Prove that \(\left[ \begin{matrix} \cos { \theta } & -\sin { \theta } \\ \sin { \theta } & \cos { \theta } \end{matrix} \right] \) is orthogonal.

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.

Solve the following system of homogenous equations.
2x + 3y − z = 0, x − y − 2z = 0, 3x + y + 3z = 0

Evaluate the following if z = 5−2i and w = −1+3i
z + w

If the area of the triangle formed by the vertices z, iz and z + iz is 50 square units, find the value of |z|

Represent the complex number −1−i

Evaluate the following if z = 5−2i and w = −1+3i
z w

Obtain the Cartesian form of the locus of z = x + iy in each of the following cases:
\(\overline { z } =z^{ -1 }\)

If α, β, γ  and \(\delta\) are the roots of the polynomial equation 2x⁴ + 5x³ − 7x² + 8 = 0, find a quadratic equation with integer coefficients whose roots are α + β + γ + \(\delta\) and αβ૪\(\delta\).

Solve: \(2\sqrt { \frac { x }{ a } } +3\sqrt { \frac { a }{ x } } =\frac { b }{ a } +\frac { 6a }{ b } \)

Find all values of x such that -6\(\pi\le x \le 6\pi\) and cos x = 0 

Simplify \({ cos }^{ -1 }\left( cos\left( \frac { 13\pi }{ 3 } \right) \right) \)

Find the period and amplitude of y = 4sin(−2x)

Find the principal value of \({sin }^{ -1 }\left( sin\left( -\frac { \pi }{ 3 } \right) \right) \)

If y = 4x + c is a tangent to the circle x² + y² = 9, find c 

Find centre and radius of the following circles.
x²+y²−x+2y−3 = 0

Show that the vectors \(\hat { i } +\hat { 2j } -\hat { 3k } \), \(\hat { 2i } -\hat { j } +\hat { 2k } \) and \(\hat { 3i } +\hat { j } -\hat { k } \)

Show that the points (2, 3, 4),(−1, 4, 5) and (8,1, 2) are collinear.

Find the acute angle between the planes \(\vec { r } .(2\hat { i } +2\hat { j } +2\hat { k } )\) and 4x-2y+2z = 15.

Find the angle of intersection of the curve y = sin x with the positive x -axis.

Using the Rolle’s theorem, determine the values of x at which the tangent is parallel to the x -axis for the following functions:
\(f(x)=\sqrt{x}-\frac{x}{3}, x\in [0,9]\)

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

Evaluate :\(\int _{ 0 }^{ 1 }{ [2x] } dx\) where [⋅] is the greatest integer function

Evaluate the following:
\(\int _{ 0 }^{ \infty }{ { x }^{ 5 }{ e }^{ -3x }dx } \)

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

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

Show that y = mx + \(\frac{7}{m}\), m ≠ 0 is a solution of the differential equation xy'+7\(\frac{1}{y'}\)-y = 0.

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 commuter train arrives punctually at a station every half hour. Each morning, a student leaves his house to the train station.Let X denote- the amount of time, in minutes that the student waits for the train from the time he reaches the train station. It is known  that the pdf of X is 
\(f(x)= \begin{cases}\frac{1}{30} & 0

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'.

Determine the truth value of each of the following statements
(i) If 6 + 2 = 5 , then the milk is white.
(ii) China is in Europe or \(\sqrt3\) is an integer
(iii) It is not true that 5 + 5 = 9 or Earth is a planet
(iv) 11 is a prime number and all the sides of a rectangle are equal

Construct the truth table for the following statements.
( p V q) V ¬q