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

If A = \(\left[ \begin{matrix} 4 & 3 \\ 2 & 5 \end{matrix} \right] \), find x and y such that A² + xA + yI₂ = O₂. Hence, find A^-1.

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 z₁= 2 + 5i, z₂ = -3 - 4i, and z₃ = 1 + i, find the additive and multiplicate inverse of z₁, z₂ and z₃

If p and q are the roots of the equation 1x²+ nx + n = 0, show that \(\sqrt { \frac { p }{ q } } +\sqrt { \frac { q }{ p } } +\sqrt { \frac { n }{ l } } \) = 0.

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 \(\frac { q-q' }{ p'-p } \).

Find the domain of f(x) = sin^-1 \((\frac{|x|-2}{3})+ \) cos^-1 \((\frac{1-|x|}{4})\)

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

Find the equation of the ellipse whose eccentricity is \(\frac { 1 }{ 2 } \), one of the foci is(2, 3) and a directrix is x = 7. Also find the length of the major and minor axes of the ellipse.

Find the foci, vertices and length of major and minor axis of the conic 4x² + 36y² + 40x − 288y + 532 = 0 

By vector method, prove that cos(α + β) = cos α cos β -  sin α sin β

With usual notations, in any triangle ABC, prove by vector method that \(\frac { a }{ sinA } =\frac { b }{ sinB }=\frac { c }{ sinc }\)

A particle is fired straight up from the ground to reach a height of s feet in t seconds, where s(t) = 128t −16t².
(1) Compute the maximum height of the particle reached.
(2) What is the velocity when the particle hits the ground?

A particle moves along a horizontal line such that its position at any time t ≥ 0 is given by s(t) = t³ − 6t² +9 t +1, where s is measured in metres and t in seconds?
(1) At what time the particle is at rest?
(2) At what time the particle changes its direction?
(3) Find the total distance travelled by the particle in the first 2 seconds.

The radius of a circular plate is measured as 12.65 cm instead of the actual length 12.5 cm. Find the following in calculating the area of the circular plate:
Relative error

The radius of a circular plate is measured as 12.65 cm instead of the actual length 12.5 cm.find the following in calculating the area of the circular plate:
Percentage error

Show that \(\int ^\frac{\pi}{2}_0\) \(\frac {dx}{4+5 sin x}\) = \(\frac {1}{3}\) log_e 2.

Evaluate : \(\int ^\frac{\pi}{4}_{0} \frac{1}{sin x+cos x}\) dx

Show that y = ae^-3x + b, where a and b are arbitary constants, is a solution of the differential equation\(\frac { { d }^{ 2 }y }{ { dx }^{ 2 } } +3\frac { dy }{ dx } =0\)

Show that y = ax + \(\frac { b }{ x } \), x ≠ 0 is a solution of the differential equation x² y" + xy' - y = 0.

In a pack of 52 playing cards, two cards are drawn at random simultaneously. If the number of black cards drawn is a random variable, find the values of the random variable and number of points in its inverse images.

An urn contains 5 mangoes and 4 apples. Three fruits are taken at randaom. If the number of apples taken is a random variable, then find the values of the random variable 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.

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.

Define an operation \(*\)on Q as follows: a * b =\(\left( \frac { a+b }{ 2 } \right) \); a,b ∈Q. Examine the closure, commutative, and associative properties satisfied by \(*\)on Q.

If \({ tan }^{ -1 }\left( \frac { \sqrt { 1+{ x }^{ 2 } } -\sqrt { 1-{ x }^{ 2 } } }{ \sqrt { 1+{ x }^{ 2 } } +\sqrt { 1-{ x }^{ 2 } } } \right) =a\) than prove that x² = sin 2a

Solve for \(x: \tan ^{-1} x+2 \cot ^{-1} x=\frac{2 \pi}{3}\)

In a murder investigation, a corpse was found by a detective at exactly 8 p.m. Being alert, the detective also measured the body temperature and found it to be 70^oF. Two hours later, the detective measured the body temperature again and found it to be 60^oF. If the room temperature is 50^oF, and assuming that the body temperature of the person before death was 98.6^oF, at what time did the murder occur? [log(2.43) = 0.88789; log(0.5)=-0.69315]

Water at temperature 100^oC cools in 10 minutes to 80^oC in a room temperature of 25^oC.
Find
(i) The temperature of water after 20 minutes
(ii) The time when the temperature is 40^oC
\(\left[ { log }_{ e }\frac { 11 }{ 15 } =-0.3101;{ log }_{ e }5=1.6094 \right] \)

Solve the differential equation \({ ye }^{ \frac { x }{ y } }dx=\left( { xe }^{ \frac { x }{ y } }+y \right) dy\)