Visualise 2.665 on the number line,
Question: Visualise 2.665 on the number line, using successive magnification. Solution: We know that 2.665 lies between 2 and 3. So, we divide the number line into 10 equal partsand mark each point of division. The first mark on the right of 2 will be 2.1 followed by 2.2 and so on.The point left of 3 will be 2.9. Now, the magnified view of this will show that 2.665 lies between 2.6 and2.7. So, our focus will be now 2.6 and 2.7. We divide this again into 10 equal parts. The first part will be2.61...
Read More →Evaluate the Given limit:
Question: Evaluate the Given limit: $\lim _{x \rightarrow-2} \frac{\frac{1}{x}+\frac{1}{2}}{x+2}$ Solution: $\lim _{x \rightarrow-2} \frac{\frac{1}{x}+\frac{1}{2}}{x+2}$ At $x=-2$, the value of the given function takes the form $\frac{0}{0}$ Now, $\lim _{x \rightarrow-2} \frac{\frac{1}{x}+\frac{1}{2}}{x+2}=\lim _{x \rightarrow-2} \frac{\left(\frac{2+x}{2 x}\right)}{x+2}$ $=\lim _{x \rightarrow-2} \frac{1}{2 x}$ $=\frac{1}{2(-2)}=\frac{-1}{4}$...
Read More →Evaluate the Given limit:
Question: Evaluate the Given limit: $\lim _{x \rightarrow 1} \frac{a x^{2}+b x+c}{c x^{2}+b x+a}, a+b+c \neq 0$ Solution: $\lim _{x \rightarrow 1} \frac{a x^{2}+b x+c}{c x^{2}+b x+a}=\frac{a(1)^{2}+b(1)+c}{c(1)^{2}+b(1)+a}$ $=\frac{a+b+c}{a+b+c}$ $=1$ $[a+b+c \neq 0]$...
Read More →Represent 3.5−−−√, 9.4−−−√, 10.5−−−−√ on the real number line.
Question: Represent $\sqrt{3.5}, \sqrt{9.4}, \sqrt{10.5}$ on the real number line. Solution: We are asked to represent the real numbers $\sqrt{3.5}, \sqrt{9.4}$ and $\sqrt{10.5}$ on the real number line We will follow a certain algorithm to represent these numbers on real number line (a) $\sqrt{3.5}$ We will takeAas reference point to measure the distance (1) Draw a sufficiently large line and mark a pointAon it (2) Take a pointBon the line such that (3) Mark a pointCon the line such that (4) Fi...
Read More →Evaluate the Given limit:
Question: Evaluate the Given limit: $\lim _{z \rightarrow 1} \frac{z^{\frac{1}{3}}-1}{z^{\frac{1}{6}}-1}$ Solution: $\lim _{z \rightarrow 1} \frac{z^{\frac{1}{3}}-1}{z^{\frac{1}{6}}-1}$ At $z=1$, the value of the given function takes the form $\frac{0}{0}$. Put $z^{\frac{1}{6}}=x$ so that $z \rightarrow 1$ as $x \rightarrow 1$. Accordingly, $\lim _{z \rightarrow 1} \frac{z^{\frac{1}{3}}-1}{z^{\frac{1}{6}}-1}=\lim _{x \rightarrow 1} \frac{x^{2}-1}{x-1}$ $=\lim _{x \rightarrow 1} \frac{x^{2}-1^{2}...
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Question: If $f(x)=|x|^{3}$, show that $f^{\prime \prime}(x)$ exists for all real $x$, and find it. Solution: It is known that, $|x|= \begin{cases}x, \text { if } x \geq 0 \\ -x, \text { if } x0\end{cases}$ Therefore, when $x \geq 0, f(x)=|x|^{3}=x^{3}$ In this case, $f^{\prime}(x)=3 x^{2}$ and hence, $f^{\prime \prime}(x)=6 x$ When $x0, f(x)=|x|^{3}=(-x)^{3}=-x^{3}$ In this case, $f^{\prime}(x)=-3 x^{2}$ and hence, $f^{\prime \prime}(x)=-6 x$ Thus, for $f(x)=|x|^{3}, f^{\prime \prime}(x)$ exist...
Read More →Evaluate the Given limit:
Question: Evaluate the Given limit: $\lim _{x \rightarrow 0} \frac{a x+b}{c x+1}$ Solution: $\lim _{x \rightarrow 0} \frac{a x+b}{c x+1}=\frac{a(0)+b}{c(0)+1}=b$...
Read More →Evaluate the Given limit:
Question: Evaluate the Given limit: $\lim _{x \rightarrow 3} \frac{x^{4}-81}{2 x^{2}-5 x-3}$ Solution: At $x=2$, the value of the given rational function takes the form $\frac{0}{0}$. $\therefore \lim _{x \rightarrow 3} \frac{x^{4}-81}{2 x^{2}-5 x-3}=\lim _{x \rightarrow 3} \frac{(x-3)(x+3)\left(x^{2}+9\right)}{(x-3)(2 x+1)}$ $=\lim _{x \rightarrow 3} \frac{(x+3)\left(x^{2}+9\right)}{2 x+1}$ $=\frac{(3+3)\left(3^{2}+9\right)}{2(3)+1}$ $=\frac{6 \times 18}{7}$ $=\frac{108}{7}$...
Read More →Evaluate the Given limit:
Question: Evaluate the Given limit: $\lim _{x \rightarrow 2} \frac{3 x^{2}-x-10}{x^{2}-4}$ Solution: At $x=2$, the value of the given rational function takes the form $\frac{0}{0}$. $\therefore \lim _{x \rightarrow 2} \frac{3 x^{2}-x-10}{x^{2}-4}=\lim _{x \rightarrow 2} \frac{(x-2)(3 x+5)}{(x-2)(x+2)}$ $=\lim _{x \rightarrow 2} \frac{3 x+5}{x+2}$ $=\frac{3(2)+5}{2+2}$ $=\frac{11}{4}$...
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Question: If $x=a(\cos t+t \sin t)$ and $y=a(\sin t-t \cos t)$, find $\frac{d^{2} y}{d x^{2}}$ Solution: It is given that, $x=a(\cos t+t \sin t)$ and $y=a(\sin t-t \cos t)$ $\therefore \frac{d x}{d t}=a \cdot \frac{d}{d t}(\cos t+t \sin t)$ $=a\left[-\sin t+\sin t \cdot \frac{d}{d x}(t)+t \cdot \frac{d}{d t}(\sin t)\right]$\ $=a[-\sin t+\sin t+t \cos t]=a t \cos t$ $\frac{d y}{d t}=a \cdot \frac{d}{d t}(\sin t-t \cos t)$ $=a\left[\cos t-\left\{\cos t \cdot \frac{d}{d t}(t)+t \cdot \frac{d}{d t}(...
Read More →Evaluate the Given limit:
Question: Evaluate the Given limit: $\lim _{x \rightarrow 0} \frac{(x+1)^{5}-1}{x}$ Solution: $\lim _{x \rightarrow 0} \frac{(x+1)^{5}-1}{x}$ Put $x+1=y$ so that $y \rightarrow 1$ as $x \rightarrow 0$ Accordingly, $\lim _{x \rightarrow 0} \frac{(x+1)^{5}-1}{x}=\lim _{y \rightarrow 1} \frac{y^{5}-1}{y-1}$ $=\lim _{y \rightarrow 1} \frac{y^{5}-1^{5}}{y-1}$ $=5 \cdot 1^{5-1}$ $\left[\lim _{x \rightarrow a} \frac{x^{n}-a^{n}}{x-a}=n a^{n-1}\right]$ $=5$ $\therefore \lim _{x \rightarrow 0} \frac{(x+5...
Read More →Represent
Question: Represent $\sqrt{6}, \sqrt{7}, \sqrt{8}$ on the number line. Solution: We are asked to represent $\sqrt{6}, \sqrt{7}$ and $\sqrt{8}$ on the number line We will follow certain algorithm to represent these numbers on real line We will consider pointAas reference point to measure the distance (1) First of all draw a lineAXandYYperpendicular toAX (2) Consider $A O=2$ unit and $O B=1$ unit, so $A B=\sqrt{2^{2}+1^{2}}$ $=\sqrt{5}$ (3) TakeAas center andABas radius, draw an arc which cuts lin...
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Question: If $\cos y=x \cos (a+y)$, with $\cos a \neq \pm 1$, prove that $\frac{d y}{d x}=\frac{\cos ^{2}(a+y)}{\sin a}$ Solution: It is given that, $\cos y=x \cos (a+y)$ $\therefore \frac{d}{d x}[\cos y]=\frac{d}{d x}[x \cos (a+y)]$ $\Rightarrow-\sin y \frac{d y}{d x}=\cos (a+y) \cdot \frac{d}{d x}(x)+x \cdot \frac{d}{d x}[\cos (a+y)]$ $\Rightarrow-\sin y \frac{d y}{d x}=\cos (a+y)+x \cdot[-\sin (a+y)] \frac{d y}{d x}$ $\Rightarrow[x \sin (a+y)-\sin y] \frac{d y}{d x}=\cos (a+y)$ ...(1) Since $...
Read More →Evaluate the Given limit:
Question: Evaluate the Given limit: $\lim _{x \rightarrow-1} \frac{x^{10}+x^{5}+1}{x-1}$ Solution: $\lim _{x \rightarrow-1} \frac{x^{10}+x^{5}+1}{x-1}=\frac{(-1)^{10}+(-1)^{5}+1}{-1-1}=\frac{1-1+1}{-2}=-\frac{1}{2}$...
Read More →Evaluate the Given limit: $lim _{x ightarrow 4} rac{4 x+3}{x-2}$
Question: Evaluate the Given limit: $\lim _{x \rightarrow 4} \frac{4 x+3}{x-2}$ Solution: $\lim _{x \rightarrow 4} \frac{4 x+3}{x-2}=\frac{4(4)+3}{4-2}=\frac{16+3}{2}=\frac{19}{2}$...
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Question: If $(x-a)^{2}+(y-b)^{2}=c^{2}$, for some $c0$, prove that $\frac{\left[1+\left(\frac{d y}{d x}\right)^{2}\right]^{\frac{3}{2}}}{\frac{d^{2} y}{d x^{2}}}$ is a constant independent of $a$ and $b$. Solution: It is given that, $(x-a)^{2}+(y-b)^{2}=c^{2}$ Differentiating both sides with respect tox, we obtain $\frac{d}{d x}\left[(x-a)^{2}\right]+\frac{d}{d x}\left[(y-b)^{2}\right]=\frac{d}{d x}\left(c^{2}\right)$ $\Rightarrow 2(x-a) \cdot \frac{d}{d x}(x-a)+2(y-b) \cdot \frac{d}{d x}(y-b)=...
Read More →Find whether the following statement are true or false.
Question: Find whether the following statement are true or false.(i) Every real number is either rational or irrational.(ii)is an irrational number.(iii) Irrational numbers cannot be represented by points on the number line. Solution: (i) True, because rational or an irrational number is a family of real number. So every real number is either rational or an irrational number. (ii) True, because the decimal representation of an irrational is always non-terminating or non-repeating. Sois an irrati...
Read More →Complete the following sentences:
Question: Complete the following sentences:(i) Every point on the number line corresponds to a .... number which many be either ... or ...(ii) The decimal form of an irrational number is neither ... nor ...(iii) The decimal representation of a rational number is either ... or ...(iv) Every real number is either ... number or ... number. Solution: (i) Every point on the number line corresponds to arealnumber which may be eitherrationalor anirrationalnumber. (ii) The decimal form of an irrational ...
Read More →Evaluate the Given limit:
Question: Evaluate the Given limit: $\lim _{r \rightarrow 1} \pi r^{2}$ Solution: $\lim _{r \rightarrow 1} \pi r^{2}=\pi(1)^{2}=\pi$...
Read More →Prove that 3–√+5–√ is an irrational number.
Question: Prove that $\sqrt{3}+\sqrt{5}$ is an irrational number. Solution: Given that $\sqrt{3}+\sqrt{5}$ is an irrational number Now we have to prove $\sqrt{3}+\sqrt{5}$ is an irrational number Let $x=\sqrt{3}+\sqrt{5}$ is a rational Squaring on both sides $\Rightarrow x^{2}=(\sqrt{3}+\sqrt{5})^{2}$ $\Rightarrow x^{2}=(\sqrt{3})^{2}+(\sqrt{5})^{2}+2 \sqrt{3} \times \sqrt{5}$ $\Rightarrow x^{2}=3+5+2 \sqrt{15}$ $\Rightarrow x^{2}=8+2 \sqrt{15}$ $\Rightarrow \frac{x^{2}-8}{2}=\sqrt{15}$ Now is r...
Read More →Evaluate the Given limit: $lim _{x ightarrow pi}left(x- rac{22}{7} ight)$
Question: Evaluate the Given limit: $\lim _{x \rightarrow \pi}\left(x-\frac{22}{7}\right)$ Solution: $\lim _{x \rightarrow \pi}\left(x-\frac{22}{7}\right)=\left(\pi-\frac{22}{7}\right)$...
Read More →Evaluate the Given limit: $lim _{x ightarrow 3} x+3$
Question: Evaluate the Given limit: $\lim _{x \rightarrow 3} x+3$ Solution: $\lim _{x \rightarrow 3} x+3=3+3=6$...
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Question: If $x \sqrt{1+y}+y \sqrt{1+x}=0$, for, $-1x1$, prove that $\frac{d y}{d x}=-\frac{1}{(1+x)^{2}}$ Solution: It is given that, $x \sqrt{1+y}+y \sqrt{1+x}=0$ $\Rightarrow x \sqrt{1+y}=-y \sqrt{1+x}$ Squaring both sides, we obtain $x^{2}(1+y)=y^{2}(1+x)$ $\Rightarrow x^{2}+x^{2} y=y^{2}+x y^{2}$ $\Rightarrow x^{2}-y^{2}=x y^{2}-x^{2} y$ $\Rightarrow x^{2}-y^{2}=x y(y-x)$ $\Rightarrow(x+y)(x-y)=x y(y-x)$ $\therefore x+y=-x y$ $\Rightarrow(1+x) y=-x$ $\Rightarrow y=\frac{-x}{(1+x)}$ Differen...
Read More →If A and B be the points (3, 4, 5) and (–1, 3, –7),
Question: If A and B be the points $(3,4,5)$ and $(-1,3,-7)$ respectively, find the equation of the set of points P such that PA $^{2}+P B^{2}=k^{2}$, where $k$ is a constant. Solution: The coordinates of points A and B are given as (3, 4, 5) and (1, 3, 7) respectively. Let the coordinates of point P be (x,y,z). On using distance formula, we obtain $\mathrm{PA}^{2}=(x-3)^{2}+(y-4)^{2}+(z-5)^{2}$ $=x^{2}+9-6 x+y^{2}+16-8 y+z^{2}+25-10 z$ $=x^{2}-6 x+y^{2}-8 y+z^{2}-10 z+50$ $\mathrm{PB}^{2}=(x+1)...
Read More →Find two irrational numbers lying between 0.1 and 0.12.
Question: Find two irrational numbers lying between 0.1 and 0.12. Solution: Let $a=0.1$ $b=0.12$ Hereaandbare rational number. So we observe that in first decimal placeaandbhave same digit. Soab. Hence two irrational numbers are $0.1010010001 \ldots$ and $0.11010010001 \ldots$ lying between $0.1$ and $0.12$....
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