The ages of two friends Ani and Biju differ by 3 years. Ani’s father is twice as old as Ani and Biju is twice as old as his sister Cathy. The ages of Cathy and Ani’s father differ by 30 years. Find the ages of Ani and Biju

 10th mathematics chapter 3 pair of linear equations in two variables optional exercise 3.7 for ncert cbse 

 The ages of two friends Ani and Biju differ by 3 years. Ani’s father  is twice as old as Ani and Biju is twice as old as his sister Cathy. The ages of Cathy and Ani’s father differ by 30 years. Find the ages of Ani and Biju

let x = age of Ani

y = age of Biju

So age of Ani's father  = 2x

 note that Biju is elder to Cathy.

Age of Cathy = (y/2) 

First assume x > y

x - y =3   -----------------(1)

[ 2x  -(y/2) =30 ] *2  

4x - y = 60-----------------(2)

solve (1) and (2)

 x  - y = 3

4x - y  =60

----------------subtracting

-3x      = (-57)

x = (-57) /(-3)

x=19

substitute in  x  - y = 3 

19 - y =3

y = 19-3

y = 16

Alternately

assume y > x

so that (1) should be changed to  y - x = 3

or (-x) + y =3 -------------(3)

solving (3)  and (2)

(-x) + y =3 

4x   - y  =60

------------------------adding

3x         = 63 

x = 63/3

x=21

substitute in (-x) + y =3 to get (-21) + y = 3

y = 3 + 21 =24

Age of Ani = 19 years and Age of Biju = 16

or

Age of Ani =21  and age of Biju =24

 

2. One says, “Give me a hundred, friend! I shall then become twice as rich as you”. The other replies, “If you give me ten, I shall be six times as rich as you”. Tell me what is the amount of their (respective) capital?

x = original amount with the first person

y = original amount with the second person

When the second person gives the first person a 100

the first person now has (x+100) while the second person now has only (y-100)

with this in mind,

x+100 = 2(y-100)

x+100 = 2y-200

x - 2y = -200 - 100

x - 2y = (-300) -----------------(1)

when the first person gives 10 to the second person

the second person will have (y+10) while the first person will be left with (x-10)

so that

(y+10) = 6(x-10)

or

y+10 = 6x-60

-6x +y =(-70)  ------------------(2)

solving (1) and (2)

x - 2y = (-300)

[-6x +y =(-70)]  *2

 

    x - 2y  = (-300)

-12x +2y = (-140)

------------------------------ adding

-11x         =(-440)

x = (-440)/(-11)

x = 40

substitute in x - 2y  = (-300)

40 -2y = (-300) 

2y = 40 +300

2y = 340

y =340/2

y=170

 

 

 

=================================================

ncert cbse 10th mathematics chapter 3 optional exercise 3.7 

 The ages of two friends Ani and Biju differ by 3 years. Ani’s father  is twice as old as Ani and Biju is twice as old as his sister Cathy. The ages of Cathy and Ani’s father differ by 30 years. Find the ages of Ani and Biju

solution

 

2. One says, “Give me a hundred, friend! I shall then become twice as rich as you”. The other replies, “If you give me ten, I shall be six times as rich as you”. Tell me what is the amount of their (respective) capital? 

solution

 

 

  ncert cbse 10th  mathematics chapter 2 optional exercise
 If the zeroes of the polynomial (x^3) – 3(x^2) + x + 1 are a – b, a, a + b, find a and b.
solution
 
2. Find a cubic polynomial with the sum, sum of the product of its zeroes taken two at a time, and the product of its zeroes as 2, –7, –14 respectively
solution



4. If two zeroes of the polynomial (x^4) – 6(x^3) – 26(x^2) + 138x – 35 are 
 [2 ±sqrt(3) ] , find other zeroes

 solution
 
5. If the polynomial (x^4) – 6(x^3) + 16(x^2) – 25x + 10 is divided by another polynomial (x^2) – 2x + k, the remainder comes out to be x + a, find k and a
solution  
 
exercise 2.3
 

3. obtain all other zeroes of 3(x^4)+6(x^3)-2(x^2)-10x-5 if two of its zeroes are sqrt(5/3) and [sqrt(5/3)]
solution
 
4. On dividing (x^3) – 3(x^2) + x + 2 by a polynomial g(x), the quotient and remainder were x – 2
and –2x + 4, respectively. Find g(x).   
solution
 
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obtain all other zeroes of 3(x^4)+6(x^3)-2(x^2)-10x-5 if two of its zeroes are sqrt(5/3) and [sqrt(5/3)]

ncert cbse 10th chapter 2  exercise 2.3

3. obtain all other zeroes of 3(x^4)+6(x^3)-2(x^2)-10x-5 if two of its zeroes are sqrt(5/3) and [sqrt(5/3)]

factors are

[x-sqrt(5/3)][x-sqrt(5/3)] = [(x^2) - (5/3)]=(1/3) [3(x^2) -5]

now divide the given expression 3(x^4)+6(x^3)-2(x^2)-10x-5 with [3(x^2) -5]

using long division

                           (x^2) +2x +1                                

                       ---------------------------------------------------------------------------

3(x^2)+0x -5 |  3(x^4)+6(x^3)- 2(x^2) - 10x  -5 

                      |  3(x^4)+0(x^3) -5(x^2)

                      -------------------------------------------------------------------

                     |              6(x^3) +3(x^2)- 10x 

                     |              6(x^3) +0(x^2) - 10x

                      -----------------------------------------------------------------

                    |                            3(x^2) +0x -5

                    |                             3(x^2) +0x -5

                    -------------------------------------------------------------

                   |                                        0   

 

 

 

 

 

 

 

 

we get the quotient as  (x^2) +2x +1      

on factoring we get (x+1)(x+1) whose zeroes are (-1) and (-1)

 

4. On dividing (x^3) – 3(x^2) + x + 2 by a polynomial g(x), the quotient and remainder were x – 2
and –2x + 4, respectively. Find g(x). 

 

let p(x)  = (x^3) – 3(x^2) + x + 2

quotient =q(x)  = (x-2)

remainder r(x) = (-2x+4)

p(x) = g(x)*q(x) +r(x)

g(x) = [p(x) - r(x)] / [q(x)]

g(x) = [{(x^3) – 3(x^2) + x + 2} - {(-2x+4)}] / [(x-2)]

g(x) = [( x^3) – 3(x^2) +3x – 2 ] / [x-2]

using long division

g(x) =  (x^2) -1x+1

 

=================================================

  ncert cbse 10th  mathematics chapter 2 optional exercise
 If the zeroes of the polynomial (x^3) – 3(x^2) + x + 1 are a – b, a, a + b, find a and b.
solution
 
2. Find a cubic polynomial with the sum, sum of the product of its zeroes taken two at a time, and the product of its zeroes as 2, –7, –14 respectively
solution



4. If two zeroes of the polynomial (x^4) – 6(x^3) – 26(x^2) + 138x – 35 are 
 [2 ±sqrt(3) ] , find other zeroes

 solution
 
5. If the polynomial (x^4) – 6(x^3) + 16(x^2) – 25x + 10 is divided by another polynomial (x^2) – 2x + k, the remainder comes out to be x + a, find k and a
solution  
 
exercise 2.3
 

3. obtain all other zeroes of 3(x^4)+6(x^3)-2(x^2)-10x-5 if two of its zeroes are sqrt(5/3) and [sqrt(5/3)]
solution
 
4. On dividing (x^3) – 3(x^2) + x + 2 by a polynomial g(x), the quotient and remainder were x – 2
and –2x + 4, respectively. Find g(x).   
solution
 
disclaimer:
There is no guarantee about the data/information on this site. You use the data/information at your own risk. You use the advertisements displayed on this page at your own risk.We are not responsible for the content of external internet sites. Some of the links may not work. Your internet usage may be tracked by the advertising networks and other organizations using tracking cookie and / or using other means  

 

Find a cubic polynomial with the sum, sum of the product of its zeroes taken two at a time, and the product of its zeroes as 2, –7, –14 respectively.

 cbse ncert 10th mathematics chapter 2 optional exercise

2. Find a cubic polynomial with the sum, sum of the product of its zeroes taken two at a time, and the product of its zeroes as 2, –7, –14 respectively.

S1 = 2

S2 = (-7)

S3 = (-14)

required expression is of the form

(x^3) - S1(x^2) + S2 x - S3

or

(x^3) - 2(x^2) + (-7) x - (-14)

 (x^3) - 2(x^2) -7x +14

 

5. If the polynomial (x^4) – 6(x^3) + 16(x^2) – 25x + 10 is divided by another polynomial (x^2) – 2x + k, the remainder comes out to be x + a, find k and a.

using long division, divide  {(x^4) – 6(x^3) + 16(x^2) – 25x + 10} by 

{(x^2) – 2x + k}  and then use the remainder obtained

                                      (x^2) -4 x +(8-k)

                               -----------------------------------------------------------------------

 (x^2) – 2x + k,      |     (x^4) – 6(x^3) + 16(x^2) – 25x + 10

                               |      (x^4) – 2(x^3) +  k (x^2)  

                               | -------------------------------------------------------------------------     

                               |                -4(x^3)+(16-k)(x^2) – 25x 

                               |                -4(x^3) +  8(x^2)     -4kx

                               |---------------------------------------------------------------------- 

                               |                             +(8-k)(x^2)+(-25+4k)x +10

                               |                             +(8-k)(x^2) +(-16+2k)x +[8k-(k^2)] 

                               |-----------------------------------------------------------------------------------

                               |                                                  (-9+2k)x +[(k^2)-8k+10]

 

 

 

using the given remainder (x+a)

 

(-9+2k)x +[(k^2)-8k+10] = 1x +a 

 

using the coefficients of x

 (-9+2k) = 1

2k = 1 +9

2k =10

k = 10/2

k=5

using constant term

a = [(k^2)-8k+10] =(5^2) -8*5 +10 =25 -40 + 10 =(-5)

a = (-5)

=================================================

  ncert cbse 10th  mathematics chapter 2 optional exercise

 If the zeroes of the polynomial (x^3) – 3(x^2) + x + 1 are a – b, a, a + b, find a and b.

solution

 

2. Find a cubic polynomial with the sum, sum of the product of its zeroes taken two at a time, and the product of its zeroes as 2, –7, –14 respectively

solution

4. If two zeroes of the polynomial (x^4) – 6(x^3) – 26(x^2) + 138x – 35 are 

 [2 ±sqrt(3) ] , find other zeroes

 solution

 

5. If the polynomial (x^4) – 6(x^3) + 16(x^2) – 25x + 10 is divided by another polynomial (x^2) – 2x + k, the remainder comes out to be x + a, find k and a

solution  

 

 

 chapter 1 miscellaneous sets ncert cbse

 16. In a survey it was found that 21 people liked product A, 26 liked product B and 29 liked product C. If 14 people liked products A and B, 12 people liked products C and A, 14 people liked products B and C and 8 liked all the three products. Find how many liked product C only.

solution

15. In a survey of 60 people, it was found that 25 people read newspaper H, 26 read newspaper T,  26 read newspaper I, 9 read both H and I, 11 read both H and T, 8 read both T and I, 3 read all three newspapers. Find:
(i) the number of people who read at least one of the newspapers.
(ii) the number of people who read exactly one newspaper

solution

 

14. In a group of students, 100 students know Hindi, 50 know English and 25 know both. Each of the students knows either Hindi or English. How many students are there in the group? 

solution

 

 

13. In a survey of 600 students in a school, 150 students were found to be taking tea and 225 taking coffee, 100 were taking both tea and coffee. Find how many
students were taking neither tea nor coffee?

solution

 
ncert cbse 11th mathematics chapter 1 sets exercise 1.6

8.In a committee, 50 people speak French, 20 speak Spanish and 10 speak both
Spanish and French. How many speak at least one of these two languages? 

solution  

7. In a group of 65 people, 40 like cricket, 10 like both cricket and tennis. How many like tennis only and not cricket? How many like tennis?

solution

In a group of 70 people, 37 like coffee, 52 like tea and each person likes at least one of the two drinks. How many people like both coffee and tea?

solution 

 

5. If X and Y are two sets such that X has 40 elements, X ∪ Y has 60 elements and
X ∩ Y has 10 elements, how many elements does Y have? 

solution

 

11th cbse ncert chapter 2 relations and functions miscellaneous exercise

 

12. Let A = {9,10,11,12,13} and let f : A → N be defined by f (n) = the highest prime factor of n. Find the range of f.

solution

 

11. Let f be the subset of Z × Z defined by f = {(ab, a + b) : a, b ∈ Z}. Is f a
function from Z to  Z? Justify your answer.

solution

 

10. Let A ={1,2,3,4}, B = {1,5,9,11,15,16} and f = {(1,5), (2,9), (3,1), (4,5), (2,11)}
Are the following true?
(i) f is a relation from A to B
(ii) f is a function from A to B.
Justify your answer in each case.

solution 

 

9. Let R be a relation from N to N defined by

 R = {(a, b) : a, b ∈ N and a = (b^2) }. 

Are the following true?
(i) (a,a) ∈ R, for all a ∈ N
(ii) (a,b) ∈ R, implies (b,a) ∈ R
(iii) (a,b) ∈ R, (b,c) ∈ R implies (a,c) ∈ R.

 solution 

 

 8. Let f = {(1,1), (2,3), (0,–1), (–1, –3)} be a function from Z to Z defined by
f(x) = ax + b, for some integers a, b. Determine a, b.

 solution  

---------------------------------------------------------------------------------------

4.The function ‘t’ which maps temperature in degree Celsius into temperature in degree Fahrenheit is defined by t(C) = (9C/5) +32 .Find (i) t(0) (ii) t(28)
(iii) t(-10) (iv)the value of C, when t(C) = 212

solution

 

A function f is defined by f(x) = 2x –5. Write down the values of

(i) f (0), (ii) f (7), (iii) f (–3).

solution

 

exercise 2.2

Let A = {1, 2, 3,...,14}. Define a relation R from A to A by 

R = {(x, y) : 3x – y = 0, where x, y ∈ A}. Write down its domain, co domain and  

range.

  solution

2. Define a relation R on the set N of natural numbers by R = {(x, y) : y = x + 5,
x is a natural number less than 4; x, y ∈N}. Depict this relationship using roster
form. Write down the domain and the range.  

  solution

disclaimer:
There is no guarantee about the data/information on this site. You use the data/information at your own risk. You use the advertisements displayed on this page at your own risk.We are not responsible for the content of external internet sites. Some of the links may not work. Your internet usage may be tracked by the advertising networks and other organizations using tracking cookie and / or using other means

If the zeroes of the polynomial x 3 – 3x 2 + x + 1 are a – b, a, a + b, find a and b.

 ncert cbse 10th chapter 2  optional exercise

 If the zeroes of the polynomial (x^3) – 3(x^2) + x + 1 are a – b, a, a + b, find a and b.

 

given expression is (x^3) – 3(x^2) + x + 1

sum of the roots  = -(-3)/1 = 3

so

(a-b) + (a) +(a+b) = 3

cancelling off b

3a = 3 or

a = 1

product of the roots  = -(1)/1 = -1

so 

(a-b) * (a) * (a+b) = (-1)

use a =1

(1-b)*1*(1+b)=(-1)

[1-(b^2)] = (-1)

(b^2) = 2

b =  +sqrt(2) or [-sqrt(2)]

4. If two zeroes of the polynomial (x^4) – 6(x^3) – 26(x^2) + 138x – 35 are 

 [2 ±sqrt(3) ] , find other zeroes

given two zeroes are  [2 +sqrt(3) ] , [2 - sqrt(3) ] 

sum of the given zeroes =  [2 +sqrt(3) ] + [2 - sqrt(3) ] = 4

product of the given zeroes =[2 +sqrt(3) ] * [2 - sqrt(3) ] 

[(2^2) - {sqrt(3)}^2 ]= 4 -3 =1

corresponding quadratic factor is (x^2) - (sum)(x)  + product

=(x^2) - 4x +1

 

or 

use the  factors {x -[2 +sqrt(3) } {x- [2 -sqrt(3) ] }

={(x-2)-sqrt(3) }{(x-2)+sqrt(3) } = {(x-2)^2 - [sqrt(3)]^2 }

= (x^2) - 4x +4 -3 =(x^2) - 4x +1

 

 

divide the given expression with
[ (x^2) - 4x +1]

 

 

divide [(x^4) – 6(x^3) – 26(x^2) + 138x – 35 ] with  [(x^2) - 4x +1]

to get the remaining factor [(x^2) - 2x -35] which can be factored as

(x-7)(x+5) so that the remaining zeroes are 7 , (-5)

=================================================

  ncert cbse 10th  mathematics chapter 2 optional exercise

 If the zeroes of the polynomial (x^3) – 3(x^2) + x + 1 are a – b, a, a + b, find a and b.

solution

4. If two zeroes of the polynomial (x^4) – 6(x^3) – 26(x^2) + 138x – 35 are 

 [2 ±sqrt(3) ] , find other zeroes

 solution

 chapter 1 miscellaneous sets ncert cbse

 16. In a survey it was found that 21 people liked product A, 26 liked product B and 29 liked product C. If 14 people liked products A and B, 12 people liked products C and A, 14 people liked products B and C and 8 liked all the three products. Find how many liked product C only.

solution

15. In a survey of 60 people, it was found that 25 people read newspaper H, 26 read newspaper T,  26 read newspaper I, 9 read both H and I, 11 read both H and T, 8 read both T and I, 3 read all three newspapers. Find:
(i) the number of people who read at least one of the newspapers.
(ii) the number of people who read exactly one newspaper

solution

 

14. In a group of students, 100 students know Hindi, 50 know English and 25 know both. Each of the students knows either Hindi or English. How many students are there in the group? 

solution

 

 

13. In a survey of 600 students in a school, 150 students were found to be taking tea and 225 taking coffee, 100 were taking both tea and coffee. Find how many
students were taking neither tea nor coffee?

solution

 
ncert cbse 11th mathematics chapter 1 sets exercise 1.6

8.In a committee, 50 people speak French, 20 speak Spanish and 10 speak both
Spanish and French. How many speak at least one of these two languages? 

solution  

7. In a group of 65 people, 40 like cricket, 10 like both cricket and tennis. How many like tennis only and not cricket? How many like tennis?

solution

In a group of 70 people, 37 like coffee, 52 like tea and each person likes at least one of the two drinks. How many people like both coffee and tea?

solution 

 

5. If X and Y are two sets such that X has 40 elements, X ∪ Y has 60 elements and
X ∩ Y has 10 elements, how many elements does Y have? 

solution

 

11th cbse ncert chapter 2 relations and functions miscellaneous exercise

 

12. Let A = {9,10,11,12,13} and let f : A → N be defined by f (n) = the highest prime factor of n. Find the range of f.

solution

 

11. Let f be the subset of Z × Z defined by f = {(ab, a + b) : a, b ∈ Z}. Is f a
function from Z to  Z? Justify your answer.

solution

 

10. Let A ={1,2,3,4}, B = {1,5,9,11,15,16} and f = {(1,5), (2,9), (3,1), (4,5), (2,11)}
Are the following true?
(i) f is a relation from A to B
(ii) f is a function from A to B.
Justify your answer in each case.

solution 

 

9. Let R be a relation from N to N defined by

 R = {(a, b) : a, b ∈ N and a = (b^2) }. 

Are the following true?
(i) (a,a) ∈ R, for all a ∈ N
(ii) (a,b) ∈ R, implies (b,a) ∈ R
(iii) (a,b) ∈ R, (b,c) ∈ R implies (a,c) ∈ R.

 solution 

 

 8. Let f = {(1,1), (2,3), (0,–1), (–1, –3)} be a function from Z to Z defined by
f(x) = ax + b, for some integers a, b. Determine a, b.

 solution  

---------------------------------------------------------------------------------------

4.The function ‘t’ which maps temperature in degree Celsius into temperature in degree Fahrenheit is defined by t(C) = (9C/5) +32 .Find (i) t(0) (ii) t(28)
(iii) t(-10) (iv)the value of C, when t(C) = 212

solution

 

A function f is defined by f(x) = 2x –5. Write down the values of

(i) f (0), (ii) f (7), (iii) f (–3).

solution

 

exercise 2.2

Let A = {1, 2, 3,...,14}. Define a relation R from A to A by 

R = {(x, y) : 3x – y = 0, where x, y ∈ A}. Write down its domain, co domain and  

range.

  solution

2. Define a relation R on the set N of natural numbers by R = {(x, y) : y = x + 5,
x is a natural number less than 4; x, y ∈N}. Depict this relationship using roster
form. Write down the domain and the range.  

  solution

disclaimer:
There is no guarantee about the data/information on this site. You use the data/information at your own risk. You use the advertisements displayed on this page at your own risk.We are not responsible for the content of external internet sites. Some of the links may not work. Your internet usage may be tracked by the advertising networks and other organizations using tracking cookie and / or using other means

Let A = {1, 2, 3,...,14}. Define a relation R from A to A by R = {(x, y) : 3x – y = 0, where x, y ∈ A}. Write down its domain, codomain and range.

 ncert cbse 11th chapter 2 relations and functions exercise 2.2

 

Let A = {1, 2, 3,...,14}. Define a relation R from A to A by 

R = {(x, y) : 3x – y = 0, where x, y ∈ A}. Write down its domain, co domain and  

range.

note that the relation is A to A so that the images have to be in A

  3x – y = 0

means y = 3x

put x  = 1 , y = 3*1 = 3 so (1,3) is in R

put x = 2 , y = 3*2 = 6 so (2,6) is in R

put x = 3 , y = 3*3 = 9 so (3,9) is in R

put x = 4 , y = 3*4 = 12 so (4,12) is in R

put x = 5 , y = 3*5 = 15 is not in A  

so we stop here

R = { (1,3) ,(2,6),(3,9),(4,12) } 

domain = {1,2,3,4} ------------- values taken by x

range = {3,6,9,12} --------------values taken by y

codomain = A ={1,2,3,...,14}

 

2. Define a relation R on the set N of natural numbers by R = {(x, y) : y = x + 5,
x is a natural number less than 4; x, y ∈N}. Depict this relationship using roster
form. Write down the domain and the range. 

restriction is for x

so x = 1,2,3

put x = 1 y =x+5 = 1+5 = 6 so (1,6) is in R

put x = 2 y =x+5 = 2+5 = 7 so (2,7) is in R

put x = 3 y =x+5 = 3+5 = 8 so (3,8) is in R

 

roster form

R = {(1,6),(2,7), (3,8)} 

domain = {1,2,3}

range = {6,7,8}

 

=================================================

 

 chapter 1 miscellaneous sets ncert cbse

 16. In a survey it was found that 21 people liked product A, 26 liked product B and 29 liked product C. If 14 people liked products A and B, 12 people liked products C and A, 14 people liked products B and C and 8 liked all the three products. Find how many liked product C only.

solution

15. In a survey of 60 people, it was found that 25 people read newspaper H, 26 read newspaper T,  26 read newspaper I, 9 read both H and I, 11 read both H and T, 8 read both T and I, 3 read all three newspapers. Find:
(i) the number of people who read at least one of the newspapers.
(ii) the number of people who read exactly one newspaper

solution

 

14. In a group of students, 100 students know Hindi, 50 know English and 25 know both. Each of the students knows either Hindi or English. How many students are there in the group? 

solution

 

 

13. In a survey of 600 students in a school, 150 students were found to be taking tea and 225 taking coffee, 100 were taking both tea and coffee. Find how many
students were taking neither tea nor coffee?

solution

 
ncert cbse 11th mathematics chapter 1 sets exercise 1.6

8.In a committee, 50 people speak French, 20 speak Spanish and 10 speak both
Spanish and French. How many speak at least one of these two languages? 

solution  

7. In a group of 65 people, 40 like cricket, 10 like both cricket and tennis. How many like tennis only and not cricket? How many like tennis?

solution

In a group of 70 people, 37 like coffee, 52 like tea and each person likes at least one of the two drinks. How many people like both coffee and tea?

solution 

 

5. If X and Y are two sets such that X has 40 elements, X ∪ Y has 60 elements and
X ∩ Y has 10 elements, how many elements does Y have? 

solution

 

11th cbse ncert chapter 2 relations and functions miscellaneous exercise

 

12. Let A = {9,10,11,12,13} and let f : A → N be defined by f (n) = the highest prime factor of n. Find the range of f.

solution

 

11. Let f be the subset of Z × Z defined by f = {(ab, a + b) : a, b ∈ Z}. Is f a
function from Z to  Z? Justify your answer.

solution

 

10. Let A ={1,2,3,4}, B = {1,5,9,11,15,16} and f = {(1,5), (2,9), (3,1), (4,5), (2,11)}
Are the following true?
(i) f is a relation from A to B
(ii) f is a function from A to B.
Justify your answer in each case.

solution 

 

9. Let R be a relation from N to N defined by

 R = {(a, b) : a, b ∈ N and a = (b^2) }. 

Are the following true?
(i) (a,a) ∈ R, for all a ∈ N
(ii) (a,b) ∈ R, implies (b,a) ∈ R
(iii) (a,b) ∈ R, (b,c) ∈ R implies (a,c) ∈ R.

 solution 

 

 8. Let f = {(1,1), (2,3), (0,–1), (–1, –3)} be a function from Z to Z defined by
f(x) = ax + b, for some integers a, b. Determine a, b.

 solution  

---------------------------------------------------------------------------------------

4.The function ‘t’ which maps temperature in degree Celsius into temperature in degree Fahrenheit is defined by t(C) = (9C/5) +32 .Find (i) t(0) (ii) t(28)
(iii) t(-10) (iv)the value of C, when t(C) = 212

solution

 

A function f is defined by f(x) = 2x –5. Write down the values of

(i) f (0), (ii) f (7), (iii) f (–3).

solution

 

exercise 2.2

Let A = {1, 2, 3,...,14}. Define a relation R from A to A by 

R = {(x, y) : 3x – y = 0, where x, y ∈ A}. Write down its domain, co domain and  

range.

  solution

2. Define a relation R on the set N of natural numbers by R = {(x, y) : y = x + 5,
x is a natural number less than 4; x, y ∈N}. Depict this relationship using roster
form. Write down the domain and the range.  

  solution

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There is no guarantee about the data/information on this site. You use the data/information at your own risk. You use the advertisements displayed on this page at your own risk.We are not responsible for the content of external internet sites. Some of the links may not work. Your internet usage may be tracked by the advertising networks and other organizations using tracking cookie and / or using other means

 

The function ‘t’ which maps temperature in degree Celsius into temperature in degree Fahrenheit is defined by

 cbse ncert 11th mathematics exercise 2.3 relations and functions

4.The function ‘t’ which maps temperature in degree Celsius into temperature in degree Fahrenheit is defined by t(C) = (9C/5) +32 .Find (i) t(0) (ii) t(28)
(iii) t(-10) (iv)the value of C, when t(C) = 212

t(C) = (9C/5) +32

 

 (i) 

put   C =0

t(0) = [9*0/5] + 32 = 32

 

(ii)

put C =28

t(28) =[9*28/5] + 32 =50.4 + 32 = 82.4

or  

t(28) =[9*28/5] + 32 = [252 + 160] / 5 = 412 / 5

 

(iii) 

put C = (-10)

t(-10) = [9*(-10)/5] + 32 = (-18) +32 = 14

 

(iv) 

given t(C) = 212

t(C) = (9C/5) +32

212 = (9C/5) +32

212-32 = 9C/5

180 = 9C/5

C =180*5/9

C =100 

A function f is defined by f(x) = 2x –5. Write down the values of

(i) f (0), (ii) f (7), (iii) f (–3).

(i)

put x = 0

f(0) =2(0) - 5 = (-5)

(ii)

put x=7

f(7) = (2*7) -5 =14-5 = 9

(iii)

put x =(-3) 

f(-3) =[2*(-3)] -5 = (-6) -5 = (-11)

=================================================

 

 chapter 1 miscellaneous sets ncert cbse

 16. In a survey it was found that 21 people liked product A, 26 liked product B and 29 liked product C. If 14 people liked products A and B, 12 people liked products C and A, 14 people liked products B and C and 8 liked all the three products. Find how many liked product C only.

solution

15. In a survey of 60 people, it was found that 25 people read newspaper H, 26 read newspaper T,  26 read newspaper I, 9 read both H and I, 11 read both H and T, 8 read both T and I, 3 read all three newspapers. Find:
(i) the number of people who read at least one of the newspapers.
(ii) the number of people who read exactly one newspaper

solution

 

14. In a group of students, 100 students know Hindi, 50 know English and 25 know both. Each of the students knows either Hindi or English. How many students are there in the group? 

solution

 

 

13. In a survey of 600 students in a school, 150 students were found to be taking tea and 225 taking coffee, 100 were taking both tea and coffee. Find how many
students were taking neither tea nor coffee?

solution

 
ncert cbse 11th mathematics chapter 1 sets exercise 1.6

8.In a committee, 50 people speak French, 20 speak Spanish and 10 speak both
Spanish and French. How many speak at least one of these two languages? 

solution  

7. In a group of 65 people, 40 like cricket, 10 like both cricket and tennis. How many like tennis only and not cricket? How many like tennis?

solution

In a group of 70 people, 37 like coffee, 52 like tea and each person likes at least one of the two drinks. How many people like both coffee and tea?

solution 

 

5. If X and Y are two sets such that X has 40 elements, X ∪ Y has 60 elements and
X ∩ Y has 10 elements, how many elements does Y have? 

solution

 

11th cbse ncert chapter 2 relations and functions miscellaneous exercise

 

12. Let A = {9,10,11,12,13} and let f : A → N be defined by f (n) = the highest prime factor of n. Find the range of f.

solution

 

11. Let f be the subset of Z × Z defined by f = {(ab, a + b) : a, b ∈ Z}. Is f a
function from Z to  Z? Justify your answer.

solution

 

10. Let A ={1,2,3,4}, B = {1,5,9,11,15,16} and f = {(1,5), (2,9), (3,1), (4,5), (2,11)}
Are the following true?
(i) f is a relation from A to B
(ii) f is a function from A to B.
Justify your answer in each case.

solution 

 

9. Let R be a relation from N to N defined by

 R = {(a, b) : a, b ∈ N and a = (b^2) }. 

Are the following true?
(i) (a,a) ∈ R, for all a ∈ N
(ii) (a,b) ∈ R, implies (b,a) ∈ R
(iii) (a,b) ∈ R, (b,c) ∈ R implies (a,c) ∈ R.

 solution 

 

 8. Let f = {(1,1), (2,3), (0,–1), (–1, –3)} be a function from Z to Z defined by
f(x) = ax + b, for some integers a, b. Determine a, b.

 solution  

---------------------------------------------------------------------------------------

4.The function ‘t’ which maps temperature in degree Celsius into temperature in degree Fahrenheit is defined by t(C) = (9C/5) +32 .Find (i) t(0) (ii) t(28)
(iii) t(-10) (iv)the value of C, when t(C) = 212

solution

 

A function f is defined by f(x) = 2x –5. Write down the values of

(i) f (0), (ii) f (7), (iii) f (–3).

solution

 

 

disclaimer:
There is no guarantee about the data/information on this site. You use the data/information at your own risk. You use the advertisements displayed on this page at your own risk.We are not responsible for the content of external internet sites. Some of the links may not work. Your internet usage may be tracked by the advertising networks and other organizations using tracking cookie and / or using other means

In a group of 65 people, 40 like cricket, 10 like both cricket and tennis. How many like tennis only and not cricket? How many like tennis?

ncert cbse chapter 1 sets exercise 1.6

7. In a group of 65 people, 40 like cricket, 10 like both cricket and tennis. How many like tennis only and not cricket? How many like tennis?

 A = set of people who like cricket

B = set of people who like tennis

n( A union B ) =65

n(A) = 40

n(B) = ?

n( A intersection B ) =10

 

n( B - A)  = n( A union B ) - n( A ) 

               = 65 - 40 = 25

25  people like tennis only and not cricket.

 

 

n( A union B ) = n(A) + n(B) - n( A intersection B )

 65 = 40 + n(B) -10

n(B) = 65 - 40 +10 =35

35 people like tennis. 

 

5. If X and Y are two sets such that X has 40 elements, X ∪ Y has 60 elements and
X ∩ Y has 10 elements, how many elements does Y have? 

n(X) =40

n( X ∪ Y  ) = 60

n(X ∩ Y) = 10

n(Y)  = ?

n( X ∪ Y  )= n(X) +n(Y) -n(X ∩ Y) 

60 =40 + n(Y) -10

n(Y) = 60- 40 +10 = 30

Y has 30 elements.

=================================================

 

 chapter 1 miscellaneous sets ncert cbse

 16. In a survey it was found that 21 people liked product A, 26 liked product B and 29 liked product C. If 14 people liked products A and B, 12 people liked products C and A, 14 people liked products B and C and 8 liked all the three products. Find how many liked product C only.

solution

15. In a survey of 60 people, it was found that 25 people read newspaper H, 26 read newspaper T,  26 read newspaper I, 9 read both H and I, 11 read both H and T, 8 read both T and I, 3 read all three newspapers. Find:
(i) the number of people who read at least one of the newspapers.
(ii) the number of people who read exactly one newspaper

solution

 

14. In a group of students, 100 students know Hindi, 50 know English and 25 know both. Each of the students knows either Hindi or English. How many students are there in the group? 

solution

 

 

13. In a survey of 600 students in a school, 150 students were found to be taking tea and 225 taking coffee, 100 were taking both tea and coffee. Find how many
students were taking neither tea nor coffee?

solution

 
ncert cbse 11th mathematics chapter 1 sets exercise 1.6

8.In a committee, 50 people speak French, 20 speak Spanish and 10 speak both
Spanish and French. How many speak at least one of these two languages? 

solution  

7. In a group of 65 people, 40 like cricket, 10 like both cricket and tennis. How many like tennis only and not cricket? How many like tennis?

solution

In a group of 70 people, 37 like coffee, 52 like tea and each person likes at least one of the two drinks. How many people like both coffee and tea?

solution 

 

5. If X and Y are two sets such that X has 40 elements, X ∪ Y has 60 elements and
X ∩ Y has 10 elements, how many elements does Y have? 

solution

 

11th cbse ncert chapter 2 relations and functions miscellaneous exercise

 

12. Let A = {9,10,11,12,13} and let f : A → N be defined by f (n) = the highest prime factor of n. Find the range of f.

solution

 

11. Let f be the subset of Z × Z defined by f = {(ab, a + b) : a, b ∈ Z}. Is f a
function from Z to  Z? Justify your answer.

solution

 

10. Let A ={1,2,3,4}, B = {1,5,9,11,15,16} and f = {(1,5), (2,9), (3,1), (4,5), (2,11)}
Are the following true?
(i) f is a relation from A to B
(ii) f is a function from A to B.
Justify your answer in each case.

solution 

 

9. Let R be a relation from N to N defined by

 R = {(a, b) : a, b ∈ N and a = (b^2) }. 

Are the following true?
(i) (a,a) ∈ R, for all a ∈ N
(ii) (a,b) ∈ R, implies (b,a) ∈ R
(iii) (a,b) ∈ R, (b,c) ∈ R implies (a,c) ∈ R.

 solution 

 

 8. Let f = {(1,1), (2,3), (0,–1), (–1, –3)} be a function from Z to Z defined by
f(x) = ax + b, for some integers a, b. Determine a, b.

solution  

disclaimer:
There is no guarantee about the data/information on this site. You use the data/information at your own risk. You use the advertisements displayed on this page at your own risk.We are not responsible for the content of external internet sites. Some of the links may not work. Your internet usage may be tracked by the advertising networks and other organizations using tracking cookie and / or using other means