Let * be a binary operation on N given by a * b = LCM(a,b) for all a,b \(\in \) N. Find 5*7

Use elementary column operation \({ C }_{ 2 }\rightarrow { C }_{ 2 }+2{ C }_{ 1 }\) in the following matrix equation :
\(\left( \begin{matrix} 4 & 2 \\ 3 & 3 \end{matrix} \right) =\left( \begin{matrix} 1 & 2 \\ 0 & 3 \end{matrix} \right) \left( \begin{matrix} 2 & 0 \\ 1 & 1 \end{matrix} \right) \)

Evaluate the integral: \(\int {(1-x)\sqrt x\ dx}\)

Find \(\lambda\) when the projection of \(\overrightarrow { a } =\lambda \hat { i } +\hat { j } +4\hat { k } \ on\ \overrightarrow { b } =2\hat { i } +6\hat { j } +3\hat { k } \) is 4 units.

Define Reflexive. Give one example.

Write in the simplest form: \(({ tan }^{ -1 }\left[ \frac { \sqrt { 1+sin\quad x } +{ \sqrt { 1-sin\quad x } }\quad }{ \sqrt { 1+sin\quad x } +{ \sqrt { 1-sin\quad x } } } \right] ,0<x<\frac { \pi }{ 2 } \)

If \(A=\left| \begin{matrix} 2 & 3 & -1 \\ 4 & 1 & 0 \\ 3 & 3 & 2 \end{matrix} \right| \) , find \(M_{ 12 }\times M_{ 21 }+C_{ 21 }\times C_{ 12 }\) when M_jj called minor and C_jj called co-factors of A.

If y = a^x +x^a+x^x+a^a, find dy/dx

The sides of an equilateral triangle are increasing at the rate of 2 cm/see, Find the rate at which its area increases, when side is 10 cm long.

Find the transpose of each of the following matrices:
\((i)\ \left[ \begin{matrix} 5 \\ \frac { 1 }{ 2 } \\ -1 \end{matrix} \right] \)
\((ii)\ \begin{bmatrix} 1 & -1 \\ 2 & 3 \end{bmatrix}\)
\((iii)\ \left[ \begin{matrix} -1 & 5 & 6 \\ \sqrt { 3 } & 5 & 6 \\ 2 & 3 & -1 \end{matrix} \right] .\)
 

\(If\ cos\ y=x\ cos\left( a+y \right) ,\ prove\ that\ \frac { dy }{ dx } =\frac { { cos }^{ 2 }\left( a+y \right) }{ sin\ y } \)

Evaluate \(\int _{ 0 }^{ 1 }{ \frac { \tan ^{ -1 }{ x } }{ 1+{ x }^{ 2 } } } dx.\)

Let set A be the set of people of different age groups associated with 'CLEANLINESS' derive, Let * be a binary operation o set A defined by a*b=older of(a,b),a,b\(\in\)A. Is * commutative, associative in A? Are you also a part of set A?

Prove that:

\({ sin }^{ -1 }\frac { 3 }{ 5 } +{ sin }^{ -1 }\frac { 8 }{ 17 } ={ sin }^{ -1 }\frac { 77 }{ 85 } \)

If \(A=\begin{bmatrix} 4 & 3 \\ 2 & 5 \end{bmatrix}\), find values of x and y such that A²- xA+yl = 0 where l is a \(2\times 2\) unit matrix and O is a \(2\times 2\) zero matrix.

If \(A=\left[ \begin{matrix} \frac { 2 }{ 3 } & 1 & \frac { 5 }{ 3 } \\ \frac { 1 }{ 3 } & \frac { 2 }{ 3 } & \frac { 4 }{ 3 } \\ \frac { 7 }{ 3 } & 2 & \frac { 2 }{ 3 } \end{matrix} \right] \)and \(B=\left[ \begin{matrix} \frac { 2 }{ 5 } & \frac { 3 }{ 5 } & 1 \\ \frac { 1 }{ 5 } & \frac { 2 }{ 5 } & \frac { 4 }{ 5 } \\ \frac { 7 }{ 5 } & \frac { 6 }{ 5 } & \frac { 2 }{ 5 } \end{matrix} \right] \) , then compute 3A-5B.

Find the inverse of the matrix \(A=\left[\begin{matrix}a&b\\c&{1+bc\over a}\end{matrix}\right]\) and show that : aA^-1 = (a²+bc+1)I-aA

Find the derivative of each of the following function w.r.t. x, or find \(\frac { dy }{ dx } \):

y=tan^-1\(\left( \frac { \sqrt { 1+sin\quad x } +\sqrt { 1-sin\quad x } }{ \sqrt { 1+sin\quad x } -\sqrt { 1-sin\quad x } } \right) \)

Find the points of local maxima and local minima, if any, of the function: f(x) = 3x⁴-2x³-6x²+6x+1. Also, find the local maximum and local minimum value.

Prove that: \(\int _{ 0 }^{ \pi /4 }{ (\sqrt { \tan { x } } +\sqrt { \cot { x } } ) } dx=\sqrt { 2 } \frac { \pi }{ 2 } .\)

Show that the differential equation x\(\frac {dy}{dx}\) sin\((\frac {x}{y})\)+ x - y sin = \((\frac {x}{y})\) = 0 is homogeneous. Find the particular solution of this differential equation, given that x =  0 when y = \(\frac{\pi}{2}\).

Find the position vectors of the point which divide the join of the points \(2\overrightarrow { a } -3\overrightarrow { b } \ and \ 3\overrightarrow { a } -2\overrightarrow { b } \) internally and externally in the ratio 2:3.

Find the value of \(\lambda \) for which the points with position vectors \(\hat { i } -\hat { j } +3\hat { k } 3\hat { i } +\lambda \hat { j } +3\hat { k } \) are equidistant from the plane \(\vec { r } .(5\hat { i } +2\hat { j } -7\hat { k } )+9=0\)

(Manufacturing Problem) A small firm manufactures gold rings and chains. The total number of rings and chains manufactured per day is atmost 24. It takes 1 hour to make a ring and 30 minutes to make a chain. The maximum number of hours available per day is 16. If the profit on a ring is Rs. 300 and that on a chain is Rs. 190, find the number of rings and chains that should be manufactured per day, so as to earn the maximum profit. Make it as an LPP and solve it graphically.

The probability of a student A passing an examination is \(3\over5\) and of student B is \(4\over5\).Assuming that the two events "A passes", "B passes" as independent. Find the probability of:
(i) Both the students passing the examination  
(ii) Only A passing the examination  
(iii) Only of them passing the examination  
(iv) none of them passing the examination.

Find the area of the region included between the parabola  \(y=\frac { 3 }{ 4 } { x }^{ 2 }\) and the line 3x - 2y + 12 =0.

Using elementary column operations, find the inverse of the following matrix :
\(\left[ \begin{matrix} -1 & 1 & 2 \\ 1 & 2 & 3 \\ 3 & 1 & 1 \end{matrix} \right] \)

Using matrices, solve the following system of linear equations:
2x + 3y + 3z = 5, x - 2y + z = -4, 3x - y - 2z = 3

Find : \(\int { \frac { \sqrt { x^{ 2 }+1 } \left\{ log({ x }^{ 2 }+1)-2logx \right\} }{ x^{ 4 } } } dx\)

Find the particular solution satisfying the given condition : \({ x }^{ 2 }dy+\left( xy+{ y }^{ 2 } \right) dx=0\); y = 1, when x = 1.

Find the vector and cartesian equations of the plane passing through the line of intersection of the planes. \(\vec { r } .\left( 2\hat { i } +2\hat { j } -3\hat { k } \right) =7,\quad \vec { r } .\left( 2\hat { i } +5\hat { j } +3\hat { k } \right) =9\)where x and z intercept are equal.

A dealer in rural area wishes to purchase a number of sewing machines. He has only Rs. 5,760 to invest and has space for at most 20 items for storage. An electronic sewing machine cost him Rs. 360 and a manually operated sewing machine Rs. 240. He can sell an electronic sewing machine at a profit of Rs. 22 and a manually operated sewing machine a profit of Rs. 18. Assuming that he can sell all the items that he can buy, how should he invest his money in order to maximize his profit? Make it as a LPP and solve it graphically.