If A = \(\left[ \begin{matrix} 8 & -4 \\ -5 & 3 \end{matrix} \right] \), verify that A(adj A) = (adj A)A = |A|I₂.

Find the rank of the matrix \(\left[ \begin{matrix} 2 \\ \begin{matrix} -3 \\ 6 \end{matrix} \end{matrix}\begin{matrix} -2 \\ \begin{matrix} 4 \\ 2 \end{matrix} \end{matrix}\begin{matrix} 4 \\ \begin{matrix} -2 \\ -1 \end{matrix} \end{matrix}\begin{matrix} 3 \\ \begin{matrix} -1 \\ 7 \end{matrix} \end{matrix} \right] \) by reducing it to an echelon form.

Solve the following system of linear equations, by Gaussian elimination method : 4x + 3y + 6z = 25, x + 5y + 7z = 13, 2x + 9y + z = 1.

Find the values of the real numbers x and y, if the complex numbers (3−i)x−(2−i)y+2i +5 and 2x+(−1+2i)y+3+ 2i are equal.

If \(\frac { 1+z }{ 1-z } =cos2\theta +isin2\theta \), show that z = i tan\(\theta\)

If z₁ = 3, z₂ = -7i, and z₃ = 5 + 4i, show that (z₁ + z₂)z₃ = z₁z₃ + z₂ + z₃

If α and β are the roots of the quadratic equation 2x²−7x+13 = 0 , construct a quadratic equation whose roots are α² and β².

Find the roots of 2x³ + 3x² + 2x + 3 = 0

Find the value of tan⁻¹(−1 ) + cos^-1\((\frac{1}{2})+sin^-1(-\frac{1}{2})\)

Find the value of
\(cos\left( { sin }^{ -1 }\left( \frac { 4 }{ 5 } \right) -{ tan }^{ -1 }\left( \frac { 3 }{ 4 } \right) \right) \)

Find the equation of the circle described on the chord 3x + y + 5 = 0 of the circle x² + y² = 16 as diameter.

Find the equation of the hyperbola in each of the cases given below:
passing through (5, −2) and length of the transverse axis along x axis and of length 8 units.

A search light has a parabolic reflector (has a cross-section that forms a ‘bowl’). The parabolic bowl is 40 cm wide from rim to rim and 30 cm deep. The bulb is located at the focus.
(1) What is the equation of the parabola used for reflector?
(2) How far from the vertex is the bulb to be placed so that the maximum distance covered?

With usual notations, in any triangle ABC, prove the following by vector method.
(i) a = b cos C + c cos B
(ii) b = c cos A + a cos C
(iii) c = a cos B + b cos A

Prove by vector method that the diagonals of a rhombus bisect each other at right angles.

For any four vectors \(\vec { a } ,\vec { b } ,\vec { c } ,\vec { d } \) we have \((\vec { a } \times \vec { b } )\times (\vec { c } \times \vec { d } )=[\vec { a } ,\vec { b } ,\vec { d } ]\vec { c } -[\vec { a } ,\vec { b } ,\vec { c } ]\vec { d } =[\vec { a } ,\vec { c } ,\vec { d } ]\vec { b } -[\vec { b } ,\vec { c } ,\vec { d } ]\vec { a } \)

If the straight lines \(\frac { x-5 }{ 5m+2 } =\frac { 2-y }{ 5 } =\frac { 1-z }{ -1 } \) and \(x=\frac { 2y+1 }{ 4m } =\frac { 1-z }{ -3 } \) are perpendicular to each other, find the value of m.

Find the equation of the plane which passes through the point (3, 4, -1) and is parallel to the plane 2x - 3y + 5z = 0. Also, find the distance between the two planes.

If the mass m(x) (in kilograms) of a thin rod of length x (in metres) is given by, m(x) = \(\sqrt { 3 } x\) then what is the rate of change of mass with respect to the length when it is x = 3 and x = 27 metres.

Find the absolute extrema of the following functions on the given closed interval.
\(f(x)=6x^{ \frac { 3 }{ 4 } }-3x^{ \frac { 1 }{ 3 } };\left[ -1,1 \right] \)

A race car driver is racing at 20^th km. If his speed never exceeds 150 km/hr, what is the maximum kilometer he can reach in the next two hours.

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

The radius of a circular plate is measured as 12.65 cm instead of the actual length 12.5 cm. find the following in calculating the area of the circular plate:
Absolute error

A coat of paint of thickness 0.2 cm is applied to the faces of a cube whose edge is 10 cm. Use the differentials to find approximately how many cubic centimeters of paint is used to paint this cube. Also calculate the exact amount of paint used to paint this cube.

A firm produces two types of calculators each week, x number of type A and y number of type B. The weekly revenue and cost functions (in rupees) are R(x, y) = 80x + 90y + 0.04xy − 0.05x² − 0.05y² and C(x, y) = 8x + 6y + 2000 respectively
(i) Find the profit function P(x, y) 
(ii) Find \(\frac { { \partial P } }{ \partial { x } } \) (1200, 1800) and \(\frac { \partial v }{ \partial y} \) (1200, 1800)

Show that \(\int ^\frac{2\pi}{0}_{0}\) g(cos x)dx = 2 \(\int ^{\pi}_{0}\) g(cosx)dx where g(cos x) is a function of cos x

Evaluate: \(\int _{ 0 }^{ 2\pi }{ { x }^{ 2 }sin\ nx\ dx } \) where n is a positive integer.

Find the volume of the solid formed by revolving the region bounded by the ellipse \(\frac { { x }^{ 2 } }{ { a }^{ 2 } } +\frac { { y }^{ 2 } }{ { b }^{ 2 } } =1\), a>b about the major axis.

Show that y = a cos(log x) + bsin (log x), x > 0 is a solution of the differential equation x² y" + xy'+y = 0.

Solve \({ y }^{ 2 }+{ x }^{ 2 }\frac { dy }{ dx } =xy\frac { dy }{ dx } \)

Suppose a discrete random variable can only take the values 0, 1, and 2. The probability mass function is defined by
\(\\ \\ \\ \\ \\ f(x)=\begin{cases} \begin{matrix} \frac { { x }^{ 2 }+1 }{ k } & forx=0,1,2 \end{matrix} \\ \begin{matrix} 0 & otherwise \end{matrix} \end{cases}\\ \\ \\ \\ \\ \\ \)
Find
(i) the value of k
(ii) cumulative distribution function
(iii) P(X ≥ 1).

A lottery with 600 tickets gives one prize of Rs. 200, four prizes of Rs.100, and six prizes of Rs. 50. If the ticket costs is Rs. 2, find the expected winning amount of a ticket.

If X is the random variable with distribution function F(x) given by,
\(F(x)=\begin{cases} \begin{matrix} 0 & x<0 \end{matrix} \\ \begin{matrix} x & 0\le x<1 \end{matrix} \\ \begin{matrix} 1 & 1\le x \end{matrix} \end{cases}\) 
then find
(i) the probability density function f(x)
(ii) P(0.2 ≤ X ≤ 0.7)

Consider p→q : If today is Monday, then 4 + 4 = 8.

Prove that q ➝ p ≡ ¬p ➝ ¬q