Show that the relation R:{1, 2, 3}\(\rightarrow\){1, 2, 3} given by R={(1, 1), (2, 2), (3, 3), (1, 2), (2, 3)} is reflexive but neither symmetric nor transitive.

If \(A=\begin{bmatrix} cos\alpha - & sin\alpha \\ sin\alpha & cos\alpha \end{bmatrix}\), then for what value of \(\alpha \), A is an identify matrix?

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

Write the value of the area of the parallelogram determined by the vectors \(2\hat { i } \ and\ 3\hat { j } \)

Let f and g be real function be \(f(x)=\sqrt { x+4 } ,x\ge 4\) find the function fg, \(\frac { f }{ g } \)

Show that : \({ tan }^{ -1 }\left( \frac { 3a^{ 2 }x-{ x }^{ 3 } }{ { a }^{ 3 }-3a{ x }^{ 2 } } \right) =3tan^{ -1 }\left( \frac { x }{ a } \right) \)

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.

\(y={ tan }^{ -1 }\frac { 5x }{ 1-6{ x }^{ 2 } } \),\(-\frac { 1 }{ \sqrt { 6 } }  then prove that \(\frac { dy }{ dx } =\frac { 2 }{ 1+4{ x }^{ 2 } } +\frac { 3 }{ 1+9{ x }^{ 3 } } \) .

Use differentials find the approximate value of \((127)^{1/3}\) .

If A,B are symmetric matrices of same order, then AB-BA is a:
(A) Skew-symmetric matrix
(B) Symmetric matrix
(C) Zero matrix
(D) Identity matrix.

If \(y=log\left( x+\sqrt { { x }^{ 2 }+1 } \right) \)prove that \(\left( { x }^{ 2 }+1 \right) \frac { { d }^{ 2 }y }{ { dx }^{ 2 } } +x\frac { dy }{ dx } =0\)

Evaluate:\(\int _{ 0 }^{ \pi /2 }{ \frac { \sin ^{ 4 }{ x } }{ \sin ^{ 4 }{ x } +\cos ^{ 4 }{ x } } } dx.\)

Let f be a real valued function defined by f(x)=4x+3. Find the real valued function g such that gof=fog=I_R

Prove that \(tan^{-1}\left(cos\ x\over1+sin\ x\right)={\pi \over 4}x\epsilon \left(-{\pi\over2},{\pi\over2}\right)\)

If \(\left[ \begin{matrix} 2x & 3 \end{matrix} \right] \left[ \begin{matrix} 1 & 2 \\ -3 & 0 \end{matrix} \right] \left[ \begin{matrix} x \\ 3 \end{matrix} \right] =0\), then find x.

Two farmers X and Y cultivate only three varieties of rice namely Basmati, Permal and Naura. The sale in rupees of these varieties of rice by both the farmers in the months of September and October are given by the following matrices A and B.
September Sales in rupees
\(A=\left[ \begin{matrix} \overset { Basmati }{ 10,000 } & \overset { Permal }{ 20,000 } & \overset { Naura }{ 30,000 } \\ 50,000 & 30,000 & 10,000 \end{matrix} \right] \begin{matrix} X \\ Y \end{matrix}\)
October Sales in rupees
\(A=\left[ \begin{matrix} \overset { Basmati }{ 5,000 } & \overset { Permal }{ 10,000 } & \overset { Naura }{ 6,000 } \\ 20,000 & 10,000 & 10,000 \end{matrix} \right] \begin{matrix} X \\ Y \end{matrix}\)
Find :
(i) What were the combined sales is September and October for each farmer in each variety?
(ii) If both farmers decided to donate 2% of the gross rupees sales in October, for the welfare of their workers, compute the total amount paid by each farmer for the welfare of the warkers.
(iii) Which values are depicted in this problem?

Using the properties of determinants, prove the following:

\(\begin{vmatrix} x & x^{ 2 } & 1+px^{ 3 } \\ y & y^{ 2 } & 1+py^{ 3 } \\ z & z^{ 2 } & 1+pz^{ 3 } \end{vmatrix}=(1+pxyz)(x-y)(y-z)(z-x),\) where p is any scalar.

Find the value of \(\frac{dy}{dx}\) at \(\theta=\frac{\pi}{4}\), if \(x=ae^{ \theta }(\sin\theta-\cos\theta)\) and \(y=ae^{ \theta }(\sin\theta+\cos\theta)\)

The side of an equilateral triangle is increasing at the rate of 2 cm/s. At what rate is its area increasing when the side of the triangle is 20 cm?

Evaluate the integral: \(\int x^2\ tan^{-1}\ x\ dx\)

Solve the following differential equation: \(xy\quad log\left( \frac { y }{ x } \right) dx+\left( { y }^{ 2 }-{ x }^{ 2 }log\left( \frac { y }{ x } \right) \right) dy=0\)

Show that \(\left| \overrightarrow { a } \right| \overrightarrow { b } +\left| \overrightarrow { b } \right| \overrightarrow { a } \) is perpendicular to \(\left| \overrightarrow { a } \right| \overrightarrow { b } -\left| \overrightarrow { b } \right| \overrightarrow { a } \) for any two non-zero vectors \(\overrightarrow { a } \ and\ \overrightarrow { b } \)

Find the co-ordinates of the point where the line through the points A(3, 4, 1) and B(5, 1, 6) crosses the XY-plane.

A manufacturer considers that men and women workers are equally efficient and so he pays them at the same rate. He has 30 and 17 units of workers (male and female) and capital respectively, which he uses to produce two types of goods A and B. To produce one unit of A, 2 workers and 3 units of capital are required while 3 workers and 1 unit of capital is required to produce one unit of B. If A and B are priced at Rs. 100 and Rs. 120 per unit respectively, how should he use his resources to maximise the total revenue ? Form the above as an LPP and solve graphically. Do you agree with this view of the manufacturer that men and women workers are equally efficient and so should be paid at the same rate?

Let d₁,d₂,d₃ be three mutually exclusive diseases. Let S = [S₁, S₂,...,S ₆] be the set of observable symptoms of these diseases. For example, S₁is the shortness of breath, S₂ is the loss of weight, S₃ 3,500 with disease d₂ and 3,300 with disease d₂. Also, 3,100 patients with disease d₁, 3,300 with disease d₂ and 3,000 with disease d₃ . Show the symptoms S. Knowing that the patient has symptoms S, the doctor wishes to determine the patient's illness. On the basis of this informations, what should the doctor conclude?

Using integration find the area of the following region: 
\(\left\{ \left( x,y \right) :\left| x-1 \right| \le y\le \sqrt { 5-{ x }^{ 2 } } \right\} \)

If \(A=\left( \begin{matrix} 1 & 0 & 2 \\ 0 & 2 & 1 \\ 2 & 0 & 3 \end{matrix} \right) \) and A³ - 6A² + 7A + kI³ = 0, find k.

Using properties of determinants, prove that:
\(\left| \begin{matrix} { (y+z) }^{ 2 } & xy & zx \\ xy & { (x+z) }^{ 2 } & yz \\ xz & yz & { (x+y) }^{ 2 } \end{matrix} \right| =2xyz{ (x+y+z) }^{ 3 }.\)

Evaluate : \(\int _{ 1 }^{ 3 }{ \left( { 3x }^{ 2 }+1 \right) } dx\) by the method of limit of sums.

Show that the differential equation
\(x\frac { dy }{ dx } \sin { \left( \frac { y }{ x } \right) } +x-y\sin { \left( \frac { y }{ x } \right) } =0\)is homogeneous. Find particular solution of this differential equation, given that x = 1 when y = \(\frac { \pi }{ 2 } \).

Find the equation of the plane passing through the line of intersection of the planes x + y + z = 1 and 2x + 3y + 4z = 5 which is perpendicular to the plane x - y + z = 0. Also, find the distance of the plane obtained above from the origin.

If a young man rides his motor-cycle at 25km per hour, he had to spend Rs. 2 per km on petrol with very little pollution in the air. If he rides it at a faster speed of 40km per hour, the petrol cost increases to Rs. 5 per km and rate of pollution also increases. He has Rs. 100 to spend on petrol and wishes to find what is the maximum distance he can travel within one hour. Express this problem as an LPP. Solve it graphically to find the distance to be covered with different speeds. What value is indicated in this question?