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.

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.)

If ax² + bx + c is divided by x + 3, x − 5, and x − 1, the remainders are 21, 61 and 9 respectively. Find a, b and c. (Use Gaussian elimination method.)

Investigate for what values of λ and μ the system of linear equations x  +  2y  +  z  =  7 ,   x  +  y  +  λz   =  μ ,   x  +  3y  −  5z   =  5 has
(i) no solution 
(ii) a unique solution 
(iii) an infinite number of solutions

If the system of equations px + by + cz = 0, ax + qy + cz = 0, ax + by + rz = 0 has a non-trivial solution and p ≠ a, q ≠ b, r ≠ c, prove that \(\frac { p }{ p-a } +\frac { q }{ q-b } +\frac { r }{ r-c } =2\).

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

Solve the equation z³+ 8i = 0, where \(z \in \mathbb{C}\)

If 2+i and 3-\(\sqrt{2}\) are roots of the equation x⁶-13x⁵+ 62x⁴-126x³+ 65x²+127x-140 = 0, find all roots.

Discuss the maximum possible number of positive and negative roots of the polynomial equations x²−5x+6 and x²−5x+16 . Also draw rough sketch of the graphs

Solve \(tan^{ -1 }\left( \frac { x-1 }{ x-2 } \right) +tan^{ -1 }\left( \frac { x+1 }{ x+2 } \right) =\frac { \pi }{ 4 } \)

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.

A concrete bridge is designed as a parabolic arch. The road over bridge is 40 m long and the maximum height of the arch is 15 m. Write the equation of the parabolic arch.

An engineer designs a satellite dish with a parabolic cross section. The dish is 5 m wide at the opening, and the focus is placed 1.2 m from the vertex
(a) Position a coordinate system with the origin at the vertex and the x -axis on the parabola’s axis of symmetry and find an equation of the parabola.
(b) Find the depth of the satellite dish at the vertex.

Find the vertex, focus, equation of directrix and length of the latus rectum of the following: y²−4y−8x+12 = 0

Using vector method, prove that cos(α − β ) = cos α cos β +sin α sin β

If the Cartesian equation of a plane is 3x - 4y + 3z = -8, find the vector equation of the plane in the standard form.

Show that the lines \(\vec { r } =(\hat {- i } -3\hat { j } -5\hat { k } )+s(3\hat { i } +5\hat { j } +7\hat { k } )\) and \(\vec { r } =(2\hat { i } +4\hat { j } +6\hat { k } )+t(\hat { i } +4\hat { j } +7\hat { k } )\) are coplanar. Also, find the non-parametric form of vector equation of the plane containing these lines

A camera is accidentally knocked off an edge of a cliff 400 ft high. The camera falls a distance of s = 16t² in t seconds.
(i) How long does the camera fall before it hits the ground?
(ii) What is the average velocity with which the camera falls during the last 2 seconds?
(iii) What is the instantaneous velocity of the camera when it hits the ground?

A ladder 17 metre long is leaning against the wall. The base of the ladder is pulled away from the wall at a rate of 5 m/s. When the base of the ladder is 8 metres from the wall.
(i) How fast is the top of the ladder moving down the wall?
(ii) At what rate, the area of the triangle formed by the ladder, wall and the floor is changing?

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

Find the equation of tangent and normal to the curve given by x = 7 cos t and y = 2sin t, t ∈ R at any point on the curve.

Find intervals of concavity and points of inflexion for the following function:
f(x) = sin x + cos x, 0 < x < 2

Prove that among all the rectangles of the given perimeter, the square has the maximum area.

Find the asymptotes of the following curves :\(f(x)=\frac { { x }^{ 2 }+6x-4 }{ 3x-6 } \)

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

For the function f{x) = 4x³ + 3x² - 6x + 1 find the intervals of monotonicity, local extrema, intervals of concavity and points of inflection.

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 V(x,y) = e^x(x cos y - y siny), then prove that \(\frac { { \partial }^{ 2 }V }{ \partial { x }^{ 2 } } =\frac { { \partial }^{ 2 }V }{ \partial { y }^{ 2 } } \) = 0

Evaluate\(\int _{ 1 }^{ 4 }{ ({ 2x }^{ 2 }+3) } \) dx, as the limit of a sum

Evaluate \(\int ^{\pi}_{-\pi} \frac{cos ^2 x}{1+ a^x}\) dx

Evaluate the following integrals using properties of integration:
\(\int _{ 0 }^{ \pi }{ x\left[ { sin }^{ 2 }(sinx)+{ cos }^{ 2 }(cosx) \right] } dx\)

Using integration find the area of the region bounded by triangle ABC, whose vertices A, B, and C are (−1, 1), (3, 2), and (0, 5) respectively

A watermelon has an ellipsoid shape which can be obtained by revolving an ellipse with major-axis 20 cm and minor-axis 10 cm about its major-axis. Find its volume using integration.

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

Solve \(\left( y+\sqrt { { x }^{ 2 }+{ y }^{ 2 } } \right) dx-xdy=0,\ y(1)=0\)

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

Solve the Linear differential equation \((1+x+{ xy }^{ 2 })\frac { dy }{ dx } +(y+{ y }^{ 3 })=0\)

The equation of electromotive force for an electric circuit containing resistance and self inductance is E = Ri + L\(\frac{di}{dt},\) Where E is the electromotive force is given to the circuit, R the resistance and L, the coefficient of induction. Find the current i at time t when E = 0.

A tank initially contains 50 litres of pure water. Starting at time t = 0 a brine containing with 2 grams of dissolved salt per litre flows into the tank at the rate of 3 litres per minute. The mixture is kept uniform by stirring and the well-stirred mixture simultaneously flows out of the tank at the same rate. Find the amount of salt present in the tank at any time t > 0.

If X is the random variable with probability density function f(x) given by,
\(f(x)=\begin{cases} \begin{matrix} x+1 & -1\le x<0 \end{matrix} \\ \begin{matrix} -x+1 & 0\le x<1 \end{matrix} \\ \begin{matrix} 0 & otherwise \end{matrix} \end{cases}\) 
then find
(i) the distribution function F(x)
(ii) P( -0.5 ≤X ≤ 0.5)

The mean and standard deviation of a binomial variate X are respectively 6 and 2.
Find
(i) the probability mass function
(ii) P(X = 3)
(iii) P(X\(\ge \)2).

Find the constant C such that the function \(f(x)= \begin{cases}C x^{2} & 1 is a density function, and compute
(i) P(1.5 < X < 3.5)
(ii) P(X ≤ 2)
(iii) P(3 < X )

Two balls are chosen randomly from an urn containing 8 white and 4 black balls. Suppose that we win Rs. 20 for each black ball selected and we lose Rs. 10 for each white ball selected. Find the expected winning amount and variance 

Establish the equivalence property connecting the bi-conditional with conditional: p ↔️ q ≡ (p ➝ q) ∧ (q⟶ p)

Let A be Q\{1}. Define ∗ on A by x*y = x + y − xy. Is ∗ binary on A? If so, examine the commutative and associative properties satisfied by ∗ on A.