If sinθ + 2 cosθ = 1,
Question: If sin + 2 cos = 1, then prove that 2 sin cos = 2. Solution: Given, $\quad \sin \theta+2 \cos \theta=1$ On squaring both sides, we get $(\sin \theta+2 \cos \theta)^{2}=1$ $\Rightarrow \quad \sin ^{2} \theta+4 \cos ^{2} \theta+4 \sin \theta \cdot \cos \theta=1$ $\Rightarrow\left(1-\cos ^{2} \theta\right)+4\left(1-\sin ^{2} \theta\right)+4 \sin \theta \cdot \cos \theta=1$ $\left[\because \sin ^{2} \theta+\cos ^{2} \theta=1\right]$ $\Rightarrow \quad-\cos ^{2} \theta-4 \sin ^{2} \theta+4 ...
Read More →Let A = {2, 3, 4, 5} and B = {3, 6, 7, 10}.
Question: Let A = {2, 3, 4, 5} and B = {3, 6, 7, 10}. Let $R=\{(x, y): x \in A, y \in B$ and $x$ is relatively prime to $y\} .$ (i) Write $\mathbf{R}$ in roster form. (ii) Find dom (R) and range (R). Solution: Given: A = {2, 3, 4, 5} and B = {3, 6, 7, 10} (i) $R=\{(x, y),: x \in A, y \in B$ and $x$ is relatively prime to $y\}$ So, R in Roster Form, R = {(2, 3), (2, 7), (3, 7), (3, 10), (4, 3), (4, 7), (5, 3), (5, 6), (5, 7)} (ii) $\operatorname{Dom}(\mathrm{R})=\{2,3,4,5\}$ Range(R) = {3, 6, 7, ...
Read More →Is |sin x| differentiable? What about cos |x|?
Question: Is |sinx| differentiable? What about cos |x|? Solution: Let, $f(x)=|\sin x|$ $\sin x=0$, for $x=n \pi$ $|\sin x|=\left\{\begin{array}{cc}-\sin x (2 m-1) \pix2 m \pi, w \text { here } m \in \mathrm{Z} \\ \sin x 2 m \pix(2 m+1) \pi, w \text { here } m \in \mathrm{Z} \\ -\sin x (2 m+1) \pix2(m+1) \pi, w \text { here } m \in \mathrm{Z}\end{array}\right.$ $(\mathrm{LHD}$ at $x=2 m \pi)=\lim _{x \rightarrow 2 m \pi^{-}} \frac{f(x)-f(2 m \pi)}{x-2 m \pi}$ $=\lim _{x \rightarrow 2 m \pi^{-}} \...
Read More →Solve the following
Question: If $1+\sin ^{2} \theta=3 \sin 0 \cos 0$, then prove that $\tan 0=1$ or $\frac{1}{2}$ Solution: Given, $1+\sin ^{2} A=3 \sin A \cdot \cos A$ On dividing by $\sin ^{2} \theta$ on both sides, we get $\frac{1}{\sin ^{2} \theta}+1=3 \cdot \cot \theta \quad\left[\because \cot \theta=\frac{\cos \theta}{\sin \theta}\right]$ $\Rightarrow \quad \operatorname{cosec}^{2} \theta+1=3 \cdot \cot \theta$ $\left[\operatorname{cosec} \theta=\frac{1}{\sin \theta}\right]$ $\Rightarrow \quad 1+\cot ^{2} \t...
Read More →Let A = {2, 4, 5, 7} and b = {1, 2, 3, 4, 5, 6, 7, 8}.
Question: Let A = {2, 4, 5, 7} and b = {1, 2, 3, 4, 5, 6, 7, 8}. Let $R=\{(x, y) x \in A, y \in B$ and $x$ divides $y\} .$ (i) Write R in roster form. (ii) Find dom (R) and range (R). Solution: Given: A = {2, 4, 5, 7} and b = {1, 2, 3, 4, 5, 6, 7, 8} (i) $R=\{(x, y) x \in A, y \in B$ and $x$ divides $y\}$ So, R in Roster Form, $\mathrm{R}=\{(2,2),(2,4),(2,6),(2,8),(4,4),(4,8),(5,5),(7,7)\}$ (ii) $\operatorname{Dom}(R)=\{2,4,5,7\}$ Range $(R)=\{2,4,5,7,6,7,8\}$...
Read More →The sum of two rational numbers is -4.
Question: The sum of two rational numbers is $-4$. If one of them is $\frac{-11}{5}$, find the other. Solution: Let the other number be $x$. Thus, we have : $x+\frac{-11}{5}=-4$ $\Rightarrow x-\frac{11}{5}=-4$ $\Rightarrow x=-4+\left(\right.$ Additive inverse of $\left.\frac{-11}{5}\right)$ $\Rightarrow x=-4+\frac{11}{5}$ $\Rightarrow x=\frac{-4}{1}+\frac{11}{5}$ $\Rightarrow x=\frac{(-4 \times 5)+(11 \times 1)}{5}$ $\Rightarrow x=\frac{-20+11}{5}$ $\Rightarrow x=\frac{-9}{5}$...
Read More →Find the additive inverse of:
Question: Find the additive inverse of: (i) $\frac{7}{-10}$ (ii) $\frac{8}{5}$. Solution: (i) $\frac{7}{-10}=\frac{7 \times-1}{-10 \times-1}=\frac{-7}{10}$ Additive inverse of $\frac{-7}{10}$ is $\frac{7}{10}$. (ii) Additive inverse of $\frac{8}{5}$ is $\frac{-8}{5}$....
Read More →The angle of elevation of the top of a tower
Question: The angle of elevation of the top of a tower from certain point is 30. If the observer moves 20 m towards the tower, the angle of elevation of the top increases by 15. Find the height of the tower. Solution: Let the height of the tower be $h$. also, $\quad S R=x \mathrm{~m}, \angle P S R=\theta$ Given that, $Q S=20 \mathrm{~m}$ and $\angle P Q R=30^{\circ}$ Now, in $\triangle P S R$, $\tan \theta=\frac{P R}{S R}=\frac{h}{x}$ $\Rightarrow$ $\tan \theta=\frac{h}{x}$ $\Rightarrow$ $x=\fra...
Read More →Let A = {1, 3, 5, 7} and B = {2, 4, 6, 8}.
Question: Let $A=\{1,3,5,7\}$ and $B=\{2,4,6,8\}$ Let $R=\{(x, y),: x \in A, y \in B$ and $xy\} .$ (i) Write $\mathbf{R}$ in roster form. (ii) Find dom (R) and range (R). (iii) Depict $\mathbf{R}$ by an arrow diagram. Solution: Given: A = {1, 3, 5, 7} and B = {2, 4, 6, 8} (i) $R=\{(x, y),: x \in A, y \in B$ and $xy\}$ So, R in Roster Form, R = {(3, 2), (5, 2), (5, 4), (7, 2), (7, 4), (7, 6)} (ii) $\operatorname{Dom}(\mathrm{R})=\{3,5,7\}$ Range(R) = {2, 4, 6} (iii)...
Read More →Tick (✓) the correct answer
Question: Tick (✓) the correct answer The reciprocal of a negative rational number (a) is a positive rational number (b) is a negative rational number (c) can be either a positive or a negative rational number (d) does not exist Solution: (b) is a negative rational numberThe reciprocal of a negative rational number is a negative rational number....
Read More →Discuss the continuity and differentiability of
Question: Discuss the continuity and differentiability of $f(x)= \begin{cases}(x-c) \cos \left(\frac{1}{x-c}\right), x \neq c \\ 0 , x=c\end{cases}$ Solution: Given: $f(x)= \begin{cases}(x-c) \cos \left(\frac{1}{x-c}\right), x \neq c \\ 0 , x=c\end{cases}$ Continuity: $(\mathrm{LHL}$ at $x=\mathrm{c})$ $\lim _{x \rightarrow c^{-}} f(x)$ $=\lim _{h \rightarrow 0} f(c-h)$ $=\lim _{h \rightarrow 0}(c-h-c) \cos \left(\frac{1}{c-h-c}\right)$ $=\lim _{h \rightarrow 0}-h \cos \left(\frac{1}{h}\right)$ ...
Read More →Tick (✓) the correct answer
Question: Tick (✓) the correct answer A rational number between $\frac{-2}{3}$ and $\frac{1}{4}$ is (a) $\frac{5}{12}$ (b) $\frac{-5}{12}$ (c) $\frac{5}{24}$ (d) $\frac{-5}{24}$ Solution: (d) $\frac{-5}{24}$ Rational number between $\frac{-2}{3}$ and $\frac{1}{4}=\frac{1}{2}\left(\frac{-2}{3}+\frac{1}{4}\right)$ $=\frac{1}{2}\left(\frac{-8+3}{12}\right)$ $=\frac{1}{2} \times \frac{-5}{12}$ $=\frac{-5}{24}$...
Read More →Discuss the continuity and differentiability of
Question: Discuss the continuity and differentiability of $f(x)= \begin{cases}(x-c) \cos \left(\frac{1}{x-c}\right), x \neq c \\ 0 , x=c\end{cases}$ Solution: Given: $f(x)= \begin{cases}(x-c) \cos \left(\frac{1}{x-c}\right), x \neq c \\ 0 , x=c\end{cases}$ Continuity: $(\mathrm{LHL}$ at $x=\mathrm{c})$ $\lim _{x \rightarrow c^{-}} f(x)$ $=\lim _{h \rightarrow 0} f(c-h)$ $=\lim _{h \rightarrow 0}(c-h-c) \cos \left(\frac{1}{c-h-c}\right)$ $=\lim _{h \rightarrow 0}-h \cos \left(\frac{1}{h}\right)$ ...
Read More →Discuss the continuity and differentiability of
Question: Discuss the continuity and differentiability of $f(x)= \begin{cases}(x-c) \cos \left(\frac{1}{x-c}\right), x \neq c \\ 0 , x=c\end{cases}$ Solution: Given: $f(x)= \begin{cases}(x-c) \cos \left(\frac{1}{x-c}\right), x \neq c \\ 0 , x=c\end{cases}$ Continuity: $(\mathrm{LHL}$ at $x=\mathrm{c})$ $\lim _{x \rightarrow c^{-}} f(x)$ $=\lim _{h \rightarrow 0} f(c-h)$ $=\lim _{h \rightarrow 0}(c-h-c) \cos \left(\frac{1}{c-h-c}\right)$ $=\lim _{h \rightarrow 0}-h \cos \left(\frac{1}{h}\right)$ ...
Read More →Find the domain and range of each of the relations given below:
Question: Find the domain and range of each of the relations given below: (i) $R=\{(-1,1),(1,1),(-2,4),(2,4),(2,4),(3,9)\}$ (ii) $\mathrm{R}=\left\{\left(\mathrm{x}, \frac{1}{\mathrm{x}}\right): \mathrm{x}\right.$ is an interger, $\left.0\mathrm{x}5\right\}$ (iii) $R=\{(x, y): x+2 y=8$ and $x, y \in N\}$ (iv) $R=\{(x, y),: y=|x-1|, x \in Z$ and $|x| \leq 3\}$ Solution: (i) Given: $R=\{(-1,1),(1,1),(-2,4),(2,4),(2,4),(3,9)\}$ $\operatorname{Dom}(R)=\{x:(x, y) \in R\}=\{-2,-1,1,2,3\}$ Range $(R)=\...
Read More →Tick (✓) the correct answer
Question: Tick (✓) the correct answer Reciprocal of $\frac{-3}{4}$ is (a) $\frac{4}{3}$ (b) $\frac{3}{4}$ (c) $\frac{-4}{3}$ (d) 0 Solution: (c) $\frac{-4}{3}$ Reciprocal of $\frac{-3}{4}$ is $\frac{4}{-3}$, i. e.,$\frac{-4}{3}$....
Read More →Tick (✓) the correct answer
Question: Tick (✓) the correct answer Additive inverse of $\frac{-5}{9}$ is (a) $\frac{-9}{5}$ (b) 0 (c) $\frac{5}{9}$ (d) $\frac{9}{5}$ Solution: (c) $\frac{5}{9}$ Additive inverse of $\frac{-5}{9}$ is $\frac{5}{9}$....
Read More →Prove the following
Question: Prove that $\sqrt{\sec ^{2} \theta+\operatorname{cosec}^{2} \theta}=\tan \theta+\cot \theta$. Solution: $L H S=\sqrt{\sec ^{2} \theta+\operatorname{cosec}^{2} \theta}$ $=\sqrt{\frac{1}{\cos ^{2} \theta}+\frac{1}{\sin ^{2} \theta}}$ $\left[\because \sec \theta=\frac{1}{\cos \theta}\right.$ and $\left.\operatorname{cosec} \theta=\frac{1}{\sin \theta}\right]$ $=\sqrt{\frac{\sin ^{2} \theta+\cos ^{2} \theta}{\sin ^{2} \theta \cdot \cos ^{2} \theta}}=\sqrt{\frac{1}{\sin ^{2} \theta \cdot \c...
Read More →Tick (✓) the correct answer
Question: Tick (✓) the correct answer $\frac{4}{9} \div ?=\frac{-8}{15}$ (a) $\frac{-32}{45}$ (b) $\frac{-8}{5}$ (C) $\frac{-9}{10}$ (d) $\frac{-5}{6}$ Solution: (d) $\frac{-5}{6}$ Let $\frac{4}{9} \div \frac{a}{b}=\frac{-8}{15}$ Now, $\frac{4}{9} \times \frac{b}{a}=\frac{-8}{15}$ $\Rightarrow \frac{b}{a}=\frac{-8}{15} \times \frac{9}{4}$ $=\frac{-6}{5}$ $\Rightarrow \frac{a}{b}=\frac{5}{-6}$ $=\frac{-5}{6}$ Hence, the missing number is $\frac{-5}{6}$...
Read More →Solve the following
Question: If $\operatorname{cosec} \theta+\cot \theta=p$, then prove that $\cos \theta=\frac{p^{2}-1}{p^{2}+1}$ Solution: Given, $\quad \operatorname{cosec} \theta+\cot \theta=p$ $\Rightarrow \quad \frac{1}{\sin \theta}+\frac{\cos \theta}{\sin \theta}=p$ $\left[\because \operatorname{cosec} \theta=\frac{1}{\sin \theta}\right.$ and $\left.\cot \theta=\frac{\cos \theta}{\sin \theta}\right]$ $\Rightarrow$ $\frac{1+\cos \theta}{\sin \theta}=\frac{p}{1}$ $\Rightarrow$ $\frac{(1+\cos \theta)^{2}}{\sin...
Read More →Discuss the continuity and differentiability
Question: Discuss the continuity and differentiability of $f(x)=e^{|x|}$. Solution: Given: $f(x)=e^{|x|}$ $\Rightarrow f(x)= \begin{cases}e^{x}, x \geq 0 \\ e^{-x}, x0\end{cases}$ Continuity: (LHL atx= 0) $\lim _{x \rightarrow 0^{-}} f(x)$ $=\lim _{h \rightarrow 0} f(0-h)$ $=\lim _{h \rightarrow 0} e^{-(0-h)}$ $=\lim _{h \rightarrow 0} e^{h}$ $=1$ $(\mathrm{RHL}$ at $x=0)$ $\lim _{x \rightarrow 0^{+}} f(x)$ $=\lim _{h \rightarrow 0} f(0+h)$ $=\lim _{h \rightarrow 0} e^{(0+h)}$ $=1$ and $f(0)=e^{...
Read More →Tick (✓) the correct answer
Question: Tick (✓) the correct answer $\left(\frac{-5}{9} \div \frac{2}{3}\right)=?$ (a) $\frac{-5}{2}$ (b) $\frac{-5}{6}$ (C) $\frac{-10}{27}$ (d) $\frac{-6}{5}$ Solution: (d) $\frac{-5}{6}$ $\frac{-5}{9} \div \frac{2}{3}=\frac{-5}{9} \times \frac{3}{2}$ $=\frac{-5 \times 3}{9 \times 2}$ $=\frac{-15}{18}$ $=\frac{-5}{6}$...
Read More →Let A and B be two nonempty sets.
Question: Let A and B be two nonempty sets. (i) What do you mean by a relation from A to B? (ii) What do you mean by the domain and range of a relation? Solution: (i) If $A$ and $B$ are two nonempty sets, then any subset of the set $(A \times B)$ is said to a relation R from set A to set B. That means, if $R$ be a relation from $A$ to $B$ then $R \subseteq(A \times B)$. Therefore, $(x, y)^{\in} \mathrm{R} \Rightarrow(x, y)^{\in}(A \times B)$ That means x is in relation to y. Or we can write xRy....
Read More →Discuss the continuity and differentiability
Question: Discuss the continuity and differentiability of $f(x)=e^{|x|}$. Solution: Given: $f(x)=e^{|x|}$ $\Rightarrow f(x)= \begin{cases}e^{x}, x \geq 0 \\ e^{-x}, x0\end{cases}$ Continuity: (LHL atx= 0) $\lim _{x \rightarrow 0^{-}} f(x)$ $=\lim _{h \rightarrow 0} f(0-h)$ $=\lim _{h \rightarrow 0} e^{-(0-h)}$ $=\lim _{h \rightarrow 0} e^{h}$ $=1$ $(\mathrm{RHL}$ at $x=0)$ $\lim _{x \rightarrow 0^{+}} f(x)$ $=\lim _{h \rightarrow 0} f(0+h)$ $=\lim _{h \rightarrow 0} e^{(0+h)}$ $=1$ and $f(0)=e^{...
Read More →Tick (✓) the correct answer
Question: Tick (✓) the correct answer $\left(\frac{-9}{16} \times \frac{8}{15}\right)=?$ (a) $\frac{-3}{10}$ (b) $\frac{-4}{15}$ (C) $\frac{-9}{25}$ (d) $\frac{-2}{5}$ Solution: (a) $\frac{-3}{10}$ $\left(\frac{-9}{16} \times \frac{8}{15}\right)=\frac{-9 \times 8}{16 \times 15}$ $=\frac{-72}{240}$ $=\frac{-3}{10}$...
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