Write each of the following numbers in usual form:
Question: Write each of the following numbers in usual form: (i) 3.74 105 (ii) 6.912 108 (iii) 4.1253 107 (iv) 2.5 104 (v) 5.17 106 (vi) 1.679 109 Solution: (i) $3.74 \times 10^{5}=\frac{374}{100} \times 10^{5}=\frac{374 \times 10^{5}}{10^{2}}=374 \times 10^{(5-2)}=374 \times 10^{3}=374000$ (ii) $6.912 \times 10^{8}=\frac{6912}{1000} \times 10^{8}=\frac{6912 \times 10^{8}}{10^{3}}=6912 \times 10^{(8-3)}=6912 \times 10^{5}=691200000$ (iii) $4.1253 \times 10^{7}=\frac{41253}{10000} \times 10^{7}=\...
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Question: If $f(x)=x^{2}+\frac{x^{2}}{1+x^{2}}+\frac{x^{2}}{\left(1+x^{2}\right)}+\ldots+\frac{x^{2}}{\left(1+x^{2}\right)}+\ldots$ then at $x=0, f(x)$ (a) has no limit (b) is discontinuous (c) is continuous but not differentiable (d) is differentiable Solution: (b) is discontinuous We have, $f(x)=x^{2}+\frac{x^{2}}{1+x^{2}}+\frac{x^{2}}{\left(1+x^{2}\right)}+\ldots+\frac{x^{2}}{\left(1+x^{2}\right)}+\ldots$ When $x=0$ then $x^{2}=0$ and $\frac{x^{2}}{1+x^{2}}=0$ $\therefore f(0)=0+0+0+0 \ldots ...
Read More →Write each of the following numbers in standard form:
Question: Write each of the following numbers in standard form: (i) 57.36 (ii) 3500000 (iii) 273000 (iv) 168000000 (v) 4630000000000 (vi) 345 105 Solution: (i) $57.36=5.736 \times 10$ (ii) $3500000=3.5 \times 10^{6}$ (iii) $273000=2.73 \times 10^{5}$ (iv) $168000000=1.68 \times 10^{8}$ (v) $4630000000000=4.63 \times 10^{12}$ (vi) $345 \times 10^{5}=3.45 \times 10^{7}$...
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Question: If 52x+ 1 25 = 125, find the value ofx. Solution: Given: $5^{2 x+1} \div 25=125$ We have: $25=5 \times 5=5^{2}$ $125=5 \times 5 \times 5=5^{3}$ $\therefore \frac{5^{2 x+1}}{5^{2}}=5^{3} \Rightarrow 5^{[(2 x+1)-2]}=5^{3}$ $\therefore \frac{5^{2 x+1}}{5^{2}}=5^{3} \Rightarrow 5^{[(2 x+1)-2]}=5^{3}$ or $5^{[(2 x+1)-2]}=5^{[2 x-1]}=5^{3}$ $\Rightarrow 2 x-1=3$ $2 x=3+1=4$ $x=\frac{4}{2}=2$ $\therefore x=2$...
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Question: If $f(x)=x^{2}+\frac{x^{2}}{1+x^{2}}+\frac{x^{2}}{\left(1+x^{2}\right)}+\ldots+\frac{x^{2}}{\left(1+x^{2}\right)}+\ldots$ then at $x=0, f(x)$ (a) has no limit (b) is discontinuous (c) is continuous but not differentiable (d) is differentiable Solution: (b) is discontinuous We have, $f(x)=x^{2}+\frac{x^{2}}{1+x^{2}}+\frac{x^{2}}{\left(1+x^{2}\right)}+\ldots+\frac{x^{2}}{\left(1+x^{2}\right)}+\ldots$ When $x=0$ then $x^{2}=0$ and $\frac{x^{2}}{1+x^{2}}=0$ $\therefore f(0)=0+0+0+0 \ldots ...
Read More →The lower window of a house is at a height of 2 m
Question: The lower window of a house is at a height of 2 m above the ground and its upper window is 4 m vertically above the lower window. At certain instant the angles of elevation of a balloon from these windows are observed to be 60 and 30, respectively. Find the height of the balloon above the ground. Solution: Let the height of the balloon from above the ground is H. A and OP=w2R=w1Q=x Given that, height of lower window from above the ground = w2P = 2 m = OR Height of upper window from abo...
Read More →By what number should
Question: By what number should $\left(\frac{-2}{3}\right)^{-3}$ be divided so that the quotient may be $\left(\frac{4}{27}\right)^{-2} ?$ Solution: Let the number be $x$. $\therefore\left(\frac{-2}{3}\right)^{-3} \div x=\left(\frac{4}{27}\right)^{-2}$ $\Rightarrow\left(\frac{3}{-2}\right)^{3} \div x=\left(\frac{27}{4}\right)^{2}$ $\Rightarrow\left(\frac{-3}{2}\right)^{3} \div x=\left(\frac{27}{4}\right)^{2}$ $\Rightarrow\left(\frac{-3}{2}\right)^{3} \times \frac{1}{x}=\left(\frac{27}{4}\right)^...
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Question: If $f(x)=a|\sin x|+b e^{|x|}+c|x|^{3}$ and if $f(x)$ is differentiable at $x=0$, then (a) $a=b=c=0$ (b) $a=0, b=0 ; c \in R$ (c) $b=c=0, a \in R$ (d) $c=0, a=0, b \in R$ Solution: (b) $a=0, b=0 ; c \in R$ We have, $f(x)=a|\sin x|+b e^{|x|}+c|x|^{3}$ $= \begin{cases}a \sin x+b e^{x}+c x^{3} 0x\frac{\pi}{2} \\ -a \sin x+b e^{-x}-c x^{3} -\frac{\pi}{2}x0\end{cases}$ Here, $f(x)$ is differentiable at $x=0$ Therefore, $(\mathrm{LHD}$ at $x=0)=(\mathrm{RHD}$ at $x=0)$ $\Rightarrow \lim _{x \...
Read More →By what number should
Question: By what number should $(-6)^{-1}$ be multiplied so that the product becomes $9^{-1} ?$ Solution: Let the required number be $x$. $\therefore x \times(-6)^{-1}=9^{-1}$ $x \times \frac{1}{-6}=\frac{1}{9} \Rightarrow \frac{x}{-6}=\frac{1}{9}$ or $x=\frac{-6}{9}$ The greatest common divisor for the numerator and the denominator is 3. $\therefore x=\frac{-6}{9}=\frac{(-6) \div 3}{9 \div 3}=\frac{-2}{3}$...
Read More →A window of a house is h m above the ground.
Question: A window of a house is h m above the ground. Form the window, the angles of elevation and depression of the top and the bottom of another house situated on the opposite side of the lane are found to be a and p, respectively. Prove that the height of the other house is h(1+tan cot )m. Solution: Let the height of the other house $=O Q=H$ and $\quad O B=M W=x \mathrm{~m}$ Given that, height of the first house $=W B=h=M O$ and $\angle Q W M=\alpha, \angle O W M=\beta=\angle W O B$ [alterna...
Read More →Find the value of x for which
Question: Find the value of $x$ for which $\left(\frac{4}{9}\right)^{4} \times\left(\frac{4}{9}\right)^{-7}=\left(\frac{4}{9}\right)^{2 x-1}$ Solution: Given: $\left(\frac{4}{9}\right)^{4} \times\left(\frac{4}{9}\right)^{-7}=\left(\frac{4}{9}\right)^{2 x-1}$ $\therefore\left(\frac{4}{9}\right)^{(4-7)}=\left(\frac{4}{9}\right)^{-3}=\left(\frac{4}{9}\right)^{2 x-1}$ $\Rightarrow 2 x-1=-3$ $2 x=-3+1=-2$ $\Rightarrow x=-1$...
Read More →Find the value of x for which
Question: Find the value of $x$ for which $\left(\frac{5}{3}\right)^{-4} \times\left(\frac{5}{3}\right)^{-5}=\left(\frac{5}{3}\right)^{3 x}$. Solution: Consider the left side: $\left(\frac{5}{3}\right)^{-4} \times\left(\frac{5}{3}\right)^{-5}=\left(\frac{5}{3}\right)^{(-4+(-5))}=\left(\frac{5}{3}\right)^{-9}$ Given: $\left(\frac{5}{3}\right)^{-9}=\left(\frac{5}{3}\right)^{3 x}$ Comparing the powers: $-9=3 x \Rightarrow x=-3$...
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Question: Let $R=\{(a, b): a, b \in Z$ and $(a-b)$ is even $\}$ Then, show that R is an equivalence relation on Z Solution: (i) Reflexivity: Let $a \in Z, a-a=0 \in Z$ which is also even. Thus, $(a, a) \in R$ for all $a \in Z$. Hence, it is reflexive (ii) Symmetry: Let $(a, b) \in R$ $(a, b) \in R$ $a-b$ is even $-(b-a)$ is even $(b-a)$ is even $(b, a) \in R$ Thus, it is symmetric (iii) Transitivity: Let $(a, b) \in R$ and $(b, c) \in R$ Then, $(a-b)$ is even and $(b-c)$ is even. $[(a-b)+(b-c)]$...
Read More →Find the value of:
Question: Find the value of: (i) (20+ 31) 32 (ii) (21 31) 23 (iii) $\left(\frac{1}{2}\right)^{-2}+\left(\frac{1}{3}\right)^{-2}+\left(\frac{1}{4}\right)^{-2}$ Solution: (i) $\left(2^{0}+3^{-1}\right) \times 3^{2}=\left(1+\frac{1}{3}\right) \times 3^{2} \quad$ (because $2^{0}=1$ and $\left.3^{-1}=\frac{1}{3}\right)$ $=\left(\frac{1 \times 3}{1 \times 3}+\frac{1 \times 1}{3 \times 1}\right) \times 3^{2}=\left(\frac{3}{3}+\frac{1}{3}\right) \times 3^{2}=\left(\frac{4}{3}\right) \times 3^{2}=4 \time...
Read More →The angle of elevation of the top of a vertical
Question: The angle of elevation of the top of a vertical tower from a point on the ground is 60 From another point 10 m vertically above the first, its angle of elevation is 45. Find the height of the tower. Solution: Let the height the vertical tower, OT = H and $\quad O P=A B=x \mathrm{~m}$ Given that, $A P=10 \mathrm{~m}$ and $\quad \angle T P O=60^{\circ}, \angle T A B=45^{\circ}$ Now, in $\triangle T P O$, $\tan 60^{\circ}=\frac{O T}{O P}=\frac{H}{r}$ $\Rightarrow$$\sqrt{3}=\frac{H}{x}$ $\...
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Question: Evaluate $\left[\left(5^{-1} \times 3^{-1}\right)^{-1} \div 6^{-1}\right]$ Solution: $\left[\left(5^{-1} \times 3^{-1}\right)^{-1} \div 6^{-1}\right]=\left[\left(\frac{1}{5} \times \frac{1}{3}\right)^{-1} \div \frac{1}{6}\right]=\left[\left(\frac{1}{15}\right)^{-1} \div \frac{1}{6}\right]=[15 \times 6]=90$...
Read More →What is an equivalence relation?
Question: What is an equivalence relation? Show that the relation of 'similarity' on the set $S$ of all triangles in a plane is an equivalence relation. Solution: An equivalence relation is one which possesses the properties of reflexivity, symmetry and transitivity. (i) Reflexivity: A relation $\mathrm{R}$ on $\mathrm{A}$ is said to be reflexive if $(\mathrm{a}, \mathrm{a}) \in \mathrm{R}$ for all a $ \mathrm{~A}$. (ii) Symmetry: A relation $R$ on $A$ is said to be symmetrical if $(a, b) \in R$...
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Question: Evaluate $\left\{\left(\frac{4}{3}\right)^{-1}-\left(\frac{1}{4}\right)^{-1}\right\}^{-1}$ Solution: $\left\{\left(\frac{4}{3}\right)^{-1}-\left(\frac{1}{4}\right)^{-1}\right\}^{-1}=\left\{\left(\frac{3}{4}\right)^{1}-\left(\frac{4}{1}\right)^{1}\right\}^{-1}=\left\{\left(\frac{3}{4}\right)-\left(\frac{4}{1}\right)\right\}^{-1}$ The L.C.M. of 4 and 1 is 4 . $\therefore\left\{\left(\frac{3 \times 1}{4 \times 1}\right)-\left(\frac{4 \times 4}{1 \times 4}\right)\right\}^{-1}$ $=\left\{\fr...
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Question: Evaluate $\left\{\left(\frac{1}{3}\right)^{-3}-\left(\frac{1}{2}\right)^{-3}\right\} \div\left(\frac{1}{4}\right)^{-3}$ Solution: $\left\{\left(\frac{1}{3}\right)^{-3}-\left(\frac{1}{2}\right)^{-3}\right\} \div\left(\frac{1}{4}\right)^{-3}=\left\{3^{3}-2^{3}\right\} \div 4^{3}=\{27-8\} \div 64=\frac{19}{64}$...
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Question: If $f(x)=\sqrt{1-\sqrt{1-x^{2}}}$, then $f(x)$ is (a) continuous on $[-1,1]$ and differentiable on $(-1,1)$ (b) continuous on $[-1,1]$ and differentiable on $(-1,0) \cup(0,1)$ (c) continuous and differentiable on $[-1,1]$ Solution: (b) continuous on $[-1,1]$ and differentiable on $(-1,0) \cup(0,1)$ We have, $f(x)=\sqrt{1-\sqrt{1-x^{2}}}$ Here, function will be defined for those values of $x$ for which $1-x^{2} \geq 0$ $\Rightarrow 1 \geq x^{2}$ $\Rightarrow x^{2} \leq 1$ $\Rightarrow|x...
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Question: Evaluate: (i) $\left\{\left(\frac{-2}{3}\right)^{2}\right\}^{-2}$ (ii) $\left[\left\{\left(\frac{-1}{3}\right)^{2}\right\}^{-2}\right]^{-1}$ (iii) $\left\{\left(\frac{3}{2}\right)^{-2}\right\}^{2}$ Solution: (i) $\left\{\left(\frac{-2}{3}\right)^{2}\right\}^{-2}=\left(\frac{-2}{3}\right)^{2 \times(-2)}=\left(\frac{-2}{3}\right)^{-4}=\left(\frac{3}{-2}\right)^{4}=\frac{3^{4}}{(-2)^{4}}=\frac{3^{4}}{2^{4}}=\frac{81}{16}$ (ii) $\left[\left\{\left(\frac{-1}{3}\right)^{2}\right\}^{-2}\right...
Read More →If R is a binary relation on a set A define
Question: If $R$ is a binary relation on a set $A$ define $R^{-1}$ on $A$. Let $R=\{(a, b): a, b \in W$ and $3 a+2 b=15\}$ and $3 a+2 b=15\}$, where $W$ is the set of whole numbers. Express $R$ and $R^{-1}$ as sets of ordered pairs. Show that (i) dom $(R)=$ range $\left(R^{-1}\right)$ (ii) range $(R)=$ dom $\left(R^{-1}\right)$ Solution: 3a + 2b = 15 $b=\frac{15-3 a}{2}$ $a=1$ $b=6$ $a=3$ $b=3$ $a=5$ $b=0$ $R=\{(1,6),(3,3),(5,0)\}$ $R^{-1}=\{(6,1),(3,3),(0,5)\}$ The domain of R is the set of fir...
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Question: Evaluate: (i) $\left(\frac{5}{9}\right)^{-2} \times\left(\frac{3}{5}\right)^{-3} \times\left(\frac{3}{5}\right)^{0}$ (ii) $\left(\frac{-3}{5}\right)^{-4} \times\left(\frac{-2}{5}\right)^{2}$ (iii) $\left(\frac{-2}{3}\right)^{-3} \times\left(\frac{-2}{3}\right)^{-2}$ Solution: (i) $\left(\frac{5}{9}\right)^{-2} \times\left(\frac{3}{5}\right)^{-3} \times\left(\frac{3}{5}\right)^{0}=\left(\frac{5}{9}\right)^{-2} \times\left(\frac{3}{5}\right)^{-3+0}$ $=\left(\frac{5}{9}\right)^{-2} \times...
Read More →A ladder against a vertical wall at an inclination
Question: A ladder against a vertical wall at an inclination a to the horizontal. Sts foot is pulled away from the wall through a distance p, so that its upper end slides a distance q down the wall and then the ladder makes an angle B to the horizontal. Show that $\frac{p}{q}=\frac{\cos \beta-\cos \alpha}{\sin \alpha-\sin \beta}$ Solution: Iet $\quad O O=x$ and $O A=v$ Given that, $B Q=q, S A=P$ and $A B=S Q=$ Length of ladder Also, $\angle B A O=\alpha$ and $\angle Q S O=\beta$ Now, in $\triang...
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Question: Evaluate: (i) $\left(\frac{5}{3}\right)^{2} \times\left(\frac{5}{3}\right)^{2}$ (ii) $\left(\frac{5}{6}\right)^{6} \times\left(\frac{5}{6}\right)^{-4}$ (iii) $\left(\frac{2}{3}\right)^{-3} \times\left(\frac{2}{3}\right)^{-2}$ (iv) $\left(\frac{9}{8}\right)^{-3} \times\left(\frac{9}{8}\right)^{2}$ Solution: (i) $\left(\frac{5}{3}\right)^{2} \times\left(\frac{5}{3}\right)^{2}=\left(\frac{5}{3}\right)^{4}=\frac{5^{4}}{3^{4}}=\frac{625}{81}$ (ii) $\left(\frac{5}{6}\right)^{6} \times\left(\...
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