Samacheer Kalvi solutions for Mathematics [English] Class 9 TN Board chapter 3 - Algebra [Latest edition]

Identify the following expression is polynomial. If not give reason:

`1/x^2 + 3x - 4`

Identify the following expression is polynomial. If not give reason:

 x²(x – 1)

Identify the following expression is polynomial. If not give reason:

`1/x(x + 5)`

Identify the following expression is polynomial. If not give reason:

`1/(x^(-2)) + 1/(x^(-1)) + 7`

Identify the following expression is polynomial. If not give reason:

`sqrt(5)x^2 + sqrt(3)x + sqrt(2)`

Identify the following expression is polynomial. If not give reason:

`"m"^2 - root(3)("m") + 7"m" - 10`

Write the coefficient of x² and x in the following polynomials

`4 + 2/5x^2 - 3x`

Write the coefficient of x² and x in the following polynomials

`6 - 2x^2 + 3x^3 - sqrt(7)x`

Write the coefficient of x² and x in the following polynomials

πx² – x + 2

Write the coefficient of x² and x in the following polynomials

`sqrt(3)x^2 + sqrt(2)x + 0.5`

Find the degree of the following polynomial

`2sqrt(5)"p"^4 - (8"p"^3)/sqrt(3) + (2"p"^2)/7`

Write the coefficient of x² and x in the following polynomials

`x^2 - 7/2 x + 8`

Find the degree of the following polynomial

`1 - sqrt(2)y^2 + y^7`

Find the degree of the following polynomial

`(x^3 - x^4 + 6x^6)/x^2`

Find the degree of the following polynomial

x³(x² + x)

Find the degree of the following polynomial

3x⁴ + 9x² + 27x⁶ 

Rewrite the following polynomial in standard form

`x - 9 + sqrt(7)x^3 + 6x^2`

Rewrite the following polynomial in standard form

`sqrt(2)x^2 - 7/2 x^4 + x - 5x^3`

Rewrite the following polynomial in standard form

`7x^3 - 6/5x^2 + 4x - 1`

Rewrite the following polynomial in standard form

`y^2 + sqrt(5)y^3 - 11 - 7/3y + 9y^4`

Add the following polynomials and find the degree of the resultant polynomial

p(x) = 6x² – 7x + 2, q(x) = 6x³ – 7x + 15

Add the following polynomials and find the degree of the resultant polynomial

h(x) = 7x³ – 6x + 1, f(x) = 7x² + 17x – 9

Add the following polynomials and find the degree of the resultant polynomial

f(x) = 16x⁴ – 5x² + 9, g(x) = −6x³ + 7x – 15

Subtract the second polynomial from the first polynomial and find the degree of the resultant polynomial

p(x) = 7x² + 6x – 1, q(x) = 6x – 9

Subtract the second polynomial from the first polynomial and find the degree of the resultant polynomial

f(y) = 6y² – 7y + 2, g(y) = 7y + y³ 

Subtract the second polynomial from the first polynomial and find the degree of the resultant polynomial

h(z) = z⁵ – 6z⁴ + z, f(z) = 6z² + 10z – 7

What should be added to 2x³ + 6x² – 5x + 8 to get 3x³ – 2x² + 6x + 15?

What must be subtracted from 2x⁴ + 4x² – 3x + 7 to get 3x³ – x² + 2x + 1?

Multiply the following polynomials and find the degree of the resultant polynomial:

p(x) = x² – 9, q(x) = 6x² + 7x – 2

Multiply the following polynomials and find the degree of the resultant polynomial:

f(x) = 7x + 2, g(x) = 15x – 9

Multiply the following polynomials and find the degree of the resultant polynomial:

h(x) = 6x² – 7x + 1, f(x) = 5x – 7

The cost of a chocolate is Rs. (x + y) and Amir bought (x + y) chocolates. Find the total amount paid by him in terms of x and y. If x = 10, y = 5 find the amount paid by him

The length of a rectangle is (3x + 2) units and it’s breadth is (3x – 2) units. Find its area in terms of x. What will be the area if x = 20 units

p(x) is a polynomial of degree 1 and q(x) is a polynomial of degree 2. What kind of the polynomial is p(x) × q(x)?

Find the value of the polynomial f(y) = 6y – 3y² + 3 at y = 1

Find the value of the polynomial f(y) = 6y – 3y² + 3 at y = –1

Find the value of the polynomial f(y) = 6y – 3y² + 3 at y = 0

If p(x) = `x^2 - 2sqrt(2)x + 1`, find `"p"(2sqrt(2))`

Find the zero of the polynomial of the following:

p(x) = x – 3

Find the zero of the polynomial of the following:

p(x) = 2x + 5

Find the zero of the polynomial of the following:

q(y) = 2y – 3

Find the zero of the polynomial of the following :

f(z) = 8z

Find the zero of the polynomial in the following:

h(x) = ax + b, a ≠ 0, a, b ∈ R

Find the zero of the polynomial of the following:

p(x) = ax when a ≠ 0

Find the roots of the polynomial equation

5x – 6 = 0

Find the roots of the polynomial equation

x + 3 = 0

Find the roots of the polynomial equation

10x + 9 = 0

Find the roots of the polynomial equation

9x – 4 = 0

Verify whether the following are zeros of the polynomial indicated against them, or not

p(x) = 2x − 1, x = `1/2`

Verify whether the following are zeros of the polynomial, indicated against them, or not

p(x) = x³ – 1, x = 1

Verify whether the following are zeros of the polynomial, indicated against them, or not

p(x) = ax + b, x = `(-"b")/"a"`

Verify whether the following are zeros of the polynomial, indicated against them, or not

p(x) = (x + 3) (x – 4), x = −3, x = 4

Find the number of zeros of the following polynomial represented by their graph

Find the number of zeros of the following polynomial represented by their graph

Find the number of zeros of the following polynomial represented by their graph

Find the number of zeros of the following polynomial represented by their graph

Find the number of zeros of the following polynomial represented by their graph

Check whether p(x) is a multiple of g(x) or not

p(x) = x³ – 5x² + 4x – 3, g(x) = x – 2

By remainder theorem, find the remainder when, p(x) is divided by g(x) where, p(x) = x³ – 2x² – 4x – 1; g(x) = x + 1

By remainder theorem, find the remainder when, p(x) is divided by g(x) where, p(x) = 4x³ – 12x² + 14x – 3; g(x) = 2x – 1

By remainder theorem, find the remainder when, p(x) is divided by g(x) where, p(x) = x³ – 3x² + 4x + 50; g(x) = x – 3

Find the remainder when 3x³ – 4x² + 7x – 5 is divided by (x + 3)

What is the remainder when x²⁰¹⁸ + 2018 is divided by x – 1

For what value of k is the polynomial p(x) = 2x³ – kx² + 3x + 10 exactly divisible by (x – 2)

If two polynomials 2x³ + ax² + 4x – 12 and x³ + x² – 2x + a leave the same remainder when divided by (x – 3), find the value of a and also find the remainder.

Determine whether (x – 1) is a factor of the following polynomials:

x³ + 5x² – 10x + 4

Determine whether (x – 1) is a factor of the following polynomials:

x⁴ + 5x² – 5x + 1

Using factor theorem, show that (x – 5) is a factor of the polynomial

2x³ – 5x² – 28x + 15

Determine the value of m, if (x + 3) is a factor of x³ – 3x² – mx + 24

If both (x − 2) and `(x - 1/2)` is the factors of ax² + 5x + b, then show that a = b

If (x – 1) divides the polynomial kx³ – 2x² + 25x – 26 without remainder, then find the value of k

Check if (x + 2) and (x – 4) are the sides of a rectangle whose area is x² – 2x – 8 by using factor theorem

Expand the following:

(2x + 3y + 4z)² 

Expand the following:

(−p + 2q + 3r)²

Expand the following:

(2p + 3)(2p – 4)(2p – 5)

Expand the following:

(3a + 1)(3a – 2)(3a + 4)

Using algebraic identity, find the coefficients of x², x and constant term without actual expansion

(x + 5)(x + 6)(x + 7)

Using algebraic identity, find the coefficients of x², x and constant term without actual expansion

(2x + 3)(2x – 5)(2x – 6)

If (x + a)(x + b)(x + c) = x³ + 14x² + 59x + 70, find the value of a + b + c

If (x + a)(x + b)(x + c) = x³ + 14x² + 59x + 70, find the value of `1/"a" + 1/"b" + 1/"c"`

If (x + a)(x + b)(x + c) = x³ + 14x² + 59x + 70, find the value of a² + b² + c² 

If (x + a)(x + b)(x + c) = x³ + 14x² + 59x + 70, find the value of `"a"/"bc" + "b"/"ac" + "c"/"ab"`

Expand: (3a – 4b)³ 

Expand: `[x + 1/y]^3`

Evaluate the following by using identities:

98³

Evaluate the following by using identities:

1001³

If (x + y + z) = 9 and (xy + yz + zx) = 26, then find the value of x² + y² + z² 

Find 27a³ + 64b³, if 3a + 4b = 10 and ab = 2

Find x³ – y³, if x – y = 5 and xy = 14.

If `"a" + 1/"a"` = 6, then find the value of `"a"^3 + 1/"a"^3`

If `x^2 + 1/x^2` = 23, then find the value of `x + 1/x` and `x^3 + 1/x^3`

If `(y - 1/y)^3` = 27, then find the value of `y^3 - 1/y^3`

Simplify: (2a + 3b + 4c) (4a² + 9b² + 16c² – 6ab – 12bc – 8ca)

Simplify: (x – 2y + 3z) (x² + 4y² + 9z² + 2xy + 6yz – 3xz)

By using identity evaluate the following:

7³ – 10³ + 3³ 

By using identity evaluate the following:

`1 + 1/8 - 27/8`

If 2x – 3y – 4z = 0, then find 8x³ – 27y³ – 64z³ 

Factorise the following expression:

2a² + 4a²b + 8a²c

Factorise the following expression:

ab – ac – mb + mc

Factorise the following:

x² + 4x + 4

Factorise the following:

3a² – 24ab + 48b²

Factorise the following:

x⁵ – 16x

Factorise the following:

`"m"^2 + 1/"m"^2 - 23`

Factorise the following:

6 – 216x² 

Factorise the following:

`"a"^2 + 1/"a"^2 - 18`

Factorise the following:

4x² + 9y² + 25z² + 12xy + 30yz + 20xz

Factorise the following:

25x² + 4y² + 9z² – 20xy + 12yz – 30xz

Factorise the following:

8x³ + 125y³

Factorise the following:

 27x³ – 8y³

Factorise the following:

a⁶ – 64

Factorise the following:

x³ + 8y³ + 6xy – 1

Factorise the following:

l³ – 8m³ – 27n³ – 18lmn

Factorise the following:

x² + 10x + 24

Factorise the following:

z² + 4z – 12

Factorise the following:

p² – 6p – 16

Factorise the following:

t² + 72 – 17t

Factorise the following:

y² – 16y – 80

Factorise the following:

a² + 10a – 600

Factorise the following:

2a² + 9a + 10

Factorise the following:

5x² – 29xy – 42y²

Factorise the following:

9 – 18x + 8x²

Factorise the following:

6x² + 16xy + 8y²

Factorise the following:

12x² + 36x²y + 27y²x²

Factorise the following:

(a + b)² + 9(a + b) + 18

Factorise the following:

(p – q)² – 6(p – q) – 16

Factorise the following:

m² + 2mn – 24n²

Factorise the following:

`sqrt(5)"a"^2 + 2"a" - 3sqrt(5)`

Factorise the following:

a⁴ – 3a² + 2

Factorise the following:

8m³ – 2m²n – 15mn²

Factorise the following:

`1/x^2 + 1/y^2 + 2/(xy)`

Find the quotient and remainder of the following.

(4x³ + 6x² – 23x + 18) ÷ (x + 3)

Find the quotient and remainder of the following.

(8y³ – 16y² + 16y – 15) ÷ (2y – 1)

Find the quotient and remainder of the following.

(8x³ – 1) ÷ (2x – 1)

Find the quotient and remainder of the following.

(−18z + 14z² + 24z³ + 18) ÷ (3z + 4)

The area of a rectangle is x² + 7x + 12. If its breadth is (x + 3), then find its length

The base of a parallelogram is (5x + 4). Find its height if the area is 25x² – 16

The sum of (x + 5) observations is (x³ + 125). Find the mean of the observations

Find the quotient and remainder for the following using synthetic division:

(x³ + x² – 7x – 3) ÷ (x – 3)

Find the quotient and remainder for the following using synthetic division:

(x³ + 2x² – x – 4) ÷ (x + 2)

Find the quotient and remainder for the following using synthetic division:

(3x³ – 2x² + 7x – 5) ÷ (x + 3)

Find the quotient and remainder for the following using synthetic division:

(8x⁴ – 2x² + 6x + 5) ÷ (4x + 1)

If the quotient obtained on dividing (8x⁴ – 2x² + 6x – 7) by (2x + 1) is (4x³ + px² – qx + 3), then find p, q and also the remainder

If the quotient obtained on dividing 3x³ + 11x² + 34x + 106 by x – 3 is 3x² + ax + b, then find a, b and also the remainder

Factorise the following polynomials using synthetic division:

x³ – 3x² – 10x + 24

Factorise the following polynomials using synthetic division:

2x³ – 3x² – 3x + 2

Factorise the following polynomials using synthetic division:

 – 7x + 3 + 4x³

Factorise the following polynomials using synthetic division:

x³ + x² – 14x – 24

Factorise the following polynomials using synthetic division:

x³ – 7x + 6

Factorise the following polynomials using synthetic division:

x³ – 10x² – x + 10

Find the G.C.D for the following:

P⁵, P¹¹, P⁹ 

Find the G.C.D for the following:

4x³, y³, z³

Find the G.C.D for the following:

9a²b²c³, 15a³b²c⁴

Find the G.C.D for the following:

 64x⁸, 240x⁶

Find the G.C.D for the following:

ab²c³, a²b³c, a³bc²

Find the G.C.D for the following:

35x⁵y³z⁴, 49x²yz³, 14xy²z²

Find the G.C.D for the following:

25ab³c, 100a²bc, 125ab

Find the G.C.D for the following:

3abc, 5xyz, 7pqr

Find the G.C.D of the following:

(2x + 5), (5x + 2)

Find the G.C.D of the following:

a^m+1, a^m+2, a^m+3  

Find the G.C.D of the following:

2a² + a, 4a² – 1

Find the G.C.D of the following:

3a², 5b³, 7c⁴ 

Find the G.C.D of the following:

x⁴ – 1, x² – 1

Find the G.C.D of the following:

a³ – 9ax², (a – 3x)² 

Draw the graph for the following

y = 2x

Draw the graph for the following

y = 4x – 1

Draw the graph for the following

y = `(3/2)x + 3`

Draw the graph for the following

3x + 2y = 14

Solve graphically

x + y = 7, x – y = 3

Solve graphically

3x + 2y = 4, 9x + 6y – 12 = 0

Solve graphically

`x/2 + y/4` = 1, `x/2 + y/4` = 2

Solve graphically

x – y = 0, y + 3 = 0

Solve graphically

y = 2x + 1, y + 3x – 6 = 0

Solve graphically

x = −3, y = 3

Two cars are 100 miles apart. If they drive towards each other they will meet in 1 hour. If they drive in the same direction they will meet in 2 hours. Find their speed by using graphical method.

Solve, using the method of substitution

2x – 3y = 7, 5x + y = 9

Solve, using the method of substitution

1.5x + 0.1y = 6.2, 3x – 0.4y = 11.2

Solve, using the method of substitution

10% of x + 20% of y = 24, 3x – y = 20

Solve, using the method of substitution

`sqrt(2)x - sqrt(3)y` = 1, `sqrt(3)x - sqrt(8)y` = 0

Raman’s age is three times the sum of the ages of his two sons. After 5 years his age will be twice the sum of the ages of his two sons. Find the age of Raman.

The middle digit of a number between 100 and 1000 is zero and the sum of the other digit is 13. If the digits are reversed, the number so formed exceeds the original number by 495. Find the number

Solve by the method of elimination

2x – y = 3, 3x + y = 7

Solve by the method of elimination

x – y = 5, 3x + 2y = 25

Solve by the method of elimination

`x/10 + y/5` = 14, `x/8 + y/6` = 15

Solve by the method of elimination

3(2x + y) = 7xy, 3(x + 3y) = 11xy

Solve by the method of elimination

`4/x + 5y` = 7, `3/x + 4y` = 5

Solve by the method of elimination

13x + 11y = 70, 11x + 13y = 74

The monthly income of A and B are in the ratio 3 : 4 and their monthly expenditures are in the ratio 5 : 7. If each saves ₹ 5,000 per month, find the monthly income of each

Five years ago, a man was seven times as old as his son, while five year hence, the man will be four times as old as his son. Find their present age

Solve by cross-multiplication method

8x – 3y = 12, 5x = 2y + 7

Solve by cross-multiplication method

6x + 7y – 11 = 0, 5x + 2y = 13

Solve by cross-multiplication method

`2/x + 3/y` = 5,  `3/x - 1/y + 9` = 0

Akshaya has 2 rupee coins and 5 rupee coins in her purse. If in all she has 80 coins totalling ₹ 220, how many coins of each kind does she have.

It takes 24 hours to fill a swimming pool using two pipes. If the pipe of larger diameter is used for 8 hours and the pipe of the smaller diameter is used for 18 hours. Only half of the pool is filled. How long would each pipe take to fill the swimming pool

The sum of a two digit number and the number formed by interchanging the digits is 110. If 10 is subtracted from the first number, the new number is 4 more than 5 times the sums of the digits of the first number. Find the first number.

The sum of the numerator and denominator of a fraction is 12. If the denominator is increased by 3, the fraction becomes `1/2`. Find the fraction

ABCD is a cyclic quadrilateral such that ∠A = (4y + 20)°, ∠B = (3y − 5)°, ∠C = (4x)° and ∠D = (7x + 5)°. Find the four angles

On selling a T.V. at 5% gain and a fridge at 10% gain, a shopkeeper gains ₹ 2000. But if he sells the T.V. at 10% gain and the fridge at 5% loss, he gains Rs. 1500 on the transaction. Find the actual price of the T.V. and the fridge.

Two numbers are in the ratio 5 : 6. If 8 is subtracted from each of the numbers, the ratio becomes 4 : 5. Find the numbers.

4 Indians and 4 Chinese can do a piece of work in 3 days. While 2 Indians and 5 Chinese can finish it in 4 days. How long would it take for 1 Indian to do it? How long would it take for 1 Chinese to do it?

If x³ + 6x² + kx + 6 is exactly divisible by (x + 2), then k = ?

The root of the polynomial equation 2x + 3 = 0 is

The type of the polynomial 4 – 3x³ is

If x⁵¹ + 51 is divided by x + 1, then the remainder is

The zero of the polynomial 2x + 5 is

The sum of the polynomials p(x) = x³ – x² – 2, q(x) = x² – 3x + 1

Degree of the polynomial (y³ – 2)(y³ + 1) is

Let the polynomials be
(A) −13q⁵ + 4q² + 12q

(B) (x² + 4)(x² + 9)

(C) 4q⁸ – q⁶ + q²

(D) `– 5/7 y^12 + y^3 + y^5` 

Then ascending order of their degree is

If p(a) = 0 then (x – a) is a ___________ of p(x)

Zeros of (2 – 3x) is ___________

Which of the following has x – 1 as a factor?

If x – 3 is a factor of p(x), then the remainder is

(x + y)(x² – xy + y²) is equal to

(a + b – c)² is equal to __________

If (x + 5) and (x – 3) are the factors of ax² + bx + c, then values of a, b and c are

Cubic polynomial may have maximum of ___________ linear factors

Degree of the constant polynomial is __________

Find the value of m from the equation 2x + 3y = m. If its one solution is x = 2 and y = −2

Which of the following is a linear equation?

Which of the following is a solution of the equation 2x – y = 6?

If (2, 3) is a solution of linear equation 2x + 3y = k then, the value of k is

Which condition does not satisfy the linear equation ax + by + c = 0

Which of the following is not a linear equation in two variable

The value of k for which the pair of linear equations 4x + 6y – 1 = 0 and 2x + ky – 7 = 0 represents parallel lines is

A pair of linear equations has no solution then the graphical representation is

If `"a"_1/"a"_2 ≠ "b"_1/"b"_2` where a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0 then the given pair of linear equation has __________ solution(s)

If `"a"_1/"a"_2 = "b"_1/"b"_2 ≠ "c"_1/"c"_2` where a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0 then the given pair of linear equation has __________ solution(s)

G.C.D of any two prime numbers is __________

The G.C.D of x⁴ – y⁴ and x² – y² is