-1 is not the reciprocal of any
Question: $-1$ is not the reciprocal of any rational number. Solution: False. The reciprocal of- 1 is $-1$....
Read More →Find the conjugate of each of the following:
Question: Find the conjugate of each of the following: $\frac{(1+i)(2+i)}{(3+i)}$ Solution: Given: $\frac{(1+i)(2+i)}{(3+i)}$ Firstly, we calculate $\frac{(1+i)(2+i)}{(3+i)}$ and then find its conjugate $\frac{(1+i)(2+i)}{(3+i)}=\frac{1(2)+1(i)+i(2)+i(i)}{(3+i)}$ $=\frac{2+i+2 i+i^{2}}{3+i}$ $=\frac{2+3 i-1}{3+i}\left[\because i^{2}=-1\right]$ $=\frac{1+3 i}{3+i}$ Now, we rationalize the above by multiplying and divide by the conjugate of 3 + i $=\frac{1+3 i}{3+i} \times \frac{3-i}{3-i}$ $=\frac...
Read More →Find the conjugate of each of the following:
Question: Find the conjugate of each of the following: $\frac{(1+i)(2+i)}{(3+i)}$ Solution: Given: $\frac{(1+i)(2+i)}{(3+i)}$ Firstly, we calculate $\frac{(1+i)(2+i)}{(3+i)}$ and then find its conjugate $\frac{(1+i)(2+i)}{(3+i)}=\frac{1(2)+1(i)+i(2)+i(i)}{(3+i)}$ $=\frac{2+i+2 i+i^{2}}{3+i}$ $=\frac{2+3 i-1}{3+i}\left[\because i^{2}=-1\right]$ $=\frac{1+3 i}{3+i}$ Now, we rationalize the above by multiplying and divide by the conjugate of 3 + i $=\frac{1+3 i}{3+i} \times \frac{3-i}{3-i}$ $=\frac...
Read More →1 is the only number
Question: 1 is the only number which is its own reciprocal. Solution: False. Because, the reciprocal of $-1$ is $-1$ and reciprocal of 1 is 1 ....
Read More →A quadrilateral has three acute angles, each measuring 75°.
Question: A quadrilateral has three acute angles, each measuring 75. Find the measure of the fourth angle. Solution: Sum of the four angles of a quadrilateral is $360^{\circ}$. If the unknown angle is $x^{\circ}$, then: $75+75+75+x=360$ $x=360-225=135$ The fourth angle measures $135^{\circ}$....
Read More →For all rational numbers
Question: For all rational numbers $a, b$ and $c, a(b+c)=a b+b c$. Solution: False. Because, for every rational numbers $a, b$ and $c,[a \times(b+c)=(a \times b)+(a \times c)]$...
Read More →The angles of a quadrilateral are in the ratio 3 : 5 : 7 : 9.
Question: The angles of a quadrilateral are in the ratio 3 : 5 : 7 : 9. Find the measure of each of these angles. Solution: Let the measures of the angles of the given quadrilateral be $(3 x)^{\circ},(5 x)^{\circ},(7 x)^{\circ}$ and $(9 x)^{\circ}$. Sum of all the angles of a quadrilateral is $360^{\circ}$. $\therefore 3 x+5 x+7 x+9 x=360$ $24 x=360$ $x=15$ Angles measure: $(3 \times 15)^{\circ}=45^{\circ}$ $(5 \times 15)^{\circ}=75^{\circ}$ $(7 \times 15)^{\circ}=105^{\circ}$ $(9 \times 15)^{\c...
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Question: Find $\frac{\mathrm{dy}}{\mathrm{dx}}$, when If $x=\left(t+\frac{1}{t}\right)^{a}, y=a^{t+\frac{1}{t}}$, find $\frac{d y}{d x}$ Solution: $a s x=\left(t+\frac{1}{t}\right)^{a}$ Differentiating it with respect to $t$ using chain rule, $\frac{d x}{d t}=\frac{d}{d t}\left(\left(t+\frac{1}{t}\right)^{a}\right)$ $=a\left(\left(t+\frac{1}{t}\right)^{a-1}\right) \frac{d}{d t}\left(t+\frac{1}{t}\right)$ $\frac{d x}{d t}=a\left(\left(t+\frac{1}{t}\right)^{a-1}\right)\left(1-\frac{1}{t^{2}}\righ...
Read More →The three angles of a quadrilateral are 76°, 54° and 108°.
Question: The three angles of a quadrilateral are 76, 54 and 108. Find the measure of the fourth angle. Solution: Sum of all the four angles of a quadrilateral is 360. Let the unknown angle be $x^{\circ}$. $76+54+108+x=360$ $238+x=360$ $x=122$ The fourth angle measures 122....
Read More →Find the conjugate of each of the following:
Question: Find the conjugate of each of the following: $\frac{(1+i)^{2}}{(3-i)}$ Solution: Given: $\frac{(1+i)^{2}}{(3-i)}$ Firstly, we calculate ${ }^{\frac{(1+i)^{2}}{(3-i)}}$ and then find its conjugate $\frac{(1+i)^{2}}{(3-i)}=\frac{1+i^{2}+2 i}{(3-i)}\left[\because(a+b)^{2}=a^{2}+b^{2}+2 a b\right]$ $=\frac{1+(-1)+2 i}{3-i}\left[\because{\mathrm{i}}^{2}=-1\right]$ $=\frac{2 i}{3-i}$ Now, we rationalize the above by multiplying and divide by the conjugate of 3 i $=\frac{2 i}{3-i} \times \fra...
Read More →Prove that the sum of the angles of a quadrilateral is 360°.
Question: Prove that the sum of the angles of a quadrilateral is 360. Solution: LetABCDbe a quadrilateral. JoinAandC. Now, we know that the sum of the angles of a triangle is 180. For $\triangle A B C$. $\angle 2+\angle 4+\angle B=180^{\circ} \quad \ldots$ (1) For $\triangle A D C:$ $\angle 1+\angle 3+\angle D=180^{\circ} \quad \cdots(2)$ Adding (1) and (2): $(\angle 1+\angle 2+\angle 3+\angle 4)+\angle B+\angle D=360^{\circ}$ or $\angle A+\angle B+\angle C+\angle D=360^{\circ}$ Hence, the sum o...
Read More →In the adjoining figure, ABCD is a quadrilateral.
Question: In the adjoining figure,ABCDis a quadrilateral. (i) How many pairs of adjacent sides are there? Name them. (ii) How many pairs of opposite sides are there? Name them. (iii) How many pairs of adjacent angles are there? Name them. (iv) How many pairs of opposite angles are there? Name them. (v) How many diagonals are there? Name them. Solution: (i) There are four pairs of adjacent sides, namely $(A B, B C),(B C, C D),(C D, D A)$ and $(D A, A B)$. (ii) There are two pairs of opposite side...
Read More →Fill in the blanks:
Question: Fill in the blanks: (i) A quadrilateral has ......... sides. (ii) A quadrilateral has ......... angles. (iii) A quadrilateral has ......... vertices, no three of which are ......... (iv) A quadrilateral has ......... diagonals. (v) A diagonal of a quadrilateral is a line segment that joins two ......... vertices of the quadrilateral. (vi) The sum of the angles of a quadrilateral is......... Solution: (i) 4 (ii) 4 (iii) 4, co-linear (iv) 2 (v) opposite (vi) 360...
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Question: Find $\frac{d y}{d x}$, when If $x=\frac{\sin ^{3} t}{\sqrt{\cos 2 t}}, y=\frac{\cos ^{3} t}{\sqrt{\cos 2 t}}$, find $\frac{d y}{d x}$ Solution: $\operatorname{as} x=\frac{\sin ^{3} t}{\sqrt{\cos 2 t}}$ Then $\frac{d x}{d t}=\frac{d}{d t}\left[\frac{\sin ^{3} t}{\sqrt{\cos 2 t}}\right]$ $=\frac{\sqrt{\cos 2 t} \cdot \frac{d}{d t}\left(\sin ^{3} t\right)-\sin ^{3} t \cdot \frac{d}{d t} \sqrt{\cos 2 t}}{\cos 2 t}$ $=\frac{\sqrt{\cos 2 t} \cdot 3 \sin ^{2} t \frac{d}{d t}(\sin t)-\sin ^{3...
Read More →Find the conjugate of each of the following:
Question: Find the conjugate of each of the following: $\frac{1}{(4+3 i)}$ Solution: Given: $\frac{1}{4+3 i}$ First, we calculate $\frac{1}{4+3 i}$ and then find its conjugate Now, rationalizing $=\frac{1}{4+3 i} \times \frac{4-3 i}{4-3 i}$ $=\frac{4-3 i}{(4+3 i)(4-3 i)}$ Now, we know that, $(a+b)(a-b)=\left(a^{2}-b^{2}\right)$ So, eq. (i) become $=\frac{4-3 i}{(4)^{2}-(3 i)^{2}}$ $=\frac{4-3 i}{16-9 i^{2}}$ $=\frac{4-3 i}{16-9(-1)}\left[\because i^{2}=-1\right]$ $=\frac{4-3 i}{16+9}$ $=\frac{4-...
Read More →For every rational numbers
Question: For every rational numbers $x, y$ and $z, x+(y \times z)=(x+y) \times(x+z)$. Solution: False. For every rational numbers $a, b$ and $c,[a \times(b+c)=(a \times b)+(a \times c)]$...
Read More →Solve this
Question: Find $\frac{d y}{d x}$, when If $x=\sin ^{-1}\left(\frac{2 t}{1+t^{2}}\right)$ and $y=\tan ^{-1}\left(\frac{2 t}{1-t^{2}}\right),-1t1$, prove that $\frac{d y}{d x}=1$ Solution: as $\mathrm{x}=\sin ^{-1}\left(\frac{2 \mathrm{t}}{1+\mathrm{t}^{2}}\right)$ Put $t=\tan \theta$ $x=\sin ^{-1}\left(\frac{2 \tan \theta}{1+\tan ^{2} \theta}\right)$ $=\sin ^{-1} \sin 2 \theta$ $=2 \theta\left[\right.$ since, $\left.\sin 2 \theta=\frac{2 \tan \theta}{1+\tan ^{2} \theta}\right]$ $x=2\left(\tan ^{-...
Read More →Tick (✓) the correct answer:
Question: Tick (✓) the correct answer: The interior angle of a regular polygon exceeds its exterior angle by 108. How many sides does the polygon have? (a) 16 (b) 14 (c) 12 (d) 10 Solution: (d) 10 Each exterior angle of a regular polygon $=\frac{360}{n}$ Each interior angle of a regular polygon $=180-\frac{360}{n}$ $180-\frac{360}{n}-108=\frac{360}{n}$ $\frac{720}{n}=180-108=72$ $n=\frac{720}{72}=10$...
Read More →For every rational number
Question: For every rational number $x, x \times 0=x$ Solution: False. Let $\mathrm{x}=2$ Then, For every rational number $x$ $(x) \times(0)=0$ $2 \times 0=0$...
Read More →For all rational numbers
Question: For all rational numbers $x$ and $y,(x) \times(y)=(y) \times(x)$ Solution: True. Let $x=2, y=3$ Then, LHS $=2 \times 3$ $=6$ $\mathrm{RHS}=3 \times 2$ By comparing $\mathrm{LHS}$ and RHS $6=6$ $\mathrm{LHS}=\mathrm{RHS}$...
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Question: Tick (✓) the correct answer: The sum of all interior angles of a regular polygon is 1080. What is the measure of each of its interior angles? (a) 135 (b) 120 (c) 156 (d) 144 Solution: (a) 135 $(2 n-4) \times 90=1080$ $(2 n-4)=12$ $2 n=16$ or $n=8$ Each interior angle $=180-\frac{360}{n}=180-\frac{360}{8}=180-45=135^{\circ}$...
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Question: Tick (✓) the correct answer: The sum of all interior angles of a hexagon is (a) 6 right s (b) 8 right s (c) 9 right s (d) 12 right s Solution: (b) 8 right s Sum of all the interior angles of a hexagon is $(2 n-4)$ right angles. For a hexagon: $n=6$ $\Rightarrow(2 \mathrm{n}-4)$ right $\angle \mathrm{s}=(12-4)$ right $\angle \mathrm{s}=8$ right $\angle \mathrm{s}$...
Read More →Find the conjugate of each of the following:
Question: Find the conjugate of each of the following: $(-5-2 i)$ Solution: Given: $z=(-5-2 i)$ Here, we have to find the conjugate of $(-5-2 i)$ So, the conjugate of $(-5-2 i)$ is $(-5+2 i)$...
Read More →Tick (✓) the correct answer:
Question: Tick (✓) the correct answer: Each interior angle of a regular decagon is (a) 60 (b) 120 (c) 144 (d) 180 Solution: (c) 144 Each interior angle of a regular decagon $=180-\frac{360}{10}=180-36=144^{\circ}$...
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Question: Find $\frac{\mathrm{dy}}{\mathrm{dx}}$, when If $x=a\left(t+\frac{1}{t}\right)$ and $y=a\left(t-\frac{1}{t}\right)$, prove that $\frac{d y}{d x}=\frac{x}{y}$ Solution: as $x=a\left(t+\frac{1}{t}\right)$ Differentiating it with respect to $t$, $\frac{\mathrm{dx}}{\mathrm{dt}}=\frac{\mathrm{ad}}{\mathrm{dt}}\left(\mathrm{t}+\frac{1}{\mathrm{t}}\right)$ $=\mathrm{a}\left(1-\frac{1}{\mathrm{t}^{2}}\right)$ $\frac{\mathrm{dx}}{\mathrm{dt}}=\mathrm{a}\left(\frac{\mathrm{t}^{2}-1}{\mathrm{t}^...
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