The number 0.318564318564318564 ........ is:

Question: The number 0.318564318564318564 ........ is:(a) a natural number(b) an integer(c) a rational number(d) an irrational number Solution: Since the given number $0.318564318564318564 \ldots=0 . \overline{318564}$ is repeating, so it is rational number because rational number is always either terminating or repeating Hence the correct option is....

if

Question: If $y=e^{a \cos ^{-1} x},-1 \leq x \leq 1$, show that $\left(1-x^{2}\right) \frac{d^{2} y}{d x^{2}}-x \frac{d y}{d x}-a^{2} y=0$ Solution: It is given that, $y=e^{a \cos ^{-1} x}$ Taking logarithm on both the sides, we obtain $\log y=a \cos ^{-1} x \log e$ $\log y=a \cos ^{-1} x$ Differentiating both sides with respect to $x$, we obtain $\frac{1}{y} \frac{d y}{d x}=a \times \frac{-1}{\sqrt{1-x^{2}}}$ $\Rightarrow \frac{d y}{d x}=\frac{-a y}{\sqrt{1-x^{2}}}$ By squaring both the sides, ...

$lim _{x ightarrow rac{pi}{2}} rac{ an 2 x}{x-rac{pi}{2}}$

Question: $\lim _{x \rightarrow \frac{\pi}{2}} \frac{\tan 2 x}{x-\frac{\pi}{2}}$ Solution: $\lim _{x \rightarrow \frac{\pi}{2}} \frac{\tan 2 x}{x-\frac{\pi}{2}}$ At $x=\frac{\pi}{2}$, the value of the given function takes the form $\frac{0}{0}$ Now, put $x-\frac{\pi}{2}=y$ so that $x \rightarrow \frac{\pi}{2}, y \rightarrow 0$ $\therefore \lim _{x \rightarrow \frac{\pi}{2}} \frac{\tan 2 x}{x-\frac{\pi}{2}}=\lim _{y \rightarrow 0} \frac{\tan 2\left(y+\frac{\pi}{2}\right)}{y}$ $=\lim _{y \rightarr...

Which of the following is rational?

Question: Which of the following is rational? (a) $\sqrt{3}$ (b) $\pi$ (C) $\frac{4}{0}$ (d) $\frac{0}{4}$ Solution: Given that $\sqrt{3}, \pi, \frac{4}{0}$ and $\frac{0}{4}$ Here, $\frac{0}{4}=0$, this is the form of $\frac{p}{q}$. So this is a rational number Hence the correct option is....

Question: If $y=e^{a \cos ^{-1} x},-1 \leq x \leq 1$, show that $\left(1-x^{2}\right) \frac{d^{2} y}{d x^{2}}-x \frac{d y}{d x}-a^{2} y=0$ Solution: It is given that, $y=e^{a \cos ^{-1} x}$ Taking logarithm on both the sides, we obtain $\log y=a \cos ^{-1} x \log e$ $\log y=a \cos ^{-1} x$ Differentiating both sides with respect to $x$, we obtain $\frac{1}{y} \frac{d y}{d x}=a \times \frac{-1}{\sqrt{1-x^{2}}}$ $\Rightarrow \frac{d y}{d x}=\frac{-a y}{\sqrt{1-x^{2}}}$ By squaring both the sides, ...

Evaluate the Given limit:

Question: Evaluate the Given limit: $\lim _{x \rightarrow 0}(\operatorname{cosec} x-\cot x)$ Solution: At $x=0$, the value of the given function takes the form $\infty-\infty$. Now, $\lim _{x \rightarrow 0}(\operatorname{cosec} x-\cot x)$ $=\lim _{x \rightarrow 0}\left(\frac{1}{\sin x}-\frac{\cos x}{\sin x}\right)$ $=\lim _{x \rightarrow 0}\left(\frac{1-\cos x}{\sin x}\right)$ $=\lim _{x \rightarrow 0} \frac{\left(\frac{1-\cos x}{x}\right)}{\left(\frac{\sin x}{x}\right)}$ $=\frac{\lim _{x \right...

Which of the following is irrational?

Question: Which of the following is irrational? (i) $0.14$ (ii) $0.14 \overline{16}$ (iii) $0 . \overline{1416}$ (iv) $0.1014001400014 \ldots$ Solution: Given that $0.14$ $0.141 \overline{6}$ $0 . \overline{416}$ $0.1014001400014 \ldots$ Here $0.1014001400014 \ldots .$ is non-terminating or non-repeating. So it is an irrational number. Hence the correct option is $d$....

Evaluate the Given limit:

Question: Evaluate the Given $\operatorname{limit} \lim _{x \rightarrow 0} \frac{\sin a x+b x}{a x+\sin b x} a, b, a+b \neq 0$ Solution: At $x=0$, the value of the given function takes the form $\frac{0}{0}$. Now, $\lim _{x \rightarrow 0} \frac{\sin a x+b x}{a x+\sin b x}$ $=\lim _{x \rightarrow 0} \frac{\left(\frac{\sin a x}{a x}\right) a x+b x}{a x+b x\left(\frac{\sin b x}{b x}\right)}$ $=\frac{\left(\lim _{a x \rightarrow 0} \frac{\sin a x}{a x}\right) \times \lim _{x \rightarrow 0}(a x)+\lim...

Which of the following is irrational?

Question: Which of the following is irrational? (a) $\sqrt{\frac{4}{9}}$ (b) $\sqrt{\frac{4}{5}}$ (c) $\sqrt{7}$ (d) $\sqrt{81}$ Solution: Given that $\sqrt{\frac{4}{9}}=\frac{2}{3}$ $\frac{4}{5}=0.8$ $\sqrt{81}=9$ And 7 is not a perfect square. Hence the correct option is....

Which of the following statements is true?

Question: Which of the following statements is true?(a) Product of two irrational numbers is always irrational(b) Product of a rational and an irrational number is always irrational(c) Sum of two irrational numbers can never be irrational(d) Sum of an integer and a rational number can never be an integer Solution: Since we know that the product of rational and irrational number is always an irrational. For example: Let $\frac{1}{2}, \sqrt{3}$ are rational and irrational numbers respectively and ...

Evaluate the Given limit:

Question: Evaluate the Given $\operatorname{limit}_{x \rightarrow 0} x \lim _{x \rightarrow 0} x$ Solution: $\lim _{x \rightarrow 0} x \sec x=\lim _{x \rightarrow 0} \frac{x}{\cos x}=\frac{0}{\cos 0}=\frac{0}{1}=0$...

Which of the following is a correct statement?

Question: Which of the following is a correct statement?(a) Sum of two irrational numbers is always irrational(b) Sum of a rational and irrational number is always an irrational number(c) Square of an irrational number is always a rational number(d) Sum of two rational numbers can never be an integer Solution: The sum of irrational number and rational number is always irrational number. Letabe a rational number andbbe an irrational number. Then, $(a+b)^{2}=a^{2}+b^{2}+2 a b$ $=\left(a^{2}+b^{2}\...

If, prove that

Question: If $y=\left|\begin{array}{ccc}f(x) g(x) h(x) \\ l m n \\ a b c\end{array}\right|$, prove that $\frac{d y}{d x}=\left|\begin{array}{ccc}f^{\prime}(x) g^{\prime}(x) h^{\prime}(x) \\ l m n \\ a b c\end{array}\right|$ Solution: $y=\left|\begin{array}{ccc}f(x) g(x) h(x) \\ l m n \\ a b c\end{array}\right|$ $\Rightarrow y=(m c-n b) f(x)-(l c-n a) g(x)+(l b-m a) h(x)$ Then, $\frac{d y}{d x}=\frac{d}{d x}[(m c-n b) f(x)]-\frac{d}{d x}[(I c-n a) g(x)]+\frac{d}{d x}[(l b-m a) h(x)]$ $=(m c-n b) ...

Evaluate the Given limit:

Question: Evaluate the Given $\operatorname{limit:} \lim _{x \rightarrow 0} \frac{a x+x \cos x}{b \sin x}$ Solution: $\lim _{x \rightarrow 0} \frac{a x+x \cos x}{b \sin x}$ At $x=0$, the value of the given function takes the form $\frac{0}{0}$ Now, $\lim _{x \rightarrow 0} \frac{a x+x \cos x}{b \sin x}=\frac{1}{b} \lim _{x \rightarrow 0} \frac{x(a+\cos x)}{\sin x}$ $=\frac{1}{b} \lim _{x \rightarrow 0}\left(\frac{x}{\sin x}\right) \times \lim _{x \rightarrow 0}(a+\cos x)$ $=\frac{1}{b} \times \f...

Evaluate the Given $operatorname{limit}: lim _{x ightarrow 0} rac{cos 2 x-1}{cos x-1}$

Question: Evaluate the Given $\operatorname{limit}: \lim _{x \rightarrow 0} \frac{\cos 2 x-1}{\cos x-1}$ Solution: $\lim _{x \rightarrow 0} \frac{\cos 2 x-1}{\cos x-1}$ At $x=0$, the value of the given function takes the form $\frac{0}{0}$. Now, $\lim _{x \rightarrow 0} \frac{\cos 2 x-1}{\cos x-1}=\lim _{x \rightarrow 0} \frac{1-2 \sin ^{2} x-1}{1-2 \sin ^{2} \frac{x}{2}-1} \quad\left[\cos x=1-2 \sin ^{2} \frac{x}{2}\right]$ $=\lim _{x \rightarrow 0} \frac{\sin ^{2} x}{\sin ^{2} \frac{x}{2}}=\li...

Which one of the following statements is true?

Question: Which one of the following statements is true?(a) The sum of two irrational numbers is always an irrational number(b) The sum of two irrational numbers is always a rational number(c) The sum of two irrational numbers may be a rational number or an irrational number(d) The sum of two irrational numbers is always an integer Solution: Since, $-\sqrt{2}$ and $\sqrt{2}-1$ are two irrational number and $-\sqrt{2}+(\sqrt{2}+1)=1$ Therefore, sum of two irrational numbers may be rational Now, l...

Does there exist a function which is continuos everywhere but not differentiable at exactly two points? Justify your answer ?

Question: Does there exist a function which is continuos everywhere but not differentiable at exactly two points? Justify your answer ? Solution: $y=\left\{\begin{array}{lc}|x| -\inftyx \leq 1 \\ 2-x 1 \leq x \leq \infty\end{array}\right.$ It can be seen from the above graph that, the given function is continuos everywhere but not differentiable at exactly two points which are 0 and 1....

Mark the correct alternative in each of the following:

Question: Mark the correct alternative in each of the following:1. Which one of the following is a correct statement?(a) Decimal expansion of a rational number is terminating(b) Decimal expansion of a rational number is non-terminating(c) Decimal expansion of an irrational number is terminating(d) Decimal expansion of an irrational number is non-terminating and non-repeating Solution: The decimal expansion of an irrational number is non-terminating and non- repeating. Thus, we can say that a num...

Evaluate the given limit:

Question: Evaluate the given $\operatorname{limit:} \lim _{x \rightarrow 0} \frac{\cos x}{\pi-x}$ Solution: $\lim _{x \rightarrow 0} \frac{\cos x}{\pi-x}=\frac{\cos 0}{\pi-0}=\frac{1}{\pi}$...

Evaluate the Given limit:

Question: Evaluate the Given limit: $\lim _{x \rightarrow \pi} \frac{\sin (\pi-x)}{\pi(\pi-x)}$ Solution: $\lim _{x \rightarrow \pi} \frac{\sin (\pi-x)}{\pi(\pi-x)}$ It is seen that $x \rightarrow \pi \Rightarrow(\pi-x) \rightarrow 0$ $\therefore \lim _{x \rightarrow \pi} \frac{\sin (\pi-x)}{\pi(\pi-x)}=\frac{1}{\pi} \lim _{(\pi-x) \rightarrow 0} \frac{\sin (\pi-x)}{(\pi-x)}$ $=\frac{1}{\pi} \times 1 \quad\left[\lim _{y \rightarrow 0} \frac{\sin y}{y}=1\right]$ $=\frac{1}{\pi}$...

Using the fact that

Question: Using the fact that $\sin (A+B)=\sin A \cos B+\cos A \sin B$ and the differentiation, obtain the sum formula for cosines. Solution: $\sin (A+B)=\sin A \cos B+\cos A \sin B$ Differentiating both sides with respect tox, we obtain $\frac{d}{d x}[\sin (A+B)]=\frac{d}{d x}(\sin A \cos B)+\frac{d}{d x}(\cos A \sin B)$ $\begin{aligned} \Rightarrow \cos (A+B) \cdot \frac{d}{d x}(A+B)= \cos B \cdot \frac{d}{d x}(\sin A)+\sin A \cdot \frac{d}{d x}(\cos B) \\ +\sin B \cdot \frac{d}{d x}(\cos A)+\...

Evaluate the Given limit:

Question: Evaluate the Given limit: $\lim _{x \rightarrow 0} \frac{\sin a x}{\sin b x}, a, b \neq 0$ Solution: $\lim _{x \rightarrow 0} \frac{\sin a x}{\sin b x}, a, b \neq 0$ At $x=0$, the value of the given function takes the form $\frac{0}{0}$ Now, $\lim _{x \rightarrow 0} \frac{\sin a x}{\sin b x}=\lim _{x \rightarrow 0} \frac{\left(\frac{\sin a x}{a x}\right) \times a x}{\left(\frac{\sin b x}{b x}\right) \times b x}$ $=\left(\frac{a}{b}\right) \times \frac{\lim _{a x \rightarrow 0}\left(\fr...

Evaluate the Given limit:

Question: Evaluate the Given limit: $\lim _{x \rightarrow 0} \frac{\sin a x}{b x}$ Solution: $\lim _{x \rightarrow 0} \frac{\sin a x}{b x}$ At $x=0$, the value of the given function takes the form $\frac{0}{0}$. Now, $\lim _{x \rightarrow 0} \frac{\sin a x}{b x}=\lim _{x \rightarrow 0} \frac{\sin a x}{a x} \times \frac{a x}{b x}$ $=\lim _{x \rightarrow 0}\left(\frac{\sin a x}{a x}\right) \times\left(\frac{a}{b}\right)$ $=\frac{a}{b} \lim _{x \rightarrow 0}\left(\frac{\sin a x}{a x}\right)$ $[x \...

Visualise the representation of 5.37¯ on the number line upto 5 decimal places,

Question: Visualise the representation of $5.3 \overline{7}$ on the number line upto 5 decimal places, that is upto $5.37777$. Solution: We know that $5.3 \overline{7}$ will lie between 5 and 6 . So, we locate $5.3 \overline{7}$ between 5 and 6 . We divide this portion of the number line between 5 and 6 into 10 equal parts and use a magnifying glass to visualize $5.3 \overline{7}$. $5.3 \overline{7}$ lies between $5.37$ and $5.38$. To visualize $5.3 \overline{7}$ more accurately we use a magnify...

Using mathematical induction prove that

Question: Using mathematical induction prove that $\frac{d}{d x}\left(x^{n}\right)=n x^{n-1}$ for all positive integers $n$. Solution: To prove: $\mathrm{P}(n): \frac{d}{d x}\left(x^{n}\right)=n x^{n-1}$ for all positive integers $n$ Forn= 1, $\mathrm{P}(1): \frac{d}{d x}(x)=1=1 \cdot x^{1-1}$ $\therefore \mathrm{P}(n)$ is true for $n=1$ Let P(k) is true for some positive integerk. That is, $\mathrm{P}(k): \frac{d}{d x}\left(x^{k}\right)=k x^{k-1}$ It has to be proved thatP(k+ 1) is also true. C...