State whether the following sets are finite or infinite.
{x \(\in \) Z : x is even and less than 10}

Discuss the following relations for reflexivity, symmetricity and transitivity :
Let A be the set consisting of all the female members of a family. The relation R defined by "aRb if a is not a sister of b".

Discuss the following relations for reflexivity, symmetricity and transitivity :
On the set of natural numbers, the relation R is defined by "xRy if x + 2y = 1".

Find the range of the following functions given by f(x) = 1 + 3 cos 2x.

Simplify \(\left( 125 \right) ^{ \frac { 2 }{ 3 } }\)

Solve \(2\left| x+1 \right| -6\le 7\) and graph the solution set in a number line.

Solve 23x < 100 when
(i) x is a natural number
(ii) x is an integer.

Simplify \(\sqrt{x^2-10x+25}\)

If one of the roots of a quadratic equation is \((1-\sqrt{5})\) find the quadratic equation.

Find the principal value of sec^-1\(\left( -\sqrt { 2 } \right) \)

Find the values of other five trigonometric functions for the following
Sec \(\theta\) = \(\frac { 13 }{ 5 },\) \(\theta\) lies in the IV quadrant

Express each of the following as a product.
sin 75^o - sin 35^o

Find the value of cos 135°.

Find the value of tan 120°.

There are six periods in each working day of a school. In how many ways can one arrange 5 subjects such that each subject is allowed atleast one period?

Find the general term in the expansion of \({ \left( \frac { 4x }{ 5 } -\frac { 5 }{ 2x } \right) }^{ 9 }\)

Evaluate 98⁴ .

Find the angle between the lines 3x² + 10xy + 8y² + 14x + 22y + 15 = 0.

Find the angle between the pair of straight lines given by
(a² - 3b²)x² + 8ab xy+(b² -3a²)y² =0.

If A =\(\begin{bmatrix} 4 & 6 & 2 \\ 0 & 1 & 5 \\ 0 & 3 & 2 \end{bmatrix}\) and B = \(\begin{bmatrix} 0 & 1 & -1 \\ 3 & -1 & 4 \\ -1 & 2 & 1 \end{bmatrix}\) verify (A - B)^T = A^T - B^T

In problems 1-6, using the table estimate the value of the limit.
\(lim_{x\rightarrow 2}{x-2\over x^2-x-2}\)

x	1.9	1.99	1.999	2.001	2.01	2.1
f(x)	0.344820	0.33444	0.33344	0.333222	0.33222	0.332258

Find the derivation  : x² e^x sin x

Integrate the function with respect to x : \(a{ sec }^{ 2 }\left( bx+c \right) +\cfrac { q }{ { e }^{ 1-mx } } \)

If A and B are two events such that \(P(A\cup B)=0.7 ,\) \(P(A\cap B)=0.2\) ,\(P(\bar { B } )=0.5,\) show that A and B are independent.