Show that
Question: Show that (i) $\left\{\frac{(3+2 i)}{(2-3 i)}+\frac{(3-2 i)}{(2+3 i)}\right\}$ is purely real, (ii) $\left\{\frac{(\sqrt{7}+i \sqrt{3})}{(\sqrt{7}-i \sqrt{3})}+\frac{(\sqrt{7}-i \sqrt{3})}{(\sqrt{7}+i \sqrt{3})}\right\}$ is purely real. Solution: Given: $\frac{3+2 i}{2-3 i}+\frac{3-2 i}{2+3 i}$ Taking the L.C.M, we get $=\frac{(3+2 i)(2+3 i)+(3-2 i)(2-3 i)}{(2-3 i)(2+3 i)}$ $=\frac{3(2)+3(3 i)+2 i(2)+2 i(3 i)+3(2)+3(-3 i)-2 i(2)+(-2 i)(-3 i)}{(2)^{2}-(3 i)^{2}}$ $\left[\because(a+b)(a-...
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Question: Tick (✓) the correct answer: How many diagonals are there in an octagon? (a) 8 (b) 16 (c) 18 (d) 20 Solution: (d) 20 For a regular $\mathrm{n}$-sided polygon: Number of diagonals $=: \frac{n(n-3)}{2}$ For an octagon: $n=8$ $\frac{8(8-3)}{2}=\frac{40}{2}=20$...
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Question: If $a \neq 0$, the multiplicative inverse of $a / b$ is $b / a$. Solution: True....
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Question: Find $\frac{\mathrm{dy}}{\mathrm{dx}}$, when $\mathrm{x}=\frac{1-\mathrm{t}^{2}}{1+\mathrm{t}^{2}}$ and $\mathrm{y}=\frac{2 \mathrm{t}}{1+\mathrm{t}^{2}}$ Solution: as $x=\frac{1-t^{2}}{1+t^{2}}$ Differentiating it with respect to $t$ using quotient rule, $\frac{d x}{d t}=\left[\frac{\left(1+t^{2}\right) \frac{d}{d t}\left(1-t^{2}\right)-\left(1-t^{2}\right) \frac{d}{d t}\left(1+t^{2}\right)}{\left(1+t^{2}\right)^{2}}\right]$ $=\left[\frac{\left(1+t^{2}\right)(-2 t)-\left(1-t^{2}\right...
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Question: Tick (✓) the correct answer: How many diagonals are there in a hexagon? (a) 6 (b) 8 (c) 9 (d) 10 Solution: (c) 9 Number of diagonals in an $\mathrm{n}$-sided polygon $=\frac{n(n-3)}{2}$ For a hexagon: $n=6$ $\therefore \frac{n(n-3)}{2}=\frac{6(6-3)}{2}$ $=\frac{18}{2}=9$...
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Question: Tick (✓) the correct answer: How many diagonals are there in a pentagon? (a) 5 (b) 7 (c) 6 (d) 10 Solution: (a) 5 For a pentagon: n = 5 Number of diagonals $=\frac{n(n-3)}{2}=\frac{5(5-3)}{2}=5$...
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Question: $9 / 6$ lies between 1 and $2 .$ Solution: True. Express each of the given rational numbers with 6 as the common denominator. Now, $1 / 1=[(1 \times 6) /(1 \times 6)]=(6 / 6)$ $2 / 1=[(2 \times 6) /(1 \times 6)]=(12 / 6)$ Then, $6 / 69 / 612 / 6$ $19 / 62$ So, $9 / 6$ is lies between 1 and 2 ....
Read More →Find the angle measure x in the given figure.
Question: Find the angle measurexin the given figure. Solution: For a regular n-sided polygon: Each interior angle $=180-\left(\frac{360}{n}\right)$ In the given figure: $n=5$ $x^{\circ}=180-\frac{360}{5}$ $=180-72$ $=108^{\circ}$ $\therefore \mathrm{x}=108$...
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Question: Find $\frac{\mathrm{dy}}{\mathrm{dx}}$, when $\mathrm{x}=\cos ^{-1} \frac{1}{\sqrt{1+\mathrm{t}^{2}}}$ and $\mathrm{y}=\sin ^{-1} \frac{\mathrm{t}}{\sqrt{1+\mathrm{t}^{2}}}, \mathrm{t} \in \mathrm{R}$ Solution: as $x=\cos ^{-1} \frac{1}{\sqrt{1+t^{2}}}$ Differentiating it with respect to $t$ using chain rule, $\frac{d x}{d t}=-\frac{1}{\sqrt{1-\left(\frac{1}{\sqrt{1+t^{2}}}\right)^{2}}} \frac{d}{d t}\left(\frac{1}{\sqrt{1+t^{2}}}\right)$ $=-\frac{1}{\sqrt{1-\frac{1}{1+t^{2}}}}\left\{-\...
Read More →Simplify each of the following and express it in the form (a + ib):
Question: Simplify each of the following and express it in the form (a + ib): Solution: Given: $\left(\frac{5}{-3+2 i}+\frac{2}{1-i}\right)\left(\frac{4-5 i}{3+2 i}\right)$ $=\left[\frac{5(1-i)+2(-3+2 i)}{(-3+2 i)(1-i)}\right]\left(\frac{4-5 i}{3+2 i}\right)$ [Taking the LCM] $=\left[\frac{5-5 i-6+4 i}{(-3)(1-i)+2 i(1-i)}\right]\left(\frac{4-5 i}{3+2 i}\right)$ $=\left[\frac{-1-i}{-3+3 i+2 i-2 i^{2}}\right]\left(\frac{4-5 i}{3+2 i}\right)$ $=\left[\frac{-(1+i)}{-3+5 i-2(-1)}\right]\left(\frac{4-...
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Question: $-7 / 2$ lies between $-3$ and $-4 .$ Solution: True. Express each of the given rational numbers with 2 as the common denominator. Now. $-3 / 1=[(-3 \times 2) /(1 \times 2)]=(-6 / 2)$ $-4 / 1=[(-4 \times 2) /(1 \times 2)]=(-8 / 2)$ Then, $-8 / 2-7 / 2-6 / 2$ $-4-7 / 2-3$ So, $-7 / 2$ is lies between $-3$ and $-4$....
Read More →In the given figure, find the angle measure x.
Question: In the given figure, find the angle measurex. Solution: Sum of all the interior angles of an $\mathrm{n}$-sided polygon $=(n-2) \times 180^{\circ}$ $m \angle A D C=180-50=130^{\circ}$ $m \angle D A B=180-115=65^{\circ}$ $m \angle B C D=180-90=90^{\circ}$ $m \angle A D C+m \angle D A B+m \angle B C D+m \angle A B C=(n-2) \times 180^{\circ}=(4-2) \times 180^{\circ}=2 \times 180^{\circ}=360^{\circ}$ $\Rightarrow m \angle A D C+m \angle D A B+m \angle B C D+m \angle A B C=360^{\circ}$ $\Ri...
Read More →Find the number of sides of a regular polygon whose each exterior angle measures:
Question: Find the number of sides of a regular polygon whose each exterior angle measures: (i) 40 (ii) 36 (iii) 72 (iv) 30 Solution: Sum of all the exterior angles of a regular polygon is $360^{\circ}$. (i) Each exterior angle $=40^{\circ}$ Number of sides of the regular polygon $=\frac{360}{40}=9$ (ii) Each exterior angle $=36^{\circ}$ Number of sides of the regular polygon $=\frac{360}{36}=10$ (iii) Each exterior angle $=72^{\circ}$ Number of sides of the regular polygon $=\frac{360}{72}=5$ (...
Read More →What is the number of diagonals in a
Question: What is the number of diagonals in a (i) heptagon (ii) octagon (iii) polygon of 12 sides? Solution: Number of diagonal in an $\mathrm{n}$-sided polygon $=\frac{n(n-3)}{2}$ (i) For a heptagon: $n=7 \Rightarrow \frac{n(n-3)}{2}=\frac{7(7-3)}{2}=\frac{28}{2}=14$ (ii) For an octagon: $n=8 \Rightarrow \frac{n(n-3)}{2}=\frac{8(8-3)}{2}=\frac{40}{2}=20$ (iii) For a 12-sided polygon: $n=12 \Rightarrow \frac{n(n-3)}{2}=\frac{12(12-3)}{2}=\frac{108}{2}=54$...
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Question: $5 / 10$ lies between $1 / 2$ and 1 . Solution: False. Express each of the given rational numbers with 10 as the common denominator. Now. $1 / 2=[(1 \times 5) /(2 \times 5)]=(5 / 10)$ $(1)=[(1 \times 10) /(1 \times 10)]=(10 / 10)$ Then, $1 / 2$ is equal to $5 / 10$ So, $5 / 6$ does not lies between $1 / 2$ and 1 ....
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Question: Find $\frac{d y}{d x}$, when $\mathrm{x}=\frac{2 \mathrm{t}}{1+\mathrm{t}^{2}}$ and $\mathrm{y}=\frac{1-\mathrm{t}^{2}}{1+\mathrm{t}^{2}}$ Solution: $a s, x=\frac{2 t}{1+t^{2}}$ Differentiating it with respect to $t$ using quotient rule, $\frac{d x}{d t}=\left[\frac{\left(1+t^{2}\right) \frac{d}{d t}(2 t)-2 t \frac{d}{d t}\left(1+t^{2}\right)}{\left(1+t^{2}\right)^{2}}\right]$ $=\left[\frac{\left(1+t^{2}\right)(2)-2 t(2 t)}{\left(1+t^{2}\right)^{2}}\right]$ $=\left[\frac{2+2 t^{2}-4 t^...
Read More →What is the sum of all interior angles of a regular
Question: What is the sum of all interior angles of a regular (i) pentagon (ii) hexagon (iii) nonagon (iv) polygon of 12 sides? Solution: Sum of the interior angles of an $n$-sided polygon $=(n-2) \times 180^{\circ}$ (i) For a pentagon: $n=5$ $\therefore(n-2) \times 180^{\circ}=(5-2) \times 180^{\circ}=3 \times 180^{\circ}=540^{\circ}$ (ii) For a hexagon: $n=6$ $\therefore(n-2) \times 180^{\circ}=(6-2) \times 180^{\circ}=4 \times 180^{\circ}=720^{\circ}$ (iii) For a nonagon: $n=9$ $\therefore(n-...
Read More →Prove the following
Question: $\frac{5}{6}$ lies between $\frac{2}{3}$ and $1 .$ Solution: True. Express each of the given rational numbers with 6 as the common denominator. Now, $(2 / 3)=[(2 \times 3) /(2 \times 2)]=(4 / 6)$ $(1)=[(1 \times 6) /(1 \times 6)]=(6 / 6)$ Then, $=4 / 65 / 66 / 6$ $=2 / 35 / 61$ So, $5 / 6$ lies between $2 / 3$ and 1 ....
Read More →Is it possible to have a regular polygon each of
Question: Is it possible to have a regular polygon each of whose interior angles is 100? Solution: Each interior angle of a regular polygon having $n$ sides $=180-\left(\frac{360}{n}\right)=\frac{180 n-360}{n}$ If each interior angle of the polygon is $100^{\circ}$, then: $100=\frac{180 n-360}{n}$ $\Rightarrow 100 n=180 n-360$ $\Rightarrow 180 n-100 n=360$ $\Rightarrow 80 n=360$ $\Rightarrow n=\frac{360}{80}=4.5$ Sincenis not an integer, it is not possible to have a regular polygon with each int...
Read More →Find the measure of each interior angle of a regular polygon having
Question: Find the measure of each interior angle of a regular polygon having (i) 10 sides (ii) 15 sides. Solution: For a regular polygon with n s Each interior angle $=180-\{$ Each exterior angle $\}=180-\left(\frac{360}{\mathrm{n}}\right)$ (i) For a polygon with 10 sides: Each exterior angle $=\frac{360}{10}=36^{\circ}$ $\Rightarrow$ Each interior angle $=180-36=144^{\circ}$ (ii) For a polygon with 15 sides: Each exterior angle $=\frac{360}{15}=24^{\circ}$ $\Rightarrow$ Each interior angle $=1...
Read More →If r/s is a rational number,
Question: If $r / s$ is a rational number, then $s$ cannot be equal to zero. Solution: True...
Read More →Express each of the following in the form (a + ib):
Question: Express each of the following in the form (a + ib): $\frac{(1+2 i)^{3}}{(1+i)(2-i)}$ Solution: Given: $\frac{(1+2 i)^{3}}{(1+i)(2-i)}$ We solve the above equation by using the formula $(a+b)^{3}=a^{3}+b^{3}+3 a^{2} b+3 a b^{2}$ $=\frac{(1)^{3}+(2 i)^{3}+3(1)^{2}(2 i)+3(1)(2 i)^{2}}{1(2)+1(-i)+i(2)+i(-i)}$ $=\frac{1+8 i^{3}+6 i+12 i^{2}}{2-i+2 i-i^{2}}$ $=\frac{1+8 i \times i^{2}+6 i+12(-1)}{2+i-(-1)}\left[\because i^{2}=-1\right]$ $=\frac{1+8 i(-1)+6 i-12}{2+i+1}$ $=\frac{1-8 i+6 i-12}...
Read More →Is it possible to have a regular polygon each of whose exterior angles is 50°?
Question: Is it possible to have a regular polygon each of whose exterior angles is 50? Solution: Each exterior angle of an $n$-sided polygon $=\left(\frac{360}{n}\right)^{\circ}$ If the exterior angle is $50^{\circ}$, then: $\frac{360}{n}=50$ $\Rightarrow n=7.2$ Sincenis not an integer, we cannot have a polygon with each exterior angle equal to 50....
Read More →If p/q is a rational number,
Question: If $p / q$ is a rational number, then $p$ cannot be equal to zero. Solution: False. If $p / q$ is a rational number, $p$ can be equal to zero $(0)$ or any integer....
Read More →Find the measure of each exterior angle of a regular
Question: Find the measure of each exterior angle of a regular (i) pentagon (ii) hexagon (iii) heptagon (iv) decagon (v) polygon of 15 sides. Solution: Exterior angle of an $n$-sided polygon $=\left(\frac{360}{n}\right)^{\circ}$ (i) For a pentagon: $n=5$ $\therefore\left(\frac{360}{n}\right)=\left(\frac{360}{5}\right)=72^{\circ}$ (ii) For a hexagon: $n=6$ $\therefore\left(\frac{360}{n}\right)=\left(\frac{300}{6}\right)=60^{\circ}$ (iii) For a heptagon: $n=7$ $\therefore\left(\frac{360}{n}\right)...
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