If A\(\left[ \begin{matrix} 1 & -2 \\ 1 & 4 \end{matrix} \right] =\left[ \begin{matrix} 6 & 0 \\ 0 & 6 \end{matrix} \right] \), then A = 

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 =

If A is a square matrix that IAI = 2, than for any positive integer n, |Aⁿ| = _______

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

.If a = 3+i and z = 2-3i, then the points on the Argand diagram representing az, 3az and - az are _________

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

If sin^-1 x+sin^-1 y+sin^-1 \(z = \frac{3\pi}{2}\), the value of x²⁰¹⁷+y²⁰¹⁸+z²⁰¹⁹\(-\frac { 9 }{ { x }^{ 101 }+{ y }^{ 101 }+{ z }^{ 101 } } \)is

If cot^-1 2 and cot^-1 3 are two angles of a triangle, then the third angle is

The number of real solutions of the equation \(\sqrt { 1+cos2x } ={ 2sin }^{ -1 }\left( sinx \right) ,-\pi  is ___________

The circle x² + y² = 4x + 8y +5 intersects the line 3x−4y = m at two distinct points if

If \([\vec{a}, \vec{b}, \vec{c}]=1\), then the value of \(\frac{\vec{a} \cdot(\vec{b} \times \vec{c})}{(\vec{c} \times \vec{a}) \cdot \vec{b}}+\frac{\vec{b} \cdot(\vec{c} \times \vec{a})}{(\vec{a} \times \vec{b}) \cdot \vec{c}}+\frac{\vec{c} \cdot(\vec{a} \times \vec{b})}{(\vec{c} \times \vec{b}) \cdot \vec{a}}\) is

The position of a particle moving along a horizontal line of any time t is given by s(t) = 3t² -2t- 8. The time at which the particle is at rest is

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

The value of \(\int _{ 0 }^{ \infty }{ { e }^{ -3x }{ x }^{ 2 }dx } \) is

The order and degree of the differential equation \(\sqrt { sinx } (dx+dy)=\sqrt { cos x } (dx-dy)\) is

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

Let X have a Bernoulli distribution with mean 0.4, then the variance of (2X - 3) is

The operation * defined by \(a * b =\frac{ab}{7}\) is not a binary operation on

If a * b=\(\sqrt { { a }^{ 2 }+{ b }^{ 2 } } \) on the real numbers then * is

Prove that \(\left[ \begin{matrix} \cos { \theta } & -\sin { \theta } \\ \sin { \theta } & \cos { \theta } \end{matrix} \right] \) is orthogonal.

If z₁= 3 - 2i and z₂ = 6 + 4i, find \(\frac { { z }_{ 1 } }{ z_{ 2 } } \) in the rectangular form.

Discuss the nature of the roots of the following polynomials:
x²⁰¹⁸+1947x¹⁹⁵⁰+15x⁸+26x⁶+2019

If \(\vec{ a } =\hat { -3i } -\hat { j } +\hat { 5k } \), \(\vec{b}=\hat{i}-\hat{2j}+\hat{k} \),  \(\vec{c}=\hat{4j}-\hat{5k} \ \) find\( \ {\vec a } .(\vec { b } \times \vec { c } )\)

Find the slope of the tangent to the following curves at the respective given points
y = x⁴ + 2x² − x at x = 1

If w(x, y, z) = x² y + y²z + z²x, x, y, z∈R, find the differential dw .

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)
\(y\left( \frac { dy }{ dx } \right) =\frac { x }{ \left( \frac { dy }{ dx } \right) +{ \left( \frac { dy }{ dx } \right) }^{ 3 } } \)

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.

Determine whether ∗ is a binary operation on the sets given below.
a*b = min (a, b) on A = {1, 2, 3, 4, 5}

Verify the property (A^T)^-1 = (A^-1)^T with A = \(\left[ \begin{matrix} 2 & 9 \\ 1 & 7 \end{matrix} \right] \).

If α, β and γ are the roots of the cubic equation x³+2x²+3x+4 = 0, form a cubic equation whose roots are, 2α, 2β, 2γ

Find the domain of sin⁻¹(2−3x²)

Find the equation of the tangent and normal to the circle x²+y²−6x+6y−8 = 0 at (2, 2) .

Prove by vector method that the area of the quadrilateral ABCD having diagonals AC and BD is \(\frac { 1 }{ 2 } \left| \vec { AC } \times \vec { BD } \right| \).

Find the points on the curve y² - 4xy = x² + 5 for which the tangent is horizontal.

Let F(x, y) = x³ y + y²x + 7 for all (x, y)∈ R². Calculate \(\frac { \partial F }{ \partial x } \)(-1, 3) and \(\frac { \partial F }{ \partial y } \)(-2, 1).

Solve \(\frac { dy }{ dx } +2y={ e }^{ -x }\)

If X~ B(n, p) such that 4P(X = 4) = P(X = 2) and n = 6. Find the distribution, mean and standard deviation of X.

Construct the truth table for \((p\overset { \_ \_ }{ \vee } q)\wedge (p\overset { \_ \_ }{ \vee } \neg q)\)

If F(\(\alpha\)) = \(\left[ \begin{matrix} \cos { \alpha } & 0 & \sin { \alpha } \\ 0 & 1 & 0 \\ -\sin { \alpha } & 0 & \cos { \alpha } \end{matrix} \right] \), show that [F(\(\alpha\))]^-1 = F(-\(\alpha\)).

If \(2cos\alpha=x+\frac { 1 }{ x } \) and \(2cos\ \beta =y+\frac { 1 }{ y } \), show that ​\({ x }^{ m }{ y }^{ n }+\frac { 1 }{ { x }^{ m }{ y }^{ n } } =2cos(m\alpha +n\beta )\)​

Solve the equation (x-2) (x-7) (x-3) (x+2)+19 = 0

If \(\vec { a } =-2\hat { i } +3\hat { j } -2\hat { k } ,\vec { b } =3\hat { i } -\hat { j } +3\hat { k } ,\vec { c } =2\hat { i } -5\hat { j } +\hat { k } \) find \((\vec { a } \times \vec { b } )\times \vec { c } \) and \((\vec { a } \times \vec { b } )\times \vec { c } \). State whether they are equal.

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.

For each of the following functions find the f_x, f_y, and show that f_xy = f_yx
f(x, y) = \(\frac { 3x }{ y+sinx \ } \) 

Evaluate: \(\int _{ 0 }^{ \frac { \pi }{ 2 } }{ (\sqrt { tan\ x } +\sqrt { cot\ x } )dx } \)

Solve the Linear differential equation:
\(\frac { dy }{ dx } +\frac { 3y }{ x } =\frac { 1 }{ { x }^{ 2 } } \), given that y = 2 when x = 1 

Suppose the amount of milk sold daily at a milk booth is distributed with a minimum of 200 Iitres and a maximum of 600 litres with probability density function 
\(\begin{cases} \begin{matrix} k & 200\le x\le 600 \end{matrix} \\ \begin{matrix} 0 & otherwise \end{matrix} \end{cases}\) 
Find
(i) the value of k
(ii) the distribution function
(iii) the probability that daily sales will fall between 300 litres and 500 litres?

Let M = \(\left\{ \left( \begin{matrix} x & x \\ x & x \end{matrix} \right) :x\in R-\{ 0\} \right\} \) and let * be the matrix multiplication. Determine whether M is closed under ∗. If so, examine the commutative and associative properties satisfied by ∗ on M.