CLASS- VIII
Cube, Cube root
11) Fill in the blanks:
a) Cube of an even natural number is ____
b) Cubes of natural number is called ____
c) Cube of natural number whose unit digit is 7, ends with digit____
d) If a natural number has one zero at its end, then it's cube will contain____zeros at the end.
e) Finding cube root is the inverse operation of finding ____
f) If a number can be expressed as the triples of equal prime factors, then it is called
12) State Whether the following statements are true or false
a) Cube of any odd number is even.
b) A perfect cubes does not end with zeros.
c) The cube of a two digit number may be a three digit number.
d) The cube of a single digit number may be a single digit number.
e) If a number is multiplied by 2 then it's cube is multiplied by 8.
CHECK YOUR PROGRESS
1) Show that each of the following numbers is a perfect cube. Also find the number whose cube is the given number:
a) 74088
b) 15625
2) Find the cube of the following numbers:
a) -17
b) -31/9
CLASS - IX
RATIONAL NUMBERS
P-1
1) Express 0.145 as a fraction in its lowest terms. 29/200
2) Express 5/48 as a decimal fraction correct to 4 decimal places. 0.1042
2) Construct each the following fraction to a decimal :
a) 15/11. 1.363636...
b) 23/55. 0.4181818...
3) Express
a) 1.3333....
b) 0.4565656...
as a vulgar fraction. 4/3, 226/495
4) Insert a rational number between 3/5 and 5/7. 23/35
5) Insert three rational numbers between 3 and 4. 13/4,7/2,15/4
6) Insert 3 rational numbers between 2 and 2.5. 2.125,2.25,2.375
7) Insert four rational numbers between 1/4 and 1/3. 25/96,13/48,7/24,5/16
8) Find ten rational numbers between -2/3, 1/4. -8/12, -7/12, -6/12, -5/12, -4/12, -3/12, -2/12, -1/12, 0, 1/12, 2/12, 3/12,
9) Find 20 rational numbers between 0 and 0.1. 0.004, 0.008, 0.012, 0.016, 0.020, 0.024, 0.028, 0.032, 0.036, 0.040, 0.044, 0.048, 0.052, 0.056, 0.060, 0.064, 0.068, 0.072, 0.076, 0.080
IRRATIONAL NUMBERS
P-1
1) Every non terminating and non-repeating decimal is irrational.
2) The square root of every non-perfect square is irrational .
3) The cube root of non-perfect cubes are irrational.
4) π is rational or irrational. Irrational
5) Show that √2 is not rational number.
6) Show that √3 is not a rational number.
7) Show that ³√6 is not a rational number.
8) Show that the negative of an irrational number is irrational.
9) Show that the sum of a rational and an irrational is irrational.
10) Show that the product of a non-zero rational with an irrational is irrational.
11) Prove (√2+ √3) is rational .
12) Show with the help of examples that :
a) the sum of two rational need not be an irrational.
b) the difference of two irrationals need not be an irrational.
c) the product of two rational need not be irrational.
13) Examin whether the following numbers are rational or irrational
a) (3+ √2)². Irrational
b) (2- √2)(2+ √2). Rational
c) 2/√2. Irrational
14) Find two rational numbers between 2 and 2.5.
15) Simplify 2/√3 by rationalizing the denominator.
16) Rationalise the denominator each of the following:
a) 1/√7. √7/7
b) 5/(2+ √3). 5(2- √3)
c) 4/(√5 - √3). 2(√5+ √3)
d) 1/(2√3 + √5). (2√3 - √5)/7
e) 1/(1+ √2+ √3). (√2+ 2 - √6)/4
f) √2/(√2+ √3 - √5). (√6+ 3 + √15)/6
17) If a and b are rational numbers and (2+√3)/(2- √3)= a+ b √3, find the values of a and b. 7,4
18) Simplify: {3√2/(√6 + √3) + √6/(√3 +√2) - 4√3/(√6 + √2). 0
EXPANSION
P-1
1) Expand:
a) (3a + 5b)². 9a²+ 25b²+ 30ab
b) (x/2 +2y/3)². x²/4 + y²/9 + 2xy/3
c) (2a + 3b + 4c)². 4a²+ 9b²+ 16c²+ 12ab + 24bc + 16ac
2) Expand:
a) (4a - 3b)². 16a²+ 9b²- 24ab
b) (2x/3 - 3y/4)². 4x²/9 + 9y²/16 - xy
c) (3a - 2b + 5c)². 9a²+ 4b²+ 25c²- 12ab - 20bc + 30ac
3) If x+ 1/x = 4, find
a) x²+ 1/x². 14
b) x⁴+ 1/x⁴. 194
4) If x - 1/2x = 3, find
a) x²+ 1/4x². 10
b) x⁴+ 1/16x⁴. 199/2
5) If a²+ 1/a²= 14, find
a) a+ 1/a. ±4
b) a - 1/a. ±2√3
c) a²- 1/a². ±8√3
6) If a²- 3a +1= 0 and a≠ 0, find
a) a+ 1/a. 3
b) a²+ 1/a². 7
7) Without multiplying evaluate
a) (107)². 11449
b) (20.6)². 424.36
c) (996)². 992016
d) (9.7)². 94.09
8) Using the formula, (a+ b)(a - b)= a²- b², evaluate (124 x 116). 14384
9) Complete each of the following expressions, making it a perfect square
a) 25a²+ 9b²+ _____. 30ab
b) 16a²- 8ab + ____. b²
10) If (a²+ b²+ c²)= 125 and (ab + bc + ca)= 50, find the value of (a+ b + c). ±15
1) Expand:
a) (3a + 2b)³. 27a³+ 8b³+ 54a²b + 36ab²
b) (5x - 3y)³. 125x³- 27y³ - 225x²y + 135xy²
2) If (x²+1)/x = 5/2, find the value of
a) x - 1/x. ±3/2
b) x³ - 1/x³. ± 63/8
3) If x² + 1/25x²= 43/5, find the value of x³+ 1/125x³. ±126/5
1) Find the following products by inspection:
a) (x +5)(x +3). x²+8x +15
b) (x -7)(x +2). x²- 5x -14
c) (2- x)(6- x). x²- 8x +12
d) (5x -3)(3x -2). 15x²- 19x +6
FACTORISATION
P-1
1) Factorise:
a) x²+ y - xy - x. (x -1)(x - y)
b) ab + bc + ax + cx. (a+ c)(b + x)
c) a²+ 4a+ a+ 4. (a+ 4)(a+ 1)
d) 6ab - b²+ 12ac - 2bc. (b + 2c)(6a - b)
2) Factorise
a) (ax + by)²+ (bx - ay)². (a²+ b²)(x²+ y²)
b) (x²+ 2x)²- 3(x²+ 2x) - y(x²+ 2x) + 3y. (x²+ 2x -3)(x²+2x- y)
3) x²+ 1/x² + 2 - 2x - 2/x. (x + 1/x)(x + 1/x -2)
1) Factorise:
a) 25a²- 9b². (5a+ 3b)(5a - 3b)
b) 1- (b - c)². (1+ b - c)(1- b + c)
2) a) 2x³- 50x. 2x(x +5)(x -5)
b) 16a³- 4a. 4a(2a +1)(2a -1)
c) 3x⁵ - 48x. 3x(x²+4)(x +2)(x -2)
d) x³- 3x² - x +3. (x -3)(x+1)(x -1)
3) a) x²- 2y + xy - 4. (x -2)(x + 2+ y)
b) a(a -1) - b(b -1). (a- b)(a + b -1)
4)a) a²+ b²- c² - 2ab. (a- b +c)(a- b - c)
b) 16x²- y²+ 4yz - 4z². (4x + y - 2z)(4x - y + 2z)
5) (x²+ y²- z²)² - 4x²y². (x + y+z)(x + y-z)(x - y+z)(x - y-z)
6) 2(ab + cd) - a² - b² + c² + d². (c +d + a -b)((c +d - a + b)
7) (1- x²)(1- y²) + 4xy. (1+xy + x -y)(1+ xy - x +y)
8) x⁴+ y⁴ - 11x²y². (x²- y²+ 3xy)(x²- y²- 3xy)
9) Express (x²- 3x +5)((x²+ 3x -5). {(x²)² - (3x -5)²}
10) Evaluate:
a) 968²- 32². 936000
b) (98.7)² - (1.3)². 9740
11) Factorise:
a) x⁴+4. (x²+ 2+ 2x)(x²+ 2- 2x)
b) x⁴+ x²+ 1. (x²+ 1+ x)(x²+ 1 -x)
1) Factorise:
a) 8a³+ 125b³. (2a +5b)(4a²- 10ab + 25b²)
b) a³+ b³+ a + b. (a+ b)(a²- ab + b²+1)
c) 24a⁴+ 81a. 3a(2a +3)(4a²- 6a+9)
2) a) x³- 343. (x -7)(x²+ 7x +49)
b) 27x³- 125/x³. (3x - 5/x)(9x²+ 15+ 25/x²)
c) a³- b³- a + b. (a- b)(a²+ ab + b²-1)
3) 64a⁶ - b⁶. (2a+b)(2a - b)(4a²- 2ab + b²)(4a²+ 2ab + b²)
4) 64 - a³b³+ 8 - 2ab. (4- ab)(18+ 4ab + a²b²)
5) x³p² - 8y³p² - 4x³q²+ 32y³q². (x -2y)(x²+ 2xy + 4y²)(p + 2q)(p - 2q)
6) 27(x + y)³+ 8(2x - y)³. (7x +y)(13x²+ 19y² - 4xy)
1) Factorise:
a) x²+ 11x +30. (x +6)(x +5)
b) x²+ 6x - 27. (x +9)(x -3)
c) x² -5x -24. (x -8)(x +3)
2) Factorise:
a) 3x² + 14x +8. (x +4)(3x +2)
b) 2x² -9x -26. (2x -13)(x +2)
c) 7 - 12x - 4x². (1- 2x)(7+2x)
3) 3x²- xy - 14y². (3x -7y)(x +2y)
4) 2x³+ 5x²y - 12xy². x(x + 4y)(2x - 3y)
5) 3(a - 2b)² - 2(a - 2b) - 8. (a -2b-2)(3a - 6b + 4)
SIMULTANEOUS EQUATIONS
P-1
1) Show that x= 4, y= 1 is a solution of the system of equation 2x + 3y=11, x - 2y = 2.
2) Show that x= 3, y= 2 is not a solution of the system of equation 4x - y=10, 2x + 3y = 11.
3) Solve 3x - y =23, 4x +3y = 48 (by Sub). 9,4
4) 4x - 18= 3y, 6x + 7y = 4. 3,2
5) 5/x + 6y= 13, 3/x + 4y =7. 1/5,-2
6) 1/7x + 1/6y = 3, 1/2x - 1/3y =5.. 1/14,1/6
7) 3x + 2y= 2xy, 6x + 2y = 3xy. 2,3
8) 3/(x + y) + 2/(x - y) =3, 2/(x + y) + 3/(x - y) = 11/3. 2,1
9) 37+ 41y=70, 41x +36y = 86. 3,-1
P-2
EXERCISE - A
1) 2x + 3y - 17= 0, 3x - 2y - 6= 0. 4,3
2) 2x - y - 3= 0, 4x + y - 3= 0. 1,-1
3) x/a + y/b = a+ b, x/a² + y/b²= 2. a², b²
4) ax + by = a - b, bx - ay - 6= a+ b.
5) a/x - b/y = 0, ab²/x + a¹b/y = a² + b². Where x, y≠0. a, b
6) ax + by = 1, bx + ay= (a+ b)²/(a²+ b²) -1 or bx + ay = 2ab/(a²+ b²). a/(a²+b²), b/(a²+ b²)
7) a(x + y)+ b(x - y)= a²- ab + b²,
a(x + y) - b(x - y)= a² +ab + b², b²/2a, (2a²+ b²/2a
8) ax + by = c, bx - ay = 1+ c. c/(a+b) - b/(a²- b²), c/(a+b) + a/(a²- b²)
9) x + y = a - b, ax - by = a² + b². a, - b
CONDITION FOR SOLVABILITY (OR CONSISTENCY)
P-2
Consider the system of equations
a₁x+ b₁y + c₁ = 0.......(i)
a₂x + b₂y + c₂ =0.......(ii)
is consistent with unique solution if
a₁/a₂ ≠ b₁/b₂
i.e., lines represented by equations (i) and (ii) are not parallel
is consistent with infinitely many solutions if
a₁/a₂ = b₁/b₂ = a₁/a₂
i,e., lines represented by equation (i) and (ii) are coincident
is inconsistent, i.e, it has no solution
a₁/a₂ ≠ b₁/b₂≠ c₁/c₂
i.e., lines represented by equation (i) and (ii) are parallel and non-coincident
EXERCISE - B
1) In each of the following systems of equations determine whether the system has a unique solution, no solution or infinite many solutions , In case there is a unique solution, find it.
a) 2x + 3y =7; 6x + 5y =11. Unique , -1/4,5/2
b) 6x + 5y =11; 9x + 15y/2 =21. no solution i.e, it is inconsistent
c) -3x + 4y =5; 9x/2 - 6y + 15/2 =0. Infinitely many solutions
2) For each of the following systems of equation determine the value of k for which the given system equation has a unique solution.
a) x - ky =2; 3x + 2y =- 5. Unique, -2/3
b) 2x - 3y =1; kx + 5y = 7. Consistent with a unique solution, -10/3
c) 2x + 3y - 5=0; kx - 6y - 8 =0. Unique , -4
d) 2x + ky =1; 5x - 7y =5. Unique , -14/5
3) For each of the following systems of equations determine the value of k for which the given system of system of equations has infinitely many solutions.
a) 5x + 2y =k; 10x + 4y =3. Infinitely many solutions, 3/2
b) (k -3)x + 3y =k; kx + ky =12. Infinitely many solutions, 6
c) kx + 3y = k - 3; 12x + ky = k. Infinitely many solutions, 6
4) For each of the following system of equations determine the value of k for which the given system has no solution :
a) 3x - 4y + 7 =0; kx + 3y - 5=0. No solution , -9/4
b) 2x - ky +3 =0; 3x + 2y -1=0. No solution , -4/3
5) Find the value/s of k for which the system of equations
kx - y =2; 6x - 2y = 3.
a) has a unique solution. Uniquely solution, if k≠ 3
b) no solution. No value of k, due to infinitely many solutions
Is there a value of k for which the system has infinity many solutions?
6) For what value of k will the equations x + 2y + 7 =0; 2x + ky + 14=0 represent coincident lines ? If k= 4
7) For what value of k, will the following system equations have infinitely many solutions ?
2x + 3y =4; (k +2)x + 6y =3k +2. When k= 2
8) Determine the values of a and b for which the following system of linear equations has infinite solutions :
2x - (a - 4)y =2b +1; 4x - (a -1)y = 5b -1. 7,3
10) Find the value of k will the following system of linear equation has no solutions ?
3x + y =1; (2k -1)x + (k -1)y =2k + 1. 2
11) Find the values of k for which the following system of linear equations has infinite solutions.
x + (k +1)y =5; (k +1)x + 9y =8k - 1. 2
12) Find the values of p and q for which the following system of equation has infinite number of solutions:
2x + 3y =7; (p + q)x + (2p - q)y =21. 5, 1
13) For what value of k, will the system of equations
x + 2y =5; 3x + ky + 15= 0. 6
14) Find the value of m,n for which the following systemic linear equations has infinite number of solutions:
2x + 3y =7; 2mx + (m + n)y =28. 4,8
15) Determine the values of m,n so that the following system of linear equation have infinite number of solution:
(2m -1)x + 3y -5 =0 ; 3x + (n -1)y - 2 =.0 17/4,11/5
16) Determine the value of k so that the following linear equations have no solution:
(3k +1)x + 3y -2=0; (k²+ 1)x + (k - 2)y -5 =0 -1
PROBLEM ON SIMULTANEOUS EQUATIONS
P-2
1) 4 chairs and 3 tables cost Rs 2100 and 5 chairs and 2 tables cost Rs 1750. Find the cost of a chair and a table separately. 150, 500
2) 37 pens and 53 pencils together cost Rs 320, while 53 pens and 37 pencils together cost Rs 400. Find the cost of a pen and that of a pencil. 6.50,1.50
3) 3 chairs and 2 tables cost Rs 2000 and 2 chairs and 3 tables cost Rs 2500. Find the cost of a chair and a table separately. 200, 1700
4) A and B each have certain number of oranges. A says to B, " if you give me 10 of your oranges, I will have twice the number of oranges left with you". B replies, "if you give me 10 of your oranges, I will have the same number of oranges as left with you". Find the number of oranges with A and B separately. 70,50
5) A man has only 20 paise coins and 25 paise coins in his purse. If he has 50 coins in all totalling Rs 11.25, How many coins of each kind does he have? 25 each
P-2
1) The sum of two numbers is 69 and their difference is 17. Find the numbers. 43,26
2) A two digit number is seven times the sum of its digits. The number formed by reversing the digits is 18 less than the original number. Find the original number. 42
3) The result of dividing a number of two digits by the number with digits reversed is 5/6. If the difference of digits is 1. Find the number. 45
4) If the numerator of a fraction is increased by 2 and its denominator decreased by 1, then it becomes 2/3. If the numerator is increased by 1 and denominator increased by 2, then it becomes 1/3. Find the fractions. 2/7
5) The monthly incomes of A and B are in the ratio 3:4 and their monthly expenditures are in the ratio 5:7. If each saves Rs 2500 per month, find their monthly incomes. 15000, 20000
6) Five years hence, a man's age will be three times his son's age and five years ago, he was seven times as old as his son. Find their present ages. 40,10
7) If the length and breadth of a room are increased by 1m each, its area is increased by 21 m¹. If the length is increased by 1m and breadth decreased by 1 m, the area is decreased by 5m². Find the area of the room. 96 m²
8) The students of a class are made to stand in complete rows. If one student is extra in each row, there would be 2 rows less, and if one student is less in each row, there would be 3 rows more. Find the number of students in the class. 60
9) A and B, each has some money. If A gives Rs 50 to B, then B will have twice the money left with A. But, if B gives Rs 20 to A, then A will have thrice as much as left with B. How much money does each have? Rs106, Rs 62
10) A jeweller has bars of 18-carat gold and 12-carat gold. How much of each must be melted together to obtain a bar of 16 carat gold, weighing 120 gm ? It is given that pure gold is 24-carat. 80,40gm
11) 4 men and 4 boys can do a piece of work in 3 days; while 2 men and 5 boys can finish it in 4 days. How long would it take 1 boy to do it? How long would it take 1 man to do it? 18,36,
12) A train covered a certain distance at a uniform speed. If the train had been 6 kmph faster, it would have taken 4 hours less than the scheduled time. And, if the train had been slower by 6 kmph, it would have taken 6 hours more than the scheduled time. Find the length of the journey. 24 hours.
13) A sailor goes 8 km downstream in 40 minutes and returns back to the starting point in 1 hour. Find the speed of the sailor in still water and the speed of the current. 10,2
14) In an examination, the ratio of passes to failure was 4:1. Had 30 less appeared and 20 less passed, the ratio of passes to failure would have 5:1. How many students appeared for the examination ? 150
GRAPH (SIMULTANEOUS EQUATIONS)
P-1
1) Solve graphically the simultaneous equations:
a) x -2y = 1; x + y = 4.
b) 3x - 5y = - 1; 2x - y = -4.
c) x +3= 0; y -2 = 0, 2x + 3y = 12.
INDICES/EXPONENTS
P-1
1) Evaluate:
a) (64)¹⁾³. 4
b) (8)⁵⁾³. 32
c) (9)³⁾². 27
d) (125)⁻¹⁾³. 1/5
e) (243)⁻³⁾⁵. 1/27
f) (9/16)⁻¹⁾². 4/3
g) (0.01)⁻¹⁾². 10
h) (64/25)⁻³⁾². 125/512
2) Simplify:
a) (81)³⁾⁴ - (32)⁻²⁾⁵ + (8)²⁾³ x (1/2)⁻¹ x 3⁰ - (1/81)⁻¹⁾². 22
b)(64/125)⁻²⁾³ + 4⁰ x 9⁵⁾² x 3⁻⁴ - √25/³√64 x (1/3)⁻¹. 13/16
3) Evaluate:
a) √36. 6
b) ³√125. 5
c) ⁴√81. 3
c) ⁵√32. 2
d) √50. 5√2
e) ³√49. 2 ³√5
4) If √2= 1.414 and √3= 1.732, find the value of
a) √72. 8.484
b) √147 + √27. 17.32
c) 5 √125 + 2 √75 - 5 √108. 21.92
5) If 2160= 2ᵃ x 3ᵇ x 5ᶜ, find the values of a, b, c. Hence, find the value of 3ᵃ x 2⁻ᵇ x 5⁻ᶜ. 81/40
6) Simplify: (5ⁿ⁺³ - 6 x 5ⁿ⁺¹)/(9 x 5ⁿ - 2² x 5ⁿ(. 19
7) Simplify:
a) (xᵇ/xᶜ)ᵃ . (xᶜ/xᵃ)ᵇ . (xᵃ/xᵇ)ᶜ. 1
b) (xᵃ/xᵇ)^(a²+ ab+ b²) . (xᵇ/xᶜ)^ (b²+ bc+ c²). (xᶜ/xᵃ)^(c²+ ca+ a²). 1
8) If aˣ = bʸ = cᶻ and b²= ac, show that y= 2xz/(z + x).
9) Solve:
a) 3³ˣ= 1/9 . -2/3
b) 2³(6⁰ + 3²ˣ)= 224/27. -3/2
c) √(3/5)¹⁻²ˣ = 125/27. 3.5
d) √(5⁰+ 2/3)= (0.6)²⁻³ˣ. 5/6
LOGARITHMS
P-1
1) Express each of the following in logarithmic form:
a) (10)³ = 1000. log₁₀(1000)= 3
b) (0.2)²= 0.04 . Log₀.₂(0.04)= 2
c) 3⁻³= 1/27. Log₃(1/27)= -3
2) Convert each of the following tp exponential form:
a) log₂16= 4. 2⁴= 16
b) log₈32= 5/3. (8)⁵⁾³= 32
c) log₁₀(0.1)= -1. (10)⁻¹= 0.1
3) By converting to exponential form, find the value of each of the following:
a) log₃243. 5
b) log₇(1/343). -3
c) log₂(0.0625). -4
4) Solve for x:
a) log₃x = -2. 1/9
b) log₈x = 2/3. 4
c) log√₃ (x +1)= 2. 2
d) log₂(x²-1)= 3. ±3
e) logₓ(9/16)= -1/2. 256/81
f) log₉27= 2x +3. -3/4
5) Given log₁₀x = 2a and log₁₀y = b/2.
a) Write 10ᵃ in terms of x. √x
b) Write 10²ᵇ⁺¹ in terms of y. 10y⁴
c) If log₁₀P= 2a - b, express P in terms of x and y. x/y²
1) Evaluate:
a) log5 + log 20+ log24+ log25 - log 60. 3
b) log6 + 2 log5 + log 4 - log 3 - log2. 2
c) 7log(16/15) + 5 log(25/24) + 3 log (81/80). log 2
2) Evaluate :
a) +log32)/(log4) . 5/2
b) (log 27)/(log 9). 3/2
3) Express {2log3 - (1/2) log 16 + log 12} as a single logarithm . Log27
4) If log 2 = x, log 3 = y and log7 = z, express log(4 ³√63) in terms of x, y and z. 2x + z/3 + 2y/3
5) Solve:
a) log₁₀x - log₁₀(2x -1) = 1. 10/19
b) log₅(x²+ x) - log₅(x +1)= 2. 25
c) log₁₀5+ log₁₀(5x +1)= log₁₀(x +5)= 1. 3
6) If log₁₀y + 2 log₁₀x = 2, express y in terms of x. y= 100/x²
7) If log2= 0.3010 and log3 = 0.4771, find the values of
a) log 6. 0.7781
b) log48. 1.6811
c) log√24. 0.69005
d) log ⁵√108. 0.40666
LINES AND ANGLES
P-1
1) Find the complement of:
a) 32°. 58°
b) 46°30'. 43°38'
c) 34° 54' 18". 55°5'42"
2) Find the supplement of each of the angles measuring:
a) 30°. 150°
b) 83°18'24". 96°42'36"
3) Find the measure of an angle which is 25° more than two thirds of its complement. supplement . 51°
4) Find the measure of an angle, if 5 times its complement is 30° less than twice its complement. 40°
5) Two supplementary angles are in the ratio 2:3. Find the angles. 72,108
Write true or false
6) a) A line has no definite length.
b) A line has no end points.
c) A ray has a definite length.
d) A line segment has one end point.
e) Only one line segment can be drawn to pass through two given points.
f) An infinite number of line can be drawn to pass through a given point.
g) Two lines can Intersect at the most in one point.
SOME ANGLE RELATION
P-1
1) In the given figure,
AOB is a straight line. If angle AOC=54° and angle BOC= x°, find the value of x and hence find angle BOC.
2) In the given figure,
what value of x will make AOB a straight line?
3) In the given figure,
AOB is a straight line, angle COD = 90° and angle BOE= 75°. If angle AOC= x°, angle BOD= y° and angle AOE= (3x)°, find the valus of x and y. Hence, find angle AOC, BOD, AOE.
4) In the given figure,
two straight lines AB and CD intersect at a point O. It is being given that angle BOD= x° and angle AOD= (5x - 12)°. Find the value of x.
Hence, find angle BOD, AID, AOC, BOC.
PARALLEL LINES
P-1
1) In the adjoining figure,
AB|| CD. Find the values of x, y and z.
2) In the adjoining figure,
AB|| CD and EF|| GH. Find the values of x,y,z and t.
3) In each of the following figure,
AB|| CD. Find the values of x in each case:
4) In the given figure,
AB|| CD. If angle ABE= 105°, angle DCE= 115° and angle BEC= x°, calculate the value of x.
5) In the given figure,
AB|| CD. If angle ECD= 100°, angle AEC= 30° and angle BAE= x°, find the value of x.
TRIANGLES
P-1
1) The angle of a triangle are in the ratio 3:5:7. Find the measure of each angle of the triangle.
2) In a ∆ ABC, if 3 times angle A= 4 times angle B= 6 times angle C, find angle A,B,C.
3) In a ∆ ABC, Angle A+ angle B= 110° and angle B+ angle C = 115°. Calculate angle A,B,C.
4) In a ∆ ABC, angle A - angle B =12° and angle B - angle C = 27°. Calculate angle A, B, C.
5) In the adjoining figure,
find the value of x.
6) In the adjoining figure,
BA|| CE and GI || BC. If angle GAH= 30° and EIJ= 80°. Find angle ACB
7) In the adjoining figure,
show that δ = α + β + γ.
8) In ∆ ABC,
the bisectors of angle B and C intersect each other at a point O. Show that: angle BOC= 90° + angle A/2.
the bisectors of angle B and C intersect each other at a point O. Show that: angle BOC= 90° + angle A/2.
CONGRUENCY OF TRIANGLES
P-1
1) In ∆ ABC, AB= AC.
If P is a point on AB and Q is a point on AC such that AP= AQ. Show that:
a) ∆ APC ≡ ∆AQB
b) ∆BPC ≡ CQB.
2) In the given figure,
AB= AC and AD is the bisector of angle A. Show that that
a) D is the midpoint of BC
b) AD perpendicular to BC.
3) In ∆ ABC,
the internal bisectors of angle B and C meet at O. Show that OA is the internal bisectors of angle A.
4) Id D is the midpoint of the hypotenuse AC of a right angled ∆ ABC,
show that BD= AC/2.
INEQUALITIES
P-1
CONSTRUCTION OF TRIANGLES
P-1
MID-POINT AND INTERCEPT THEOREMS
P-1
CONSTRUCTION
P-1
PYTHAGORAS
P-1
POLYGON
P-1
CONSTRUCTION
P-1
QUADRILATERALS
P-1
PERIMETER AND AREA OF PLANE FIGURES
1) Find the area of a triangle with base 24cm and height 15 cm.
2) The base of a triangular field is three times its altitude. If the cost of cultivating the field at Rs 38 per hectare is Rs 513, find the base and height.
3) Find the area of a triangle whose sides are 42cm, 34cm and 20 cm. Hence, find the height corresponding to the longest side.
4) Calculate the area of an equilateral triangle of side 12 cm, correct to two decimal places.
5) Calculate the area of an equilateral triangle whose height is 6cm. (Take √3= 1.73).
6) The perimeter of an isosceles triangle is 42 cm and base is 3/2 times each of the equal sides.
7) The base of an isosceles triangle is 24cm and its area is 192 cm². Find its perimeter.
8) The difference between the sides of a right angled triangle containing the right angle is 7cm and its area is 60 cm². Calculate the perimeter of the triangle.
MENSURATION (RECTANGLE)
1) The perimeter of a rectangular plot is 120m. If the length of the plot is twice its width, find the area of the plot. 800m²
2) How many square tiles of side 20cm will be needed to pave a footpath which is 2 meters wide and surrounds a rectangular plot 40m long and 22m wide? 6600
3) The area of a square plot is 1764m². Find the length of its one side and one diagonal. 42, 59.39m
4) Two adjacent sides of a parallelogram are 24cm and 18cm. If the distance between the longer sides is 12cm, find the distance between shorter sides. 16cm
5) If the length of a rectangle is increased by 10cm and breadth is decreased by 5cm, the area is unaltered. If the length is decreased by 5cm and breadth is increased by 4cm, even then the area is unaltered. Find the dimensions of the rectangle. 30,20
6) The sides of a square exceeds the side of another square by 3cm and the sum of the areas of the two squares is 549 cm². Find the perimeters of the squares. 60,72
7) If the sides of a square are lengthened by 3cm, the area becomes 121 cm². Find the perimeter of the original square. 32cm
TRIANGLE AND RECTANGULE/SQUARE (Mixed)
1) A triangle and a parallelogram have the same base and the same area. If the sides of the triangle are 15, 14 and 13cm and the parallelogram stands on the base 15cm, find the height of the parallelogram. 5.6cm
2) A kite in the shape of a square with diagonal 32cm and an isosceles triangle of base 8cm and sides 6cm each is to be made of three different shades as shown in the figure. How much paper of each shade has been used in it ? 17.9cm² (app)
3) ABCD is a square with sides of length of 6cm. Find point M on BC such that area of ∆ ABM: area of trapezium ADCM = 1:3.
4) In the adjoining figure, ABCD is a square. E is a point DC such that area of ∆ AED: area of the trapezium ABCE = 1:5, find the ratio of the perimeters of ∆ AED and trapezium ABCE. (4+ √10): (8+ √10)
PERIMETER AND AREA OF QUADRILATERAL
1) In a four sided field, the longer diagonal is 108m. The lengths of the perpendiculars from the opposite vertices upon this diagonal are 17.6 m and 12.5 m respectively. Find the area of the field.
2) Find the area of a quadrilateral whose sides are 9m, 40m, 28m and 15m respectively and the angle between first two sides is a right angle.
3) The length of the rectangular plot is twice its breath. If the perimeter of the plot is 270 m, find its area.
4) Find the area of a rectangular plot of land one of whose sides measure 35m and the length of the diagonal 37 m.
5) A rectangular carpet has an area of 60m². if its diagonal and longer side together equal 5 times the shorter side, find the length of the carpet .
6) The length of the diagonal of a square is 36 cm. Find
a) the area of the square
b) its perimeter upto 2 decimals places.
7) Find the area of a parallelogram one of whose sides is 34 cm and the corresponding height is 8cm.
8) Two adjacent sides of a parallelogram are 24cm and 18 cm. If the distance between the longer sides is 12cm, find the distance between the shorter sides.
9) The diagonals of a rhombus are 30cm and 16 cm.
a) Find the area of the rhombus .
b) the perimeter of the Rhombus.
10) Find the area of a trapezium whose parallel sides are 25cm and 18 cm and the distance between them is 8 cm.
11) Find the area of a trapezium ABCD in which AB || DC, AB= 77cm, BC= 25cm, CD= 60cm and DA= 26cm.
12) The length and breadth of a rectangular grassy plot are in the ratio 7:4. A path 4 m wide running all around outside it has an area of 416m². Find the dimensions of the grassy plot.
13) A rectangular lawn 60m by 40m has two roads, each 5m wide, running in the middle of it, one parallel to length and the other parallel to breadth. Find the cost of the gravelling them at Rs 3.60 per m².
14) if the length and breadth of a rectangular room are each increased by 1 m, then the area of floor is increased by 21 m². If the length is increased by 1 m and breadth is decreased by 1m, then the area is decrease by 7 m². Find the perimeter of the floor.
Multiple Choice Questions
1) If the perimeter of a square is 80cm, then its area is
a) 800 cm² b) 600cm² c) 400cm² d) 200 cm²
2)
VOLUME AND SURFACE AREA OF SOLIDS
TRIGONOMETRICAL RATIOS
1) In a ∆ ABC, angle B= 90°, AB= 5 units and AC= 13 units, find
a) sinA.
b) tanA
c) cosec²A - cot²A.
d) CosC
d) CosC
e) cotC
f) cosecC
2) In a right ∆ ABC, if angle A is acute and tanA= 3/4, find the remaining trigonometric ratios of angle A.
3) If x is an acute angle such that sin x = √3/2, then find the value of (cosec x + cotx).
4) If tan A= 4, find the value of (5sinA - 3 cosA)/(5 sinA + 3 cosA).
5) If sinx + cosecx = 2, find the value of sin²x+ cosec²x.
T-RATIOS
1) Find the value of:
a) (cos0°+ sin45°+ sin30°)(sin90° - cos45°+ cos60°).
b) (3 cos²30°+ sec²30°+ 2 cos0°+ 3 sin90° - tan²60°).
2) If A= 60°, verify that
a) sin²A + cos²A = 1
b) sec²A - tan²A = 1
c) cosec²A - cot²A = 1.
3) Taking A= 30°, verify that
a) sin2A= 2 sinA cosA.
b) cos2A = (cos²A - sin²A).
c) tan2A= 2tanA/(1- tan²A).
d) sinA= √{(1- cos2A)/2}.
4) Taking A= 30°, verify that:
a) sin3A = (3 sinA - 4 sin³A).
b) cos3A= 4cos³ - 3 cosA.
5) If A= 60° and B= 30°, show that
Sin(A - B)= sinA cosB - cosA sinB.
6) If A and B are acute angles such that cos(A + B)= 1/2 = sun(A - B), find the valus of A and B.
7) Verify: sin²x + cos²x = 1 for x= 30°.
8) Verify that: cos60°= (1- 2 sin²30°)= (2 cos² 30° -1).
9) Find the valus of
a) tan²30°
b) sun⅗30°.
c) cos⁴45°.
d) (cos²45° + sin²60°+ sin²30°).
COMPLEMENTARY ANGLES
1) Evaluate:
a) sin53°/cos37°.
b) cos49°/sin41°.
c) tan66°/cot24°.
2) Evaluate: cos35° cos55° - sin35° sin 55°= 0.
3) Express (sin85° + cosec85°) in terms of trigometric ratios of angles between 0° and 45°.
4) Show that
a) sin35° sin55° - cos35° cos55 = 0.
b) cos80°/sin10° + cos59° cosec31° = 2
c) sinθ/sin(90°- θ) + cosθ/cos(90° - θ)= secθ cosecθ.
°°°°°°°°°°°°°°
Trigonometry Ratios
1) If tanθ= a/b, find the value of expression (cosθ + sinθ)/(cosθ - sinθ).
2) Given tanA= 12/5 (0°≤A ≤ 90°), find the value sinA and cosA.
3) If sin²θ + 3 cos²θ= 4, show that tanθ = 1/√3; (0°≤θ< 90°).
4) If secθ = 5/4, verify that tanθ/(1+ tan²θ)= sinθ/secθ.
5)
CLASS - X
BANKING
1) Vivek deposits Rs 500 per month in a recurring deposit account for 12 months. Find the amount he will receive at thr time of maturity at the rate of 10% p.a.
2) Akshay deposits Rs 2000 per month in a recurring account for 5 years, at the 12 p.a. find the amount, Akshay will get after 5 years at the time of maturity.
LINEAR INEQUATIONS
1) Solve:
a) 2x - 3< 5x - 3 ≤ 12, x belongs to N. Hence illustrate on the number line.
b) 3(x -2)≥ 2x - 3, x belongs to R. Hence draw a diagram to illustrate this inequation.
c) (x -2)/(2x +5) < 1/3, x belongs to R, hence draw a diagram for this inequation.
d) Given A: {x: -8< 5x +2≤ 17, x∈ I}
B:{ x: -2≤ 7 + 3x < 17, x ∈R}
Represent A and B on two different number lines. Write down the elements of A ∩ B.
5) The diagram represents two inequations A and B on real number lines.
a) Write down A and B in the set builder notation.
b) Represent A ∩B and A ∩B' on two different number lines.
QUADRATIC EQUATIONS
1) 2x²+ 2= 5x.
2) (x +3)/(x +2)= (3x -7)/(2x -3).
3) x²+ 2√2 x -6=0.
4) x/(x +1) + (x +1)/x = 34/15, x≠0, x ≠ -1.
5) Determine the value of k for which the given solution of the equation 3x²+ 2kx -3=0; x = -1/2.
6) 2x²+ 3x -20=0.
7) 4x²- 12x +9=9.
8) 3x²- 8x +2=0. Leave your answer in radical form.
9) x² - 3√3x +6=0. Write your answer correct to 2 places.
10) The value of k for which 2x²- Kx +1=0 has real and equal roots.
11) Find the values of k for which kx²- 6x -2=0 has real roots.
1) Find a possible number which when decreased by 20 is equal to 69 times the reciprocal of the number.
2) The sides of a right angled triangle are 2x -1, 2x, and 2x +1. Find x and hence the area of the triangle.
3) A lawn 50m long and 34m broad has a path uniform width around it. if the area of the path is 540 m², find its width.
4) If a train travelled 5 kmph faster, it would take one hour less to travel 210 km. Find the speed of the train.
REFLECTION
1) A Triangle ABC is such that the coordinates A, B and C are (2,0),(1,1) and (0,2) respectively. Write down the coordinates of the triangle obtained by reflecting ∆ ABC in the line y=0. Also reflect (2,0) in the line x=0.
2) Draw the unit square, whose vertices are (2,2),( 4,2),(4,4) and (2,4). Reflect the square in the y-axis and then reflect the image in the origin. What single transformation would give the same final result ?
3) A man leaving point A must take water from a river and deliver it to a man at point B. Use reflection to find the shortest path.
RATIO
1)
HEIGHT AND DISTANCES
1) Two boys are an opposite sides of a tower. They measure the angles of elevation of the top of the tower is 30° and 60° respectively. If the height of the tower is 100m, Find the distance between the two boys.
2) The upper part of a tree broken over by the wind makes an angle of 30° with the ground and the horizontal distance from the root of the tree to the point where the top of the tree meets the ground is 20 m. Find the height of the tree before it was broken to the nearest metre .
3) A man on the deck of a ship is 10m above water level. He observes that the angle of elevation of the top of a cliff is 45° and the angle of depression of the basis is 30°. Calculate the distance of the cliff from the ship and the height of the cliff.
4) From the top of the tower 50m high the angles of depression of the top and bottom of a pole are observed to be 45° and 60° respectively. Find the height of the pole.
5) The angle of elevation of the top of a tower from two points a and b from the base and in the same straight line with it are complementary. Prove that the height of the tower is √(ab).
6) Two ships are sailing in the sea on either side of a lighthouse, the angles of depression of two ships as observed from the top of the lighthouse are 60° and 30° respectively . If the lighthouse is 200m high, find the distance between the ships.
7) The shadow of a tower, when the angle of elevation is 45° is found to 10 metres longer than when it is 60°. Find the height of the tower.
8) The angle of depression at a point on the level ground viewed from a 20m high window and the top of the building are 30° and 45°. Calculate the height of the building.
9) Two poles of equal height are standing opposite to each other on either side of a road, which is 100m wise. From a point between them on the road, the angle of elevation of the tops are 30° and 60°. Find the point and also the height of the poles.
10) The tallest tower in a city is 100m high and a multistoreyed hotel at the city centre is 20 m high. The angle of elevation of the top of the tower at the top of the hotel is 30°. A building h meter high, is situated on the road connecting the tower with the city centre at a distance of 1km from the tower. Find the value of h if the top of the hotel, the top of the building and the top of the tower are in a straight line. Also find the distance between the tower from the city centre.
Booster - B
1) A Fire at a building B is reported on telephone to two fire stations X and Y 10 km apart from each other. X observes that the fire is a triangle of 60° from it and Y observes that it is an angle of 45° from it. Which station should send his team and how much will it have to travel ?
2) The angle of elevation of a cliff from a fixed point A is θ. After going up a distance of k metres towards the top of the cliff at an angle of φ, it is found that the angle of elevation of α. Show that the height of the cliff, in metres, is k(cosφ - sinφ cotα)/(cotθ - cotα) metre.
3) A boy is standing on the ground and flying a kite with 100m of strings at an elevation of 30°. Another boy is standing on the roof of a 20m high building and is flying of both the kites. Find the length of the string that the second boy must have so that the two kites meet.
4) A vertical tower stands on a horizontal plane and is surmounted by a vertical flagstaff of height h. At a point on the plane, the angle of elevation of the bottom of the flagstaff is α and that at the top of the flagstaff is β. Prove that the height of the tower is (h tanα)/(tanβ - tanα).
5) An aeroplane flying horizontally 1 km above the ground is observed at an elevation of 60°. After 10 seconds , its elevation is observed to be 30°. Find the speed of the aeroplane in km/hr.
6) A man on the top of a tower, standing in the seashore, find that a boat coming towards him take 10 minutes for the angle of depression to change from 30° to 60°. How soon will the boat reach the seashore ?
7) Determine the height of a mountain if the elevation of its top at an unknown distance from the base is 30° and at a distance 10km further off form the mountain, along the same line, the angle of elevation is 30°.
8) The angle of elevation of a cliff from a fixed point A is 45°. After going up a distance of 600m towards the top of the cliff at an inclination of 30°, it is found that the angle of elevation is 60°. Find the height of the cliff.
9)
GRAPHICAL REPRESENTATION
1) The time taken in seconds to solve a problem for reach of 25 pupils is as follows:
16, 20, 26, 27, 28, 30, 33, 37, 38, 40, 42, 43, 46, 46, 46, 48, 49, 50, 53, 58, 59, 60, 64, 52, 20.
a) Construct the frequency distribution for these data using a class-interval of 10 seconds ,
b) Draw a histogram to represent the frequency distribution.
2) Draw a histogram of the following data if the wages are given only in rupees.
Weekly wages no of workers
01-10 4
11-20 8
21-30 10
31-40 24
41-50 36
51-60 12
61-70 6
3) In q study diabetes patients the following data were obtained.
Age(in years ) no of patients
10-19 1
20-29 0
30-39 1
40-49 10
50-59 17
60-69 38
70-79 9
80-89 3
Display it with the help of Histogram and Frequency polygon on the same scale.
4) Draw a less than ogive and a more than ogive from the following data:
Age:0-10 10-20 20-30 30-40 40-50 50-60
F: 8 16 30 35 15 26
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