If A = \(\left[ \begin{matrix} 2 & 0 \\ 1 & 5 \end{matrix} \right] \) and B = \(\left[ \begin{matrix} 1 & 4 \\ 2 & 0 \end{matrix} \right] \) then |adj (AB)| = 

If A^TA⁻¹ is symmetric, then A² =

Which of the following is/are correct?
(i) Adjoint of a symmetric matrix is also a symmetric matrix.
(ii) Adjoint of a diagonal matrix is also a diagonal matrix.
(iii) If A is a square matrix of order n and λ is a scalar, then adj(λA) = λⁿ adj(A).
(iv) A(adjA) = (adjA)A = |A| I

The area of the triangle formed by the complex numbers z, iz and z+iz in the Argand’s diagram is

The conjugate of a complex number is \(\cfrac { 1 }{ i-2 } \). Then the complex number is

If z is a non zero complex number, such that 2iz² = \(\bar { z } \) then |z| is

According to the rational root theorem, which number is not possible rational zero of 4x⁷ + 2x⁴ - 10x³ - 5?

If a, b, c ∈ Q and p +√q (p, q ∈ Q) is an irrational root of ax²+bx+c = 0 then the other root is ___________

Let a > 0, b > 0, c >0. Theh n both the root of the equation ax²+bx+c = 0 are _________

A stone is thrown up vertically. The height it reaches at time t seconds is given by x = 80t -16t². The stone reaches the maximum height in time t seconds is given by

The maximum value of the function \(x^{2} e^{-2 x}, x>0\) is

If f (x, y) = e^xy then \(\frac { { \partial }^{ 2 }f }{ \partial x\partial y } \) is equal to

If we measure the side of a cube to be 4 cm with an error of 0.1 cm, then the error in our calculation of the volume is

If p and q are the order and degree of the differential equation \(y=\frac { dy }{ dx } +{ x }^{ 3 }\left( \frac { { d }^{ 2 }y }{ { dx }^{ 2 } } \right) +xy=cosx,\) When

A binary operation * is defined on the set of positive rational numbers Q⁺ by a*b = \(\frac { ab }{ 4 } \). Then 3 * \(\left( \frac { 1 }{ 5 } *\frac { 1 }{ 2 } \right) \) is _____________

Find the modulus of the following complex numbers
\(\frac { 2i }{ 3+4i } \)

Find the modulus of the following complex numbers
(1-i)¹⁰

Find the modulus and principal argument of the following complex numbers.
\(-\sqrt { 3 } +i\)

Find the value of the complex number (i²⁵)³.

If α, β, γ  and \(\delta\) are the roots of the polynomial equation 2x⁴ + 5x³ − 7x² + 8 = 0, find a quadratic equation with integer coefficients whose roots are α + β + γ + \(\delta\) and αβ૪\(\delta\).

Find a polynomial equation of minimum degree with rational coefficients, having 2-\(\sqrt{3}\) as a root.

Find a polynomial equation of minimum degree with rational coefficients, having 2i+3 as a root.

Examine for the rational roots of x⁸- 3x + 1 = 0

Find the eccentricity of the hyperbola with foci on the x-axis if the length of its conjugate axis is \({ \left( \frac { 3 }{ 4 } \right) }^{ th }\) of the length of its tranverse axis.

Solve :\(\frac { dy }{ dx } =\frac { 2x }{ { x }^{ 2 }+1 } \)

Given the complex number z = 2 + 3i, represent the complex numbers in Argand diagram z, iz , and z+iz

Given the complex number z = 2 + 3i, represent the complex numbers in Argand diagram z, −iz , and z−iz

For the hyperbola 3x² - 6y² = -18, find the length of transverse and conjugate axes and eccentricity.

Find the equation of the plane passing through the line of intersection of the planes \(\vec { r } .(2\hat { i } -7\hat { j } +4\hat { k } )=3\) and 3x - 5y + 11 = 0, and the point (-2, 1, 3)

Find the equation of the plane which passes through the point (3, 4, -1) and is parallel to the plane 2x - 3y + 5z = 0. Also, find the distance between the two planes.

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).

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

A right circular cylinder has radius r =10 cm. and height h = 20 cm. Suppose that the radius of the cylinder is increased from 10 cm to 10. 1 cm and the height does not change. Estimate the change in the volume of the cylinder. Also, calculate the relative error and percentage error.

Assuming log₁₀e = 0.4343, find an approximate value of log₁₀ 1003

Find the partial derivatives of the following functions at the indicated point
h (x, y, z) = x sin (xy) + z²x, \(\left( 2,\frac { \pi }{ 4 }, 1\right) \) 

A firm produces two types of calculators each week, x number of type A and y number of type B. The weekly revenue and cost functions (in rupees) are R(x, y) = 80x + 90y + 0.04xy − 0.05x² − 0.05y² and C(x, y) = 8x + 6y + 2000 respectively
(i) Find the profit function P(x, y) 
(ii) Find \(\frac { { \partial P } }{ \partial { x } } \) (1200, 1800) and \(\frac { \partial v }{ \partial y} \) (1200, 1800)

Let U(x, y, z) = xyz, x = e^-t, y = e^-t cos t, z = sin t, t ∈ R. Find \(\frac{dU}{dt}\)

Evaluate the following definite integrals:
\(\int _{ 0 }^{ 1 }{ \sqrt { \frac { 1-x }{ 1+x } } } dx\)

For what value of λ, the system of equations x + y + z = 1, x + 2y + 4z = λ, x + 4y + 10z = λ² is consistent.

Find a polynomial equation of minimum degree with rational coefficients, having \(\sqrt{5}\)−\(\sqrt{3}\) as a root.

Find the domain of the following functions
(i) f(x) = sin^-1(2x - 3)
(ii) f(x) = sin^-1x + cos x

Find the ratio of the area between the curves y=cosx and y=cos2x and x- axis from x=0 to \(x=\frac { \pi }{ 3 } \)

Find the area bounded by the curve y2(2a-x)=x² and the line x=2a.

Solve : \({ e }^{ \frac { dy }{ dx } }=x+1,y(0)=5\)

Solve :x²dy+y(x+y)dx=0 given that y=1 when x=1.

Solve :\(x\frac { dy }{ dx } sin\left( \frac { y }{ x } \right) +x-ysin\left( y\frac { y }{ x } \right) =,y(1)=\frac { \pi }{ 2 } \)