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