If A = \(\frac { 1 }{ 7 } \left[ \begin{matrix} 6 & -3 & a \\ b & -2 & 6 \\ 2 & c & 3 \end{matrix} \right] \) is orthogonal, find a, b and c , and hence A⁻¹.

In a T20 match, a team needed just 6 runs to win with 1 ball left to go in the last over. The last ball was bowled and the batsman at the crease hit it high up. The ball traversed along a path in a vertical plane and the equation of the path is y = ax² + bx + c with respect to a xy-coordinate system in the vertical plane and the ball traversed through the points (10, 8), (20, 16) (40, 22) can you conclude that the team won the match?
Justify your answer. (All distances are measured in metres and the meeting point of the plane of the path with the farthest boundary line is (70, 0).)

Test for consistency of the following system of linear equations and if possible solve:
4x − 2y + 6z = 8, x + y − 3z = −1, 15x − 3y + 9z = 21.

Investigate the values of λ and μ the system of linear equations 2x + 3y + 5z = 9, 7x + 3y - 5z = 8, 2x + 3y + λz = μ, have
(i) no solution
(ii) a unique solution
(iii) an infinite number of solutions.

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

Show that \(\left( \frac { \sqrt { 3 } }{ 2 } +\frac { i }{ 2 } \right) ^{ 5 }+\left( \frac { \sqrt { 3 } }{ 2 } -\frac { i }{ 2 } \right) ^{ 5 }=-\sqrt { 3 } \)

Simplify: \(\left( -\sqrt { 3 } +3i \right) ^{ 31 }\)

Solve the equation (x-2) (x-7) (x-3) (x+2)+19 = 0

Solve: (x - 4)(x - 7)(x - 2)(x + 1) = 16

Solve \(tan^{ -1 }\left( \frac { x-1 }{ x-2 } \right) +tan^{ -1 }\left( \frac { x+1 }{ x+2 } \right) =\frac { \pi }{ 4 } \)

Find the vertex, focus, directrix, and length of the latus rectum of the parabola x²−4x−5y−1 = 0.

The maximum and minimum distances of the Earth from the Sun respectively are 152 × 10⁶ km and 94.5 × 10⁶ km. The Sun is at one focus of the elliptical orbit. Find the distance from the Sun to the other focus.

An engineer designs a satellite dish with a parabolic cross section. The dish is 5 m wide at the opening, and the focus is placed 1.2 m from the vertex
(a) Position a coordinate system with the origin at the vertex and the x -axis on the parabola’s axis of symmetry and find an equation of the parabola.
(b) Find the depth of the satellite dish at the vertex.

On lighting a rocket cracker it gets projected in a parabolic path and reaches a maximum height of 4 m when it is 6 m away from the point of projection. Finally it reaches the ground 12 m away from the starting point. Find the angle of projection.

With usual notations, in any triangle ABC, prove by vector method that \(\frac { a }{ sinA } =\frac { b }{ sinB }=\frac { c }{ sinc }\)

Find the point of intersection of the lines \(\frac { x-1 }{ 2 } =\frac { y-2 }{ 3 } =\frac { z-3 }{ 4 } \) and \(\frac { x-4 }{ 5 } =\frac { y-1 }{ 2 } =z\)

Find the non-parametric form of vector equation, and Cartesian equations of the plane passing through the points (2, 2, 1), (9, 3, 6) and perpendicular to the plane 2x + 6y + 6z = 9

Show that the lines \(\vec { r } =(\hat {- i } -3\hat { j } -5\hat { k } )+s(3\hat { i } +5\hat { j } +7\hat { k } )\) and \(\vec { r } =(2\hat { i } +4\hat { j } +6\hat { k } )+t(\hat { i } +4\hat { j } +7\hat { k } )\) are coplanar. Also, find the non-parametric form of vector equation of the plane containing these lines

A road running north to south crosses a road going east to west at the point P. Car A is driving north along the first road, and car B is driving east along the second road. At a particular time car A 10 kilometres to the north of P and traveling at 80 km/hr, while car B is 15 kilometres to the east of P and traveling at 100 km/hr. How fast is the distance between the two cars changing?

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

Prove that the ellipse x² + 4y² = 8 and the hyperbola x²-2y² = 4 intersect orthogonally.

Find the intervals of monotonicities and hence find the local extremum for the following function:
\(f(x)=\frac { { e }^{ x } }{ 1-{ e }^{ x } } \)

Find the smallest possible value x²+y² given that x +y = 10.

Find the dimensions of the largest rectangle that can be inscribed in a semi circle of radius r cm.

Find the asymptotes of the following curve \(f(x)=\frac { { x }^{ 2 }-6x-1 }{ x+3 } \)

Sketch the graphs of the following function 
\(y=\frac { 1 }{ 1+{ e }^{ -x } } \)

For the function f{x) = 4x³ + 3x² - 6x + 1 find the intervals of monotonicity, local extrema, intervals of concavity and points of inflection.

Find the local maximum and minimum of the function x² y² on the line x + y = 10

Let w(x, y) = xy+\(\frac { { e }^{ y } }{ { y }^{ 2 }+1 } \) for all (x, y) ∈ R². Calculate \(\frac { { \partial }^{ 2 }w }{ { \partial y\partial x } } \) and \(\frac { { \partial }^{ 2 }w }{ { \partial x\partial y } } \)

If z(x, y) = x tan-¹ (xy), x = t², y = se^t, s, t ∈ R. Find \(\frac { \partial z }{ \partial s } \) and \(\frac { \partial z }{ \partial t } \) at s = t = 1

Evaluate the following integrals as the limits of sums.
\(\int _{ 0 }^{ 1 }{ (5x+4)dx } \)

Prove that \(\int ^\frac {\pi}{4}_{0} \frac{dx}{a^2 sin^2 x+b^2 cos^2 x}\) = \(\frac{1}{ab} tan^{-1} (\frac{a}{b})\) where a, b > 0

Evaluate the following integrals using properties of integration:
\(\int _{ 0 }^{ 1 }{ \frac { log(1+x) }{ 1+{ x }^{ 2 } } } dx\)

Evaluate the following:
\(\int _{ 0 }^{ \frac { \pi }{ 2 } }{ { x }^{ 2 }cos2x\ dx } \)

Using integration, find the area of the region which is bounded by x-axis, the tangent and normal to the circle x² +  y² = 4 drawn at (1, \(\sqrt 3\))

Find the area of the region common to the circle x² +  y² = 16 and the parabola y² = 6x.

Solve \(\frac { dy }{ dx } =\frac { x-y+5 }{ 2(x-y)+7 } .\)

Solve the differential equation (y²-2xy) dx = (x²-2xy) dy

Solve the Linear differential equation:
(2x- 10y³) dy + ydx = 0

The rate of increase in the number of bacteria in a certain bacteria culture is proportional to the number present. Given that the number triples in 5 hours, find how many bacteria will be present after 10 hours?

A tank initially contains 50 litres of pure water. Starting at time t = 0 a brine containing with 2 grams of dissolved salt per litre flows into the tank at the rate of 3 litres per minute. The mixture is kept uniform by stirring and the well-stirred mixture simultaneously flows out of the tank at the same rate. Find the amount of salt present in the tank at any time t > 0.

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)

The mean and standard deviation of a binomial variate X are respectively 6 and 2.
Find
(i) the probability mass function
(ii) P(X = 3)
(iii) P(X\(\ge \)2).

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 the operation +₅ on Z₅ using table corresponding to addition modulo 5.

Using truth table check whether the statements ¬(p V q) V (¬p ∧ q) and ¬p are logically equivalent.