The function for exchanging American dollars for Singapore Dollar on a given day is f(x) = 1.23x, where x represents the number of American dollars. On the same day function for exchanging Singapore dollar to Indian Rupee is g(y) = 50.50y, Where y represents the number of Singapore dollars. Write a function which will give the exchange rate of American dollars in terms of Indian rupee

A simple cipher takes a number and codes it, using the function f(x) = 3x - 4. Find the inverse of this function, determine whether the inverse is also a function and verify the symmetrical property about the line y = x(by drawing the lines)

For the given curve, \(y=x^{1\over 3}\)given in  figure draw
(i) \(y=-x^{ \left( \frac { 1 }{ 3 } \right) }\)
(ii) ​\(y=x^{ \left( \frac { 1 }{ 3 } \right) }+1\)
(iii)​ \(y=x^{ \left( \frac { 1 }{ 3 } \right) }-1\)
(iii) ​\(y=(x+1)^{1\over 3}\)​

Discuss the following relations for reflexivity, symmetricity and transitivity:
Let P denote the set of all straight lines in a plane. The relation R defined by "lRm if l is perpendicular to m".

Let A = {a, b, c, d}, B = {a, c, e}, C = {a, e}.
Show that A ∩ (B ∩ C) = (A ∩ B) ∩ C

Check whether the following functions are one-to-one and onto.
(i)  \(f:N\rightarrow N\) defined by f(n) = n + 2.
(ii) \(f: \mathbb{N} \cup\{-1,0\} \rightarrow \mathbb{N}\) defined by \(f(n)=n+2\)

Let f = {(1, 2), (3, 4), (2, 2)} and g = {(2, 1), (3, 1), (4, 2)}. Find g o f and f o g.

If a²+b² = 7ab. Show that log \(\ \frac { a+b }{ 3 } =\frac { 1 }{ 2 } \)  (log a + log b)

Solve: \(\sqrt{x+5}+\sqrt{x+21}=\sqrt{6x+40}\)

Find all pairs of consecutive odd natural numbers both of which are larger than 10 and their sum is less than 40.

Resolve the following rational expressions into partial fractions.
\({{1}\over{x^4-1}}\)

Resolve the following rational expressions into partial fractions.
\({{6x^2-x+1}\over{x^3+x^2+x+1}}\)

Show that \(\frac { sin8x\ cosx-sin6x\ cos3x }{ cos2x\ cosx-sin3x\ sin4x } =tan2x\)

Prove that \(\frac { sinx+sin3x+sin5x+sin7x }{ cosx+cos3x+cos5x+cos7x } =tan4x\)

If A + B + C = \(\frac { \pi }{ 2 } \), prove the following sin 2A + sin 2B + sin 2C = 4 cos A cos B cos C

In \(\triangle\)ABC, Prove the following a (cos B + cos C) = 2 (b+c)sin² \(\frac { A }{ 2 } \)

Express each of the following as a product.
cos 35^o - cos 75^o

If \(sin x=\frac{4}{5}\) (in I quadrant) and \(cos\ y=\frac{-12}{13}\) (in II quadrant), then find
(i) sin (x - y) 

Solve \(\sqrt{3}tan^2\theta+(\sqrt{3}-1)tan\theta-1=0\)

In a \(\triangle\) ABC, prove that \(({B-C\over 2})={b-c\over a} cos {A\over 2}\)

Prove that \(\frac{2cos2\theta+1}{2cos2\theta-1}=tan(60°+\theta)(tan60°-\theta)\)

How many strings can be formed using the letters of the word LOTUS if the word
(i) either starts with L or ends with S?
(ii) neither starts with L nor ends with S?

By the principle of mathematical induction, prove that for n > 1,
\(1^2 + 3^2 + 5^2 + ... + (2n-1)^2 = {n(2n-1)(2n+1)\over 3}\)

By the principle of mathematical induction, prove that, for all integers n ≥ 1,
1² + 2² + 3²+...n² = \(\frac { n(n+1)(2n+1) }{ 6 } \).

Compute the sum of first n terms of the following series 8 + 88 + 888 + .......

Find the value of n if the sum to n terms of the series \(\sqrt { 3 } +\sqrt { 75 } +\sqrt { 243 } +....is\quad 435\sqrt { 3 } .\)

Find the value of \(\sum _{ n=1 }^{ \infty }{ \frac { 1 }{ { 2 }^{ n-1 } } \left( \frac { 1 }{ { 9 }^{ n-1 } } +\frac { 1 }{ { 9 }^{ 2n-1 } } \right) } \)

If \(\alpha ,\beta \)are the roots of the equation x²-px + q = 0, then prove that \(\log { (1+px+q{ x }^{ 2 }) } =(\alpha +\beta )x=\frac { { \alpha }^{ 2 }+{ \beta }^{ 2 } }{ 2 } { x }^{ 2 }+\frac { { \alpha }^{ 2 }+{ \beta }^{ 2 } }{ 3 } { x }^{ 3 }-....\infty \)

Find the sum of the first 20-terms of the arithmetic progression having the sum of first 10 terms as 52 and the sum of the first 15 terms as 77.

If θ is a parameter, find the equation of the locus of a moving point, whose coordinates are x = a cos³ θ, y = a sin³ θ.

Find the equations of straight lines which are perpendicular to the line 3x + 4y - 6 = 0 and are at a distance of 4 units from (2, 1).

Show that the equation 2x² - xy - 3y² - 6x + 19y - 20 = 0 represents a pair of intersecting lines. Show further that the angle between them is \(\tan ^{ -1 }{ \left( 5 \right) } \)

Find p and q, if the following equation represents a pair of perpendicular lines 6x² + 5xy - py² + 7x + qy - 5 = 0.

If P and p¹ be the perpendicular from the original upon the straight lines \(x\sec\theta+y cosec\theta=a\)and \(x\cos\theta-y\sin\theta=a\cos2\theta\) , prove that 4p² +p¹ = a²

Find the equation of the line through the intersection of the lines 3x + 2y + 5 = 0 and 3x - 4y + 6 = 0 and the point (1, 1).

Express the matrix A =\(\begin{bmatrix} 1 & 3 & 5 \\ -6 & 8 & 3 \\ -4 & 6 & 5 \end{bmatrix}\)as the sum of a symmetric and a skew-symmetric matrices.

If A = \(\begin{bmatrix} 1 & 2 & 2 \\ 2 & 1 & -2 \\ x & 2 & y \end{bmatrix}\) is a matrix such that AA^T = 9I, find the values of x and y.

Without expanding the determinants, show that | B | = 2| A |.
Where B =\(\begin{bmatrix} b+c & c+a & a+b \\ c+a & a+b &b+c \\a+b & b+c & c+a \end{bmatrix}\)and A =\(\begin{bmatrix} a& b & c \\ b & c & a \\ c & a & b \end{bmatrix}\)

If ABCD is a quadrilateral and E and F are the midpoints of AC and BD respectively, then prove that \(\overrightarrow{AB}\) + \(\overrightarrow{AD}\) + \(\overrightarrow{CB}\) +\(\overrightarrow{CD}\) = 4 \(\overrightarrow{EF}\). 

The position vectors of the vertices of a triangle are \(\hat{i}+2\hat{j}+3\hat{k};3\hat{i}-4\hat{j}+5\hat{k}\) and\(-2\hat{i}+3\hat{j}-7\hat{k}\).Find the perimeter of the triangle.

Check if \(lim_{x\rightarrow-58}f(x)\)exists or not, where \(f(x)=\left\{\begin{array}{cc} \frac{|x+5|}{x+5} & , \text { for } x \neq-5 \\ 0, & \text { for } x=-5 \end{array}\right.\)

Evaluate the following limits :
\(lim_{x\rightarrow0}{\sqrt{1-x}-1\over x^2}\)

Evaluate the following limits :\(lim_{x\rightarrow0}{e^x-e^{-x}\over sin x}\)

Evaluate the following limits :\(lim_{x\rightarrow 0}{sin \ x(1-cos \ x)\over x^3}\)

Differentiate the following: \(y=\frac{e^{3 x}}{1+e^x}\)

If y = e^{tan-1 x}, Show that (1 + x²) y" + (2x - 1) y' = 0.

Integrate the following with respect to x : \(\left(1-x^2\right)^{-\frac{1}{2}}\)

Integrate the following functions with respect to x : \(x+1\over (x+2)(x+3)\)

If P(A) = 0.5, P(B) = 0.8 and P(B/A) = 0.8, find P(A/B) and P(A\(\cup \)B)

A factory has two machines I and II. Machine I produces 40% of items of the output and Machine II produces 60% of the items. Further 4% of items produced by Machine I are defective and 5% produced by Machine II are defective. An item is drawn at random. If the drawn item is defective, find the probability that it was produced by Machine II. (See the previous example, compare the questions).