If A, B and C are overlapping sets, then draw Venn diagram for the following sets:
(i) (A-B)\(\cap \)C
(ii) (A\(\cup \)C)-B
(iii) A-(A\(\cap \)C)
(iv) (B\(\cup \)C)-A
(v) A\(\cap \)B\(\cap \)C

Verify \(n\left( A\cup B\cup C \right) =n(A)+n(B)+n(C)-n(A\cap B)-n(B\cap C)-n(A\cap C)+n(A\cap B\cap C)\)for the following sets.
(i) A = {a, c, e, f , h} , B = {c, d, e, f } and C = {a, b, c, f }
(ii) A = {1, 3, 5} , B = {2, 3, 5, 6} and C = {1, 5, 6, 7}

A soap company interviewed 800 people in a city. It was found out that \(\frac{3}{8}\) use brand A soap, \(\frac{1}{5}\)use brand B soap, 70 use brand A and B soap, 55 use brand B and C soap, 60 use brand A and C soap and \(\frac{1}{40}\) use all the three brands Find,
(i) Number of people who use exactly two branded soaps,
(ii) Number of people who use atleast one branded soap,
(iii) Number of people who do not use any one of these brands.

Verify \(n\left( A\cup B\cup C \right) \) = n(A) + n(B) +n(C) - \(n\left( A\cap B \right) -n\left( B\cap C \right) -n\left( A\cap C \right) -n\left( A\cap C \right) +\left( A\cap B\cap C \right) \) for the following A = {1,3,5,6,8} C = {1,2,3,6}

In a party of 45 people, each one likes tea or coffee or both. 35 people like tea and 20 people like coffee. Find the number of people who
(i) like both tea and coffee.
(ii) do not like Tea.
(iii) do not like coffee

Find any two irrational numbers between \(\sqrt { 2 } \) and \(\sqrt { 3 } \)

Find the 5^th root of 100000

Simplify the following using multiplication and division properties of surds:
(i) \(\sqrt { 3 } \times \sqrt { 5 } \times \sqrt { 2 } \)
(ii) \(\sqrt { 35 } \div \sqrt { 7 } \)
(iii) \(\sqrt [ 3 ]{ 27 } \times \sqrt [ 3 ]{ 8 } \times \sqrt [ 3 ]{ 125 } \)
(iv) \((7\sqrt { a } -5\sqrt { b } )(7\sqrt { a } +5\sqrt { b } )\)
(v) \(\left[ \sqrt { \frac { 225 }{ 729 } } -\sqrt { \frac { 25 }{ 144 } } \right] \div \sqrt { \frac { 16 }{ 81 } } \)

Without actual division, find which of the following rational numbers have terminating decimal expansion
(i) \(\frac { 7 }{ 128 } \)
(ii) \(\frac { 21 }{ 15 } \)
 (iii) \(4\frac { 9 }{ 35 } \)
(iv) \(\frac { 219 }{ 2200 } \)

If the polynomials f(x) = ax³ + 4x² + 3x –4 and g(x) = x³ – 4x + a leave the same remainder when divided by x–3, find the value of a. Also find the remainder.

If both (x - 2) and \(\left( x-\frac { 1 }{ 2 } \right) \) are the factors of ax²+ 5x + b, then show that a = b.

The sum of the digits of a given two digit number is 5. If the digits are reversed, the new number is reduced by 27. Find the given number.

Draw a triangle ABC, where AB = 8 cm, BC = 6 cm and ㄥB = 70⁰ and locate its circumcentre and draw the circumcircle.

Diagonal AC of a parallelogram ABCD bisects ㄥA. Show that
(i) it bisects ㄥC also
(ii) ABCD is a rhombus.

Find the value of x° in the following figures:

Show that the point A (3,7) B (6, 5) and C (15, -1) are collinear.

Find the coordinates of the point which divides the line segment joining the points (3, 5) and (8, −10) internally in the ratio 3:2.

In the class, weight of students is measured for the class records. Caculate mean weight of the students using direct method.

Weight in kg	15-25	25-35	35-45	45-55	55-65	56-75
No.of students	4	11	19	14	0	2

A farmer has a field in the shape of a rhombus. The perimeter of the field is 400 m and one of its diagonal is 120 m. He wants to divide the field into two equal parts to grow two different types of vegetables. Find the area of the field.