exercise 3.1 math solution ncert class 10th

  1. Form the pair of linear equations in the following problems, and find their solutions graphically:
    (i) 10 students of Class X took part in a Mathematics quiz. If the number of girls is 4 more than the number of boys, find the number of boys and girls who took part in the quiz.
    Let,
    number of girls = x
    number of boys = y
    according to question,
    π‘₯+𝑦=10,
    π‘₯βˆ’π‘¦=4
    Adding, 2π‘₯=14β‡’π‘₯=7. Hence 𝑦=10βˆ’7=3.
    Girls=7,Boys=3.
    (Graphically, the lines 𝐴(π‘₯+𝑦)=10 and π‘₯βˆ’π‘¦=4 intersect at (7,3).

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2. On comparing the ratios​​, ​​, find out whether the lines representing the following pairs of linear equations intersect at a point, are parallel or coincident:

(i) 5xβˆ’4y+8=0

7x+6yβˆ’9=0

Since , the lines intersect at one point.

(ii) 9x+3y+12=

18x+6y+24=0

Since

All equal lines are coincident (infinitely many solutions).

(iii) 6xβˆ’3y+10=0

2xβˆ’y+9=0

Since

First two equal, third different parallel (no solution).

3. On comparing the ​​, find out whether the following pairs of linear equations are consistent, or inconsistent:

(i) 3x+2y=5, 2xβˆ’3y=7

Since

the lines intersect at one point, so consistent with a unique solution.

(ii) 2xβˆ’3y=8, 4xβˆ’6y=9

Since

First two equal, third different So, inconsistent (parallel).

(iii)9xβˆ’10y=14

Since

the lines intersect at one point so consistent with a unique solution.
(iv)5xβˆ’3y=11,10x+6y=βˆ’22

Since

All equal consistent, coincident (infinitely many solutions).

(v) 2x+3y=12

Since

All equal consistent, coincident (infinitely many solutions).

4. Which of the following pairs of linear equations are consistent/inconsistent? If consistent, obtain the solution graphically:

(i) x+y=5, 2x+2y=10

Since

All equal consistent, coincident (infinitely many solutions).


(ii) xβˆ’y=8, 3xβˆ’3y=16

Since

First two equal, third different So, inconsistent (parallel).

(iii) 2x+yβˆ’6=0, 4xβˆ’2yβˆ’4=0

. Substitute into second:

and hence

Since

the lines intersect at one point so consistent with a unique solution.

(iv) 2xβˆ’2yβˆ’2=0, 4xβˆ’4yβˆ’5=0

Since

First two equal, third different So, inconsistent (parallel).

5. Half the perimeter of a rectangular garden, whose length is 4 m more than its width, is 36 m. Find the dimensions of the garden.

Let,

width , length . Then

Add: , and .

6. Given the linear equation 2x+3yβˆ’8=0, write another linear equation in two variables such that the geometrical representation of the pair so formed is:
(i) Intersecting lines

Here,

So for ,intersecting where

Equation 5x+6y-17

(ii) Parallel lines

Here,

So for , Parallel where

Equation
(iii) Coincident lines

Here,

So for , Coincident lines where

Equation

7. Draw the graphs of the equations xβˆ’y+1=0and 3x+2yβˆ’12=0. Determine the coordinates of the vertices of the triangle formed by these lines and the x-axis, and shade the triangular region.