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.

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).)

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 systems of linear equations by Cramer’s rule:
 3x + 3y − z = 11, 2x − y + 2z = 9, 4x + 3y + 2z = 25.

For what value of λ, the system of equations x + y + z = 1, x + 2y + 4z = λ, x + 4y + 10z = λ² is consistent.

Solve the equation z³+ 8i = 0, where \(z \in \mathbb{C}\)

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.

If c ≠ 0 and \(\frac { p }{ 2x } =\frac { a }{ x+x } +\frac { b }{ x-c } \) has two equal roots, then find p. 

Find the domain of the following functions
(i) f(x) = sin^-1(2x - 3)
(ii) f(x) = sin^-1x + cos x

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

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.

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

The girder of a railway bridge is a parabola with its vertex at the highest point 15 m above the ends. If the span is 120 m, find the height of the bridge at 24 m from the middle point.

A kho-kho player In a practice session while running realises that the sum of tne distances from the two kho-kho poles from him is always 8m. Find the equation of the path traced by him of the distance between the poles is 6m.

Find the non-parametric form of vector equation of the plane passing through the point (1, −2, 4) and perpendicular to the plane x + 2y −3z = 11 and parallel to the line \(\frac { x+7 }{ 3 } =\frac { y+3 }{ -1 } =\frac { z }{ 1 } \)

Show that the points A, B, C with position vector \(2\overset { \wedge }{ i } -\overset { \wedge }{ j } +\overset { \wedge }{ k } ,\overset { \wedge }{ i } -3\overset { \wedge }{ j } -5\overset { \wedge }{ k } \) and \(3\overset { \wedge }{ i } -4\overset { \wedge }{ j } +4\overset { \wedge }{ k } \) respectively are the vector of a right angled, triangle. Also, find the remaining angles of the triangle.

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points on the plane

Using differential find the approximate value of cos 61; if it is given that sin 60° = 0.86603 and 1⁰ = 0.01745 radians.

Prove that \(\int _{ 0 }^{ \frac { \pi }{ 2 } }{ sin2x\ log(tan\ x)dx } =0\)

A population grows at the rate of 2% per year. How long does it take for the population to double?

Verify (p ∧ ~p) ∧ (~q ∧ p) is a tautlogy, contradiction or contingency.