Find the rank of the matrix \(\left[ \begin{matrix} 2 \\ \begin{matrix} -3 \\ 6 \end{matrix} \end{matrix}\begin{matrix} -2 \\ \begin{matrix} 4 \\ 2 \end{matrix} \end{matrix}\begin{matrix} 4 \\ \begin{matrix} -2 \\ -1 \end{matrix} \end{matrix}\begin{matrix} 3 \\ \begin{matrix} -1 \\ 7 \end{matrix} \end{matrix} \right] \) by reducing it to an echelon form.

Find the adjoint of the following:
\(\left[ \begin{matrix} 2 & 3 & 1 \\ 3 & 4 & 1 \\ 3 & 7 & 2 \end{matrix} \right] \)

If z_i = 2− i and z₂ = -4+3i , find the inverse of z₁z₂ and \(\frac { { z }_{ 1 } }{ { z }_{ 2 } } \)

If \(cos\alpha +cos\beta +cos\gamma =sin\alpha +sin\beta +sin\gamma =0\) then show that 
(i) \(cos3\alpha +cos3\beta +cos3\gamma =3cos(\alpha +\beta +\gamma )\)
(ii) \(sin3\alpha +sin3\beta +sin3\gamma +sin3\gamma =3sin\left( \alpha +\beta +\gamma \right) \)

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

If the equations x² + px + q = 0 and x² + p'x + q' = 0 have a common root, show that it must  be equal to \(\frac { pq'-p'q }{ q-q' } \) or \(\frac { q-q' }{ p'-p } \).

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

Find tan(tan^-1(2019))

Find the equation of the tangent and normal to the circle x²+y²−6x+6y−8 = 0 at (2, 2) .

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

With usual notations, in any triangle ABC, prove the following by vector method.
(i) a² = b² + c² − 2bc cos A
(ii) b² = c² + a² − 2ca cos B
(iii) c² = a² + b² − 2ab cos C

If \(\vec { a } =\hat { i } -\hat { k } ,\vec { b } =x\hat { i } +\hat { j } +(1-x)\hat { k } ,\vec { c } =y\hat { i } +x\hat { j } +(1+x+y)\hat { k } \) show that \([\vec { a } ,\vec { b } ,\vec { c } ]\) depends on neither x nor y.

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 
(i) \((\vec { a } \times \vec { b } )\times \vec { c } \)
(ii) \(\vec { a } \times (\vec { b } \times \vec { c } )\)

Find the direction cosines of the straight line passing through the points (5, 6, 7) and (7, 9, 13). Also, find the parametric form of vector equation and Cartesian equations of the straight line passing through two given points.

Find the equations of tangent and normal to the curve y = x² + 3x − 2 at the point (1, 2)

Find the values in the interval \((\frac{1}{2},2)\) satisfied by the Rolle's theorem for the function \(f(x)=x+\frac{1}{x}, x\in[\frac{1}{2},2]\)

Write down the Taylor series expansion, of the function log x about x =1 upto three nonzero terms for x > 0.

Use linear approximation to find an approximate value of \(\sqrt { 9.2 } \) without using a calculator.

A coat of paint of thickness 0.2 cm is applied to the faces of a cube whose edge is 10 cm. Use the differentials to find approximately how many cubic centimeters of paint is used to paint this cube. Also calculate the exact amount of paint used to paint this cube.

Let f (x, y) = 0 if xy ≠ 0 and f (x, y) = 1 if xy = 0.
Calculate: \(\frac { \partial f }{ \partial x } (0,0),\frac { \partial f }{ \partial y } (0,0).\)

For each of the following functions find the g_xy, g_xx, g_yy and g_yx. 
g(x, y) = x² + 3xy − 7y + cos(5x)

A firm produces two types of calculators each week, x number of type A and y number of type B. The weekly revenue and cost functions (in rupees) are R(x, y) = 80x + 90y + 0.04xy − 0.05x² − 0.05y² and C(x, y) = 8x + 6y + 2000 respectively

If f (x) = f (a + x), then \(\int _{ 0 }^{ 2a }{ f(x)dx=2\int _{ 0 }^{ a }{ f(x)dx } } \)

Evaluate the following definite integrals:
\(\int _{ 0 }^{ 1 }{ \sqrt { \frac { 1-x }{ 1+x } } } dx\)

If \(\int _{ 0 }^{ \infty }{ { e }^{ -a{ x }^{ 2 } }{ x }^{ 3 }dx=32,\alpha >0,\ find\ \alpha } \)