Given A = \(\left[ \begin{matrix} 1 & -1 \\ 2 & 0 \end{matrix} \right] \), B = \(\left[ \begin{matrix} 3 & -2 \\ 1 & 1 \end{matrix} \right] \) and C = \(\left[ \begin{matrix} 1 & 1 \\ 2 & 2 \end{matrix} \right] \), find a matrix X such that A X B = C.

Find the rank of the following matrices by row reduction method:
\(\left[ \begin{matrix} 1 \\ \begin{matrix} 2 \\ 5 \end{matrix} \end{matrix}\begin{matrix} 1 \\ \begin{matrix} -1 \\ -1 \end{matrix} \end{matrix}\begin{matrix} 1 \\ \begin{matrix} 3 \\ 7 \end{matrix} \end{matrix}\begin{matrix} 3 \\ \begin{matrix} 4 \\ 11 \end{matrix} \end{matrix} \right] \) 

A chemist has one solution which is 50% acid and another solution which is 25% acid. How much each should be mixed to make 10 litres of a 40% acid solution? (Use Cramer’s rule to solve the problem).

Find the adjoint of the following:
\(\frac { 1 }{ 3 } \left[ \begin{matrix} 2 & 2 & 1 \\ -2 & 1 & 2 \\ 1 & -2 & 2 \end{matrix} \right] \)

The complex numbers u, v, and w are related by \(\frac { 1 }{ u } =\frac { 1 }{ v } +\frac { 1 }{ w } \) If v = 3−4i and w = 4+3i, find u in rectangular form.

If \(\left| z-\frac { 2 }{ z } \right| =2\) show that the greatest and least value of |z| are \(\sqrt { 3 } +1\) and \(\sqrt { 3 } -1\) respectively.

Show that the equation \({ z }^{ 3 }+2\bar { z } =0\) has five solutions

Show that \(\left( \frac { 19-7i }{ 9+i } \right) ^{ 12 }+\left( \frac { 20-5i }{ 7-6i } \right) ^{ 12 }\) is real

Obtain the Cartesian form of the locus of z = x + iy in each of the following cases:
 Im[(1−i)z+1] = 0

Obtain the Cartesian equation for the locus of z = x + iy in each of the following cases:
|z - 4|²- |z -1 |² = 16

Solve the equation 2x³+11x²−9x−18 = 0.

Solve the cubic equation : 2x³−x²−18x + 9 = 0 if sum of two of its roots vanishes.

Solve the cubic equations: 8x³ - 2x² - 7x + 3 = 0

For what value of x, the inequality \(\frac { \pi }{ 2 } <{ cos }^{ -1 }(3x-1)<\pi \) holds?

Find tan(tan^-1(2019))

Prove that \({ tan }^{ -1 }x+{ tan }^{ -1 }\frac { 2x }{ 1-{ x }^{ 2 } } ={ tan }^{ -1 }\frac { 3x-{ x }^{ 3 } }{ 1-{ 3x }^{ 2 } } ,|x|<\frac { 1 }{ \sqrt { 3 } } \)

Find the value of the expression in terms of x, with the help of a reference triangle.
tan\(\left( { sin }^{ -1 }\left( x+\frac { 1 }{ 2 } \right) \right) \)

Find the equations of the tangent and normal to the circle x² + y² = 25 at P(-3, 4).

A circle of area 9π square units has two of its diameters along the lines x + y = 5 and x−y = 1. Find the equation of the circle.

Find the equation of the hyperbola with vertices (0, ±4) and foci(0, ±6).

Find the equation of the hyperbola in each of the cases given below:
passing through (5, −2) and length of the transverse axis along x axis and of length 8 units.

Find the equation of the tangent to the parabola y² = 16x perpendicular to 2x + 2y + 3 = 0.

Find the equations of the tangent and normal to hyperbola 12x²−9y² = 108 at \(\theta =\frac { \pi }{ 3 } \) (Hint: use parametric form)

If D is the midpoint of the side BC of a triangle ABC, then show by vector method that \({ \left| \vec { AB } \right| }^{ 2 }+{ \left| \vec { AC } \right| }^{ 2 }=2({ \left| \vec { AD} \right| }^{ 2 }+{ \left| \vec { BD } \right| }^{ 2 })\)

The vector equation in parametric form of a line is \(\vec { r } =(3\hat { i } -2\hat { j } +6\hat { k } )+t(2\hat { i } -\hat { j } +3\hat { k } )\). Find
(i) the direction cosines of the straight line
(ii) vector equation in non-parametric form of the line
(iii) Cartesian equations of the line.