##### Class 10^{th} Mathematics Tamilnadu Board Solution

**Exercise 10.1**- y = 3x^2 Draw the graph of the following function.
- y = - 4x^2 Draw the graph of the following function.
- y = (x + 2) (x + 4) Draw the graph of the following function.
- y = 2x^2 - x + 3 Draw the graph of the following function.
- x^2 - 4 = 0 Solve the following equation graphically
- x^2 - 3x - 10 = 0 Solve the following equation graphically
- (x - 5) (x - 1) = 0 Solve the following equation graphically
- (2x + 1) (x - 3) = 0 Solve the following equation graphically
- Draw the graph of y = x^2 and hence solve x^2 - 4x - 5 = 0.
- Draw the graph of y = x^2 + 2x - 3 and hence find the roots of x^2 - x - 6 = 0.…
- Draw the graph of y = 2x^2 + x - 6 and hence solve 2x^2 + x - 10 = 0.…
- Draw the graph of y = x^2 - x - 8 and hence find the roots of x^2 - 2x - 15 =…
- Draw the graph of y = x^2 + x - 12 and hence solve x^2 + 2x + 2 = 0.…

**Exercise 10.2**- A bus travels at a speed of 40 km/hr. Write the distance-time formula and draw…
- The following table gives the cost and number of notebooks bought. Draw the…
- |c|c|c|c|c|c| x&1&3&5&7&8 y&2&6&10&14&16 Draw the graph for the above table and…
- The cost of the milk per liter is ₹15. Draw the graph for the relation between…
- Draw the Graph of xy = 20, x, y 0. Use the graph to find y when x = 5, and to…
- |c|c|c|c|c|c|c| &3&4&6&8&9&16 &96&72&48&36&32&18 Draw graph for the data given…

**Exercise 10.1**

- y = 3x^2 Draw the graph of the following function.
- y = - 4x^2 Draw the graph of the following function.
- y = (x + 2) (x + 4) Draw the graph of the following function.
- y = 2x^2 - x + 3 Draw the graph of the following function.
- x^2 - 4 = 0 Solve the following equation graphically
- x^2 - 3x - 10 = 0 Solve the following equation graphically
- (x - 5) (x - 1) = 0 Solve the following equation graphically
- (2x + 1) (x - 3) = 0 Solve the following equation graphically
- Draw the graph of y = x^2 and hence solve x^2 - 4x - 5 = 0.
- Draw the graph of y = x^2 + 2x - 3 and hence find the roots of x^2 - x - 6 = 0.…
- Draw the graph of y = 2x^2 + x - 6 and hence solve 2x^2 + x - 10 = 0.…
- Draw the graph of y = x^2 - x - 8 and hence find the roots of x^2 - 2x - 15 =…
- Draw the graph of y = x^2 + x - 12 and hence solve x^2 + 2x + 2 = 0.…

**Exercise 10.2**

- A bus travels at a speed of 40 km/hr. Write the distance-time formula and draw…
- The following table gives the cost and number of notebooks bought. Draw the…
- |c|c|c|c|c|c| x&1&3&5&7&8 y&2&6&10&14&16 Draw the graph for the above table and…
- The cost of the milk per liter is ₹15. Draw the graph for the relation between…
- Draw the Graph of xy = 20, x, y 0. Use the graph to find y when x = 5, and to…
- |c|c|c|c|c|c|c| &3&4&6&8&9&16 &96&72&48&36&32&18 Draw graph for the data given…

###### Exercise 10.1

**Question 1.**Draw the graph of the following function.

y = 3x^{2}

**Answer:**The given function is y = 3x^{2}

for various values of X we can write the values of Y

The obtained data can be given in form of graph

**Question 2.**Draw the graph of the following function.

y = – 4x^{2}

**Answer:**The given function is y = –4x^{2}

for various values of X we can write the values of Y

**Question 3.**Draw the graph of the following function.

y = (x + 2) (x + 4)

**Answer:**The given function is y = (x + 2) (x + 4)

Lets assign values of X from –5 to 0 so we will find the values of Y as follows using the expression

**Question 4.**Draw the graph of the following function.

y = 2x^{2} – x + 3

**Answer:**The given function is y = 2x^{2} – x + 3

Lets assign values of X from –2 to 2 so we will find the values of Y as follows using the expression

**Question 5.**Solve the following equation graphically

x^{2} – 4 = 0

**Answer:**Here y = x^{2} – 4

Taking the values of X from –2 to 2 we will find different values of Y which we can use to find the values of roots to the equation given.

Here the curve touches the X- axis at (–2,0) and (2,0) so the two roots for the equation is (–2,2)

**Question 6.**Solve the following equation graphically

x^{2} – 3x – 10 = 0

**Answer:**Here y = x^{2} – 3x – 10

Taking the values of X between –2 to 5 we will find different values of Y which we can use to find the values of roots to the equation given.

Here the curve touches the X- axis at (–2,0) and (5,0) so the two roots for the equation is (–2,5)

**Question 7.**Solve the following equation graphically

(x – 5) (x – 1) = 0

**Answer:**Here y = (x – 5) (x – 1) Taking the values of X from 0 to 5 we will find different values of Y which we can use to find the values of roots to the equation given.

Here the curve touches the X- axis at (0,0), (1,0) and (5,0) so the two roots for the equation is (1,5)

**Question 8.**Solve the following equation graphically

(2x + 1) (x – 3) = 0

**Answer:**Here y = (2x + 1) (x – 3) Taking the values of X from –1 to 4 we will find different values of Y which we can use to find the values of roots to the equation given.

Here the curve touches the X- axis at (– 1\2, 0) and (3,0) so the two roots for the equation is (–1\2, 3)

**Question 9.**Draw the graph of y = x^{2} and hence solve x^{2} – 4x – 5 = 0.

**Answer:**We are given y = x^{2}

So taking the values of X in between –2 and 5 we can find various value of y and put them in graph.

Now in the eqution we are given that

x^{2} – 4x – 5 = 0

we know from the first expression that y = x^{2} so putting this value in the above equation we will find

x^{2} – 4x – 5 = 0

⇒ y–4x–5 = 0

⇒ y = 4x + 5

Now taking the values of X in between –1 and 3 we can find another set of values for y and can put them in the graph to find the solution for the eqution.

Now we can draw the graph

Here the lines intersects each other at (–1, 1) and (5,25) so the two roots for the equation is (–1,5)

**Question 10.**Draw the graph of y = x^{2} + 2x – 3 and hence find the roots of x^{2} – x – 6 = 0.

**Answer:**We are given y = x^{2} + 2x – 3

So taking the values of X in between –3 and 3 we can find various value of y and put them in graph

Again we are given

x^{2} – x – 6 = 0

we can write the above expression as

x^{2} + (2x–2x)–x –3–3 = 0

⇒ x^{2} + 2x–3–2x–x–3 = 0

⇒ y–3x–3 = 0 (as x^{2} – x – 6 = 0 )

⇒ y = 3x + 3

Now we can draw another pair of value using the above data

Now we can draw the graph and the intersection of the two lines will give the roots for the equation.

So the lines intersects at {–2,3} and (3,12).

**Question 11.**Draw the graph of y = 2x^{2} + x – 6 and hence solve 2x^{2} + x – 10 = 0.

**Answer:**We are given y = 2x^{2} + x – 6

So taking the values of X in between –3 to 4 we can find various value of y and put them in graph

Again we are given

2x^{2} + x – 10 = 0

we can write the above expression as

⇒ 2x^{2} + x–6–4 = 0

⇒ y–4 = 0 (as y = 2x^{2} + x – 6)

⇒ y = 4

Now we can draw another pair of value using the above data

Now we can draw the graph and the intersection of the two lines will give the roots for the equation. Where Y_{1} and Y_{2} are two lines showing y = 2x^{2} + x – 6 and y = 4 respectively

here from the graph its clear that the point of intersection are (–2.5,4) and (2,4) so the roots are (–2.5,2)

**Question 12.**Draw the graph of y = x^{2} – x – 8 and hence find the roots of x^{2} – 2x – 15 = 0.

**Answer:**We are given y = x^{2} – x – 8

So taking the values of X in between –3 and 5 we can find various value of y and put them in graph

Again we are given

x^{2} – 2x – 15 = 0

we can write the above expression as

x^{2} –x–x–8–7 = 0

⇒ x^{2}–x–8–x–7 = 0

⇒ y–x–7 = 0 (as x^{2}–x–8 = y )

⇒ y = x + 7

Now we can draw another pair of value using the above data

Now we can draw the graph and the intersection of the two lines will give the roots for the equation.

here from the graph its clear that the point of intersection are (–3,4) and (5,12) so the roots are (–3,5)

**Question 13.**Draw the graph of y = x^{2} + x – 12 and hence solve x^{2} + 2x + 2 = 0.

**Answer:**We are given y = x^{2} + x – 12

So taking the values of X in between –3 and 3 we can find various value of y and put them in graph

Again we are given

x^{2} + 2x + 2 = 0

we can write the above expression as

x^{2} + x + x–14–12 = 0

⇒ x^{2} + x–12 + x + 14 = 0

⇒ y + x + 14 = 0 (as x^{2} – x – 6 = 0 )

⇒ y = –x–14

Now we can draw another pair of value using the above data

Now we can draw the graph and the intersection of the two lines will give the roots for the equation.

From the above graph it’s clear that the line doesn’t intersect each other so there is no solution.

**Question 1.**

Draw the graph of the following function.

y = 3x^{2}

**Answer:**

The given function is y = 3x^{2}

for various values of X we can write the values of Y

The obtained data can be given in form of graph

**Question 2.**

Draw the graph of the following function.

y = – 4x^{2}

**Answer:**

The given function is y = –4x^{2}

for various values of X we can write the values of Y

**Question 3.**

Draw the graph of the following function.

y = (x + 2) (x + 4)

**Answer:**

The given function is y = (x + 2) (x + 4)

Lets assign values of X from –5 to 0 so we will find the values of Y as follows using the expression

**Question 4.**

Draw the graph of the following function.

y = 2x^{2} – x + 3

**Answer:**

The given function is y = 2x^{2} – x + 3

Lets assign values of X from –2 to 2 so we will find the values of Y as follows using the expression

**Question 5.**

Solve the following equation graphically

x^{2} – 4 = 0

**Answer:**

Here y = x^{2} – 4

Taking the values of X from –2 to 2 we will find different values of Y which we can use to find the values of roots to the equation given.

Here the curve touches the X- axis at (–2,0) and (2,0) so the two roots for the equation is (–2,2)

**Question 6.**

Solve the following equation graphically

x^{2} – 3x – 10 = 0

**Answer:**

Here y = x^{2} – 3x – 10

Taking the values of X between –2 to 5 we will find different values of Y which we can use to find the values of roots to the equation given.

Here the curve touches the X- axis at (–2,0) and (5,0) so the two roots for the equation is (–2,5)

**Question 7.**

Solve the following equation graphically

(x – 5) (x – 1) = 0

**Answer:**

Here y = (x – 5) (x – 1) Taking the values of X from 0 to 5 we will find different values of Y which we can use to find the values of roots to the equation given.

Here the curve touches the X- axis at (0,0), (1,0) and (5,0) so the two roots for the equation is (1,5)

**Question 8.**

Solve the following equation graphically

(2x + 1) (x – 3) = 0

**Answer:**

Here y = (2x + 1) (x – 3) Taking the values of X from –1 to 4 we will find different values of Y which we can use to find the values of roots to the equation given.

Here the curve touches the X- axis at (– 1\2, 0) and (3,0) so the two roots for the equation is (–1\2, 3)

**Question 9.**

Draw the graph of y = x^{2} and hence solve x^{2} – 4x – 5 = 0.

**Answer:**

We are given y = x^{2}

So taking the values of X in between –2 and 5 we can find various value of y and put them in graph.

Now in the eqution we are given that

x^{2} – 4x – 5 = 0

we know from the first expression that y = x^{2} so putting this value in the above equation we will find

x^{2} – 4x – 5 = 0

⇒ y–4x–5 = 0

⇒ y = 4x + 5

Now taking the values of X in between –1 and 3 we can find another set of values for y and can put them in the graph to find the solution for the eqution.

Now we can draw the graph

Here the lines intersects each other at (–1, 1) and (5,25) so the two roots for the equation is (–1,5)

**Question 10.**

Draw the graph of y = x^{2} + 2x – 3 and hence find the roots of x^{2} – x – 6 = 0.

**Answer:**

We are given y = x^{2} + 2x – 3

So taking the values of X in between –3 and 3 we can find various value of y and put them in graph

Again we are given

x^{2} – x – 6 = 0

we can write the above expression as

x^{2} + (2x–2x)–x –3–3 = 0

⇒ x^{2} + 2x–3–2x–x–3 = 0

⇒ y–3x–3 = 0 (as x^{2} – x – 6 = 0 )

⇒ y = 3x + 3

Now we can draw another pair of value using the above data

Now we can draw the graph and the intersection of the two lines will give the roots for the equation.

So the lines intersects at {–2,3} and (3,12).

**Question 11.**

Draw the graph of y = 2x^{2} + x – 6 and hence solve 2x^{2} + x – 10 = 0.

**Answer:**

We are given y = 2x^{2} + x – 6

So taking the values of X in between –3 to 4 we can find various value of y and put them in graph

Again we are given

2x^{2} + x – 10 = 0

we can write the above expression as

⇒ 2x^{2} + x–6–4 = 0

⇒ y–4 = 0 (as y = 2x^{2} + x – 6)

⇒ y = 4

Now we can draw another pair of value using the above data

Now we can draw the graph and the intersection of the two lines will give the roots for the equation. Where Y_{1} and Y_{2} are two lines showing y = 2x^{2} + x – 6 and y = 4 respectively

here from the graph its clear that the point of intersection are (–2.5,4) and (2,4) so the roots are (–2.5,2)

**Question 12.**

Draw the graph of y = x^{2} – x – 8 and hence find the roots of x^{2} – 2x – 15 = 0.

**Answer:**

We are given y = x^{2} – x – 8

So taking the values of X in between –3 and 5 we can find various value of y and put them in graph

Again we are given

x^{2} – 2x – 15 = 0

we can write the above expression as

x^{2} –x–x–8–7 = 0

⇒ x^{2}–x–8–x–7 = 0

⇒ y–x–7 = 0 (as x^{2}–x–8 = y )

⇒ y = x + 7

Now we can draw another pair of value using the above data

Now we can draw the graph and the intersection of the two lines will give the roots for the equation.

here from the graph its clear that the point of intersection are (–3,4) and (5,12) so the roots are (–3,5)

**Question 13.**

Draw the graph of y = x^{2} + x – 12 and hence solve x^{2} + 2x + 2 = 0.

**Answer:**

We are given y = x^{2} + x – 12

So taking the values of X in between –3 and 3 we can find various value of y and put them in graph

Again we are given

x^{2} + 2x + 2 = 0

we can write the above expression as

x^{2} + x + x–14–12 = 0

⇒ x^{2} + x–12 + x + 14 = 0

⇒ y + x + 14 = 0 (as x^{2} – x – 6 = 0 )

⇒ y = –x–14

Now we can draw another pair of value using the above data

Now we can draw the graph and the intersection of the two lines will give the roots for the equation.

From the above graph it’s clear that the line doesn’t intersect each other so there is no solution.

###### Exercise 10.2

**Question 1.**A bus travels at a speed of 40 km/hr. Write the distance-time formula and draw the graph of it. Hence, find the distance travelled in 3 hours.

**Answer:**Suppose a body X travels a at a speed of S km/h for T hours then the distance covered by him will be D and its given as

D = ST km

Here speed of bus, s = 40 km/hr

Travel time , t = 3hrs

So using this we can find a set of distance and put them in graph to find the exact distance travelled by the bus

to know the distance covered by the bus at three hours draw a line parallel to Y-axis from X = 3 to the curve and from the intersecting point of that line and the curve draw another line parallel to X-axis and the point at which the line intersect the Y-axis will give the distance coveredso from the graph it’s clear that as the bus was running for 3 hrs. and it covered a distance of 120 km.

**Question 2.**The following table gives the cost and number of notebooks bought.

Draw the graph and hence (i) find the cost of seven note books.

(ii) How many note books can be bought for ₹165.

**Answer:**

From the above it’s clear that y = 15 × X

Putting the above value in graph we will find

(i) To find the cost of 7 books draw a line parallel to Y-axis from X = 7 to the curve and from the intersecting point of that line and the curve draw another line parallel to X-axis and the point at which the line intersect the Y-axis will give the cost of 7 books.so from the graph cost of 7 books is found to be ₹105

(ii) to find the number of book that can be purchased at a cost of 165 draw a line parallel to X-axis from Y = 165.from the point at which the line intersect the curve draw another line parallel to Y-axis and the point at which the line intersect x-axis will give the number of book.so we can buy 11 note books

**Question 3.**

Draw the graph for the above table and hence find

(i) the value of y if x = 4

(ii) the value of x if y = 12

**Answer:**Using the data we can draw a graph

(I) TO FIND THE VALUE OF the value of y if x = 4, draw a line parallel to Y-axis from X = 4 to the curve and from the intersecting point of that line and the curve draw another line parallel to X-axis and the point at which the line intersect the Y-axis will give the value of Y ay X = 4.so from the graph at X = 4 Y is found to be 8

(II) to find the value of X at Y = 12 draw a line parallel to X-axis from Y = 12 and from the point at which the line intersect the curve draw another line parallel to Y-axis and the point at which the line intersect x-axis will give the value of X.so we find at Y = 12 Xis found to be 6

**Question 4.**The cost of the milk per liter is ₹15. Draw the graph for the relation between the quantity and cost. Hence find

(i) the proportionality constant.

(ii) the cost of 3 litres of milk.

**Answer:**(i) k = 15 (ii) ₹45

Let the cost of milk be Y and X be the quantity of milk in liter. As it’s given that cost per one liter is 15rs so increase in quantity will increase the cost of milk. So it’s a direct proportion

Mathematically,

Y = KX

Where K is the constant of proportionality

So we can draw a table

so from the graph it’s clear that y = kx = 15x

Proportionality, k = 15

We can draw a graph using the table given

To find the cost of 3 liter milk we have to draw a line parallel to Y-axis at X = 3 then from the intersection point of that parallel line and the curve draw another line parallel to X-axis so the point of intersection of that line and Y-axis will give the cost of 3 liter of milk.

From the graph its clear that cost of 3 liter is 45 RS.

**Question 5.**Draw the Graph of xy = 20, x, y > 0. Use the graph to find y when x = 5, and to find x when y = 10.

**Answer:**We are given that xy

= 20

⇒ y = (20/x)

We can find different value of Y by taking values of x between 1 and 10

From the graph we get that at x = 5, y = 4, and to find y = 10, x = 2

**Question 6.**

Draw graph for the data given in the table. Hence find the number of days taken by 12 workers to complete the work.

**Answer:**We can construct a graph using the data given

To find the number of days taken by 12 worker to complete the work draw a line parallel to Y-axis from X = 12 to the curve and from the intersecting point of that line and the curve draw another line parallel to X-axis and the point at which the line intersect the Y-axis will give the value of Y at X = 12.so from the graph at X = 12 Y is found to be 26

**Question 1.**

A bus travels at a speed of 40 km/hr. Write the distance-time formula and draw the graph of it. Hence, find the distance travelled in 3 hours.

**Answer:**

Suppose a body X travels a at a speed of S km/h for T hours then the distance covered by him will be D and its given as

D = ST km

Here speed of bus, s = 40 km/hr

Travel time , t = 3hrs

So using this we can find a set of distance and put them in graph to find the exact distance travelled by the bus

to know the distance covered by the bus at three hours draw a line parallel to Y-axis from X = 3 to the curve and from the intersecting point of that line and the curve draw another line parallel to X-axis and the point at which the line intersect the Y-axis will give the distance coveredso from the graph it’s clear that as the bus was running for 3 hrs. and it covered a distance of 120 km.

**Question 2.**

The following table gives the cost and number of notebooks bought.

Draw the graph and hence (i) find the cost of seven note books.

(ii) How many note books can be bought for ₹165.

**Answer:**

From the above it’s clear that y = 15 × X

Putting the above value in graph we will find

(i) To find the cost of 7 books draw a line parallel to Y-axis from X = 7 to the curve and from the intersecting point of that line and the curve draw another line parallel to X-axis and the point at which the line intersect the Y-axis will give the cost of 7 books.so from the graph cost of 7 books is found to be ₹105

(ii) to find the number of book that can be purchased at a cost of 165 draw a line parallel to X-axis from Y = 165.from the point at which the line intersect the curve draw another line parallel to Y-axis and the point at which the line intersect x-axis will give the number of book.so we can buy 11 note books

**Question 3.**

Draw the graph for the above table and hence find

(i) the value of y if x = 4

(ii) the value of x if y = 12

**Answer:**

Using the data we can draw a graph

(I) TO FIND THE VALUE OF the value of y if x = 4, draw a line parallel to Y-axis from X = 4 to the curve and from the intersecting point of that line and the curve draw another line parallel to X-axis and the point at which the line intersect the Y-axis will give the value of Y ay X = 4.so from the graph at X = 4 Y is found to be 8

(II) to find the value of X at Y = 12 draw a line parallel to X-axis from Y = 12 and from the point at which the line intersect the curve draw another line parallel to Y-axis and the point at which the line intersect x-axis will give the value of X.so we find at Y = 12 Xis found to be 6

**Question 4.**

The cost of the milk per liter is ₹15. Draw the graph for the relation between the quantity and cost. Hence find

(i) the proportionality constant.

(ii) the cost of 3 litres of milk.

**Answer:**

(i) k = 15 (ii) ₹45

Let the cost of milk be Y and X be the quantity of milk in liter. As it’s given that cost per one liter is 15rs so increase in quantity will increase the cost of milk. So it’s a direct proportion

Mathematically,

Y = KX

Where K is the constant of proportionality

So we can draw a table

so from the graph it’s clear that y = kx = 15x

Proportionality, k = 15

We can draw a graph using the table given

To find the cost of 3 liter milk we have to draw a line parallel to Y-axis at X = 3 then from the intersection point of that parallel line and the curve draw another line parallel to X-axis so the point of intersection of that line and Y-axis will give the cost of 3 liter of milk.

From the graph its clear that cost of 3 liter is 45 RS.

**Question 5.**

Draw the Graph of xy = 20, x, y > 0. Use the graph to find y when x = 5, and to find x when y = 10.

**Answer:**

We are given that xy

= 20

⇒ y = (20/x)

We can find different value of Y by taking values of x between 1 and 10

From the graph we get that at x = 5, y = 4, and to find y = 10, x = 2

**Question 6.**

Draw graph for the data given in the table. Hence find the number of days taken by 12 workers to complete the work.

**Answer:**

We can construct a graph using the data given

To find the number of days taken by 12 worker to complete the work draw a line parallel to Y-axis from X = 12 to the curve and from the intersecting point of that line and the curve draw another line parallel to X-axis and the point at which the line intersect the Y-axis will give the value of Y at X = 12.so from the graph at X = 12 Y is found to be 26