BUDS PUBLIC SCHOOL
WORKSHEET 1* MATHEMATICS GRADE 9
Number System
1. Every rational number is:
a) natural number b) an integer c) a real number d) a whole number
2. To rationalize the denominator of 1/(√3 - √2) , we multiply it by the number:
a)√3 - √2 b) 1/(√3 + √2) c) (√3 + √2)/(√3 + √2) d) (√3 - √2)/(√3 - √2)
3. Find a rational number between 5 and 6.
4. Locate the following numbers on the number line a) √3 b) √5
5. Express the following in the rational form a) 2.3 b) 4.67 c) 7.543 d) 9.2123
6. Visualise the following numbers on the number line using successive magnification. a) 5.38 b) 1.85
7. Rationalise the denominators of: i) 3/√5 ii) 4/3√2 iii) 6/5√3 iv) √3/√2
8. State which of the following are true:
i) √3 + √2 = √5 ii) 2√5 + 3√5 = 5√5 iii) √8 - √3 = √5 iv) 8√2 - 3√2 = 5√2 v) 6√3 x 3√5 = 18√15
vi) 3/√3 = √3 vii) √7/7 = 1/√7 viii) √12 = 2√3
9. Evaluate: i)24 x 2-2 ii) 34 x 76 x 3-2 x 7-5 iii) 55 x 5-6 x 57 iv) 18 x 30 x 53 x 22 v) 57 ÷ 55
WORKSHEET 1**
1. Between 1/7 and 8/7 we can insert:
a) no rational number b) infinitely many rational numbers
c) infinitely many integers c) Only one rational number
2. The decimal representation of a rational number cannot be:
a) terminating b) non terminating
c) non terminating recurring d) non terminating non -recurring
3. On simplifying 6/(3√2 - 2√3), we get:
a) 3√2 + 2√3 b)3√3 + 2√2) c) 3√2 d) 2√3
4. Find the rational numbers of the following a) between 1 and 2 b)between 3/2 and 2 c)between 2/3 and 5/6 d)between 1/10 and 4/5
5. Locate the following numbers on the number line a) √5 b) √6 c) √7
6. Express the following in the rational form a) 1.9999….. b) 3.2 c) 0.06 d) 2.37
7. Visualise the following numbers on the number line using successive magnification. a) 2.847 b) 3.7
8. Rationalise the denominators of:i) 1/ (√3 + 1) ii) 3/ (√5 - √2) iii) 1 /(√3 + √2) iv) 2/(√3 - 2)
9. Simplify i) (5 + √3) (3 + √2) ii) (5 + √3) (3 - √2) iii) (5 - √3) (3 + √2) iv) (5 + √3) (3 + √2)
10. Evaluate: i) (625)-3/4 ii) (27/64)-2/3 iii) (1/32)-2/5 iv) (81)1/2 v) (10000)1/4
WORKSHEET 1***
1. Find 5 rational numbers between 5/8 and ¾.
2. Locate the following numbers on the number line a) √11 b) √18
3. Express the following in the rational form a) 0.123 b) 0.68 c) 0.235
4.Visualise the following numbers on the number line using successive magnification. a) 1.36 b)2. 07
5. Rationalise the denominators of : i) 5/(√13 - 2√2) ii) 30/(5√3 - 3√5) iii) 5/(4√3 - 3√2) iv) 7√3/(√10+√3)
6. Evaluate: i) (47)2 x (4-3)4 ii)( 1253)0 x (-37)0 iiii) (2-9 ÷ 2-11)3
7. Find the value of a in the following: 6/(3√2 -2√3) = 3√2 - a√3
8. If x = 2 + √3, then find x + 1/x .
9. If (5 + 2√3)/(7 + 4√3) = a + b√3, find the values of a and b.
10. Find the value of (13 + 23 + 33)-3/2
11. Simplify i) (2 + √3) x (2 - √3) ii) (4 - √13) x (4 +√13) iii) (5 - 2√3) (5 + 2√3) iv) (√7 + √5)(√7 - √5)
v) (3√2 + 2√3)(3√2 -2√3) vi) (7√3 + 2√2)(5√3 -3√2) vii) (2√7 - 3√5)(6√5 + 5√7)
viii) (16-1/5)5/2 ix) 3√(343)-2 x) (0.001)1/3 xi)[ (25)3/2 x (243)3/5]/(16)5/4 x (8)4/3
BUDS PUBLIC SCHOOL
WORKSHEET 2* MATHEMATICS GRADE 9
Area of Triangles- Heron’s formula
1. Find the area of a triangle with base 4 cm and altitude 6 cm.
2. Find the area of a triangle whose sides are 12 cm,16 cm and 20 cm.
3. Find the area of a triangular field whose sides are 50 m, 45m and 35 m.
4. The length of sides of a triangular park are 90 m, 70 m and 40m. Find its area.
WORKSHEET 2**
1. Find the area of an equilateral triangle of side 4√3 cm.
2. Find the area of an isosceles triangle whose one of the equal sides measures 8 cm and the third side is 4 cm long.
3. Find the base of an isosceles triangle whose area is 48 sq.cm and length of one of its equal sides is 10 cm.
4. The length of the sides of a triangle are in the ratio 4:3:5. If the perimeter of the triangle is 96 cm, find its area.
WORKSHEET 2***
1. Find the area of the quadrilateral ABCD in which AB = 3 cm, BC = 4 cm, CD =6 cm, DA = 5 cm and the diagonal AC = 5 cm.
2. Find the area of the quadrilateral ABCD in which AB = 10cm, BC =24 cm, CD =26 cm, DA = 20 cm and the diagonal AC = 26 cm.
3. Find the area of the quadrilateral ABCD in which AB = 16 cm, BC = CD = 26 cm, DA = 12 cm and the diagonal BD = 20 cm.
4.Find the area of the quadrilateral ABCD in which AB = BC = 20 cm, CD = 16 cm, DA = 12 cm and <D = 900.
5. A rhombus shaped platform is to be plastered. If each side of the platform is 15 m and its longer diagonal is 24 m, find the cost of plastering it at the rate of Rs. 36 per m2
BUDS PUBLIC SCHOOL
WORKSHEET 3* MATHEMATICS GRADE 9
Co-ordinate Geometry
1. The two quadrants where the abscissa are positive are
A) I and II B) II and III C) I and IV D) III and IV
2. A line l is parallel to x-axis and it is 4 units away from x- axis. Find the coordinates of that point.
3. The point (-2, 8) lies in ….quadrant. 4. The point (2,- 8) lies in ….quadrant.
5.The point (-2, -8) lies in ….quadrant. 6. If the line parallel to x-axis, then y is a ….
7. If a line is parallel to y –axis , then x is a …. 8. The equation of y – axis is ….
9. The equation of x- axis is ….. 10. Point (-10, 0) lies ………
WORKSHEET 3**
6. By taking suitable scale on both the axes, plot the following points in the coordinate plane:
A(-6, 3) ,B(3, -6) ,C (-3, -6), D (3, 6), E (0, 4), F(4, 0), G(0, -3) and H( - 5, 0).
7. In which quadrant or on which axis does each of the points A (4, 7), B (-4, 7), C (-8, -3), D(5,-8) E(7, 0) , F(0, -3) and G(0, 8) lie?
8. Plot the points (x, y) given in the following table, as points on the Cartesian plane, choosing suitable units of distance on the axes:
x / 3 / 5 / 0 / -4 / -3 / -5 / 0 / 4y / 0 / 2 / 4 / 2 / 0 / -4 / -2 / -4
WORKSHEET 3** *
1. A(-3, 4), C(1, -4) and D(-3,-4) are the vertices of the rectangle ABCD. Plot the given points on the Cartesian plane .Complete the figure to get the rectangle ABCD and then use it to find the coordinates of vertex B. Find the area of the rectangle ABCD
2. Plot the points P (3, 2), Q(3, -3) and R(6, -3) in a xy – plane. Name the figure obtained on the joining these points and, if possible, find the area of this figure.
BUDS PUBLIC SCHOOL
WORKSHEET 4* MATHEMATICS GRADE 9
Polynomials
1. Find the degree of polynomial of the following polynomials: a) 6 – x2 + 8x b) √6 c) 16
d) 3x e) 9 – y f) 7x6 + 2x3 – 1 g) 2y8 – 5y 10 + 10y6 h) -2 + m i) 3x + x3 – 5x2
2. Write the co-efficient of: a) y in -3xy b) xy in 5xyz2 c) a2 in a3 - 5a2 +7 d)y0 in 2y +5
3. From the given expressions separate monomials, binomials, trinomials
3/2 x, -6 + 3x, 4 – 2y +y3 , 8 + 2x , 3 x 2x -10 +y , 4+ 2z/5 , 2a – 3b, 4a +2x – 3y,
7x2 –8x +3, 2 + 5y – y2
4. Which of the algebraic expression is a polynomial: i) 2 + 4x – 7x2 ii) 3y – 1/y
iii) 5√x + 8x + 2 iv) 5√z + z2 v) 3x + 5/x2 – x/7 vi) y3/8 – y/7 +2
5. Find the value of the polynomial 3x2 – 2x + 8 at i) x = 0 ii) x = 2 iii) x = -2
6. Find the zero of the polynomial in each of the following cases: i) f(x) = x – 3 ii) f(y) = 2y + 5
iii) f(z) = z +7 iv) p(x) = 4x – 5 v) p(y) = 2y – 1
7. Find the remainder when polynomial f(x) = 3x2 – 2x + 1 is divided by :
i) x – 2 ii) x + 1 iii) x iv) x + 3 v) x + 4 vi) 2x + 1 vii) 3x – 2 viii) x – 3
8. Determine (x – 1) is a factor of which of the following polynomials:
i) x3 – 3x2 + 3x – 1 ii) 4x2 – 3x + 2 iii) 3x3 + 4x2 – 7x + 2
9. Factorise: i) x2 – 4 ii) 9 – x2 iii) 9m2 – 25n2 iv) a2b2 – 121 v) x2 + x – 6
10. Expand: i) (3x + 2y)2 ii)(2a + 5b)2 iii) (4x + 3)2 iv) (10x + 3)2 v) (2x - 5y)2 vi)(5x - 8y)2
vii) (x + 7) (x + 12) viii) (y - 8) (y + 35) ix) (8x + 3y)(5x + 2y) x) (4a + 5) (2a + 9)
xi) (2x + 3y + 4z)2 xii)(5a – 3b + 2c)2 xiii) (x + 3)3 xiv) (2a – 5)3
WORKSHEET 4**
1. Verify whether the indicated numbers are zeros of the polynomial corresponding to them in the following cases:
i) f(x) = 2x2 – x- 3 ;x = - 1 ii) p(m) = m(m-4); m = 2 iii) f(y) = 2y2 – y – 1; y= - ½
iv) q(x) = x2 – 9 ; x = -3, 3 v) p(x) = x3 – 4x2 ; x = 0, 4 vi) r(x) = 3s2 – 8; s = 3
2. Use the remainder theorem to find the remainder when x3 – 3x2 + 3x – 1 is divided by:
i) x - 1 ii) x + 1 iii) x – ½ iv) 2x + 1
3. Using factor theorem to determine whether g(x) is a factor of p(x) in each of the following cases: i) p(x) = x3 + 5x2 + 7x + 3 ; g(x) = x + 1 ii) p(x) = x3 - 2x2 – 5x - 3
4. Factorise : i) x2 + 11x +30 ii) a2 – 16a +63 iii) x2 + 2x -35 iv) 15 – 2x – x2
5. Factorise, using remainder theorem, each of the following polynomials:
i) x2 – 3x – 10 ii) y2 + 6y – 16 iii) 2z2 + 3z + 1 iv) 3x2 – 4x – 4 v) 6x2 + x - 1
6. Factorisei) a3 – b3 + c3 + 3abc ii) x3 + y3 + 9xy – 27 iii) 8x3 – 125 y3 – z3 – 30 xyz iv) 125x3 – 8 + 27y3 + 90xy v) 27x3 + y3 vi) 500a3 – 4b3 vii) 8a3 +36a2b + 54 ab2 + 27b3 viii) 8x3 – y3 -12x2y + 6xy2 ix) a3 – 27b3 +2a2b – 6ab2
7. Expand i) (2x – 1/x)2 ii) (2x + y)(2x – y) iii) (a + 2b + c)2 iv) (2a – 3b – c)2
v) (-3x +y + z)2 vi) (m + 2n – 5p)2 vii) (1/x + y/3)3 viii) (4 – 1/3x)3
8. If a-b = 10 and ab = 21, find the value of a3 + b3.
9. If 2x + 3y = 13 and xy = 6, find the value of 8x3 + 27y3.
10. Simplify i) (x + 3)3 + (x - 3)3 ii) (x + 2/x)3 + (x- 2/x)3 iii) (2x – 5y)3 – (2x + 5y)3
11. Simplify i) (a + b + c)2 + (a – b + c)2 ii) (2x + p – c)2 - (2x – p +c)2
12. If a + b+ c = 0 and a2 + b2 +c2 = 16, find the value of ab + bc + ca.
13. Find the value of 4x2 + y2 + 25z2 + 4xy – 10yz – 20zx when x = 4, y= 3, and z = 2.
14. If a + b = 10 andab = 21, find the value of a3 + b3.
15. If a + b+ c = 0 then prove that a3 + b3+ c3 = 3abc.
16. Evaluate i) 233 – 173 ii) 293 - 113
WORKSHEET 4***
1. If zero of the polynomial p(x) = x + a is x = -3, find ‘a’.
2. If zero of the polynomial f(y) = 2y - m is y = 2, find ‘m’.
3. If zero of the polynomial q(z) = az + 7 is x = -1, find ‘a’.
4. If zero of the polynomial p(y) = c – 3y is y = -2, find ‘c’.
5. Find out whether (x + 4) is a factor of x3 + x2-5x + 2 or not.
6. Use remainder theorem to show that (x + 1) is a factor of 9x3 + 15x2 – 6x - 12
7. Show that:i) (2x – 3) is a factor of 2x3 – 9x2 + x + 12 ii) (x+ 2) is a factor of x4 – x2– 12
8. Find the value of k, if x + 1 is a factor of p(x) in each of the following cases:
i) p(x) = x3 – 3x2+ kx ii) p(x) = 3x2 – kx + √3 iii) p(x) = kx3 – 9x2 + x + 6k
9. (x – 2) and (x + 3) are the factors of p(x) = ax3 + 3x2 –bx - 12. Find the values of a and b.
10. Factorise: i) 3x2 – 4x +1 ii) 3x2 + x - 2 iii) 12a2 + 11a – 5 iv) 24m2 + m – 23
v) x2 + 6√3x + 24 vi) 6x2 + 7√2x + 4 vii) 14 + 29mn – 15m2n2 viii) 15x2 + 3√3x - 8
11. Factorise, using factor theorem: i) x3 + 3x2 – 4x - 12 ii) 2x3 – 9x2 + x + 12 iii) x3 – 3x2 -10x +24 iv) 2x3 – 7x2 -3x +18 v) 3x3 + 10x2 + x - 6
12. Evaluate, using appropriate identities: i) (103)2 ii)(207)2 iii) (97)2 iv)(23)3 v) (103)3 vi) (97)3 vii) (999)3 viii) 104 x 107 ix) 115 x 95 x) 97 x 112
13. Expand: i) (1/3 a + ½ b – 1)2 ii) (x – 1/x + 1)2 iii)( 2x – 1/x – 3)2 iv) (5 – 3x – 1/3x)2
14. Without actually calculating the cubes, find the value of the following:
i) (8)3 + (-5)3 + (-3)3 ii) (27)3 + (15)3 + (-42)3 iii) (-15)3 + (-13)3 + (28)3
15. Give possible expressions for length and breadth of each of the following rectangles, whose areas are: i) (x2 – 6x – 7) sq. unit ii) (2x2 + 5x -3) sq. unit