Let F₁ be the set of parallelograms, F₂ be the set of rectangles, F₃ be the set of rhombus and F₄ be the set of squares. Then, show that F₁ is equal to the union of all sets.

Write the following sets in roster form:
B = {x : x ∈ N and -1 < x - 3 < 6}

Prove that the sum of first n even numbers is n(n+1).

The marks obtained by a student of class XI in first and second terminal examinations are 62 and 48, respectively. Find the minimum marks he should get in the annual examination to have an average of atleast 60 marks.

How many 4-digit numbers are there, when a digit may be repeated any number of times?

Expand the following expansions.
\(\frac { 1 }{ \sqrt [ 3 ]{ \left( 27-6x \right) ^{ 2 } } } \)

The 6th and 17th terms of an AP are 19 and 41 respectively, find the 40th term.

Find the equation of hyperbola, having foci \((\pm 4,0)\)  and length of latusrectum is 12.

In a single throw of two die, find the probability of getting a total of 7 or 9.

If A = { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, then insert appropriate symbol ∈ or ∉ in each of the following blank space.
-4 ... A

If z = x + iy, w = \(\frac { 1-iz }{ z-i } \) and \(|w|=1\) then show that z is purely real.

Solve the following system of inequalities graphically.
\(x+y\le 4,2x-y>0,x\ge 0\quad and\quad y\ge 0.\)

If the integers r (>1), n (>2) and coefficients of (3r)th and (r + 2)nd terms in the expansion of (1 + x)²ⁿ are equal, then prove that n = 2r.

If a,b,c are in AP and b,c,d are in GP and \(\frac { 1 }{ c } ,\frac { 1 }{ d } ,\frac { 1 }{ e } \) are in AP, then prove that a,c,e are in GP.

Find the equation of line passing through the origin and the intersection of the line x - y - 7 = 0 and 2x + y - 2 = 0

Find the equation of the parabola, whose focus is the point (4,0) and whose directrix is x = -4. Also, find the length of latusrectum

Find the equation of hyperbola, when foci are at \((\pm 5,0)\) and transvere axis is of length 8.

The mid-point of the sides of a triangle are (1, 5, -1), (0, 4, -2) and (2, 3, 4) find its vertices and also find the centroid of the triangle.

Find the mean deviation from the mean for the following data
6, 5, 5.25, 5.5, 4.75, 4.5, 6.25, 7.75, 9.

Two dice are thrown simultaneously. Find the probability of getting a multiple of 3 as the sum.

20 cards are numbered from 1 to 20.In those of them, one card is drawn at random.What is the probability that the number on the card drawn is an odd number?

Find the intersection of each pair of sets of question 1above.

Prove the following by using the principle of mathematical induction for all n ∊ N: \(\frac { 1 }{ 2.5 } +\frac { 1 }{ 5.8 } +\frac { 1 }{ 8.11 } +...+\frac { 1 }{ (3n-1)(3n+2) } \)= \(\frac { n }{ (6n+4) } \)

If the fourth term in the expansion of \(\left( ax+\frac { 1 }{ x } \right) ^{ n }is \frac { 20 }{ 27 } \) then find the value of a and n.

Find four numbers in G.P. whose sum is 85 and product is 4096.

Three vertices of a parallelogram ABCD are A(3, – 1, 2), B (1, 2, – 4) and C (– 1, 1, 2). Find the coordinates of the fourth vertex.

State the converse and contrapositive of each of the following statements:
(i) p: A positive integer is prime only if it has
no divisors other then 1 and itself.
(ii) q: I goto a beach whenever it is a sunny day.
(iii) r: if it is not outside, then you feel thirsty.

The sum and sum of squares corresponding to length x (in cm) and weight y (in gm) of 50 plant products are given below:
\(\sum _{ i=1 }^{ 50 }{ { x }_{ i } } =212,\sum _{ i=1 }^{ 50 }{ { x }_{ i }^{ 2 } } =902.8,\sum _{ i=1 }^{ 50 }{ { y }_{ i } } =261,\) \(\sum _{ i=1 }^{ 50 }{ { y }_{ i }^{ 2 }=1457.6 } \) 
which is more varying, the length or weight?

The mean and variance of eight observations are 9 and 9.25 respectively. If six of the observations are 6, 7, 10, 12, 12 and 13, find the remaining two observations.