\({ \left[ 1+\frac { { d }^{ 2 }y }{ { dx }^{ 2 } } \right] }^{ \frac { 3 }{ 2 } }=a\frac { { d }^{ 2 }y }{ d{ x }^{ 2 } }\)
Here we eliminate the radical sign.
Squaring both sides, we get
\({ \left[ 1+\frac { { d }^{ 2 }y }{ { dx }^{ 2 } } \right] }^{ \frac { 3 }{ 2 } }=a^2(\frac { { d }^{ 2 }y }{ d{ x }^{ 2 } })^2\)
∴ order = 2, ∴ Degree = 3

Let \(A=\left( \begin{matrix} i & -1 \\ 3 & -6 \end{matrix} \right) \)
Order of A is 2 \(\times\) 2
\(\therefore \rho (A)\le 2\) [Since minimum of (2, 2) is 2]
 Consider the second order minor
\(\left| \begin{matrix} 1 & -1 \\ 3 & -6 \end{matrix} \right| =-6-(-3)\)
= -6 + 3 = -3
\(\neq 0\)
There is a minor of order 2, which is not zero
\(\therefore \rho (A)=2\)

The mathematical form of assignment problem is minimize Z = \(\overset { n }{ \underset { i=1 }{ \Sigma } } \overset { n }{ \underset { j=1 }{ \Sigma } } { { C }_{ ij } }{ x }_{ ij }\) subject to the
Constraints \(\overset { n }{ \underset { i=1 }{ \Sigma } }{ x }_{ ij } = 1, j=1,2,.....n\)
\(\overset { n }{ \underset { j=1 }{ \Sigma } }{ x }_{ ij } = 1, i=1,2,.....n\) and x_ij = 0 or 1 for all i, j

The control limits for mean chart in two different cases are :

Case (i) when \(\overline {X }\) and SD are given	Case (ii) when \(\overline {X }\) and SD are not given
(i) UCL = \({\overline{X}} + 3 \frac{\sigma}{\sqrt n}\)	(i) UCL =  \(\overline {X }\) + A2 \(\overline {R }\)
(ii) CL =  \(\overline {X }\)	(ii)  CL =  \({\overline {X }}\)
(iii) LCL =   \({\overline{X}} - \frac{3\sigma}{\sqrt n}\)	(iii) LCL =  \(\overline {X }\) - A2 \(\overline {R }\)

It is the frequency distribution which is formed with various values of a statistic computed from different samples of the same size drawn from the same population.

Let f(x) = x³ cos³x
f(-x) = (-x)³ [cos(-x)]³
= -x³ (cos x)³
[Since cos x is an even function]
= -f(x)
∴ f(-x) = -f(x) ⇒ f(x) is an odd function
By the property, \(\int _{ -a }^{ a }{ f\left( x \right) } dx=0\) if f(x) is an odd function.
\(\Rightarrow \int _{ \frac { -\pi }{ 4 } }^{ \frac { \pi }{ 4 } }{ { x }^{ 3 } } { cos }^{ 3 }x\ dx=0\)

(i) Number of lightnings per second.
(ii) Number of printing mistakes per page in a textbook.

When we perform any experiment, we expect an outcome. We associate a real numbers with each outcome of an experiment. In other words, we considering a function whose domain is the set of possible outcomes and whose range is subset of the set of real numbers such a function is called random variable.

\(\int { { \left( \sqrt { 2x } -\frac { 1 }{ \sqrt { 2x } } \right) }^{ 2 } } dx\)

\(=\int { \left( 2x-2+\frac { 1 }{ 2x } \right) } dx\)
\(=2\frac { { x }^{ 2 } }{ 2 } -2x+\frac { 1 }{ 2 } \log { \left| x \right| +c } \)
\(={ x }^{ 2 }-2x+\frac { 1 }{ 2 } \log { \left| x \right| } +c\)