Math Assignment Class X Ch-3 | Linear Equations in Two Variables

 CHAPTER 3   CLASS  X

PAIR OF LINEAR EQUATIONS IN TWO VARIABLE

Extra questions of chapter 3 class 10 : Pair of Linear Equations in Two Variable with answer and  hints . Useful math assignment for the students of class 10

Basic points of  Chapter 3 Pair of Linear Equations in two variable.

ASSIGNMENT BASED ON CH-3 CLASS 10

Question 1 

By comparing the ratios of the coefficients  find the nature of sol./ graph / consistency or inconsistency

a)   2x - 3y = 1 ;  x + 2y = 2

Ans: Unique solution, intersecting lines, consistent

b)   3x - y = 6 ;  6x - 2y = 4

Ans: No solution, parallel lines, inconsistent

c)   x + 3y = 8 ;  2x - 3y = 12

Ans: Unique solution, intersecting lines, consistent

d)   9x + 3y + 12 = 0 ;  18x + 6y = - 24

Ans: Many solution, coincident lines, consistent

e)   6x - 3y +10 = 0 ;  2x – y + 9 = 0

Ans: No solution, parallel lines, inconsistent

Question 2) 

Find the value of p so that the following equations have coincident lines

a)    2x + y - 5 = 0;  6x + 3y + p = 0      Ans [p = -15]

b)   x + 2y + 7 = 0;   2x + py + 14 = 0    Ans [p = 4]

c)  3x + 4y + 2 = 0;  9x + 12y + p = 0    Ans[p = 6]

d)    x + 5y - 7 = 0 ;  4x + 20y + p = 0    Ans [p = -28]

Question 3) Find k if the following equations are inconsistent

a)   8x + 5y = 9;   kx + 10y = 15   Ans: k =16

b) (3k + 1)x + 3y - 2 = 0;    (k² + 1)x + (k - 2)y  -  5 = 0 

c)   3x + y = 1;   (2k - 1)x + (k - 1)y = 2k + 1 

d) kx + 3y = k  –  3;  12x + ky = k     

Answer:   a)  k = 16 ,    b) k = -1,     c) k = 2,     d) k = 土 6

Question 4) Find a & b or k so that the following equations have infinitely many solutions

a)  (a + b)x  - 2by = 5a + 2b + 1;    3x - y = 14            Ans [a = 5, b =  1]

b)     2x - 3y = 7;  (a + b)x  +  (a + b)y = 4a + b          Ans [-5, -1]

c) (2a - 1)x + 3y – 5 = 0;   3x + (b - 1)y – 2 = 0          Ans [a = 17/4,   b= 11/5]

d)   kx + 3y = 2k + 1;  2(k + 1)x + 9y = 7k + 1           Ans [k = 2]

e)   2x + 3y = 7;   (k – 1)x + (k + 2)y = 3k                  Ans [k = 7]

Question 5) Solve the following equations graphically

a) x + y = 3;   2x + 5y = 12                       Ans [1, 2]

b)  2x + 3y + 5 = 0;   3x - 2y - 12 = 0       Ans [2, -3]

PROBLEMS BASED ON GRAPHICAL METHOD

Question 6 

Solve the following equations graphically. Shade the area between lines and x-axis. Also find the area of the shaded triangle.

a) x + y = 5;      2x - y = - 2                     

[Sol. (1, 4) & Area= 12]

b) x - y + 1 = 0;    3x + 2y - 12 = 0         

 [Area= 7.5]

c)  2x + y = 6;      2x - y + 2 = 0               

[Area= 8]

Question 7)

Solve 2(x - 1) = y & x + 3y = 15 graphically and find the coordinates where lines meet the y-axis. Also shade the triangle between the lines and y-axis.      Ans [(3, 4), (0, -2), (0, 5)]

Question 8)

Solve x – y = 1 & 2x + y = 8 graphically and shade the area between the lines and the y-axis. Also find the area of the triangle.      Ans [Area = 13.5]

SOLVE THE  PAIR OF LINEAR EQUATIONS BY

 ALGEBRAIC METHOD

Question 9) 

 Solve the following pair of linear equations in two variables

a)     x + y = 7    ;   3x - 2y = 1                 Ans [3, 4]

b)     3x + 2y = 10  ; 4x - y = 6                Ans [2, 2]

c)          Ans [2, -3]

d) 2x + y – 35 = 0   ;  3x + 4y - 65 = 0     Ans [15, 5]

e)  x/a - y/b = 0 ;  bx + ay = 4ab            Ans[2a, 2b]

f) - 6x + 5y = 2 ;    - 5x + 6y = 9             Ans[3, 4]

g)  23x - 29y = 98  ;   29x - 23y = 110     Ans [3, - 1]

h)   99x + 101y =  499 ; 101x +  99y = 501     Ans [x = 3, y = 2]

i)    a(x + y) + b(x - y) = a² – ab + b² ;  a(x + y) - b(x - y) = a² + ab + b²          

j) ax + by = a - b ;  bx - ay = a + b          Ans [1, - 1]

PROBLEMS BASED ON DAILY LIFE SITUATIONS

Question 10) Find two numbers whose sum is 75 and difference is 15            

Ans [45 and 30]

Question 11) Sum of two numbers is 8. Find the numbers if sum of their reciprocal is 8/15.

Question 12) 

The ratio of incomes of two persons is 9 : 7  and the ratio of their expenditure  is  4 : 3 If each of them  saves Rs 200/ month, find their monthly incomes.      

Ans [Rs1800 & Rs 1400]

Question 13

If 2 is subtracted from the numerator and 1 is added to the denominator, a fraction becomes  1/2 , but when 4 is added to the numerator and 3 is subtracted from the denominator , it becomes 3/2. Find the fractions.                                                 

Ans [8/11]

Question 14) 

37 pens and 53 pencils together cost Rs 320, while 53 pens and 37 pencils together cost Rs 400. Find the cost of a pen and that of a pencil.                         

Ans  [Rs 6.50 and Rs 1.50]

Question 15)

In a two digit number , the ten’s digit is three times the unit’s digit. When the number is decreased by 54, the digits are reversed. Find the number.                        

Ans  [93]

Question 16)

The sum of the two digit number and the number obtained by reversing the order of its digits is 121, and the two digits are differ by 3. Find the numbers.                

Ans [47 or 74]

Question 17)

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

Ans   [64]  

Question 18)

A fraction is such that if the numerator is multiplied by 3 and the denominator is reduced by 3, we get 18/11, but if the numerator is increased by 8 and the denominator is doubled , we get 2/5.  Find the fraction.                                                      

Ans  [12/25]

Question 19)

The denominator of a fraction is 4 more than twice the numerator. When both the numerator and denominator are decreased by 6, then the denominator becomes 12 times the numerator. Find the fraction                                    

Ans  [7/18]

Question 20)

If twice the son’s age in years is added to the fathers age,  the sum is 70. But if twice the father’s age is added to the son’s age, the sum is 95. Find the age of father and son.                Ans  [40,  15]

Question 21) 

Two years ago, a father was 5 times as old as his son. Two years later , his age will be 8 more than 3 times the age of his son. Find the present ages of father and son.    

Ans  [42, 10]  

Question: 22

Sum of two numbers is 105 and their difference is 45. Find the numbers.

Answer: Required Numbers are: 75, 30

Question: 23

Three years ago, Rashmi was thrice as old as Nazma. Ten years later, Rashmi will be twice as old as Nazma. How old are Rashmi and Nazma now ?

Solution Hint

Let present age of Rashmi and Nazma be x years and y years respectively

Therefore, x - 3 = 3(y - 3)

Or x - 3y + 6 = 0

And x + 10 = 2(y + 10)

Or x – 2y – 10 = 0

Solving equations to get x = 42, y = 16

Present age of Rashmi is 42 years and that of Nazma is 16 Years

Question 24

Tara 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 wrong answer, then Tara would have scored 50 marks. Assuming that Tara attempted all questions, find the total number of questions in the test.

Solution Hint

Let number of correct answers be x and

number of incorrect answers be y

3x - y = 40

4x - 2y = 50

Solving, we get x = 15, y = 5

Total number of questions = 20

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HIGHER ORDER THINKING SKILL (HOTS)

(Deleted from CBSE syllabus)

Question 1)
 

 

Ans: x = 2, y = 1]

Question 2)

 

  

Ans: x = 1, y = 1

Question 3)

Ans: x = 8, y = 3

4)    ax + by = c ;       bx + ay = 1 + c      

5) ax + by = c  ;  bx - ay = 0                     

6)   6(ax + by) = 3a + 2b ;  6(bx +  ay) = 3b + 2a     Ans [1/2, 1/3]

7)  ax + by = a² + b² ; bx - ay = 0         Ans [a, b]

 

8) (a - b)x + (a + b)y = a² - 2ab - b² ;  (a + b)(x + y)  = a² + b²    

 

9) 2(ax - by) + (a + 4b) = 0 ;  2(bx + ay) + (b - 4a) = 0     Ans [-1/2, 2]

10)  x +  y = 5xy  ;   3x + 2y = 13xy          Ans [x = 1/2 ,  y = 1/3]

 11)  

Q12) A person can row downstream 20km in 2 hours. And upstream 4km in 2 hours. Find the speed of boat in still water and that of the current.                            

Ans [6,4]

Q13) A boat covers 32 km upstream and 36 km downstream in 7 hours. Also it covers 40 km upstream and 48 km downstream in 9 hours. Find the speed of the boat in still water and that of the stream.                                                                               

Ans  [10 km/h and  2 km/h]

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