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 by minor method:
\(\left[ \begin{matrix} 1 \\ 3 \end{matrix}\begin{matrix} -2 \\ -6 \end{matrix}\begin{matrix} -1 \\ -3 \end{matrix}\begin{matrix} 0 \\ 1 \end{matrix} \right] \)

Find the rank of the following matrices by minor method:
\(\left[ \begin{matrix} 1 & -2 & 3 \\ 2 & 4 & -6 \\ 5 & 1 & -1 \end{matrix} \right] \)

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

Which one of the points i, −2 + i, and 3 is farthest from the origin?

Represent the complex number −1−i

Find the square roots of −6+8i

Show that the following equations represent a circle, and, find its centre and radius
\(\left| 2z+2-4i \right| =2\)

Find the modulus and principal argument of the following complex numbers.
\(\sqrt { 3 } \)-i

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

Show that the polynomial 9x⁹+ 2x⁵- x⁴- 7x²+ 2 has at least six imaginary roots.

Find the period and amplitude of y = sin 7x

Show that cot⁻¹\(\left( \frac { 1 }{ \sqrt { { x }^{ 2 }-1 } } \right) ={ sec }^{ -1 }x,|x|>1\)

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

Find all values of x such that
-5\(\pi\le x \le 5\pi\) and cos x =1

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

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 .

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

Identify the type of conic section for each of the equations.
2x² − y² = 7

Find the length of the tangent from (2, -3) to the circle x² + y² - 8x - 9y + 12 = 0.

Find the volume of the parallelepiped whose coterminus edges are given by the vectors \(\hat { 2i } -\hat { 3j } +\hat { 4k } \), \(\hat { i } +\hat { 2j } -\hat { k } \) and \(\hat {3 i } -\hat { j } +\hat { 2k } \)

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

A person learnt 100 words for an English test. The number of words the person remembers in t days after learning is given by W(t) = 100 × (1− 0.1t)², 0 ≤ t ≤ 10. What is the rate at which the person forgets the words 2 days after learning?

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

Find two positive numbers whose sum is 12 and their product is maximum.

Evaluate  \(\underset{x\rightarrow 1}{lim}(\frac{x^{2}-3x+2}{x^{2}-4x+3})\).

Find the asymptotes of the curve \(f(x)=\frac { { 2x }^{ 2 }-8 }{ { x }^{ 2 }-16 } \)

Find the equation of the tangent to the curve y²=4x+5 and which is parallel to y=2x+7

Evaluate the following limits, if necessary using L’Hopitalrule
(i) \(\underset { x\rightarrow 2 }{ lim } \cfrac { sin\pi x }{ 2-x } \) 
(ii) \(\cfrac { lim }{ x\rightarrow 2 } \cfrac { { x }^{ n }-{ a }^{ n } }{ x-2 } \) 
(iii) \(\underset { x\rightarrow \infty }{ lim } \cfrac { sin\frac { 2 }{ x } }{ \frac { 1 }{ x } } \)
(iv) \(\underset { x\rightarrow \infty }{ lim } \cfrac { { x }^{ 2 } }{ { e }^{ x } } \)

If the radius of a sphere, with radius 10 cm, has to decrease by 0 1. cm, approximately how much will its volume decrease?

Find differential dy for each of the following function
y = (3 + sin(2x)) ^2/3 

Evaluate \(\int _{ 0 }^{ 1 }{ \left( \frac { { e }^{ 5logx }-{ e }^{ 4logx } }{ { e }^{ 3logx }-{ e }^{ 2logx } } \right) } \)

Find the slope of the tangent to the curve \(y=\int _{ 0 }^{ x }{ \frac { dt }{ 1+{ t }^{ 3 } } stx=1 } \) 

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

Express each of the following physical statements in the form of differential equation.
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.

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

Determine the order and degree of \(\frac { \left[ 1+\left( \frac { dy }{ dx } \right) ^{ 2 } \right] ^{ \frac { 3 }{ 2 } } }{ \frac { { d }^{ 2 }y }{ { dx }^{ 2 } } } =k\)

Find the order and degree of \(\left( \frac { { d }^{ 2 }y }{ { dx }^{ 2 } } \right) ^{ 2 }+cos\left( \frac { dy }{ dx } \right) =0\)

An urn contains 5 mangoes and 4 apples. Three fruits are taken at randaom. If the number of apples taken is a random variable, then find the values of the random variable and number of points in its inverse images.

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

Let A =\(\begin{bmatrix} 0 & 1 \\ 1 & 1 \end{bmatrix},B=\begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}\)be any two boolean matrices of the same type. Find AvB and A\(\wedge\)B.

Determine whether ∗ is a binary operation on the sets given below.
a*b = min (a, b) on A = {1, 2, 3, 4, 5}

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