Exercise 10D
For given y, prove the following \[y={{\tan }^{-1}}\{\frac{\sqrt{1+x}-\sqrt{1-x}}{\sqrt{1+x}+\sqrt{1-x}}\},\frac{dy}{dx}=\frac{1}{2\sqrt{1-{{x}^{2}}}}\]
For given y, prove the following \[y=\sin \{2{{\tan }^{-1}}\sqrt{\frac{1-x}{1+x}}\},\frac{dy}{dx}=-\frac{x}{\sqrt{1-{{x}^{2}}}}\]
For given y, prove the following \[y={{\sec }^{-1}}\left( \frac{x+1}{x-1} \right)+{{\sin }^{-1}}\left( \frac{x-1}{x+1} \right),\frac{dy}{dx}=0\]
For given y, prove the following \[y={{\sin }^{-1}}\left( \frac{2x}{1+{{x}^{2}}} \right)+{{\sec }^{-1}}\left( \frac{1+{{x}^{2}}}{1-{{x}^{2}}} \right),\frac{dy}{dx}=\frac{4}{1+{{x}^{2}}}\]
For given y, prove the following \[y={{\tan }^{-1}}\left( \frac{ax-b}{a+bx} \right),\frac{dy}{dx}=\frac{1}{1+{{x}^{2}}}\]
Differentiate the following functions with respect to x: \[{{\tan }^{-1}}\left( \frac{2x}{1+15{{x}^{2}}} \right)\]
Differentiate the following functions with respect to x: \[{{\tan }^{-1}}\left( \frac{5x}{1-6{{x}^{2}}} \right)\]
Differentiate the following functions with respect to x: \[{{\tan }^{-1}}\left( \frac{3-2x}{1+6x} \right)\]
Differentiate the following functions with respect to x: \[{{\tan }^{-1}}\left( \frac{\sqrt{a}+\sqrt{x}}{1-\sqrt{ax}} \right)\]
Differentiate the following functions with respect to x: \[{{\tan }^{-1}}\left( \frac{\sqrt{x}-x}{1+{{x}^{3/2}}} \right)\]
Differentiate the following functions with respect to x: \[{{\cos }^{-1}}(2x)+1{{\cos }^{-1}}(\sqrt{1-4{{x}^{2}}})\]
Differentiate the following functions with respect to x: \[{{\tan }^{-1}}\left( \frac{{{e}^{2x}}+1}{{{e}^{2x}}-1} \right)\]
Differentiate the following functions with respect to x: \[{{\sin }^{-1}}\left( \frac{{{x}^{2}}}{\sqrt{{{a}^{4}}+{{x}^{4}}}} \right)\]
Differentiate the following functions with respect to x: \[{{\tan }^{-1}}\left( \frac{\sqrt{1+{{a}^{2}}{{x}^{2}}}-1}{ax} \right)\]
Differentiate the following functions with respect to x: \[{{\sin }^{-1}}(2ax\sqrt{1-{{a}^{2}}{{x}^{2}}})\]
Differentiate the following functions with respect to x: \[{{\tan }^{-1}}\left( \frac{x}{\sqrt{{{a}^{2}}-{{x}^{2}}}} \right)\]
Differentiate the following functions with respect to x: \[{{\cos }^{-1}}\left( \frac{1-{{x}^{2n}}}{1+{{x}^{2n}}} \right)\]
Differentiate the following functions with respect to x: \[{{\sec }^{-1}}\left( \frac{{{x}^{2}}+1}{{{x}^{2}}-1} \right)\]
Differentiate the following functions with respect to x: \[{{\sin }^{-1}}\left( \frac{1}{\sqrt{1+{{x}^{2}}}} \right)\]
Differentiate the following functions with respect to x: \[{{\sec }^{-1}}\left( \frac{1+{{x}^{2}}}{1-{{x}^{2}}} \right)\]
Differentiate the following functions with respect to x: \[\cos e{{c}^{-1}}\left( \frac{1+{{x}^{2}}}{2x} \right)\]
Differentiate the following functions with respect to x: \[{{\tan }^{-1}}\left( \frac{3x-{{x}^{3}}}{1-3{{x}^{2}}} \right)\]
Differentiate the following functions with respect to x: \[{{\cot }^{-1}}\left( \frac{1+x}{1-x} \right)\]
Differentiate the following functions with respect to x: \[{{\tan }^{-1}}\left( \frac{1+x}{1-x} \right)\]
Differentiate the following functions with respect to x: \[{{\sin }^{-1}}\left( \frac{1}{\sqrt{1+{{x}^{2}}}} \right)\]
Differentiate the following functions with respect to x: \[{{\sec }^{-1}}\left( \frac{1}{1-2{{x}^{2}}} \right)\]
Differentiate the following functions with respect to x: \[{{\cot }^{-1}}\left( \frac{\sqrt{1-{{x}^{2}}}}{x} \right)\]
Differentiate the following functions with respect to x: \[{{\tan }^{-1}}\left( \frac{x}{1+\sqrt{1-{{x}^{2}}}} \right)\]
Differentiate the following functions with respect to x: \[{{\tan }^{-1}}\left( \frac{x}{\sqrt{1-{{x}^{2}}}} \right)\]
Differentiate the following functions with respect to x: \[{{\sec }^{-1}}\left( \frac{1}{\sqrt{1-{{x}^{2}}}} \right)\]
Differentiate the following functions with respect to x: \[{{\sin }^{-1}}(1-2{{x}^{2}})\]
Differentiate the following functions with respect to x: \[{{\sin }^{-1}}(3x-4{{x}^{3}})\]
Differentiate the following functions with respect to x: \[{{\sin }^{-1}}\{2x\sqrt{1-{{x}^{2}}}\}\]
Differentiate the following functions with respect to x: \[{{\cos }^{-1}}\{\sqrt{\frac{1+x}{2}}\}\]
Differentiate the following functions with respect to x: \[{{\sin }^{-1}}\{\sqrt{\frac{1-x}{2}}\}\]
Differentiate the following functions with respect to x: \[{{\sin }^{-1}}\{\sqrt{1-{{x}^{2}}}\}\]
Differentiate the following functions with respect to x: \[{{\tan }^{-1}}(\cot x)+{{\cot }^{-1}}(\tan x)\]
Differentiate the following functions with respect to x: \[{{\cot }^{-1}}(\cos ecx+\cot x)\]
Differentiate the following functions with respect to x: \[\cos e{{c}^{-1}}\left( \frac{1+{{\tan }^{2}}x}{2\tan x} \right)\]
Differentiate the following functions with respect to x: \[{{\sin }^{-1}}\left( \frac{1-{{\tan }^{2}}x}{1+{{\tan }^{2}}x} \right)\]
Differentiate the following functions with respect to x: \[{{\sec }^{-1}}\left( \frac{1+{{\tan }^{2}}x}{1-{{\tan }^{2}}x} \right)\]
Differentiate the following functions with respect to x: \[{{\cot }^{-1}}\left( \frac{\cos x-\sin x}{\cos x+\sin x} \right)\]
If the roots of the equations $a x^{2}+2 b x+c=0$ and $b x^{2}-2 \sqrt{a c x}+b=0$ are simultaneously real then prove that $b^{2}=a c$
It is given that the roots of the equation $a x^{2}+2 b x+c=0$ are real. $\begin{array}{l} \therefore D_{1}=(2 b)^{2}-4 \times a \times c \geq 0 \\ \Rightarrow 4\left(b^{2}-a c\right) \geq 0 \\...
If the roots of the equation $\left(a^{2}+b^{2}\right) x^{2}-2(a c+b d) x+\left(c^{2}+d^{2}\right)=0$ are equal, prove that $\frac{a}{b}=\frac{c}{d}$
It is given that the roots of the equation $\left(a^{2}+b^{2}\right) x^{2}-2(a c+b d) x+\left(c^{2}+d^{2}\right)=0$ are equal. $\begin{array}{l} \therefore D=0 \\ \Rightarrow[-2(a c+b...
If a and $\mathrm{b}$ are real and $a \neq b$ then show that the roots of the equation $(a-b) x^{2}+5(a+b) x-2(a-b)=0 .$ are equal and unequal.
The given equation is $(a-b) x^{2}+5(a+b) x-2(a-b)=0$. $\begin{array}{l} \therefore D=[5(a+b)]^{2}-4 \times(a-b) \times[-2(a-b)] \\ =25(a+b)^{2}+8(a-b)^{2} \end{array}$ Since a and $\mathrm{b}$ are...
Find the values of $k$ for which the given quadratic equation has real and distinct roots:
(i) $9 x^{2}+3 k x+4=0$.
(ii) $5 x^{2}-k x+1=0$.
(i) The given equation is $9 x^{2}+3 k x+4=0$. $\therefore D=(3 k)^{2}-4 \times 9 \times 4=9 k^{2}-144$ The given equation has real and distinct roots if $D>0 .$ $\begin{array}{l} \therefore 9...
Find the values of $k$ for which the given quadratic equation has real and distinct roots:
(i) $k x^{2}+6 x+1=0$.
(ii) $x^{2}-k x+9=0$.
(i) The given equation is $k x^{2}+6 x+1=0$. $\therefore D=6^{2}-4 \times k \times 1=36-4 k$ The given equation has real and distinct roots if $D>0$. $\begin{array}{l} \therefore 36-4 k>0 \\...
Find the value of $k$ for which the roots of $9 x^{2}+8 k x+16=0$ are real and equal
Given: $\begin{array}{l} 9 x^{2}+8 k x+16=0 \\ \text { Here, } \\ a=9, b=8 k \text { and } c=16 \end{array}$ It is given that the roots of the equation are real and equal; therefore, we have:...
Find the value of a for which the equation $(\alpha-12) x^{2}+2(\alpha-12) x+2=0$ has equal roots.
$(\alpha-12) x^{2}+2(\alpha-12) x+2=0$ Here, $a=(\alpha=12), b=2(\alpha-12) \text { and } c=2$ It is given that the roots of the equation are equal; therefore, we have $\begin{array}{l} D=0 \\...
Find the value of $p$ for which the quadratic equation $2 x^{2}+p x+8=0$ has real roots.
$2 x^{2}+p x+8=0$ Here, $a=2, b=p$ and $c=8$ Discriminant $D$ is given by: $\begin{array}{l} D=\left(b^{2}-4 a c\right) \\ =p^{2}-4 \times 2 \times 8 \\ =\left(p^{2}-64\right) \end{array}$ If $D...
If the roots of the quadratic equation $\left(c^{2}-a b\right) x^{2}-2\left(a^{2}-b c\right) x+\left(b^{2}-a c\right)=0$ are real and equal, show that either $\mathrm{a}=0$ or $\left(a^{3}+b^{3}+c^{3}=3 a b c\right)$
$\left(c^{2}-a b\right) x^{2}-2\left(a^{2}-b c\right) x+\left(b^{2}-a c\right)=0$ Here, $a=\left(c^{2}-a b\right), b=-2\left(a^{2}-b c\right), c=\left(b^{2}-a c\right)$ It is given that the roots of...
If the quadratic equation $\left(1+m^{2}\right) x^{2}+2 m c x+\left(c^{2}-a^{2}\right)=0$ has equal roots, prove that $c^{2}=a^{2}\left(1+m^{2}\right)$
$\left(1+m^{2}\right) x^{2}+2 m c x+\left(c^{2}-a^{2}\right)=0$ Here, $a=\left(1+m^{2}\right), b=2 m c \text { and } c=\left(c^{2}-a^{2}\right)$ It is given that the roots of the equation are equal;...
If $-4$ is a root of the equation $x^{2}+2 x+4 p=0$. find the value of $k$ for the which the quadratic equation $x^{2}+p x(1+3 k)+7(3+2 k)=0$ has equal roots.
It is given that $-4$ is a root of the quadratic equation $x^{2}+2 x+4 p=0$ $\begin{array}{l} \therefore(-4)^{2}+2 \times(-4)+4 p=0 \\ \Rightarrow 16-8+4 p=0 \\ \Rightarrow 4 p+8=0 \\ \Rightarrow...
If 3 is a root of the quadratic equation $x^{2}-x+k=0$., find the value of $p$ so that the roots of the equation $x^{2}+2 k x+\left(k^{2}+2 k+p\right)=0$ are equal.
It is given that 3 is a root of the quadratic equation $x^{2}-x+k=0$. $\begin{array}{l} \therefore(3)^{2}-3+k=0 \\ \Rightarrow k+6=0 \\ \Rightarrow k=-6 \end{array}$ The roots of the equation...
If $-5$ is a root of the quadratic equation $2 x^{2}+p x-15=0$. and the quadratic equation $p\left(x^{2}+x\right)+k=0$ has equal roots, find the value of $\mathrm{k}$.
It is given that $-5$ is a root of the quadratic equation $2 x^{2}+p x- 15=0$ $\therefore 2(-5)^{2}+p \times(-5)-15=0$ $\begin{array}{l} \Rightarrow-5 p+35=0 \\ \Rightarrow p=7 \end{array}$ The...
Find the values of $p$ for which the quadratic equation $(p+1) x^{2}-6(p+1) x+3(p+9)=0$. , $p \neq-1$ has equal roots. Hence find the roots of the equation.
The given equation is $(p+1) x^{2}-6(p+1) x+3(p+9)=0$. This is of the form $a x^{2}+b x+c=0$, where $a=p+1, b=-6(p+1)$ and $c=3(p+9)$. $\begin{array}{l} \therefore D=b^{2}-4 a c \\ =[-6(p+1)]^{2}-4...
Find the value of $\mathrm{p}$ for which the quadratic equation $(2 p+1) x^{2}-(7 p+2) x+(7 p-3)=0$. has real and equal roots.
The given equation is $(2 p+1) x^{2}-(7 p+2) x+(7 p-3)=0$. This is of the form $a x^{2}+b x+c=0$, where $a=2 p+1, b=-(7 p+2)$ and $c=7 p-3$. $\begin{array}{l} \therefore D=b^{2}-4 a c \\ =-[-(7...
Find the values of $\mathrm{k}$ for which the quadratic equation $(3 k+1) x^{2}+2(k+1) x+1=0$. has real and equal roots.
The given equation is $(3 k+1) x^{2}+2(k+1) x+1=0$. This is of the form $a x^{2}+b x+c=0$, where $a=3 k+1, b=2(k+1)$ and $c=1$. $\begin{array}{l} \therefore D=b^{2}-4 a c \\ =[2(k+1)]^{2}-4 \times(3...
Find the nonzero value of $\mathrm{k}$ for which the roots of the quadratic equation $9 x^{2}-3 k x+k=0$. are real and equal.
The given equation is $9 x^{2}-3 k x+k=0$. This is of the form $a x^{2}+b x+c=0$, where $a=9, b=-3 k$ and $c=k$. $\therefore D=b^{2}-4 a c=(-3 k)^{2}-4 \times 9 \times k=9 k^{2}-36 k$ The given...
For what values of $p$ are the roots of the equation $4 x^{2}+p x+3=0$. real and equal?
The given equation is $4 x^{2}+p x+3=0$. This is of the form $a x^{2}+b x+c=0$, where $a=4, b=p$ and $c=3$. $\therefore D=b^{2}-4 a c=p^{2}-4 \times 4 \times 3=p^{2}-48$ The given equation will have...
For what value of $k$ are the roots of the quadratic equation $k x(x-2 \sqrt{5})+10=0$ real and equal.
The given equation is $\begin{array}{l} k x(x-2 \sqrt{5})+10=0 \\ \Rightarrow k x^{2}-2 \sqrt{5} k x+10=0 \end{array}$ This is of the form $a x^{2}+b x+c=0$, where $a=k, b=-2 \sqrt{5} k$ and $c=10$....
For what values of $k$ are the roots of the quadratic equation $3 x^{2}+2 k x+27=0$ real and equal?
Given: $3 x^{2}+2 k x+27=0$ Here, $a=3, b=2 k \text { and } c=27$ It is given that the roots of the equation are real and equal; therefore, we have: $\begin{array}{l} D=0 \\ \Rightarrow(2 k)^{2}-4...
Show that the roots of the equation $x^{2}+p x-q^{2}=0$ are real for all real values of $\mathrm{p}$ and $\mathrm{q}$.
$x^{2}+p x-q^{2}=0$ Here, $a=1, b=p$ and $c=-q^{2}$ Discriminant $D$ is given by: $\begin{array}{l} D=\left(b^{2}-4 a c\right) \\ =p^{2}-4 \times 1 \times\left(-q^{2}\right) \\ =\left(p^{2}+4...
If a and b are distinct real numbers, show that the quadratic equations $2\left(a^{2}+b^{2}\right) x^{2}+2(a+b) x+1=0 .$ has no real roots.
The given equation is $2\left(a^{2}+b^{2}\right) x^{2}+2(a+b) x+1=0$ $\begin{array}{l} \therefore D=[2(a+b)]^{2}-4 \times 2\left(a^{2}+b^{2}\right) \times 1 \\ =4\left(a^{2}+2 a...
Find the nature of roots of the following quadratic equations:
(i) $12 x^{2}-4 \sqrt{15} x+5=0$
(ii) $x^{2}-x+2=0$.
(i) The given equation is $12 x^{2}-4 \sqrt{15} x+5=0$ This is of the form $a x^{2}+b x+c=0$, where $a=12, b=-4 \sqrt{15}$ and $c=5$. $\therefore$ Discriminant, $D=b^{2}-4 a c=(-4 \sqrt{15})^{2}-4...
Find the nature of roots of the following quadratic equations:
(i) $5 x^{2}-4 x+1=0$
(ii) $5 x(x-2)+6=0$
(i) The given equation is $5 x^{2}-4 x+1=0$ This is of the form $a x^{2}+b x+c=0$, where $a=5, b=-4$ and $c=1$. $\therefore$ Discriminant, $D=b^{2}-4 a c=(-4)^{2}-4 \times 5 \times 1=16-20=-4<0$...
Find the nature of roots of the following quadratic equations:
(i) $2 x^{2}-8 x+5=0$
(ii) $3 x^{2}-2 \sqrt{6} x+2=0$
(i) The given equation is $2 x^{2}-8 x+5=0$ This is of the form $a x^{2}+b x+c=0$, where $a=2, b=-8$ and $c=5$. $\therefore$ Discriminant, $D=b^{2}-4 a c=(-8)^{2}-4 \times 2 \times 5=64-40=24>0$...