REAL NUMBER
MAIN CONCEPTS AND RESULTS
**Euclid’s Division Lemma : Given two positive integers a and b, there exist unique integers q and r
satisfying a = bq + r, 0 ≤ r < b.
** Fundamental Theorem of Arithmetic : Every composite number can be expressed as a product of
primes, and this expression (factorisation) is unique, apart from the order in which the prime factors occur.
** Let p be a prime number. If p divides a2, then p divides a, where a is a positive integer.
** 2 , 3 , 5 are irrational numbers.
** The sum or difference of a rational and an irrational number is irrational.
** The product or quotient of a non-zero rational number and an irrational number is irrational.
**For any two positive integers a and b, HCF (a, b) × LCM (a, b) = a × b.
p
**Let x = , q , p and q are co-prime, be a rational number whose decimal expansion terminates. Then, the
q
prime factorisation of q is of the form 2m.5n; m, n are non-negative integers.
p
** Let x = , q be a rational number such that the prime factorisation of q is not of the form 2m.5n; m, n
q
being non-negative integers. Then, x has a non-terminating repeating decimal expansion.
QUESTIONS FROM NCERT BOOKS
1. Consider the numbers 4n, where n is a natural number. Check whether there is any value of n for which
4n ends with the digit zero.
2. Find the HCF of 96 and 404 by the prime factorisation method. Hence, find their LCM.
3. Find the HCF and LCM of 6, 72 and 120, using the prime factorisation method.
4. Find the LCM and HCF of the pair of integers 336 and 54 and verify that LCM × HCF = product of the
two numbers.
5. Given that HCF (306, 657) = 9, find LCM (306, 657).
6. Explain why 7 × 11 × 13 + 13 and 7 × 6 × 5 × 4 × 3 × 2 × 1 + 5 are composite numbers.
7. There is a circular path around a sports field. Sonia takes 18 minutes to drive one round of the field, while
Ravi takes 12 minutes for the same. Suppose they both start at the same point and at the same time, and go
in the same direction. After how many minutes will they meet again at the starting point?
8. Prove that 3 is irrational.
9. Show that 5 – 3 is irrational.
10. Show that 3 2 is irrational.
11. Prove that 3 + 2 5 is irrational.
, 1
12. Prove that is irrational.
2
ANSWERS
1. There is no natural number n for which 4n ends with the digit zero.
2. 96 = 25 × 3, 404 = 22 × 101, LCM (96, 404) = 9696
3. HCF (6, 72, 120) = 6, LCM (6, 72, 120) = 360 4. 22338
ADDITIONAL QUESTIONS
1. Find the HCF and LCM of 612 and 1314 using prime factorisation method.
2. Find the HCF and LCM of 108, 120 and 252 using prime factorisation method.
3. Find the largest number which divides 245 and 1037, leaving remainder 5 in each case.
4. Find the least number which when divided by 35, 56 and 91 leaves the same remainder 7 in each case.
5. Find the smallest number which when divided by 28 and 32 leaves remainders 8 and 12 respectively.
6. Find the greatest number of four digits which is exactly divisible by 15, 24 and 36.
7. Find the least number which should be added to 2497 so that the sum is exactly divisible by 5, 6, 4 and 3
8. Find the greatest possible length which can be used to measure exactly the lengths 7 m, 3 m 85 cm and
12 m 95 cm.
9. Three measuring rods are 64 cm, 80 cm and 96 cm in length. Find the least length of cloth that can be
measured an exact number of times, using any of the rods.
10. Prove that 5 is irrational.
11. Prove that ( 2 3 ) is irrational.
12. Prove that 4 5 2 is an irrational number.
ANSWERS
1. HCF(612, 1314) = 18, LCM(612, 1314) = 44676
2. HCF(108, 120, 252) = 12, LCM(108, 120, 252) = 7560. 3. 24
4. 3647 5. 204 6. 9720
7. 23 8. 35 cm 9. 9.6 m
, POLYNOMIALS
MAIN CONCEPTS AND RESULTS
** Polynomials of degrees 1, 2 and 3 are called linear, quadratic and cubic polynomials respectively.
** A quadratic polynomial in x with real coefficients is of the form ax2 + bx + c, where a, b, c are real
numbers with a ≠ 0.
** The zeroes of a polynomial p(x) are precisely the x-coordinates of the points, where the graph of y = p(x)
intersects the x -axis.
** A quadratic polynomial can have at most 2 zeroes and a cubic polynomial can have at most 3 zeroes.
b c
** If α and β are the zeroes of the quadratic polynomial ax2 + bx + c, then α + β = − , αβ = .
a a
** The division algorithm states that given any polynomial p(x) and any non-zero polynomial g(x), there are
polynomials q(x) and r(x) such that p(x) = g(x) q(x) + r(x), where r(x) = 0 or degree r(x) < degree g(x).
QUESTIONS FROM NCERT BOOKS
1. Find the zeroes of the quadratic polynomial 4x2 – 4x + 1, and verify the relationship between the zeroes
and the coefficients.
2. Find the zeroes of the polynomial x2 – 3 and verify the relationship between the zeroes and the
coefficients.
1
3. Find a quadratic polynomial, the sum and product of whose zeroes are 2 and , respectively.
3
4. Find a quadratic polynomial, the sum and product of whose zeroes are 0 and 5 , respectively.
ANSWERS
3. 3x2 − 3 2 x + 1 4. x2 + 5
ADDITIONAL QUESTIONS
1. Find the zeros of the polynomial 6x2 − 3 − 7x and verify the relationship between the zeros and the
coefficients.
2. Obtain the zeros of the quadratic polynomial 3 x2 − 8x + 4 3 and verify the relation between its zeros
and coefficients.
3. If the product of the zeros of the polynomial (ax2 − 6x − 6 )is 4, find the value of a.
4. If one zero of the polynomial (a2 + 9)x2 + 13x + 6a is reciprocal of the other, find the value of a.
ANSWERS
3 1 2 3
1. Zeros of are , 2. Zeros of are 2 3 and 3. 4. a = 3.
2 3 3 2
, PAIR OF LINEAR EQUATIONS IN TWO VARIABLES
MAIN CONCEPTS AND RESULTS
** Two linear equations in the same two variables are called a pair of linear equations in two variables.
**The most general form of a pair of linear equations is a1x + b1y + c1 = 0, a2x + b2y + c2 = 0
where a1, a2, b1, b2, c1, c2 are real numbers, such that a12 b12 0, a 22 b22 0 .
** A pair of linear equations is consistent if it has a solution – either a unique or infinitely many.
In case of infinitely many solutions, the pair of linear equations is also said to be dependent. Thus, in this
case, the pair of linear equations is dependent and consistent.
**A pair of linear equations is inconsistent, if it has no solution.
** A pair of linear equations in two variables can be represented, and solved, by the:
(i) graphical method (ii) algebraic method
** Graphical Method : The graph of a pair of linear equations in two variables is represented by two lines.
(i) If the lines intersect at a point, then that point gives the unique solution of the two equations. In this
case, the pair of equations is consistent.
(ii) If the lines coincide, then there are infinitely many solutions- each point on the line being a solution. In
this case, the pair of equations is dependent (consistent).
(iii) If the lines are parallel, then the pair of equations has no solution. In this case, the pair of equations is
inconsistent.
** Algebraic Methods : We have discussed the following methods for finding the solution(s) of a pair of
linear equations :
(i) Substitution Method (ii) Elimination Method (iii) Cross-multiplication Method
** Let a pair of linear equations is given by a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0,
a1 b1
(i) the pair of linear equations is consistent and the graph will be a pair of lines intersecting at a
a 2 b2
unique point, which is the solution of the pair of equations.
a1 b1 c1
(ii) the pair of linear equations is inconsistent and the graph will be a pair of parallel lines
a 2 b2 c2
and so the pair of equations will have no solution.
a1 b1 c1
(iii) the pair of linear equations is dependent and consistent and the graph will be a pair of
a 2 b 2 c2
coincident lines. Each point on the lines will be a solution, and so the pair of equations will have
infinitely many solutions.
MAIN CONCEPTS AND RESULTS
**Euclid’s Division Lemma : Given two positive integers a and b, there exist unique integers q and r
satisfying a = bq + r, 0 ≤ r < b.
** Fundamental Theorem of Arithmetic : Every composite number can be expressed as a product of
primes, and this expression (factorisation) is unique, apart from the order in which the prime factors occur.
** Let p be a prime number. If p divides a2, then p divides a, where a is a positive integer.
** 2 , 3 , 5 are irrational numbers.
** The sum or difference of a rational and an irrational number is irrational.
** The product or quotient of a non-zero rational number and an irrational number is irrational.
**For any two positive integers a and b, HCF (a, b) × LCM (a, b) = a × b.
p
**Let x = , q , p and q are co-prime, be a rational number whose decimal expansion terminates. Then, the
q
prime factorisation of q is of the form 2m.5n; m, n are non-negative integers.
p
** Let x = , q be a rational number such that the prime factorisation of q is not of the form 2m.5n; m, n
q
being non-negative integers. Then, x has a non-terminating repeating decimal expansion.
QUESTIONS FROM NCERT BOOKS
1. Consider the numbers 4n, where n is a natural number. Check whether there is any value of n for which
4n ends with the digit zero.
2. Find the HCF of 96 and 404 by the prime factorisation method. Hence, find their LCM.
3. Find the HCF and LCM of 6, 72 and 120, using the prime factorisation method.
4. Find the LCM and HCF of the pair of integers 336 and 54 and verify that LCM × HCF = product of the
two numbers.
5. Given that HCF (306, 657) = 9, find LCM (306, 657).
6. Explain why 7 × 11 × 13 + 13 and 7 × 6 × 5 × 4 × 3 × 2 × 1 + 5 are composite numbers.
7. There is a circular path around a sports field. Sonia takes 18 minutes to drive one round of the field, while
Ravi takes 12 minutes for the same. Suppose they both start at the same point and at the same time, and go
in the same direction. After how many minutes will they meet again at the starting point?
8. Prove that 3 is irrational.
9. Show that 5 – 3 is irrational.
10. Show that 3 2 is irrational.
11. Prove that 3 + 2 5 is irrational.
, 1
12. Prove that is irrational.
2
ANSWERS
1. There is no natural number n for which 4n ends with the digit zero.
2. 96 = 25 × 3, 404 = 22 × 101, LCM (96, 404) = 9696
3. HCF (6, 72, 120) = 6, LCM (6, 72, 120) = 360 4. 22338
ADDITIONAL QUESTIONS
1. Find the HCF and LCM of 612 and 1314 using prime factorisation method.
2. Find the HCF and LCM of 108, 120 and 252 using prime factorisation method.
3. Find the largest number which divides 245 and 1037, leaving remainder 5 in each case.
4. Find the least number which when divided by 35, 56 and 91 leaves the same remainder 7 in each case.
5. Find the smallest number which when divided by 28 and 32 leaves remainders 8 and 12 respectively.
6. Find the greatest number of four digits which is exactly divisible by 15, 24 and 36.
7. Find the least number which should be added to 2497 so that the sum is exactly divisible by 5, 6, 4 and 3
8. Find the greatest possible length which can be used to measure exactly the lengths 7 m, 3 m 85 cm and
12 m 95 cm.
9. Three measuring rods are 64 cm, 80 cm and 96 cm in length. Find the least length of cloth that can be
measured an exact number of times, using any of the rods.
10. Prove that 5 is irrational.
11. Prove that ( 2 3 ) is irrational.
12. Prove that 4 5 2 is an irrational number.
ANSWERS
1. HCF(612, 1314) = 18, LCM(612, 1314) = 44676
2. HCF(108, 120, 252) = 12, LCM(108, 120, 252) = 7560. 3. 24
4. 3647 5. 204 6. 9720
7. 23 8. 35 cm 9. 9.6 m
, POLYNOMIALS
MAIN CONCEPTS AND RESULTS
** Polynomials of degrees 1, 2 and 3 are called linear, quadratic and cubic polynomials respectively.
** A quadratic polynomial in x with real coefficients is of the form ax2 + bx + c, where a, b, c are real
numbers with a ≠ 0.
** The zeroes of a polynomial p(x) are precisely the x-coordinates of the points, where the graph of y = p(x)
intersects the x -axis.
** A quadratic polynomial can have at most 2 zeroes and a cubic polynomial can have at most 3 zeroes.
b c
** If α and β are the zeroes of the quadratic polynomial ax2 + bx + c, then α + β = − , αβ = .
a a
** The division algorithm states that given any polynomial p(x) and any non-zero polynomial g(x), there are
polynomials q(x) and r(x) such that p(x) = g(x) q(x) + r(x), where r(x) = 0 or degree r(x) < degree g(x).
QUESTIONS FROM NCERT BOOKS
1. Find the zeroes of the quadratic polynomial 4x2 – 4x + 1, and verify the relationship between the zeroes
and the coefficients.
2. Find the zeroes of the polynomial x2 – 3 and verify the relationship between the zeroes and the
coefficients.
1
3. Find a quadratic polynomial, the sum and product of whose zeroes are 2 and , respectively.
3
4. Find a quadratic polynomial, the sum and product of whose zeroes are 0 and 5 , respectively.
ANSWERS
3. 3x2 − 3 2 x + 1 4. x2 + 5
ADDITIONAL QUESTIONS
1. Find the zeros of the polynomial 6x2 − 3 − 7x and verify the relationship between the zeros and the
coefficients.
2. Obtain the zeros of the quadratic polynomial 3 x2 − 8x + 4 3 and verify the relation between its zeros
and coefficients.
3. If the product of the zeros of the polynomial (ax2 − 6x − 6 )is 4, find the value of a.
4. If one zero of the polynomial (a2 + 9)x2 + 13x + 6a is reciprocal of the other, find the value of a.
ANSWERS
3 1 2 3
1. Zeros of are , 2. Zeros of are 2 3 and 3. 4. a = 3.
2 3 3 2
, PAIR OF LINEAR EQUATIONS IN TWO VARIABLES
MAIN CONCEPTS AND RESULTS
** Two linear equations in the same two variables are called a pair of linear equations in two variables.
**The most general form of a pair of linear equations is a1x + b1y + c1 = 0, a2x + b2y + c2 = 0
where a1, a2, b1, b2, c1, c2 are real numbers, such that a12 b12 0, a 22 b22 0 .
** A pair of linear equations is consistent if it has a solution – either a unique or infinitely many.
In case of infinitely many solutions, the pair of linear equations is also said to be dependent. Thus, in this
case, the pair of linear equations is dependent and consistent.
**A pair of linear equations is inconsistent, if it has no solution.
** A pair of linear equations in two variables can be represented, and solved, by the:
(i) graphical method (ii) algebraic method
** Graphical Method : The graph of a pair of linear equations in two variables is represented by two lines.
(i) If the lines intersect at a point, then that point gives the unique solution of the two equations. In this
case, the pair of equations is consistent.
(ii) If the lines coincide, then there are infinitely many solutions- each point on the line being a solution. In
this case, the pair of equations is dependent (consistent).
(iii) If the lines are parallel, then the pair of equations has no solution. In this case, the pair of equations is
inconsistent.
** Algebraic Methods : We have discussed the following methods for finding the solution(s) of a pair of
linear equations :
(i) Substitution Method (ii) Elimination Method (iii) Cross-multiplication Method
** Let a pair of linear equations is given by a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0,
a1 b1
(i) the pair of linear equations is consistent and the graph will be a pair of lines intersecting at a
a 2 b2
unique point, which is the solution of the pair of equations.
a1 b1 c1
(ii) the pair of linear equations is inconsistent and the graph will be a pair of parallel lines
a 2 b2 c2
and so the pair of equations will have no solution.
a1 b1 c1
(iii) the pair of linear equations is dependent and consistent and the graph will be a pair of
a 2 b 2 c2
coincident lines. Each point on the lines will be a solution, and so the pair of equations will have
infinitely many solutions.
