If A = \(\left[ \begin{matrix} 4 & 3 \\ 2 & 5 \end{matrix} \right] \), find x and y such that A² + xA + yI₂ = O₂. Hence, find A^-1.

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

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

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.

Determine the values of λ for which the following system of equations (3λ − 8)x + 3y + 3z = 0, 3x + (3λ − 8)y + 3z = 0, 3x + 3y + (3λ − 8)z = 0. has a non-trivial solution.

Solve the following system of linear equations by matrix inversion method:
x + y + z − 2 = 0, 6x − 4y + 5z − 31 = 0, 5x + 2y + 2z = 13.

Show that \(\left( \frac { 19+9i }{ 5-3i } \right) ^{ 15 }-\left( \frac { 8+i }{ 1+2i } \right) ^{ 15 }\) is purely imaginary.

If \(2cos\alpha=x+\frac { 1 }{ x } \) and \(2cos\ \beta =y+\frac { 1 }{ y } \), show that ​\({ x }^{ m }{ y }^{ n }+\frac { 1 }{ { x }^{ m }{ y }^{ n } } =2cos(m\alpha +n\beta )\)​

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

Solve : (x - 5) (x - 7) (x + 6) (x + 4) = 504

Find all zeros of the polynomial x⁶- 3x⁵- 5x⁴ + 22x³- 39x²- 39x + 135, if it is known that 1+2i and \(\sqrt{3}\) are two of its zeros.

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

 A road bridge over an irrigation canal have two semi circular vents each with a span of 20m and the supporting pillars of width 2m. Use Figure to write the equations that represents the semi-verticular vents

Find the equation of the circle through the points (1, 0),(-1, 0) , and (0, 1) 

Cross section of a Nuclear cooling tower is in the shape of a hyperbola with equation\(\frac { { x }^{ 2 } }{ { 30 }^{ 2 } } -\frac { { y }^{ 2 } }{ { 44 }^{ 2 } } =1\). The tower is 150 m tall and the distance from the top of the tower to the centre of the hyperbola is half the distance from the base of the tower to the centre of the hyperbola. Find the diameter of the top and base of the tower.
 

A rod of length 1.2 m moves with its ends always touching the coordinate axes. The locus of a point P on the rod, which is 0.3 m from the end in contact with x -axis is an ellipse. Find the eccentricity.

Using vector method, prove that cos(α − β ) = cos α cos β +sin α sin β

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) \((\vec { a } \times \vec { b } )\times (\vec { c } \times \vec { d } )=[\vec { a } ,\vec { b } ,\vec { d } ]\vec { c } -[\vec { a } ,\vec { b } ,\vec { c } ]\vec { d } \)

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

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 intervals of monotonicities and hence find the local extremum for the following function:
f(x) = 2x³+ 3x²-12x

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

We have a 12 square unit piece of thin material and want to make an open box by cutting small squares from the corners of our material and folding the sides up. The question is, which cut produces the box of maximum volume?

Evaluate the following:
\(\int _{ 0 }^{ 1 }{ \frac { sin(3{ tan }^{ -1 }x){ tan }^{ -1 }x }{ 1+{ x }^{ 2 } } } dx\)