Find a matrix A if adj(A) = \(\left[ \begin{matrix} 7 & 7 & -7 \\ -1 & 11 & 7 \\ 11 & 5 & 7 \end{matrix} \right] \).

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 each of the following matrices:
\(\left[ \begin{matrix} 4 & 3 \\ -3 & -1 \\ 6 & 7 \end{matrix}\begin{matrix} 1 & -2 \\ -2 & 4 \\ -1 & 2 \end{matrix} \right] \)

If z₁= 3 - 2i and z₂ = 6 + 4i, find \(\frac { { z }_{ 1 } }{ z_{ 2 } } \) in the rectangular form.

Obtain the Cartesian equation for the locus of z = x + iy in each of the following cases:
|z - 4| = 16

Simplify the following
\({ i }^{ 59 }+\frac { 1 }{ { i }^{ 59 } } \)

Find the modulus of the following complex number \(\frac { 2-i }{ 1+i } +\frac { 1-2i }{ 1-i } \)

Find the following \(\left| \overline { (1+i) } (2+3i)(4i-3) \right| \)

Find a polynomial equation of minimum degree with rational coefficients, having 2 +√3 i as a root.

Construct a cubic equation with roots 1, 1 and −2

Find the period and amplitude of y = sin 7x

If cot^-1\(\frac{1}{7}=\theta\), find the value of cos \(\theta\).

Show that \(cot(sin^{ -1 }x)=\frac { \sqrt { 1-x^{ 2 } } }{ x } -1\le x\le 1\)and x \(\neq \) 0

Find the value of \({ cos }^{ -1 }\left( cos\frac { \pi }{ 7 } cos\frac { \pi }{ 17 } -sin\frac { \pi }{ 7 } sin\frac { \pi }{ 17 } \right) .\)

Simplify \({ tan }^{ -1 }\left( tan\left( \frac { 3\pi }{ 4 } \right) \right) \)

Obtain the equation of the circles with radius 5 cm and touching x-axis at the origin in general form.

Identify the type of conic section for each of the equations.
x² + y² + x − y = 0

Find the volume of the parallelepiped whose coterminous edges are represented by the vectors \(-6\hat { i } +14\hat { j } +10\hat { k } ,14\hat { i } -10\hat { j } -6\hat { k } \) and \(2\hat { i } +4\hat { j } -2\hat { k } \)

Find the vector and Cartesian equations of the plane passing through the point with position vector \(4\hat { i } +2\hat { j } -3\hat { k } \) and normal to vector \(2\hat { i } -\hat { j } +\hat { k } \)

Find the length of the perpendicular from the point (1, -2, 3) to the plane x - y + z = 5.

If the volume of a cube of side length x is v = x³. Find the rate of change of the volume with respect to x when x = 5 units.

Using the Rolle’s theorem, determine the values of x at which the tangent is parallel to the x -axis for the following functions:
f(x) = x² − x, x ∈ [0, 1]

Evaluate the following limit, if necessary use l ’Hôpital Rule
\(\underset { x\rightarrow 0 }{ lim } \frac { 1-cosx }{ { x }^{ 2 } } \)

Let us assume that the shape of a soap bubble is a sphere. Use linear approximation to approximate the increase in the surface area of a soap bubble as its radius increases from 5 cm to 5.2 cm. Also, calculate the percentage error.

Find differential dy for each of the following function
y = _ex^2_-5x+7 cos (x² - 1)

In each of the following cases, determine whether the following function is homogeneous or not. If it is so, find the degree.
f(x, y) = x²y + 6x³ + 7

Evaluate the following definite integrals:
\(\int _{ 3 }^{ 4 }{ \frac { dx }{ { x }^{ 2 }-4 } } \)

Evaluate the following \(\int _{ 0 }^{ \pi /2 }{ { cos}^{ 7}x\quad dx } \)

Find the area of the region bounded by the curve y = sin x and the ordinate x=0 \(x=\frac { \pi }{ 3 } \) 

For each of the following differential equations, determine its order, degree (if exists)
\(\sqrt { \frac { dy }{ dx } } -4\frac { dy }{ dx } -7x=0\)

Express each of the following physical statements in the form of differential equation.
(i) Radium decays at a rate proportional to the amount Q present.
(ii) The population P of a city increases at a rate proportional to the product of population and to the difference between 5,00,000 and the population.
(iii) 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.
(iv) A saving amount pays 8% interest per year, compounded continuously. In addition, the income from another investment is credited to the amount continuously at the rate of Rs. 400 per year.

Find value of m so that the function y = e^mx is a solution of the given differential equation, y''− 5y' + 6y = 0

Find the differential equation of the family of parabolas y² = 4ax, where a is an arbitrary constant.

Three fair coins are tossed simultaneously. Find the probability mass function for number of heads occurred.

If the probability that a fluorescent light has a useful life of at least 600 hours is 0.9, find the probabilities that among 12 such lights
(i) exactly 10 will have a useful life of at least 600 hours;
(ii) at least 11 will have a useful life of at I least 600 hours;  
(iii) at least 2 will not have a useful life of at : least 600 hours.

Examine the binary operation (closure property) of the following operations on the respective sets (if it is not, make it binary)
a*b = a + 3ab − 5b²; ∀a,b∈Z

Write each of the following sentences in symbolic form using statement variables p and q.
(i) 19 is not a prime number and all the angles of a triangle are equal.
(ii) 19 is a prime number or all the angles of a triangle are not equal
(iii) 19 is a prime number and all the angles of a triangle are equal
(iv) 19 is not a prime number

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