If A = \(\left[ \begin{matrix} a & b \\ c & d \end{matrix} \right] \) is non-singular, find A⁻¹.

Find the rank of the matrix \(\left[ \begin{matrix} 1 & 2 & 3 \\ 2 & 1 & 4 \\ 3 & 0 & 5 \end{matrix} \right] \) by reducing it to a row-echelon form.

Find the following \(\left| \frac { 2+i }{ -1+2i } \right| \) 

If \(\omega \neq 1\) is a cube root of unity, then the show that \(\cfrac { a+b\omega +c{ \omega }^{ 2 } }{ b+c\omega +{ a\omega }^{ 2 } } +\cfrac { a+b\omega +{ c\omega }^{ 2 } }{ c+a\omega +b{ \omega }^{ 2 } } =-1\)

Find the square roots of −6+8i

Write in polar form of the following complex numbers
\(3-i\sqrt { 3 } \)

If α, β, and γ are the roots of the equation x³ + px² + qx + r = 0, find the value of  \(\Sigma \frac { 1 }{ \beta \gamma } \) in terms of the coefficients.

A 12 metre tall tree was broken into two parts. It was found that the height of the part which was left standing was the cube root of the length of the part that was cut away. Formulate this into a mathematical problem to find the height of the part which was left standing.

If x²+2(k+2)x+9k = 0 has equal roots, find k.

Solve: (2x-1) (x+3) (x-2) (2x+3)+20 = 0

Examine for the rational roots of x⁸- 3x + 1 = 0

Find the principal value of sin^-1\(\left( -\frac { 1 }{ 2 } \right) \)(in radians and degrees).

Solve tan^-1\(\left( \frac { 1-x }{ 1+x } \right) =\frac { 1 }{ 2 } { tan }^{ -1 }\) x for x > 0

Prove that 
\({ tan }^{ -1 }(\frac { 2 }{ 11 }) +{ tan }^{ -1 }(\frac { 7 }{ 24 }) ={ tan }^{ -1 }(\frac { 1 }{ 2 } )\)

Find the principal value of \({sin }^{ -1 }\left( sin\left( \frac { 5\pi }{ 6 } \right) \right) \)

Find the general equation of a circle with centre (-3, -4) and radius 3 units.

Examine the position of the point (2, 3) with respect to the circle x² + y² − 6x − 8y + 12 = 0.

Obtain the equation of the circles with radius 5 cm and touching x-axis at the origin in general form.

If y = 2\(\sqrt2\)x + c is a tangent to the circle x² + y² = 16, find the value of c.

Identify the type of conic section for each of the equations.
3x²+3y²−4x+3y+10 = 0

If \(\vec { a } =\hat { i } -2\hat { j } +3\hat { k }, \vec { b } =2\hat { i } +\hat { j } -2\hat { k }, \vec { c } =3\hat { i } +2\hat { j } +\hat { k } \)  find \(\vec { a } .(\vec { b } \times \vec { c } )\).

If \(\vec { a } =2\hat { i } +3\hat { j } -\hat { k } ,\vec { b } =3\hat { i } +5\hat { j } +2\hat { k } ,\vec { c } =-\hat { i } -2\hat { j } +3\hat { k } \), verify that
(i) \((\vec { a } \times \vec { b } )\times \vec { c } =(\vec { a } .\vec { c } )\times \vec { b } -(\vec { b } .\vec { c } )\vec { a } \)
(ii) \(\vec { a } \times (\vec { b } \times \vec { c } )=(\vec { a } .\vec { c } )\times \vec { b } -(\vec { a } .\vec { b } )\vec { c } \)

Find the slope of the tangent to the following curves at the respective given points
y = x⁴ + 2x² − x at x = 1

Evaluate the following limit, if necessary use  l ’Hôpital Rule
\(\underset { x\rightarrow \infty }{ lim } \frac { { 2x }^{ 2 }-3 }{ { x }^{ 2 }-5x+3 } \)

Find the partial derivatives of the following functions at the indicated point
g(x, y) = 3x² + y² + 5x + 2, (1, -2)