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 is a non-singular matrix such that A^-1 = \(\left[ \begin{matrix} 5 & 3 \\ -2 & -1 \end{matrix} \right] \), then (A^T)⁻¹ =

The solution of the equation |z| - z = 1 + 2i is

If \(\omega \neq 1\) is a cubic root of unity and \(\left| \begin{matrix} 1 & 1 & 1 \\ 1 & { -\omega }^{ 2 }-1 & { \omega }^{ 2 } \\ 1 & { \omega }^{ 2 } & { \omega }^{ 7 } \end{matrix} \right| \) = 3k, then k is equal to 

If sin⁻¹x = 2sin⁻¹ \(\alpha\) has a solution, then

If \(\cot ^{-1}(\sqrt{\sin \alpha})+\tan ^{-1}(\sqrt{\sin \alpha})=u\), then cos2u is equal to

Tangents are drawn to the hyperbola  \(\frac { { x }^{ 2 } }{ 9 } -\frac { { y }^{ 2 } }{ 4 } =1\) parallel to the straight line 2x − y = 1. One of the points of contact of tangents on the hyperbola is

If \(\vec { a } \) and \(\vec { b } \) are unit vectors such that \([\vec { a } ,\vec { b },\vec { a } \times \vec { b } ]=\frac { 1}{ 4 } \), then the angle between \(\vec { a } \) and \(\vec { b } \) is

If \(\vec { a } =\hat { i } +\hat { j } +\hat { k } \), \(\vec { b } =\hat { i } +\hat { j } \), \(\vec { c } =\hat { i } \) and \((\vec { a } \times \vec { b } )\times\vec { c } \) = \(\lambda \vec { a } +\mu \vec { b } \), then the value of \(\lambda +\mu \) is

If the direction cosines of a line are \(\frac { 1 }{ c } ,\frac { 1 }{ c } ,\frac { 1 }{ c } \), then

The number given by the Mean value theorem for the function \(\frac { 1 }{ x } \), x ∈ [1, 9] is

Linear approximation for g(x) = cos x at \(x=\frac{\pi}{2}\) is

The integrating factor of the differential equation \(\frac { dy }{ dx } +y=\frac { 1+y }{ \lambda } \) is

The number of arbitrary constants in the particular solution of a differential equation of third order is

If cosx is an integrating factor of the differential equation \(\frac{dy}{dx}+Py= Q\), then P = ___________

The I.F of \(\frac{dy}{dx}-y\) tan x = cos x is _________

A random variable X has binomial distribution with n = 25 and p = 0.8 then standard deviation of X is

In eight throws of a die, 1 or 3 is considered a success. Then the mean number of success is _____________

If x is a continuous random variable then P(x ≥ a) = ________

If * is defined by a * b = a² + b² + ab + 1, then (2 * 3) * 2 is _____________

For the matrix A, if A³ = I, then find A^-1.

Show that the system of equations is inconsistent. 2x + 5y= 7, 6x + 15y = 13.

If y = 4x + c is a tangent to the circle x² + y² = 9, find c 

Verify whether the line \(\frac { x-3 }{ -4 } =\frac { y-4 }{ -7 } =\frac { z+3 }{ 12 } \) lies in the plane 5x-y+z = 8.

Using the Rolle’s theorem, determine the values of x at which the tangent is parallel to the x -axis for the following functions:
\(f(x)=\sqrt{x}-\frac{x}{3}, x\in [0,9]\)

Evaluate the following integrals using properties of integration:
\(\int _{ -\frac { \pi }{ 4 } }^{ \frac { \pi }{ 4 } }{ { sin }^{ 2 }xdx } \)

For each of the following differential equations, determine its order, degree (if exists)
\(y\left( \frac { dy }{ dx } \right) =\frac { x }{ \left( \frac { dy }{ dx } \right) +{ \left( \frac { dy }{ dx } \right) }^{ 3 } } \)

Show that y = e^−x + mx + n is a solution of the differential equation e^x \(\left( \frac { { d }^{ 2 }y }{ { dx }^{ 2 } } \right) \) -1 = 0

A random variable X has the following probability mass function.

Find
(i) the value of k
(ii) P(2 \(\le\) X < 5)
(iii) P(3 < X )

Write the statements in words corresponding to ¬p, p ∧ q , p ∨ q and q ∨ ¬p, where p is ‘It is cold’ and q is ‘It is raining'.

Prove, using mean value theorem, that \(|sin \alpha-sin\beta|\le |\alpha-\beta|, \alpha, \beta \in R\)

A thermometer was taken from a freezer and placed in a boiling water. It took 22 seconds for the thermometer to raise from −10°C to 100°C. Show that the rate of change of temperature at some time t is 5°C per second.

Explain why Lagrange’s mean value theorem is not applicable to the following functions in the respective intervals:
f(x) = \(\frac { x+1 }{ x } \), x ∈ [-1, 2]

A race car driver is racing at 20^th km. If his speed never exceeds 150 km/hr, what is the maximum kilometer he can reach in the next two hours.

Write the Maclaurin series expansion of the following function
cos x

Evaluate the following limit, if necessary use l ’Hôpital Rule
\(\underset { x\rightarrow 0 }{ lim } \left( \frac { 1 }{ sinx } -\frac { 1 }{ x } \right) \)

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

Find the intervals of monotonicity and hence find the local extrema for the function \(f(x)=x^{\frac{2}{3}}\).

Determine the intervals of concavity of the curve y = 3+ sin x .

Find the slant (oblique) asymptote for the function \(f(x)=\frac { { x }^{ 2 }-6x+7 }{ x+5 } \)