Solve the following system of equations, using matrix inversion method:
2x₁ + 3x₂ + 3x₃ = 5, x₁ - 2x₂ + x₃ = -4, 3x₁ - x₂ - 2x₃ = 3.

If A = \(\left[ \begin{matrix} -4 & 4 & 4 \\ -7 & 1 & 3 \\ 5 & -3 & -1 \end{matrix} \right] \) and B = \(\left[ \begin{matrix} 1 & -1 & 1 \\ 1 & -2 & -2 \\ 2 & 1 & 3 \end{matrix} \right] \), find the products AB and BA and hence solve the system of equations x - y + z = 4, x - 2y - 2z = 9, 2x + y + 3z = 1.

Test for consistency and if possible, solve the following systems of equations by rank method.
2x - y + z = 2, 6x - 3y + 3z = 6, 4x - 2y + 2z = 4

Find the value of k for which the equations
kx - 2y + z = 1, x - 2ky + z = -2, x - 2y + kz = 1 have
(i) no solution
(ii) unique solution
(iii) infinitely many solution

Solve the equation z³+ 27 = 0

Solve the equation x³− 9x²+14x + 24 = 0 if it is given that two of its roots are in the ratio 3:2.

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

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

Find the equation of the ellipse whose eccentricity is \(\frac { 1 }{ 2 } \), one of the foci is(2, 3) and a directrix is x = 7. Also find the length of the major and minor axes of the ellipse.

Find the equations of tangents to the hyperbola \(\frac { { x }^{ 2 } }{ 16 } -\frac { { y }^{ 2 } }{ 64 } \) = 1 which are parallel to10x − 3y + 9 = 0.

At a water fountain, water attains a maximum height of 4 m at horizontal distance of 0.5 m from its origin. If the path of water is a parabola, find the height of water at a horizontal distance of 0.75 m from the point of origin.

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.

Find the vertex, focus, equation of directrix and length of the latus rectum of the following: y²−4y−8x+12 = 0

If \(\vec { a } =\vec { i } -\vec { j } ,\vec { b } =\hat { i } -\hat { j } -4\hat { k } ,\vec { c } =3\hat { j } -\hat { k } \) and \(\vec { d } =2\hat { i } +5\hat { j } +\hat { k } \)
(i) \((\vec { a } \times \vec { b } )\times (\vec { c } \times \vec { d } )=[\vec { a } ,\vec { b } ,\vec { d } ]\vec { c } -[\vec { a } ,\vec { b } ,\vec { c } ]\vec { d } \)

Salt is poured from a conveyer belt at a rate of 30 cubic metre per minute forming a conical pile with a circular base whose height and diameter of base are always equal. How fast is the height of the pile increasing when the pile is 10 metre high?

A conical water tank with vertex down of 12 metres height has a radius of 5 metres at the top. If water flows into the tank at a rate 10 cubic m/min, how fast is the depth of the water increases when the water is 8 metres deep?

A ladder 17 metre long is leaning against the wall. The base of the ladder is pulled away from the wall at a rate of 5 m/s. When the base of the ladder is 8 metres from the wall.
(i) How fast is the top of the ladder moving down the wall?
(ii) At what rate, the area of the triangle formed by the ladder, wall and the floor is changing?

Find the equations of the tangents to the curve y = 1 + x³ for which the tangent is orthogonal with the line x +12y = 12.

Find the equation of tangent and normal to the curve given by x = 7 cos t and y = 2sin t, t ∈ R at any point on the curve.

Write the Maclaurin series expansion of the following function:
cos² x

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

Discuss the monotonicity and local extrema of the function \(f(x)=log(1+x)-\frac{x}{1+x},x>-1\) and hence find the domain where, \(log(1+x)>\frac{x}{1+x}\)

Find the intervals of monotonicity and local extrema of the function \(f(x)=\frac{1}{1+x^{2}}\)

Evaluate the following integrals as the limits of sums.
\(\int _{ 1 }^{ 2 }{( 4x^2-1)dx } \)

Find the area of the region in the first quadrant bounded by the parabola y² = 4x, the line x + y = 3 and y-axis.