The rank of m x n matrix whose elements are unity is ________.

The system of linear equations x + y + z = 2, 2x + y − z = 3, 3x + 2y + k = 4 has unique solution, if k is not equal to _______.

ഽ\(\frac { { 2x }^{ 3 } }{ 4+{ x }^{ 4 } } \)dx is _______.

\(\int _{ 0 }^{ \frac { \pi }{ 3 } }\)tanx dx is _______.

If the marginal revenue MR = 35 + 7x − 3x², then the average revenue AR is ________.

The particular integral of the differential equation is \(\frac { { d }^{ 2 }y }{ d{ x }^{ 2 } } -8\frac { dy }{ dx } \) + 16y = 2e^4x ______.

E f (x)= _______.

If A is a square matrix of order n, then |Adj A|=______

If A is a square matrix such that A² = I, then A^-1=_______

The system of linear equations x + y + Z = 2, 2x + Y - z = 3, 3x + 2y + kz = 4 has a unique solution if k is_______

A set of values of the variable x₁,x₂,...x_n satisfying all the equations simultaneously is called__________ of the system

Find the rank of each of the following matrices.
\(\left( \begin{matrix} 5 & 6 \\ 7 & 8 \end{matrix} \right) \)

If f'(x) = x + b, f(1)= 5 and f(2) = 13, then find f(x)

Evaluate the following using properties of definite integrals:
\(\int _{ 0 }^{ 1 }{ \log\left( \frac { 1 }{ x } -1 \right) dx } \)

Calculate the producer’s surplus at x = 5 for the supply function p = 7 + x.

Find the order and degree of the following differential equations.
\({ \left( \frac { dy }{ dx } \right) }^{ 3 }+y=x-\frac { dx }{ dy } \)

From the following table obtain a polynomial of degree y in x

Define critical value.

Find the rank of the matrix \(\begin{pmatrix} 1 & 5 \\ 3 & 9 \end{pmatrix}\)

Evaluate ഽ\(\frac { dx }{ x^{ 2 }-3x+2 } \)

Solve : (D²−4D−1)y = e^−3x

From the following table find the missing value

Assume the mean height of children to be 69.25 cm with a variance of 10.8 cm. How many children in a school of 1,200 would you expect to be over 74 cm tall?

Solve by Cramer’s rule x + y + z = 4, 2x − y + 3z = 1, 3x + 2y − z = 1

Evaluate
\(\int _{ 0 }^{ \infty }{ { e }^{ -2x }{ x }^{ 5 }dx } \) 

The normal lines to a given curve at each point(x,y) on the curve pass through the point (1, 0). The curve passes through the point (1, 2). Formulate the differential equation representing the problem and hence find the equation of the curve.