Prove the following
Question: $\frac{\sin \theta}{1+\cos \theta}+\frac{1+\cos \theta}{\sin \theta}=2 \operatorname{cosec} \theta$ Solution: $\mathrm{LHS}=\frac{\sin \theta}{1+\cos \theta}+\frac{1+\cos \theta}{\sin \theta}=\frac{\sin ^{2} \theta+(1+\cos \theta)^{2}}{\sin \theta(1+\cos \theta)}$ $=\frac{\sin ^{2} \theta+1+\cos ^{2} \theta+2 \cos \theta}{\sin \theta(1+\cos \theta)} \quad\left[\because(a+b)^{2}=a^{2}+b^{2}+2 a b\right]$ $=\frac{1+1+2 \cos \theta}{\sin \theta(1+\cos \theta)} \quad\left[\because \sin ^{2...
Read More →Solve this
Question: If $f(x)=\left\{\begin{array}{ll}a x^{2}-b, \text { if }|x|1 \\ \frac{1}{|x|}, \text { if }|x| \geq 1\end{array}\right.$ is differentiable at $x=1$, find $a, b$. Solution: Given: $f(x)= \begin{cases}a x^{2}+b, |x|1 \\ \frac{1}{|x|}, |x| \geq 1\end{cases}$ $\Rightarrow f(x)= \begin{cases}-\frac{1}{x}, x-1 \\ a x^{2}-b, -1x1 \\ \frac{1}{x}, x \geq 1\end{cases}$ It is given that the given function is differentiable atx= 1. We know every differentiable function is continuous. Therefore it ...
Read More →If the height of a tower and the distance
Question: If the height of a tower and the distance of the point of observation from its foot, both.are increased by 10%, then the angle of elevation of its top remains unchanged. Solution: True Case I Let the height of a tower be $h$ and the distance of the point of observation from its foot is $x$. In $\triangle A B C$, $\tan \theta_{1}=\frac{A C}{B C}=\frac{h}{x}$ $\Rightarrow \quad \theta_{1}=\tan ^{-1}\left(\frac{h}{x}\right)$ $\ldots$ (i) Case II Now, the height of a tower increased by $10...
Read More →Solve this
Question: If $f(x)=\left\{\begin{array}{ll}a x^{2}-b, \text { if }|x|1 \\ \frac{1}{|x|}, \text { if }|x| \geq 1\end{array}\right.$ is differentiable at $x=1$, find $a, b$. Solution: Given: $f(x)= \begin{cases}a x^{2}+b, |x|1 \\ \frac{1}{|x|}, |x| \geq 1\end{cases}$ $\Rightarrow f(x)= \begin{cases}-\frac{1}{x}, x-1 \\ a x^{2}-b, -1x1 \\ \frac{1}{x}, x \geq 1\end{cases}$ It is given that the given function is differentiable atx= 1. We know every differentiable function is continuous. Therefore it ...
Read More →For any sets A and B, prove that
Question: For any sets A and B, prove that $(A \times B) \cap(B \times A)=(A \cap B) \times(B \cap A)$ Solution: Given: A and B two sets are given. Need to prove: $(A \times B) \cap(B \times A)=(A \cap B) \times(B \cap A)$ Let us consider, $(x, y)^{\in}(A \times B) \cap(B \times A)$ $\Rightarrow(x, y)^{\in}(A \times B)$ and $(x, y)^{\in}(B \times A)$ $\Rightarrow\left(x \in_{A}\right.$ and $\left.y \in B\right)$ and $\left(x \in_{B}\right.$ and $\left.y \in_{A}\right)$ $\Rightarrow\left(x^{\in} ...
Read More →Show that the function
Question: Show that the function $f(x)= \begin{cases}|2 x-3|[x], x \geq 1 \\ \sin \left(\frac{\pi x}{2}\right), x1\end{cases}$is continuous but not differentiable atx = 1. Solution: Given: $f(x)=\left\{\begin{array}{l}|2 x-3|[x], \quad x \geq 1 \\ \sin \left(\frac{\pi \mathrm{x}}{2}\right), \quad x1\end{array}\right.$ Continuity atx= 1: $(\mathrm{LHL}$ at $x=1)=\lim _{x \rightarrow 1^{-}} f(x)=\lim _{h \rightarrow 0} f(1-h)=\lim _{h \rightarrow 0} \sin \left(\frac{\pi(1-h)}{2}\right)=\sin \frac{...
Read More →Tick (✓) the correct answer
Question: Tick (✓) the correct answer $\left(\frac{-5}{16}+\frac{7}{12}\right)=?$ (a) $-\frac{7}{48}$ (b) $\frac{1}{24}$ (c) $\frac{13}{48}$ (d) $\frac{1}{3}$ Solution: (c) $\frac{13}{48}$ The denominators of the given rational numbers are 16 and 12 , respectively. LCM of 16 and 12 is $(4 \times 4 \times 3)$, that is, 48 Now, we have: $\left(\frac{-5}{16}+\frac{7}{12}\right)=\frac{3 \times(-5)+4 \times 7}{48}$ $=\frac{(-15)+28}{48}$ $=\frac{13}{48}$...
Read More →Show that the function
Question: Show that the function $f(x)= \begin{cases}|2 x-3|[x], x \geq 1 \\ \sin \left(\frac{\pi x}{2}\right), x1\end{cases}$is continuous but not differentiable atx = 1. Solution: Given: $f(x)=\left\{\begin{array}{l}|2 x-3|[x], \quad x \geq 1 \\ \sin \left(\frac{\pi \mathrm{x}}{2}\right), \quad x1\end{array}\right.$ Continuity atx= 1: $(\mathrm{LHL}$ at $x=1)=\lim _{x \rightarrow 1^{-}} f(x)=\lim _{h \rightarrow 0} f(1-h)=\lim _{h \rightarrow 0} \sin \left(\frac{\pi(1-h)}{2}\right)=\sin \frac{...
Read More →At a cricket test match
Question: At a cricket test match $\frac{2}{7}$ of the spectators were in a covered place while 15000 were in open. Find the total number of spectators. Solution: Ratio of spectators in the open $=1-\frac{2}{7}$ $=\frac{5}{7}$ Total number of spectators in the open $=x$ Then, $\frac{5}{7} \times x=15000$ $\Rightarrow x=15000 \div \frac{5}{7}$ $=15000 \times \frac{7}{5}$ $=\frac{15000}{1} \times \frac{7}{5}$ $=\frac{15000 \times 7}{1 \times 5}$ $=\frac{10500}{5}$ $=21000$ Hence, the total number ...
Read More →The angle of elevation of the top of a tower is 30°.
Question: The angle of elevation of the top of a tower is 30. If the height of the tower is doubled, then the angle of elevation of its top will also be doubled. Solution: False Let the height of the tower is $h$ and $B C=x \mathrm{~m}$ In $\triangle A B C$, $\tan 30^{\circ}=\frac{A C}{B C}=\frac{h}{x}$ $\Rightarrow \quad \frac{1}{\sqrt{3}}=\frac{h}{x}$ ......(i) Case II By condition, the height of the tower is doubled. i.e., $P R=2 h$. In $\triangle P Q R, \quad \tan \theta=\frac{P R}{Q R}=\fra...
Read More →Find the values of a and b so that the function
Question: Find the values of $a$ and $b$ so that the function $f(x) \begin{cases}x^{2}+3 x+a, \text { if } x \leq 1 \\ b x+2, \text { if } x1\end{cases}$is differentiable at eachxR. Solution: Given: $f(x)=\left\{\begin{array}{l}x^{2}+3 x+a, \quad x \leq 1 \\ b x+2, \quad x1\end{array}\right.$ It is given that the function is differentiable at each $x \in R$ and every differentiable function is continuous. So, $f(x)$ is continuous at $x=1$. Therefore, $\lim _{x \rightarrow 1^{-}} f(x)=\lim _{x \r...
Read More →Find the values of a and b so that the function
Question: Find the values of $a$ and $b$ so that the function $f(x) \begin{cases}x^{2}+3 x+a, \text { if } x \leq 1 \\ b x+2, \text { if } x1\end{cases}$is differentiable at eachxR. Solution: Given: $f(x)=\left\{\begin{array}{l}x^{2}+3 x+a, \quad x \leq 1 \\ b x+2, \quad x1\end{array}\right.$ It is given that the function is differentiable at each $x \in R$ and every differentiable function is continuous. So, $f(x)$ is continuous at $x=1$. Therefore, $\lim _{x \rightarrow 1^{-}} f(x)=\lim _{x \r...
Read More →find the number.
Question: If $\frac{3}{5}$ of a number exceeds its $\frac{2}{7}$ by 44, find the number. Solution: Let $x$ be the required number. We know that $\frac{3}{5}$ of the number exceeds its $\frac{2}{7}$ by 44 . That is, $\frac{3}{5} \times x=\frac{2}{7} \times x+44$ $\frac{3}{5} \times x-\frac{2}{7} \times x=44$ $\left(\frac{3}{5}-\frac{2}{7}\right) \times x=44$ $\left(\frac{3}{5}+\right.$ Additive inverse of $\left.\frac{2}{7}\right) \times x=44$ $\left(\frac{21-10}{35}\right) \times x=44$ $\frac{11...
Read More →Prove the following
Question: $\cos \theta=\frac{a^{2}+b^{2}}{2 a b}$ where $\mathrm{a}$ and $\mathrm{b}$ are two distinct numbers such that $\mathrm{ab}0$. Solution: False Given, a and b are two distinct numbers such that ab 0. Using, AM GM [since, AM and GM of two number a and $b$ are $\frac{a+b}{2}$ and $\sqrt{a b}$, respectively] $\Rightarrow$$\frac{a^{2}+b^{2}}{2}\sqrt{a^{2} \cdot b^{2}}$ $\Rightarrow$ $a^{2}+b^{2}2 a b$ $\Rightarrow$ $\frac{a^{2}+b^{2}}{2 a b}1$ $\left[\because \cos \theta=\frac{a^{2}+b^{2}}{...
Read More →Amit earns ₹ 32000 per month.
Question: Amit earns ₹ 32000 per month. He spends $\frac{1}{4}$ of his income of food; $\frac{3}{10}$ of the remainder on house rent and $\frac{5}{21}$ of the remainder on the education of children. How much money is still left with him? Solution: Amit's income per month $=₹ 32,000$ Money spent on food $=\frac{1}{4}$ of ₹ $32,000=\frac{1}{4} \times ₹ 32,000=₹ 8,000$ Remaining amount $=₹ 32,000-₹ 8,000=₹ 24,000$ Money spent on house rent $=\frac{3}{10} \times ₹ 24,000=$ Rs 7,200 Money left = ₹24,...
Read More →The value of
Question: The value of $2 \sin \theta$ can be $a+\frac{1}{a}$ where $a$ is a positive number and $a \neq 1$. Solution: False Given, $a$ is a positive number and $a \neq 1$, then $A MG M$ $\Rightarrow \quad \frac{a+\frac{1}{a}}{2}\sqrt{a \cdot \frac{1}{a}} \Rightarrow\left(a+\frac{1}{a}\right)2$ [since, $\mathrm{AM}$ and $\mathrm{GM}$ of two number's $\mathrm{a}$ and $\mathrm{b}$ are $\frac{(a+b)}{2}$ and $\sqrt{a b}$, respectively] $\Rightarrow \quad 2 \sin \theta2$ $\left[\because 2 \sin \theta...
Read More →Rita had Rs 300 .
Question: Rita had Rs 300 . She spent $\frac{1}{3}$ of her money on notebooks and $\frac{1}{4}$ of the remainder on stationery items. How much money is left with her? Solution: Amount of money spent on notebooks $=300 \times \frac{1}{3}$ $=\frac{300}{1} \times \frac{1}{3}$ $=\frac{300}{3}$ $=100$ $\therefore$ Money left after spending on notebooks $=300-100$ = 200 Amount of money spent on stationery items from the remainder $=200 \times \frac{1}{4}$ $=\frac{200}{1} \times \frac{1}{4}$ $=\frac{20...
Read More →If a man standing on a plat form 3 m above
Question: If a man standing on a plat form 3 m above the surface of a lake observes a cloud and its reflection in the lake, then the angle of elevation of the cloud is equal to the angle of depression of its reflection Solution: False From figure, we observe that, a man standing on a platform at point P, 3 m above the surface of a lake observes a cloud at point C. Let the height of the cloud from the surface of the platform is h and angle of elevation of the cloud is1. Now at same point P a man ...
Read More →How many pages are there in the book?
Question: After reading $\frac{7}{9}$ of a book, 40 pages are left. How many pages are there in the book? Solution: Ratio of the read book $=\frac{7}{9}$ Ratio of the unread book $=1-\frac{7}{9}$ $=\frac{2}{9}$ Letxbe the total number of pages in the book. Thus, we have: $\frac{2}{9} \times x=40$ $\Rightarrow x=40 \div \frac{2}{9}$ $=40 \times \frac{9}{2}$ $=\frac{40}{1} \times \frac{9}{2}$ $=\frac{40 \times 9}{1 \times 2}$ $=\frac{360}{2}$ $=180$ Hence, the total number of pages in the book is ...
Read More →Show that the function
Question: Show that the function $f(x)= \begin{cases}x^{m} \sin \left(\frac{1}{x}\right) , x \neq 0 \\ 0 , x=0\end{cases}$ (i) differentiable at $x=0$, if $m1$ (ii) continuous but not differentiable at $x=0$, if $0m1$ (iii) neither continuous nor differentiable, if $m \leq 0$ Solution: Given: $f(x)= \begin{cases}x^{m} \sin \left(\frac{1}{x}\right) \\ 0 \mathrm{x} \neq 0, \mathrm{x}=0\end{cases}$ (i) Let $\mathrm{m}=2$, then the function becomes $f(x)= \begin{cases}x^{2} \sin \left(\frac{1}{x}\ri...
Read More →Show that the function
Question: Show that the function $f(x)= \begin{cases}x^{m} \sin \left(\frac{1}{x}\right) , x \neq 0 \\ 0 , x=0\end{cases}$ (i) differentiable at $x=0$, if $m1$ (ii) continuous but not differentiable at $x=0$, if $0m1$ (iii) neither continuous nor differentiable, if $m \leq 0$ Solution: Given: $f(x)= \begin{cases}x^{m} \sin \left(\frac{1}{x}\right) \\ 0 \mathrm{x} \neq 0, \mathrm{x}=0\end{cases}$ (i) Let $\mathrm{m}=2$, then the function becomes $f(x)= \begin{cases}x^{2} \sin \left(\frac{1}{x}\ri...
Read More →If there are 240 girls, find the number of boys in the school.
Question: In a school, $\frac{5}{8}$ of the students are boys. If there are 240 girls, find the number of boys in the school. Solution: If $\frac{5}{8}$ of the students are boys, then the ratio of girls is $1-\frac{5}{8}$, that is, $\frac{3}{8}$. Now, let $x$ be the total number of students. Thus, we have: $\frac{3}{8} x=240$ $\Rightarrow x=240 \div \frac{3}{8}$ $=240 \times \frac{8}{3}$ $=\frac{240}{1} \times \frac{8}{3}$ $=\frac{240 \times 8}{1 \times 3}$ $=\frac{1920}{3}$ $=640$ Hence, the to...
Read More →If the length of the shadow of a tower is increasing,
Question: If the length of the shadow of a tower is increasing, then the angle of elevation of the Sun is also increasing. Solution: False To understand the fact of this question, consider the following example I. A tower $2 \sqrt{3} \mathrm{~m}$ high casts a shadow $2 \mathrm{~m}$ long on the ground, then the Sun's elevation is $60^{\circ}$. In $\triangle A C B$, $\tan \theta=\frac{A B}{B C}=\frac{2 \sqrt{3}}{2}$ $\Rightarrow \quad \tan \theta=\sqrt{3}=\tan 60^{\circ}$ $\therefore \quad \theta=...
Read More →The product of two fractions is
Question: The product of two fractions is $9 \frac{3}{5}$. If one of the fractions is $9 \frac{3}{7}$, find the other. Solution: Let the other fraction bex. Now, we have: $9 \frac{3}{7} \times x=9 \frac{3}{5}$ $\Rightarrow x=9 \frac{3}{5} \div 9 \frac{3}{7}$ $=\left(9+\frac{3}{5}\right) \div\left(9+\frac{3}{7}\right)$ $=\frac{48}{5} \div \frac{66}{7}$ $=\frac{48}{5} \times \frac{7}{66}$ $=\frac{48 \times 7}{5 \times 66}$ $=\frac{336}{330}$ $=\frac{56}{55}$ $=1 \frac{1}{55}$ Hence, the other frac...
Read More →The area of a room is
Question: The area of a room is $65 \frac{1}{4} \mathrm{~m}^{2}$. If its breadth is $5 \frac{7}{16}$ metres, what is its length? Solution: Area of a room $=$ Length $\times$ Breadth Thus, we have: $65 \frac{1}{4}=$ Length $\times 5 \frac{7}{16}$ Length $=65 \frac{1}{4} \div 5 \frac{7}{16}$ $=\left(65+\frac{1}{4}\right) \div\left(5+\frac{7}{16}\right)$ $=\frac{261}{4} \div \frac{87}{16}$ $=\frac{261}{4} \times \frac{16}{87}$ $=\frac{261 \times 16}{4 \times 87}$ $=\frac{4176}{348}$ $=12 \mathrm{~m...
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