**JEE Main Mathematics Syllabus**: All three of the JEE Main 2022 papers—the B.E./B. Tech paper, the B. Arch, and the B. Planning Paper—share the same JEE Main Mathematics Syllabus. The mathematics curriculum combines topics that are both straightforward and challenging. Top engineering schools in the nation can be accessed with a systematic approach that takes topic weighting into account when approaching the most crucial section of the JEE Main 2022 exam.

The majority of test-takers report that 25% of the questions are simple, 25% are challenging, and the remaining 50% are rated as being of a medium difficulty level. We have addressed a variety of topics to help candidates with the JEE Main Mathematics Syllabus. Read the article for information on the mathematics curriculum by unit, topic weighting, how to maximize your math grade, and advice on how to answer questions.

Table of Contents

## JEE Main Mathematics Syllabus Unit-wise by NTA

The mathematics curriculum for JEE Main 2022 is broken up into 16 different units.

Topic | Sub-topics |
---|---|

Sets, Relations, and Functions | Sets and their representation; Union, intersection, and complement of sets and their algebraic properties; Powerset; Relation, Types of relations, equivalence relations; Functions; one-one, into and onto functions, the composition of functions. |

Trigonometry | Identities of Trigonometry and Trigonometric equations; Functions of Trigonometry; Properties of Inverse trigonometric functions. Problems on Heights and Distances. |

Complex Numbers and Quadratic Equations | Complex numbers as ordered pairs of reals. Representation of complex numbers in the form (a+ib) and their representation in a plane, Argand diagram; Algebra of complex numbers, modulus and argument (or amplitude) of a complex number, square root of a complex number. Triangle inequality; Quadratic equations in real and complex number systems and their solutions; The relation between roots and coefficients, nature of roots, the formation of quadratic equations with given roots. |

Vector Algebra | Scalars and Vectors. Addition, subtraction, multiplication and division of vectors; Vector’s Components in 2D and 3D space; Scalar products and vector products, triple products. |

Permutations and Combinations | The fundamental principle of counting; Permutation as an arrangement and combination as a selection; The meaning of P (n,r) and C (n,r). Simple applications. |

Differential Equations | Ordinary differential equations, their order, and degree; Formation of differential equations; The solution of differential equations by the method of separation of variables; The solution of homogeneous and linear differential equations. |

Binomial Theorem | Binomial theorem for a positive integral index; General term and middle term; Properties of Binomial coefficients and simple applications. |

Coordinate Geometry | Cartesian system of rectangular coordinates in a plane, distance formula, section formula, locus and its equation, translation of axes, the slope of a line, parallel and perpendicular lines, intercepts of a line on the coordinate axes; Straight lines: Various forms of equations of a line, intersection of lines, angles between two lines, conditions for concurrence of three lines; Distance of a point from a line, equations of internal and external bisectors of angles between two lines, coordinates of the centroid, orthocentre, and circumcentre of a triangle, equation of the family of lines passing through the point of intersection of two lines; Circles, conic sections: Standard form of equation of a circle, general form of the equation of a circle, its radius and centre, equation of a circle when the endpoints of a diameter are given, points of intersection of a line and a circle with the centre at the origin and condition for a line to be tangent to a circle, equation of the tangent; Sections of cones, equations of conic sections (parabola, ellipse, and hyperbola) in standard forms, condition for y = mx + c to be a tangent and point (s) of tangency. |

Integral Calculus | Integral as an antiderivative; Fundamental integrals involving algebraic, trigonometric, exponential, and logarithmic functions; Integration by substitution, by parts, and by partial fractions; Integration using trigonometric identities. Integral as limit of a sum; Evaluation of simple integrals; Fundamental Theorem of Calculus; Properties of definite integrals, evaluation of definite integrals, determining areas of the regions bounded by simple curves in standard form. |

Sequence and Series | Arithmetic and Geometric progressions, insertion of arithmetic; Geometric means between two given numbers; The relation between A.M. and G.M; Sum up to n terms of special series: Sn, Sn2, Sn3; Arithmetic Geometric progression. |

Limit, Continuity and Differentiability | Real-valued functions, algebra of functions, polynomials, rational, trigonometric, logarithmic and exponential functions, inverse functions; Graphs of simple functions; Limits, continuity, and differentiability. Differentiation of the sum, difference, product, and quotient of two functions; Differentiation of trigonometric, inverse trigonometric, logarithmic, exponential, composite and implicit functions; derivatives of order up to two; Rolle’s and Lagrange’s Mean Value Theorems; Applications of derivatives: Rate of change of quantities, monotonic increasing and decreasing functions, Maxima, and minima of functions of one variable, tangents, and normals. |

3D Geometry | Coordinates of a point in space, the distance between two points; Section formula, direction ratios and direction cosines, the angle between two intersecting lines; Skew lines, the shortest distance between them and its equation; Equations of a line and a plane in different forms, the intersection of a line and a plane, coplanar lines. |

Mathematical Induction | The principle of Mathematical Induction and its simple applications. |

Statistics and Probability | Measures of Dispersion: Calculation of mean, mode, median, variance, standard deviation, and mean deviation of ungrouped and grouped data; Probability: Probability of events, multiplication theorems, addition theorems, Bayes theorem, Bernoulli trials, Binomial distribution and probability distribution. |

Matrices and Determinants | Matrices: Algebra of matrices, types of matrices, and matrices of order two and three; Determinants: Properties of determinants, evaluation of determinants, the area of triangles using determinants; Adjoint and evaluation of inverse of a square matrix using determinants and elementary transformations; Test of consistency and solution of simultaneous linear equations in two or three variables using determinants and matrices. |

Mathematical Reasoning | Statements and logical operations: or, and, implied by, implies, only if and if; Understanding of contradiction, tautology, contrapositive and converse. |

## Topic-wise Weightage for JEE Main Mathematics Syllabus 2022

The following is a breakdown of the topic-wise weightage for the JEE Main mathematics exam, based on trends from the previous year:

JEE Main Mathematics Topics | Weightage |
---|---|

Coordinate Geometry | 17% |

Binomial Theorem | 3% |

Differential Calculus | 17% |

Statistics | 3% |

Matrices and Determinants | 7% |

Permutation and Combination | 3% |

Trigonometry | 7% |

Probability | 3% |

Quadratic Equation | 3% |

Mathematical Reasoning | 3% |

Sequence and Series | 7% |

Algebra | 3% |

Integral Calculus | 14% |

Complex Numbers | 3% |

## Maximize Your Score in JEE Main 2022 Mathematics

- Prioritize Calculus and Algebra: Since these two subjects make up a big part of the JEE Main Mathematics Question Paper, candidates must be done with them by practicing and getting any questions answered.
- Algebra is one of the best and easiest parts of the JEE Main math curriculum. All the candidates need to do well in this section is to know the basics well and be fast at math. Most algebra questions are grouped with questions from other units, so studying algebra will help you prepare for the rest.
- Calculus is one of the most important parts of modern math. It has two main parts: differential calculus and integral calculus. There are two ways to ask a question: based on theory or on how it is used.

- Coordinate geometry: Since most of the questions in this section are based on formulas, it may seem like the easiest section to some. But this section shouldn’t be taken lightly. It takes a lot of practice to avoid making silly mistakes in this area.
- Look over all the formulas and make a list: A strong understanding of math formulas will help you a lot on the test. Candidates must make sure they understand all of the formulas in geometry, trigonometry, probability, and, of course, calculus. During the time they are getting ready, they can keep making a list of formulas to review later.
- Applying ideas: Candidates should review how geometry and differential equations can be used. This is true for both the properties of definite integrals and the results of algebraic calculations about conic properties. Before the day of the JEE Main exam, these applications should be carefully gone over again.
- Previous year’s question papers and practice tests: Before you make a plan for how you’ll prepare, looking at previous year’s question papers and practice tests will help you make a plan that will get you where you want to go.
- Accurate References: Before you start studying, make sure that the materials you are using are correct. Also, don’t try to read all the books; you should have enough to help you understand the ideas and put them into practice.
- Early Review: Most people wait until they’ve covered everything on the JEE Main Mathematics Syllabus before they start to review. But in a subject like math, you need to review often to keep your understanding of the ideas fresh. This will also save you time during full rounds of studying. So, when making a plan to study, candidates must leave enough time for reviewing.

## Question-Solving Tips for JEE Main 2022 Mathematics

- Solve Everyday: Mathematics is one subject you must practice every day, not just one. Candidates must ensure that they practice 20 to 25 math questions each day.
- By the Clock: Applicants must make very sure that they are solving mathematical problems according to the clock. Try to set a time limit for each question if you are unable to finish them all within the allotted time.
- Sort Your Problems: After you’ve answered a question, mark it as “E, M, or D.” E should be used for questions that you found simple and that you don’t need to revisit during the revision period. M for the questions that you found to be a little difficult and that you should review once while doing your revision. D for the questions you had to answer correctly to maintain your understanding of the material.
- Support your strategy with a logical flow: It is best to support your strategy while logically resolving the issue; avoid guesswork at every turn as much as you can. Conceptual clarity is achieved by following a precise logical order.
- Day-by-day goals: The mathematics curriculum is extensive and dynamic; don’t get overwhelmed by how much still needs to be covered. Candidates can plan ahead and establish daily goals they must adhere to.
- Finish the day by analyzing: You need to keep track of your errors and weak points; simply practicing at random won’t help. Analyze everything you’ve learned today, and work to overcome your weaknesses.

## Best Reference Books to Cover JEE Main Mathematics Syllabus 2022

Books and Authors | Best Suitable For |
---|---|

Maths for Class XI and XII by R.D. Sharma | Understanding of Basics |

Differential Calculus by Amit M Agarwal (Arihant Publications) | Calculus |

Maths for Class XI and XII by R.S. Agarwal | Understanding of Basics |

Geometry by Dr Gorakh Prasad | Geometry |

The Elements Of Coordinate Geometry by S L Loney | Coordinate Geometry |

Algebra by Dr S K Goyal Arihant Publications | Algebra |

Play with Graphs by Amit M Agarwal (Arihant Publications) | Practicing Problems |

Plane Trigonometry by S L Loney | Trigonometry |

Integral Calculus by Amit M Agarwal (Arihant Publications) | Calculus |

Candidates should make sure to thoroughly go over the Class XI and XII NCERT, as test-takers frequently cite it as the best resource for preparing for the JEE Main Question Papers.

## Video Lectures by IIT Faculty for JEE Main Mathematics Preparation

Online video lectures on math given by IIT professors can be very helpful for candidates. On the NTA website, you can find links to the video lectures. Here are the steps to get to the video lectures:

- Step 1: Go to nta.ac.in, which is the NTA website.
- Step 2: Click the “CONTENT BASED LECTURES – FOR JEE MAIN AND NEET-UG BY IIT PROFESSORS/SUBJECT EXPERTS” tab.
- Step 3: You’ll be taken to a page with the names of various subjects.
- Step 4: Click on the subject to choose which video lecture to watch.

## Difficulty Level Analysis for JEE Main Mathematics

The level of challenge presented by the Mathematics section of the JEE Main varies significantly from session to session. Test takers frequently report that it is the section that presents the greatest amount of difficulty and demands the most amount of their time. The following table provides a basic distribution of the number of questions in terms of difficulty level for the B.E./B. Tech paper for the JEE Main 2020 January session. This is intended to assist candidates in better understanding the exam.

Sections | Tough | Medium | Easy |
---|---|---|---|

Chemistry | 4 | 11 | 10 |

Physics | 2 | 7 | 16 |

Mathematics | 2 | 10 | 13 |

Sections | Tough | Medium | Easy |
---|---|---|---|

Mathematics | 2 | 10 | 13 |

Chemistry | 4 | 11 | 10 |

Physics | 2 | 7 | 16 |

Candidates need to keep in mind that the level of difficulty is a subjective concern and that different applicants can have a different perspectives on the question paper. With that said, the mathematics section of the January B.E./B.Tech paper can be defined as being of a medium difficulty level overall. Good luck to all of you who are taking the exam!

## Topic-wise Tips by IITians and Experts for JEE Main Mathematics

For the purpose of enhancing your knowledge of certain JEE Main mathematics topics, we have compiled some helpful contributions made by IITians:

### Complex Numbers and Quadratic Equations

- The chapter on complex numbers and quadratic equations is important not just in school, but at any point in a technical person’s life. In the last three or four years, each complex number and quadratic equation have been used in one question on the JEE Main 2022 syllabus.
- Math is all about practice, but you need a plan to get ready for tests. First, you should read through the theory in a standard book and try out the worked-out examples to get a feel for the subject.
- As soon as you feel comfortable with the basic questions, you can look at the complex number and quadratic equation questions from last year’s exam, whether it’s for school or JEE Main.
- Now it’s time to practice. Choose a question bank book and keep working on it until you can answer any kind of question that can be made from the chapters.
- If you are getting ready for a competitive exam, you can also sign up for an all-India test series to see how you compare to other people.
- I suggest that you practice questions at home with a timer if you are getting ready for school tests. But don’t overdo it. If you can answer 10 questions about complex numbers and quadratic equations every day, that’s enough because you also have to study other things.

**Here are some basic sample questions from Complex Numbers and quadratic equations**

- (x + yi) / i = ( 7 + 9i ) , where x and y are real, what is the value of (x + yi)(x – yi)?
- Determine all complex numbers z that satisfy the equation: z + 3z’ = 5 – 6i, where z’ is the complex conjugate of z
- How many real roots does the equation have? : x^2 + 3x + 4 = 0
- Evaluate 1/(i^78)

These are the **best books **to help you work with complex numbers and quadratic equations:

- NCERT: First, fill this out for school.
- RD Sharma: If you want to get into IIT, do all the questions in this book after you finish NCERT.
- Now, about JEE For this, you’ll need high-level books like those by ML Khanna, A Dasgupta, the Arihant Series, and so on.
- Keep going over the tests from the past.

### Coordinate Geometry

- Given enough question practice, coordinate geometry falls into the category of a subject where you could easily get good grades.
- It’s a really encouraging topic when you can answer a question by quickly sketching a rough figure.
- Coordinate geometry is a practical subject where you can actually “see” what is happening, unlike other subjects like calculus or combinatorics where you have to think and do. The JEE Main syllabus only covers cartesian coordinates, though a brief introduction to polar coordinates is also covered.
- Straight lines, pairs of straight lines, different types of straight lines, the intersection of lines and angles between two lines, conditions for the concurrence of three lines, a distance of a point from the line, bisectors; conic sections, requirements for the standard conic equation to be true for particular conics like a parabola, hyperbola, ellipse, and circle, equation of tangents and normals to conic sections, various forms of equations; are all included in a thorough and concise syllabus for
- Expected questions from this topic in JEE Main typically range from 4 to 6.
- Out of the 16 questions that were asked from the various topics of the class 11th syllabus, there were 4 questions in the JEE Main 2019 question paper that was specifically about coordinate geometry. (The 11th-grade curriculum is weighted 25% and the overall percentage is 13.33%)
- The average number of questions from this topic in JEE Advanced is 6 to 7, with an easy to moderate difficulty level (both paper 1 and paper 2 combined).

These are the** best books** to study for Coordinate Geometry:

- SK Goyal has some questions that are interesting and hard.
- The book starts with the basics and takes you up to the advanced level step by step. There are also some corollaries to the important theorems, which could be very helpful for the JEE Advanced. There are a lot of worked-out examples, which could help you understand some tough ideas.
- You can also read books and articles by SL Loney, Cengage, Balaji, and Disha, among others.
- Aside from NCERT questions for boards, SK Goyal is recommended. Questions from last year’s JEE Main and Advanced are highly recommended.
- If you have board exams, make sure you do NCERT well. Also, answer the questions from your board’s tests from the past. For JEE Main and JEE Advanced, you should do more than just study SK Goyal or any other book. You should also do questions from past years. Give some tests on the topic to help you learn more about it and understand it better. Because there are a lot of equations in this topic, it’s important to keep going over it.

#### The following is a selection of the frequently asked questions pertaining to the subject:

**Ques 1. The locus of the point of intersection of the lines xcost+(1-cost)y=asint and xsint-(1+cost)y+asint=0 is **

**1) x2-y2=a2 3) y2 = ax**

**2)x2+y2=a2 4) x2 = ay**

**Ans. **Rearranging the above equations, we get

1-cost/sint=a-x/y and 1+cost/sint=a+x/y

multiplying the above equations, we get x2+y2=a2

**Ques 2. If the normal chord at a point ‘t’ on the parabola y2=4axsubtends a right angle at the vertex, then the value of t is**

**1)4 3) 3**

**2)1 4) 2**

**Ans. **Equation of the normal at ‘t’ to the parabola y2=4ax is y= -tx+2at+at3

The joint equation of the lines joining the vertex to the points of the intersection of the parabola and the normal is

y2=4ax[y+tx/2at+t3]

4tx2-(2t+t3)y2+4xy=0

Since these lines are at right angles so the coefficient of x2+y2=0

So, t comes out to be 2.

**Ques 3. If the area of the triangle formed by the equation 8×2-6xy+y2=0 and the line 2x+3y=a is 7 then the value of a is ?**

**1) 14 3) 7**

**2) 28 4) 17**

**Ans. **Equations of the sides of the given triangle are y=2x, y=4x, and 2x+3y=a

So, vertices of the triangle are (a/8,a/4);(a/14,2a/7) and (0,0)

By the determinant method, the area of the triangle formed by these coordinates comes out to be a2/112 which is equal to 7

This gives the value of as 28.

**Ques 4. If the point (3,4) lies on the locus of the point of intersection of the lines xcost+ysint =a and xsint-ycost=b, the point (a,b) lies on the line 3x-4y=0 then |a+b|=?**

**1) 1 3) 7**

**2) 3 4) 12**

**Ans. **Squaring and adding the given equations of the lines we get

x2+y2=a2+b2 as the locus of the point of intersection of these lines.

Since (3,4) lies on the locus we get

9+16=a2+b2

Also (a,b) lies on 3x-4y=0 so 3a-4b=0

Solving the two equations for a and b, we get |a+b|= 7

**Ques 5. Lines ax+by+c=0, where 3a+2b+4c=0 & a,b,c all belong to the set of real numbers that are concurrent at the point?**

**1) (3,2) 3) (3,4) **

**2) (2,4) 4) (3/4, 1/2)**

**Ans. **We know that,

3a+2b+4c =0

which equals (3/4)a+(1/2)b+c=0

So, the line passes through (3/4,1/2).

**Ques 6. If the line x=k; k= 1,2,3,…..,n meet the line y=3x+4 at the points Ak(xk,yk), k= 1,2,3…..,n then the ordinate of the centre of the mean position of points Ak, k= 1,2,3,…..,n is**

**1) n+1/2 3) 3(n+1)/2**

**2) 3n+11/2 4) none of the above**

**Ans. **We have yk=3k+4, the ordinate of the intersection of x=k and y=3x+4. So the ordinate of the mean position of the points Ak k= 1,2,3,…..,n is

(1/n){sum of all yk’s} which comes out to be 3n+11/2

**Ques 7. Equation of the circle with center (-4,3) touching internally and containing the circle x2+y2=1is**

**1) x2+y2+8x-6y+9=0 3) x2+y2-8x+6y+9=0**

**2)x2+y2+8x-6y+11=0 4) x2+y2-8x+6y-11=0**

**Ans. **Let the equation of the required circle be (x+4)2+(y-3)2=r2

If the above circle touches the circle x2+y2=1 internally, then the distance between the centers of the circles is equal to the difference in their radii

42+32=r-1

Which implies r=6

So the equation of the circle is x2+y2+8x-6y-11=0

### Vectors and 3D geometry

- This subject is similar to coordinate geometry. All of the ideas from basic coordinate geometry are exactly the same here, except that there is room for a third coordinate, called the “z coordinate,” in 3D geometry.
- The topic more or less tests your ability to see and imagine the problem, so you need to have a mental picture of it.

It talks about 3D shapes like a sphere, tetrahedron, and parallelepiped. - In coordinate geometry, all calculations are based on two coordinates, x, and y. However, in 3D geometry, a coordinate is represented by three unit vectors, i, j, and k.
- JEE Main recently increased the number of questions about this topic, but it’s easy to get good marks in this area. Even though it takes a while to get used to the third coordinate at first, once the idea is clear, solving 3D geometry problems is easy.
- About 4 to 5 questions about 3D geometry and vectors are expected on the JEE Main. (Based on what happened in previous years). If you look at the JEE Main papers from previous years, on average there were 16 questions from the 12th-grade curriculum, and 4 of them were about this topic (25 percent weightage in the 12th syllabus and 13.3 percent weightage overall).
- The JEE syllabus for these units includes the following topics: 3D geometry: the coordinates of a point in space, the distance between two points, the section formula, direction ratios, and direction cosines, the angle between two lines that cross, skew lines, the shortest distance between them, and their equation. Equations of a line and a plane in different forms, the intersection of a line and a plane, coplanar lines, tetrahedron, parallelepiped, and sphere; Vectors and the math behind them – Vector addition, the parts of vectors in two-dimensional and three-dimensional space, scalar and vector products, and scalar and vector triple products are all things that can be done with vectors. In vectors and vector algebra, the JEE mostly asks questions based on triple products.
- In the JEE Advanced syllabus, the expected number of questions from these topics varies a lot from year to year. Most of the questions on the JEE Advanced come from the 12th-grade curriculum. You can expect about 15 to 20 percent of the questions to come from these topics, which is about 4 questions.

The** best books** for learning Vectors and 3D Geometry are:

- For this topic, it’s a good idea to look at JEE Main question papers from previous years. They have all kinds of questions about 3-D geometry and are enough to test what you know.
- Apart from this, Vector and 3D Geometry for JEE Main and Advanced by Amit M. Agarwal is a great book because it has a lot of questions and solved examples.

#### The following are some of the most frequently asked questions relating to the topics discussed above:

**Ques 1. The angle between a diagonal of a cube and one of its edges is,**

**1) cos-1(1/3) 3) /3**

**2) /4 4) /6**

**Ans.** Let a= a1i, b=a1j ,c=a1k.

Then the vector d= a1(i+j+k) is a diagonal of the cube. The angle between one of the edges a, b or c and the diagonal d is given by,

cos=a.d/|a||d| which comes out to be cos-1(1/3).

**Ques 2. Let N be the foot of the perpendicular of length p from the origin to a plane and l,m,n be the direction cosines of ON, the equation of the plane is **

**1) px+my+nz=l 3) lx+my+pz=n**

**2) lx+py+nz =m 4) lx+my+nz=p**

**Ans.** The coordinates of N are (pl,pm,pn) and let P(x,y,z) be any point on the plane. The direction cosines of PN are proportional to x-pl, y-pm and z-pm. Since ON is perpendicular to the plane, it is the perp. To PN

Hence, l(x-pl)+m(y-pm)+n(z-pm)=0

lx+my+nz=p(l2+m2+n2)=p, which is the locus of P and is the required equation of the plane.

**Ques 3. If A,B,C, and D are four points in space and |ABxCD+BCxAD+CAxBD|=(area of triangle ABC). Then the value is-**

**1) 1 3) 3**

** 2) 2 4) 4**

Ans. Let D be the origin of reference and DA=a, DB= b, DC= c

So, |ABxCD+BCxAD+CAxBD|= |(b-a) x (-c) +(c-b) x (-a) + (a-c) x (-b)|

= 2|axb+bxc+cxa|

=2(2 area of ABC)

Hence equals 4

**Ques 4. Volume of the tetrahedron with vertices P(-1,2,0) ; Q(2,1,-3) ; R(1,0,1) and S (3,-2,3) is **

**1) 1/3 3) 1/4**

**2) 2/3 4) ¾**

**Ans**. The volume of the tetrahedron is given by a scalar triple product,

The volume of tetrahedron= ⅙|PQ.(PR x PS)|

Here PQ, PR, and PS are three vectors made from the above-provided coordinates.

So, the volume of the tetrahedron comes out to be ⅔ after solving the triple product.

**Ques 5. The image of the point (-1,3,4) in the plane x-2y = 0 is **

**1) 8,4,4 3) 15,11,4**

**2) 9/5, -13/5,4 4) 4,4,1**

**Ans.** Required image of the line lies on the line through A(-1,3,4) and perpendicular to x-2y=0 that is on the line

x+1/1 = y-3/-2 = z-4/ 0 =t (say)

So, the coordinates of the image is (t-1,-2t+3,4)

This point also lies on the plane, so t comes out to be 14/5

So, the required image is (9/5, -13/5,4).

**Ques 6. A unit tangent vector at t=2 on the curve x=t2+2,y=4t3-5,z=2t2-6t is?**

Ans. The position vector of any point at t is r=(t2+2)i+(4t3-5)j+(2t2-6t) k

dr/dt=2ti+12t2j+(4t-6)k

At t=2, the above comes out to be 4i+48j+2k, and the unit vector comes out to be (1/580) 2i+24j+k

**Ques 7. If (2,3,5) is one end of the diameter of the spherex2+y2+z2-6x-12y-2z+20=0 then the coordinates of the other end are**

**1) 4,9,-3 3) 4,3,3**

**2) 4,-3,3 4) 4,3,5**

**Ans.** Let the other end of the diameter be (a,b,c) , then the equation of the sphere is (x-2)(x-a)+(y-3)(y-b)+(z-5)(z-c)=0

Which equals x2+y2+z2-(2+a)x-(3+b)y-(5+c)z+2a+3b+5c=0

Comparing the above equations of the sphere with the equation given and comparing the corresponding terms we get,

The required coordinates as (4,9,-3).

### Statistics and Probability

- Probability is a popular part of the IIT JEE math curriculum, but it gives many people nightmares. This chapter can test the patience and skills of even the smartest student. Probability is a bit different from other math topics like calculus, trigonometry, and coordinate geometry because it focuses more on how a person thinks and uses their imagination. Instead of memorization, math, solving equations, etc.
- Statistics aren’t on the JEE Advanced exam, but it is on the JEE Main exam, and it’s easy to get a perfect score on questions about statistics.
- Statistics also don’t take long to learn as other subjects do.
- People often ask about variance and standard deviation, so the formulas for variance and standard deviation should be easy to find.
- Probability is an important part of both the JEE Advanced and JEE Main Papers, and questions about it come up every year. Most of the time, 2-3 questions on the JEE Advanced are about probability, which makes up 6-7 percent of the paper. While 9–10% of the questions on JEE Main come from Statistics and Probability.
- In recent years, most probability questions have been in the form of paragraphs. These paragraphs usually cover Bayes Theorem, Total Probability, and Binomial Distribution, so you should pay extra attention to these topics.
- The best way to learn probability is to practice it over and over again, but it’s also important to understand the basics. Before you can start with probability, you need to know about permutations, combinations, and sets.
- When answering Probability Questions, people often use ideas like the Principle of Counting, Combinations, Group Formation, Selection among r elements, and the Venn Diagram. So, before you start with Probability, make sure you understand PnC. If you don’t, you’ll get knocked out.

Best Reference Books for JEE Main Probability and Statistics Preparation:

- NCERT and NCERT Exemplar
- Mathematics for JEE Advanced Algebra Cengage Algebra by G..Tewani
- Problem Plus in Mathematics by A Das Gupta
- Skills in Mathematics for JEE Mains and Advanced Algebra by S.K Goyal
- Previous Year JEE Question Papers

#### The following are some very important questions that deserve your attention:

**Example 1: Four Persons independently solve a certain problem correctly with probabilities ½, ¾, ¼ and 1/8. Then the probability that the problem is solved correctly by at least one of them. [JEE Advanced 2013]**

**Solution:** P(Problem solved by at least one of them)=1-P(Problem solved by none of them)

=1-½*¼*¾*7/8

=1-21/256

=235/256

**Example 2: Sixteen players P1, P2, ….. P16 play in a tournament. They are divided into eight pairs at random. From each pair, a winner is decided on the basis of a game played between the two players of the pair. Assuming that all the players are of equal strength, the probability that exactly one of the players P1 and P2 is among the eight winners is**

**(a) 4/15**

**(b) 5/9**

**(c) 3/8**

**(d) 8/15**

**Solution:** Let E1 and E2 denote the event that P1 and P2 are paired or not paired together. Let A denote the event that one of the two players P1 and P2 is among the winners.

Since P1 can be paired with any of the remaining 15 players, so P(E1) = 1/15

and P(E2) = 1 – P(E1) = 1 – 1/15 = 14/15

In case E1 occurs, it is certain that one of P1 and P2 will be among the winners. In case E2 occurs, the probability that exactly one of P1 and P2 is among the winners is

P[(P1 ∩ P2C) ∪ (P1C ∩ P2)] = P(P1 ∩ P2C) + P(P1C ∩ P2)

= P(P1) P(P2C) + P(P1C) P(P2)

= ½ (1 – 1/2) + (1 – 1/2)1/2

= ¼ + ¼

= ½

i.e. P(A/E1) = 1 and P(A/E2) = ½

By the total probability rule,

P(A) = P(E1). P(A/E1) + P(E2) P(A/E2)

= 1/15 (1) + 14/15(1/2)

= 8/15

**Example 3: A speaks truth 3 out of 4 times. He reported that Mohan Bagan has won the match. Find the probability that his report was correct. **

**Solution:**

**Method 1:**

Let T : A speaks the truth

B : Mohan Bagan won the match

Given, P(T) = 3/4

.·. P(TC) = 1 – 1/3 = 1/4

A match can be won, drawn or loosen

.·. P(B/T) = 1/3 P(B/TC) = 2/3.

Using Baye’s theorem we get

P(T/B) = (P(T).P(B/T))/(P(T).P(B/T) + P(TC)P(B/TC))

= 3/4×1/3)/(3/4×1/3 + 1/4×2/3) = (1/4)/(5/12)

=3/5

**Method 2:**

Let, T : The man speaks the truth

A : Mohan Bagan won the match

B : He reported that Mohan Bagan has won.

P(A) = 1/3(the match may also end in a draw)

P(T) = ¾

P(B) = P(A) P(T) + P(AC) P(TC)

= 1/3×3/4 + 2/3×1/4

= ¼ + 1/6

= (3+2)/12 = 5/12

P(T/B) = (P(B/T).P(T))/(P(B)) = (1/3×3/4)/(5/12)

= 3/5 (Ans.)

**Example 4: An unbiased coin is tossed. If the result is head, a pair of unbiased dice is rolled, and the number obtained by adding the number on the two faces is noted. If the result is a tail, a card from a well-shuffled pack of 11 cards numbered 2, 3, 4…. .., 12 is picked & the number on the card is noted. What is the probability that the number noted is 7 or 8?**

**Solution:** Let us define the events:

A : head appears.

B : Tail appears

C : 7 or 8 is noted.

We have to find the probability of C i.e. P (C)

P(C) = P(A) P (C/A) + P(B) P(C/B)

Now we calculate each of the constituents one by one P(A) = probability of appearing head=½

P(C/A) = Probability that event C takes place i.e. 7 or 8 being noted when the head has already appeared. (If something has already happened then it becomes certain, i.e. now it is certain that head has appeared we have to certainly roll a pair of unbiased dice).

= 11/36 (since (6, 1) (1, 6) (5, 2) (2, 5) (3, 4) (4, 3) (6, 2) (2, 6) (3, 5) (5, 3) (4, 4)

i.e. 11 favorable cases and of course 6 × 6 = 36 total number of cases)

Similarly, P(B) = 1/2

P(B/C) = 2/11 (Two favorable cases (7 and 8) and 11 total number of cases).

Hence, P(C) = ½ × 11/36 + ½ × 2/11 = 193/792 (Ans.)

**Example 5: In a test,** an examinee either guesses or copies or knows the answer to a multiple choice question that** has 4 choices. The probability that he makes a guess is 1/3 and the probability that he copies is 1/6. The probability that his answer is correct, given the copied it, is 1/8. Find the probability that he knew the answer to the question, given that he answered it correctly.**

**Solution: **

P(g) = probability of guessing = 1/3

P(c) = probability of copying = 1/6

P(k) = probability of knowing = 1 – 1/3 – 1/6 = ½

(Since the three-event g, c and k are mutually exclusive and exhaustive)

P(w) = probability that the answer is correct

P(k/w) = (P(w/k).P(k))/(P(w/c)P(c) + P(w/k)P(k) + P(w/g)P (g)) (using Baye’s theorem)

= (1×1/2)/((1/8,1/6) + (1×1/2) + (1/4×1/3) )

= 24/29(Ans.)

## JEE Main Mathematics Syllabus: FAQs

**Are definite integrals part of the JEE Main Mathematics syllabus?**

The answer is yes; you can find definite integrals in the section of the curriculum devoted to integral calculus.

**What topic has the highest weightage in the JEE Main mathematics syllabus?**

According to the patterns that have emerged over the course of the previous years, both differential calculus and coordinate geometry have the highest weightage, with each carrying 17 percent of the total marks available on the exam.

**How do I tackle the large JEE Main Mathematics Syllabus properly?**

The most effective way to study for the mathematics portion of the JEE Main exam is to break the material down into smaller sections, choose a few of those sections to focus on each day for practice, and use practice questions to identify areas in which you need more work.

**Is the JEE Main Mathematics Syllabus different for BE/BTech, BArch and BPlanning?**

No, the mathematics course outline for the JEE Main 2022 examination is the same for all three papers.