10 Maths Key Concept-2

8/3/2019 10 Maths Key Concept-2

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5. Division Algorithm for Polynomial

For any two polynomials p(x) and g(x); g(x) 0

We can find two polynomials q(x) and r(x) such that p(x) =g(x) x q(x) + r(x)

Where r(x) = 0 or degree of r(x) < degree of g(x)

Chapter 2

1 Mark Questions

1. Verify that one and two are the zeroes of the polynomial p(x) = (x-1) (x

2)

2. Show that 3 is a zero of the polynomial x3 - 8x2 + 8x + 21

3. Verify whether

3

1=x

is a zero of the polynomial p(x) = 3x +1

4. Find the zeroes of the polynomial x2-3

5. Find the zeroes of the quadratic polynomial 4u2 + 8u

2 Mark Questions

Example : Find a quadratic polynomial in which sum of zero = -1 and product of zeros

is4

1

Let the quadratic polynomial be ax2+bx+c with zeroes and

According to given 111 ==

=+a

b

a

b

4

1

4

1==

a

c

Take a = L.C.M. of (1,4) = 4

b=4 and c=

14

4=

so required polynomial is 4x2 + 4x +1


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6. Find a quadratic polynomial each with the given numbers as the sum and

product of its zeroes

503

12 ,)ii(,)i(

7. Prove that x2+6x+15 has no zeroes.

8. Find the zeroes of 3x2-x-4 and verify the relationship between the zeroes and

the co-efficient.

9. Divide 2x2 + 3x+1 by (x+2), find the quotient and the remainder.

10. Check whether the first polynomial is a factor of the second polynomial by division

method.

i) p(x) = x4 -5x + 6; g(x) = 2-x2

ii) p(x) = 42 - 19x -5x2 +2x3; g(x) = (x-2)

3 Marks Questions

Example : On dividing x3 -3x2 + x + 2 by polynomial g(x), the quotient and the remainder

were (x-2) and -2x+4 respectively. Find g(x)

Here p(x) = x3 - 3x2 +x + 2, q (x) = x-2, r(x) = -2x +4 By using division alogorithm.

g(x) x (x-2) + (-2x+4) = x3 - 3x2 + x + 2

g(x) x(x-2) = x3 - 3x2 + x + 2 + 2x - 4

g(x) x (x-2) = x3 - 3x2 + 3x-2


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Dividing x3-3x2+3x-2 by (x-2) we get g(x) = x2-x+1

11. Find all the zeroes of 2x4-3x3 - 3x2 + 6x -2 if two of its zeroes are 22 and

12. Find all the zeroes of the polynomial x3+6-7x if one of its zero is -3.

13. Divide x3-3x2-x+3 by (x+1) and verify the division algorithm.

14. If 5 is a zero of the polynomial x3-6x2+3x+10, find the other two zeroes.

Answers :

Q.4 33 , Q. 5 0, - 2; Q.6 i) 1233 2 + xx ii) 52 +x

Q.8 13

4, Q.9 Quotient 2x-1, Remainder 3

Q.10 (i) No (ii) Yes Q.11 and 1 Q.12 1,2 Q. 14 -1,2,5


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CHAPTER - 2

POLYNOMIAL

2 Marks Questions

Q.1 P(x) = (x-1) (x-2)

One or two are the Zeroes of the polynomial

P(1) = (1-1) (1-2)

= 2121 /+///

= 0

P(2) = (2-1) (2-2)

=2244 /+///

= 0

Q.2 Let x3

- 8x2

+ 8x + 21 = P(n)

Put x=3 in this Equation

P(3)=(3)3 - 8(3)2 + 8 (3) + 21

= 27 - 8 x 9 + 24 + 21

= 27 - 72 + 24 + 21

= 72 - 72

= 0

3 is zero of the polynomial x3 - 8x2 + 8x + 21


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Q.3 P(x) = 3x + 1

Put

3

1=x

in this Equation

13

13

3

1+

=

P

= -1 + 1

= 0

Yes x=3

1 is zero of P(x)

Hence Proved

Q.4 x2 - 3 = 0

x2 = 3

3=x

(Zeroes of the Polynomial)

Q.5 4u2 + 8u = 0

4u ( u+2) = 0

4u= 0 or u + 2 = 0

u = 0 or u = -2

(Zeroes of the quadratic Polynomial)

Q.6 (i)

3

1

2,

Let the quadratic polynomial be ax2 + bx + c with zeroes and

x2 - (sum of roots) x + Product of roots = 0


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22

==+a

b

3

1

3

1==

ac

If a = 3,

3

32x

a

b =

a=3, 23=b and c=1

So required polynomial is 1233 2 +x

ii) 0, 5

Let the quadratic polynomial be ax2+bx+c with zero and

).........(b

11

0

10

=

=+

).........(a

c2

1

55 ==+

By taking equation (1) and (2)

a=1, b=-0, c= 5

Sorequired polynomial is 52 +x

Q.8 3x2 - x - 4

3x2 + 3x - 4x - 4

3x (x+1) - 4 (x+1)

(3x -4) (x+1)

So, the value of 3x2 - x - 4 is zero when x+1 = 0 or 3x - 4 = 0 i.e. when x = -1 or

x3

4=


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therefore, the zeros of 3x2 - x - 4 are -1 and3

4

Sum of zeros =3

1

3

43

3

41 =+=++

23

1

3

1

xofCofficient

)xofCofficient(

a

b==

=

Product of Zeroes =a

cxx =

=

=

3

4

3

41

23

4

xofCofficient

termttanCons=

Q.9 Divide 2x2+3x+1 by (x+2)

1213222 +++ x(xxx

2x2

4x--------------------

- x + 1

x

2------------------

3------------------

quotient = 2x - 1

Remainder = 3

Q.10 P(x)= 42 - 19x - 5x2 + 2x3, g(x) = x-2

(xxxx 4219522 23

2x2-x-21

2x3

4x2

-------------------x2 - 19x

x2

2x---------------------

- 21x + 42

21x

42----------------

0---------------- Yex, g(x) is a factor of P(x)


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Q.11 Since two zeroes are

( ) 22222 2=+ xx)x(,and

is a factor of given

polynomial

13226332222342 ++ xx(xxxxx

2x4

4x2

---------------------------------3x3 + x2 + 6x - 2

3x3+

6x-----------------------------

x2 - 2

x2

2-----------------

0-----------------

So, 2x2 - 3x +1

by splitting the middle term

11222 + xxx

)x()x(x 1112

)X)(x( 112

so, its zeroes are given by

12

1== xandx

Therefore, the Zeroes of the given polynomial are 12

122 and,,

Q.12 x3 + 6 - 7x

if one of its Zero is -3

3

2

1 xofCofficient)xofCofficient(o

ab ===++

31

6

xofCofficient

termttanCons

a

dxx =

=

=


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0=++

03 =++ )..(.......... 13=+ 6= xx

63 +=+ )(

3

6

/

/=

2=

)(.......... 22

=

Put the value of in equation (1)

=+=

+ 3232 2

=+ 322

termmiddletheSplitingBy,0232 =+

022

2

=++( ) ( ) 0212 =

( ) ( )21

=0

So, the value of

232 +

is zero when 01=

or

2

i.e. when

21 == or

Put the value of in Eq. (2)

21

2==

Zeroes of the polynomial are 3= ,

2

1

=

=


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Q.13 Divide

x3 - 3x2 - x + 3 by (x+1)

34331223 +++ xx(xxxx

23 xx

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

xx

xx

44

34

2

2

+

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

33

33

+

x

xquotiend = x2 -4x+3

----------------- Remainder = 0

0

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

By division algorithm,

Dividend = Divisior x quotient + Remainder

(x3-3x2-x+3) = (x+1) (x2-4x+3) + 0

(x3-3x2-x+3) = x3-4x2+3x+x2-4x+3+0

(x3-3x2-x+3) = x3-4x2+x2+3x-4x+3+0

(x3-3x2-x+3) = x3-3x2-x+3

Hence, the division algorithm is verified

Q.14 x3-6x2+3x+10

If 5 is a zero of the polynomial

3

2

1

6

1

6

xofCofficient

)xofCofficient()(

a

b ==

=

=++


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31

10

xofCofficient

termCofficient

a

dxx =

=

=

65

6

=++

=++

)..(..........

xx

)(..........

22

2

5

10

105

10

11

56

=

=

=

=

=

=+

=+Put the value of

in Eq. (1)

12

=

+

1

22

=

022 =

022 = , by spliting the middle term

02122 =+

0212 =+ )()(

)()( 22 + = 0

So, the value of

22

is Zero when 02 = i.e. when

21 == or

Put the value of 2.Eqin


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21

2=

/

/=

Zero of Polynomial are 5= ,

B = -1

2=

Chapter 3

One Mark Questions

Q.1 If x=-1, y=5 is a solution of the equation 4x-3y = 11

Q.2 Write the condition for the following system of linear equations to have a unique

solution.

ax+by = c; px+qy=r

Q.3 Write T or F

(i) -3x+4y = 5;

02

156

2

9=+ y

x

having no solution.

Q.4 Fill up the blank spaces

(i) When l1and l

2are parallel lines then the system has ......................

(ii) When l1and l

2are coincident then the system has .........................

Q.5 (i) Write the standard form of linear equation in two variables.

Ans. : (1) No (2)q

b

p

a (3) True (4) (i) no solution 4(ii) infinite number of solutions

(5) ax+by =c

SOLVED EXAMPLES

Ex. 1 Find the value of K for which the system of equations 3x+5y=0, kx+10y=0 has

a non-zero solution.


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Solution Here,10

53

2

1

2

1 ==b

b,

ka

a

for a non-zero Solution, we have

30510

53

2

1

2

1 kor,k

.e.i,b

b

a

a

6k

Ex.2 Solve the following pair of linear equation :

3x-y=3, 9x-3y=9

Solution 3x-y=3 .........(1)

9x-3y= 9 .......(2)

from (1)

y = 3x -3 ........ (3)

put (3) in (2)

9x - 3(3x-3) = 9

9x - 9x + 9 = 9

9 = 9, which is true hence, the given system of equations has

infinitely many solutions.

Two marks question

Q.1 On comparing the ratios2

1

2

1

b

b,

a

aand

2

1

c

c, find out whether the following pair of

linear equations intersect at a point or parallel or coincident.

2x+3y = 4

3x+5y = 5


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Q.2 Solve the following system of linear equations by substitution method.

5x+2y = 2 and 2x+3y = -8

Q.3 Solve the following pair of linear equations by elimination method.

5x + 10 y = 28

15x=20y -121

Q.4 Check whether the following system of equations has unique solution, no

solution or infinitely many soultion.

4x-3y=1

x - 2y = 4

Ans. (1) consistent at a point

(2) x = 2, y = 4

(3)10

41

5

13 == yx

(4) unique solution

Three Marks questions

Q.1 Draw the graph of the equations 4x - y = 4 and 4x + y=12

Determine the vertices of the triangle formed by the lines, representing these

equations, and the x-axis. Shade the triangular region so formed.


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Q.2 For what value of k, will the system of equations :

x+2y = 5

and, 3x+ky -15=0 has (i) a unique solution (ii) no solution.

Q.3 Find two numbers whose sum is 28 and seven times their difference is equal to

four times of their sum.

Q.4 Solve : 5x + 2y = 2 and 2x+3y = -8 and hence find the value of m for which

y=mx+4

Ans. (1) (1,0), (2,4) and (3,0)

(2) (i) 6k (ii) no value of k

(3) 22 and 6(4) x=2, y = -4, m= -4

Six Marks Questions

Q.1 Solve the following pair of equations by reducing them to a pair of linear

equations :

6

13

2

2

3

12

3

1

2

1=+=+

yx;

yx

Q.2 A bag contains 94 coins of 50 paise and 25 paise denominations. If the total

worth of these coins be Rs. 29.75. Find the number of coins in each kind.

Q.3 A boat goes 24km. upstream and 28km. downstream in 6hrs. It goes 30 km

upstream and 21km. downstream in 6 hrs. Find the speed of the boat in still

water and also speed of stream.


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Q.4 Solve the following systems of equations :

62 ==+xy

yx,xy

yx

Ans. (1)3

1

2

1== y,x

(2) No. of 50 paise coins = 25

No. of 25 paise coins = 69

(3) Speed of stream = 4km/hr

Speed of boat = 10 km/hr

(4)4

1

2

1== y;x


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Chapter 3

Linear Equations in two variables

Q.1 4x - 3y = 11

x = -1, y = 5

4(-1) - 3(5) = 11

-4 - 15 = 11

-19 11

x = -1, y = 5 is not a soltuion of the equation

Q.2 ax + by = c

a1

= a, b1

= b, c1

= c

px + qy = n

a2

= p b2=q c

2=n

r

c

q

b

p

a

c

c

b

b

a

a

2

1

2

1

2

1

Q.3 -3x + 4y = 5;2

15

2

9+byx

having no solution = True

Q.4 i) When l1 and l2 are parallel lines the system has no solution.

ii) When l1and l

2are coincident then the system has infinte many solution.


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Q.5 a1x + b

1y +c

1=0

a2x + b2y+c2=0

Q.1 2x + 3y = 4

a1

= 2 b1

= 3 c1=4

3x + 5y = 5

a2

= 3 b2

= 5 c2=5

5

4

5

3

3

2

2

1

2

1

2

1

=== c

c

b

b

a

a

2

1

2

1

2

1

c

c

b

b

a

a

linear equations intersect at a point.

Q.2 By substitution method

5x + 2y = 2

2x + 3y = -8

5x + 2y = 2

5x = 2 - 2y

5

22 yx

=

2x + 3 y = -8

835

22

2 =+

y

y

835

44=+

y

y

5

401544 =+ yy


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11y = -40 - 4

11y = -44

411

44=

=y

5

22 yx

=

5

422 )(x

=

5

82+=x

25

10==x

Q.3 By elimination method

5x + 10 y = 28 (1), 15 x = 20y -121

15x - 20y = -121 ........... (2)

15 x 5x + 10y = 28

5 x 15x -20y = -121

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

75x + 150y = 420

75x

100y =

605

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

250y = 1025

y =

10

41

250

1025=

75x + 150y = 420

75x + 150

10

41=420


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75x + 615 = 420

75x = 420 - 615

75x = -195

75

195=x

5

13=x

Q.4 4x - 3y = 1

a1 = 4 b1 = -3 c1 = 1

x -2y = 4

a2

= 1 b2

= -2 c2=4

4

1

2

3

1

4

2

1

2

1

2

1 =

==

c

c

b

b

a

a

2

1

2

1

2

1

c

c

b

b

a

a

Unique Solution

Q.2 1) x + 2y = 5

3x + ky = -15

a1

= 1 b1

= 2 c1=5

a2

=3 b2=k c

2=-15

For unique solution

2

1

2

1

2

1

c

c

b

b

a

a


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15

52

3

1

k

k

2

3

1

6k

For no solution

x + 2y = 5

a1

= 1 b1= 2 c

1=5

3x + ky = -15

a2

= 3 b2

= k c2= -15

2

1

2

1

2

1

c

c

b

b

a

a=

15

52

3

1

=

k

k

2

3

1=

6=k

15

52

k

305 k

65

305 =

k

6k

Q.4 5x + 2y = 2

2x + 3y = -8


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By Substitution method

5x = 2 - 2y

5

22 yx

=

835

222 =+

y

y

835

44=+

y

y

4 - 4y + 15y = -40

11y = -40-4

11y = -44

411

44=

=y

5

22 yx

=

5

422 )(x

=

5

82 +=x

25

10==x

y = mx + 4

-4 = m(2) + 4

-4 -4 = 2m

-8 = 2m

42

8==

=m


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Q.1 23

1

2

1=+

yx

Let by

,ax

==11

232=+

ba

3a + 2b = 12 ............ (1)

by

ax

Let ==11

6

13

2

2

3=+

ba

2a + 3b = 13 .............(2)

Multiply 1 by 2 and 2 by 3.

2 x (3a + 2b = 12)

3 x (2a + 3b = 13)

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

6a + 4b = 24

6a

9b =

39-------------------------

-5b = - 15

35

15==b

6a + 4 (3) = 246a = 24 - 12

26

12==a

111==

ya

x

1

31

1

21==

yx

2x = 1 3y = 1

3

1

2

1== yx


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Q.4 2=+

xy

yx

x + y = 2xy.............(1)

6=

xy

yx

x-y = 6xy ...............(2)

But Elimination Method :

x + y = 2xy

x - y = 6xy

----------------2x = 8xy

2 = 8y

4

1

8

2==y

4

12

4

1==+ xxx

xx2

1

4

1=+

4

1

2

1 = xx

4

1

2

=

x

24 = x

2

1= x

Q.3 Let Ist no. = n

Another no = 28-n

7 [x-(28-x)] = 4(28)


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14x - 196 = 112

14x = 196 + 112

14x = 308

14

308=x

x = 22

1st no. = x = 22

Another no = 28 - x = 28 - 22 = 6

Q.1 4x - y = 4 4x + y = 12

4

4 yx

+=

4

12 yx

=

x 1 2 3 x 3 2 1

y 0 4 8 y 0 4 8

Ans. vertices = (1,0), (2,4) & (3,0)

8 (1,8) (3,8)

6

4 (2,4)

2

(1,0) (3,0)

0 2 4 6 8 x axis


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Q.2 Let 50 paise coins = x

Let 25 paise coins = y

Acc to Ques. x+y = 94 ................ (1)

Total 50 paise coins =2100

50 x.Rs

x=

Total 25 paise coins =

4100

25 y.Rs

y=

Acc. to Ques.

752942

.y

.x

=+

2x + y = 119 .......................... (2)

By Elimination method

2 x ( x + y = 94)

1 x ( 2x + y = 119)

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

2x + 2y = 118

2xy = 119

-----------------------------------------y = 69

x + y = 94

x + 69 = 94

x = 94 - 69

x = 25

No. of 50 paise coins = x = 25

No. of 25 paise coins = y = 69


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Q.3 Let speed of boat = x km/hr

Let speed of stream = y km/hr

Speed of boat down stream = x+y

Speed of boat upstream = x - y

Case I

Time of boat in downstream =

yx +

28

Time of boat in upstream =yx

24

Acc. to Question 62428

=

++ yxyx

Let byx

ayx

=

=+

11

28a + 24b = 6 ............................(1)

Case II Time of boat in downstream yx +=

21

Time of boat in upstream =yx

=30

Acc. to Qus.2

16

3021=

+

+ yxyx

Let byx

ayx

=

=+

11

21a + 30b = 2

13

42a + 60b = 13 .................. (2)


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60 x (28a + 24b = 6)24 x (42a + 60b = 13)

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

1680a + 144b = 360

1008a

144b =

312

------------------------------------672a = 48

42

3

672

48==a

42

3=a

28a + 24b = 6

62442

328 =+

b

62421

42=+ b

21

42624 =b

21

4212624

=b

24

1

21

84xb=

21

7=b

byx

ayx

=

=+

11

21

71

42

31=

=

+ yxyx


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3x + 3y = 42...(3) 7x - 7y = 21 ......(4)


7 x (3x + 3y = 42)

3 x (7x - 7y = 21)

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

21x + 21y = 294

21x

21y =

126

------------------------------------42y = 168

442

168==y

3x + 3y = 42

3x + 3(4) = 42

3x+12 = 42

3x = 42-12

3

30=x

x=10

Speed of boat = x = 10km/h

Speed of Stream = y = 4km/h

0

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10 Maths Key Concept-2