If A = \(\left[ \begin{matrix} 5 & 3 \\ -1 & -2 \end{matrix} \right] \), show that A² - 3A - 7I₂ = O₂. Hence find A⁻¹.

The prices of three commodities A, B and C are Rs. x, y and z per units respectively. A person P purchases 4 units of B and sells two units of A and 5 units of C. Person Q purchases 2 units of C and sells 3 units of A and one unit of B. Person R purchases one unit of A and sells 3 unit of B and one unit of C. In the process, P, Q and R earn Rs. 15,000, Rs. 1,000 and Rs. 4,000 respectively. Find the prices per unit of A, B and C. (Use matrix inversion method to solve the problem.)

A boy is walking along the path y = ax² + bx + c through the points (−6, 8), (−2, −12) and (3, 8). He wants to meet his friend at P(7, 60). Will he meet his friend? (Use Gaussian elimination method.)

Find the value of k for which the equations
kx - 2y + z = 1, x - 2ky + z = -2, x - 2y + kz = 1 have
(i) no solution
(ii) unique solution
(iii) infinitely many solution

Solve the following system of homogenous equations.
3x + 2y + 7z = 0, 4x − 3y − 2z = 0, 5x + 9y + 23z = 0

Solve the following systems of linear equations by Cramer’s rule:
 3x + 3y − z = 11, 2x − y + 2z = 9, 4x + 3y + 2z = 25.

If z = x + iy and arg \(\left( \frac { z-i }{ z+2 } \right) =\frac { \pi }{ 4 } \), then show that x² + y² + 3x - 3y + 2 = 0

Find all cube roots of \(\sqrt { 3 } +i\)

Form the equation whose roots are the squares of the roots of the cubic equation x³+ ax²+ bx + c = 0.

Find all zeros of the polynomial x⁶- 3x⁵- 5x⁴ + 22x³- 39x²- 39x + 135, if it is known that 1+2i and \(\sqrt{3}\) are two of its zeros.

If a₁, a₂, a₃, ... an is an arithmetic progression with common difference d, prove that tan\( \left[ tan^{ -1 }\left( \frac { d }{ 1+{ a }_{ 1 }{ a }_{ 2 } } \right) +tan^{ -1 }\left( \frac { d }{ 1+{ a }_{ 2 }{ a }_{ 3 } } \right) +....tan^{ -1 }\left( \frac { d }{ 1+{ a }_{ n }{ a }_{ n-1 } } \right) \right] =\frac { { a }_{ n }-{ a }_{ 1 } }{ 1+{ a }_{ 1 }{ a }_{ n } } \)

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.

Certain telescopes contain both parabolic mirror and a hyperbolic mirror. In the telescope shown in figure the parabola and hyperbola share focus F₁ which is 14m above the vertex of the parabola. The hyperbola’s second focus F₂ is 2m above the parabola’s vertex. The vertex of the hyperbolic mirror is 1m below F₁. Position a coordinate system with the origin at the centre of the hyperbola and with the foci on the y-axis. Then find the equation of the hyperbola.

Assume that water issuing from the end of a horizontal pipe, 7. 5 m above the ground, describes a parabolic path. The vertex of the parabolic path is at the end of the pipe. At a position 2. 5 m below the line of the pipe, the flow of water has curved outward 3m beyond the vertical line through the end of the pipe. How far beyond this vertical line will the water strike the ground?

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

Find the parametric form of vector equation of a straight line passing through the point of intersection of the straight lines \(\vec { r } =(\hat { i } +\hat { 3j } -\hat { k } )+t(2\hat { i } +3\hat { j } +2\hat { k } )\) and \(\frac { x-2 }{ 1 } =\frac { y-4 }{ 2 } =\frac { z+3 }{ 4 } \) and perpendicular to both straight lines.

Show that the straight lines \(\vec { r } =(5\hat { i } +7\hat { j } -3\hat { k } )+s(-4\hat { i } +4\hat { j } -5\hat { k } )\) and \(\vec { r } =(8\hat { i } +4\hat { j } +5\hat { k } )+t(7\hat { i } +\hat { j } +3\hat { k } )\)are coplanar. Find the vector equation of the plane in which they lie.

If we blow air into a balloon of spherical shape at a rate of 1000 cm³ per second. At what rate the radius of the baloon changes when the radius is 7cm? Also compute the rate at which the surface area changes.

A conical water tank with vertex down of 12 metres height has a radius of 5 metres at the top. If water flows into the tank at a rate 10 cubic m/min, how fast is the depth of the water increases when the water is 8 metres deep?

Find the tangent and normal to the following curves at the given points on the curve
x = cos t, y = 2sin t² at t = \(\frac { \pi }{ 3 } \)

Find intervals of concavity and points of inflexion for the following function:
\(f(x)=\frac { 1 }{ 2 } \left( { e }^{ x }-{ e }^{ -x } \right) \)

Write the Maclaurin series expansion of the following function:
cos² x

Evaluate the following limit, if necessary use l’Hôpital Rule
\(​​​​​​\underset { x\rightarrow { 0 }^{ + } }{ lim } { x }^{ x }\)

Discuss the monotonicity and local extrema of the function \(f(x)=log(1+x)-\frac{x}{1+x},x>-1\) and hence find the domain where, \(log(1+x)>\frac{x}{1+x}\)

A steel plant is capable of producing x tonnes per day of a low-grade steel and y tonnes per day of a high-grade steel, where \(y=\frac { 40-5x }{ 10-x } \). If the fixed market price of low-grade steel is half that of high-grade steel, then what should be optimal productions in low-grade steel and high-grade steel in order to have maximum receipts.

If z(x, y) = x tan-¹ (xy), x = t², y = se^t, s, t ∈ R. Find \(\frac { \partial z }{ \partial s } \) and \(\frac { \partial z }{ \partial t } \) at s = t = 1

Evaluate: \(\int ^4_{-4}\) |x+3| dx. 

Evaluate the following integrals using properties of integration:
\(\int _{ 0 }^{ { sin }^{ 2 }x }{ { sin }^{ -1 }\sqrt { t } dt+\int _{ 0 }^{ { cos }^{ 2 }x }{ { cos }^{ -1 }\sqrt { t } dt } } \)

Evaluate the following:
\(\int _{ 0 }^{ \frac { \pi }{ 2 } }{ { x }^{ 2 }cos2x\ dx } \)

Find the area of the region bounded by the curve 2+x−x²+y = 0 , x-axis, x = −3 and x = 3.

Find the differential equation of the family of all the parabolas with latus rectum 4a and whose axes are parallel to the x-axis.

Solve the following differential equations:
\(\\ \\ \\ \frac { dy }{ dx } ={ tan }^{ 2 }(x+y)\)

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

Solve [y(1-x tan x)+x² cosx] dx-dy = 0

Solve the Linear differential equation:
\(\frac { dy }{ dx } +\frac { y }{ (1-x)\sqrt { x } } =1-\sqrt { x } \)

The rate of increase in the number of bacteria in a certain bacteria culture is proportional to the number present. Given that the number triples in 5 hours, find how many bacteria will be present after 10 hours?

At 10.00 A.M. a woman took a cup of hot instant coffee from her microwave oven and placed it on a nearby Kitchen counter to cool. At this instant the temperature of the coffee was 180^o F, and 10 minutes later it was 160^o F. Assume that constant temperature of the kitchen was 70^oF.
(i) What was the temperature of the coffee at 10.15 A.M.? \(\left[\log \frac{9}{11}=-0.6061\right]\)
(ii) The woman likes to drink coffee when its temperature is between 130^oF and 140^oF between what times should she have drunk the coffee? \(\left[\log \frac{6}{11}=-0.2006\right]\)

The probability density function of X is given
\(f(x)=\begin{cases} \begin{matrix} { Ke }^{ \frac { -x }{ 3 } } & \begin{matrix} for & x>0 \end{matrix} \end{matrix} \\ \begin{matrix} 0 & \begin{matrix} for & x\le 0 \end{matrix} \end{matrix} \end{cases}\)
Find
(i) the value of k
(ii) the distribution function.
(iii) P(X <3)
(iv) P(5 ≤X)
(v) P(X ≤ 4)

A retailer purchases a certain kind of electronic device from a manufacturer. The manufacturer, indicates that the defective rate of the device is 5%. The inspector of the retailer randomly picks 10 items from a shipment. What is the probability that there will be
(i) at least one defective item
(ii) exactly two defective items.

A random variable X has the following probability mass function

x	1	2	3	4	5	6
f(x)	k	2k	6k	5k	6k	10k

Find
(i) P(2 < X < 6)
(ii) P(2 ≤ X < 5)
(iii) P(X ≤4)
(iv) P(3 < X )

Suppose that f (x) given below represents a probability mass function

x	1	2	3	4	5	6
f(x)	c2	2c2	3c2	4c2	c	2c

Find
(i) the value of c
(ii) Mean and variance.

Identify the valid statements from the following sentences.

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.