read More →The negative of 0 does not exist.
Question: The negative of 0 does not exist. Solution: True....
Read More →The negative of the negative
Question: The negative of the negative of any rational number is the number itself. Solution: True. Let y be a positive rational number. Then, The negative of the negative of $y$ is $=-(-y)$ $=\mathrm{y}$...
Read More →Find the real values of θ for which
Question: Find the real values of $\theta$ for which $\frac{1+i \cos \theta}{1-2 i \cos \theta}$ is purely real. Solution: Since $\frac{1+i \cos \theta}{1-2 i \cos \theta}$ is purely real Firstly, we need to solve the given equation and then take the imaginary part as 0 $\frac{1+i \cos \theta}{1-2 i \cos \theta}$ We rationalize the above by multiply and divide by the conjugate of (1 -2i cos ) $=\frac{1+i \cos \theta}{1-2 i \cos \theta} \times \frac{1+2 i \cos \theta}{1+2 i \cos \theta}$ $=\frac{...
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Question: Tick (✓) the correct answer: The measure of each exterior angle of a regular polygon is 40. How many sides does it have? (a) 8 (b) 9 (c) 6 (d) 10 Solution: (b) 9 Each exterior angle of a regular $\mathrm{n}-$ sided polygon $=\frac{360}{n}=40$ $\Rightarrow n=\frac{360}{40}=9$...
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Question: Find $\frac{d y}{d x}$, when If $x=e^{\cos 2 t}$ and $y=e^{\sin 2 t}$, prove that $\frac{d y}{d x}=-\frac{y \log x}{x \log y}$ Solution: Here, $x=e^{\cos 2 t}$ Differentiating it with respect to $\theta$ using chain rule, $\frac{\mathrm{dx}}{\mathrm{dt}}=\frac{\mathrm{d}}{\mathrm{dt}}\left(\mathrm{e}^{\cos 2 \mathrm{t}}\right)$ $=e^{\cos 2 t} \frac{d}{d t}(\cos 2 t)$ $=e^{\cos 2 t}(-\sin 2 t) \frac{d}{d t}(2 t)$ $=e^{\cos 2 t}(-\sin 2 t)(2)$ $\frac{d x}{d t}=-2 \sin 2 t e^{\cos 2 t}$ ....
Read More →If x+y=0, then -y is known as the negative
Question: If $x+y=0$, then $-y$ is known as the negative of $x$, where $x$ and $y$ are rational numbers. Solution: False. If $x$ and $y$ are rational numbers, then $y$ is known as the negative of $x$...
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Question: Tick (✓) the correct answer: The angles of a pentagon arex, (x+ 20), (x+ 40), (x+ 60) and (x+ 80). The smallest angle of the pentagon is (a) 75 (b) 68 (c) 78 (d) 85 Solution: (b) 68 Sum of all the interior angles of a polygon with $n$ sides $=(n-2) \times 180^{\circ}$ $\therefore(5-2) \times 180^{\circ}=x+x+20+x+40+x+60+x+80$ $\Rightarrow 540=5 x+200$ $\Rightarrow 5 x=340$ $\Rightarrow x=68^{\circ}$...
Read More →The reciprocal of a non-zero rational
Question: The reciprocal of a non-zero rational number $\mathrm{q} / \mathrm{p}$ is the rational number $\mathrm{q} / \mathrm{p}$. Solution: False. Reciprocal of non-zero rational number $q / p$ is $p / q$....
Read More →If x / y is the additive inverse of
Question: If $x / y$ is the additive inverse of $c / d$, then, $(x / y)-(c / d)=0$ Solution: False. Let $x / y=2 / 3$ and its additive inverse $c / d=-2 / 3$ Then, $(x / y)-(c / d)$ $=(2 / 3)-(-2 / 3)$ $=(2 / 3)+(2 / 3)$ $=4 / 3$...
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Question: Tick (✓) the correct answer: A polygon has 27 diagonals. How many sides does it have? (a) 7 (b) 8 (c) 9 (d) 12 Solution: (c) 9 $\frac{n(n-3)}{2}=27$ $\Rightarrow n(n-3)=54$ $\Rightarrow n^{2}-3 n-54=0$ $\Rightarrow n^{2}-9 n+6 n-54=0$ $\Rightarrow n(n-9)+6(n-9)=0$ $\Rightarrow n=-6$ or $n=9$ Number of sides cannot be negative. $\therefore \mathrm{n}=9$...
Read More →For every rational number
Question: For every rational number $x, x+1=x$. Solution: False. Let $x=3$ Then, $3+1=4$ $3 \neq 4$ So, it is clear that $x+1 \neq x$...
Read More →Prove the following
Question: If $x / y$ is the additive inverse of $c / d$, then $(x / y)+(c / d)=0$ Solution: True. Let $x / y=1 / 2$ and its additive inverse $c / d=-1 / 2$ Then, $(x / y)+(c / d)$ $=1 / 2+(-1 / 2)$ $=1 / 2-1 / 2$ $=0$...
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Question: Find $\frac{d y}{d x}$, when If $x=2 \cos \theta-\cos 2 \theta$ and $y=2 \sin \theta-\sin 2 \theta$, prove that $\frac{d y}{d x}=\tan \left(\frac{3 \theta}{2}\right)$. Solution: $a s x=2 \cos \theta-\cos 2 \theta$ Differentiating it with respect to $\theta$ using chain rule, $\frac{\mathrm{dx}}{\mathrm{d} \theta}=2(-\sin \theta)-(-\sin 2 \theta) \frac{\mathrm{d}}{\mathrm{d} \theta}(2 \theta)$ $=-2 \sin \theta+2 \sin 2 \theta$ $\frac{d x}{d \theta}=2(\sin 2 \theta-\sin \theta)$ ....(1) ...
Read More →The additive inverse of
Question: The additive inverse of $1 / 2$ is $-2$. Solution: False. The additive inverse of $1 / 2$ is $-1 / 2$....
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Question: Tick (✓) the correct answer: How many diagonals are there in a polygon having 12 sides? (a) 12 (b) 24 (c) 36 (d) 54 Solution: (d) 54 For an n-sided polygon: Number of diagonals $=\frac{n(n-3)}{2}$ $\therefore n=12$ $\Rightarrow \frac{12(12-3)}{2}=54$...
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Question: Tick (✓) the correct answer: How many diagonals are there in a polygon having 12 sides? (a) 12 (b) 24 (c) 36 (d) 54 Solution: (d) 54 For an n-sided polygon: Number of diagonals $=\frac{n(n-3)}{2}$ $\therefore n=12$ $\Rightarrow \frac{12(12-3)}{2}=54$...
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