List of practice Questions

Prove that the Greatest Integer Function f: R \(\to\) R given by f(x) = [x], is neither one-once nor onto, where [x] denotes the greatest integer less than or equal to x.

Number of binary operations on the set {a, b} are

Consider a binary operation *on N defined as \(a*b=a^3+b^3.\)Choose the correct answer.

Let f: R-\(\{ - \frac {-4} {3} \}\)→R be a function defined as \(f (x) = \frac {4x} {3x + 4}.\) The inverse of f is map g: Range f→R- \(\{ \frac {- 4} {3}\}\) given by

If f : R→R be given by f(x)= \((3-x^3)^\frac {1} {3}\),,then fof(x) is 

Determine whether each of the following relations are reflexive, symmetric, and transitive.

1. Relation R in the set A={1,2,3,...13,14} defined as R={(x,y): 3x-y=0}.
2. Relation R in the set of N natural numbers defined as R={(x,y): y=x+5 and x<4}.
3. Relation R in the set A={1,2,3,4,5,6} defined as R={(x,y): y is divisible by x}.
4. Relation R in the set of Z integers defined as R={(x,y): x-y is an integer}
5. Relation R in the set of human beings in a town at a particular time given by
   1. R={(x,y): x and y work at same place}
   2. R={(x,y): x and y live in the same locality}
   3. R={(x,y): x is exactly 7 am taller that y}
   4. R={(x,y): x is wife of y}
   5. R={(x,y):x is father of y}

For the matrices A and B, verify that (AB)′= B'A' where
(i)A=\(\begin{bmatrix}1\\-4\\3\end{bmatrix},\,B=\begin{bmatrix}-1&2&1\end{bmatrix}\)

(ii)A=\(\begin{bmatrix}0\\1\\2\end{bmatrix},\,B=\begin{bmatrix}1&5&7\end{bmatrix}\)

For the matrix A=\(\begin{bmatrix}1&5\\6&7\end{bmatrix}\),verify that 
  I. (A+A') is a symmetric matrix
  II. (A-A') is a skew symmetric matrix

1. Show that the matrix A=\(\begin{bmatrix}1&-1&5\\-1&2&1\\5&1&3\end{bmatrix}\) is a symmetric matrix.
2. Show that the matrix A=\(\begin{bmatrix}0&1&-1\\-1&0&1\\1&-1&0\end{bmatrix}\) is a skew symmetric matrix

Find the inverse of each of the matrices, if it exists. \(\begin{bmatrix}1&-1\\2&3\end{bmatrix}\)

If A'=\(\begin{bmatrix}-2&3\\1&2\end{bmatrix}\)and B=\(\begin{bmatrix}-1&0\\1&2\end{bmatrix}\),then find (A+2B)'

If A=\(\begin{bmatrix}-1&2&3\\5&7&9\\-2&1&1\end{bmatrix}\)and B=\(\begin{bmatrix}-4&1&-5\\1&2&0\\1&3&1\end{bmatrix}\),then verify that
 (i)(A+B)'=A'+B'
 (ii)(A-B)'=A'-B'

Find the values of \(\sin(\frac{\pi}{3}-\sin^{-1}(-\frac{1}{2}))\) is equal to

Find the values of \(tan^{-1}(\tan\frac{3\pi}{4})\)

Find the values of \(\sin^{-1}(\sin^2\frac{\pi}{3})\)

Find the value of \(\cot(tan^{-1}a+\cot^{-1}a)\)

Find the value of \(\tan ^{-1}\bigg[2\cos\Big(2\sin^{-1}\frac{1}{2}\Big)\bigg]\)

Write the function in the simplest form: \(tan^{-1}\frac{x}{\sqrt{a^2-x^2}},\mid x\mid<a\)