If A is a non-singular matrix of odd order, prove that |adj A| is positive

Find the rank of the matrix \(\left[ \begin{matrix} 1 & 2 & 3 \\ 2 & 1 & 4 \\ 3 & 0 & 5 \end{matrix} \right] \) by reducing it to a row-echelon form.

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 following \(\left| \frac { 2+i }{ -1+2i } \right| \) 

Write in polar form of the following complex numbers
\(2+i2\sqrt { 3 } \)

Simplify the following
\(\sum _{ n=1 }^{ 10 }{ { i }^{ n+50 } } \).

Find the square roots of −6+8i

Represent the complex numbe \(1+i\sqrt { 3 } \) in polar form.

Find the monic polynomial equation of minimum degree with real coefficients having 2 -\(\sqrt{3}\)i as a root.

Discuss the maximum possible number of positive and negative roots of the polynomial equation 9x⁹- 4x⁸+ 4x⁷- 3x⁶+ 2x⁵+ x³+7x²+7x+2 = 0

Find all the values of x such that -10\(\pi\)\(\le x\le\)10\(\pi\) and sin x = 0 

Find the principal value of
sec^-1\((\frac{2}{\sqrt3})\)

Find the period and amplitude of
y = -sin\((\frac{1}{3}x)\)

Find the principal value of
cosec^{-1\((-\sqrt{2})\)}

Determine whether x + y − 1 = 0 is the equation of a diameter of the circle x² + y² − 6x + 4y + c = 0 for all possible values of c .

Identify the type of the conic for the following equations:
3x²+2y² = 14

If \(\hat { 2i } -\hat { j } +\hat { 3k } ,\hat { 3i } +\hat { 2j } +\hat { k } ,\hat { i } +\hat { mj } +\hat { 4k } \) are coplanar, find the value of m.

Find the vector and Cartesian form of the equations of a plane which is at a distance of 12 units from the origin and perpendicular to \(6\hat { i } +2\hat { j } -3\hat { k } \)

Find the intercepts cut off by the plane \(\vec { r } .(6\hat { i } +4\hat { j } -3\hat { k } )\) = 12 on the coordinate axes.

Find the acute angle between the following lines
2x = 3y = −z and 6x = − y = −4z.

Find the point on the curve y = x² − 5x + 4 at which the tangent is parallel to the line 3x + y = 7.

Explain why Lagrange’s mean value theorem is not applicable to the following functions in the respective intervals
f(x) = |3x + 1|, x ∈ |-1, 3|

Evaluate the following limit, if necessary use l ’Hôpital Rule
\(\underset { x\rightarrow \infty }{ lim } \frac { x }{ logx } \) 

Find df for f(x) = x² + 3x and evaluate it for
x = 3 and dx = 0.02

In each of the following cases, determine whether the following function is homogeneous or not. If it is so, find the degree.
\(U(x,y,z)=xy+sin\left( \frac { { y }^{ 2 }-2{ x }^{ 2 } }{ xy } \right) \)

Evaluate the following integrals using properties of integration:
\(\int _{ -\frac { \pi }{ 2 } }^{ \frac { \pi }{ 2 } }{ ({ x }^{ 5 }+xcos\ x+{ tan }^{ 3 }x+1)dx } \)

Evaluate the following
\(\int _{ 0 }^{ \frac { \pi }{ 2 } }{ { sin }^{ 2 }x{ cos }^{ 4 }xdx } \)

Find the area enclosed between the parabola y²=4ax and the line x=a, x=9a.

For each of the following differential equations, determine its order, degree (if exists)
\({ x }^{ 2 }\frac { { d }^{ 2 }y }{ d{ x }^{ 2 } } +{ \left[ 1+{ \left( \frac { dy }{ dx } \right) }^{ 2 } \right] }^{ \frac { 1 }{ 2 } }=0\)

Find the differential equation of the family of all non-vertical lines in a plane.

Determine the order and degree (if exists) of the following differential equations: 
\(3\left( \frac { { d }^{ 2 }y }{ { dx }^{ 2 } } \right) ={ \left[ 4+{ \left( \frac { dy }{ dx } \right) }^{ 2 } \right] }^{ \frac { 3 }{ 2 } }\)

Solve the following differential equations or show that the solution of 
\(\\ \\ \\ \frac { dy }{ dx } =\sqrt { \frac { 1-{ y }^{ 2 } }{ 1-{ x }^{ 2 } } } \)

A six sided die is marked '1' on one face, '3' on two of its faces, and '5' on remaining three faces. The die is thrown twice. If X denotes the total score in two throws, find
(i) the probability mass function
(ii) the cumulative distribution function
(iii) P(4 ≤ X < 10)
(iv) P(X ≥ 6)

A random variable X has the following probability mass function.

x	1	2	3	4	5
f(x)	k2	2k2	3k2	2k	3k

Find
(i) the value of k
(ii) P(2 \(\le\) X < 5)
(iii) P(3 < X )

The time to failure in thousands of hours of an electronic equipment used in a manufactured computer has the density function \(f(x)=\begin{cases} \begin{matrix} { 3e }^{ -3x } & x>0 \end{matrix} \\ \begin{matrix} 0 & elsewhere \end{matrix} \end{cases}\) 
Find the expected life of this electronic equipment.

Write the statements in words corresponding to ¬p, p ∧ q , p ∨ q and q ∨ ¬p, where p is ‘It is cold’ and q is ‘It is raining'.

Which one of the following sentences is a proposition?
(i) 4 + 7 =12
(ii) What are you doing?
(iii) 3ⁿ ≤ 81, n ∈ N
(iv) Peacock is our national bird
(v) How tall this mountain is!