Given sin-1 (2a/ 1+ a2) + sin-1 (2b/ 1+ b2) = 2 tan-1 x

### Prove that: (i) (ii)

$(i)$ $(ii)$

### Prove the following results: (ix) (x)

$(ix)$ $(x)$

### Prove the following results: (vii) (viii)

$(vii)$ $(viii)$

### Prove the following results: (v) (vi)

$(v)$ $(vi)$

### Prove the following results: (iii) (iv)

$(iii)$ $(iv)$

### Prove the following results: (i) (ii)

$(i)$ $(ii)$

### Evaluate the following: (iii) (iv)

$(iii)$ $(iv)$

### Evaluate the following: (i) (ii)

$(i)$ $(ii)$

Since, cos-1 (a/x) – cos-1 (b/x) = cos-1 (1/b) – cos-1 (1/a)

Given Cos (sin -1 3/5 + sin-1 5/13)

### Prove the following results: (iii)

$(iii)$

### Prove the following results: (i) (ii)

$(i)$ $(ii)$ LHS

### If , find

Since, $\cos ^{-1} x+\sin ^{-1} x=\pi / 2$ => $\cos ^{-1} x=\pi / 2-\sin ^{-1} x$ Substituting this in $\left(\sin ^{-1} x\right)^{2}+\left(\cos ^{-1} x\right)^{2}=17 \pi^{2} / 36$ $\left(\sin...

$\cot \left(\cos ^{-1} 3 / 5+\sin ^{-1} x\right)=0$ => $\begin{array}{l} \left(\cos ^{-1} 3 / 5+\sin ^{-1} x\right)=\cot ^{-1}(0) \\ \left(\operatorname{Cos}^{-1} 3 / 5+\sin ^{-1} x\right)=\pi /...

Given sin-1 x + sin-1 y = π/3 ……. (i) And cos-1 x – cos-1 y = π/6 ……… (ii)

Since, cos-1 x + cos-1 y = π/4

### Evaluate: (v)

$(v)$ $=>0$

### Evaluate: (iii) for (iv) Cot

$(iii)$ $(iv)$

### Evaluate: (i) (ii) for

$(i)$ $(ii)$

### Evaluate: (iii) cot

$(iii)$

### Evaluate: (i) (ii) Sec

$(i)$ $(ii)$

### Evaluate each of the following: (ix)

$(ix)$ .

### Evaluate each of the following: (vii) Tan (viii)

$(vii)$ $(viii)$

### Evaluate each of the following: (v) (vi)

(v) \[\begin{array}{*{35}{l}} {} \\ Let\text{ }co{{s}^{-1}}\left( 8/17 \right)\text{ }=\text{ }y \\ cos\text{ }y\text{ }=\text{ }8/17\text{ }where\text{ }y\text{ }\in \text{ }\left[ 0,\text{ }\pi...

### Evaluate each of the following: (iii) (iv)

(iii) (iv)

### Evaluate each of the following: (i) (ii)

(i) \[\begin{array}{*{35}{l}} Given\text{ }sin\text{ }\left( si{{n}^{-1}}~7/25 \right) \\ let\text{ }y\text{ }=\text{ }si{{n}^{-1}}~7/25 \\ sin\text{ }y\text{ }=\text{ }7/25\text{ }where\text{...

### Find the derivative of the function f defined by f (x) = mx + c at x = 0.

f(x) = mx + c, Checking the differentiability at x = 0 This is the derivative of a function at x = 0, and also this is the derivative of this function at every value of x.

### If f (x) =, find f’ (4).

f(x) = x3 + 7x2 + 8x – 9, => Checking the differentiability at x = 4

### If for the function Ø (x) =, Ø’ (5) = 97, find λ.

Finding the value of λ given in the real function and we are given with the differentiability of the function f(x) = λx2 + 7x – 4 at x = 5 which is f ‘(5) = 97 =>

### Show that the derivative of the function f is given by f (x) =

, at x = 1 and x = 2 are equal.

We are given with a polynomial function f(x) = 2x3 – 9x2 + 12x + 9, and we have

### If f is defined by f (x) = – 4x + 7, show that f’ (5) = 2 f’ (7/2)

### Discuss the continuity and differentiability of the function f (x) = |x| + |x -1| in the interval of (-1, 2).

Since, a polynomial and a constant function is continuous and differentiable everywhere => f(x) is continuous and differentiable for x ∈ (-1, 0) and x ∈ (0, 1) and (1, 2). Checking continuity...

### Show that the function is defined as follows Is continuous at , but not differentiable thereat.

Since, LHL = RHL = f (2) Hence, F(x) is continuous at x = 2 Checking the differentiability at x = 2 $=> 5$ Since, (RHD at x = 2) ≠ (LHD at x = 2) Hence, f (2) is not differentiable at x =...

checking differentiability of given function at x = 3 => LHD (at x = 3) = RHD (at x = 3) = 12 Since, (LHD at x = 3) = (RHD at x = 3) Hence, f(x) is differentiable at x = 3.

### Show that f (x) = is not differentiable at x = 0.

Since, LHD and RHD does not exist at x = 0 Hence, f(x) is not differentiable at x = 0

### Show that f (x) = |x – 3| is continuous but not differentiable at x = 3.

### Differentiate with respect to , if, (i) (ii)

(i) Let (ii) Let

### Differentiate with respect to if, (iii)

(iii) Let

### Differentiate with respect to if, (i) (ii)

(i) Let (ii)

### Differentiate with respect to , if (i) (ii)

(i) Given sin-1 √ (1-x2) (ii) Given sin-1 √ (1-x2)

### If and , prove that .

### Find dy/dx, when

### Find dy/dx, when

### Find dy/dx, when x = 2 t / 1+t^2 and y = 1-t^2 / 1+t^2.

Given, $x=2 t /\left(1+t^{2}\right)$ On differentiating $x$ with respect to t using quotient rule, $$ \begin{array}{l} \frac{\mathrm{dx}}{\mathrm{dt}}=\left[\frac{\left(1+\mathrm{t}^{2}\right)...

### Find dy/dx, when

### Find dy/dx, when

### Find dy/dx, when x = 3 a t / 1+t^2 and y = 3 a t^2/1+t^2

### Find dy/dx, when

### Find dy/dx, when

### Find dy/dx, when

### Find dy/dx, when

### Find dy/dx, when

### Find dy/dx, when

### Find dy/dx, when

### Differentiate the following functions with respect to x:

### Differentiate the following functions with respect to x:

(vii) (viii)

### Differentiate the following functions with respect to x:

(v) (vi)

### Differentiate the following functions with respect to x:

(iv)

### Differentiate the following functions with respect to x:

### Differentiate the following functions with respect to x:

### Differentiate the following functions with respect to x:

### Differentiate the following functions with respect to x:

### Differentiate the following functions with respect to x:

### Differentiate the following functions with respect to x:

### Differentiate the following functions with respect to x:

### Differentiate the following functions with respect to x:

### Differentiate the following functions with respect to x:

### Differentiate the following functions with respect to x:

### Differentiate the following functions with respect to x:

Let y = (log x)cos x Taking log both the sides, we get

### Differentiate the following functions with respect to x:

### Differentiate the following functions with respect to x:

### Differentiate the following functions with respect to x:

### Differentiate the following functions with respect to x:

### Differentiate the following functions with respect to x:

### If , prove that

### , prove that .

### If , prove that

### If , prove that

### Find dy/dx in each of the following:

differentiating the equation on both sides with respect to x, we get,

### Find dy/dx in each of the following:

### Find dy/dx in each of the following:

differentiating the equation on both sides with respect to x, we get,

### Find dy/dx in each of the following:

differentiating the equation on both sides with respect to x, we get,

### Find dy/dx in each of the following:

differentiating the equation on both sides with respect to x, we get,

### Find dy/dx in each of the following:

differentiating the equation on both sides with respect to x, we get,

### Find dy/dx in each of the following:

### Find dy/dx in each of the following:

differentiating the equation on both sides with respect to x, we get,

### Find dy/dx in each of the following:

differentiating the equation on both sides with respect to x, we get,

### Find dy/dx in each of the following:

differentiating the equation on both sides with respect to x, we get,

### Find dy/dx in each of the following:

### Differentiate the following functions with respect to x:

### Differentiate the following functions with respect to x:

### Differentiate the following functions with respect to x:

### Differentiate the following functions with respect to x:

### Differentiate the following functions with respect to x:

### Differentiate the following functions with respect to x:

### Differentiate the following functions with respect to x:

### Differentiate the following functions with respect to x:

### Differentiate the following functions with respect to x:

### Differentiate the following functions with respect to x:

### Differentiate the following functions with respect to x:

### Differentiate the following functions with respect to x:

### Differentiate the following functions with respect to x:

### Differentiate the following functions with respect to x:

### Differentiate the following functions with respect to x:

### Differentiate the following functions with respect to x:

### Differentiate the following functions with respect to x:

### Differentiate the following functions with respect to x:

Let,

### Differentiate the following functions with respect to x:

Let,

### Differentiate the following functions with respect to x:

### Differentiate the following functions with respect to x:

### Differentiate the following functions with respect to x:

### Differentiate the following functions with respect to x:

### Differentiate the following functions with respect to x:

### Differentiate the following functions with respect to x:

### Differentiate the following functions with respect to x:

### Differentiate the following functions with respect to x:

### Differentiate the following functions with respect to x: log (cosec x – cot x)

### Differentiate the following functions with respect to x:

### Differentiate the following functions with respect to x:

### Differentiate the following functions with respect to x: x

### Differentiate the following functions with respect to x:

### Differentiate the following functions with respect to x:

### Differentiate the following functions with respect to x:

### Differentiate the following functions with respect to x:

### Differentiate the following functions with respect to x:

### Differentiate the following functions with respect to x:

### Differentiate the following functions with respect to x:

### Differentiate the following functions with respect to x: .

Let y = (log sin x)2

### Differentiate the following functions with respect to x:

### Differentiate the following functions with respect to x:

### Differentiate the following functions with respect to x:

### Differentiate the following functions with respect to x:

### Differentiate the following functions with respect to x:

### Differentiate the following functions with respect to x:

### Differentiate the following functions with respect to x:

### Differentiate the following functions with respect to x: tan 5x

Let y = tan (5x°)

### Differentiate the following functions with respect to x:

### Differentiate the following functions with respect to x:

Let y = sin2 (2x + 1) On differentiating y with respect to x, we get

### Differentiate the following functions with respect to x:

### Differentiate the following functions with respect to x:

### Differentiate the following functions with respect to x: Sin (log x)

### Differentiate the following functions with respect to x:

### Differentiate the following functions with respect to x:

Given tan2 x

### Differentiate the following functions from the first principles:

### Differentiate the following functions from the first principles:

let f (x) = ecos x By using the first principle formula, we get,

### Differentiate the following functions from the first principles:

### Differentiate the following functions from the first principles:

A real function f is said to be continuous at x = c, where c is any point in the domain of f if A function is continuous at x = c if Function is changing its nature (or expression) at x = 2, so we...

### If for , find the value which can be assigned to at so that the function becomes continuous everywhere in .

A real function f is said to be continuous at x = c, where c is any point in the domain of f if Where h is a very small positive number. i.e. left hand limit as x → c (LHL) = right hand limit as...

### The function is defined by If f is continuous on [0, 8], find the values of a and b.

A real function f is said to be continuous at x = c, where c is any point in the domain of f if h is a very small positive number. i.e. left hand limit as x → c (LHL) = right hand limit as x → c...

### Find the values of a and b so that the function f (x) defined by

### The function Is continuous on . Find the most suitable values of and .

A real function f is said to be continuous at x = c, where c is any point in the domain of f if h is a very small positive number. i.e. left hand limit as x → c (LHL) = right hand limit as x → c...

### In the following, determine the value(s) of constant(s) involved in the definition so that the given function is continuous: (iii) (i v)

(iii) A real function f is said to be continuous at x = c, where c is any point in the domain of f if h is a very small positive number. i.e. left hand limit as x → c (LHL) = right hand limit as x →...

### In the following, determine the value(s) of constant(s) involved in the definition so that the given function is continuous: (i) (ii)

(i) A real function f is said to be continuous at x = c, where c is any point in the domain of f if h is a very small positive number. i.e. left hand limit as x → c (LHL) = right hand limit as x → c...

### Find the points of discontinuity, if any, of the following functions: (x i) (x i i)

(xi) A real function f is said to be continuous at x = c, where c is any point in the domain of f if h is a very small positive number. i.e. left hand limit as x → c (LHL) = right hand limit as x →...

### Find the points of discontinuity, if any, of the following functions: (i x) (x)

(ix) A real function f is said to be continuous at x = c, where c is any point in the domain of f if h is a very small positive number. i.e. left hand limit as x → c (LHL) = right hand limit as x →...

### Find the points of discontinuity, if any, of the following functions: (viii)

((vii) A real function f is said to be continuous at x = c, where c is any point in the domain of f if h is a very small positive number. i.e. left hand limit as x → c (LHL) = right hand limit as x...

### Find the points of discontinuity, if any, of the following functions: (i) (ii)

(i) A real function f is said to be continuous at x = c, where c is any point in the domain of f if h is a very small positive number. i.e. left hand limit as x → c (LHL) = right hand limit as x → c...

### Discuss the the continuity of the function

A real function f is said to be continuous at x = c, where c is any point in the domain of f if Since, h is a very small positive number. i.e. left hand limit as x → c (LHL) = right hand limit as x...

### Prove that the function is everywhere continuous.

A real function f is said to be continuous at x = c, where c is any point in the domain of f if A function is continuous at x = c if From definition of f(x), f(x) is defined for all real numbers....

### Find the inverse of the following matrices by using elementary row transformations:

Solution: For row transformation $\begin{array}{l} A=I A \\ \Rightarrow\left[\begin{array}{cc} 7 & 1 \\ 4 & -3 \end{array}\right]=\left[\begin{array}{ll} 1 & 0 \\ 0 & 1...

### 43.

Assuming the L.H.S and taking L.C.M and on simplifying we will get, $=\frac{(\cos ecA)(\cos ecA+1+\cos ecA-1)}{(\cos e{{c}^{2}}A-1)}$ $=\frac{(2\cos e{{c}^{2}}A)}{{{\cot }^{2}}A}$ $=\frac{2{{\sin...

### 42.

Solving the L.H.S, we will get $=\frac{\cos A}{1-\tan A}+\frac{\sin A}{1-\cot A}$ $=\frac{\cos A}{1-\frac{\sin A}{\cos A}}+\frac{\sin A}{1-\frac{\cos A}{\sin A}}$ $=\frac{{{\cos }^{2}}A}{\cos A-\sin...

### 41.

Assuming L.H.S and taking L.C.M and on simplifying we will get, $=\frac{\sec A+1+\sec A-1}{(\sec A+1)(\sec A-1)}$ $=\frac{2\sec A}{({{\sec }^{2}}A-1)}$ $=\frac{2{{\cos }^{2}}A}{(\cos A{{\sin...

### 40.

Solving L.H.S and divide the numerator and denominator with $(1-\cos A),$, we have $=\frac{(1-\cos A)(1-\cos A)}{(1+\cos A)(1-\cos A)}$ $=\frac{{{(1-\cos A)}^{2}}}{(1+{{\cos }^{2}}A)}$...

### 39.

Solving $LHS={{(\sec A-\tan A)}^{2}}$, we get $={{\left[ \frac{1}{\cos A}-\frac{\sin A}{\cos A} \right]}^{2}}$ $=\frac{{{(1-\sin A)}^{2}}}{{{\cos }^{2}}A}$ $=\frac{{{(1-\sin A)}^{2}}}{1-{{\sin...

### 38. Prove that:(iii)(iv)

Solving L.H.S and dividing the numerator and denominator with its respective conjugates, we have $=\sqrt{\frac{(1-\cos \theta )(1-\cos \theta )}{(1+\cos \theta )(1-\cos \theta...

### 38. Prove that: (i)(ii)

Solving L.H.S and divide the numerator and denominator with its respective conjugates, we have $=\sqrt{\frac{(\sec \theta -1)(\sec \theta -1)}{(\sec \theta +1)(\sec \theta -1)}}+\sqrt{\frac{(\sec...

### 37. (i) (ii)

Solving L.H.S and dividing the numerator and denominator with $\sqrt{(1+\sin A)},$we have $=\sqrt{\frac{(1+\sin A)(1+\sin A)}{(1-\sin A)(1+\sin A)}}=\sqrt{\frac{{{(1+\sin A)}^{2}}}{1-{{\sin...

### 36.

Solving L.H.S $LHS=\frac{1+\cos A}{\sin A}$ Multiply the numerator and denominator by $(1-\cos A)$ we will have $=\frac{(1+\cos A)(1-\cos A)}{\sin A(1-\cos A)}$ $=\frac{1-{{\cos }^{2}}A}{\sin...

### 35.

Solving L.H.S $LHS=\frac{\sec A-\tan A}{\sec A+\tan A}$ Dividing the denominator and numerator with $(\sec A+\tan A)$ and using ${{\sec }^{2}}\theta -{{\tan }^{2}}\theta =1,$we have, $=\frac{{{\sec...

### 34.

Solving L.H.S and using the trigonometric identity ${{\sin }^{2}}A+{{\cos }^{2}}A=1,$, we have ${{\sin }^{2}}A=1-{{\cos }^{2}}A$ $\Rightarrow {{\sin }^{2}}A=(1-\cos A)(1+\cos A)$ $LHS=\frac{1+\cos...

### Find the adjoint of each of the following matrices:

(i)

(ii) Verify that for the above matrices.

Solution: (i) Suppose $A=\left[\begin{array}{cc}-3 & 5 \\ 2 & 4\end{array}\right]$ Cofactors of $A$ are $C_{11}=4$ $C_{12}=-2$ $C_{21}=-5$ $C_{22}=-3$ Since, adj...

### 33.

First solve L.H.S and using the trigonometric identity we all know that ${{\sec }^{2}}\theta {{\tan }^{2}}\theta =1\Rightarrow 1+{{\tan }^{2}}\theta ={{\sec }^{2}}\theta $ $LHS=\frac{{{\sec...

### 32.

Using the trigonometric identity we get $\cos e{{c}^{2}}\theta +{{\cot }^{2}}\theta =1$ cubing it on both side ${{(\cos e{{c}^{6}}\theta +{{\cot }^{2}}\theta )}^{3}}=1$ $\cos e{{c}^{6}}-{{\cot...

### 31.

Using trigonometric identity, ${{\sec }^{2}}\theta -{{\tan }^{2}}\theta =1$ Cubing it on both side ${{({{\sec }^{2}}\theta -{{\tan }^{2}}\theta )}^{2}}=1$ ${{\sec }^{6}}\theta -{{\tan }^{6}}\theta...

### 30.

Solving L.H.S, we get $LHS=\frac{\tan \theta }{1-\frac{1}{\tan \theta }}+\frac{\cot \theta }{1-\tan \theta }$ $=\frac{{{\tan }^{2}}\theta }{\tan \theta -1}+\frac{\cot \theta }{1-\tan \theta }$...

### 29.

Solving L.H.S and using the trigonometric identity ${{\sin }^{2}}\theta +{{\cos }^{2}}\theta =1,$we get Multiplying by $(1-\cos \theta )$to Numerator and denominator $LHS=\frac{1+\sec \theta }{\sec...

### 28.

Solve L.H.S $\frac{1+{{\tan }^{2}}\theta }{1+{{\cot }^{2}}\theta }$ Using trigonometric identity${{\sec }^{2}}\theta -{{\tan }^{2}}\theta =1,and\cos e{{c}^{2}}\theta -{{\cot }^{2}}\theta =1$...

### 27.

Firstly we will solve L.H.S=R.H.S Then use trigonometric identity $\sin \theta +\cos \theta =1,$, we get $LHS=\frac{{{(1+\sin \theta )}^{2}}+{{(1-\sin \theta )}^{2}}}{2{{\sec }^{2}}\theta }$...

### 26.

Firstly we will solve L.H.S Using the trigonometric identity ${{\sin }^{2}}\theta +{{\cos }^{2}}\theta =1,$, we get $LHS=\frac{1+\sin \theta }{\cos \theta }+\frac{\cos \theta }{1+\sin \theta }$...

### 25.

Firstly we will solve L.H.S $LHS=\frac{1}{1+\sin A}+\frac{1}{1-\sin A}$ $=\frac{(1-\sin A)+(1+\sin A)}{(1+\sin A)(1-\sin A)}$ $=\frac{1-\sin A+1+\sin A}{1-{{\sin }^{2}}A}$ $\because (1+\sin...

### Find the principal value of the following:

(i)

(ii)

Solution: (i) Suppose $\sin ^{-1}\left(\cos \frac{3 \pi}{4}\right)=\mathrm{y}$ Therefore we can write the above equation as $\sin \mathrm{y}=\cos \frac{3 \pi}{4}=-\sin \left(\pi-\frac{3...

### Find the principal value of the following:

(i)

(ii)

Solution: (i) It is given that functions can be written as $\sin ^{-1}\left(\frac{\sqrt{3}-1}{2 \sqrt{2}}\right)=\sin ^{-1}\left(\frac{\sqrt{3}}{2 \sqrt{2}}-\frac{1}{2 \sqrt{2}}\right) $Taking $1 /...