REAL NUMBERS SHORT ANSWER TYPE...

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REAL NUMBERS TOPIC-1

RATIONAL NUMBER VERY SHORT ANSWER TYPE QUESTIONS

Q.1 Identify a rational number among the following

number: √25

6, √

20

4, 2.27̅̅̅̅ , √2, √3 [2.27̅̅̅̅ , 2014]

Q.2 Express the rational number 0.9̅ in the form 𝑝

𝑞, where

p and q are integers and q ≠ 0. [1,2014]

Q.3 Is √98

√2 is a rational number or not. [7,2012]

Q4 write the simplest form of a rational number177

413.

[3/7, 2012]

Q.5 calculate the decimal which represent the fraction 7

8

[0.875,2012]

Q.6 Identify the smallest number among the following

numbers: 1.101001….., 1.1101001….,1.011012…..,

1.011̅̅ ̅̅ ̅. [1.011 = 1.011011011…., 2012]

Q.7 What is the maximum number of digits in the

repeating block of 1

𝑛, where n is a prime number.

[ n-1, 2012]

Q.8 Write a real number which has terminating decimal

expansion. [0.248,2012]

Q.9 Calculate the value of 2.9̅ in the form of 𝑝

𝑞, where p

and q are integers and q ≠ 0. [3, 2012]

SHORT ANSWER TYPE QUESTIONS-I

Q.1 Express 0.6̅ in the form 𝑝

𝑞, where p and q are integers

and q ≠ 0. [2/3, 2013]

Q.2 If 7x = 1, then find the decimal expansion of x. [x = 0.142857̅̅ ̅̅ ̅̅ ̅̅ ̅̅ , 2012]

Q.3 Represent 0.237̅̅ ̅̅ ̅ in the form 𝑝

𝑞, where p and q are

integers and q ≠ 0. [237, 2012]

Q.4 Express 2157

625 in the decimal form and state whether

it is terminating or not. [3.4512, 2012]

Q.5 insert three rational numbers between 3

5 and

5

7.

[ 22/35,23/35,24/35, 2012]

SHORT ANSWER TYPE QUESTIONS-II

Q.1 Find three rational numbers between 5

7 and

9

11.

[386/539, 387/539,388/539, 2012]

Q.2 Find six rational numbers between 3 and 4.

[22/7,23/7, 24/7, 25/7, 26/7, 27/7 2014] LONG ANSWER TYPE QUESTIONS-I

Q.1 Arrange in descending order √23

,√54

, √76

, and √312

Q.2 Express 0.32̅̅̅̅ + 0.35̅̅̅̅ in the form 𝑝


integers and q ≠ 0. [1659/990, 2012]

TOPIC-2

IRRATIONAL NUMBER VERY SHORT ANSWER TYPE QUESTIONS

Q.1 Identify the irrational number among the following

numbers: 0.13, 0.1315̅̅̅̅ , 01315̅̅ ̅̅ ̅̅ ̅, 0.3013001300013…

Q.2 Is the product of two irrational numbers always an

irrational number? [No, 2012]

Q.3 Calculate the 𝑝

𝑞 in the form of 0.777, where p and q

are integers and q ≠ 0. [x=7/9, 2012]

Q.4 Write the sum of √52

, & √73

. [√52

+ √73

, 2012]

Q.5 Calculate the irrational number between 2 & 2.5.

Q.6 Write the sum of 0.3̅ And 0.4̅. [x=7/9, 2012]

Q.7 Simplify the decimal expansion of the number √2.

Q.8 Simplify is the number (√2 + √5)2.


Q.1 Find any two irrational numbers between 0.1 and

0.12. [2014] Q.2 Find any two irrational numbers between 0.5 and

0.55. [2014]

Q.3 Express the decimal numbers 2.218̅̅̅̅ in the form 𝑝

𝑞,

where p and q are integers and q ≠ 0. [122/55, 2013]

Q.4 Find any two irrational numbers between 1

7 and

2

7,

when it is given that 1

7 = 0. 142857̅̅ ̅̅ ̅̅ ̅̅ ̅̅ . [2012]


Q.1 Represent √4.5 on the number line. [2014]

Q.2 Represent √5 on the number line. [2014]

Q.3 Express 0.6 in the form of 𝑝


integers and q ≠ 0. [2013]


Q.4 Represent √3 on the number line. [2013]

Q.5 Represent √9.3 on the number line. [2013]

Q.6 Express 0.328̅̅̅̅ in the form of 𝑝


integers and q = 0. [2013]

TOPIC-3

nth Root of a Real Number VERY SHORT ANSWER TYPE QUESTIONS

Q.1 Simplify: (5 + √5 ) (5 - √5 ). [20, 2014]

Q.2 Write the equation of √12 x √8. [4√6, 2012]

Q.3 Calculate the value of 4√28 ÷ 3√7. [8/3, 2012]

Q.4 If x = √7

5 and

5

𝑥 = p √7, then find the value of p.

[25/7, 2012]

Q.5 If b > 0 and b2 = a, then find the value of √𝑎.

[√𝑎 = b, 2012]

Q.6 Write the equivalent of (a + √𝑏 ) (a - √𝑏 ). [a2-b, 2012]

Q.7 If x is a positive real number, then calculate the x2

value of √ √√𝑥234 .


Q.1 Simplify√50

2 , x √32

3, x √18

4 [2880√2, 2014]

Q.2 If x = 3 – 2 √2, find the value of √𝑥 + 1

√𝑥.

[-2√2,2014]

Q.3 Simplify √164

, -6 √3433

, + 18 √2435

- √196.

[0, 2014]

Q.4 Simplify the product: (4√3 + 3√2) x (4√3 - 3√2)

Q.5 √147

√75 is not a rational number as √147 and √75 are

not rational. State whether It is true or false. Justify your

answer. [7/5, 2012]

Q.6 Simplify (4√3 + 3√5)2. [3(31-8√15), 2012]

Q.7 Simplify √50 - √98 + √162. [7√2, 2012]

Q.8 Simplify 3 √403

, -4 √3203

, - √53

. [11√53

, 2012]

Q.9 Multiply 2 √33

by 3√2. [1√726

, 20122012]

Q.10 Find the product of 5√2 (3 + √2) (5 + √2).

[85√2 + 80, 2012]


Q.1 Simplify 3√45 - √125 + √200 - √50.

[4√5+5√2 2014]

Q.2 Find the value of (729)-1/6 [1/3, 2014]

Q.3 Evaluate: √5 + 2√6 + √8 − 2√15

[√2 + √5, 2012]

TOPIC-4 Laws of Exponents with Integral

Powers VERY SHORT ANSWER TYPE QUESTIONS

Q.1 Find the value of [(16)1/2]1/2. [2, 2014]

Q.2 Write the simplified value of (81) -1/4 x √814

[1]

Q.3 Calculate the value of [{(81)-1/2}-1/4]2. [3]

Q.4 Calculate the quotient obtained when √1500 is

divided by 2√15. [5]

Q.5 Calculate the value of 163/4/16-1/4 [16]


Q.1 Show that 𝑥𝑎(𝑏−𝑐)

𝑥𝑏(𝑎−𝑐) ÷ [(𝑥𝑏)

(𝑥𝑎)]c = 1. [2014]

Q.2 Find the value of 330+329+328

331+330−329 . [13/33, 2012]

Q.3 Simplify [5[81/3 + 271/3]3]1/4 .

[5, 2012]

Q.4 Simplify: (81

16) -3/4 x (

25

9)-3/2. [8/125, 2012]

Q.5 Find the value of (13 + 23 + 33)-3/2. [1/216, 2012]

Q.6 If a = 2 and b = 3, then find the value of ab + ba.

[17, 2012]

Q.7 If a = 2 and b = 3 then find the value of: (A) (ab + ba)-1

(B) (aa + bb)-1 [1/17, 1/31, 2012]


Q.1 Simplify (2√2-5)2 + (3√2 + √3)2 – ( √2 – 1)2.

[51-18√2+6√6, 2012]

Q.2 If xa = y, yb = z and zc = x, then prove that abc = 1.

[1, 2012]

Q.3 Simplify: (√𝑥)-2/3 √𝑦4 ÷ √(𝑥𝑦)−1/2. [2012]


Q.4 Simplify: (81

16) -3/4 x [(

25

9)-3/2 ÷ (

5

2)-3] [1, 2012]

Q.5 Simplify: (5−1x72

52x7−4 )7/2 x (5−2x73

53x7−5 )-5/2. [175, 2012]

Q.6 Find (2

3)x . (

3

2)2x =

18

16 . [4, 2012]

Q.7 If 2x x 4x = ((8)1/3 x (32)1/5. Find the value of x.

[2/3, 2012]

Q.8 Show that: (xa-b)a+b . (xb-c)b+c. (xc-a)c+a. = 1. [1, 2012]

Q.9 Find the value of 4

(216)−2/3 - 1

(256)−3/4 . [80, 2012]

Q.10 Prove that: a−1

a−1+b−1 + a−1

a−1−b−1 = −(2b2)

a2+b2 [2012]

Q.11 If (𝑎

𝑏)x-1 = (

𝑏

𝑎)2x-8, then find the value of x.

[X=3, 2012] Q.12 If x = 5 and y = 2, then find the value of :

(i) (xy + yx)-1 (ii) (xx + yy)-1 [1/3129, 2012]

LONG ANSWER TYPE QUESTIONS

Q.1 Find the value of 4

(216)2/3 + 1

(256)3/4 + 2

(243)1/5 .

[214, 2014]

Q.2 Evaluate: (81

16) -3/4 x [(

9

25)3/2 ÷ (

5

2)-3] [1, 2014]

Q.3 Evaluate: 4

(2187)−3/4 - 5

(256)−1/4 + 2

(13312)−1/3 .

[330, 2012]

Q.4 Simplify: (2−1 x 32

22 x 3−4) 7/2 x (2−2 x 33

23 x 3−5) -5/2 . [12, 2012]

Q.5 Show that: 1

1+x𝑎−𝑏 + 1

1+x𝑏−𝑎 = 1. [12, 2012]

Q.6 If xyz = 1, then show that

(1 + x + y-1)-1 + (1 + y + z-1)-1 + ( 1 + z + x-1)-1 = 1. [1, 2012]

Q.7 If 52x-1 – 25x-1 = 2500, then find the value of x.

[3, 2012]

TOPIC-5 Retionalisation of Real Numbers VERY SHORT ANSWER TYPE

QUESTIONS

Q.1 Write the rationalizing factor of 1

√50 . [ √2, ]


Q.1 Simplify: 6−4√3

6+4√3 by rationalizing the denominator.

[4√3-7, 2014]

Q.2 If x = 3-2√2, find the value of √𝑥 + 1

√𝑥 .

[±2√2, 2012]

Q.3 If √2 = 1.414, find the value of 1

√2+1.

[0.414, 2012]

Q.4 Taking √2 = 1.414 and π = 3.141, evaluate 1

√2 + π

upto three places of decimal. [3.848, 2012]


Q.1 If x = √2 − 1, find the value of (x - 1

𝑥)3. [-8]

Q.2 Find the values of ‘a’ and ‘b’ when

5+√6

5−√6 = a + b √6

[A=31/19, B=10/19 2012]

Q.3 If x = 1

3−2√2 and y =

1

3+2√2 , find the value of

x+y+xy. [7, 2012]

Q.4 If 3

4√5−√3+

2

4√5+√3 = a√5 + b√3 , then find the

value of a and b. [A=20/77, B= 1/77, 2013]

Q.5 Simplify: √2

√5+ 2 -

2

√10− 2√2 +

8

√2. [0, 2013]

Q.6 Rationalize the denominator: 1

2√7+3√3

[2√7 − 3√3, 2014]

Q.7 If x = √𝑝+2𝑞+√𝑝−2𝑞

√𝑝+2𝑞−√𝑝−2𝑞 then show that: qx2-px+q=0

[0, 2012]

Q.8Evaluate √5+√2

√5−√2 , given that √10 = 3.162.

[4.441, 2012]

Q.9 Simplify: (√3+ 1) (1-√12) +9

√3+√12 . [-5 2012]

Q.10 If √2 = 1.414 and√3 = 1.732, then calculate 4

3√3−2√2+

3

3√3+2√2 [2.063]

Q.11 If p = 5+2 √6 and x = 1

𝑝, then what will be the

value of p2 + x2 ? [98, 2012]

Q.12 rationalize the denominator of 30

5√3−3√5.

[5√3 + 3√5, 2012]

Q.13 If x = 9 + √5 , find the value of √𝑥 - 1

√𝑥 .

[4, 2012]

Q.14 Find the value of a and b, when a + b √15 = √5+√3

√5−√3

[1, 2012]

Q.15 Find the value of (x - 1

𝑥)3, if x = 1 + √2. [8, 2012]

Q.16 Simplify: 1

1+ √5 -

1

√2+ √3 +

2

√3+√5. [√5-1, 2012]


Q.17 Simplify: √6

√2+ √3 +

3√2

√6+ √3 +

4√3

√6+√2.. [0, 2012]

Q.18 write √43

, √3, √64

in ascending order. [2012]

Q.19 Find four rational numbers between 1

5 and

1

6.

[26/150, 27/150,28/150,29/150, 2012]

Q.20 Simplify: 5+ √3

7−4 √3 -

5+ √3

7+4 √3 . [8(3+5√3, 2012]

LONG ANSWER TYPE QUESTIONS-I

Q.1 Rationalize the denominator of 1

( √2+ √3)−√4) .

[ +5√2+3√3+4√6

23, 2014]

Q2 Prove that: 1

3− √8 -

1

√8− √7 +

1

√7−√6-

1

√6−√5+

1

√5−2=5

Q.3 If x = 4 - √15, find the value of ( x + 1

𝑥)2. [64, 2014]

Q.4 Prove that: 1

3− √7 +

1

√7− √5 +

1

√5−√3+

1

√3+ 1 = 1.

Q.5 If x=√3+√2

√3−√2 and y =

√3−√2

√3+√2 , find x2+y2. [98, 2014]

Q.6 Find a and b if 2√5+√3

2√5−√3 +

2√5−√3

2√5+√3 = a + √15 b[2013]

[b=0, 2013]

Q.7 Prove that:

1

√4+√5 +

1

√5+√6 +

1

√6+√7+

1

√7+ √8+

1

√8+ √9= 1.

Q.8 If x = √3+1

√3−1, y =

√3−1

√3+1 then find the value of

x2+y2+xy. [15, 2012]

Q.9 If x = 1

2−√3 , find the value of 2x3-2x2+7x+5. [3]

Q.10 Evaluate: 1

√10+√20+√40−√5−√80 , given that √5=2.2

and 3.2. [5.4, 2012]

Q.11 If x=3-2√2, find tha velue of x4-1

x4.

[-816√2, 2012]

Q.12 If a = 2− √5

2+ √5 , b =

2+ √5

2− √5 then find (a+b)3.

[-5832, 2012]

Q.13 Find the value of a and b in 3− √5

3+2 √5 = a √5-

𝑏

11.

[19, 2012]


POLYNOMIALS TOPIC-1

POLYNOMIALS VERY SHORT ANSWER TYPE QUESTIONS Q.1 Find p (0) if p(y) = y2-y+1. [1, 2014]

Q.2 Find the value of m, if x=4 is a factor of the polynomial x2+3x+m. [-4, 2014]

Q.3 Write the expression which represents a polynomial.

[√3x2-x-1]

Q.4 What is the degree of a zero polynomial? [not define]

Q.5 Name the polynomial of the expression -3x+2.

[Linear polynomial] Q.6 What is the degree of a zero polynomial (x3+5)(4-x5).

[8]

Q.7 What is x- 1

x .

[Not a polynomial]

Q.8 write an example of a constant polynomial.

[7] Q.9 Name the polynomial containing two non-zero

terms. [Binomial]

Q.10 What is the degree of polynomial √3? [0]

Q.11 What is the zero of the zero polynomials?

[zero]

Q.12 write the numbers of zeroes in a cubic polynomial.

[3] Q.13 write the factor (x+1)3 – (x+1).

[(x+1)]

Q.14 If – 14 is the zero of the polynomial p(x) = x2 + 11x + k, then calculate the value of k. [k=28]

Q.15 Write the polynomial in one variable.

Q.16 In the expression x2 + π

2 x – 7 , what is the

coefficient of x?

SHORT ANSWER TYPE QUESTIONS I

Q.1 Find the value of k, so that polynomial x2+3x2-kx-3. [k=1,2014]

Q.2 For what value of k, is the polynomial

p (x) = 2x3 – kx2 + 3x + 10. [k=-3, 2014]

Q.3 Find the value of k, if (x-1) is a factor of

p(x) = 2x2 + kx + √2 [-2-√2, 2014]

Q.4 Find the value of polynomial

P(x) = x3 – 3x2 – 2x + 6 at x = √2 [k0, 2014]

Q.5 Classify the following as linear, quadratic and cubic polynomial:

(a) x2+x (b) x-x2 (c ) 1+x (d) 7x3


Q.1 If f(x) = 3x + 5, evaluate f(7) – f(5). [6]

Q.2 Find the value of polynomial x2 – 3x + 6 at

(i) x = √2 (ii) x = 3. [(i) √2, (ii) 6, ]


Q.1 If f(x) = x2 – 5x + 7, evaluate f(2) – f(1

3).[

−59

9, 2014]

TOPIC-2

Remainder Theorem VERY SHORT ANSWER TYPE QUESTIONS

Q.1 On dividing 5x3-2y2-7y +1 by y, what remainder do

we get? [1]

Q.2 If x11+101 is divided by x+1, then what remainder

do we get? [100]

Q.3 write the zeroes of the polynomial

p(x) = x (x-2) (x-3). [x=0,2,3]

Q.4 If p(x) = x2 – 3x + 2, then what is the value of

p (0) + p(2)? [2]


Q.1 Find the remainder when x3+x2+x+1 is divided by

x + 1

2 , using remainder theorem. [15/8,2012]

Q.2 Find the remainder when x4+x3-2x2+x+1is divided by x-1.

Q.3 Find the remainder when x3+6x-ax2-a is divided by

x-a.

SHORT ANSWER TYPE QUESTIONS II Q.1Using remainder theorem, factories:

6 x3-25x2 + 32x -12. [(x-2) (2x-3) (3x-2)]

Q.2 polynomial 3x3-5x2+kx-2 and -x3-x2+7x+k leave the

same remainder when divided by x+2. Find the value of k. [k = -12, 2013]


Q.1 The polynomial ax3-3x2-13 and 2x3-5x+a leave the same remainder in each case, when divided by (x-4).

Find the value of a. [a = 1, 2014]


Q.2 Find the quotient and remainder obtained on dividing p(x) = 4x4+11x3+2x2-11x+6 by x2+2x+2 and

verify remainder by using remainder theorem.

[4x4 + 8x3 + 2x2 – 11x + 6, 2014]

Q.3 Find the value of ‘a’ if remainder is same when

polynomial p(x) = x3+8x2+17x+a is divided by (x+2) and

(x+1). [a = 0, 2014]

Q.4 If f(x) = x4-2x3-ax+b is a polynomial such that when

it is divided by x-1 and x+1, the remainder are 5 and 19 respectively. Determine the remainder when f(x) is

divided by x-2. [10, 2013]

Q.5 Find the quotient when p(x) = x3+3x2+3x+5 is divided by g(x) = (x+2). Also find the remainder.

[Quotient x2 + x + 1, Reminder =3]

Q.6 Find the quotient and remainder when

6x4+11x3+13x2-3x2-3x+27 is divided by 3x+4. Also

check the remainder obtained by using remainder theorem. [47, 2012]

Q.7 The polynomial bx3+3x2-3 and 2x3-5x+b when

divided by x-4, leave the remainder R1 and R2 respectively. Find the value of b, if 2 R1-R2=0.

[b = 18/127, 2012]

Q.8 Divide x3+4x2-3x-10 by x+1 and verify your

remainder by remainder theorem. [-4, 2012]

Q.9 Divide 3x3-8x2+3x+2 by x2-3x+2 and verify the

division algorithm. [2012]

Q.10 If the polynomial ax3+4x2+3x-4 and x2-4x+a leave the same remainder when divided by (x-3). Find the

value of a. [a = -1, 2012]

Q11 Find the value of p(x) = x4-2x3+3x2-px+3p-7 when

divided by (x+1) leave the remainder 19. Also find the

remainder when p(x) is divided by x+2. [62, 2012]

Q.12 What must be subtracted from x4+1 so that x4+1 is

exactly divisible by (x-1). Write the resultant polynomial

which is exactly divisible by (x-1). [x4-1, 2012]

Q.13 If x=2 and x=0 are zeroes of the polynomial

2x3-5x2+px+b, then find the value of p and b. [p= 2 b=0, 2012]

Q.14 If p(x) = x3+3x2-2x+4, then find the value of

p(2) + p(-2) – p(0). [36, 2012]

Q.15 If the polynomials f(x) = x4-2x3+3x2-9x+3a-7,

when divided by x+1leave the remainder 20, then find

the value of a. Also find the remainder when f(x) is

divided by x+2. [67, 2012]

Q.16 The polynomials x3+2x2-5ax-8 and x3+ax2-12x-6,

when divided by (x-2) and (x-3) leave remainders p and q respectively. If q-p=10, find the value of a.

[33/19, 2011]

Q.17 The polynomials ax3-3x2+4 and 2x3-5x+a when divided by (x-2), leave the remainders p and q

respectively. If p-2q =4, find the value of a. [a=4, 2011]

TOPIC-3

Factor Theorem VERY SHORT ANSWER TYPE QUESTIONS

Q.1 Factories: x2-3x. [x(x-3), 2014]

Q.2 Factories 12a2b-6ab2. [6ab(2a-b), 2014]

Q.3 Find the value of k, if x-2 is a factor of

p(x) = 2x2+3x-k. [k=14, 2011]

Q.4 If f(x) be a polynomial such that f(−1

3) =0, then

calculate one factor of f(x). [3x+1, 2011]

Q.5 Write the factors of a3-1. [a2+1+a,]

Q.6 Write the factors of polynomial: x2+5√2𝑥+12.

[(x+3√2) (x+2√2) ]

Q.7 Write the factors of polynomial:

4x4+y2+4xy+8x+4y+4. [(a6 + b6)]


Q.1 Factories: 9x2+6xy+y2 [(3x+y)22014]

Q.2 Factories: 8a3+8b3 [2014]

Q.3 Factories: (x+3y)3 + (3x-y)3 [2014]

Q.4 Factories: 64a3-27b3-144a2b+108ab2 [(4a-3b)3 2014]

Q.5 Factories: a9+b9+3a6b3+3a3b6 [(a3+b3)3 2014]

Q.6 Factories (x+2)2+p2+2p(x+2) [(x+2+p)2 2012]

Q.7 Factories: 8-27a3-36a+54a2 [(2-3a)3 2012]

Q.8 Factories: x4y4-256z4 [2012]

Q.9 Factories: m (m-1)-n (n-1). [(m-n) (m+n-1) 2012].

Q.10 Factories: x4-125xy3.


[x(x-5y) (x2+25y2 + 5xy, 2012]

Q.11 2y3+y2-2y-1. [(y-1) (y+1) (2y+1), 2012]

Q.12 Factories: Show that (x-1) is a factor of the

polynomial f(x) =2x3-3x2+7x-6. [0, 2012]

Q.13 For what value of k, (x+1) is a factor of

p(x) = kx2-x-4? [3]

Q.14 Factories: x4-y4 [(x-y) (x+y) (x2+y2), 2012]

Q.15 Factories: 64x3+ √125y3

[(4x+√5𝑦 ) (16x2-4√5xy + 5y2), 2012]

SHORT ANSWER TYPE QUESTIONS II

Q.1 Factories: p3q3 + 343

729. [(pq+

7

9) (p2q2+

49

81 -

7𝑝𝑞

9), 2012]

Q.2 Factories: a6-b6.

[(a-b) (a+b) (a2+b2+2ab) (a2+b2-ab), 2012]

Q.3 Factories: (5𝑎 +2

3)2 - (2𝑎 +

1

3)2 [(7a+

1

3)(3a+1)]

Q.4 Factories: 9x2+y2+z2-6xy+2yz-6xz. Hence, find its

value when x=1, y=2 and z=-1. [4, 2012]

Q.5 Factories: 125x3-27y3+z3+45xyz. [2012]

Q.6 Find the value of a for which (x-a) is a factor of the

polynomial x6-ax5+x4-ax3+3x-a+2. [-1, 2012]

Q.7 Factories: a7+ab6 [a(a2+b2) (a4+b4-a2b2), 2012]

Q.8 Factories: (x2-4x) (x2-4x-1) -20.

[(x-5) (x+1) (x-2)2, 2012]

Q.9 If x-a is the factor of 3x2-mx-nx, then prove that

a = 𝑚+𝑛

3 [2012]

Q.10 Factories: 250x3-432y3 [2(5x-6x) (25x2+36y2+30xy), 2012]

Q.11 Factories: x3-3x2-9x-5. [2012]

Q.12 Factories: x2 + 1

x2 + 2 - 2x - 2

𝑥

[(𝑥 +1

𝑥)2 -2 (𝑥 +

1

𝑥), 2012]

Q.13 If (3x-2) is a factor of 3x3+x2-20x+12, find other

factors. [(x+3) (x-2), 2012]

Q.14 Factories: 3-12(a-b)2. [2012]

Q.15 Factories: 27p3- 1

216 -

9

2 p2 +

1

4 p.

[(3𝑝 −1

6) (3𝑝 −

1

6) (3𝑝 −

1

6), 2012]


Q.1 Factories: x3-12x2+47x-60.

[(x-3) (x-4) (x-5), 2014]

Q.2 Using factor theorem, find the value of ‘a’ if

2x4-ax3+4x2-x+2 is divisible by 2x+1. [a=-29, 2014]

Q.3 Show by long division method that x-3 is a factor of

2x4+3x3-26x2-5x+6. [2014]

Q.4 Factories: (m+2n)2+101(m+2n)+100.

[(m+2n+1) (m+2n+100), 2014]

Q.5 Factories: x3-3x2-9x-5. [(x-5) (x+1) (x+1), 2014]

Q.6 Factories: 125a3-27b3+75a2b-45ab2

[(5a+3b) (5a+3b) (5a-3b), 2014]

Q.7 Factories: 1

64 x3-8y3+

3

16 x2y -

3

2 xy2.

[(𝑥

4− 2𝑦) (

𝑥

4+ 𝑦) (

𝑥

4+ 4𝑦), 2014]

Q.8 Factories: x3+13x2+32x+20.

[(x+2) (x+1) (x+10), 2014]

Q.9 Find the value of p for which the polynomial

x3+4x2-px+8 is exactly divisible by x-2. Hence factories

the polynomial. [(x-2) (x2+6x-4), 2014]

Q.10 Without actual division, show that f(x) = 2x4-

6x3+3x2+3x-2 is exactly divisible by x2-3x+2. [(x-1) and (x+2) are the factors of f(x).

F(x) is already divisible by g(x). 2013]

Q.11 Factories: 2x3-9x2+x+12. [x=4, 2013]

Q.12 Factories: (p + q)2 -20 (p + q)-125

[(p+q-25) (p+q+5), 2013]

Q.13 Factories completely: x8-y8

[(x4+y4) (x2+y2) (x+y) (x-y), 2013]

Q.14 Find the value of a and b, if x2-4 is a factor of

ax4+2x3-3x2+bx-4 and hence factories it completely.

[(x-2) (x+2) (x+1)2, 2012] Q.15 Simplify and factories (a+b+c)2 – (a-b-c)2 + 4b2-4c2

[4(b+c) (a+b-c), 2012]

Q.16 Factories: x4+2x3y-2xy3-y4

[(x-y) (x+y)3, 2012]


Q.17 The volume of a cuboid is polynomial p(x) = 8x3+12x2-2x-3. Find possible expression for dimension

of the cuboid. Verify the result by taking x=5 units.

[1287 cuboid units, 2012] Q.18 Verify x3-y3= (x-y) (x2+y2+xy). Hence factories

216x3-125y3. [(6x-5y) (36x2+25y2+30xy), 2012]

Q.19 If x+a is a factor of the polynomial x2+px+q and

x2+mx+n, prove that a= 𝑛−𝑞

𝑚−𝑝 . [2012]

Q.20 If (x-2) and (𝑥 −1

2) are factors of px2+5x+r, then

show that p = r [2012]

Q.21 Verify that (x-1), (x-2) and (2x+1) are the factors

of the polynomial 2x3-5x2+x+2. [2012]

Q.22 Using factor theorem, factories x3-2x2-5x+6.

Q.23 Factories: (a) 4a2-9b2-2a-3b. (b) a2 – b2 -2(ab-ac-bc).

[(a) (2a+3b) (2a-3b-1), (b) (a-b) (a-b+2c). 2012]

Q.24 Factories: (a2-2a)2 – 23(a2-2a) + 120. [(a-5) (a+3) (a-4) (a+2), 2012]

TOPIC-4

Algebraic Identities VERY SHORT ANSWER TYPE QUESTIONS

Q.1 Simplify (𝑥 +1

2) (𝑥 −

3

2) [x2+2x+

3

2, 2012]

Q.2 If 𝑥

𝑦 +

𝑦

𝑥 = -1, (x≠y, y≠0,, x≠0), then what is the

value of x3-y3 [(x-y) x 0 = 0, 2012]

Q.3 if (a+b+c) =0, then write the equivalent of a3+b3+c3.

[3abc, 2012]

Q.4 Write the coefficient of x2 in the expansion of (x-2)3.

[-6, 2012]

Q.5 If 𝑥 +1

𝑥 = 4, then calculate the value of x2 +

1

x2 .

[14]

Q.6 Calculate the value of 833+173

832−83 X 17+172 [100]


Q.1 Explain by using identity (2x-y+z)2.

[x2 +y2+z2-4xy-2yz+4xz, 2012] 2014]

Q.2 Expand: (1

3𝑥 −

2

3𝑦)3 [2013]

Q.3 Find the value of the polynomial x2-9, for x = 97. [9400, 2012]

Q.4 If x and y are two positive real numbers such that

x2+4y2 = 17 and xy = 2, then find the value of (x+2y). [5, 2012]

Q.5 Give possible expression for the length and breadth

of a rectangle whose are is given by 25a2-35a+12.

[(5a-3) (5a-4), 2012] Q.6 Find the value of 8x3+27y3, if

2x + 3y = 8 and xy = 2. [224, 2012]

Q.7 Evaluate 103 x 107 without multiply directly

[11021, 2012]

Q.8 Evaluate 249 x 251 by using an identity. [62499, 2012]

Q.9 If a,b,c are all non zero and a+b+c = 0, prove

a2

bc+

b2

ac +

c2

ab =3. [3, 2012]

Q.10 Factories: 12(x2+7)2 -8)x2+7) (2x-1) -15 (2x-1)2. [(2x2-6x+17) (6x2+10x+37), 2012]

Q.11 Without actually calculating the cubes. Evaluate 143+133-273

[-14742, 2012]

Q.12 Expand using suitable identity (2x-3y+z)2. [4x2+9y2+z2-12xy-6yz+4xz, 2012]

Q.13 If (18

15)3- (

1

3)3-(

1

5)3=

𝑥

75, find x. [8, 2012]

Q.14 simplify (𝑥 +1

𝑥) (𝑥 −

1

𝑥) (x2 +

1

x2) (x4 +

1

x4)

[(x8 +1

x8), 2012]

Q.15 Using suitable identity evaluate (103)3

[1092727, 2012]

Q.16 Expand (𝑎

4−

𝑏

2+ 1)2 using identity.

[a8

16+

b8

4+ 1 −

ab

4 -b +

a8

2 , 2012]

Q.17 If a = 3+b, then what is the value of a3-b3-9ab.

[27]

Q.18 Find the value of x2 + 1

x2 , if x - 1

𝑥 =√3 . [5, 2011]


Q.1 Evaluate (√2 + √3)2 + (√5 + √2)2.

[2(6+√6 - √10 ), 2013]

Q.2 Find the product (x+y+2z) (x2+y2+4z2-xy-2yz-2zx). [x3+y3+8z3-6xyz, 2012]


Q.3 Find the value of

(x-a)3 + (x-b)3 + (x-c)3 - 3(x-a) (x-b) (x-c), if a+b+c= 3x. [0, 2013]

Q.4 Find the value of x3+y3+15xy-125, when x+y = 5. [0]

Q.5 If 𝑥 +1

𝑥 = 4, find x2 +

1

x2. [14, 2013]

Q.6 Find the value of ab+bc+ca, if a+b+c = 9 and a2+b2+c2 = 35. [23, 2012]

Q.7 If x = √3+√2

√3−√2 and y =

√3−√2

√3+√2, find the value of

x2-y2+xy, if √6 = 2.4. [97, 2012]

Q.8 Simplify (a2−b2)3+(b2−c2)3+(c2−a2)3

(a−b)3+(b−c)3+(c−a)3 .

[(a+b) (b+c) (c+a), 2012]

Q.9 If a2+b2+c2 = 280 and ab + bc + ca = 9

2 , then find

the value of (a+b+c)3. [4913, 2012]

Q.10 If x + y +4 = 0, then find the value of

x3 + y3-12xy+64. [0, 2012]

Q.11 Simplify ( a+2b+3c)2 - ( a-2b-3c)2 – 6b2-9bc

[92b+3c) (4a-3b), 2012]

Q.12 If 𝑥 +1

𝑥 = 3, then find x3 +

1

x3 . [18, 2012]

Q.13 Simplify (𝑥

3+

𝑦

5)3 - (

𝑥

3−

𝑦

5)3.

[2y

5 (

x2

3

y2

25) ]


Q.1 If 𝑥 +1

𝑥 = 5, evaluate x2 +

1

x2. [23, 2014]

Q.2 Evaluate by using identities

(i) 103 x 107 (ii) (102)3 [(i) 11021, (ii) 1061208, 2014]

Q.3 If a-b = 7 and a2 + b2 = 85, find a3 – b3 [721, 2014]

Q.4 Verify that

x3+y3+z3-3xyz = 1

2 (x + y + z) [(x-y)2 + (y-z)2 + (z-x)2]

[2012] Q.5 Prove that

2x3 +2y3 +2z3 – 6xyz = (x+y+z) [(x-y)2 + (y-z)2 + (z-x)2].

[252, 2013] Q.6 If x + y + z = 10 and x2+y2+z2 = 40.

Find xy + yz + zx and x3 + y3 + z3-3xyz. [100, 2012]

Q.7 Find the value of x3 – 8y3- 36xy-216, when x = 2y+6 [0, 2012]

Q.8 Simplify: (a+b)3 + (a-b)3 + 6a(a2-b2). [8a3, 2012]

Q.9 If a+b+c = 6 and ab + bc + ca = 11, find the value of a3+b3+c3 – 3abc. [18, 2012]

Q.10 If x2 + 1

x2 = 7, find the value of x3 + 1

x3 . [18, 2012]

Q.11 Find the value of p3-q3, if p-q = 10

9 and

5

3.

[5050/729, 2012]

Q.12 If a = 5 + 2 √6 and b = 1

a , then what will be the

value of a2 + b2 and a3 + b3. [970, 2012]

Q.13 Prove that (x+y)3 + (y+z)3 + (z+x)3 -3(x+y) (y+z)

(z+x) = 2 (x3+y3+z3 – 3xyz). [2(x3+y3+z3-3xyz). 2012]

Q.14 If x and y two positive real numbers such that 8x3 +

2y3 = 730 and 2x2y +3xy2 = 15, then evaluate 2x+3y. [10, 2012]


LINES & ANGLES TOPIC-1

Different Types of Angles

VERY SHORT ANSWER TYPE QUESTIONS

Q.1 In the fig. below, AOB is a straight line. Calculate

the measure of ∠COD.

D

C 2x-200

X+200 600

A B

Q.2 What is the measure of an angle which is

compliment of itself. [450]

Q.3 Write a complimentary angle of 650. [250]

Q.4 Two angles (300 – a) and (1250 + 2a). If each one is

the supplement of the other, then find the value of a.

[250]

Q.5 Write the complement of (900 – a). [a]

Q.6 Write the angle which is one fifth of its complement. [150]

Q.7 In the triangle, two angle, measure (550 + 3a) and

(1150 – 2a). If each is supplement of the other, then calculate the value of a. [100]

Q.8 In the given fig. what is the value of x? [500]

C

D

x X+100 X+200

A B

Q.9 Calculate the value of x in the figure below. [200]

5x 4x

A O B

Q.10 In the figure below, Calculate the value of y.

[280]

3y

400 2y A B

SHORT ANSWER TYPE QUESTIONS I Q.1 In the figure, lines XY and MN intersect at O. If

∠POY = 900 and a:b = 2:3, find the value of c. [1260]

P

M

a

b O

X Y

c

Q.2 If ∠ AOP = 5y, ∠ QOD = 2y and ∠BOC = 5y in the

given figure, find the value of y. [150]

A D

5y O 2y

P Q 5y

C B

Q.3 In the figure, if x + y = w + z, then prove that AOB

is a line C [1800] B

x

y w

z

A D

Q.4 In the given figure, ∠AOC and ∠ BOC form a line AB. If a-b = 800, find the value of a and b. [500]

C

A b A B


Q.5 In the given figure, ∠AOB: ∠BOC = 2:3. If ∠AOC

= 750, then find the measure of ∠AOB and ∠BOC. [450] A

B

O 750 C

Q.6 In figure, prove that , ∠AOB + ∠BOC + ∠COD +

∠DOA = 3600.

B A

O

D

C

Q.7 In figure ∠DOB = 870 and ∠COA = 820. If ∠BOA

= 350, then find ∠COB and ∠COD. [400]

C D

B 820

350 A

Q.8 In the figure, a is greater than b, by 1

6 of a straight

angle. Find the values of a and b. [750]

a b

Q.9 Two supplementary angles are in the ratio 2:3, find

the angles. [720 , 1080]

SHORT ANSWER TYPE QUESTIONS II

Q.1 In the given figure, PO Ʇ AB. If x:y:z = 1:3:5, then

find the degree measure of x,y and z.

[x=100, y=300,z=500] P

Q

R y x

z

A O B

Q.2 Prove that if two lines intersect each other, then the

bisectors of vertically opposite angles are in the same line.

Q.3 In figure, if AB || CF and CD || FE, then find the value of x. [750]

B

F x E 400

650

A C D

Q.4 I the given figure ∠3 and ∠4 are exterior angles of quadrilateral ABCD at point D and B respectively. And

∠A = ∠2, ∠C = ∠1. Prove that ∠3 + ∠4 = ∠1 + ∠2. D C

3 1

2 4

A B

Q.5 Lines PQ and RS intersect each others at O (see

figure). If ∠POR: ∠ROQ = 5:7, find all the angles a, b, c, and d. [1050]

p S

a

d O b c

R Q

Q.6 In the given figure, AC||DE and AD||CE, find x and

y, when it is given that ∠BAC = 700 and ∠DEC = 550. [550]

A x D

700

Y 550

B C E


Q.7 In figure, POQ is a line. Ray or is Ʇ to PQ. OS is

another ray lying between OP and OR. Prove that:

∠ROS = 1

2 [∠QOS - ∠POS]

R

S

P O Q

Q.8 It is given that ∠XYZ = 640 and XY is produced to a

point P. Draw a figure from the given information.

If ray YQ bisect ∠ZYP, find ∠XYQ and reflex ∠QYP.

[1220, 3020]

Q x Z x 640

P Y X


Q.1 In the given figure, two straight lines PQ and RS

intersect each other at O. If ∠POT = 700, find the value of a, b, c. [c=480]

Q

R

2c a

4b

750 b S

P T

TOPIC-2

Transversal Line VERY SHORT ANSWER TYPE QUESTIONS

Q.1 From the given figure, identify the incorrect statement, given that l || m and t is the transversal.

5 6

1 2 8 7 3

4

l m

Q.2 In fig. below, calculate the value of angle q.

P

A C

500 x 900

a y

D B

Q Q.3 In the given fig. AB and CD are parallel to each

other then calculate the value of x. [1000]

A B

1200

x E 1400

C D

Q.4 In the given figure, AB || CD and ‘l’ is transversal, then calculate the value of ‘x’. [260]

l

A 3x+350 B

y

2x+150

C D

Q.5 In fig., If m || n and ∠a:∠b = 2:3, then what will be

the measure of ∠h? [1080]

b a

m

n

h

Q.6 In the given fig. PQ || RS and EF || QS. If ∠PQS =

600, then what will be the measure of ∠REF? [1200]

P R

E F

600


Q S

Q.7 In fig. PQ || RS, ∠QPR = 700, ∠ROT = 200, find the

value of x. [500]

P Q 700

T

1200

O


Q.1 In the fig. AB || DE, ∠ABC = 1400 and ∠CDE =

1000. Find ∠BCD. D E

A B 1000

1400

[600] C

Q.2 In the given fig. m || n and p || q. If ∠1 = 750 Prove

that ∠2 = ∠1 1

3 of right angle.

m

1

2 n

P q

Q.3 In the given fig., if l1 || l2 and l3 || l4, what is y in

term of x? [900 - 𝑥

2]

L3 l4

x l1

2

y l2

y

Q.4 In given fig. ΔLMN is an isosceles triangle with ∠m

= ∠n and LP bisects ∠NLQ. Prove that LP || MN.

Q

L

P

M N

Q.5 In the fig. AB || DC. Determine x. [2850]

A B

450

x O

300

D C

Q.6 In the given figure find x and y and then show that

l || m. t

500 l

x

y m

1300

Q.7 In the given figure, show that AB || EF. A

B

600

E F

1550

350

250

C D

Q.8 In the figure AB || CD, find the value of z, ∠DNM

and ∠CNM. [570]

M A B

(3z-420)

(2z+130) C N D

Q.9 In figure in AB || CD || EF and x:y = 3:2, find z.

[1080]

A B

x

A y B

A z B


Q.10 Find the supplement of 4

3 of the right angle. [600]

Q.11 In the given figure, state which lines are parallel and why.

B D A 700

G E 800

800 C H

F

Q.13 If a transversal intersects two parallel lines, then the bisectors of any pair of alternate angles are parallel.

Prove it.

A M B

l Q x

y P

C N D m


Q.1 In figure, a transversal l cuts two lines AB and CD

at E and F respectively. EG is the bisector of ∠AEF and

FH is the bisector of ∠EFD such that ∠a = ∠b. Show that

EG || FH and AB || CD. [2014]

l

A E B

a

G H

b

C F D

Q.2 In figure, AB || CD, then find x. [1000]

A G B

1350 E x

1250 C D

Q.3 If a transversal intersects two lines such that the bisectors of a pair of corresponding angles are parallel,

then prove that the two lines are parallel. [2013]

M Q

1

A 2 B P

S

3 4

C R D

N

Q.4 In the figure, AB || CD, EF Ʇ CD and ∠GFC = 1300.

Find x, y and z. [500]

A E G B

y x

1300 z

C F D

Q.5 In the given figure, QP || ML, find the value of x.

[300]

Q A M

1350

150 100

x

P B L

Q.6 Prove that the angle between internal bisector of one

base angle of a triangle is equal to one-half of the

vertical angle. [2013] A E

B C D


Q.7 If two parallel lines are intersected by a transversal, prove that the bisector of the interior angles on the same

side of transversal intersect each other at right angles.

E [2012]

A M B

1

3 p

2

C N D

F

Q.8 In the given figure, AB || CD and EF is a transversal

cutting them at G and H respectively. If ∠EGB = 350 and

QP Ʇ EF, then find ∠PQH [2012]

E

A 350 B G

C Q H D

P

F

Q.9 In the given figure, if AB || CD, ∠BPQ = (5x-200) and ∠PQD – (2x-100), find the value of y and z. [2012]

[Y=500, z= 500]

A y B

P 5x-200

Q 2x-100

C z D

Q.10 In the figure, AB || CD, EF || DQ. Determine

∠PDQ, ∠AED and ∠DEF. [720]

P Q

430

C D

F

650

A E B Q.11 In the given figure l || m || n. From the figure, find

the ratio of ( x = y) : (y – x). [2012]

A B

l

1000 400

x0 E m

C D

y0

300 200 n

E


Q.1 In the given figure, AB || DC, ∠BDC = 350 and

∠BAD = 800. Find x, y and z. [350, 650, 1100]

D C

350 z

y

y-30

800

x

A B

Q.2 In the above figure ABCD is a quadrilateral in

which ∠ABC = 730, ∠C = 970 and ∠D = 1100. If AE ||

DC and BE || AD and AE intersect BC at F, find the

measure of ∠EBF. [270]

D C

1100 970

A F E

730

B


TOPIC-3

Angle Sum Property of triangle VERY SHORT ANSWER TYPE QUESTIONS

Q.1 What is the value of x in the figure given below? [400]

1100 600

Q.2 In the figure below, if x, y and z are exterior angles

of ΔABC, then calculate the value of x + y + z. [3600]

A

z

y B

x C

Q.3 In the figure below, if ∠A + ∠B + ∠C + ∠D + ∠E + ∠F = k right angles, then what is the value of k? [k=4] A

F E

B C

D

Q.4 An exterior angle of a triangle is 800 and two interior opposite angles are equal. What will be the measure of

each? [400]

Q.5 In the given figure, calculate the value of ∠PQR. [300]

750 P

1050

Q R

Q.6 In ΔABC, ∠A = ∠B/2 = ∠c/6, then what will be the measure of ∠A? [200] Q.7 In the given figure, ABC is an isosceles triangle

with AB = AC and ∠A = 500. Calculate ∠B. [650]

A

500

B C

Q.8 In the fig. below, in ΔABC, AB = AC, then calculate

the value of x A [1300]

800

B C

x

Q.9 In the given figure below, what will be the value of

x? [1300]

A

1100

1200 B

x C


Q.1 In the given figure, if ∠A = 600 and ∠B = 700, then

find∠ ACD. [1300]

A

600

700 1200

B C D


Q.2 In ΔABC, ∠A + ∠B = 650 and ∠B + ∠C = 1400, find the value of ∠B and ∠C. [B=250, C=1150] Q.3 In figure, if AB || CD, ∠APQ = 400 and ∠PRD = 1180, find x and y. [780] A B 400

y

x 1180

C Q R D

Q.4 In ΔABC, if ∠A = (2x-50), ∠B = (5x+50), ∠C =

(3x+500), then find the value of x, ∠A, ∠B and ∠C. [210, 700, 890]

Q.5 Prove that if one angle of a triangle is equal to the

sum of the other two angles, then the triangle is right

angled triangle. [2012]

Q.6 In ΔABC, ∠A = 600, ∠B = 400. Which side of this triangle is the smallest? Give reason for your answer? [2012] A

600

400

B C

Q.7 An exterior angle of a triangle is 1100 and one of the

interior opposite angles is 300. Find the measure of

another two angles of the triangle. [800]

Q.8 In ΔABC, ∠B = 450, ∠c = 550, AD bisects ∠A. Find ∠ADB and ∠ADC. [850] Q.9 Find the value of x in the given figure, where ∠A = 400 and ∠BED = 1200. [100]

A

400

E

F 1200

x B C D

Q.10 In the given figure AP and DP are bisectors of ∠A

and ∠D. Prove that 2∠APD = ∠B + ∠C. [2012]

A B

P

C D

Q.11 In figure, If lines PQ and RS intersect at point T,

such that ∠PRT = 500, ∠TSQ = 600 and ∠RPT = 1000,

find ∠SQT. [900] P

1000 500 T S

R 600

Q

Q.12 In the given figure AF || BE, AC Ʇ BE and AF

bisects ∠GAD. If ∠GAD = 700 then find the measure of

∠ABC and ∠ADE. [350]

G

A 700 F

B C D E

Q.13 Prove that if in a triangle, its sides are produced in

order, then sum of the exterior angles so formed is 3600.

[2012]

Q.14 The degree measure of three of a triangle are x0, y0,

and z0. If = x0+x0

2, then find the value of z0. [600]



Q.1 In given figure DE Ʇ AB. Find the value of x

and y. [500, 600]

A

E

1100 F

x y 400

D C B

Q.2 If the figure QT Ʇ PR, ∠TQR = 400 & ∠SRP = 300 Find x and y. [800] P

300

T

400 y x

Q S R

Q.3 In figure, PQ Ʇ QR. QP || RL, ∠RQT = 380 and

∠QTL = 750. Find x and y. [530, 370]

P Q

y

380

x 750

R T L

Q4 In the given figure, ∠CAB : ∠BAD = 1:2, find all the internal angles of ΔABC. [2012] B

x

D

690 x+130

E A C

Q.5 In the given figure. BO and CO are bisectors of

∠DBC and ∠ECB respectively. If ∠BAC = 700 and

∠ABC = 400, find the measure of ∠BOC. [550] A

700

400

B C

?

D E O

Q.6 In the given figure, find a + b [1270]

B C

5( 𝑥

2-10)

1

2 x x + 90

A b D

a

Q.7 In figure AB || CD, then find the measure of x.

[180]

B D Q

x

P 1100 880

R A C

Q.8 In ΔABC, AD and CE are the bisectors of ∠A and

∠C respectively. If ∠ABC = 900 then find ∠AOC. [1350]

A

E

O

B D C

Q.9 In the given figure, AB || CD, ∠BAC = 720 and ∠CEF = 400. Find ∠CFE. [320]


E

D

400

B C

720 ?

A F Q.10 In the given figure, find the value of x0. [980]

A

230

D x0

400 350

Q.11 Prove that the sum of angles of a triangle is 1800

[2012]

Q.12 In the given figure, if ∠BCD = 250, ∠BAQ = 1100 and ∠ACR = 1250, then find the value of x, y and z. [550, 300, 800] Q 1100 A

D z

y 1250 x 250

B C R

Q.13 In the figure, ∠BAC = 500, ∠GBD = 700 and l and m are parallel lines. Find x, y and z. [1200, 700, 600] A

G 500 x

B C l 700

y z

D E m

Q.14 In figure, triangle ABC is right angled at A. AL is

drawn perpendicular to BC. Prove that ∠BAL = ∠ACB.

[2012] A

B C

L

LONG SHORT ANSWER TYPE QUESTIONS

Q.1 Prove that the sum of three angles of a triangle is 1800. Using this result, find the value of x and all the

three angles of the triangle, if the angles are:

(2x-7)0, (x+25)0 and (3x+12)0. [x = 250] A

B C

Q.2 Prove that the sum of all the angles of a triangle is

1800. Also find the angle of a triangle if they are in ratio 5:6:7.

Q3 The sides AB and AC of ΔABC are produced to

points P and Q respectively. If bisectors BO and CO of

∠CBP and ∠BCQ respectively meet a point o, then

prove that ∠BOC = 900 - 1

2 x.

Q.4 In the given figure, AB || CD, ∠ECD = 240, ∠EDC = 420 and AC = CE. Find x and y. [420, 720]

A B

x

y

E

z

C 240 420 D

Q.5 In the given figure, ∠ACD = ∠ABC and CP bisects ∠BCD. Prove that ∠APC = ∠ACP. A

D

P

B C

Q.6 In the given figure, find the value of x and y if AB ||

CD. [420, 380]


l

Q

A B

200 y

P x

220 C D

580 R


TRIANGLES TOPIC-1

Criteria for Congruence of

Triangle

VERY SHORT ANSWER TYPE QUESTIONS Q.1 In ΔABC and ΔDEF, AB = DE, ∠A = ∠D. What will be the condition in which the two triangles will be

congruent by SAS axiom?

Q.2 Given ΔOAP = ΔOBP in the figure given below.

Prove the criteria by which the triangles are congruent.

A

P

O

B

Q.3 In The figure given below, if AB = QR, BC = PR

and CA = PQ, then ……………………

A P

B C Q R

Q.4 In the given figure AD = BC and ∠BAD = ∠ABC, then prove that ∠ACB ≈ ∠BDA. D C

A B

Q.5 The exterior angle of a triangle is equal to the sum

of two……………….

Q.6 What do we call a triangle if the angle are in the ratio 5:3:7?

Q.7 In the figure below, it is given that ΔABC ≅ ΔBAC.

What criteria is used to prove that the triangles are congruent? D C

A B

Q8 In the given figure, if AB = DC, ∠ABD = ∠CBD,

which congruent rule would you apply to prove

ΔABD ≅ ΔCBD?

D C

A B

Q.9 Among the following which is not a criteria for

congruence of two triangles?

Q.10 In ΔAOC and ΔXZY, ∠A = ∠X, AO = XZ, AC = XY, then by which congruence rule ΔAOC ≅ ΔXZY ? A X

O C Y Z

Q.11 ΔABC ≅ ΔPQR, AB = PQ. Which statement has been followed in this?


Q.1 In the figure ΔABC and ΔDBC are two isosceles

triangles on the same base BC. Prove that ∠ABD =

∠ACD. A

B C

D Q.2 Prove that the sum of the four angles of a

quadrilateral ABCD is 3600, using properties of

triangles. A B

6 1

5 2

3 7

4 8


C D

Q.3 In the figure below, ABC is a triangle in which AB = AC. X and Y are points on AB and AC such that AX =

AY. Prove that ΔABY ≅ ΔACX

A

X Y

B C

Q.4 In the figure below. ABCD is a square and P is the

mid-point of AD and CP are joined. Prove that ∠PCB =

∠PBC.

A B

P

D C

Q.5 In the figure below, O is the mid-point of AB and CD, prove that AC = BD.

A

D

O

C

B

Q.6 In the figure below the diagonal AC of quadrilateral

ABCD bisects ∠BAD and ∠BCD. Prove that BC = CD.

D

C

A

B

Q.7 In the figure, AO = OB and OD = OC. Show that:

(i) ΔAOD ≅ ΔBOC, (ii) AD || BC.

B

C

O

D

A

Q.8 In the given figure, D is the mid-point of base BC. DE and DF are perpendiculars of AB and AC

respectively such that DE = DF. Prove that ∠B = ∠C. A

E F

B C

Q.9 In figure ∠B = ∠E, BD = CE and ∠1 = ∠2. Show ΔABC ≅ ΔAED. A

B D C E

Q.10 In figure, AX = BY and AX || BY, prove that

ΔAPX ≅ ΔBPY

X B

P

A Y



Q.1 In figure, PQRS is a square and SRT is an equilateral triangle. Prove that:

(i) PT = QT T

(ii) ∠TQR = 150

S R

P Q

Q.2 In the given figure AD = BC and BD = AC. Prove

that, ∠ADB = ∠BCA and ∠DAB = ∠CBA.

A B

D C

Q.3 In the given figure, if ∠ACD = ∠ACE and AB = BC, then prove that AE = CD.

B D A

x

E y

C

Q.4 In the given figure, AD is the bisector of ∠BAC and

∠CPD = ∠BPD. Prove that ΔCAP ≅ ΔBAP and CP = BP.

C

A P D

B

Q.5 ABCD is a quadrilateral in which AD = BC and

∠DAB = ∠CBA. Prove that BD = AC.

D

A

B

C

Q.6 In the given figure, prove that: CD + DA + AB + BC > 2AC.

C B

D A

Q.7 In the given figure, BA⊥CA, RP⊥QP, AB = PQ and

BR = CQ. Prove that AC = PR. A

R B Q C

P

Q.8 In the given figure, ABCD is a square and M is the

mid-point of AB. PQ⊥CM meets AD at P and CB produced at Q. Prove that PA = BQ.

D C

P


A M B

Q

Q.9 In figure, AB = EF, BC = ED, AB⊥BD, FE⊥EC.

Prove that ΔABD ≅ ΔFEC.

A F

B E

C D


Q.1 If D is the mid-point of the hypotenuse AC of a right

triangle ABC, prove that BD = 1/2AC.

A

E D

B C

Q.2 In the given figure, if AC = BC, ∠DCA = ∠ECB and

∠DCB = ∠EAC, then prove that BD = AE.

D E

A C B

Q.3 In figure ΔABC and ΔABD are such that AD = BC,

∠1 = ∠2 and ∠3 = ∠4. Prove that BD = AC.

D C

1 2

3 4

Q.4 Prove that two triangles are congruent if any two

angles and the included side of one triangle in equal to

any two angles and the included side of the other

triangle.

Q.5 If two parallel lines are intersected by a transversal,

then prove that bisectors of the interior angles from a rectangle.

Q.8 In the given figure, PQRS is a quadrilateral and T

and U are points on PS and RS respectively, such that

PQ = RQ, ∠PQT = ∠RQU and ∠TQS = ∠UQS. Prove that: QT = QU P T

S

Q

U R

Q.9 In the given figure, AM⊥BC and AN is the bisector

of ∠A. If ∠ABC = 700 and ∠ACB = 200, find the value

of ∠MAN. [250] A

700 200

B M N C

Q.10 In figure ABCD is a quadrilateral in which AD =

BC and ∠DAB = ∠CBA. Prove that:

(i) ΔABD ≅ ΔBAC

(ii) BD = AC

(iii) ∠ABD = ∠BAC

A

D

B


C

Q.11ΔABC and ΔDBC are two isosceles triangles on the

same base BC and vertices A and D are on the same side

of BC. If AD is extended to intersect BC at P, show that:

(i) ΔABD ≅ ΔACD

(ii) ΔABP ≅ ΔACP

(iii) AP bisects ∠A as well as ∠D.

Q.12 In the figure BL⊥AC, MC⊥LN, AL = CN and BL

= CM. Prove that ΔABC ≅ ΔNML.

M

A L

C N

B

TOPIC-2

Some Properties of Triangle

SHORT ANSWER TYPE QUESTIONS

Q.1 Triangle ABC is an isosceles triangles such that AB

= AC. Side BA is produced to D, such that AD = AB.

Show that ∠BCD is a right angle.

Q.2 Prove that each angle of an equilateral triangle is

600.

Q.5 PQR is a triangle in which PQ = PQ. S is any point

on the side PQ. Through S, a line is drawn parallel to QR intersecting PR at T. Prove that PS = PT.

LONG SHORT ANSWER TYPE QUESTIONS Q.1 In figure ABCD is a square and EF is parallel to

diagonal BD and EM = FM. Prove that:

(i) DF = BE

(ii) (ii) AM bisects ∠BAD. [2012]

(iii)

D A

F

M

C E B

Q.2 In figure, it is given that RT = TS. ∠1 = 2∠2 and ∠4 = 2∠3. Prove that T

A B

1 4

2 3

R S

TOPIC-3

Inequalities of a Triangle

SHORT ANSWER TYPE QUESTIONS

Q.1 In ΔPQR, if S is any point on the side QR. Show that PQ + QR + RP > 2PS.

Q.2 In the given figure, PQR is a triangle and S is any point in its interior. Show that SQ + SR < PQ + PR.

P

S

Q R

Q.3 In ΔABC, if AB is the greatest side, then prove that

∠C > 600

LONG SHORT ANSWER TYPE QUESTIONS Q.1 Prove that the sum of two sides of a triangle is

greater than twice the median with respect to the third

side.

Q.2 ABCD is a quadrilateral in which AB and CD are

smallest and longest sides respectively. Prove that ∠A > ∠C and ∠B > ∠D.


Q3 Show that the difference of any two sides of a

triangle is less than the third side.


Co- ordinate Geometry TOPIC-1

Cartesian System


Q.1 Locate and write the co-ordinates of a point:

(A) Above x-axis lying in y-axis at a distance of 5 units from origin.

(B) Below x-axis lying in y-axis at a distance of 3 units

from origin. (C ) lying on x-axis to the right of origin at a distance of

5 units.

(D) Lying on x-axis to the left of origin at a distance of 2

units.

Q.2 In which quadrant do the following points lie?

(A) (-6, 2) (B) (-5, -4) (C ) (3, -2) (D) (9,6)

Q.3 Find the co-ordinates of the point which lies on y-

axis at a distance of 4 units in negative direction of y-axis.

(A) (-4, 0) (B) (4, 0)

(C ) (0, -4) (D) (0,4)

Q.4 In which quadrants do the following points lie?

(A) (2. -1) (B) (-1, 7) (C ) (-2, -3) (D) (4,5)

Q.5 In which quadrants will the points lie, if:

(A) The ordinate is 2 and the abscissa is -3? (B) the abscissa is -4 and the ordinate is -2?

(C ) The ordinate is 3 and the abscissa is 4?

(D) The ordinate is 3 and the abscissa is -2?


Q.1 Plot the point A (1,3), B(1,-1), C(-1,-1). What must

be the coordinates of the point D, if ABCD is a rectangle?

Q.2 In which quadrilateral or on which axis does each of

the following points lie. (-5, 3), (4, -3), (5, 0), (6, 6), (-5, -4)?

Q.3 Find the co-ordinates of a point: (A) Which lies on x and y-axis, both.

(B) Whose abscissa is 5 and ordinate is 6.

(C ) Whose ordinate is 6 and which lies on y-axis.

(D) Whose ordinate is -4 and abscissa is -7. (E) Whose abscissa is 3 and which lies on x-axis.

(F) Whose abscissa is 4 and ordinate is 4.

Q.4 Plot the following points and write the name of the figure, thus obtained. A (3,0), B(5,0), C(5,3) and D(3,3).

Q.5 Write the coordinates of the vertices of a rectangle whose length and breadth are 6 and 3 units respectively,

one vertex at the origin, the longer side lies on the x-axis

and one of the vertices lies in the III quadrant.


Q.1 If the coordinates of a point M are (-2,9) which can

also be expressed as (1 + x, y2) and y > 0, then find in

which quadrant do the following points lie:

P(y,x), Q(2,x), R(x2,y-1), S(2x,-3y). [2012]

Q.2 Observed the points plotted in the figure and find the

following: [2012] (i) The co-ordinates of E

(ii) The point with the co-ordinates (-4, -1).

(iii)The abscissa of A – abscissa of B

(iv)The ordinates of C + ordinate of F.

Q.3 Plot (-3.0), (5,0) and (0,4) on Cartesian plane. Name

the figure formed by joining these points and find its area. [16 sq. units, 2012]

Q.4 In which quadrant or on which axis do each of the points (-2,4), (3,-1), (-1,0), (-3,-5), and (1,2) lie?

Verify your answer by locating them on the Cartesian

plane.

Q.5 Plot the following points in the Cartesian plane:

A(5,0), B(3,2), C(0,-5), D(-6,1) E(-4,-4), F(2,-3).

Q.6 Plot the points A(-2,3), B(-2,0), C(2,0) and D(2,6)

on the graph paper. Join them consecutively and find the

length of BC and AB. Also find the area of ΔABC.

Q.7 Write the quadrant in which each of the following

points lie: (i) (-3,-5) (ii) (2,-5) (iii) (-3,5)

TOPIC-2

Plotting a Point in a Plane


Q.1 Plot the points (5,-3), (-6,0) (-2,-3) and (-4,3) on the

graph.

Q.2 Plot two points A (23,3) and B(3,23) on the graph

paper. Draw line segment AB and find its mid-point.

[13,13]


Q.3 Plot the points A(3,10), B(-3,5) and C(-1,-6) on the

graph paper. Join them in pairs and identify the figure so formed. [2014]

Q.4 In which quadrant or on which axis do the points

(-2,-4), (2,4) (0,-2) and (4,6) lie? Verify your answer by locating them on the Cartesian plane. [2014]

Q.5 Plot the following points on the graph sheet:

A(-3,-4), B(-2,0), C(-1,4), D(1,0). These points lie in which quadrant axes? [2012]

Q.6 Plot the following points on the graph sheet and join them in order: [2012]

B(-5,3) E(-3,-2) S(4,-2) T(1,3)

Also mention the quadrant in which the points lie.

Q.7 Write the co-ordinates of A, B, C and D from the

following figure: Y

C 4

3 B 2

1

X A X

-2-1 1 1 2 3 4 5 2

3

D

Y Q.8 In the figure given below, ABCD is a rectangle with

length 6 cm and breadth 3cm. O is the mid-point of AB.

Find the co-ordinates of A, B, C, and D. [2011]

Y

D C

X’ X’’ A O B

Y

Q.9 A point lies on x-axis at a distance of 9 units from y-axis. What are its co-ordinates? What will be the co-

ordinates of a point if it lies on y-axis at a distance of 9

units from x-axis? [2011]

Q.10 From the given figure, write the following:

(A) The co-ordinates of P.

(B) The abscissa of the point Q

(C ) The ordinates of the point R. (D) The Points whose abscissa is 0.

Y’’

4

3

M 2 Q 1

R S

X’-6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 X’’ 2

P 3

Y’

Q.11 Plot the point A(0,3), B(5,3), C(4,0) and D(-1,0) on the graph paper. Identify the figure ABCD and find

whether the point (2,2) lies inside the figure or not?


Q.1 Plot the points A(4,0) and B(0,4). Join AB to the origin O. Find the area of ΔAOB. [8sq. unit]

Q.2 Plot the point A(-3,-3), B(3,-3), C(3,3) and D(-3,3) in the Cartesian plane. Also find the length of line

segment AB. [22011]

Q.3 Plot the points given in the table below in the Cartesian plane: [2011]

X -1 3 0 -8 5 -3

Y 7 -4 7 0 -2 -3

Q.4 (i) Plot the point A(0,4), B(-3,0), C(0,-4) and D(3,0)

(ii) Name the figure obtained by joining the point A,B,C,D.

(ii) Also name the quadrant in which sides AB and AD

lie.

Q.5 In the figure PQR is an equilateral triangle in the co-

ordinates of Q and R as (0,4) and (0,-4). Find the co-

ordinates of the vertex P. [√48 𝑜𝑟 √34

]

Q.6 (i) Plot the point M(4,3) N(4,0) O(0,0) P(0,3).

(ii) Name the figure obtained by joining the point MNOP.

(iii ) Find the perimeter of the figure. [2011]


Q.7 In figure ΔABC and ΔADC are equilateral triangles. Find the co-ordinates of point C and D.

Y [c are (0,a√3, D are (0, - a√3,)]

C

O

(-a,0) O (a,0)

X’ X

A B

D

Y’

Q.8 In the given figure, PQR is an equilateral triangle

with co-ordinates of Q and R as (-2,0) and (2,0) respectively. Find the co-ordinates of the vertex P.

Y [(0, 2√3)]

P

X’ O R X Q -2 -1 1 2

(-2,0) 1 (2,0)

2 3

Y’


Q.1 (i) Plot the points M(5,-3) and N(-3,-3).

(ii) What is the length of MN?

(iii ) Find the co-ordinate of point A,B and C lying on

MN, such that:

MA = AB = BC = CN.

Q.2 See figure and write the following:

(i) The co-ordinate of B.

(ii) The point identified by the coordinates (-3,-2).

(iii) The abscissa of the point D. (iv) The ordinate of the point C.

Y’’

6

5

B 4 3

2 D

1

X’-6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 X’’

1 C E 2

3 A

Q.3 Find the co-ordinates of the vertices of a rectangle placed in III quadrant in the Cartesian plane with length

‘p’ units on x-axis and breadth ‘q’ units on y-axis.

Q.4 Three vertices of a rectangle are (3,2), (-4,2) and (-

4,5). Plot these points and find the coordinates of the

fourth vertex.

Q.5 In the given figure ΔABC and ΔADC are equilateral

triangles on common base AC, each side of triangles

being 2a units. Vertices A and C lies on x-axis, vertices B and D lies on y-axis. O is the mid-point of AC and

BD. Find the co-ordinates of the point B.

Y

B

O

X’ X

C A

D Y’


TOPIC-3

Graph and Linear Equation


Q.1 Draw a graph of line y = x.

Q.2 Draw a graph of line y = 2x + 1.


Q.1 The following table gives the number of pairs of

shoes and their corresponding price.

No. of pairs of

shoes

1 2 3 4 5 6

Corresponding

prices

5 10 15 20 25 30

Plot these as ordered pairs and join them. What type of

graph do you get? [2011]

Q.2 Draw the graph of the equation = y 3x – 2


Q.1 Plot the point A(1,-1) and B(4,5).

(i) Draw a line segment and write the co-ordinates of a

point on this line segment between the points A and B.

(ii) Extend this line segment and write the co-ordinates

of a point on this line which lies outside the line segment

AB.


AREAS TOPIC-1

Area of Triangles


Q.1 Find the area of an equilateral triangle whose

perimeter is 60 cm. [100√3cm2, 2012]

Q.2 If the area of an equilateral triangle is 81√3 cm2 Find the perimeters. [54 cm. 2012]


Q.1 For an isosceles right triangle having each of equal

sides a, find the perimeter.

Q.2 The base of hypotenuse of a right triangle are

respectively 8 cm and 10 cm long. Find its area. [24 cm2]

LONG ANSWER TYPE QUESTIONS Q.1 From a point in the interior of an equilateral triangle,

perpendiculars are drawn on the three sides. The lengths

of the perpendiculars are 14 cm, 10 cm, and 6 cm. Find

the area of the triangle.

TOPIC-2

Heron’s Formula

SHORT ANSWER TYPE QUESTIONS-I Q.1 The sides of a triangle are 12 cm, 16 cm and 20 cm.

find its area. [96 cm2, 2014]

Q.2 The sides of a triangle are 70 cm, 80 cm and 90 cm.

find its area. [2676 cm2, 2014]

Q.3 The perimeter of a Δ is 120 cm and its sides are in

ratio 5:12:13. Find the area of the triangle. [480 cm2]

Q.4 Find the area of an isosceles triangle whose equal

sides are of length 12 cm each and third side is 12 cm.

[18√21cm2, 2012]

Q.5 Using Heron’s formula, find the area of a triangle

whose sides measure 20 cm, 30 cm and 40 cm. [290.48 cm2, 2012]


Q.1 An umbrella is made by stitching ten triangular

pieces of cloth, each measuring 60 cm, 60 cm and 20

cm. Find the area of the cloth required for the umbrella.

[100√35cm2, 2014]

Q.2 Black and white colored triangular sheets are used to make a toy as shown in figure. Find total area of black

and white colors sheets used for making the toy.

[16√2cm2, 2012]

4 cm white

6 cm

6 cm 6 cm

4 cm Black 4 cm Black

6 cm 6 cm

4 cm White

Q.3 The sides of a triangular field are 51 m, 37 m and 20

m. Find the number of rose beds that can be prepared in

the field if each rose bed occupies a space of 6 sq. m.

[51, 2012] Q.4 Find the area of an isosceles triangle whose one side

is 10 cm greater than its equal side and its perimeter is

100 cm (Take √5 = 2.23) [446 cm2, 2012]

Q.5 The sides of triangle are 120m, 170m ad 250m. Find

its area and height of the triangle if base is 250m.

[72m. 2012] Q.6 The base of an isosceles triangle is 12 cm and its

perimeter is 32 cm. Find its area. [48 cm2, 2012]

Q.7 The sides of a triangle are x, x+1, 2x-1 and its area

is x√10. What is the value of x? [6, 2012]

Q.8 Find the percentage increase in the area of a triangle,

if its each side is doubled. [300%, 2012]



Q.1 The sides of a triangle are in the ratio of 13:14:15

and its perimeter is 84 cm. Find the area of the triangle.

[336 cm2]

Q.2 Two identical circles with same inside design as

shown in the given figure are to be made at the entrance. The identical triangular leaves are to be painted green.

Find the total area to be painted red. [1512cm2]

41cm 15cm

28 cm

Q.3 A triangular park ABC has sides 120 m, 80m, and

50m (see the fig.) A gardener Dhania has to put a fence all around it and also plant grass inside. How much area

does she need to plant? Find the cost of fencing it with

barded wire at the rate of ₹20 per meter leaving a space 3m wide for a gate on one side. [₹4940]

A

50m 80m

3m

B 120m C

TOPIC-3 Application of Heron’s Formula in

finding Area of Quadrilaterals


Q.1 Find the area of a rhombus whose perimeter is 200m

and one of the diagonal is 80m. [2400m2]

Q.2 In the given figure, ABCD is a rectangle, where AB

= 8cm, BC = 6cm and the diagonals bisects each other at

O. Find the area of shaded region by Heron’s formula. [12cm2]

D C

O

6 cm

A 8 cm B

Q.3 Compute the area of the trapezium shown in the

figure. [150 cm2]

D C

17 cm

A 6cm O 8 cm B

Q.4 Find the area of a rhombus whose sides is 20 cm and

one of its diagonal is 24 cm.

SHORT ANSWER TYPE QUESTIONS-II Q.1 Find the area of a quadrilateral field ABCD in which

AB = 50m, BC =- 82m, DA = 50m and ∠CBD = 900

[1920 m2]

Q.2 A kite is in the shape of a square with side 16 cm

and an isosceles Triangle of base 4 cm and equal side of 6 cm each. It is made up of two colors as shown in the

figure. Find the area of paper of each color used in it.

[133.64 cm2]


Q.3 The perimeter of a rhombus is 52 cm. One of the

diagonals is 24 cm. Find the area of the rhombus. [120cm2]

Q.4 The adjacent sides of a parallelogram are 34 cm, 20

cm and a diagonal is 43 cm. Find the area of the

parallelogram. [672 cm2]

Q.5 A triangle and a parallelogram have the same base

and the same area. If the sides of the triangle are 26 cm,

28 cm, and 30 cm and the parallelogram stands on the base 28 cm. Find the height of the parallelogram.

[12 cm]

Q.6 A rhombus field has green grass for 20cows to

graze. If each side of the rhombus is 52m and longer diagonal is 96m, how much area of the grass field will

each cow be getting? [96m2]

Q.7 The shape of cross-selection of a canal is a

trapezium. If the canal is 10 m wide at the top and 6m

wide at the bottom and the area of the cross-section is 72m2, find its depth. [9m]


Q.1 A field is in the shape of a trapezium whose parallel

sides are 35m and 10m. The non-parallel sides are 14m and 13m. Find the area of the field. [26m]

REAL NUMBERS SHORT ANSWER TYPE...

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