Solve the equations
x⁴+ 3x³- 3x - 1 = 0

Evaluate :\(\int _{ 0 }^{ 1 }{ \frac { 2x+7 }{ { 5x }^{ 2 }+9 } } dx\)

Evaluate: \(\int _{ 0 }^{ \frac { \pi }{ 2 } }{ \frac { cos\theta }{ (1+sin\theta )(2+sin\theta ) } } d\theta \)

Show that \(\int _{ 0 }^{ \pi }{ g(sinx)dx=2 } \int _{ 0 }^{ \frac { \pi }{ 2 } }{ g(sinx)dx, } \) where g(sin x) is a function of sin x.

Evaluate \(\int _{ 0 }^{ x }{ { x }^{ 2 } } \)cos nx dx, where n is a positive integer.

Find the differential equation of the family of parabolas with vertex at (0, −1) and having axis along the y-axis.

Find the differential equation of the curve represented by xy = ae^x + be^−x + x².

Solve the following differential equations:
\(sin\frac { dy }{ dx } =a,y(0)=1\)

Solve the following differential equations:
\(\frac { dy }{ dx } -x\sqrt { 25-{ x }^{ 2 } } =0\)

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

Solve the Linear differential equation:
\(x\frac { dy }{ dx } +y=xlogx\)

Solve the Linear differential equation:
\(x\frac { dy }{ dx } +2y-x^2logx=0\)

The engine of a motor boat moving at 10 m/s is shut off. Given that the retardation at any subsequent time (after shutting off the engine) equal to the velocity at that time. Find the velocity after 2 seconds of switching off the engine.

The probability density function of X is 
\(f(x)=\left\{\begin{array}{cc} x & 0
find P(0.2 ≤ X< 0.6) 

For the random variable X with the given probability mass function as below, find the mean and variance \(f(x)= \begin{cases}2(x-1) & 1

The probability that a certain kind of component will survive a electrical test is \(\frac { 3 }{ 4 } \). Find the probability that exactly 3 of the 5 components tested survive.

Suppose a pair of unbiased dice is rolled once. If X denotes the total score of two dice, write down
(i) the sample space
(ii) the values taken by the random variable X,
(iii) the inverse image of 10, and
(iv) the number of elements in inverse image of X.

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.

Two fair coins are tossed simultaneously (equivalent to a fair coin is tossed twice). Find the probability mass function for number of heads occurred.

Find the mean and variance of a random variable X , whose probability density function is \(f(x)=\begin{cases} \begin{matrix} { \lambda e }^{ -2x } & for\ge 0 \end{matrix} \\ \begin{matrix} 0 & otherwise \end{matrix} \end{cases}\)

Verify the
(i) closure property,
(ii) commutative property,
(iii) associative property
(iv) existence of identity and
(v) existence of inverse for the arithmetic operation - on Z.

How many rows are needed for following statement formulae?
\(p \vee \neg t \wedge(p \vee \neg s)\)

Let \(A=\left( \begin{matrix} 1 & 0 \\ 0 & 1 \\ 1 & 0 \end{matrix}\begin{matrix} 1 & 0 \\ 0 & 1 \\ 0 & 1 \end{matrix} \right) ,B=\left( \begin{matrix} 0 & 1 \\ 1 & 0 \\ 1 & 0 \end{matrix}\begin{matrix} 0 & 1 \\ 1 & 0 \\ 0 & 1 \end{matrix} \right) ,C=\left( \begin{matrix} 1 & 1 \\ 0 & 1 \\ 1 & 1 \end{matrix}\begin{matrix} 0 & 1 \\ 1 & 0 \\ 1 & 1 \end{matrix} \right) \)be any three boolean matrices of the same type.
Find (A∨B)∧C 

Prove that q ➝ p ≡ ¬p ➝ ¬q

Show that ¬(p↔️q) ≡ p↔️¬q