If the function f(x) satisfies
Question: If the function $1(x)$ satisfies $\lim _{x \rightarrow 1} \frac{f(x)-2}{x^{2}-1}=\pi$, evaluate $\lim _{x \rightarrow 1} f(x)$. Solution: $\lim _{x \rightarrow 1} \frac{f(x)-2}{x^{2}-1}=\pi$ $\Rightarrow \frac{\lim _{x \rightarrow 1}(f(x)-2)}{\lim _{x \rightarrow 1}\left(x^{2}-1\right)}=\pi$ $\Rightarrow \lim _{x \rightarrow 1}(f(x)-2)=\pi \lim _{x \rightarrow 1}\left(x^{2}-1\right)$ $\Rightarrow \lim _{x \rightarrow 1}(f(x)-2)=\pi\left(1^{2}-1\right)$ $\Rightarrow \lim _{x \rightarrow...
Read More →If
Question: If $f(x)= \begin{cases}|x|+1, x0 \\ 0, x=0 \\ |x|-1, x0\end{cases}$ For what value (s) of a does $\lim _{x \rightarrow a} f(x)$ exists? Solution: The given function is $f(x)= \begin{cases}|x|+1, x0 \\ 0, x=0 \\ |x|-1, x0\end{cases}$ When $a=0$, $\lim _{x \rightarrow 0^{-}} f(x)=\lim _{x \rightarrow 0^{-}}(|x|+1)$ $=\lim _{x \rightarrow 0}(-x+1) \quad[$ If $x0,|x|=-x]$ $=-0+1$ $=1$ $\lim _{x \rightarrow 0^{+}} f(x)=\lim _{x \rightarrow 0^{+}}(|x|-1)$ $=\lim _{x \rightarrow 0}(x-1) \quad...
Read More →An irrational number between 2 and 2.5 is
Question: An irrational number between 2 and 2.5 is (a) $\sqrt{11}$ (b) $\sqrt{5}$ (c) $\sqrt{22.5}$ (d) $\sqrt{12.5}$ Solution: Let $a=2$ $b=2.5$ Here $a$ and $b$ are rational numbers. So we observe that in first decimal place $a$ and $b$ have distinct. According to question $ab$.so an irrational number between 2 and $2.5$ is $2.236067978$ OR $\sqrt{5}$ Hence the correct answer is....
Read More →The radius of a circle is increasing at the rate of 0.7 cm/s.
Question: The radius of a circle is increasing at the rate of $0.7 \mathrm{~cm} / \mathrm{s}$. What is the rate of increase of its circumference? Solution: The circumferenceof a circle (C) with radius (r)is given by $C=2 \pi r .$ Therefore, the rateof change of circumference (C) with respect to time (t)is given by, $\frac{d C}{d t}=\frac{d C}{d r} \cdot \frac{d r}{d t}$ (By chain rule) $=\frac{d}{d r}(2 \pi r) \frac{d r}{d t}$ $=2 \pi \cdot \frac{d r}{d t}$ It is given that $\frac{d r}{d t}=0.7 ...
Read More →A stone is dropped into a quiet lake and waves move in circles at the speed of 5 cm/s.
Question: A stone is dropped into a quiet lake and waves move in circles at the speed of 5 cm/s. At the instant when the radius of the circular wave is 8 cm, how fast is the enclosed area increasing? Solution: The area of a circle $(A)$ with radius $(r)$ is given by $A=\pi r^{2}$. Therefore, the rate of change of area (A)with respect to time (t)is given by, $\frac{d A}{d t}=\frac{d}{d t}\left(\pi r^{2}\right)=\frac{d}{d r}\left(\pi r^{2}\right) \frac{d r}{d t}=2 \pi r \frac{d r}{d t}$ [By chain ...
Read More →Letbe fixed real numbers and define a function
Question: Let $a_{1}, a_{2}, \ldots ., a_{n}$ be fixed real numbers and define a function $f(x)=\left(x-a_{1}\right)\left(x-a_{2}\right) \ldots\left(x-a_{n}\right)$ What is $\lim _{x \rightarrow a} f(x)$ ? For some $a \neq a_{1}, a_{2} \ldots, a_{n,}$ compute $\lim _{x \rightarrow a} f(x)$. Solution: The given function is $f(x)=\left(x-a_{1}\right)\left(x-a_{2}\right) \ldots\left(x-a_{n}\right)$ $\lim _{x \rightarrow a_{1}} f(x)=\lim _{x \rightarrow a_{1}}\left[\left(x-a_{1}\right)\left(x-a_{2}\...
Read More →The value of 0.23¯¯¯¯ + 0.22¯¯¯¯ is
Question: The value of $0.23+0.22$ is (a) $0 . \overline{45}$ (b) $0 . \overline{43}$ (c) $0 . \overline{45}$ (d) $0.45$ Solution: Given that $0 . \overline{23}+0 . \overline{22}$ Let $x=0 . \overline{23}+0 . \overline{22}$ Now we have to find the value of $x$ $\Rightarrow x=0 . \overline{23}+0 . \overline{22}$ $\Rightarrow x=\frac{23}{99}+\frac{22}{99}$ $\Rightarrow x=\frac{23+22}{99}$ $\Rightarrow x=\frac{45}{99}$ $\Rightarrow x=\frac{5}{11}$ $0 . \overline{23}+\overline{0.22}=0 . \overline{45...
Read More →Suppose f(x) = and iff(x) = f(1) what are possible values of a and b?
Question: Suppose $f(x)=\left\{\begin{array}{ll}a+b x, x1 \\ 4, x=1 \\ b-a x x1\end{array}\right.$ and if $\lim _{x \rightarrow 1} f(x)=f(1)$ what are possible values of $a$ and $b ?$ Solution: The given function is $f(x)= \begin{cases}a+b x, x1 \\ 4, x=1 \\ b-a x x1\end{cases}$ $\lim _{x \rightarrow 1^{-}} f(x)=\lim _{x \rightarrow 1}(a+b x)=a+b$ $\lim _{x \rightarrow 1^{+}} f(x)=\lim _{x \rightarrow 1}(b-a x)=b-a$ $f(1)=4$ It is given that $\lim _{x \rightarrow 1} f(x)=f(1)$. $\therefore \lim ...
Read More →0.001¯¯¯¯¯ when expressed in the form
Question: 0. $\overline{001}$ when expressed in the form $\frac{p}{q}$ ( $p, q$ are integers, $q \neq 0$ ), is (a) $\frac{1}{1000}$ (b) $\frac{1}{100}$ (C) $\frac{1}{1999}$ (d) $\frac{1}{999}$ Solution: Given that $0 . \overline{001}$ Now we have to express this number into $\frac{p}{q}$ form Let $x=0 . \overline{001}$ $=0+\frac{1}{999}$ $=\frac{1}{999}$ The correct option is...
Read More →An edge of a variable cube is increasing at the rate of 3 cm/s.
Question: An edge of a variable cube is increasing at the rate of 3 cm/s. How fast is the volume of the cube increasing when the edge is 10 cm long? Solution: Letxbe the length of a side andVbe the volume of the cube. Then, $V=x^{3}$ $\therefore \frac{d V}{d t}=3 x^{2} \cdot \frac{d x}{d t}$ (By chain rule) It is given that, $\frac{d x}{d t}=3 \mathrm{~cm} / \mathrm{s}$ $\therefore \frac{d V}{d t}=3 x^{2}(3)=9 x^{2}$ Thus, whenx= 10 cm, $\frac{d V}{d t}=9(10)^{2}=900 \mathrm{~cm}^{3} / \mathrm{s...
Read More →Findf(x), where f(x) =
Question: Find $\lim _{x \rightarrow 5} 1(x)$, where $t(x)=|x|-5$ Solution: The given function is $f(x)=|x|-5$. $\lim _{x \rightarrow 5} f(x)=\lim _{x \rightarrow 5^{-}}[|x|-5]$ $=\lim _{x \rightarrow 5}(x-5) \quad[$ When $x0,|x|=x]$ $=5-5$ $=0$ $\lim _{x \rightarrow 5^{+}} f(x)=\lim _{x \rightarrow 5^{+}}(|x|-5)$ $=\lim _{x \rightarrow 5}(x-5) \quad[$ When $x0,|x|=x]$ $=5-5$ $=0$ $\therefore \lim _{x \rightarrow 5^{-}} f(x)=\lim _{x \rightarrow 5^{+}} f(x)=0$ Hence, $\lim _{x \rightarrow 5} f(x...
Read More →23.43¯¯¯¯ when expressed in the form
Question: 23. $\overline{43}$ when expressed in the form $\frac{p}{q}(p, q$ are integers $q \neq 0)$, is (a) $\frac{2320}{99}$ (b) $\frac{2343}{100}$ (c) $\frac{2343}{999}$ (d) $\frac{2320}{199}$ Solution: Given that $23 . \overline{43}$ Now we have to express this number into the form of $\frac{p}{q}$ Let $x=23.43$ $x=23+0.4343 \ldots$ $x=23+\frac{43}{99}$ $x=\frac{2277+43}{99}=\frac{2320}{99}$ $\Rightarrow 23 . \overline{43}=\frac{2320}{99}$ The correct option is...
Read More →The radius of a circle is increasing uniformly at the rate of 3 cm/s.
Question: The radius of a circle is increasing uniformly at the rate of 3 cm/s. Find the rate at which the area of the circle is increasing when the radius is 10 cm. Solution: The area ofa circle (A)with radius(r) is given by, $A=\pi r^{2}$ Now, the rate of change of area (A) with respect to time (t)is given by, $\frac{d A}{d t}=\frac{d}{d t}\left(\pi r^{2}\right) \cdot \frac{d r}{d t}=2 \pi r \frac{d r}{d t}$ [By chain rule] It is given that, $\frac{d r}{d t}=3 \mathrm{~cm} / \mathrm{s}$ $\ther...
Read More →Findf(x), where f(x) =
Question: Find $\lim _{x \rightarrow 0} f(x)$, where $f(x)= \begin{cases}\frac{x}{|x|}, x \neq 0 \\ 0, x=0\end{cases}$ Solution: The given function is $f(x)= \begin{cases}\frac{x}{|x|}, x \neq 0 \\ 0, x=0\end{cases}$ $\lim _{x \rightarrow 0^{-}} f(x)=\lim _{x \rightarrow 0^{-}}\left[\frac{x}{|x|}\right]$ $=\lim _{x \rightarrow 0}\left[\frac{x}{-x}\right] \quad[$ When $x0,|x|=-x]$ $=\lim _{x \rightarrow 0}(-1)$ $=-1$ $\lim _{x \rightarrow 0^{+}} f(x)=\lim _{x \rightarrow 0^{+}}\left[\frac{x}{|x|}...
Read More →0.32¯ when expressed in the form
Question: 0. 32 when expressed in the form $\frac{p}{q}(p, q$ are integers $q \neq 0)$, is (a) $\frac{8}{25}$ (b) $\frac{29}{90}$ (C) $\frac{32}{99}$ (d) $\frac{32}{199}$ Solution: Given that $0.3 \overline{2}$ Now we have to express this number into $\frac{p}{q}$ form Let $X=0.3 \overline{2}$ $10 x=3+0.2222$ $\Rightarrow 10 x=3+\frac{2}{9}$ $\Rightarrow 10 x=\frac{29}{9}$ $\Rightarrow x=\frac{29}{90}$ The correct option is...
Read More →Evaluate $lim _{x ightarrow 0} f(x)$, where $f(x)=$
Question: Evaluate $\lim _{x \rightarrow 0} f(x)$, where $f(x)= \begin{cases}\frac{|x|}{x}, x \neq 0 \\ 0, x=0\end{cases}$ Solution: The given function is $f(x)= \begin{cases}\frac{|x|}{x}, x \neq 0 \\ 0, x=0\end{cases}$ $\lim _{x \rightarrow 0^{-}} f(x)=\lim _{x \rightarrow 0^{-}}\left[\frac{|x|}{x}\right]$ $=\lim _{x \rightarrow 0}\left(\frac{-x}{x}\right) \quad[$ When $x$ is negaitve, $|x|=-x]$ $=\lim _{x \rightarrow 0}(-1)$ $=-1$ $\lim _{x \rightarrow 0^{+}} f(x)=\lim _{x \rightarrow 0^{+}}\...
Read More →The volume of a cube is increasing at the rate of 8 cm3/s.
Question: The volume of a cube is increasing at the rate of 8 cm3/s. How fast is the surface area increasing when the length of an edge is 12 cm? Solution: Letxbe the length of a side,Vbe the volume, andsbe the surface area of the cube. Then, $V=x^{3}$ and $S=6 x^{2}$ where $x$ is a function of time $t$. It is given that $\frac{d V}{d t}=8 \mathrm{~cm}^{3} / \mathrm{s}$. Then, by using the chain rule, we have: $\therefore 8=\frac{d V}{d t}=\frac{d}{d t}\left(x^{3}\right)=\frac{d}{d x}\left(x^{3}...
Read More →The number 0.3¯ in the form
Question: The number $0 .$ a in the form $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$, is (a) $\frac{33}{100}$ (b) $\frac{3}{10}$ (C) $\frac{1}{3}$ (d) $\frac{3}{100}$ Solution: Given number is $0 . \overline{3}$ $0 . \overline{3}=0+\frac{3}{9}$ $=\frac{3}{9}$ $=\frac{1}{3}$ The correct option is...
Read More →Find the rate of change of the area of a circle with respect to its radius r when
Question: Find the rate of change of the area of a circle with respect to its radiusrwhen (a) $r=3 \mathrm{~cm}$ (b) $r=4 \mathrm{~cm}$ Solution: The area of a circle (A)with radius (r) is given by, $A=\pi r^{2}$ Now, the rate of change of the area with respect to its radius is given by, $\frac{d A}{d r}=\frac{d}{d r}\left(\pi r^{2}\right)=2 \pi r$ 1. When $r=3 \mathrm{~cm}$, $\frac{d A}{d r}=2 \pi(3)=6 \pi$ Hence, the area of the circle is changing at the rate of 6 cm when its radius is 3 cm. 2...
Read More →Which of the following is irrational?
Question: Which of the following is irrational? (a) $0.15$ (b) $0.01516$ (c) $0 . \overline{1516}$ (d) $0.5015001500015$. Solution: Given decimal numbers are $0.15,0.1516,0 . \overline{1516}$ and $0.501500115000115 \ldots$ Here the number $0.501500115000115 \ldots .$ is non terminating or non-repeating. Hence the correct option is $d$....
Read More →Every point on a number line represents
Question: Every point on a number line represents(a) a unique real number(b) a natural number(c) a rational number(d) an irrational number Solution: In basic mathematics, number line is a picture of straight line on which every point is assumed to correspond to real number. Hence the correct option is....
Read More →Find $lim _{x ightarrow 1} f(x)$, where $f(x)=
Question: Find $\lim _{x \rightarrow 1} f(x)$, where $f(x)= \begin{cases}x^{2}-1, x \leq 1 \\ -x^{2}-1, x1\end{cases}$ Solution: The given function is $f(x)=\left\{\begin{array}{l}x^{2}-1, x \leq 1 \\ -x^{2}-1, x1\end{array}\right.$ $\lim _{1} f(x)=\lim \left[x^{2}-1\right]=1^{2}-1=1-1=0$ $\lim _{x \rightarrow 1^{\prime}} f(x)=\lim _{x \rightarrow 1}\left[-x^{2}-1\right]=-1^{2}-1=-1-1=-2$ It is observed that $\lim _{x \rightarrow 1^{-}} f(x) \neq \lim _{x \rightarrow 1^{+}} f(x)$. Hence, $\lim _...
Read More →Which of the following numbers can be represented as non-terminating, repeating decimals?
Question: Which of the following numbers can be represented as non-terminating, repeating decimals? (a) $\frac{39}{24}$ (b) $\frac{3}{16}$ (C) $\frac{3}{11}$ (d) $\frac{137}{25}$ Solution: Given that $\frac{39}{24}=1.625$ $\frac{3}{16}=0.1875$ $\frac{3}{11}=0.272727272$ $\frac{1.37}{25}=0.0548$ Here $\frac{3}{11}$ is repeating but non-terminating. Hence the correct option is....
Read More →Find f(x) andf(x), where f(x) =
Question: Find $\lim _{x \rightarrow 0} f(x)$ and $\lim _{x \rightarrow 1} f(x)$, where $f(x)= \begin{cases}2 x+3, x \leq 0 \\ 3(x+1), x0\end{cases}$ Solution: The given function is $f(x)= \begin{cases}2 x+3, x \leq 0 \\ 3(x+1), x0\end{cases}$ $\lim _{x \rightarrow 0^{-}} f(x)=\lim _{x \rightarrow 0}[2 x+3]=2(0)+3=3$ $\lim _{x \rightarrow 0^{+}} f(x)=\lim _{x \rightarrow 0} 3(x+1)=3(0+1)=3$ $\therefore \lim _{x \rightarrow 0^{-}} f(x)=\lim _{x \rightarrow 0^{+}} f(x)=\lim _{x \rightarrow 0} f(x)...
Read More →If n is a natural number, then
Question: If $n$ is a natural number, then $\sqrt{n}$ is (a) always a natural number(b) always an irrational number(c) always an irrational number(d) sometimes a natural number and sometimes an irrational number Solution: The term "natural number" refers either to a member of the set of positive integer $1,2,3$. And natural number starts from one of counting digit. Thus, if $n$ is a natural number then sometimes $n$ is a perfect square and sometimes it is not. Therefore, sometimes $\sqrt{n}$ is ...
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