If A = \(\left[ \begin{matrix} 3 & 2 \\ 7 & 5 \end{matrix} \right] \) and B = \(\left[ \begin{matrix} -1 & -3 \\ 5 & 2 \end{matrix} \right] \), verify that (AB)^-1 = B^-1A^-1

Find the inverse of the non-singular matrix A = \(\left[ \begin{matrix} 0 & 5 \\ -1 & 6 \end{matrix} \right] \), by Gauss-Jordan method.

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

Find the values of the real numbers x and y, if the complex numbers (3−i)x−(2−i)y+2i +5 and 2x+(−1+2i)y+3+ 2i are equal.

If |z| = 1, show that \(2\le \left| { z }^{ 2 }-3 \right| \le 4\)

Find the quotient \(\frac { 2\left( cos\frac { 9\pi }{ 4 } +isin\frac { 9\pi }{ 4 } \right) }{ 4\left( cos\left( \frac { -3\pi }{ 2 } + \right) isin\left( \frac { -3\pi }{ 2 } \right) \right) } \) in rectangular form

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

Find the sum of squares of roots of the equation 2x⁴- 8x³+ 6x²-3 = 0.

It is known that the roots of the equation x³- 6x²- 4x + 24 = 0 are in arithmetic progression. Find its roots.

Sketch the graph of y = sin\((\frac{1}{3}x)\) for 0\(\le x <6\pi\).

Find the value of
\(tan\left( { cos }^{ -1 }\left( \frac { 1 }{ 2 } \right) -{ sin }^{ -1 }\left( -\frac { 1 }{ 2 } \right) \right) \)

Find the domain of the following functions
\(\frac{1}{2}tan^{-1}(1-x^2)-\frac{\pi}{4}\)

Find the equation of the circle described on the chord 3x + y + 5 = 0 of the circle x² + y² = 16 as diameter.

Find the equation of the ellipse in each of the cases given below:
length of latus rectum 4, distance between foci 4 \( \sqrt{ 2}\) , centre (0, 0) and major axis as y - axis.

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

Identify the type of conic and find centre, foci, vertices, and directrices of each of the following :
\(\frac { { \left( x+1 \right) }^{ 2 } }{ 100 } +\frac { { \left( y-2 \right) }^{ 2 } }{ 64 } =1\)

Prove by vector method that if a line is drawn from the centre of a circle to the midpoint of a chord, then the line is perpendicular to the chord.

Let \(\vec { a } =\hat { i } +\hat { j } +\hat { k } \),  \(\vec { b } =\hat { i } \)  and \(\vec { c } ={ c }_{ 1 }\hat { i } +{ c }_{ 2 }\hat { j } +{ c }_{ 3 }\hat { k } \). If \({ c }_{ 1 }=1\) and \({ c }_{ 2 }=2\), find \({ c }_{ 3 }\) such that \(\vec { a } ,\vec { b } \) and \(\vec { c } \) are coplanar.

Find the vector equation in parametric form and Cartesian equations of the line passing through (-4, 2, -3) and is parallel to the line  \(\frac { -x-2 }{ 4 } =\frac { y+3 }{ -2 } =\frac { 2z-6 }{ 3 } \)

Show that the straight lines x + 1=  2y = −12z and x = y + 2 = 6z − 6 are skew and hence find the shortest distance between them.

Find the equation of the plane passing through the intersection of the planes \(\vec { r } .(\hat { i } +\hat { j } +\hat { k } )+1=0\) and \(\vec { r } .(2\hat { i } -3\hat { j } +5\hat { k } )=2\) and the point (-1, 2, 1).

A particle moves along a straight line in such a way that after t seconds its distance from the origin is s = 2t² + 3t metres.

Find the absolute extrem of the following function on the given closed interval
f(x) = x² -12x + 10; [1, 2]

Does there exist a differentiable function f(x) such that f(0) = -1, f(2) = 4 and f'(x) ≤ 2 for all x. Justify you answer.

Evaluate: \(\underset{x\rightarrow \infty}{lim}(\frac{x^{2}+17x+29}{x^{4}})\).

Find the asymptotes of the function f(x) = \(\frac{1}{x}\)

Find a linear approximation for the following functions at the indicated points.
f(x) = x³ - 5x + 12, x₀ = 2

The trunk of a tree has diameter 30 cm. During the following year, the circumference grew 6cm.
(i) Approximately, how much did the tree's diameter grow?
(ii) What is the percentage increase in area of the tree's cross-section?

Evaluate \(\begin{matrix} lim \\ (x,y)\rightarrow (0,0) \end{matrix}cos=\left( \frac { { e }^{ x }siny }{ y } \right) \), if the limit exists.

If v(x, y, z) = x³ + y³ + z³ + 3xyz, show that \(\frac { { \partial }^{ 2 }v }{ \partial y\partial z } =\frac { { \partial }^{ 2 }v }{ \partial z\partial y } \)

If w (x, y, z) = x² + y² + y², x = e^t, y = e^t sin t, z = e^t cos t, find \(\frac{dw}{dt}\)

Evaluate :\(\int _{ 0 }^{ 9 }{ \frac { 1 }{ x+\sqrt { x } } dx } \)

Evaluate \(\\ \int _{ 0 }^{ 1 }{ { e }^{ -2x }(1+x-{ 2x }^{ 3 })dx } \)

Show that Γ(n) = 2\(\int _{ 0 }^{ \infty }{ { e }^{ -{ x }^{ 2 } }{ x }^{ 2n-1 }dx } \)

Find, by integration, the volume of the solid generated by revolving about y-axis the region bounded between the parabola x = y² +1, the y-axis, and the lines y = 1 and y = −1.

Form the differential equation by eliminating the arbitrary constants A and B from y = A cos x + B sin x.

Solve \({ y }^{ 2 }+{ x }^{ 2 }\frac { dy }{ dx } =xy\frac { dy }{ dx } \)

A six sided die is marked '2' on one face, '3' on two ofits faces, and '4' on remaining three faces. The die is thrown twice. If X denotes the total score in two throws, find the values of the random variable and number of points in its inverse images.

The probability density function random variable X is given by \(f(x)=\begin{cases} \begin{matrix} { 16xe }^{ -4x } & for\quad x>0 \end{matrix} \\ \begin{matrix} 0 & for\quad x\le 0 \end{matrix} \end{cases}\) find the mean and variance of X.

Two balls are chosen randomly from an urn containing 6 white and 4 black balls. Suppose that we win Rs. 30 for each black ball selected and we lose Rs. 20 for each white ball selected. If X denotes the winning amount, then find the values of X and number of points in its inverse images.

Verify the
(i) closure property,
(ii) commutative property,
(iii) associative property
(iv) existence of identity and
(v) existence of inverse for the arithmetic operation - on Z.

Let \(A=\left( \begin{matrix} 1 & 0 \\ 0 & 1 \\ 1 & 0 \end{matrix}\begin{matrix} 1 & 0 \\ 0 & 1 \\ 0 & 1 \end{matrix} \right) ,B=\left( \begin{matrix} 0 & 1 \\ 1 & 0 \\ 1 & 0 \end{matrix}\begin{matrix} 0 & 1 \\ 1 & 0 \\ 0 & 1 \end{matrix} \right) ,C=\left( \begin{matrix} 1 & 1 \\ 0 & 1 \\ 1 & 1 \end{matrix}\begin{matrix} 0 & 1 \\ 1 & 0 \\ 1 & 1 \end{matrix} \right) \)be any three boolean matrices of the same type. Find AVB

Let \(A=\left( \begin{matrix} 1 & 0 \\ 0 & 1 \\ 1 & 0 \end{matrix}\begin{matrix} 1 & 0 \\ 0 & 1 \\ 0 & 1 \end{matrix} \right) ,B=\left( \begin{matrix} 0 & 1 \\ 1 & 0 \\ 1 & 0 \end{matrix}\begin{matrix} 0 & 1 \\ 1 & 0 \\ 0 & 1 \end{matrix} \right) ,C=\left( \begin{matrix} 1 & 1 \\ 0 & 1 \\ 1 & 1 \end{matrix}\begin{matrix} 0 & 1 \\ 1 & 0 \\ 1 & 1 \end{matrix} \right) \)be any three boolean matrices of the same type.
Find (A∨B)∧C