Find the inverse of the matrix \(\left[ \begin{matrix} 2 & -1 & 3 \\ -5 & 3 & 1 \\ -3 & 2 & 3 \end{matrix} \right] \).

Find the inverse (if it exists) of the following:
\(\left[ \begin{matrix} 5 & 1 & 1 \\ 1 & 5 & 1 \\ 1 & 1 & 5 \end{matrix} \right] \)

Find the rectangular form of the complex numbers
\(\left( cos\frac { \pi }{ 6 } +isin\frac { \pi }{ 6 } \right) \left( cos\frac { \pi }{ 12 } +isin\frac { \pi }{ 12 } \right) \)

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

Solve the equation
2x³ - 9x² + 10x = 3

Let p and q be rational numbers such that \(\sqrt{q}\) is irrational. If p + \(\sqrt{q}\) is a root of a quadratic equation with rational coefficients, then p − \(\sqrt{q}\) is also a root of the same equation.

Find the value of
\(cos\left( { sin }^{ -1 }\left( \frac { 4 }{ 5 } \right) -{ tan }^{ -1 }\left( \frac { 3 }{ 4 } \right) \right) \)

The position of a point P(x₁ , y ₁)  with respect to a given circle x² + y² + 2gx + 2 fy + c = 0 in the plane containing the circle is outside or on or inside the circle according as
\(x_{1}^{2}+y_{1}^{2}+2 g x_{1}+2 f y_{1}+c \text { is } \begin{cases}>0 & \text { or } \\ =0 & \text { or } \\ <0 & \end{cases}\)

Find the acute angle between the planes \(\vec { r } .(2\hat { i } +2\hat { j } +2\hat { k } )\) and 4x-2y+2z = 15.

The acute angle \(\theta\) between the two planes \(\vec{r} \cdot \vec{n}_{1}=p_{1} \text { and } \vec{r} \cdot \vec{n}_{2}=p_{2}\) is given by \(\theta=\cos ^{-1}\left(\frac{\left|\vec{n}_{1} \cdot \vec{n}_{2}\right|}{\left|\vec{n}_{1}\right|\left|\vec{n}_{2}\right|}\right)\)

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 limit, if necessary use l ’Hôpital Rule
\(​​​​​​\underset { x\rightarrow \infty }{ lim } \ { \left( 1+\frac { 1 }{ x } \right) }^{ x }\)

If A is non-singular, then 
\((i)\ \left|A^{-1}\right|=\frac{1}{|A|} \)
\((ii)\ \left(A^{T}\right)^{-1}=\left(A^{-1}\right)^{T} \)
\((iii)\ (\lambda A)^{-1}=\frac{1}{\lambda} A^{-1},\)
where is \(\lambda\) non-zero scalar

Let \(f(x)=\sqrt [ 3 ]{ x } \). Find the linear approximation at x = 27. Use the linear approximation to approximate \(\sqrt [ 3 ]{ 27.2 } \)

If α and β are the roots of the quadratic equation 17x²+43x−73 = 0 , construct a quadratic equation whose roots are α + 2 and β + 2.

Show that \(\int ^\frac{2\pi}{0}_{0}\) g(cos x)dx = 2 \(\int ^{\pi}_{0}\) g(cosx)dx where g(cos x) is a function of cos x

Evaluate the following:
\(\int _{ 0 }^{ \frac { 1 }{ 2 } }{ \frac { { e }^{ { a\ sin }^{ -1x } }{ sin }^{ -1 }x }{ \sqrt { 1-{ x }^{ 2 } } } dx } \)

Find the differential equation of the family of circles passing through the points (a, 0) and (−a, 0).

Two balls are chosen randomly from an urn containing 6 white and 4 black balls. Suppose that we win Rs. 30 for each black ball selected and we lose Rs. 20 for each white ball selected. If X denotes the winning amount, then find the values of X and number of points in its inverse images.

Write each of the following sentences in symbolic form using statement variables p and q.
(i) 19 is not a prime number and all the angles of a triangle are equal.
(ii) 19 is a prime number or all the angles of a triangle are not equal
(iii) 19 is a prime number and all the angles of a triangle are equal
(iv) 19 is not a prime number

Solve: 3x+ay = 4, 2x + ay = 2, a ≠ 0 by Cramer's rule.

Evaluate \(\int _{ 0 }^{ \frac { \pi }{ 2 } }{ { cos }^{ 10 }xdx } \)

By using the properties of definite integrals, I evaluate \(\int_{0}^{1}|x-1| d x\)

Solve:\(\frac { dy }{ dx } =\frac { 1-cosx }{ 1+cosx } \)

In a continuous distribution the p.d.f of x is \(f(x)=\left\{\begin{array}{c} \frac{3}{4} x(2-x), 0, Find the mean and variance of the distribution.

The p.d.f. of X is given by 

x	-2	2	5
P(X-x)	\( \frac{1}{4} \)	\( \frac{1}{4} \)	\( \frac{1}{2}\)

then find 4E(X²) - var(2X)

The probability that an event A happens in one treat of an experiment is 0.4. Three independent treats of the experiment are performed. Find the probability that the event A happens atleast once?

Show that f(x, y) = \(\frac { { x }^{ 2 }-{ y }^{ 2 } }{ { y }^{ 2 }+1 } \) is continuous at every (x, y) ∈ R²

Let U(x, y, z) = x² − xy + 3 sin z, x, y, z ∈ R Find the linear approximation for U at (2,−1,0).

Find ∆f and df for the function f for the indicated values of x, ∆x and compare
(1) f(x) = x³ - 2x² ; x = 2, ∆ x = dx = 0.5
(2) f(x) = x² + 2x + 3; x = -0.5, ∆x = dx = 0.1