Find the inverse of each of the following by Gauss-Jordan method:
\(\left[ \begin{matrix} 1 & 2 & 3 \\ 2 & 5 & 3 \\ 1 & 0 & 8 \end{matrix} \right] \)

Determine k and solve the equation 2x³-6x²+3x+k = 0 if one of its roots is twice the sum of the other two roots.

For the ellipse 4x² + y² + 24x − 2y + 21 = 0, find the centre, vertices and the foci. Also prove that the length of latus rectum is 2  

Parabolic cable of a 60m portion of the roadbed of a suspension bridge are positioned as shown below. Vertical Cables are to be spaced every 6m along this portion of the roadbed. Calculate the lengths of first two of these vertical cables from the vertex.

Prove by vector method that the perpendiculars (attitudes) from the vertices to the opposite sides of a triangle are concurrent.

Prove by vector method that sin(α + β ) = sin α cos β + cos α sin β

A particle moves along a line according to the law s(t) = 2t³ − 9t² +12t − 4, where t ≥ 0.
(i) At what times the particle changes direction?
(ii) Find the total distance travelled by the particle in the first 4 seconds.
(iii) Find the particle’s acceleration each time the velocity is zero.

A particle moves along a line according to the law s(t) = 2t³ − 9t² +12t − 4, where t ≥ 0.

A garden is to be laid out in a rectangular area and protected by wire fence. What is the largest possible area of the fenced garden with 40 metres of wire.

Determine the intervals of concavity of the curve f (x) = (x −1)³. (x − 5), x∈R and, points of inflection if any.

Evaluate the following:
\(\int _{ 0 }^{ \frac { 1 }{ 2 } }{ \frac { { e }^{ { a\ sin }^{ -1x } }{ sin }^{ -1 }x }{ \sqrt { 1-{ x }^{ 2 } } } dx } \)

Father of a family wishes to divide his square field bounded by x = 0, x = 4, y = 4 and y = 0 along the curve y² = 4x and x² = 4y into three equal parts for his wife, daughter and son. Is it possible to divide? If so, find the area to be divided among them.

Find, by integration, the volume of the container which is in the shape of a right circular conical frustum.

Find the equation of the curve whose slope is \(\frac { y-1 }{ { x }^{ 2 }+x } \) and which passes through the point (1, 0).

Solve (x² -3y²) dx + 2xydy = 0.

Solve the following differential equations
(x²+y²)dy = xy dx. It is given that y(1) = 1 and y(x₀) = e. Find the value of x₀.

Solve the Linear differential equation:
\(\frac { dy }{ dx } +\frac { 3y }{ x } =\frac { 1 }{ { x }^{ 2 } } \), given that y = 2 when x = 1 

A pot of boiling water at 100^o C is removed from a stove at time t = 0 and left to cool in the kitchen. After 5 minutes, the water temperature has decreased to 80^o C , and another 5 minutes later it has dropped to 65^oC. Determine the temperature of the kitchen.

If X is the random variable with probability density function f(x) given by,
\(f(x)=\begin{cases} \begin{matrix} x+1 & -1\le x<0 \end{matrix} \\ \begin{matrix} -x+1 & 0\le x<1 \end{matrix} \\ \begin{matrix} 0 & otherwise \end{matrix} \end{cases}\) 
then find
(i) the distribution function F(x)
(ii) P( -0.5 ≤X ≤ 0.5)

If X is the random variable with probability density function f(x) given by,

\(f(x)=\begin{cases} \begin{matrix} x-1 & 1\le x<2 \end{matrix} \\ \begin{matrix} -x+3 & 2\le x<3 \end{matrix} \\ \begin{matrix} 0 & Otherwise \end{matrix} \end{cases}\) 
find
(i) the distribution function F(x)
(ii) P(1.5 ≤ X ≤ 2.5)

Two balls are chosen randomly from an urn containing 8 white and 4 black balls. Suppose that we win Rs. 20 for each black ball selected and we lose Rs. 10 for each white ball selected. Find the expected winning amount and variance 

Verify 
(i) closure property 
(ii) commutative property
(iii) associative property
(iv) existence of identity, and
(v) existence of inverse for following operation on the given set m*n = m + n - mn; m, n ∈Z

Verify 
(i) closure property 
(ii) commutative property 
(iii) associative property 
(iv) existence of identity and
(v) existence of inverse for the operation +₅ on Z₅ using table corresponding to addition modulo 5.

Let M = \(\left\{ \left( \begin{matrix} x & x \\ x & x \end{matrix} \right) :x\in R-\{ 0\} \right\} \) and let * be the matrix multiplication. Determine whether M is closed under ∗. If so, examine the commutative and associative properties satisfied by ∗ on M.

Let A be Q\{1}. Define ∗ on A by x*y = x + y − xy. Is ∗ binary on A? If so, examine the commutative and associative properties satisfied by ∗ on A.