Q1. Write 3 Rational Numbers Between 3 & 4

NUMBER SYSTEM

Q1. Write 3 rational numbers between 3 & 4.

Q2. Write 3 rational numbers between -3/4 & -1/2.

Q3. Write 5 rational numbers between 3/5 & 4/5.

Q4. Write three numbers whose decimal expansions are non-terminating non - recurring.

Q5. Find two irrational numbers between 2 & √3.

Q6. Find two irrational numbers between 1.12 & 1.13.

Q7. Find three irrational numbers between the rational numbers 5/7 & 9/11.

Q8. Which is smaller in each of the following.

a) √2 or ∛3 b) ∜5 or ∛4 c) ∛5 or √10 d) √ 10 or ∜ 9

Q9. Arrange in ascending order: ∜3, ∛4, √2.

Q10. Simplify: a) ∛24 + ∛ 81 - ∛192 b) ∜81 - 8 ∛216 + 15 ∜16 + √225

Q11. Rationalise the denominator.

a) 11+ 2- √3 b) 13+2- √5 c) 156+5- √11

Q12. If a = 6 - √35, find the value of a2 + 1/ a2.

Q13. If a = 1 - √2, find the value of (a - 1/a)3.

Q14. If x = 3 + 2√2, find the value of x4 + 1/ x4.

Q15. If x = 7 + Ö40, find the value of Öx + 1/ Öx.

Q16. Simplify: 2√62+√3 + 6√26+ √3 - 8√36+ √2

Q17. If 3+2√23- √2 = x + y√2, find x & y where x & y are rational numbers.

Q18. Find the continued product of

( √2 + √3 + √5) (√2 - √3 + √5) (√2 + √3 - √5 ) ( - √2 + √3 + √5)

Q19. If √2 =1.414 & √3 = 1.732 , find the value of 12+3 .

Q20. Prove: 11+√2 + 12+√3 + 13+√4 + ...... + 18+√9 = 2.

Q21. If x = 12- √3 , find the value of x3 - 2x2 - 7x + 5.

POLYNOMIALS

Q1. Factorise each of the following.

a) 4x2 + 28x + 49 b) 24x2 - 41x +12

c) x2 - x - 6 d) (x +1)2 - (y - 1)2

e) 25x2 - 10x + 1 - 36y2 f) 1 - x2 - y2 - 2xy

g) 27x3 + 64 h) 8x3 - (2x - y)3

i) 4(x - y)2 - 12(x - y)(x + y) + 9(x + y)2 j) 4x2 + 9y2 + z2 - 12xy + 4xz - 6yz

k) a4 - a l) 18a3 + 14a2b + 16ab2 + 127b3

m) (2a + 3)3 + (3a - 2)3 - (5a +1)3

Q2. Use remainder theorem to find the remainder when p(x) is divided by q(x) in each of the following.

i) p(x) = x3 - 2x2 + 5x - 1, q(x) = x - 2 ii) p(x) = x3 - 2x2 + 1, q(x) = x + 1

iii) p(x) =4x3 -12x2 + 11x - 5, q(x) = x - 12

Q3. Use factor theorem to verify that q(x) is a factor of p(x) in each of the following.

i) p(x) = x2 -5x + 6 , q(x) = x - 2 ii) p(x) = 7x2 - 2√8 x - 6, q(x) = x - √2

Q4. Find the value of 'a' so that x + 6 is a factor of x3 + 3x2 + 4x + a.

Q5. Show that x + a is a factor of xn + an for any odd positive integer n.

Q6. For what value of 'm' is the polynomial 2x4 - mx3 +4x2 + 2x + 1 divisible by 1 - 2x?

Q7. The polynomial ax3 + 3x2 - 3 and 2x3 - 5x + a when divided by x - 4 leave the same remainder in

each case. Find the value 'a'.

Q8. Find the integral zeroes of the polynomial 2x3 + 5x2 - 5x - 1.

Q9. Find the values of m & n so that x - 1 & x + 2 are factors of the polynomial x3 + 10x2 + mx + n.

Q10. If x - 3 & x - 1/3 are factors of ax2 + 5x + b, show that a = b.

Q11. Find the value of a3 - 8b3 -36ab - 216 when a = 2b + 6.

Q12. If A & B be the remainders when the polynomials x3 + 2x2 - 5ax - 7 and x3 + ax2 - 12x + 6 are divided by x + 1 & x - 2 respectively. If 2A + B = 6, find the value of a.

GEOMETRY

Q1. In the adjoining figure, if AC = BD, show that AB = CD . . . .

State the Euclid's postulate/axiom used for the same. A B C D

Q2. In the adjoining figure, Ð CAB =400, ÐACB = 300, C

Ð EDB = 450 & ÐCED = x. find the value of x. E

Q3. In triangle ABC, the bisector AD of angle A

is perpendicular to side BC. Show that A B D

triangle ABC is an isosceles triangle. D

Q4. In the adjoining figure, ABCDE is a pentagon in which E C

ÐAEB = a, ÐEBD = b & ÐEDB = c.

Find the relation between a, b & c. A B

Q5. Prove that two distinct lines cannot have more than one point in common.

Q6. If a point lies between two points A & B such that AC = BC, then prove that AC = ½ AB.

Q7. Prove that every line segment has one and only one end point.

COORDINATE GEOMETRY A

Q1. In the adjoining figure, ABC is an equilateral triangle.

The coordinates of vertices B and C are (3, 0) and (-3, 0)

respectively. Find the coordinates of its vertex A. C O B

Q2. Plot the points A(3, 0), B(3, 3) and C(0, 3) in a Cartesian plane. Join OA, AB, BC and CO. Name

the figure formed and write its one property.

Q3. A point lies on x-axis at a distance of 9 units from y-axis. What are its coordinates? What will be its

coordinates if it lies on y-axis at a distance of -9 units from x-axis?

Q4. Name the quadrants in which the following points lie:

i) (-4, -5) ii) (-3, 5) iii) (2, 2) iv) (4, -1)

Q5. The lengths of perpendiculars PM & PN drawn from a point P, on x-axis & y-axis are 3 and 2 units

respectively. Find the coordinates of points P,M & N.

Q6. Plot the points A(-5, 3), B(3, 3) , C(3, 0) & D(-5, 0) in a Cartesian plane. Name the figure ABCD.

Find the ratio of areas of two parts of ABCD in the I quadrant & II quadrant.

HERON'S FORMULA

Q1. Find the area of a triangle whose perimeter is 180cm & two of its sides are 80cm & 18cm. Also calculate the altitude of the triangle corresponding to the shortest side.

Q2. The sides of a triangle are 8cm & 11cm and its perimeter is 32cm. Find the area of the triangle.

Q3. The lengths of the sides of a triangle are in the ratio 3 : 4 : 5 & its perimeter is 144cm. Find the area of the triangle.

Q4. Find the area of rhombus whose perimeter is 200cm &one of the diagonals is 80cm.

Q5. The perimeter of a triangle is 60cm. Its hypotenuse is 26cm. Find the other two sides & the area of the triangle.

Q6. Two parallel sides of a trapezium are 60cm &77cm and other sides are 25cm & 26cm. Find the area of the trapezium.

Q7. From a point in the interior of an equilateral triangle perpendiculars drawn to three sides are 8cm, 10cm & 11cm. Find the area of the triangle to the nearest cm. (Take Ö3 = 1.73)