If the function
Question: If the function $f(x)=\frac{\sin 10 x}{x}, x \neq 0$ is continuous at $x=0$, find $f(0)$. Solution: Given: $f(x)=\frac{\sin 10 x}{x}, x \neq 0$ is continuous at $x=0$. $\lim _{x \rightarrow 0} f(x)=f(0)$ $\Rightarrow \lim _{x \rightarrow 0} \frac{\sin 10 x}{x}=f(0)$ $\Rightarrow \lim _{x \rightarrow 0} \frac{10 \sin 10 x}{10 x}=f(0)$ $\Rightarrow 10 \lim _{x \rightarrow 0} \frac{\sin 10 x}{10 x}=f(0)$ $\Rightarrow f(0)=10$...
Read More →By what number should
Question: By what number should $\frac{-33}{8}$ be divided to get $\frac{-11}{2} ?$ Solution: Let the number be $\mathrm{x}$. Now, $\frac{-33}{8} \div \mathrm{x}=\frac{-11}{2}$ $\Rightarrow \frac{-33}{8} \times \frac{1}{\mathrm{x}}=\frac{-11}{2}$ $\Rightarrow \frac{1}{\mathrm{x}}=\frac{-11}{2} \div \frac{-33}{8}$ $\Rightarrow \frac{1}{\mathrm{x}}=\frac{-11}{2} \times \frac{8}{-33}$ $\Rightarrow \frac{1}{\mathrm{x}}=\frac{88}{66}$ $\Rightarrow \frac{1}{\mathrm{x}}=\frac{4}{3}$ $\Rightarrow \mathr...
Read More →Solve this
Question: If $f(x)=\left\{\begin{array}{cl}\frac{x}{\sin 3 x}, x \neq 0 \\ k , x=0\end{array}\right.$ is continuous at $x=0$, then write the value of $k$. Solution: If $f(x)$ is continuous at $x=0$, then $\lim _{x \rightarrow 0} f(x)=f(0)$ $\Rightarrow \lim _{x \rightarrow 0} \frac{x}{\sin 3 x}=k$ $\Rightarrow \lim _{x \rightarrow 0} \frac{1}{\frac{\sin 3 x}{x}}=k$ $\Rightarrow \lim _{x \rightarrow 0} \frac{1}{\frac{3 \sin 3 x}{3 x}}=k$ $\Rightarrow \frac{1}{3}\left(\frac{1}{\lim _{x \rightarrow...
Read More →By what rational number should
Question: By what rational number should $\frac{-8}{39}$ be multiplied to obtain $\frac{1}{26} ?$ Solution: Let the number be $\mathrm{x}$. Now, $\mathrm{x} \times \frac{-8}{39}=\frac{1}{26}$ $\Rightarrow \mathrm{x}=\frac{1}{26} \div \frac{-8}{39}$ $\Rightarrow \mathrm{x}=\frac{1}{26} \times \frac{39}{-8}$ $\Rightarrow \mathrm{x}=\frac{39}{-208}$ $\Rightarrow \mathrm{x}=\frac{39 \times-1}{-208 \times-1}$ $\Rightarrow \mathrm{x}=\frac{-39}{208}$ $\Rightarrow \mathrm{x}=\frac{-39 \div 13}{208 \div...
Read More →By what rational number should we multiply
Question: By what rational number should we multiply $\frac{-15}{56}$ to get $\frac{-5}{7} ?$ Solution: Let the number be $\mathrm{x}$. Now, $\mathrm{x} \times \frac{-15}{56}=\frac{-5}{7}$ $\Rightarrow \mathrm{x}=\frac{-5}{7} \div \frac{-15}{56}$ $\Rightarrow \mathrm{x}=\frac{-5}{7} \times \frac{56}{-15}$ $\Rightarrow \mathrm{x}=\frac{280}{105}$ $\Rightarrow \mathrm{x}=\frac{280 \div 5}{105 \div 5}$ $\Rightarrow \mathrm{x}=\frac{56}{21}$ $\Rightarrow \mathrm{x}=\frac{56 \div 7}{21 \div 7}$ $\Rig...
Read More →The product of two rational numbers is
Question: The product of two rational numbers is $\frac{-16}{9}$. If one of the numbers is $\frac{-4}{3}$, find the other. Solution: Let the number be $\mathrm{x}$. Now, $\mathrm{x} \times \frac{-4}{3}=\frac{-16}{9}$ $\Rightarrow \mathrm{x}=\frac{-16}{9} \div \frac{-4}{3}$ $\Rightarrow \mathrm{x}=\frac{-16}{9} \times \frac{3}{-4}$ $\Rightarrow \mathrm{x}=\frac{-16 \times 3}{9 \times(-4)}$ $\Rightarrow \mathrm{x}=\frac{48}{36}$ $\Rightarrow \mathrm{x}=\frac{4}{3}$...
Read More →Find f (0), so that
Question: Find $f(0)$, so that $f(x)=\frac{x}{1-\sqrt{1-x}}$ becomes continuous at $x=0$. Solution: If $f(x)$ is continuous at $x=0$, then $\lim _{x \rightarrow 0} f(x)=f(0)$ ....(1) Given: $f(x)=\frac{x}{1-\sqrt{1-x}}$ $\Rightarrow f(x)=\frac{x(1+\sqrt{1-x})}{(1-\sqrt{1-x})(1+\sqrt{1-x})}$ $\Rightarrow f(x)=\frac{x(1+\sqrt{1-x})}{1-(1-x)}$ $\Rightarrow f(x)=(1+\sqrt{1-x})$ $\lim _{x \rightarrow 0}(1+\sqrt{1-x})=f(0) \quad$ [From eq. (1)] $\Rightarrow f(0)=2$ So, for $f(0)=2$, the function $f(x)...
Read More →The product of two rational numbers is −9.
Question: The product of two rational numbers is 9. If one of the numbers is 12, find the other. Solution: Let the number be $\mathrm{x}$. Now, $\mathrm{x} \times(-12)=-9$ $\Rightarrow \mathrm{x}=-9 \div(-12)$ $\Rightarrow \mathrm{x}=(-9) \times \frac{1}{-12}$ $\Rightarrow \mathrm{x}=\frac{-9}{-12}$ $\Rightarrow \mathrm{x}=\frac{3}{4}$...
Read More →Ayush starts walking from his house to office.
Question: Ayush starts walking from his house to office. Instead of going to the office directly, he goes to a bank first, from there to his daughters school and then reaches the office. What is the extra distance travelled by Ayush in reaching his office? (Assume that all distance covered are in straight lines). If the house is situated at (2, 4), bank at (5, 8), school at (13,14) and office at (13, 26) and coordinates are in km. Solution: By given condition, we drawn a figure in which every pl...
Read More →What happens to a function
Question: What happens to a function $f(x)$ at $x=a$, if $\lim _{x \rightarrow a} f(x)=f(a)$ ? Solution: If $f(x)$ is a function defined in its domain such that $\lim _{x \rightarrow \mathrm{a}} f(x)=f(a)$, then $f(x)$ becomes continuous at $x=a$....
Read More →Verify whether the given statement is true or false:
Question: Verify whether the given statement is true or false: (i) $\left(\frac{5}{9} \div \frac{1}{3}\right) \div \frac{5}{2}=\frac{5}{9} \div\left(\frac{1}{3} \div \frac{5}{2}\right)$ (ii) $\left\{(-16) \div \frac{6}{5}\right\} \div \frac{-9}{10}=(-16) \div\left\{\frac{6}{5} \div \frac{-9}{10}\right\}$ (iii) $\left(\frac{-3}{5} \div \frac{-12}{35}\right) \div \frac{1}{14}=\frac{-3}{5} \div\left(\frac{-12}{35} \div \frac{1}{4}\right)$ Solution: (i) $\left(\frac{5}{9} \div \frac{1}{3}\right) \di...
Read More →Define continuity of a function at a point.
Question: Define continuity of a function at a point. Solution: Continuity at a point: A function $f(x)$ is said to be continuous at a point $x=a$ of its domain, iff $\lim _{x \rightarrow a} f(x)=f(a)$. Thus, $f(x)$ is continuous at $x=a$. $\Leftrightarrow \lim _{x \rightarrow a} f(x)=f(a) \Leftrightarrow \lim _{x \rightarrow a^{-}} f(x)=\lim _{x \rightarrow a^{+}} f(x)=f(a)$...
Read More →The set of points of discontinuity
Question: The set of points of discontinuity of $f(x)=\frac{1}{x-[x]}$ is______________ Solution: The function $f(x)=\frac{1}{x-[x]}$ is discontinuous when $x-[x]=0$. $x-[x]=0$ $\Rightarrow x=[x]$ $\Rightarrow x$ is an integer So, the functionf(x) is discontinuous for allxZi.e. the set of integers. Thus, the set of points of discontinuity of $f(x)=\frac{1}{x-[x]}$ is the set of integers i.e. $\mathbf{Z}$. The set of points of discontinuity of $f(x)=\frac{1}{x-[x]}$ is the set of integers i.e. Z...
Read More →Students of a school are standing in rows
Question: Students of a school are standing in rows and columns in their playground for a drill practice. A, B, C and D are the positions of four students as shown in figure. Is it possible to place Jaspal in the drill in such a way that he is equidistant from each of the four students A, B, C and D? If so, what should be his position? Solution: Yes, from the figure we observe that the positions of four students A, B, C and D are (3, 5), (7, 9), (11, 5) and (7,1) respectively i.e., these are fou...
Read More →If A = {1, 3, 5) B = {3, 4} and C = {2, 3}, verify that:
Question: If A = {1, 3, 5) B = {3, 4} and C = {2, 3}, verify that: (i) $A \times(B \cup C)=(A \times B) \cup(A \times C)$ (ii) $A \times(B \cap C)=(A \times B) \cap(A \times C)$ Solution: (i) Given: A = {1, 3, 5}, B = {3, 4} and C = {2, 3} L. H. S $=A \times(B \cup C)$ By the definition of the union of two sets, $(B \cup C)=\{2,3,4\}$ $=\{1,3,5\} \times\{2,3,4\}$ Now, by the definition of the Cartesian product, Given two non empty sets P and Q. The Cartesian product P Q is the set of all ordered...
Read More →The set of points of discontinuity
Question: The set of points of discontinuity off(x) = [x] is ___________. Solution: The graph off(x) = [x] is shown below. It can be seen that, the functionf(x) = [x] is discontinuous at all integral values ofxi.e.xZ.Thus, the set of points of discontinuity off(x) = [x] is the set of integers i.e.Z.The set of points of discontinuity off(x) = [x] is__the set of integers i.e.Z___....
Read More →Verify whether the given statement is true or false:
Question: Verify whether the given statement is true or false: (i) $\frac{13}{5} \div \frac{26}{10}=\frac{26}{10} \div \frac{13}{5}$ (ii) $-9 \div \frac{3}{4}=\frac{3}{4} \div(-9)$ (iii) $\frac{-8}{9} \div \frac{-4}{3}=\frac{-4}{3} \div \frac{-8}{9}$ (iv) $\frac{-7}{24} \div \frac{3}{-16}=\frac{3}{-16} \div \frac{-7}{24}$ Solution: (i) $\frac{13}{5} \div \frac{26}{10}=\frac{26}{10} \div \frac{13}{5}$ LHS $\frac{13}{5} \div \frac{26}{10}$ $=\frac{13}{5} \times \frac{10}{26}$ $=\frac{130}{130}$ $=...
Read More →The set of points of discontinuity
Question: The set of points of discontinuity off(x) = tanxis ___________. Solution: The given function is $f(x)=\tan x$ $f(x)=\tan x=\frac{\sin x}{\cos x}$ The function $f(x)$ is discontinuous when $\cos x=0$. $\cos x=0$ $\Rightarrow x=(2 n+1) \frac{\pi}{2}, n \in \mathbf{Z}$ Thus, the set of point of discontinuity of $f(x)=\tan x$ is $\left\{(2 n+1) \frac{\pi}{2}: n \in \mathbf{Z}\right\}$. The set of points of discontinuity of $f(x)=\tan x$ is $\left\{(2 n+1) \frac{\pi}{2}: n \in \mathbf{Z}\ri...
Read More →If the points A (1, – 2), B (2, 3), C(a, 2)
Question: If the points A (1, 2), B (2, 3), C(a, 2) and D(- 4, 3) form a parallelogram, then find the value of a and height of the parallelogram taking AB as base. Solution: In parallelogram, we know that, diagonals are bisects each other i.e., mid-point of AC = mid-point of BD $\Rightarrow$$\left(\frac{1+a}{2}, \frac{-2+2}{2}\right)=\left(\frac{2-4}{2}, \frac{3-3}{2}\right)$ $\Rightarrow$ $\frac{1+a}{2}=\frac{2-4}{2}=\frac{-2}{2}=-1$ $\left[\right.$ since, mid-point of a line segment having poi...
Read More →Solve this
Question: If $f(x)=\left\{\begin{array}{cl}\frac{\sin 3 x}{x}, \text { if } x \neq 0 \\ \frac{k}{2}, \text { if } x=0\end{array}\right.$ is continuous at $x=0$, then $k$ is equal to_____________ Solution: The given function $f(x)=\left\{\begin{array}{cl}\frac{\sin 3 x}{x}, \text { if } x \neq 0 \\ \frac{k}{2}, \text { if } x=0\end{array}\right.$ is continuous at $x=0$ $\therefore f(0)=\lim _{x \rightarrow 0} f(x)$ $\Rightarrow \frac{k}{2}=\lim _{x \rightarrow 0} \frac{\sin 3 x}{x}$ $\Rightarrow ...
Read More →If f(x) is continuous at x = a
Question: If $f(x)$ is continuous at $x=a$ and $\lim _{x \rightarrow a^{-}} f(x)=\lim _{x \rightarrow a^{+}} f(x)=k$, then $k$ is equal to____________ Solution: It is given that,f(x) is continuous atx=a. $\therefore f(a)=\lim _{x \rightarrow a^{-}} f(x)=\lim _{x \rightarrow a^{+}} f(x)$ ....(1) Also, $\lim _{x \rightarrow a^{-}} f(x)=\lim _{x \rightarrow a^{+}} f(x)=k$ .....(2) From (1) and (2), we have $f(a)=k$ Thus, the value ofkisf(a). If $f(x)$ is continuous at $x=a$ and $\lim _{x \rightarro...
Read More →If A = {x ϵ N : x ≤ 3} and {x ϵ W : x < 2}, find (A × B) and (B × A).
Question: If A = {x ϵ N : x 3} and {x ϵ W : x 2}, find (A B) and (B A). Is (A B) = (B A)? Solution: Given: $A=\{x \in N: x \leq 3\}$ Here, N denotes the set of natural numbers. $\therefore \mathrm{A}=\{1,2,3\}$ $[\because$ It is given that the value of $x$ is less than 3 and natural numbers which are less than 3 are 1 and 2] and $B=\{x \in W: x2\}$ Here, W denotes the set of whole numbers (non negative integers). $\therefore B=\{0,1\}$ $[\because$ It is given that $x2$ and the whole numbers whic...
Read More →Solve this
Question: If $f(x)=\left\{\begin{array}{cl}a x+1, \text { if } x \geq 1 \\ x+2, \text { if } x1\end{array}\right.$ is continuous, then 'a' should be equal to____________ Solution: It is given that, the function $f(x)=\left\{\begin{array}{cl}a x+1, \text { if } x \geq 1 \\ x+2, \text { if } x1\end{array}\right.$ is continuous. So, the functionf(x) is continuous atx= 1. $\therefore f(1)=\lim _{x \rightarrow 1} f(x)$ $\Rightarrow f(1)=\lim _{x \rightarrow 1^{-}} f(x)=\lim _{x \rightarrow 1^{+}} f(x...
Read More →Simplify:
Question: Simplify: (i) $\frac{4}{9} \div \frac{-5}{12}$ (ii) $-8 \div \frac{-7}{16}$ (iii) $\frac{-12}{7} \div(-18)$ (iv) $\frac{-1}{10} \div \frac{-8}{5}$ (v) $\frac{-16}{35} \div \frac{-15}{14}$ (vi) $\frac{-65}{14} \div \frac{13}{7}$ Solution: (i) $\frac{4}{9} \div \frac{-5}{12}$ $=\frac{4}{9} \times \frac{12}{-5}$ $=\frac{4 \times 12}{9 \times-5}$ $=\frac{48}{-45}$ $=\frac{-48}{45}$ $=\frac{-16}{15}$ (ii) $-8 \div \frac{-7}{16}$ $=-8 \times \frac{16}{-7}$ $=\frac{8 \times 16}{7}$ $=\frac{12...
Read More →The set of points at which the function
Question: The set of points at which the function $f(x)=\frac{1}{\log |x|}$ is not continuous, is_________ Solution: The given function $f(x)=\frac{1}{\log |x|}$ is discontinuous when $\log |x|=0$. Also, $x \neq 0 \quad$ (log 0 is not defined) Now, $\log |x|=0$ $\Rightarrow|x|=1$ $\Rightarrow x=\pm 1$ Thus, the given function is not continuous atx= 0,x=1 andx= 1. Hence, the set of points at which the given function is not continuous is {1, 0, 1}. The set of points at which the function $f(x)=\fr...
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