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NCERT Solutions for Class 9 Maths Chapter 12 Heron's Formula

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NCERT Solutions for Class 9 Maths Chapter 12 Heron's Formula

Edited By Ramraj Saini | Updated on May 08, 2023 02:50 PM IST

NCERT Solutions for Class 9 Maths Chapter 12 Heron's Formula

Heron's Formula Class 9 Questions And Answers are provided here. These NCERT solutions are created by expert team at Careers360 keeping in mind latest CBSE syllabus 2023-23. Practicing these solutions give you confidence during the exam which ultimately lead to score well in the exam. In this particular NCERT book chapter 12 maths class 9, you will learn a special technique to find the area of any triangle. NCERT Solutions for class 9 maths chapter 12 Heron's Formula is covering the explanation of all questions using Heron's formula.

Class 9 herons formula NCERT solutions are covering the solutions to each and every question present in the practice exercises in detail. If three sides of triangles are a, b, and c then the area in terms of a, b and c is given by the Heron's Formula class 9. The area is given by the formula \sqrt{s(s-a)(s-b)(s-c)} where s is half the perimeter. That is s=\frac{a+b+c}{2} . You may think about the application of Heron's Formula. This formula is used for calculating the area of irregular plots. If you have a land whose shape is not regular we will divide the plots into different triangles and then apply Heron's Formula class 9 to calculate the area. Here you will get NCERT solutions for class 9 also.

Also read:

Herons Formula Class 9 Questions And Answers PDF Free Download

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Herons Formula Class 9 Solutions - Important Formulae

Triangle:

Semi-perimeter of a triangle = s = (a + b + c)/2,

Where 's' is the semi-perimeter, and 'a', 'b', and 'c' are the lengths of its sides.

Area = √[s(s - a)(s - b)(s - c)]

Equilateral Triangle:

For an equilateral triangle with side length 'a':

  • Its perimeter is given by: Perimeter = 3a units.

  • The altitude (height) of an equilateral triangle is equal to √3/2 times its side length, Altitude = √3/2 * units.

  • The area of an equilateral triangle is equal to √3/4 times the square of its side length, Area = √3/4 * a^2 units. These formulas are specific to equilateral triangles and are based on their unique properties.

Free download NCERT Solutions for Class 9 Maths Chapter 12 Heron's Formula for CBSE Exam.

Herons Formula Class 9 NCERT Solutions (Intext Questions and Exercise)

Class 9 maths chapter 12 question answer - exercise: 12.1

Q1 A traffic signal board, indicating ‘SCHOOL AHEAD’, is an equilateral triangle with side ‘ a ’. Find the area of the signal board, using Heron’s formula. If its perimeter is 180 cm, what will be the area of the signal board?

Answer:

Given the perimeter of an equilateral triangle is 180cm.

So, 3a = 180\ cm or a = 60\ cm .

Hence, the length of the side is 60cm.

Now,

Calculating the area of the signal board by the Heron's Formula:

A = \sqrt{s(s-a)(s-b)(s-c)}

Where, s is the half-perimeter of the triangle and a, b and c are the sides of the triangle.

Therefore,

s = \frac{1}{2}Perimeter = \frac{1}{2}180cm = 90cm

a =b=c = 60cm as it is an equilateral triangle.

Substituting the values in the Heron's formula, we obtain

\implies A = \sqrt{90(90-60)(90-60)(90-60)} = 900\sqrt{3}\ cm^2 .

Q2 The triangular side walls of a flyover have been used for advertisements. The sides of the walls are 122 m, 22 m and 120 m (see Fig. 12.9). The advertisements yield an earning of Rs. 5000 per m 2 per year. A company hired one of its walls for 3 months. How much rent did it pay?

1640600558297 Answer:

From the figure,

The sides of the triangle are:

a = 122m,\ b = 120m\ and\ c = 22m

The semi perimeter, s will be

s = \frac{a+b+c}{2} = \frac{122+120+22}{2} = \frac{264}{2} = 132m

Therefore, the area of the triangular side wall will be calculated by the Heron's Formula,

A = \sqrt{s(s-a)(s-b)(s-c)}

= \sqrt{132(132-122)(132-120)(132-22)}\ m^2

= \sqrt{132(10)(12)(110)}\ m^2

= \sqrt{(12\times11)(10)(12)(11\times10)}\ m^2 = 1320\ m^2

Given the rent for 1 year (i.e., 12 months) per meter square is Rs. 5000.

Rent for 3 months per meter square will be:

Rs.\ 5000\times \frac{3}{12}

Therefore, for 3 months for 1320 m 2 :

Rs.\ 5000\times \frac{3}{12}\times 1320 = Rs.\ 16,50,000.


Q3 There is a slide in a park. One of its side walls has been painted in some colour with a message “KEEP THE PARK GREEN AND CLEAN” (see Fig. 12.10 ). If the sides of the wall are 15 m, 11 m and 6 m, find the area painted in colour.

1640600584782 Answer:

Given the sides of the triangle are:

a = 15 m,\ b= 11m\ and\ c= 6m.

So, the semi perimeter of the triangle will be:

s = \frac{a+b+c}{2} = \frac{15+11+6}{2} = \frac{32}{2} = 16m

Therefore, Heron's formula will be:

A = \sqrt{s(s-a)(s-b)(s-c)}

= \sqrt{16(16-15)(16-11)(16-6)}

= \sqrt{16(1)(5)(10)}

= \sqrt{(4\times 4)(1)(5)(5\times 2 )}

= 4\times 5 \sqrt2 = 20\sqrt2\ m^2

Hence, the area painted in colour is 20\sqrt2\ m^2 .

Q4 Find the area of a triangle two sides of which are 18cm and 10cm and the perimeter is 42cm.

Answer:

Given the perimeter of the triangle is 42cm and the sides length a= 18cm and b= 10cm

So, a+b+c = 42cm

Or, c = 42 - 18-10 = 14cm

So, the semi perimeter of the triangle will be:

s = \frac{P}{2} = \frac{42cm}{2} = 21cm

Therefore, the area given by the Heron's Formula will be,

A = \sqrt{s(s-a)(s-b)(s-c)}

= \sqrt{21(21-18)(21-10)(21-14)}

= \sqrt{(7\times3 )(3)(11)(7)}

= 21\sqrt{11}\ cm^2

Hence, the area of the triangle is 21\sqrt{11}\ cm^2.

Q5 Sides of a triangle are in the ratio of 12 : 17 : 25 and its perimeter is 540cm. Find its area.

Answer:

Given the sides of a triangle are in the ratio of 12:17:25 and its perimeter is 540cm

Let us consider the length of one side of the triangle be a = 12x

Then, the remaining two sides are b = 17x and c = 25x.

So, by the given perimeter, we can find the value of x:

Perimeter = a+b+c = 12x+17x+25x = 540cm

\implies 54x = 540cm

\implies x = 10

So, the sides of the triangle are:

a = 12\times10 =120 cm

b = 17\times10 =170 cm

c = 25\times10 =250 cm

So, the semi perimeter of the triangle is given by

s = \frac{540cm}{2} = 270cm

Therefore, using Heron's Formula, the area of the triangle is given by

A = \sqrt{s(s-a)(s-b)(s-c)}

= \sqrt{270(270-120)(270-170)(270-250)}

= \sqrt{270(150)(100)(20)}

= \sqrt{81000000} = 9000cm^2

Hence, the area of the triangle is 9000cm^2 .

Q6 An isosceles triangle has perimeter 30 cm and each of the equal sides is 12 cm. Find the area of the triangle.

Answer:

The perimeter of an isosceles triangle is 30 cm (Given).

The length of the sides which are equal is12 cm.

Let the third side length be 'a cm'.

Then, Perimeter = a+b+c

\Rightarrow 30= a+12+12

\Rightarrow a = 6cm

So, the semi-perimeter of the triangle is given by,

s= \frac{1}{2}Perimeter =\frac{1}{2}\times30cm = 15cm

Therefore, using Herons' Formula, calculating the area of the triangle

A = \sqrt{s(s-a)(s-b)(s-c)}

= \sqrt{15(15-6)(15-12)(15-12)}

= \sqrt{15(9)(3)(3)}

= 9\sqrt{15}\ cm^2

Hence, the area of the triangle is 9\sqrt{15}cm^2.

Class 9 maths chapter 12 NCERT solutions - exercise : 12.2

Q1 A park, in the shape of a quadrilateral ABCD, has \angle C = 90\degree , AB = 9 m , BC = 12 m , CD = 5 m and AD = 8 m . How much area does it occupy?

Answer:

From the figure:

1640600614678

We have joined the BD to form two triangles so that the calculation of the area will be easy.

In triangle BCD, by Pythagoras theorem

BD^2 = BC^2+CD^2

\Rightarrow BD^2 = 12^2+5^2 = 144+25 = 169

\Rightarrow BD = 13\ cm

The area of triangle BCD can be calculated by,

Area_{(BCD)} = \frac{1}{2}\times BC\times DC = \frac{1}{2}\times 12\times 5 = 30\ cm^2

and the area of the triangle DAB can be calculated by Heron's Formula:

So, the semi-perimeter of the triangle DAB,

s = \frac{a+b+c}{2} = \frac{9+8+13}{2} = \frac{30}{2} = 15\ cm

Therefore, the area will be:

A = \sqrt{s(s-a)(s-b)(s-c)}

where, a = 9 cm,\ b = 8cm\ and\ c = 13cm .

= \sqrt{15(15-9)(15-8)(15-13)}

= \sqrt{12(6)(7)(2)} = \sqrt{1260} = 35.5\ cm^2 (Approximately)

Then, the total park area will be:

= Area\ of\ triangle\ BCD + Area\ of\ triangle\ DAB

\Rightarrow Total\ area\ of\ Park = 30+35.35 = 65.5\ cm^2

Hence, the total area of the park is 65.5\ cm^2.

Q2 Find the area of a quadrilateral ABCD in which AB = 3 cm, BC = 4 cm, CD = 4 cm, DA = 5 cm and AC = 5 cm.

Answer:

From the figure:

1640600632801

We have joined the AC to form two triangles so that the calculation of the area will be easy.

The area of the triangle ABC can be calculated by Heron's formula:

So, the semi-perimeter, where a = 3 cm,\ b =4cm\ and\ c = 5cm.

s = \frac{a+b+c}{2} = \frac{3+4+5}{2} = 6cm

Heron's Formula for calculating the area:

A = \sqrt{s(s-a)(s-b)(s-c)}

= \sqrt{6(6-3)(6-4)(6-5)}= \sqrt{6(3)(2)(1)} = \sqrt{36} = 6\ cm^2

And the sides of the triangle ACD are a' =4cm,\ b' = 5cm\ and\ c' = 5cm.

So, the semi-perimeter of the triangle:

s' = \frac{a'+b'+c'}{2} = \frac{4+5+5}{2} = \frac{14}{2} = 7cm

Therefore, the area will be given by, Heron's formula

A = \sqrt{s'(s'-a')(s'-b')(s'-c')} .

= \sqrt{7(7-4)(7-5)(7-5)}

= \sqrt{7(3)(2)(2)} = 2\sqrt{21} = 9.2\ cm^2\ \ \ \ (Approx.)

Then, the total area of the quadrilateral will be:

= Area\ of\ triangle\ ABC + Area\ of\ triangle\ ACD

\Rightarrow Total\ area\ of\ quadrilateral\ ABCD = 6+9.2 = 15.2\ cm^2

Hence, the area of the quadrilateral ABCD is 15.2 \ cm^2.

Q3 Radha made a picture of an aeroplane with coloured paper as shown in Fig 12.15. Find the total area of the paper used.

1640600656571 Answer:

The total area of the paper used will be the sum of the area of the sections I, II, III, IV, and V. i.e., Total\ area = I +II+III+IV+V

For section I:

Here, the sides are a = 1cm\ and\ b =c = 5cm.

So, the Semi-perimeter will be:

s = \frac{a+b+c}{2} = \frac{5+5+1}{2} = 5.5\ cm

Therefore, the area of section I will be given by Heron's Formula,

A = \sqrt{s(s-a)(s-b)(s-c)}

= \sqrt{5.5(5.5-1)(5.5-5)(5.5-5)}

= \sqrt{5.5(4.5)(0.5)(0.5)} = \sqrt{6.1875} = 2.5\ cm^2\ \ \ \ \ \ (Approx.)

For section II:

Here the sides of the rectangle are l =6.5\ cm and b =1 \ cm.

Therefore, the area of the rectangle is = l\times b = 6.5\times 1 = 6.5\ cm^2.

For section III:

1640600697563

From the figure:

Drawing the parallel line AF to DC and a perpendicular line AE to BC.

We have the quadrilateral ADCF,

AF || DC ...........................by construction.

AD || FC ...........................[ \because ABCD is a trapezium]

So, ADCF is a parallelogram.

Therefore, AF = DC = 1\ cm and AD = FC = 1\ cm

\left [ \because Opposite\ sides\ of\ a\ parallelogram \right ]

Therefore, BF = BC -FC =2-1 = 1\ cm.

\implies ABF is an equilateral triangle. \left [ \because AB = BF =AF = 1\ cm \right ]

Then, the area of the equilateral triangle ABF is given by:

\implies \frac{\sqrt3}{4}a^2 = \frac{\sqrt3}{4}1^2 = \frac{\sqrt3}{4}

= \frac{1}{2}\times BF \times AE

= \frac{1}{2}\times 1cm \times AE = \frac{\sqrt3}{4}

\implies AE = \frac{\sqrt3}{2} = \frac{1.732}{2} = 0.866 \approx 0.9

Hence, the area of trapezium ABCD will be:

= \frac{1}{2}\times(AD+BC)\times AE

= \frac{1}{2}\times(1+2)\times 0.9

=1.35 =1.4\ cm^2\ \ \ \ (Approx.)

For Section IV:

Here, the base is 1.5 cm and the height is 6 cm.

Therefore, the area of the triangle is :

= \frac{1}{2}\times base\times height

= \frac{1}{2}\times 1.5\times 6 = 4.5\ cm^2

For section V:

The base length = 1.5cm and the height is 6cm.

Therefore, the area of the triangle will be:

= \frac{1}{2}\times 1.5\times 6 = 4.5\ cm^2

Hence, the total area of the paper used will be:

Total\ area = I +II+III+IV+V

= 2.5+6.5+1.4+4.5+4.5 = 19.4\ cm^2


Q4 A triangle and a parallelogram have the same base and the same area. If the sides of the triangle are 26 cm, 28 cm and 30 cm, and the parallelogram stands on the base 28 cm, find the height of the parallelogram.

Answer:

From the figure:

1640600726893 The sides of the triangle are a = 26\ cm,b= 28\ cm\ and\ c =30\ cm.

Then, calculating the area of the triangle:

So, the semi-perimeter of triangle ABE,

s = \frac{a+b+c}{2} = \frac{28+26+30}{2} = 42\ cm.

Therefore, its area will be given by the Heron's formula:

A = \sqrt{s(s-a)(s-b)(s-c)}

= \sqrt{42(42-28)(42-26)(42-30)}

= \sqrt{42(14)(16)(12)} = \sqrt{112896} = 336\ cm^2

Given that the area of the parallelogram is equal to the area of the triangle:

Area\ of\ Parallelogram = Area\ of\ Triangle

\implies base\times corresponding\ height = 336\ cm^2

\implies 28\times corresponding\ height = 336\ cm^2

\implies height = \frac{336}{28} = 12\ cm.

Hence, the height of the parallelogram is 12 cm.

Q5 A rhombus shaped field has green grass for 18 cows to graze. If each side of the rhombus is 30 m and its longer diagonal is 48 m, how much area of grass field will each cow be getting?

Answer:

To find the area of the rhombus:

We first join the diagonal AC of quadrilateral ABCD. (See figure)

1640600753242 Here, the sides of triangle ABC are,

a = 30\ m,\ b = 30\ m\ and\ c = 48\ m.

So, the semi-perimeter of the triangle will be:

s = \frac{a+b+c}{2} = \frac{30+30+48}{2} = \frac{108}{2} = 54\ m

Therefore, the area of the triangle given by the Heron's formula,

Area = \sqrt{s(s-a)(s-b)(s-c)}

= \sqrt{54(54-30)(54-30)(54-48)}

= \sqrt{54(24)(24)(6)} = \sqrt{186624} = 432\ m^2

Hence, the area of the quadrilateral will be:

= 2\times 432\ m^2 = 864\ m^2

Therefore, the area grazed by each cow will be given by,

= \frac{Total\ area}{Number\ of\ cows} = \frac{864}{18} = 48\ m^2.

Q6 An umbrella is made by stitching 10 triangular pieces of cloth of two different colours (see Figure), each piece measuring 20 cm, 50 cm and 50 cm. How much cloth of each colour is required for the umbrella?

1640600779979 Answer:

The sides of the triangle are:

a = 20\ cm,\ b = 50\ cm\ and\ c =50\ cm.

So, the semi-perimeter of the triangle is given by,

s = \frac{a+b+c}{2} = \frac{20+50+50}{2} = \frac{120}{2} = 60\ cm.

Therefore, the area of the triangle can be found by using Heron's formula:

Area = \sqrt{s(s-a)(s-b)(s-c)}

= \sqrt{60(60-20)(60-50)(60-50)}

= \sqrt{60(40)(10)(10)} = 200\sqrt{6}\ cm^2

Now, for the 10 triangular pieces of cloths, the area will be,

= 10\times200\sqrt6 = 2000\sqrt6\ cm^2

Hence, the area of cloths of each colour will be:

= \frac{2000\sqrt6}{2} = 1000\sqrt6\ cm^2.

Q7 A kite in the shape of a square with a diagonal 32 cm and an isosceles triangle of base 8 cm and sides 6 cm each is to be made of three different shades as shown in Figure How much paper of each shade has been used in it?

1640600801509 Answer:

From the figure:

1640600823417 Calculation of the area for each shade:

The shade I: Triangle ABD

Here, base BD = 32\ cm and the height AO = 16\ cm.

Therefore, the area of triangle ABD will be:

= \frac{1}{2} \times base\times height = \frac{1}{2}\times 32\times 16

= 256\ cm^2

Hence, the area of paper used in shade I is 256\ cm^2.

Shade II: Triangle CBD

Here, base BD = 32\ cm and height CO = 16\ cm.

Therefore, the area of triangle CBD will be:

= \frac{1}{2} \times base\times height = \frac{1}{2}\times 32\times 16

= 256\ cm^2

Hence, the area of paper used in shade II is 256\ cm^2.

Shade III: Triangle CEF

Here, the sides are of lengths, a = 6\ cm,\ b = 6\ cm\ and\ c = 8\ cm.

So, the semi-perimeter of the triangle:

s = \frac{a+b+c}{2} = \frac{6+6+8}{2} = \frac{20}{2} = 10\ cm.

Therefore, the area of the triangle can be found by using Heron's formula:

Area = \sqrt{s(s-a)(s-b)(s-c)}

= \sqrt{10(10-6)(10-6)(10-8)}

= \sqrt{10(4)(6)(2)}

= 8\sqrt{5}\ cm^2

Hence, the area of the paper used in shade III is 8\sqrt{5}\ cm^2 .

Q8 A floral design on a floor is made up of 16 tiles which are triangular, the sides of the triangle being 9 cm, 28 cm and 35 cm (see Fig. 12.18). Find the cost of polishing the tiles at the rate of 50p per cm 2 .

1640600847170 Answer:

GIven the sides of the triangle are:

a = 9\ cm,\ b = 28\ cm\ and\ c= 35\ cm.

So, its semi-perimeter will be:

s = \frac{a+b+c}{2} = \frac{9+28+35}{2} = 36\ cm

Therefore, the area of the triangle using Heron's formula is given by,

Area = \sqrt{s(s-a)(s-b)(s-c)}

= \sqrt{36(36-9)(36-28)(36-35)} = \sqrt{36(27)(8)(1)}

= \sqrt{7776} \approx 88.2\ cm^2

So, we have the area of each triangle tile which is 88.2\ cm^2 .

Therefore, the area of each triangular 16 tiles will be:

= 16\times 88.2\ cm^2 = 1411.2\ cm^2

Hence, the cost of polishing the tiles at the rate of 50 paise per cm 2 will be:

= Rs.\ 0.50\times1411.2\ cm^2 =Rs.\ 705.60


Q9 A field is in the shape of a trapezium whose parallel sides are 25 m and 10 m. The non-parallel sides are 14 m and 13 m. Find the area of the field.

Answer:

The trapezium field is shown below in figure:

Drawing line CF parallel to AD and a line perpendicular to AB, we obtain

1640600870807 Then in quadrilateral ADCF,

CF || AD ............................ \left [ \because\ by\ construction \right ]

CD || AF ............................ \left [ \because ABCD\ is\ a\ trapezium \right ]

Therefore, ADCF is a parallelogram.

So, AD = CF = 13\ m and CD = AF = 10\ m

\left ( \because Opposite\ sides\ of\ a\ parallelogram \right )

Therefore, BF = AB-AF = 25-10 = 15\ m

Now, the sides of the triangle;

a = 13\ m,\ b =14\ m\ and\ c = 15\ m.

So, the semi-perimeter of the triangle will be:

s= \frac{a+b+c}{2} = \frac{13+14+15}{2} = \frac{42}{2} = 21\ m

Therefore, the area of the triangle can be found by using Heron's Formula:

Area = \sqrt{s(s-a)(s-b)(s-c)}

= \sqrt{21(21-13)(21-14)(21-15)}

= \sqrt{21(8)(7)(6)}

= \sqrt{7056} = 84\ m^2.

Also, the area of the triangle is given by,

Area = \frac{1}{2}\times BF\times CG

\Rightarrow \frac{1}{2}\times BF\times CG = 84\ m^2

\Rightarrow \frac{1}{2}\times 15\times CG = 84\ m^2

Or,

\Rightarrow CG = \frac{84\times2}{15} = 11.2\ m

Therefore, the area of the trapezium ABCD is:

= \frac{1}{2} \times (AB+CD)\times CG

= \frac{1}{2} \times (25+10)\times 11.2

= 35\times5.6

= 196\ m^2

Hence, the area of the trapezium field is 196\ m^2.

Summary Of NCERT Solutions for Class 9 Maths Chapter 12 Heron's Formula

Chapter 12 of Class 9 Maths, "Heron’s Formula," is part of Unit 5: Mensuration, which carries a weightage of 13 marks out of 100 in the examination. Due to this, it is an essential chapter that requires thorough study. The chapter covers critical topics, including the area of a triangle using Heron’s formula and the application of Heron’s formula in finding areas of quadrilaterals. Heron's formula is used to calculate the area of a triangle using its three side lengths. Students who are interested in class 9 maths ch 12 question answer, can practice exercise using link given below.

NCERT solutions for class 9 maths - chapter wise

Chapter No.
Chapter Name
Chapter 1
Chapter 2
Chapter 3
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Chapter 10
Chapter 11
Chapter 12
Heron’s Formula
Chapter 13
Chapter 14
Chapter 15

Key Features of NCERT Solutions for Class 9 Maths Chapter 12 Heron’s Formula

Comprehensive coverage: The maths chapter 12 class 9 solutions provide complete coverage of the chapter's topics, including the derivation and application of Heron's formula for finding the area of a triangle and quadrilaterals.

Easy-to-understand language: The ch 12 maths class 9 solutions use simple and easy-to-understand language, making it easier for students to comprehend the mathematical concepts and solve problems.

Helps in scoring good marks: Practising these herons formula class 9 solutions can greatly enhance a student's chances of scoring well in their exams, as the solutions provide a comprehensive understanding of the chapter's concepts.

NCERT solutions for class 9 subject wise

How to use NCERT solutions for class 9 maths chapter 12 Heron's Formula?

  • Learn some basics about the triangles.
  • Read Heron's formula.
  • Observe its application to mathematical problems.
  • Now, It's time to apply the application of this formula to the practice exercises' problems.
  • While practicing the exercises, you can take the help of NCERT solutions for class 9 maths chapter 12 Heron's Formula.

Keep working hard and happy learning!

Also Check NCERT Books and NCERT Syllabus here:

Frequently Asked Question (FAQs)

1. What are the important topics in chapter Heron's Formula

Area of a Triangle – by Heron’s Formula and Application of Heron’s Formula in finding Areas of Quadrilaterals are two important topics of this chapter. Students can prioritize important topics from the NCERT syllabus and study accordingly to score well in exams. for ease you can study heron's formula class 9 pdf  both online and offline mode.

2. What are the key benefits of learning NCERT Solutions for class 9 chapter 12 maths?

Here are the rephrased key benefits of NCERT Solutions for Class 9 Maths Chapter 12:

  1. The solutions for each exercise within the chapter are easily accessible for students to refer to.

  2. The solutions are designed with graphs and illustrations that aid in providing a clear understanding of the mathematical concepts.

  3. The solutions are meticulously prepared by the expert team at Careers360, with a strong emphasis on accuracy.

3. How does the NCERT solutions are helpful ?

NCERT solutions are helpful for the students if they are no able to solve NCERT problems on their own.  Also, these solutions are provided in a very detailed manner which will give them conceptual clarity.

4. Where can I find the complete solutions of NCERT for class 9 maths ?

Here you will get the detailed NCERT solutions for class 9 maths by clicking on the link. Practicing these solutions will give you in-depth understanding of concepts which help you to score good marks in the exam.

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K/2\,

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Option 1)

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6L\, HCl_{(aq)}  is consumed for ever 3L\, H_{2(g)}      produced

Option 3)

33.6 L\, H_{2(g)} is produced regardless of temperature and pressure for every mole Al that reacts

Option 4)

67.2\, L\, H_{2(g)} at STP is produced for every mole Al that reacts .

How many moles of magnesium phosphate, Mg_{3}(PO_{4})_{2} will contain 0.25 mole of oxygen atoms?

Option 1)

0.02

Option 2)

3.125 × 10-2

Option 3)

1.25 × 10-2

Option 4)

2.5 × 10-2

If we consider that 1/6, in place of 1/12, mass of carbon atom is taken to be the relative atomic mass unit, the mass of one mole of a substance will

Option 1)

decrease twice

Option 2)

increase two fold

Option 3)

remain unchanged

Option 4)

be a function of the molecular mass of the substance.

With increase of temperature, which of these changes?

Option 1)

Molality

Option 2)

Weight fraction of solute

Option 3)

Fraction of solute present in water

Option 4)

Mole fraction.

Number of atoms in 558.5 gram Fe (at. wt.of Fe = 55.85 g mol-1) is

Option 1)

twice that in 60 g carbon

Option 2)

6.023 × 1022

Option 3)

half that in 8 g He

Option 4)

558.5 × 6.023 × 1023

A pulley of radius 2 m is rotated about its axis by a force F = (20t - 5t2) newton (where t is measured in seconds) applied tangentially. If the moment of inertia of the pulley about its axis of rotation is 10 kg m2 , the number of rotations made by the pulley before its direction of motion if reversed, is

Option 1)

less than 3

Option 2)

more than 3 but less than 6

Option 3)

more than 6 but less than 9

Option 4)

more than 9

Data Administrator

Database professionals use software to store and organise data such as financial information, and customer shipping records. Individuals who opt for a career as data administrators ensure that data is available for users and secured from unauthorised sales. DB administrators may work in various types of industries. It may involve computer systems design, service firms, insurance companies, banks and hospitals.

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Bio Medical Engineer

The field of biomedical engineering opens up a universe of expert chances. An Individual in the biomedical engineering career path work in the field of engineering as well as medicine, in order to find out solutions to common problems of the two fields. The biomedical engineering job opportunities are to collaborate with doctors and researchers to develop medical systems, equipment, or devices that can solve clinical problems. Here we will be discussing jobs after biomedical engineering, how to get a job in biomedical engineering, biomedical engineering scope, and salary. 

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Ethical Hacker

A career as ethical hacker involves various challenges and provides lucrative opportunities in the digital era where every giant business and startup owns its cyberspace on the world wide web. Individuals in the ethical hacker career path try to find the vulnerabilities in the cyber system to get its authority. If he or she succeeds in it then he or she gets its illegal authority. Individuals in the ethical hacker career path then steal information or delete the file that could affect the business, functioning, or services of the organization.

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GIS Expert

GIS officer work on various GIS software to conduct a study and gather spatial and non-spatial information. GIS experts update the GIS data and maintain it. The databases include aerial or satellite imagery, latitudinal and longitudinal coordinates, and manually digitized images of maps. In a career as GIS expert, one is responsible for creating online and mobile maps.

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Data Analyst

The invention of the database has given fresh breath to the people involved in the data analytics career path. Analysis refers to splitting up a whole into its individual components for individual analysis. Data analysis is a method through which raw data are processed and transformed into information that would be beneficial for user strategic thinking.

Data are collected and examined to respond to questions, evaluate hypotheses or contradict theories. It is a tool for analyzing, transforming, modeling, and arranging data with useful knowledge, to assist in decision-making and methods, encompassing various strategies, and is used in different fields of business, research, and social science.

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Individuals who opt for a career as geothermal engineers are the professionals involved in the processing of geothermal energy. The responsibilities of geothermal engineers may vary depending on the workplace location. Those who work in fields design facilities to process and distribute geothermal energy. They oversee the functioning of machinery used in the field.

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Database Architect

If you are intrigued by the programming world and are interested in developing communications networks then a career as database architect may be a good option for you. Data architect roles and responsibilities include building design models for data communication networks. Wide Area Networks (WANs), local area networks (LANs), and intranets are included in the database networks. It is expected that database architects will have in-depth knowledge of a company's business to develop a network to fulfil the requirements of the organisation. Stay tuned as we look at the larger picture and give you more information on what is db architecture, why you should pursue database architecture, what to expect from such a degree and what your job opportunities will be after graduation. Here, we will be discussing how to become a data architect. Students can visit NIT Trichy, IIT Kharagpur, JMI New Delhi

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Individuals who opt for a career as a remote sensing technician possess unique personalities. Remote sensing analysts seem to be rational human beings, they are strong, independent, persistent, sincere, realistic and resourceful. Some of them are analytical as well, which means they are intelligent, introspective and inquisitive. 

Remote sensing scientists use remote sensing technology to support scientists in fields such as community planning, flight planning or the management of natural resources. Analysing data collected from aircraft, satellites or ground-based platforms using statistical analysis software, image analysis software or Geographic Information Systems (GIS) is a significant part of their work. Do you want to learn how to become remote sensing technician? There's no need to be concerned; we've devised a simple remote sensing technician career path for you. Scroll through the pages and read.

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Budget Analyst

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The invention of the database has given fresh breath to the people involved in the data analytics career path. Analysis refers to splitting up a whole into its individual components for individual analysis. Data analysis is a method through which raw data are processed and transformed into information that would be beneficial for user strategic thinking.

Data are collected and examined to respond to questions, evaluate hypotheses or contradict theories. It is a tool for analyzing, transforming, modeling, and arranging data with useful knowledge, to assist in decision-making and methods, encompassing various strategies, and is used in different fields of business, research, and social science.

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Underwriter

An underwriter is a person who assesses and evaluates the risk of insurance in his or her field like mortgage, loan, health policy, investment, and so on and so forth. The underwriter career path does involve risks as analysing the risks means finding out if there is a way for the insurance underwriter jobs to recover the money from its clients. If the risk turns out to be too much for the company then in the future it is an underwriter who will be held accountable for it. Therefore, one must carry out his or her job with a lot of attention and diligence.

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Individuals in the operations manager jobs are responsible for ensuring the efficiency of each department to acquire its optimal goal. They plan the use of resources and distribution of materials. The operations manager's job description includes managing budgets, negotiating contracts, and performing administrative tasks.

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Stock Analyst

Individuals who opt for a career as a stock analyst examine the company's investments makes decisions and keep track of financial securities. The nature of such investments will differ from one business to the next. Individuals in the stock analyst career use data mining to forecast a company's profits and revenues, advise clients on whether to buy or sell, participate in seminars, and discussing financial matters with executives and evaluate annual reports.

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Researcher

A Researcher is a professional who is responsible for collecting data and information by reviewing the literature and conducting experiments and surveys. He or she uses various methodological processes to provide accurate data and information that is utilised by academicians and other industry professionals. Here, we will discuss what is a researcher, the researcher's salary, types of researchers.

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Welding Engineer Job Description: A Welding Engineer work involves managing welding projects and supervising welding teams. He or she is responsible for reviewing welding procedures, processes and documentation. A career as Welding Engineer involves conducting failure analyses and causes on welding issues. 

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A career as Transportation Planner requires technical application of science and technology in engineering, particularly the concepts, equipment and technologies involved in the production of products and services. In fields like land use, infrastructure review, ecological standards and street design, he or she considers issues of health, environment and performance. A Transportation Planner assigns resources for implementing and designing programmes. He or she is responsible for assessing needs, preparing plans and forecasts and compliance with regulations.

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Environmental Engineer

Individuals who opt for a career as an environmental engineer are construction professionals who utilise the skills and knowledge of biology, soil science, chemistry and the concept of engineering to design and develop projects that serve as solutions to various environmental problems. 

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Conservation Architect

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For an individual who opts for a career as an actor, the primary responsibility is to completely speak to the character he or she is playing and to persuade the crowd that the character is genuine by connecting with them and bringing them into the story. This applies to significant roles and littler parts, as all roles join to make an effective creation. Here in this article, we will discuss how to become an actor in India, actor exams, actor salary in India, and actor jobs. 

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Depending on the video game designer job description and experience they may also have to lead a team and do the early testing of the game in order to suggest changes and find loopholes.

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A career as radio jockey has a lot to offer to deserving candidates. If you want to know more about a career as radio jockey, and how to become a radio jockey then continue reading the article.

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Social Media Manager

A career as social media manager involves implementing the company’s or brand’s marketing plan across all social media channels. Social media managers help in building or improving a brand’s or a company’s website traffic, build brand awareness, create and implement marketing and brand strategy. Social media managers are key to important social communication as well.

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Photography is considered both a science and an art, an artistic means of expression in which the camera replaces the pen. In a career as a photographer, an individual is hired to capture the moments of public and private events, such as press conferences or weddings, or may also work inside a studio, where people go to get their picture clicked. Photography is divided into many streams each generating numerous career opportunities in photography. With the boom in advertising, media, and the fashion industry, photography has emerged as a lucrative and thrilling career option for many Indian youths.

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Producer

An individual who is pursuing a career as a producer is responsible for managing the business aspects of production. They are involved in each aspect of production from its inception to deception. Famous movie producers review the script, recommend changes and visualise the story. 

They are responsible for overseeing the finance involved in the project and distributing the film for broadcasting on various platforms. A career as a producer is quite fulfilling as well as exhaustive in terms of playing different roles in order for a production to be successful. Famous movie producers are responsible for hiring creative and technical personnel on contract basis.

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In a career as a copywriter, one has to consult with the client and understand the brief well. A career as a copywriter has a lot to offer to deserving candidates. Several new mediums of advertising are opening therefore making it a lucrative career choice. Students can pursue various copywriter courses such as Journalism, Advertising, Marketing Management. Here, we have discussed how to become a freelance copywriter, copywriter career path, how to become a copywriter in India, and copywriting career outlook. 

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In a career as a vlogger, one generally works for himself or herself. However, once an individual has gained viewership there are several brands and companies that approach them for paid collaboration. It is one of those fields where an individual can earn well while following his or her passion. 

Ever since internet costs got reduced the viewership for these types of content has increased on a large scale. Therefore, a career as a vlogger has a lot to offer. If you want to know more about the Vlogger eligibility, roles and responsibilities then continue reading the article. 

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For publishing books, newspapers, magazines and digital material, editorial and commercial strategies are set by publishers. Individuals in publishing career paths make choices about the markets their businesses will reach and the type of content that their audience will be served. Individuals in book publisher careers collaborate with editorial staff, designers, authors, and freelance contributors who develop and manage the creation of content.

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Careers in journalism are filled with excitement as well as responsibilities. One cannot afford to miss out on the details. As it is the small details that provide insights into a story. Depending on those insights a journalist goes about writing a news article. A journalism career can be stressful at times but if you are someone who is passionate about it then it is the right choice for you. If you want to know more about the media field and journalist career then continue reading this article.

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Individuals in the editor career path is an unsung hero of the news industry who polishes the language of the news stories provided by stringers, reporters, copywriters and content writers and also news agencies. Individuals who opt for a career as an editor make it more persuasive, concise and clear for readers. In this article, we will discuss the details of the editor's career path such as how to become an editor in India, editor salary in India and editor skills and qualities.

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QA Manager
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Quality Controller

A quality controller plays a crucial role in an organisation. He or she is responsible for performing quality checks on manufactured products. He or she identifies the defects in a product and rejects the product. 

A quality controller records detailed information about products with defects and sends it to the supervisor or plant manager to take necessary actions to improve the production process.

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Process Development Engineer

The Process Development Engineers design, implement, manufacture, mine, and other production systems using technical knowledge and expertise in the industry. They use computer modeling software to test technologies and machinery. An individual who is opting career as Process Development Engineer is responsible for developing cost-effective and efficient processes. They also monitor the production process and ensure it functions smoothly and efficiently.

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AWS Solution Architect

An AWS Solution Architect is someone who specializes in developing and implementing cloud computing systems. He or she has a good understanding of the various aspects of cloud computing and can confidently deploy and manage their systems. He or she troubleshoots the issues and evaluates the risk from the third party. 

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Azure Administrator

An Azure Administrator is a professional responsible for implementing, monitoring, and maintaining Azure Solutions. He or she manages cloud infrastructure service instances and various cloud servers as well as sets up public and private cloud systems. 

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Computer Programmer

Careers in computer programming primarily refer to the systematic act of writing code and moreover include wider computer science areas. The word 'programmer' or 'coder' has entered into practice with the growing number of newly self-taught tech enthusiasts. Computer programming careers involve the use of designs created by software developers and engineers and transforming them into commands that can be implemented by computers. These commands result in regular usage of social media sites, word-processing applications and browsers.

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Information Security Manager

Individuals in the information security manager career path involves in overseeing and controlling all aspects of computer security. The IT security manager job description includes planning and carrying out security measures to protect the business data and information from corruption, theft, unauthorised access, and deliberate attack 

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Automation Test Engineer

An Automation Test Engineer job involves executing automated test scripts. He or she identifies the project’s problems and troubleshoots them. The role involves documenting the defect using management tools. He or she works with the application team in order to resolve any issues arising during the testing process. 

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