Let A = \(\left[ \begin{matrix} 2 & -1 & 1 \\ -1 & 2 & -1 \\ 1 & -1 & 2 \end{matrix} \right] \) and 4B = \(\left[ \begin{matrix} 3 & 1 & -1 \\ 1 & 3 & x \\ -1 & 1 & 3 \end{matrix} \right] \). If B is the inverse of A, then the value of x is

Which of the following is not an elementary transformation?

If (1, -3) is the centre of the circle x² + y² + ax + by + 9 = 0 its radius is _________

The number of normals to the hyperbola \(\frac { { x }^{ 2 } }{ { a }^{ 2 } } +\frac { { y }^{ 2 } }{ { b }^{ 2 } } \) = 1 from an external point is ________

The two planes 3x + 3y - 3z - 1 = 0 and x + y - z + 5 = 0 are _____________

The critical points of the function f(x) = \((x-2)^{ \frac { 2 }{ 3 } }(2x+1)\) are __________

The function -3x+12 is ________ function on R.

The function f(x) = x⁹ + 3x⁷+ 64 is increasing on ________

The approximate value of (627)^\(\frac14\) is ................

The area bounded by the parabola y = x² and the line y = 2x is __________

The solution of sec²x tan y dx + sec²y tan x dy = 0 is _________

The differential equation of x²y = k is _________.

Which one of the following is not a statement?

Which of the following is a contradiction?

If p is true and q is unknown, then _________

Reduce the matrix \(\left[ \begin{matrix} 3 & -1 & 2 \\ -6 & 2 & 4 \\ -3 & 1 & 2 \end{matrix} \right] \) to a row-echelon form.

Find the rank of the following matrices which are in row-echelon form :
\(\left[ \begin{matrix} 2 & 0 & -7 \\ 0 & 3 & 1 \\ 0 & 0 & 1 \end{matrix} \right] \)

Find the rank of the following matrices which are in row-echelon form :
\(\left[ \begin{matrix} -2 & 2 & -1 \\ 0 & 5 & 1 \\ 0 & 0 & 0 \end{matrix} \right] \)

Find the rank of the following matrices which are in row-echelon form :
\(\left[ \begin{matrix} 6 \\ \begin{matrix} 0 \\ \begin{matrix} 0 \\ 0 \end{matrix} \end{matrix} \end{matrix}\begin{matrix} 0 \\ \begin{matrix} 2 \\ \begin{matrix} 0 \\ 0 \end{matrix} \end{matrix} \end{matrix}\begin{matrix} -9 \\ \begin{matrix} 0 \\ \begin{matrix} 0 \\ 0 \end{matrix} \end{matrix} \end{matrix} \right] \)

Find the square root of 6−8i .

Write the following in the rectangular form:
 \(\overline { 3i } +\frac { 1 }{ 2-i } \).

Solve the equations:
6x⁴- 35x³+ 62x²- 35x + 6 = 0

Find the principal value of sin^-1(2), if it exists.

Find the principal value of
 \({ Sin }^{ -1 }\left( \frac { 1 }{ \sqrt { 2 } } \right) \)

Find the value of sec⁻¹\(\left( -\frac { 2\sqrt { 3 } }{ 3 } \right) \)

Find the value, if it exists. If not, give the reason for non-existence.
sin^-1 [sin5]

Find the distance between the parallel planes x + 2y - 2z + 1 = 0 and 2x + 4y - 4z + 5 = 0

Express each of the following physical statements in the form of differential equation.
For a certain substance, the rate of change of vapor pressure P with respect to temperature T is proportional to the vapor pressure and inversely proportional to the square of the temperature.

An urn contains 5 mangoes and 4 apples. Three fruits are taken at randaom. If the number of apples taken is a random variable, then find the values of the random variable and number of points in its inverse images.

Write the converse, inverse, and contrapositive of each of the following implication.
If x and y are numbers such that x = y, then x² = y²

Find the inverse of each of the following by Gauss-Jordan method:
\(\left[ \begin{matrix} 2 & -1 \\ 5 & -2 \end{matrix} \right] \) 

Find the inverse (if it exists) of the following:
\(\left[ \begin{matrix} 5 & 1 & 1 \\ 1 & 5 & 1 \\ 1 & 1 & 5 \end{matrix} \right] \)

Find the approximate value of f (3.02) where f(x) = 3x² + 5x +3.

Find the linear approximation to \(g(z)=\sqrt [ 4 ]{ zat } z=2\)

Evaluate : \(\underset { \left( x,y \right) \rightarrow \left( 2,0 \right) }{ lim } \frac { \sqrt { 2x-y-2 } }{ 2x-y-4 } \)

Evaluate : \(\underset { \left( x,y \right) \rightarrow \left( 0,0 \right) }{ lim } \frac { { x }^{ 2 }-xy }{ \sqrt { x } -\sqrt { y } } \)

Evaluate \(\int _{ 0 }^{ \frac { \pi }{ 2 } }{ { sin }^{ 0 }xdx } \)

Form the D.E of family of curves represented by y=c(x-c)².where c is the parameter.

Show that the function y=Acos2x-Bsin2x is a solution of the D.E y²+4y=0

Solve:\(\frac { dy }{ dx } =\frac { 1-cosx }{ 1+cosx } \)

A six sided die is marked '2' on one face, '3' on two ofits faces, and '4' on remaining three faces. The die is thrown twice. If X denotes the total score in two throws, find the values of the random variable and number of points in its inverse images.

Verify the
(i) closure property,
(ii) commutative property,
(iii) associative property
(iv) existence of identity and
(v) existence of inverse for the arithmetic operation - on Z.

Show that the matrix \(\left[ \begin{matrix} 3 & 1 & 4 \\ 2 & 0 & -1 \\ 5 & 2 & 1 \end{matrix} \right] \) is non-singular and reduce it to the identity matrix by elementary row transformations.

Find the inverse of A = \(\left[ \begin{matrix} 2 & 1 & 1 \\ 3 & 2 & 1 \\ 2 & 1 & 2 \end{matrix} \right] \) by Gauss-Jordan method.

Verify 
(i) closure property 
(ii) commutative property 
(iii) associative property
(iv) existence of identity and
(v) existence of inverse for the operation ×11 on a subset A = {1, 3, 4, 5, 9} of the set of remainders {0,1, 2, 3, 4, 5, 6, 7, 8, 9,10}

Define an operation∗ on Q as follows:  a*b = \(\left( \frac { a+b }{ 2 } \right) \); a,b ∈Q. Examine the existence of identity and the existence of inverse for the operation * on Q.

Let S be a non-empty set and 0 be a binary operation on s defined by x 0 y = x; x, Y \(\in \) s. Determine whether 0 is commutative and association.