Let A =  {1,2, 3, 4} and B = {-1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12} Let R = {(1, 3), (2, 6), (3, 10), (4, 9)} \(\subseteq \) A x B bea relation. Show that R is a function and find its domain, co-domain and the range of R.

Let A = {0, 1, 2, 3} and B = {1, 3, 5, 7, 9} be two sets. Let f: A \(\rightarrow\)B be a function given by  f(x) = 2x + 1. Represent this function as  a set of ordered pairs.

State whether the graph represent a function. Use vertical line test.

Let A = {0, 1, 2, 3} and B = {1, 3, 5, 7, 9} be two sets. Letf: A \(\rightarrow\)B be a function given by  f(x) = 2x + 1. Represent this function as  a table .

Let A = {0, 1, 2, 3} and B = {1, 3, 5, 7, 9} be two sets. Let f : A \(\rightarrow\)B be a function given by  f(x) = 2x + 1. Represent this function as an arrow .

Let A = {0, 1, 2, 3} and B = {1, 3, 5, 7, 9} be two sets. Letf: A \(\rightarrow\)B be a function given by  f(x) = 2x + 1. Represent this function as a graph.

Use Euclid's algorithm to find the HCF of 4052 and 12756.

Show that any positive odd integer is of the form 4q + 1 or 4q + 3, where q is some integer.

Find the LCM and HCF of 6 and 20 by the prime factorisation method.

Prove that \(\sqrt { 3 } \) is irrational

Solve the following system of linear equations in three variables.
x + y + z = 6; 2x + 3y + 4z = 20;
3x + 2y + Sz = 22

Using quadratic formula solve the following equations.
p²x² + (P² -q²) X - q² = 0

Using quadratic formula solve the following equations.9x²-9(a+b)x+(2a²+5ab+2b²)=0

Find the values of k for which the following equation has equal roots.
(k - 12)r + 2(k - 12)x + 2 = 0

Prove that the equation x²(a²+b²)+2x(ac+bd)+(c²+ d²) = 0 has no real root if ad≠bc.

In figure if PQ || RS Prove that \(\Delta POQ\sim \Delta SOQ\)

In figure OA· OB = OC·OD
Show that \(\angle A=\angle C\ and\ \angle B=\angle D\)

In figure the line segment xy is parallel to side AC of \(\Delta ABC\) and it divides the triangle int two parts of equal areas. Find the ratio \(\cfrac { AX }{ AB } \)

Find a relation between x and y such that the point (x, y) is equidistant from the points (7, 1) and (3, 5).

Show that the points (1, 7), (4, 2), (-1,-1) and (-4,4) are the vertices of a square.

If A (-5, 7), B (-4, -5), C (-1, -6) and D (4, 5) are the vertices of a quadrilateral, find the area of the quadrilateral ABCD.

Given tan A \(\frac { 4 }{ 3 } \) find the other trigonometric ratios of the angle A.

Prove that \(\frac { sin\theta -cos\theta +1 }{ sin\theta +cos\theta -1 } =\frac { 1 }{ sec\theta -tan\theta } \) using the identity sec²θ= 1+ tan²θ.

Prove that sec A (1 - sin A) (sec A + tan A) = 1.

In a right triangle ABC, right-angled at B, if tan A = 1, then verify that 2 sin A cos A = 1.

\(\frac { sin\quad \theta }{ 1+cos\theta } +\frac { 1+cos\theta }{ sin\quad \theta } =2cosec\theta \)

\(\frac { tan\quad A }{ 1\quad sec\quad A } ​​​​\frac { tan\quad A }{ 1\quad sec\quad A } ​​​​2cosec\quad A\)

If tan A=\(\frac{3}{4}\), then sin A cos A=\(\frac{12}{15}\)

(sin ∝+cos ∝)(tan ∝+cot ∝)=sec ∝+cosec ∝

\(\sqrt3\)1 (3-cot30^o)=tan³ 60^o-2sin60^o

\(1+\frac { { cot }^{ 2 }\alpha }{ 1+cosex\alpha } =cosec\alpha\)

tan θ+tan(90^o-θ)=secθ sec(90^o-θ)

Find the angle of elevation of the sun when the shadow of a pole h metres high is \(\sqrt3\) h metres long.

If \(\sqrt3\) tan θ=1, then find the value of sin²θ-cos²θ

A ladder 15 metres long just reaches the top of a vertical wall. If the ladder makes an angle of 60^o with the wall, finf the height of the wall.

Simplify (1+tan²θ)(1-sinθ)(1+sinθ)

If 2sin²θ-cos2θ=2, then find the value θ.

Show that \(\\ \frac { { cos }^{ 2 }({ 45 }^{ 0 }+\theta ){ +cos }^{ 2 }({ 45 }^{ 0 }-\theta ) }{ tan({ 60 }^{ 0 }+\theta )tan({ 30 }^{ 0 }-\theta ) } =1\)

An observer 1.5 metres tall is 20.5 metres away from a tower 22 metres high. Determine the angle of elevation of the top of the tower from eye of the observer.

Show that tan⁴θ+tan²θ=sec⁴θ-sec²θ.

If the radii of the circular ends of a conical bucket which is 45 cm high are 28 cm and 7 cm, find the capacity of the bucket. (Use π = \(\frac{22}{7}\))

Find the depth of a cylindrical tank of radius 28 m, if its capacity is equal to that of a rectangular tank of size 28 m x 16 m x 11 m.

Find the standard deviation of 30, 80, 60, 70, 20, 40, 50 using the direct method.

Find the standard deviation for the following data. 5, 10, 15, 20, 25. And also find the new S.D. if three is added to each value.

The marks scored by 5 students in a test for 50 marks are 20, 25, 30, 35, 40. Find the S.D for the marks. If the marks are converted for 100 marks, find the S.D. for newly obtained marks.