Verify (AB)^-1 = B^-1A^-1 with A = \(\left[ \begin{matrix} 0 & -3 \\ 1 & 4 \end{matrix} \right] \), B = \(\left[ \begin{matrix} -2 & -3 \\ 0 & -1 \end{matrix} \right] \).

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

Test for consistency of the following system of linear equations and if possible solve:
x - y + z = -9, 2x - 2y + 2z = -18, 3x - 3y + 3z + 27 = 0.

Solve the following system of linear equations by matrix inversion method :
2x  −  y  =  8 ,   3x  +  2y  =  −2.

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 z = x + iy is a complex number such that \(\left| \frac { z-4i }{ z+4i } \right| =1\) show that the locus of z is real axis.

Obtain the Cartesian form of the locus of z in each of the following cases.
|z| = |z - i|

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

If the sides of a cubic box are increased by 1, 2, 3 units respectively to form a cuboid, then the volume is increased by 52 cubic units. Find the volume of the cuboid.

If the roots of x³+ px²+ qx + r = 0 are in H.P. prove that 9pqr = 27r²+2q³.

Solve the following equations,
12x³+ 8x = 29x²- 4

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

Find the domain of
g(x) = sin⁻¹x + cos⁻¹x

Solve \({ cot }^{ -1 }x-{ cot }^{ -1 }\left( x+2 \right) =\frac { \pi }{ 12 } ,x>0\)

Find the equation of the ellipse with foci (±2, 0), vertices (±3, 0)  

Identify the type of the conic for the following equations:
(1) 16y² = −4x²+64
(2) x²+y² = −4x−y+4
(3) x²−2y = x+3
(4) 4x²−9y²−16x+18y−29 = 0

The equation of the ellipse is \(\frac { { \left( x-11 \right) }^{ 2 } }{ 484 } +\frac { { y }^{ 2 } }{ 64 } =1\). ( x and y are measured in centimeters) where to the nearest centimeter, should the patient’s kidney stone be placed so that the reflected sound hits the kidney stone?

A particle acted upon by constant forces \(\hat { 2j } +\hat { 5j } +\hat { 6k } \) and \(-\hat { i } -\hat { 2j } -\hat { k } \)  is displaced from the point (4, −3, −2) to the point (6, 1, −3). Find the total work done by the forces.

Prove that \([\vec { a } \times \vec { b } ,\vec { b } \times \vec { c } ,\vec { c } \times \vec { a } ]\) = \([{ \vec { a } ,\vec { b } ,\vec { c } }]^{ 2 }\)

Find the shortest distance between the two given straight lines \(\vec { r } =(2\hat { i } +3\hat { j } +4\hat { k } )+t(-2\hat { i } +\hat { j } -2\hat { k } )\) and \(\frac { x-3 }{ 2 } =\frac { y }{ -1 } =\frac { z+2 }{ 2 } \)

Find the coordinates of the point where the straight line \(\vec { r } =(2\hat { i } -\hat { j } +2\hat { k } )+t(3\hat { i } +4\hat { j } +2\hat { k } )\) intersects the plane x−y+z−5 = 0.

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

Find the absolute extrema of the following functions on the given closed interval.
\(f(x)=2cosx+sin2x;\left[ 0,\frac { \pi }{ 2 } \right] \)

Write the Taylor series expansion of \(\frac{1}{x}\) about x = 2 by finding the first three non-zero terms.

Evaluate the following limit, if necessary use l ’Hôpital Rule
\(\underset { x\rightarrow { 1 }^{ + } }{ lim } \left( \frac { 2 }{ { x }^{ 2 }-1 } -\frac { x }{ x-1 } \right) \)

Let \(f(x)=\sqrt [ 3 ]{ x } \). Find the linear approximation at x = 27. Use the linear approximation to approximate \(\sqrt [ 3 ]{ 27.2 } \)

An egg of a particular bird is very nearly spherical. If the radius to the inside of the shell is 5 mm and radius to the outside of the shell is 5.3 mm, find the volume of the shell approximately.

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

Let U(x, y, z) = x² − xy + 3 sin z, x, y, z ∈ R Find the linear approximation for U at (2,−1,0).

Evaluate \(\int _{ 0 }^{ 1 }{ xdx } \), as the limit of a sum.

Evaluate the following definite integrals:
\(\int _{ 0 }^{ \frac { \pi }{ 2 } }{ \sqrt { cos\theta } } { sin }^{ 3 }\theta d\theta \)

Evaluate \(\int _{ 0 }^{ 1 }{ { x }^{ 3 }{ (1-x) }^{ 4 }dx } \)

Find the area of the region bounded by x−axis, the curve y = |cos x|, the lines x = 0 and x = \(\pi\).

Find the differential equation corresponding to the family of curves represented by the equation y = Ae^8x + Be^-8x, where A and B are arbitrary constants.

Solve the following differential equations:
ydx + (1 +x²) tan^-1 xdy = 0

Solve the Linear differential equation:
\(x\frac { dy }{ dx } +2y-x^2logx=0\)

For the random variable X with the given probability mass function as below, find the mean and variance \(f(x)= \begin{cases}2(x-1) & 1

The probability that a certain kind of component will survive a electrical test is \(\frac { 3 }{ 4 } \). Find the probability that exactly 3 of the 5 components tested survive.

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.

Construct the truth table for \((p\overset { \_ \_ }{ \vee } q)\wedge (p\overset { \_ \_ }{ \vee } \neg q)\)

Prove that q ➝ p ≡ ¬p ➝ ¬q