mixture: October 2020

Friday, October 30, 2020

(v) A lending library has a fixed charge for the first three days and an additional charge for each day thereafter. Saritha paid Rs.27 for a book kept for seven days, while Susy paid Rs.21 for the book she kept for five days. Find the fixed charge and the charge for each extra day.

cbse ncert 10th mathematics chapter 3 exercise 3.4 pair of linear equations in two variables

2. Form the pair of linear equations in the following problems, and find their solutions (if they exist) by the elimination method :

(v) A lending library has a fixed charge for the first three days and an additional charge for each day thereafter. Saritha paid Rs.27 for a book kept for seven days, while Susy paid Rs. 21 for the book she kept for five days. Find the fixed charge and the charge for each extra day.

let x = fixed charge for the first 3 days

and y = charge for each extra day

Saritha paid Rs.27 for a book kept for seven days

7 - 3 = 4 extra days

x + 4y = 27--------------------(1)

Susy paid Rs. 21 for the book she kept for five days

5-3 = 2 extra days

x + 2y = 21-----------------------(2)

elimination method

x + 4y = 27--------------------(1)

x + 2y = 21-----------------------(2)

--------------------------------------------------- subtracting

2y = 6

y = 6/2

y=3

substitute in x + 2y = 21

x + 2(3) = 21

x = 21-6

x=15

fixed charge for the first 3 days ,x = Rs.15/-

charge for each extra day , y = Rs.3/-

(iv) Meena went to a bank to withdraw Rs.2000. She asked the cashier to give her
Rs. 50 and Rs.100 notes only. Meena got 25 notes in all. Find how many notes of
Rs.50 and Rs.100 she received.

let x = number of 50 rupee notes

and y = number of 100 rupee notes

Meena got 25 notes

x+y=25--------------------(1)

Meena went to withdraw Rs.2000

using value of the notes

50x+100y =2000

divide each term with 50

x +2y =40 --------------------(2)

using elimination method

x+y=25--------------------(1)

x +2y =40 --------------------(2)

---------------------------------------------------subtracting

(-1)y=(-15)

y = 15

substitute in x+y=25

x +15 =25

x = 25 - 15

x = 10

so she got 10 fifty rupee notes and 15 hundred notes

=================================================

ncert cbse 10th mathematics chapter 3 optional exercise 3.7

The ages of two friends Ani and Biju differ by 3 years. Ani’s father is twice as old as Ani and Biju is twice as old as his sister Cathy. The ages of Cathy and Ani’s father differ by 30 years. Find the ages of Ani and Biju

solution

2. One says, “Give me a hundred, friend! I shall then become twice as rich as you”. The other replies, “If you give me ten, I shall be six times as rich as you”. Tell me what is the amount of their (respective) capital?

solution

3. A train covered a certain distance at a uniform speed. If the train would have been 10 km/h faster, it would have taken 2 hours less than the scheduled time. And, if the train were slower by 10 km/h; it would have taken 3 hours more than the scheduled time. Find the distance covered by the train.

solution

4. The students of a class are made to stand in rows. If 3 students are extra in a row, there would be 1 row less. If 3 students are less in a row, there would be 2 rows more. Find the number of students in the class.

solution

5. In a ∆ ABC, ∠ C = 3 ∠ B = 2 (∠ A + ∠ B). Find the three angles.

solution

Solve the following pair of linear equations:

px + qy = p – q

qx – py = p + q

solution

(ii) ax + by = c
bx + ay = 1 + c

solution

(iii)

(x/a) -(y/b) = 0

ax +by = (a^2) + (b^2)

solution

(iv)

(a – b)x + (a + b) y = (a^2) – 2ab – (b^2)

(a + b)(x + y) = (a^2) + (b^2 )

solution

(v)

152x – 378y = – 74

–378x + 152y = – 604

solution

ABCD is a cyclic quadrilateral Find the angles of the cyclic quadrilateral,

if angles are A =(4y+20) , B =(3y-5) , C=(-4x), D=(-7x+5)

solution

exercise 3.6

Ritu can row downstream 20 km in 2 hours, and upstream 4 km in 2 hours. Find her speed of rowing in still water and the speed of the current.

solution

(ii) 2 women and 5 men can together finish an embroidery work in 4 days, while 3 women and 6 men can finish it in 3 days. Find the time taken by 1 woman alone to
finish the work, and also that taken by 1 man alone.

solution

(iii)

Roohi travels 300 km to her home partly by train and partly by bus. She takes 4 hours if she travels 60 km by train and the remaining by bus. If she travels 100 km by train and the remaining by bus, she takes 10 minutes longer. Find the speed of the train and the bus separately.

solution

exercise 3.5

A part of monthly hostel charges is fixed and the remaining depends on the
number of days one has taken food in the mess. When a student A takes food for
20 days she has to pay Rs.1000 as hostel charges whereas a student B, who takes
food for 26 days, pays Rs.1180 as hostel charges. Find the fixed charges and the
cost of food per day.

solution

(ii) A fraction changes to (1/3) when 1 is subtracted from the numerator and it changes to (1/4) when 8 is added to its denominator. Find the fraction.

solution

(iii)Y scored 40 marks in a test, getting 3 marks for each right answer and losing 1 mark for each wrong answer. Had 4 marks been awarded for each correct answer and 2 marks been deducted for each incorrect answer, then Yash would have scored 50 marks. How many questions were there in the test?

solution

(iv) Places A and B are 100 km apart on a highway. One car starts from A and another from B at the same time. If the cars travel in the same direction at different speeds,they meet in 5 hours. If they travel towards each other, they meet in 1 hour. What are the speeds of the two cars?

solution

(v) The area of a rectangle gets reduced by 9 square units, if its length is reduced by 5 units and breadth is increased by 3 units. If we increase the length by 3 units and the breadth by 2 units, the area increases by 67 square units. Find the dimensions of the rectangle

solution

3. Solve the following pair of linear equations by the substitution and cross-multiplication methods :

8x + 5y = 9
3x + 2y = 4

solution

(i) For which values of a and b does the following pair of linear equations have an
infinite number of solutions?
2x + 3y = 7
(a – b) x + (a + b) y = 3a + b – 2

solution

(ii) For which value of k will the following pair of linear equations have no solution?

3x + y = 1

(2k – 1) x + (k – 1) y = 2k + 1

solution

Which of the following pairs of linear equations has unique solution, no solution, or infinitely many solutions. In case there is a unique solution, find it by using cross multiplication method.

x – 3y – 3 = 0
3x – 9y – 2 = 0

solution

(ii)

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

solution

(iii)

3x – 5y = 20
6x – 10y = 40

solution

(iv)

x – 3y – 7 = 0
3x – 3y – 15 = 0

solution

exercise 3.4

2. Form the pair of linear equations in the following problems, and find their solutions (if they exist) by the elimination method :

solution

(iv) Meena went to a bank to withdraw Rs.2000. She asked the cashier to give her
Rs. 50 and Rs.100 notes only. Meena got 25 notes in all. Find how many notes of
Rs.50 and Rs.100 she received.

solution

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Wednesday, October 28, 2020

(iii) 3x – 5y = 20 6x – 10y = 40

cbse ncert 10th mathematics chapter 3 exercise 3.5pair of linear equations in two variables

Which of the following pairs of linear equations has unique solution, no solution, or infinitely many solutions. In case there is a unique solution, find it by using cross multiplication method.

(iii)

3x – 5y = 20
6x – 10y = 40

re writing the equations

3x-5y-20=0

6x-10y-40=0

a1=3 , b1 =(-5) c1 = (-20)

a2 =6 b2 = (-10) c2 = (-40)

a1/a2 = 3/6=1/2

b1/b2 = (-5)/(-10) =1/2

c1/c2 =(-20)/(-40) =1/2

therefore

a1/a2 =b1/b2 =c1/c2

therefore the system has infinitely many solutions

put y =t

use first equation

3x-5(t)-20=0

x = (20+5t)/3

solution is

x = (20+5t)/3

y =t

where t is any real number

(iv)

x – 3y – 7 = 0
3x – 3y – 15 = 0

a1=1 , b1 =(-3) c1 = (-7)

a2 =3 b2 = (-3) c2 = (-15)

a1/a2 = 1/3

b1/b2 = (-3)/(-3) =1

a1/a2 =/=(not equal) b1/b2

so system has unique solution

use cross multiplication method

x y 1

(-3 ) (-7) 1 (-3)

(-3) (-15) 3 (-3)

x / [(-3)(-15) - (-3)(-7)] = y / [(-7)(3)- (-15)(1)] = 1/[1(-3) - 3(-3)]

x /[45-21] = y/[-21+15] = 1/[-3+9]

x/24 = y/[-6] = 1/6

using first and last

x/24 =1/6

x = 24/6

x=4

using first and third

y/[-6] = 1/6

y = (-6)/6

y=(-1)

=================================================

ncert cbse 10th mathematics chapter 3 optional exercise 3.7

solution

5. In a ∆ ABC, ∠ C = 3 ∠ B = 2 (∠ A + ∠ B). Find the three angles.

solution

Solve the following pair of linear equations:

px + qy = p – q

qx – py = p + q

solution

(ii) ax + by = c
bx + ay = 1 + c

solution

(iii)

(x/a) -(y/b) = 0

ax +by = (a^2) + (b^2)

solution

(iv)

(a – b)x + (a + b) y = (a^2) – 2ab – (b^2)

(a + b)(x + y) = (a^2) + (b^2 )

solution

(v)

152x – 378y = – 74

–378x + 152y = – 604

solution

ABCD is a cyclic quadrilateral Find the angles of the cyclic quadrilateral,

if angles are A =(4y+20) , B =(3y-5) , C=(-4x), D=(-7x+5)

solution

exercise 3.6

Ritu can row downstream 20 km in 2 hours, and upstream 4 km in 2 hours. Find her speed of rowing in still water and the speed of the current.

solution

(iii)

solution

exercise 3.5

solution

(ii) A fraction changes to (1/3) when 1 is subtracted from the numerator and it changes to (1/4) when 8 is added to its denominator. Find the fraction.

solution

(iii)Y scored 40 marks in a test, getting 3 marks for each right answer and losing 1 mark for each wrong answer. Had 4 marks been awarded for each correct answer and 2 marks been deducted for each incorrect answer, then Yash would have scored 50 marks. How many questions were there in the test?

solution

(iv) Places A and B are 100 km apart on a highway. One car starts from A and another from B at the same time. If the cars travel in the same direction at different speeds,they meet in 5 hours. If they travel towards each other, they meet in 1 hour. What are the speeds of the two cars?

solution

3. Solve the following pair of linear equations by the substitution and cross-multiplication methods :

8x + 5y = 9
3x + 2y = 4

solution

(i) For which values of a and b does the following pair of linear equations have an
infinite number of solutions?
2x + 3y = 7
(a – b) x + (a + b) y = 3a + b – 2

solution

(ii) For which value of k will the following pair of linear equations have no solution?

3x + y = 1

(2k – 1) x + (k – 1) y = 2k + 1

solution

Which of the following pairs of linear equations has unique solution, no solution, or infinitely many solutions. In case there is a unique solution, find it by using cross multiplication method.

x – 3y – 3 = 0
3x – 9y – 2 = 0

solution

(ii)

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

solution

(iii)

3x – 5y = 20
6x – 10y = 40

solution

(iv)

x – 3y – 7 = 0
3x – 3y – 15 = 0

solution

Tuesday, October 27, 2020

Which of the following pairs of linear equations has unique solution, no solution, or infinitely many solutions. In case there is a unique solution, find it by using cross multiplication method. x – 3y – 3 = 0 3x – 9y – 2 = 0

cbse ncert 10th mathematics chapter 3 exercise 3.5pair of linear equations in two variables

Which of the following pairs of linear equations has unique solution, no solution, or infinitely many solutions. In case there is a unique solution, find it by using cross multiplication method.

x – 3y – 3 = 0
3x – 9y – 2 = 0

a1=1 , b1 =(-3) c1 = (-3)

a2 =3 b2 = (-9) c2 = (-2)

a1/a2 = 1/3

b1/b2 = (-3)/(-9) =1/3

c1/c2 =(-3)/(-2) =3/2

a1/a2 =b1/b2 =/= (not equal )c1/c2

system has no solution

(ii)

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

rewrite the equations

2x+1y-5=0

3x+2y-8=0

a1=2 , b1 =1 c1 = (-5)

a2 =3 b2 = 2 c2 = (-8)

a1/a2 = 2/3

b1/b2 = 1/2

a1/a2 =/= (not equal) b1/b2

system has unique solution

use cross multiplication method

x y 1

1 (-5) 2 1

2 (-8) 3 2

x / [1(-8) - 2(-5)] = y / [(-5)3 - (-8)2] = 1 / [2*2 -3*1]

x /[-8+10] = y / [-15+16] = 1 / [4-3]

x/2 =y/1=1/1

using the first and last

x/2 = 1/1

x =2

using the second and last

y/1 = 1/1

y=1

=================================================

ncert cbse 10th mathematics chapter 3 optional exercise 3.7

solution

5. In a ∆ ABC, ∠ C = 3 ∠ B = 2 (∠ A + ∠ B). Find the three angles.

solution

Solve the following pair of linear equations:

px + qy = p – q

qx – py = p + q

solution

(ii) ax + by = c
bx + ay = 1 + c

solution

(iii)

(x/a) -(y/b) = 0

ax +by = (a^2) + (b^2)

solution

(iv)

(a – b)x + (a + b) y = (a^2) – 2ab – (b^2)

(a + b)(x + y) = (a^2) + (b^2 )

solution

(v)

152x – 378y = – 74

–378x + 152y = – 604

solution

ABCD is a cyclic quadrilateral Find the angles of the cyclic quadrilateral,

if angles are A =(4y+20) , B =(3y-5) , C=(-4x), D=(-7x+5)

solution

exercise 3.6

Ritu can row downstream 20 km in 2 hours, and upstream 4 km in 2 hours. Find her speed of rowing in still water and the speed of the current.

solution

(iii)

solution

exercise 3.5

solution

(ii) A fraction changes to (1/3) when 1 is subtracted from the numerator and it changes to (1/4) when 8 is added to its denominator. Find the fraction.

solution

(iii)Y scored 40 marks in a test, getting 3 marks for each right answer and losing 1 mark for each wrong answer. Had 4 marks been awarded for each correct answer and 2 marks been deducted for each incorrect answer, then Yash would have scored 50 marks. How many questions were there in the test?

solution

(iv) Places A and B are 100 km apart on a highway. One car starts from A and another from B at the same time. If the cars travel in the same direction at different speeds,they meet in 5 hours. If they travel towards each other, they meet in 1 hour. What are the speeds of the two cars?

solution

3. Solve the following pair of linear equations by the substitution and cross-multiplication methods :

8x + 5y = 9
3x + 2y = 4

solution

(i) For which values of a and b does the following pair of linear equations have an
infinite number of solutions?
2x + 3y = 7
(a – b) x + (a + b) y = 3a + b – 2

solution

(ii) For which value of k will the following pair of linear equations have no solution?

3x + y = 1

(2k – 1) x + (k – 1) y = 2k + 1

solution

Which of the following pairs of linear equations has unique solution, no solution, or infinitely many solutions. In case there is a unique solution, find it by using cross multiplication method.

x – 3y – 3 = 0
3x – 9y – 2 = 0

solution

(ii)

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

solution

(ii) For which value of k will the following pair of linear equations have no solution? 3x + y = 1 (2k – 1) x + (k – 1) y = 2k + 1

cbse ncert 10th mathematics chapter 3 exercise 3.5pair of linear equations in two variables

(ii) For which value of k will the following pair of linear equations have no solution?
3x + y = 1

(2k – 1) x + (k – 1) y = 2k + 1

rewrite the equations as

3x +1y -1 =0

(2k – 1) x + (k – 1) y -(2k+1) = 0

a1=3, b1=1,c1=(-1)

a2=(2k-1), b2=(k-1), c2 =[-(2k+1)]

condition for system to have no solution is

[a1 /a2] = [b1/b2] =/=(not equal) [c1/c2]

[3/(2k – 1)]=[1/(k – 1)] =/=(not equal) [(-1)/ {-(2k+1)}]

using the first and second

[3/(2k – 1)]=[1/(k – 1)]

3(k-1) =(2k-1)

3k-3=2k-1

3k-2k =3-1

k =2

verify with the third expression

[3/(2k – 1)]=[1/(k – 1)] =/=(not equal) [(-1)/ {-(2k+1)}]

changes to

3/[2(2)-1] =[1/(2-1)]=/=(not equal) [(-1)/{-(2(2)+1)}]

3/3 = 1/1 =/=(not equal)[1/5]

1 = 1 =/=(not equal) [1/5]

is valid

so answer is k=2

(i) For which values of a and b does the following pair of linear equations have an
infinite number of solutions?
2x + 3y = 7
(a – b) x + (a + b) y = 3a + b – 2

rewrite the equations

2x +3y -7 =0

(a – b) x + (a + b) y -(3a+b-2)=0

a1=2 , b1 =3 , c1 =(-7)

a2 =(a – b), b2 =(a + b) c2 = {-(3a+b-2)}

condition for the system to have infinite number of solutions is

[a1 /a2] = [b1/b2] = [c1/c2]

[2/(a-b)] =[3/(a+b)] =[(-7)/{-(3a+b-2)}]

using first and second expression

[2/(a-b)] =[3/(a+b)]

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

2a+2b=3a-3b

2a-3a +2b+3b =0

-a +5b=0------------------------------(1)

[2/(a-b)] =[3/(a+b)] =[(-7)/{-(3a+b-2)}]

now using the first and third expression

[2/(a-b)] =[(-7)/{-(3a+b-2)}] cancelling off (-)

2 (3a+b-2) = 7(a-b)

6a+2b-4 = 7a-7b

6a-7a +2b+7b =4

-1a +9b =4--------------------(2)

-a +5b=0---------------------------(1)

-1a +9b =4--------------------(2)

-----------------------------------------------subtracting

-4b = -4

b = (-4)/(-1)

b=1

substitute in -a +5b=0

a =5b = 5(1) =5

=================================================

ncert cbse 10th mathematics chapter 3 optional exercise 3.7

solution

5. In a ∆ ABC, ∠ C = 3 ∠ B = 2 (∠ A + ∠ B). Find the three angles.

solution

Solve the following pair of linear equations:

px + qy = p – q

qx – py = p + q

solution

(ii) ax + by = c
bx + ay = 1 + c

solution

(iii)

(x/a) -(y/b) = 0

ax +by = (a^2) + (b^2)

solution

(iv)

(a – b)x + (a + b) y = (a^2) – 2ab – (b^2)

(a + b)(x + y) = (a^2) + (b^2 )

solution

(v)

152x – 378y = – 74

–378x + 152y = – 604

solution

ABCD is a cyclic quadrilateral Find the angles of the cyclic quadrilateral,

if angles are A =(4y+20) , B =(3y-5) , C=(-4x), D=(-7x+5)

solution

exercise 3.6

Ritu can row downstream 20 km in 2 hours, and upstream 4 km in 2 hours. Find her speed of rowing in still water and the speed of the current.

solution

(iii)

solution

exercise 3.5

solution

(ii) A fraction changes to (1/3) when 1 is subtracted from the numerator and it changes to (1/4) when 8 is added to its denominator. Find the fraction.

solution

(iii)Y scored 40 marks in a test, getting 3 marks for each right answer and losing 1 mark for each wrong answer. Had 4 marks been awarded for each correct answer and 2 marks been deducted for each incorrect answer, then Yash would have scored 50 marks. How many questions were there in the test?

solution

(iv) Places A and B are 100 km apart on a highway. One car starts from A and another from B at the same time. If the cars travel in the same direction at different speeds,they meet in 5 hours. If they travel towards each other, they meet in 1 hour. What are the speeds of the two cars?

solution

3. Solve the following pair of linear equations by the substitution and cross-multiplication methods :

8x + 5y = 9
3x + 2y = 4

solution

(i) For which values of a and b does the following pair of linear equations have an
infinite number of solutions?
2x + 3y = 7
(a – b) x + (a + b) y = 3a + b – 2

solution

(ii) For which value of k will the following pair of linear equations have no solution?

3x + y = 1

(2k – 1) x + (k – 1) y = 2k + 1

solution

Monday, October 26, 2020

(v) The area of a rectangle gets reduced by 9 square units, if its length is reduced by 5 units and breadth is increased by 3 units. If we increase the length by 3 units and the breadth by 2 units, the area increases by 67 square units. Find the dimensions of the rectangle

cbse ncert 10th mathematics chapter 3 exercise 3.5pair of linear equations in two variables

let x = length of the rectangle

and y = breadth of the rectangle

original area = length *breadth

original area =xy

area of a rectangle gets reduced by 9 square units, if its length is reduced by 5 units and breadth is increased by 3 units

new length =(x-5)

new breadth = (y+3)

new area =(new length) * (new breadth) =(x-5) (y+3) =xy+3x-5y-15

area of a rectangle gets reduced by 9 means

new area = original area -9

xy+3x-5y-15 = xy -9

3x -5y = -9 +15

3x -5y =6----------------------(1)

If we increase the length by 3 units and the breadth by 2 units, the area increases by 67 square units

increase the length by 3 units and the breadth by 2 units means

new length =(x+3); new breadth = (y+2)

area =(x+3)(y+2)=xy+2x+3y+6

area increases by 67 square units means

xy+2x+3y+6 = xy+67

2x +3y =67-6

2x+3y=61------------------(2)

3x -5y =6----------------------(1) *2

2x+3y=61-----------------------(2) *3

--------------------------------------------------------eliminate x

6x-10y=12----------------------(3)

6x+9y=183------------------(4)

--------------------------------------------------------------------subtracting

-19y=-171

y =(-171)/(-19)

y =9

substitute in 3x -5y =6

3x -5(9) =6

3x = 6+45

3x =51

x=51/3

x=17

length =17 units

breadth = 9 units

3. Solve the following pair of linear equations by the substitution and cross-multiplication methods :

8x + 5y = 9
3x + 2y = 4

substitution method

8x + 5y = 9-----------------(1)
3x + 2y = 4----------------(2)

using (2)

3x = 4 -2y

x = [(4-2y)/3]----------------(4)

substitute (4) in (1)

8 [(4-2y)/3] + 5y = 9

[32-16y +15y ]/3 =9

32-1y = 27

y =32-27

y =5

substitute y=5 in (4)

x = [(4-2(5))/3]

x=(4-10) /3

x = (-6)/3

x=(-2)

cross multiplication method

remember to make RHS zero before starting

8x+5y-9=0

3x+2y-4=0

x y 1

5 (-9) 8 5

2 (-4) 3 2

x / [5(-4)-2(-9)] = y / [(-9)(3)-(-4)(8)]= 1/[8(2)-3(5)]

x/[(-20)+18] = y/ [-27+32] = 1/ [16-15]

x / [-2] = y / [5] = 1/1

using first and third

x / [-2] =1

x = (-2)

using second and third

y / [5] = 1/1

y = 5

=================================================

ncert cbse 10th mathematics chapter 3 optional exercise 3.7

solution

5. In a ∆ ABC, ∠ C = 3 ∠ B = 2 (∠ A + ∠ B). Find the three angles.

solution

Solve the following pair of linear equations:

px + qy = p – q

qx – py = p + q

solution

(ii) ax + by = c
bx + ay = 1 + c

solution

(iii)

(x/a) -(y/b) = 0

ax +by = (a^2) + (b^2)

solution

(iv)

(a – b)x + (a + b) y = (a^2) – 2ab – (b^2)

(a + b)(x + y) = (a^2) + (b^2 )

solution

(v)

152x – 378y = – 74

–378x + 152y = – 604

solution

ABCD is a cyclic quadrilateral Find the angles of the cyclic quadrilateral,

if angles are A =(4y+20) , B =(3y-5) , C=(-4x), D=(-7x+5)

solution

exercise 3.6

Ritu can row downstream 20 km in 2 hours, and upstream 4 km in 2 hours. Find her speed of rowing in still water and the speed of the current.

solution

(iii)

solution

exercise 3.5

solution

(ii) A fraction changes to (1/3) when 1 is subtracted from the numerator and it changes to (1/4) when 8 is added to its denominator. Find the fraction.

solution

(iii)Y scored 40 marks in a test, getting 3 marks for each right answer and losing 1 mark for each wrong answer. Had 4 marks been awarded for each correct answer and 2 marks been deducted for each incorrect answer, then Yash would have scored 50 marks. How many questions were there in the test?

solution

(iv) Places A and B are 100 km apart on a highway. One car starts from A and another from B at the same time. If the cars travel in the same direction at different speeds,they meet in 5 hours. If they travel towards each other, they meet in 1 hour. What are the speeds of the two cars?

solution

3. Solve the following pair of linear equations by the substitution and cross-multiplication methods :

8x + 5y = 9
3x + 2y = 4

solution

Friday, October 23, 2020

Y scored 40 marks in a test, getting 3 marks for each right answer and losing 1 mark for each wrong answer. Had 4 marks been awarded for each correct answer and 2 marks been deducted for each incorrect answer, then Yash would have scored 50 marks. How many questions were there in the test?

cbse ncert 10th mathematics chapter 3 exercise 3.5pair of linear equations in two variables

(iii) Y scored 40 marks in a test, getting 3 marks for each right answer and losing 1mark for each wrong answer. Had 4 marks been awarded for each correct answer and 2 marks been deducted for each incorrect answer, then Y would have scored 50 marks. How many questions were there in the test?

let x =number of correct answers

and y =number of wrong answers

40 marks in a test, getting 3 marks for each right answer and losing 1mark for each wrong answer means

3x-1y=40 ------------------------------------(1)

50 marks , had 4 marks been awarded for each correct answer and 2 marks been deducted for each incorrect answer means

4x -2y =50

dividing by 2

2x -1y =25 ------------------------(2)

3x-1y=40 ------------------------------------(1)

2x -1y =25 ------------------------(2)

---------------------------------------------------------subtracting

x =15

substitute in 3x-1y=40

3(15) - y =40

y =45-40

y=5

so x =15 correct answers

and y =5 wrong answers

Total number of questions = x+y = 15 +5 =20

(iv) Places A and B are 100 km apart on a highway. One car starts from A and another from B at the same time. If the cars travel in the same direction at different speeds,they meet in 5 hours. If they travel towards each other, they meet in 1 hour. What are the speeds of the two cars?

Let x =speed of the car starting at A in km/hr

and y = speed of the car starting at B in km/hr

If the cars travel in the same direction at different speeds,they meet in 5 hours.

distance = speed *time

so distance travelled by the cars in 5 hours (in the same direction) are

5x km and 5y km respectively

Let the cars meet at a point P when they are travelling in the same direction

Assume that x >y which means that the first car is faster and travels a longer distance in the 5 hours

100km

*--------------------------*---------------------------------

A B P

Clearly AP = 5x and BP =5y

AP -BP = 100 km using the figure

5x - 5y =100

divide by 5

x -y =20-------------------------(1)

If they travel towards each other, they meet in 1 hour

let the cars meet at Q

As before the first car travels 1x km and the second car travels 1y km

100km

*----------------*---------*

A Q B

AQ=1x

BQ=1y

AQ+BQ = 100 using the figure

1x + 1y =100 ----------------------------(2)

x - y =20-------------------------(1)

x +y =100 ----------------------------(2)

-----------------------------------------------------------------adding

2x =120

x =60 km/hr

substitute in x +y =100

y =40 km/hr

=================================================

ncert cbse 10th mathematics chapter 3 optional exercise 3.7

solution

5. In a ∆ ABC, ∠ C = 3 ∠ B = 2 (∠ A + ∠ B). Find the three angles.

solution

Solve the following pair of linear equations:

px + qy = p – q

qx – py = p + q

solution

(ii) ax + by = c
bx + ay = 1 + c

solution

(iii)

(x/a) -(y/b) = 0

ax +by = (a^2) + (b^2)

solution

(iv)

(a – b)x + (a + b) y = (a^2) – 2ab – (b^2)

(a + b)(x + y) = (a^2) + (b^2 )

solution

(v)

152x – 378y = – 74

–378x + 152y = – 604

solution

ABCD is a cyclic quadrilateral Find the angles of the cyclic quadrilateral,

if angles are A =(4y+20) , B =(3y-5) , C=(-4x), D=(-7x+5)

solution

exercise 3.6

Ritu can row downstream 20 km in 2 hours, and upstream 4 km in 2 hours. Find her speed of rowing in still water and the speed of the current.

solution

(iii)

solution

exercise 3.5

solution

(ii) A fraction changes to (1/3) when 1 is subtracted from the numerator and it changes to (1/4) when 8 is added to its denominator. Find the fraction.

solution

(iii)Y scored 40 marks in a test, getting 3 marks for each right answer and losing 1 mark for each wrong answer. Had 4 marks been awarded for each correct answer and 2 marks been deducted for each incorrect answer, then Yash would have scored 50 marks. How many questions were there in the test?

solution

(iv) Places A and B are 100 km apart on a highway. One car starts from A and another from B at the same time. If the cars travel in the same direction at different speeds,they meet in 5 hours. If they travel towards each other, they meet in 1 hour. What are the speeds of the two cars?

solution

Thursday, October 22, 2020

A part of monthly hostel charges is fixed and the remaining depends on the number of days one has taken food in the mess. When a student A takes food for 20 days she has to pay 1000 as hostel charges whereas a student B, who takes food for 26 days, pays 1180 as hostel charges. Find the fixed charges and the cost of food per day.

cbse ncert 10th mathematics chapter 3 exercise 3.5

pair of linear equations in two variables

let x = fised charges for a month

and y = charge of food per day

student A takes food for 20 days has to pay Rs.1000 means

x + 20y =1000 ----------------------(1)

student B, who takes food for 26 days, pays Rs.1180 means

x +26 y =1180 ---------------------(2)

x + 20y =1000 ----------------------(1)

x +26 y =1180 ---------------------(2)

-----------------------------------------------------subtracting

(-6)y =-180

y = 30

substitute y =30 in x + 20y =1000

x+20(30) =1000

x =1000 - 20*30

x=400

fixed charge =Rs.400 /- per month

charge for food = Rs.30 /- per day

(ii) A fraction changes to (1/3) when 1 is subtracted from the numerator and it changes to (1/4) when 8 is added to its denominator. Find the fraction.

let fraction =(x/y)

fraction changes to (1/3) when 1 is subtracted from the numerator means

[(x-1)/y] =(1/3)

3(x-1) = 1y

3x-3 = y

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

fraction changes to (1/4) when 8 is added to its denominator means

[x /(y+8)] =(1/4)

4x =1(y+8)

4x=1y+8

4x-1y =8-----------------(2)

3x-y =3----------------------(1) eliminating y

4x-y =8-------------------(2)

----------------------------------------subtracting

(-1)x =(-5)

x = 5

substitute x=5 in 3x-y=3

3(5)-y=3

(-y)= 3-15

(-y)= =(-12)

y=12

fraction = [5/12]

=================================================

ncert cbse 10th mathematics chapter 3 optional exercise 3.7

solution

5. In a ∆ ABC, ∠ C = 3 ∠ B = 2 (∠ A + ∠ B). Find the three angles.

solution

Solve the following pair of linear equations:

px + qy = p – q

qx – py = p + q

solution

(ii) ax + by = c
bx + ay = 1 + c

solution

(iii)

(x/a) -(y/b) = 0

ax +by = (a^2) + (b^2)

solution

(iv)

(a – b)x + (a + b) y = (a^2) – 2ab – (b^2)

(a + b)(x + y) = (a^2) + (b^2 )

solution

(v)

152x – 378y = – 74

–378x + 152y = – 604

solution

ABCD is a cyclic quadrilateral Find the angles of the cyclic quadrilateral,

if angles are A =(4y+20) , B =(3y-5) , C=(-4x), D=(-7x+5)

solution

exercise 3.6

Ritu can row downstream 20 km in 2 hours, and upstream 4 km in 2 hours. Find her speed of rowing in still water and the speed of the current.

solution

(iii)

solution

exercise 3.5

solution

(ii) A fraction changes to (1/3) when 1 is subtracted from the numerator and it changes to (1/4) when 8 is added to its denominator. Find the fraction.

solution

Tuesday, October 20, 2020

(ii) 2 women and 5 men can together finish an embroidery work in 4 days, while 3 women and 6 men can finish it in 3 days. Find the time taken by 1 woman alone to finish the work, and also that taken by 1 man alone.

ncert cbse 10th mathematics chapter 3 exercise3.6 pair of linear equations in two variables

let x = number of days for a woman alone to finish the work

and y = number of days for a man alone to finish the work

This is a problem on time and work so remember to take reciprocals of x , y and time

2 women and 5 men can together finish an embroidery work in 4 days

[2/x] +[5/y]=[1/4]

3 women and 6 men can finish it in 3 days

[3/x]+[6/y] =[1/3]

equations are

[2/x] +[5/y]=[1/4]

[3/x]+[6/y] =[1/3]

to get rid of (1/x) and (1/y)

use a=(1/x) and b=(1/y)

equations change to

2a+5b= [1/4] *3

3a+6b= [1/3] *2

-------------------------------------eliminate a

6a +15b=[3/4]

6a+12b=[2/3]

------------------------------------subtracting

3b =[3/4] -[2/3]

3b =[1/12]

b=[1/36]

substitute in 3a+6b= [1/3]

3a +6*(1/36) =[1/3]

3a = [1/3] - [1/6]

3a = [1/6]

a=[1/18]

a=[1/18] ,b=[1/36]

take reciprocal

x =18 days , y = 36 days

Ritu can row downstream 20 km in 2 hours, and upstream 4 km in 2 hours. Find her speed of rowing in still water and the speed of the current.

Let x = speed of her rowing in still water in km/hr

and y = speed of the current in km/hr

speed downstream = (x+y)

speed upstream = (x-y)

time = distance / speed

downstream 20 km in 2 hours

means

2 =20/(x+y)

x+y = 20/2

x+y =10-------------------(1)

upstream 4 km in 2 hours

means

2 =4/(x-y)

x-y =4/2

x-y =2----------------------(2)

x+y =10-------------------(1)

x-y =2----------------------(2)

------------------------------------------adding

2x = 12

x= 6km/hr

substitute in x-y =2

to get

y=4km/hr

=================================================

ncert cbse 10th mathematics chapter 3 optional exercise 3.7

solution

5. In a ∆ ABC, ∠ C = 3 ∠ B = 2 (∠ A + ∠ B). Find the three angles.

solution

Solve the following pair of linear equations:

px + qy = p – q

qx – py = p + q

solution

(ii) ax + by = c
bx + ay = 1 + c

solution

(iii)

(x/a) -(y/b) = 0

ax +by = (a^2) + (b^2)

solution

(iv)

(a – b)x + (a + b) y = (a^2) – 2ab – (b^2)

(a + b)(x + y) = (a^2) + (b^2 )

solution

(v)

152x – 378y = – 74

–378x + 152y = – 604

solution

ABCD is a cyclic quadrilateral Find the angles of the cyclic quadrilateral,

if angles are A =(4y+20) , B =(3y-5) , C=(-4x), D=(-7x+5)

solution

exercise 3.6

Ritu can row downstream 20 km in 2 hours, and upstream 4 km in 2 hours. Find her speed of rowing in still water and the speed of the current.

solution

(iii)

solution

ABCD is a cyclic quadrilateral Find the angles of the cyclic quadrilateral.

10th ncert cbse mathematics chapter 3 miscellaneous / optional exercise

pair of linear equations in two variables

ABCD is a cyclic quadrilateral Find the angles of the cyclic quadrilateral,

if angles are A =(4y+20) , B =(3y-5) , C=(-4x), D=(-7x+5)

We know that the opposite angles of a cyclic quadrilateral are supplementary,

which means they add up to 180 degrees.

A+C=180 degrees

and

B+D = 180 degrees

A+C=180

gives

(4y+20) +(-4x)=180

-4x +4y =180-20

-4x +4y =160

dividing by 4 makes it

-x+y=40--------------(1)

B+D = 180

gives

(3y-5)+(-7x+5) =180

-7x+3y=180-------------(2)

eliminating y

-x+y=40--------------(1) *3

-7x+3y=180-------------(2)

-3x+3y =120

-7x+3y =180

--------------------------subtracting

4x =(-60)

x = (-60)/4

x=(-15)

substitute in

-x+y=40

-(-15) +y =40

15+y-40

y=40-15

y=25

use x=(-15) and y=25

A =(4y+20) =4*25+20=120 degrees

B =(3y-5)=3*25-5=75-5=70 degrees

C=(-4x) =(-4)*(-15) = 60 degrees

D=(-7x+5) =(-7)*(-15) +5 =105+5=110 degrees

exercise 3.6

(iii)

let x = speed of train in km/hr

and y = speed of bus in km/hr

plan 1

60 km by train and remaining 300-60 = 240 km by bus

time = distance / speed

t1=(60/x) in the train

and

t2=(240/y) hours in the bus

total of 4 hours means

t1 +t2 =4 hours

(60/x) +(240/y) =4--------------------(1)

plan 2

100 km by train and remaining 300-100 = 200 km by bus

time = distance / speed

t3=(100/x) in the train

and

t4=(200/y) hours in the bus

10 minutes longer means 4hours 10minutes

This has to be changed in term of hours

4hours 10minutes =4+(10/60)=4+(1/6) = (25/6)hours

t3 +t4 = (25/6)hours

(100/x)+(200/y) =(25/6)----------------(2)

equations are

(60/x) +(240/y) =4--------------------(1)

(100/x)+(200/y) =(25/6)----------------(2)

we have to use substitution (1/x)=u , (1/y)=v

equations change to

60u + 240v =4 ----------------------(3) *5

100u + 200v =(25/6)-------------------(4) *3

--------------------------------------------------------------------eliminate u

300u + 1200v =20

300u +600v =(25/2)

------------------------------------------subtracting

600v = 20 -(25/2)

600v = (15/2)

v = [ 15 / (2*600) ]

v = [1/80]

resubstitute v = [1/80] in 60u + 240v =4

60u+{240/80} =4

60u+3=4

60u=4-3

60u=1

u=(1/60)

v = [1/80]

take reciprocal

x=60 km/hr

y=80 km/hr

speed of train =60km/hr

speed of bus =80km/hr

=================================================

ncert cbse 10th mathematics chapter 3 optional exercise 3.7

solution

5. In a ∆ ABC, ∠ C = 3 ∠ B = 2 (∠ A + ∠ B). Find the three angles.

solution

Solve the following pair of linear equations:

px + qy = p – q

qx – py = p + q

solution

(ii) ax + by = c
bx + ay = 1 + c

solution

(iii)

(x/a) -(y/b) = 0

ax +by = (a^2) + (b^2)

solution

(iv)

(a – b)x + (a + b) y = (a^2) – 2ab – (b^2)

(a + b)(x + y) = (a^2) + (b^2 )

solution

(v)

152x – 378y = – 74

–378x + 152y = – 604

solution

ABCD is a cyclic quadrilateral Find the angles of the cyclic quadrilateral,

if angles are A =(4y+20) , B =(3y-5) , C=(-4x), D=(-7x+5)

solution

exercise 3.6

(iii)

solution

ncert cbse 10th mathematics chapter 2 optional exercise
If the zeroes of the polynomial (x^3) – 3(x^2) + x + 1 are a – b, a, a + b, find a and b.
solution

2. Find a cubic polynomial with the sum, sum of the product of its zeroes taken two at a time, and the product of its zeroes as 2, –7, –14 respectively
solution

4. If two zeroes of the polynomial (x^4) – 6(x^3) – 26(x^2) + 138x – 35 are
[2 ±sqrt(3) ] , find other zeroes

solution

5. If the polynomial (x^4) – 6(x^3) + 16(x^2) – 25x + 10 is divided by another polynomial (x^2) – 2x + k, the remainder comes out to be x + a, find k and a
solution

exercise 2.3

3. obtain all other zeroes of 3(x^4)+6(x^3)-2(x^2)-10x-5 if two of its zeroes are sqrt(5/3) and [sqrt(5/3)]
solution

4. On dividing (x^3) – 3(x^2) + x + 2 by a polynomial g(x), the quotient and remainder were x – 2
and –2x + 4, respectively. Find g(x).
solution

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Friday, October 30, 2020

(iii)Y scored 40 marks in a test, getting 3 marks for each right answer and losing 1 mark for each wrong answer. Had 4 marks been awarded for each correct answer and 2 marks been deducted for each incorrect answer, then Yash would have scored 50 marks. How many questions were there in the test?

(iv) Places A and B are 100 km apart on a highway. One car starts from A and another from B at the same time. If the cars travel in the same direction at different speeds,they meet in 5 hours. If they travel towards each other, they meet in 1 hour. What are the speeds of the two cars?

Wednesday, October 28, 2020

(iii)Y scored 40 marks in a test, getting 3 marks for each right answer and losing 1 mark for each wrong answer. Had 4 marks been awarded for each correct answer and 2 marks been deducted for each incorrect answer, then Yash would have scored 50 marks. How many questions were there in the test?

(iv) Places A and B are 100 km apart on a highway. One car starts from A and another from B at the same time. If the cars travel in the same direction at different speeds,they meet in 5 hours. If they travel towards each other, they meet in 1 hour. What are the speeds of the two cars?

Tuesday, October 27, 2020

(iii)Y scored 40 marks in a test, getting 3 marks for each right answer and losing 1 mark for each wrong answer. Had 4 marks been awarded for each correct answer and 2 marks been deducted for each incorrect answer, then Yash would have scored 50 marks. How many questions were there in the test?

(iv) Places A and B are 100 km apart on a highway. One car starts from A and another from B at the same time. If the cars travel in the same direction at different speeds,they meet in 5 hours. If they travel towards each other, they meet in 1 hour. What are the speeds of the two cars?

(iii)Y scored 40 marks in a test, getting 3 marks for each right answer and losing 1 mark for each wrong answer. Had 4 marks been awarded for each correct answer and 2 marks been deducted for each incorrect answer, then Yash would have scored 50 marks. How many questions were there in the test?

(iv) Places A and B are 100 km apart on a highway. One car starts from A and another from B at the same time. If the cars travel in the same direction at different speeds,they meet in 5 hours. If they travel towards each other, they meet in 1 hour. What are the speeds of the two cars?

Monday, October 26, 2020

(iii)Y scored 40 marks in a test, getting 3 marks for each right answer and losing 1 mark for each wrong answer. Had 4 marks been awarded for each correct answer and 2 marks been deducted for each incorrect answer, then Yash would have scored 50 marks. How many questions were there in the test?

(iv) Places A and B are 100 km apart on a highway. One car starts from A and another from B at the same time. If the cars travel in the same direction at different speeds,they meet in 5 hours. If they travel towards each other, they meet in 1 hour. What are the speeds of the two cars?

Friday, October 23, 2020

Y scored 40 marks in a test, getting 3 marks for each right answer and losing 1 mark for each wrong answer. Had 4 marks been awarded for each correct answer and 2 marks been deducted for each incorrect answer, then Yash would have scored 50 marks. How many questions were there in the test?

(iii)Y scored 40 marks in a test, getting 3 marks for each right answer and losing 1 mark for each wrong answer. Had 4 marks been awarded for each correct answer and 2 marks been deducted for each incorrect answer, then Yash would have scored 50 marks. How many questions were there in the test?

(iv) Places A and B are 100 km apart on a highway. One car starts from A and another from B at the same time. If the cars travel in the same direction at different speeds,they meet in 5 hours. If they travel towards each other, they meet in 1 hour. What are the speeds of the two cars?

Thursday, October 22, 2020

Tuesday, October 20, 2020