Q1) 3/7 lies between___ fractions

(a) 4/9,5/9

(b)43/99,4/9

(c)42/99,4/9

(d)41/99,42/99

Q2) if a= bm, then _____ (m>0;a,b>0)

&nb

Equation for the volume of spare pyramid

What is the sloution of this question ?

find the sum of the two middlemost terms of the AP: -4/3, -1 , -2/3, .... , 4 1/3.

HOW DO WE PROVE THAT A SQUARE IS A PARALELLOGRAM

sometimes when we get the area of a triangle or a quadrilateral, we get the answer in roots.

so should we take an approximate value of the root to get an approximate anwer.

Q1) 3/7 lies between___ fractions

(a) 4/9,5/9

(b)43/99,4/9

(c)42/99,4/9

(d)41/99,42/99

Q2) if a= bm, then _____ (m>0;a,b>0)

&nb

how do you find missing side of a triangle using pythagorean theorem?

Zero is a rational or an irrational number?

what is the adjective inverse of -7/19?

Is (x+y)/2 is an irrational number or a rational number?

How to find out the Pythagorean triplet whose one of the numbers is 4?

How to answer the following question?rnrnQ) There are two boxes. Which box requires the lesser amount of material to make? The measurements are as belowrna) height 30 cm, width 80 cm and length 10 cmrnb) height 20 cm,

In what ratio does the x axis divide the line segment joining the points (-4,-6) and (-1,7)? So, how to find the coordinate of the point of division?

How to answer the following question?rnAmir starts walking from his house to office. Instead of going to the office directly, he goes to the bank first, from there to his daughter's school and then reaches the office. What

What is Euclid's fifth postulate?

What are the examples of non-Euclidean theory?

rn

how to solve the following question? Tickets numbered 1 to 20 are mixed up together and then a ticket is drawn at random. What is the probability that the ticket has a number, which is the multiple of 3 or 7?

solving linear equation

How many linear equations in x and y can have a solution as (x = 1, y = 3)?

How to find the area of a quadrilateral ABCD in which AB = 8 cm, BC = 6 cm, CD = 8 cm, DA = 10 cm and AC = 10 cm. (3)

If I throw a die what is the probability that I will get a number greater than 4?

Why is Axiom 5, in the list of Euclid's axioms, considered a 'universal truth'?

How to answer the following attached question?

How to answer the following attached question?

A card is drawn from an ordinary pack and the gambler bets that it's a spade or an ace. What are the odds against winning the bet?

How to answer the following question?

Ramesh is a cashier at Canara Bank. He has notes of denominations of Rs 100. 50 and 10 respectively. The ratio of the number of these notes are is  2: 3: 5 respectively.

What is addictive inverse? and for what purposes it's used?

A sum of Rs 2700 is to be given in the form of 63 prizes. If the prize is of either Rs 100 or Rs 25, find the number of prizes of each type. Can anyone help me please that how to do it?

How to find the value of A+B+C, if IAB+CCA=697 and there is no carry-over in addition?

How to find the least value of 'y', if 56*32y is divisible by 18.

How to find whether the relation is Direct or Indirect? Can anyone help me to solve it?

a) The time taken by a train to cover a fixed distance and the speed of the train.

b) The distance travelled by CN

in this test how is the answer for the question: what is the probability of a coin getting tail and the answer came 1 i nstead of  the correct answer 1/2

How to do ex 1.1

there are 20 terms in a an as .sum of the first and last term is  88.

a)if the 10 th term is 42,what is the 11 th term?

What is 100 divided by 2?

Explain what is the formula for calculating interest rate?

What is a arithmetic expression

if x+a is  the HCF of x2+px+q and x2+lx+m.find the value of a.

Factorise:x2-1-2a-a2

Factorize: (x2-4x) (x2-4x-1)-20

What can you say about the prime factorization of the denominator of 27.142857

What can you say about the prime factorization of the denominator of 27.142857

7/5 is the correct answer of this question but I don't know how to get it can you please show the step by step method to get the final

Iam attaching the question below

Plot the points P (1, 0), Q (4, 0) and S (1, 3). Find the coordinate of the point R such that PQRS is a square.

7/5 is the correct answer of this question but I don't know how to get it can you please show the step by step method to get the final

Iam attaching the question below

what are real numbers?

How can we post a question?

How can i attach a document?

How can the expression 2 + 1/cot^2A + cot^2A be simplified?

In triangle PQR, PQ=QR and L,M and N are the mid -points of the sides PQ,QR and RP respectively. Prove that LN= MN

In the figure OPQR is a parallelogram.Find a -the coordinates of the vertex Q b-area of the parallelogram

In the figure OPQR is a parallelogram.Find a -the coordinates of the vertex Q b-area of the parallelogram

what is an algebra?

In the figure OPQR is a parallelogram.Find a -the coordinates of the vertex Q b-area of the parallelogram

In the figure OPQR is a parallelogram.Find a -the coordinates of the vertex Q b-area of the parallelogram

In the figure OPQR is a parallelogram.Find a -the coordinates of the vertex Q b-area of the parallelogram

How can we find the irrational numbers between two fractions

A man set out to travel 240km to his office. In order to save 1 h on his usual time he must travel 20km per hour faster. What is his usual time for the journey?

A sector with central angle 120 is cut and removed from a circular disc and a cone is made out of it. a) What is the slant height of the cone? b) Calculate radius of the cone.

Distance formula

squares are made using dots . How to draw it? find number of dots needed for each square

The algebraic form of an arithmetic sequence is 5n+4

A.what is the difference of its 10th and 20th terms?

B.can the differece of any terms of this sequence be 368?Justify

find the range of the given data: 26,16,3,22,24,23and 25

how can i do the problems based on eucids division lemma

What is the multiplicative inverse of rational numbers

As we made triangles using dots, assume that squares are made using dots. Find the number of dots needed for each square.Write it as a sequence.

x,y,z are in arithmetic sequence . prove that y=x+y+z/3

nth term potion

Good Evening teacher,

In Vedic Maths, adding time, at the last 1.35+3.55=490 and the last 2 digits greater than 60 we added 40 and we got the answer 530. My question how do we get "40" ?

Abhijith C J

What is a square root ? What is the square root of 25 ?

Does that mean when we multiply a number by itself , we get it's square root ?

What is formula to find the measure of Interior Angles ?

What is the formula to find the Interior Angles of a Quadrilateral ?

The algebraic expression of a sequence is X

find two rationals between 0.5 and 0.55

find zeros of (x+2) (x-3)

In an arithmetic sequence the sumof 1stnine terms 279and the sum of the 1st twenty terms is 1280.then (a)question. What is the fifth term of the sequence. (b)questions.what is the sixtenth term of the sequence. (c)question.write the sequence

Consider the arithamatic sequence numbers 171,167,163 a) 0 is term this sequence justify b) how many positive terms at in the sequence

Let ABCD be a sqare of side 2a. Find the coordinates of the vertices of this sqare when: 1: A coincide with the origin and AB and coordinayes axes are parallel to the side AB and AD respectively. 2:The centre of the sqare is at the origin amd coordinate axes are parallel to the side AB and AD respectively.

Sum of the area of two squares is 468centimetre square if the difference of their perimeter is 24meter find the side of the square

Sum of the area of two squares is 468centimetre square if the difference of their perimeter is 24meter find the side of the square

solve for x and y a2/x-b2/y=0,a2b/x+b2a/y=a+b where x and y are not equal to zero

Show graphically that this system equation 2 X + 3 y is equal to 10, 4x+ 6yis equal to 12. has no solution

If the hcf of 408 and1032is expressible inthe form 1032m-408*5 find m

Can u explain solving of rational numbers with appropriate property

Sum of the digits of a two-digit numberis9.When we interchange the digits,it is found that the resulting new number is greater than the original number by 27.What is the two digit number.

Sum of the digits of a two-digit numberis9.When we interchange the digits,it is found that the resulting new number is greater than the original number by 27.What is the two digit number.

how to identify root problems in simply?

how to identify root problems in simply?

How can we convert Percentage into Fraction ??

1) 91.66% - Convert into Fraction

How can draw a rhombus

അടുത്തടുത്ത രണ്ട് അധി ഒറ്റ സംഖ്യകളുടെ ഗുണനഫലം 399 ആയാൽ സംഖ്യകൾ ഏവ

Gushed bgxbfdgbdkh bx

Draw a square of area 5 cm square in three different ways (recall Pythagoras theorem

What is a Pythagorean Triplet ?

Give Examples.....

SUM OF ALL ANGLES OF A TRIANGLE GIVES 180. PROVE

How to solve the problems with axioms

Reliable latest lines and angles question 6

Reliable latest lines and angles question 6

#### Question No - 20

In the below figure . Find  ?

In this question from lines and angles exercise.in the solution it was said that:-

### -----------------------------X----------------------------X------------------------X------------X-----

Now, alternate angles are equal

[ corresponding angles]

Hence,

[ corresponding angles]

[ alternative angles]

Therefore,

-------------X-------------------------X--------------------------X------------------------------X-----------------------X----------

In the above said solution where did the 60 degree come from?

Mydkyrkxb.xmhr b bsym

What is ASA rule with examples

A regular Pentagon is drawn with all its vertices on a circle of radius 15 cm . calculate the length of the sides of this Pentagon

In figure a-b=80 and POQ is a straight line,then find a and b

A,B,C are the three angles of triangle ABC. if A-B=25, and B-C=400, find angles A, B, C. also what type of triangle is it?

thank you

A and B have a ceratin number of mangoes. A says to B If you give 30 of your mangoes I will have twice as many left with you. B replies, if you give me 10, I will have thrice as many as left with you. How many mangoes does each have ?

An almirah is sold at rupees 5225 after allowing discount of 5%. find its marked price

An almirah is sold at rupees 5225 after allowing discount of 5%. find its marked price

What is the difference between Direct and Inverse Proportion ?

Draw acircle of radius 3cm.Draw a triangle with angles 50° and 70° with all sides are tangents to circle?

Draw acircle of radius 3cm.Draw a triangle with angles 50° and 70° with all sides are tangents to circle?

In the figure angle m =angle n =46 degree ,Express x in the terms of a,b and c where a,b and c are lengths of LM,MN and NK respectively .

From a point P on the ground the angle of elevation of the top of the a 10m tall building and a helicopter hovering at some height of vertically over the top of the building are 30 degree and 60 degree respectively . Find the height of the helicopter above the ground.

Bczyreiygljvvngdsgr

Solve the following equations

Why are identities used ?

Ft4yg6hu6vytdr4gyhcffrcgt

Starting from Q6 i am not able to find solutions of it

I'm not getting solutions of surface area and volumes starting from q6 . And there no teach me how icon also ! What to do ?

I'm not getting solutions of surface area and volumes starting from q6 . And there no teach me how icon also ! What to do ?

The length of the sides of a triangle are 5cm, 12cmand 13cm.find it's circumradius.

Explain why 13233343563715 is a composite number.

Explain why 13233343563715 is a composite number.

#### Question No - 15

Draw these figures, Four equal rhombuses #### Question No - 16

Draw these figures., Five equal rhombuses: #### Question No - 17

Draw these figures, Four rhombuses around a square: #### Question No - 18

Parallelograms on two sides of a square : What all theorems are there in the lesson triangles, quadrilaterals and areas of parallelograms and triangles

What all theorems are there in the lesson triangles, quadrilaterals and areas of parallelograms and triangles

When will I get the video lessons of physics , chemistry & biology.

Find x from the figure

A metallic sphere of diameter 3cm is melted and recast into three spheres . The radius of one of these is 0.75cm and that of another is1.25cm . Determine the radius of the third sphere.

Find the value of a if x-a is a factor of x^3-2ax^2+x-a+1

how can we find a number if it is a inverse proportion

What is a circle ?

Is the gas acidic or basic?

Find the hcf of 81 and 237 and express it in linear combination

If x/y + y/x = -1 ( x, y=o) the value of x2-y2 is

## Is zero (0) a rational number ? justify your answer ?

Yes, zero is a rational number

zero can be expressed as      etc

which are in the form of,   where p and q are integers and q  0.

## Write the simplest form of a rational number

## Is     a rational number or not ?

so it is rational number.

## Find the decimal expansion of

=   0.058 ( Decimal points is shifted  three places to the left)

## Identify an irrational number among the following numbers: 0.13 , ,  , 0.3013001300013.....

0.13 is a terminating number. So, it is not an irrational number.

= 0.131515......, 15 is repeating continuously so it is not an irrational number.

= 0.13151315........is repeating continuously so it is not an irrational number.

0.3013001300013....., non terminating and non recurring decimal. Hence, it is an irrational number. So, 0.3013001300013 is an irrational number.

## Is the product of two irrational number is always an irrational number ?

No, it may be rational or irrational.

sum of  and

## Insert three rational numbers between    and   .

So three rational numbers are

## Find two rational numbers between 4 and 5 .

The numbers are  and

## Express the rational numbers    in the form   ,  where p and q are integers and

Let  X  =  0.999............

10x  =  9.999.......

10x - x  =  (9.999....) – (0.999....)

9x  =  9

X = 1

## Calculate the value of    in form of     where  p and q are integers  and

Let,  x   =    =   2.9999.....

10x   =   29.999.......

10x –x  =  (29.999....) – (2.999....)

9 x  =  27

= 3

## Write the sum of and .

= 0.777...

x  = 0.777

10x  =  7.777

10 x – x  = ( 7.777.... ) – ( 0.777....)

9 x  =  7

## Calculate the irrational number 2 and 2.5

Since

hence, the irrationl number between2 and 2.5 is

## Simplify the number

## Find the rational numbers between 0.121221222122221...and  0.141441444144441... in the p form, where p and q integers and

Two rational numbers between0.121221222122221...and 0.141441444144441...are 0.13 and 0.14

## Find four rational numbers between      and

Since LCM of 5 and 6 is 30

Hence , four rational numbers between      and

are

## Find six rational numbers between 3 and 4.

let  a = 3 , and b = 4

Here, we find six rational numbers i,e n = 6

1st rational number =

2nd rational number =

3 rd rational number =

4th rational number  =

5th rational number  =

6th rational number  =

So six rational numbers are

## Insert three rational numbers between  and     and

LCM of 5 and 7 is  35

and

so

The required three rational numbers are

## simlify

## is not a rational number as   and are not rational. State whether it is true or false. Justify your answer.

false

justification  :

Which is a rational number

## simplify      by rationalizing the denominator

## if  = 1.414, then, find the value of

Alternative method

:

## Find three rational numbers between    and

Since LCM of 7 and 11 is 77

Hence three rational numbers between    and    are

## Rationalize the denominator

## Rationalize the denominator of

Alternative Method

## Simplify

## Give two rational numbers whose  : (1)  Difference is a rational number (2)  Sum is a rational number (3)  Product is a rational number (4)  Division is a rational  number               Justify also.

Any example and verification of example

product          =       4/5 x 9/2 = 36/10 (Rational number)

Division         =       9/2   4/5 = 45/8  ( Rational number)

## Rationalize the denominator of

## Simplify

Given expression  =

= 0

Alternative Method

=

=

=

## Simplify

Alternative Method

## Express  the decimal  number    in the form of  , where p and q are integers and

....................

10 x  = 22.18181818...........

1000 x = 2218.181818......

1000x – 10x = (2218.181818....) – (22.181818......)

990 x = 2196.00

## Given two rational numbers whose (1) difference is a rational number (2) sum is a rational number (3) product is a rational number (4) division is a rational number

Any example and verification of example

let m = 4/5, n= 9/2

## show that

## Simplify

## Find any two irrational numbers between 0.1 and 0.12.

Required to irrational number are :

i)  0.10100100010000......

ii) 0.1020020002000.......

## Simplify

## Simplify

## Simplify

## Simplify

## Evaluate      given that

## if

Alternative Method

## If x=     and y =      then find the value of x+y+xy.

## if  x = 2+  then find the value of

Squaring both sides we get

## If    and      then what will be the value of

Alternative Method

## if   then find the value of

=-8

## Find the value of

Alternative Method

## if      then find the value of

Alternative Method

## Find the values of a and b when

Compairing the rational and irrational parts  of both sides we get

## Find the values of a and b when

Alternative method

Compairing both sides we get

a= 4, b= 1

## if         find the value of a .

On compairing

## Height of students of  class x are given in the following distribution : Find the model of height  Height (in cm)                  150- 155            155 -160           160- 165            165 - 170            170-175 Number of students              15                      8                     20                     12                      5

Class interval                                Frequency

150 -155                                          15

155 - 160                                           8

160 -165                                          20

165 -170                                         12

170- 175                                          5

Total                                              60

Here,

Modal class = 160 -165

Mode  =

Model height  = 163 cm .

## If        then find the values of a and b.

............................(1)

From eq. (1)

Compairing on both sides

,

## The following frequency distribution shows the number of runs scored by some batsmen of india in one - day cricket mathches: Run scored                      2000-4000           4000-6000           6000-8000         8000-10000           10000-12000 Number of batsmen                9                         8                       10                       2

Here,

Modal class  is  6000- 8000

Mode  =

## If       then show that

Alternative Method

Squaring the both sides we get

## Rationalize the denominator of

## Prove that :

## Prove that

## Evaluate      given that

Denominator =

Alternative Method

## Find the  values of a and b in

On Comparing both sides we get

## Find a and b if

## If      then find

Alternative Method

## If         then find the value of

## if       then find the value of

## if          then find the value of

## Prove that

Now LHS =

Thus LHS =  RHS

## if    then find the value of

Given

Now ,

## if          and        find the value  of

## Find the value of

## Find the value of

## Show that

## Find the co-ordinate 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)

The point on y- axis  has x-axis has x- coordinate 0.

Since it lies at a distance of 4 units in the negative direction of y-axis.

The point  is (0, -4)

## If    find the value of

## (i)Find the six rational  numbers bbetween 3 and 4  (ii) Which mathematical concept is used in this problem (iii) Which value is depticted in this question

(i) We known that between two rational numbers  x and y such that x< y there is a rational number .

ie,

Now a rational number between 3 and      is :

A rational nummber berween

Further a rational number between 3 and

A rational number between

A rational number between

Hence , six rational numbers between 3 and 4 are

(ii) Number syatem

(iii) Rationality is always welcomed

## Two classmates Salma and Anil simplified two different expressions during te revision hour and explained to each other their simplifications. Salma explains simplification of       and Anil explains simplification of   .  Write the both simplification. What value does it depict ?

Justify

Again

Co-operative learning among classmates with out any gender and religious bias.

Not defined

## Name the degree of the polynomial  -3x + 2.

Linear polynomial

## Write an exampleof a constant polynomial.

Constant polynomial is 7.

## What is

Not a polynomial

Binomial

## What is the degree of polynomial

Degree of a polynomial

Degree of

Degree of

Degree of

## What is the zero of the zero polynomial ?

Every real number is a zero of the zero polynomial

## Write the number of zeroes in a cubic polynomial

No of the zeroes of cubic polynomial  = 3

## In the expression      what is the co-efficient of x ?

Co -efficient of x in expression

## If -4 is a zero of the polynomial      then calculate the value of k.

Given ,

Since -4 is a zero of polynomial

## Write the zeroes of the polynomial

For zeroes , put

Putting

Thus

## Clasify the following as linear , quadratic and cubic polynomials  (a)         (b)     (c) 1+x            (d)

linear polynomial      degree 1

Cubic Polynomial      degree 3

## if y = 2  and  y = 0  are the zeroes of the polynomial    find the value of a and b

.......................(1)

## Find the value of the polynomial    (1)                 (2)

(1) When

then

(2) When

then

Then

## On dividing      by y what remainder do we get ?

Let ,

then remainder =

## If      is divided by x+1 then what remainder do we get ?

let,

Put

Then remainder is :

## Find the remainder when     is divided by      using remainder theorem .

## Find the remainder when     is divided by x - a

Here ,

and the zero of x - a is a.

so,

So, by the remainder theorem 5a is the remainder when     is divided by x-a

## Find the remainder when the polynomial     is divided by

2x -1 = 0

Hence required remainder is

## By actual division, find the quotient and remainder when    is divided by 2x-1. ## Using remainder theorem, factorize

Factors of

is a zero of    is a factor

## Polynomial    leave the same remainder when divided by   find the value at k.

Let ,    and

Put

or

and       leave the same remainder when divided by x+2.

## The polynomial    and    leave the same remainder in each case when divided by (x-2). Find the value of a

Let,

## Find the value of 'a' if remainder is same when polynomial    is divided by    and

leave the same remainder when divided by    and  .

Remainders are equal

so,

## If the polynomial       and      leave the same remainder when divided by (x - 3)  Find the value of a.

Let ,

and

Put      = 0  or    in  and

According to the  question

.

## What must be substracted from    so that     is exactly divisible by (x-1). write the resultant polynomial which is exactly divisible by (x-1).

Put,   or  x = 1 in

Hence , 2 must be substracted from  so that it is exactly divisible b (x-1).

Resultant polynomial to be divisible by (x-1)

.

## Find the value of p if the polynomial      when divided by     leaves  the remainder 19. Also find the remainder when p (x) is divided by x+2.

-

## The  polynomials     when  divided  by (X - 2) leave  the remainders  p  and  q  respectively  .   find the  value  of  a.

Let ,

When divided by

and

## If      is a polynomial such that when it is divided by  and  the remaniders are 5 and 19 respectively . Determine the remainders when    is divided by

put x-1    or   x=1  in     we get.

..........................(1)

Again put x+1  = 0    or   x = -1 in  f(x)  we get.

...........................(2)

By equation (2)

Again Put,   or    in

## The polynomials    when divided by  leave the remainders  and  respectively .  Find the value of b if

Let

Put    or    in

We get

Let,

Again  put x - 4 = 0 or x = 4 in

Given that

## If f be a polynomial such that  then calculate one factor of

Since,

is a polynomial

So,  or  is a factor of .

is a factor

## Find the value of k if x-2 is a factor of

x-2 is a factor of p(x) then p(2)=0

## Find the value of k if 2x-1 is a factor of the polynomial .

2x-1 is a  factor of

## Write the factors of

Since ,

Factors are a and

## write the factors of

Then factors of  are and

aaaa

One Factor  is

is a factor of

## Find the value of k if   is a factor of

is  a factor of

## Find the value of 'a' for which (x-1) is factor of the polynomial

(x-1) is a factor of

## For what value of k (x+1) is a factor of

Given ,

(x+1) is a factor of p(x) then

.

## Show that (x-1) is a factor of the polynomial

or   in

Thus,

Hence  (x-1)  is a factor of f(x)

## For what value of k is the polynomial  exactly divisible by (x+2)

(x+2) is a factor of

## Find the value of 'k' if (x-1) is a factor of

is a factor of  then

Let,

is a factor of

Fatots are :

## If x-a is the factor of  then prove that

If (x-a) is a factor of p(x) then

## Find the value of a for which (x-a) is   a factor of the polynomial

Let

(x-a) is a  factor of the polynomial p(x) then p (a)=0

= 0

## If (x-2) and  are factors of  then show that p=r .

Let ,

(1)=(2) as both are equal to -10

## Using factor theorem show that (m-n), (n-p) and (p-m) are factors of

Let

=0

Similarly  &

n-p is a factor of  p and p -m is a factor of p.

Factor of

is a factor of

Factors of   5

## Factorize :

factor of  6 =

is zero of p (x) of (x+1) is a factor of p (x)

## factorize

(x+2)  is a factor of

Factors are: (x+2)   (x+1)  (x+10)

Alternative method

Factors  of  20 =

is a zero of p(x), and (x+1) is a factor of p(x)

Then

## Verify if 1 and  - 3  are zeroes of the polynomial   if yes, factorize the polynomial.

is a zero of p(x)

-3 is a zero of p(x)

is a factor of p(x)

, when  we divide physically

Hence,

## Verify that (x-1), (x-2) and (2x+1) are the factors of the polynomial

Let,

Put x = 1 in p (x)

Hence, (x-1) is a factor of p (x).

Put x =  2 in p(x)

hence (x-2) is a factor of p(x)

Put,  in p (x)

## Verify that (x-1), (x-2) and (2x+1) arec the factors of the polynomial

Let ,

Put x = 1 in p (x)

Hence (x-1) is a factor of p (x)

Put x = 2

Hence (x-2) is a  factor  of p(x)

Put  in p (x)

Hence  or  (2x+1) is a factor of p (x)

## Using factor theorem, find the value of a if  is divisible be 2x+1.

If  (2x+1)is a factor of p(x) then 2x+1= 0,

is a zero  of the polynomial   p(x)

so,

## Find the value  of p for which the polynomialexactly divisible by  x-2 Hence factorize the polynomial.

p (x) is exactly divisible be x- 2

Then,

## With out actual division show that  is exactly divisible by

Let,

Zero of x -2 is 2 as  x-2=0   x = 2

Zero of x -1 is 1 as  x-1=0    x = 1

Now,

and (x-2) are the factors of f (x)

is exactly diviible by g (x)

## If  has (x-1) as a factor and leaves a remainder 2 when divided by x-2 find the relation between a and b.

Let,

is a factor of  f (x)  then by factor theorem f(1) = 0

f(1)=0

On putting  x = 1 in (1) we get

From (ii)

When f(x) is divided by (x-2)it leaves remainder 2

by remainder theorem f(2) = 2

On putting x = 2 in (i)  and using (4) we get

From (ii) and (v)

## If  x+a is a factor of the polynomial  and  prove that

Let,

and

Since x+a is a factor  of p(x) and q(x) then p(-a) = 0 and q (-a) = 0

Subtracting eqn (2) from eqn (1) we get

## Write the co efficient of   in the expression of

coefficient of in the expansion of

a

Given,

x

=

## Expand by using identity (2x-y+z)2

Alternative method

By using this identity

## Give possible expression for the length and breadth  of a rectangle whose  area is given by

Area of rectangle =

=

length and bredth are (5a-3) and (5a-4) respectively .

## Simplify

as (a+b)(a-b)= a2-b2

## Find the value of  if

Given,

On squaring both sides, we get

=3

## Factorize : 12(x2+7)2 - 8 (x2+7) (2x-1)-15 (2x-1)2.

Let  the given expression

.

## factorize

Alternative method

x

.

8-27a3-36a+54a2

## Factorize  64a3-27b3-144a2b+108ab2

64a3-27b3-144a2b+108ab2

3-12(a-b)

a

Given Exp =

x+y = -z

(i)

(ii)

## Factorize

Let,

expression

Alternative method

If   then

## Simplify  :(2a+3b)3-(2a-3b)3

where  2a + 3b = x and 2a -3b = y

## If  Find the value of  taking only the positive value of

[On taking positive value ]

Now,

= 3 x 6 = 18

= 10 x 97 = 970

## If x+ y+4 = 0, then find the value of

i,e.,

Alternative method

## Find the value of  when x+y = 5.

Given, x+y = 5 then

## Simplify

Both numerator and denominator are of the form a3+b3+c3

We  know that when a+b+c = 0

then a3+b3+c3 = 3abc

For numerator, a2-b2+b2-c2+c2-a2= 0

For denominator     a-b+b-c+c-a= 0

## If   and  find the value of  if  .

times

## Simplify :

= 2a x 2 (2b+3c)-3b(2b+3c)

= (2b+3c) (4a-3b).

Given,

a + b + c = 9

We know that

## 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.

Given, a+b+c = 3x

or  3x-a-b-c = 0

=0.

(i)

(ii)

## If evaluate

On squaring both sides , we get

## If  evaluate

Cubing both sides

=4+2 = 6

and

Now,

## Prove that  Hence evaluate :

Prove RHS =LHS as in Q.2

Now,

=42 x [1+1+4]

= 42 x 6 = 252.

=0 x

RHS =

## if a+b+c = 6 and ab+bc+ca = 11, find the value of a3+b3+c3-3abc.

Alternative method

## Find the value of p3-q3 , if p-q =  and

=

Alternative method

## If x+y+z = 10 and  find xy+yz+zx and

Expansion of  (x+y+z)2

=

=

Alternative method

and

10x 1 0 = 100

Also

Since,

Rightarrow

and

now.

RHS =

Now, 216x3-125y

(i)

(ii)

## Factorize :(p+q)2-20(p+q)-125

Let, p+q = x then

Let

=

?

## Ram has two rectangles having their areas as given below: (1) 25a2-35a+12                   (II) 35y2+13y-12 (i) Given possible expressions for the length and breadth of each rectangle. (ii) Which mathematical concept is used in this problem? (iii) Which value is depicted in this problem

(i) Possible length and breadth of the rectangle are the factors of its given area.

Area    =

So possible length and breadth are (5a -3) and (5a - 4) units, respectively.

(ii)  Area  =

So, possible length and breadth are (7y-3) and (5y+4) units respectively.

(ii) Factorization of polynomials

(iii) Expression of one's desires and news is very necessary.

## The general form of linear equation in two variables is......................

where a,b, c are real numbers and both

## Is   a linear equation

No,  for a linear equation

True .

## In a one day cricket match Raina and Dhoni scored 198 runs. Express this as a linear equation in two variables .

Let the scored by Raina and Dhoni are x & y respectively then

x+y  = 198

## Total number of legs in a heard of goats and hens is 40. Represent this in the form of linear equation of two variables.

Let the number of goats and hens in heard are x & y respectevely them

4x+2y = 40

x- 6y = 5

## Does the following equation x = 5y represent a straight line passing through a point  (0,0)

Given equation  x= 5y

The point  (0, 0 )  statifies the given equation, hence answer is yes .

No

## Find the point where equation intersects  3 x +2 y=12  y - axis

As the line intersects y- axis, put x = 0 in the given equation, we get

The required point is (0,6).

Given,

## If the point (2,3) lies on the line 4y = ax + 5, then a  = ...........................

Given point lies on the

ie,

.

## If (0,2) is a solution of the linear equation 2x+3y = k, then find the value f k.

is the solution of given equation

it satisfies the equation

(1,-1)

(0,0)

x-y = 3

## Find three solutions of the linear equation   in two variables

So, when

 x 5 0 10 y 0 -7 7

## Give equations of two lines on the same plane which are intersecting at point (2,3).

The equation of two lines on the same plane which are intersecting at point (2,3) are :

## After 5 years the age of father will be two times the age of his son. Write a linear equation in two variables to represent this statement.

Let father's present age = x years

Son's present age = y years

After 5 years father's age will be = (x+5) years

After 5 years son's age will be = (y+5) years

According to the question,

## A part of the monthly expense of a family on milk is fixed which is Rs. 700 and the remaining varies with the quantity of milk taken extra at the rate of Rs. 25 per litre. Taking the quantity of milk required extra as x litre and total expenditure on milk is Rs. y. Write a  linear equation representing the above information.

According to the question,

## Check  which of the following is (are)  the solution (s) of the equation  3y - 2x = 1.   (i) (4,3)             (ii)

Put x = 4 and  y = 3

then

So, (4,3)  is the solution of the equation

Again put

and  ,  then

So,  is not a solution of the given equation

## Find the value  of k for which the point (-1,3) lies on the graph  of the equation 2x-y+k = 0

(-1,3) lies on the graph

## If x = 2 and  y = 1 is the solution of the linear equation  find the value of k.

If x = 2 and y = 1 is the solution of the linear equation  then

## Express y in terms x from the equation 3x+2y = 8 and check wether the point (4,-2) lies on the line.

For

lies on the line

## Express y in terms of x in equation 2x-3y =12. Find the points where the line represented by this equation cuts x-axis and y-axis.

Equation

On x-axis y=0

At point (6,0) the given line cuts the x-axis. On y-axis

x = 0

## The point (3,4) lies on the graph of the equation 3y = ax+7. Find the value of 'a'

If points (3,4) lies on

## Find the co-ordinates of points where the graph of the equation  intersects x=axis and y-axis.

Equation : 4x+3y = 12

For intersection with x-axis

y =0

or

co-ordinates are (3, 0)

For intersection with y-axis

Co-ordinates are (0,4).

## Find the value of k so that x= -1 and y = -1 is a solution of the linear equation

Substituting  and y = -1 in

We get,

## Find the point at which the equation  meets the x -axis

On the x-axis, y coordinate is zero.

So, put y = 0 in

we get,

meets the x-axis at  (2,0)

Putting  We get

## Express y in terms of x, given that 2x - 5y =7. Check wether the point (-3,-2) is on the given line.

Given,

2x-5y = 7

Now, when

Hence, the point (-3,-2) does not lie on the given line.

## Give the equation of a line passing through (2,14). How many more such lines are there. Write the equation in the form

The line passing through (2,14) is

2y = 14x

y = 7x

In finitely many lines are there.

The equation in the form ax+by+c = 0 is 7x-y+0 = 0.

## Write the equation  in the form  check weather  (0,-1) and  are the solution of this equation

Putting x = 0,  y = -1

is not the solution of the given equation

Putting

Which is correct

is a solution   of the given equation

## When5 times the larger of the two numbers is divided by the smaller, the quotient and the remainder is 2 and 9 respectively. Form a linear equation in two variables. Write it in standard form

Let larger number be x, then 5 times of larger number = 5x and smaller number be y

Qutiont = 2 and remainder = 9

So, according to the question,

## ABCD is a square. Co-ordinates of A and C are (-1,-1) and (1,1) respectively. Write the coordinates of B and D. Also write the equations of all the sided of the square.

Given,  A (-1,-1) and C (1,1)

Then,   B (1,-1) and D (-1,1)

Also, equation of sides of square are

AB :  y = -1

BC :  X = 1

CD:  Y=1

DA:  X = -1

## If the point (-1,-5) lies on the graph of 3x = ay+7, then find the value of 'a'

(-1,-5) lies on the graph of

## Given the equation 2x+y= 7, (i)  What is the value of x, when the value of y is 3 (ii) What is the value of y, when the value of x is 4? (iii)Find the one more solution for the above equation ?

(i) when y =3 then

(ii) When x = 4, then

(iii) When x = 1, then

One more solution is (1,5)

## For what value of k, the linear equation 2x+ky =8 has x = 2 and y = 1  as its solution  if x=4, then find the value of y.

The linear equation is 2x+ky = 8

At  x = 2, y = 1

If x = 4 then

## Show that the points  A (1,2) , B (-1,-16) and C (0,-7) lie on the linear equation  y = 9x-7.

The equation is  y =9x-7

A (1,2)             2 = 9 (1)-7

2 = 2 True

B (-1,-16)       -16= 9(-1)-7=-9-7

= -16 : True

C (0,-7)          -7 = 9(0)-7

= 0-7 = -7 True

## Find the coordinates of the points where the line representing the equation  cut the x-axis and the y-axis.

Writing in the standared form

On x-axis, y = 0

Point on the x-axis = (4,0)

On y-axis, x = 0

Point on the y-axis = (0,6).

## For what value of p; x =2,y=3 is a solution of (p+1)x-(2p+3) y - 1 = 0 and write the equation

Given equation is

If x = 2, y = 3  is  the solution of the equation (i) then

Put the equation (i) then

is the required equation

## Determine the point on the graph of the linear equation x+ y = 6, whose ordinates is 2 times its abscissa.

Given, y = 2x

Putting y = 2x in the  equation x+y = 6, we get

Required points is (2,4).

## If the point (3,4) lie on the graph of the linear equation 3y = kx+7, then find the value  of k. Also find two more solutions of the equation

On putting (3,4) in the equation of the line

3y = kx +7

The equation becomes

9y=5x+21

Two more solutions are (-6,-1) and (12,9)

## Find k in each case if x = 2, y = 1 is a solution o f the equation :  (i) 3x+2y = k,            (ii) 2x-ky= 6 (iii)

Given,

3x+2y = k

Put x = 2, y = 1 then

(ii)

Put x = 2, y = 1 then

2(2)-k (1) = 6

(iii)

Put x = 2, y = 1  then

## Find three different solutions for the equation 3x+2y =1.

Given

(i) Put x = 1, then

(ii) Put x = 3 then

(iii) Put x = 5, then

Hence three different solutions for the equations 3x+2y = 1

 x 1 3 5 y -1 -4 -7

y = k

y = k.

y- axis

infinite

(0,4)

a>0

Parallel, 6

## The graph of the linear equation 3x+5y =15 cuts the x-axis at the point.................

x = 5, y = 0. ie., (5,0)

## Draw the graph representing the equation of x+y = 0

y = 0 ## To which linear equation does the graph represent

3x-4y+2 = 0 ## Express  x in terms of y, in the equation 7x-3y = 15. Check if the line represented by the equation intersects the y-axis at y = -5.

Given

At y- axis  x= 0

Given line intersects the y-axis at y = -5

## Find two different solutions of the equation 4x+3y = 12 from its graph .

Given equation

4x+3y = 12

Put x= 0, then 0+3y = 12

Hence point on y-axis is (0,4)

Now put y = 0, then

4x+0 = 12

x = 3

Hence the point on x-axis is (3, 0) ## The cost of a toy elephant is the same as the cost of 3 balls . Express the statement as a linear equation in two variables and plot the equation on a graph paper.

Let the cost of a toy elephant =x, ball = y

 x 3 6 9 y 1 2 3 ## Draw the graph of 2x-3y-12 = 0 on the  graph paper

 x 6 9 3 y 0 2 2 ## Solve the equation and represent the solution on the   cartesian plane. Put x = 3 on the Cartesian plane

On Cartesian plane, x = 3 is a parallel to y-axis

## Write three solutions of the equation 3x = y +3 Draw its graph and find the points where the graph intersects the axes.

three solutions are From graph it is clear that line meets x- axis  at (1,0) and y- axis at (0,-3)

## If zeroes  of the polynomial   are  and  , then find the value of a.

Products of roots (zeroes)

## Write the equation   in  standared form  and draw the graph

The given equation can be written as,

.

 x -2 4 10 y 0 -5 -10 ## ABCD  is a rectangle. Write the equation of its sides. Also find its area. Equation of the sides are,

AB :   Y = 0

BC:    X = -1

CD:   Y = -4

DA:   X = -4

Area  = 4 x 3

= 12 sq.units

## Draw a triangle whose sides are represented by x = 0, y = 0 and x+y = 3in cartesian system. Also find the co-ordinates of its verticles.

Given,

x+y = 3

On the y axis put x= 0 then

Hence on the y- axis co-ordinates of B is (0,3)

On the x-axis put y = 0then

x = 3

Hence on the x-a xis co-ordinate of Ais (3,0)

Hence a triangle whose sides are x = 0, y = 0, and x+y = 3 is as shown in fig.

Again the vertices of the triangle are A (3,0), B (0,3) and O (0,0). ## Two friends sita and gita, together contributed Rs . 200 towards prime minister relif fund. Write a linerar equation which satisfies this data. Draw the graph.

Let sita contribute = x and contribute= y

According to the question,

x+ y = 200

y = 200-x

 x 0 200 100 y 200 0 100 ## Give the geometric representation of  y = 4  as an equation in : (i) one variable  (ii) Two variable

(i) one variable (ii) Two variable

The points are (0,4) and (1,4) ## The cost of a pen Rs. 16. Taking a number of pens bought as x and total cost as y, from a linear equation in x and y and draw its graph. Find the cost of 6 pens from the graph

Total cost of value of pen  x number of pens

y = 16x

Table of value of (i) is

 x 0 1 2 3 4 5 6 y 0 16 32 48 64 80 96

From the graph, cost of 6 pens = Rs . 96. ## Give the geometrical representation of the equation 3x+15 = 0  as an equation  (i) One variable  (ii) In two variable

3x + 15 = 0

X = -5

(i) Equay=tion in one variable (Number line ): A point P at a distance of 5 units to left of O on the number line. (ii) In two variables (Cartesian plane): A line AB parallelto y-axis at a distance of 5 units to the left of y-axis. ## Draw the graph of linear equation 2x+ y = 8 on cartesian plane. Write the co-ordinates of the points where this line intersects x-axis and y-axis

2x+y = 8

y = -2x+8

 x -1 0 1 y 10 8 6

From the graph it is clear that

Line intersects x- axis at (4,0)

and y-axis at (0,8) ## Solve 5x-2 = 3x-8 and represent the solution  (i) On a  number line  (ii) In the cartesian plane

(i) Point P (-3,0)  represent the solution x = -3 on the number line (ii)Line AB represent the solution in the cartesian plane. ## Draw the graph of the linear equation x+y =7. Verify from the graph that (8,-1) is a solution of the equation x+ y = 7

x+ 7  = 7

y = 7-x

 X 5 7 4 Y 2 0 3 From graph it is clear that (8,-1) lies on the line AB.

Hence (8,-1) is a solution of the given equation.

## Draw the paragraph of     and find the points where it  cuts the co-ordinate axes.

(i) When the line cut x- axis then put y = 0

i.e,  2x = 6

x = 3

Hence points is (3,0).

(ii)  When the line cut y-axis then put x = 0

i,e., 3y+6 = 0

y = -2

Hence the point  (0,-2)

 x 0 3 6 Y -2 0 2 ## Draw the graph of the linear equation 3x+4 y = 6.Find the points where the line representing the equation 3x+ 4y = 6 cuts the axes of x  and y.

Given equation is 3x+ 4 y = 6

(i) When it cut x- axis then

put y = 0 ie., 3x = 6

x = 2

Hence point on x- axis is (2,0)

(ii) When it cut y-axis then

put x = 0 i.,e , 4y = 6

y = 3/2

Hence the point on y-axis is

3x + 4 y = 6

 X 2 -2 6 Y 0 3 -3 ## Fahrenheit (F) and Celcius (C) are two different units of temperature and the relation between them is given by  Draw the graph for this relation. At what temperature both  units read the same.

Let x = y = a ## The auto rikshaw fare in a city is charged RS . 10 for first kilometer and @Rs.4 per kilometer for subsequent distance covered. Write the linear equation to express the above statment. Draw the graph of the linear  equation.

Total distance covered =  x km.

Total fare = y km.

Fare for the first kilometer

Subsequent distance  = (x-1) km

Fare for the subsequent distance = Rs. 4(x-1)

According to the question,

y = 10+4 (x-1)

y = 10+4x -4

Table of solutions

 X 0 1 -1 Y 6 10 2 ## Draw the graph of linear equations x+ y = 10 and 2x-y = 5 and  find the point of intersection

x+y = 10   y = 10-x

 X 0 2 3 4 5 Y 10 8 7 6 5

2x-y = 5      y = 2x -5

 X 0 2 5 Y -5 -1 5

Plot these points on the graph paper. ## Write the equations of lines p and r in the given graph  A student answered the equation of a line q as x+ y  = 1. Did he answer correctly? Also, find the area of lines enclosed between p,q and r. Equation of p is  x = -1

Equation of r is y = -2

Yes, the equation of q is

x+y = 1

Area   =  Sq. units

## Draw the graph of x+y = 3 and 2x+2y = 8 on the same axes. What does the graph of these lines represent.

x +y =3

y = 3-x

 X 0 3 1 Y 3 0 2

2x+2y =8

x+y =4

y =4-x

 X 0 4 1 Y 4 0 3 From the graph, it is clear that line are parallel  to   each other.

## Shade the triangle formed by the graph of  2x- y = 4, x+ y = 2 and the y-axis. Write the coordinates of vertices of the triangle.

2x -y = 4

 X 0 2 1 Y -4 0 -2

 x 0 2 1 Y 2 0 1 From the graph, ABC is the required triangle and its verticle s are A (0,2), B (0,-4), C (2,0).

## Ramesh is driving his car with a uniform speed of  90 km per hour. Draw the time- distance graph on the graph paper. From the graph, find the distance travelled by him in: (i)   hour                            (ii)  hours

Graph of equation y = 90x,

where x be time and y be distance from the graph

(i) Distance travelled in  hour = 45 km

(ii) Distance travelled in 2 hours  = 180 km. ## The taxi fare in a city is as follows: For the kilometre, the fare is Rs.8 and for the subsequent distance, it is Rs. 5 per km. Taking distance covered as x km and the total fare as Rs. y. write a linear equation for this information and draw its graph.

The fare of first km is Rs. 8.00

Let the total distance to be covered is x km.

Fare for (x-1) kilometer at the rate of Rs.5 per km is 5 (x-1)

Total fare   y = 5 (x-1)+8

y =5x-5+8

y = 5x+3

 X 0 -1 -2 Y 3 -2 -7 ## Draw the graph of x =3y -4 Find the  (i) Value of y, when x= -1 (ii) Value of x, when y  = 5

Given,

 X 2 5 8 Y 2 3 4 ## Solve the equation 3x+4 = 5x+8  and represent the solution on (i) the number line (ii) the cartesian plane. What do you get as the representation of the solution on the cartesian plane ? In cartesian planes, how many solutions this equation has?

(i) On the number line the point P (-2,0)

represent the solution (ii) On the Cartesian plane x = -2 is a line parallel to the y-axis at a distance of 2 units to left of it.

It has infinite solutions. ## A student amit of class IX is unable to write in his examination, due to fracture in his arm. Akhil a student of a class VI writes for him. The sum of their ages is  25 years. (i) Write a linear equation for the above situation and represent it graphically.  (ii) Find the age of of Akhil from the graph, when age of Amit is 14 years.

Let  Age of Amit = x years

Age of Akhil =  Y years

(i) According to the question the linear equation  for above situation is

y = 25-x

 X 0 10 15 Y 25 15 10 (ii) From the graph when Amit's age = 14  years, then Akhil's age = 11 years.

## Force applied on a body is directly proportional to the acceleration produced in the body. Write an equation  to express the situation and plot the graph of the equation taking the constant to be 5 units

Let F be the force applied and a be the acceleration produced.

where k is a constant

Replace  a by x and F by y.

y = k(x)

y =  xk

Here, k = 5

y = 5x

 X 0 1 -1 Y 0 5 -5 ## The ratio of girls and boys in a class is  1 : 3. set up an equation between the student of a class  and boys  and then draw its graph. Also find the number of boys in a class of 40 students from the grpah.

Total students in the class  = y

Let the boys in the class = x, then equation between the students and the boys

Now

 X 0 30 60 Y 0 40 80

Now, draw a graph between these points. ## For this equation 3x-5y -15 = 0, find the points where its graph intersects  x and y  axis using these, draw the graph of the equation find its area.

Equation :

3x+ 5y-15 = 0

Substituting y = 0 in the equation

3x-0-15 = 0

x = 5

Hence the point on x- axis is (5,0)

When x = 0  in the equation, we get

Points on y-axis is (0,-3)

Plotting A (5,0) B (0,-3) We get the triangle AOB. Since OA  = 5 units and OB = 3 units

Area of a triangle AOB

## Explain why 13233343563715 is a composite number ?

The given number ends in 5. Hence it is a multiple of 5. Therefore  it is a composite  number.

## a and b are two positive integers such that the least prime factor of a is 3 and the least prime factor of b is 5. Then calculate the least prime factor of (a+b).

a and b are two positive integers such that the least prime factor of a  is 3 and the least prime factor of b is  5. Then least prime factor of (a+b) is 2.

## Find HCF of the numbers given below :  k, 2k, 3k, 4k and 5k, where k is any positive integer .

HCF of    k is k

k.2 is k

k.3 is k

k.22 is k

k.5 is k

## AB and CD are two parallel chords on the same side of the circle. AB = 6 cm, CD= 8 cm. The small chords is at a distance of 4 cm from the centre. At what distance from the centre is the other chord

In   OM = 4 cm, MB = 3 cm

OB= OM2+MB2

= 16+9

= 25 In

= 25-16

= 9

The other chord is at a distance of 3 cm from the centre.

## What is the HCF of the smallest composite number and the smallest prime number ?

The smallest prime number is  2 and the smallest composite number is 22

Hence, required HCF of (22 , 2 ) = 2.

## Calculate the HCF of  33  x 5 and 32 x 52.

HCF of   33 x 5   and   32 x 52

= 32 x 5

= 9 x 5

= 45

## If HCF (a,b) = 12 and a x b  = 1800   then find LCM (a,b).

We know that,

a x b  = HCF  (a,b)  x LCM (a,b)

## Find the HCF of the numbers given below : k, 2k, 3k, 4k, and  5k where k is any positive integer.

HCF of

k

k.2

k.3                       is k .

k.22

k.5

## Using Euclid's algoritham, find the HCF of 240 and 228.

240 = 228 x 1 + 12

228 = 12 x 19 + 0

HCF of 240 and  228= 12

## Given that HCF (306,1,314) = 18. Find LCM (306,1,314).

Given HCF  (306,1,314) = 18.

LCM (306, 1, 314) = ?

Let a = 306

b = 1,314

We know  that

a x b  = LCM (a,b) x HCF  (a,b)

## Complete the following factor tree and find the composite number x. y = 5x 13 = 65

x  = 3 x 195 = 585

## Complete the following factor tree and find the composite number x:  x = 11130

## Complete the following factor tree and find the composite number x:  Composite number, x  = 6762.

## Find the missing numbers a,b,c  and d in the given factor tree: Since 143 = 11 x 13, So, C = 11 or 13

d =13 or 11.

## Explan whether 3  x 12 x 101 + 4 is a prime number or a composite number.

=  a composite number

[Product of more than two factors]