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Class 12 Maths - 10 Most Likely Questions You Must Prepare

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Class 12 Maths - 10 Most Likely Questions You Must Prepare

Edited By Ramraj Saini | Updated on Jan 04, 2023 09:10 AM IST

Four topics from the Class 12 Mathematics syllabus for the Central Board of Secondary Education (CBSE) together account for close to 80% of all questions asked in the board exam. These four topics are Calculus, Vector and Three- Dimensional Geometry, Algebra and Probability.

Questions based on certain concepts have been asked repeatedly through the years. Once all concepts are clearly understood, it is a good idea to direct attention specifically to topics that are tested on most often.

Given below are the most important sections along with a few most commonly-asked questions along with solutions.

Calculus

This unit includes five topics which are correlated to each other. If you have grasped the fundamental concepts of integration, continuity and differentiability, then you can score high marks in the other three topics of this unit because these are nothing but the application of integrals and derivatives.

Class 12 Maths - 10 Most Likely Questions You Must Prepare
Class 12 Maths - 10 Most Likely Questions You Must Prepare

Integration

This is the reverse process of differentiation and it is used for adding or summing up the parts to find the whole. There are two types of integrals,

Definite Integrals: If an integral contains a lower and upper limit then it is a definite integral. It is represented as,

ab f(x) dx

Indefinite Integrals: If an integral does not have a lower or upper limit then it is an indefinite integral. It is represented as,

∫ f(x) dx = F(x) + C , where C is constant.

There are some methods for finding integrals,

  • Integration by substitution.

  • Integration by parts.

  • Integration using by partial fraction

Here are some often-asked questions..

Q.1 : Evaluate following integrals:

  1. ∫ 5bx(a+2x) dx

  2. ∫ 7bx/ ( c2 + a2 x2 ) dx

  3. ∫{( sin2x - cos2x)/ (sinx.cosx)} dx

  4. 13 ( 5x2 + 3x) dx

  5. -11 ( 7x3+ 3x + 11) dx

Q.2 :

(1). Evaluate given functions using the substitution method:

  1. ∫ { (sec2x ⋅ sec2x )/(tan2x)}dx

  2. ∫ (cos3x ⋅ sin2x )dx

(2).

  1. Find the area of region lying above the x-axis and bounded by the circle x2 + y2 - 12x = 0 and inside the parabola y2 = 4x.

  2. Find the area of the region bounded by y2 = 12x , x =3 , x =5 and above the x-axis.

  1. Find the area of the curve y = sin2x between 0 to 2π.

Another fundamental topic is Differentiation. It measures the instantaneous rate of change of any function with respect to its variables. Differential equations and continuity and differentiability are the applications of differential and integrals. For example, velocity is the rate of change of displacement with respect to time. If x and y are two variables then the rate of change of x with respect to y is given by dy/dx. It is also represented by f’(x).

Here are some questions from that section.

Q.1 :

a. Find the order and degree of given equations:

  1. x3 (d2y/dx2) = {1 + (dy/dx)2}3

  2. x4 (d2y/dx2)3 + (dy/dx)3 = 0

  3. d2y/dx2 + x(dy/dx)3 = 5x2 log(d2y/dx2)

b. Determine the point of discontinuity of given functions:

  1. f(x) = (x+2)/{(x-3)(x+5)}

  2. f(x) = (x2+ 5x + 6)/(x2 + 2x - 3)

Q.2 :

a. Find the differential equations representing the family of following curves:

  1. y = 5q cos( x + p ), p and q are constant.

  2. A circle having a centre on the x-axis and radius 5 unit.

b. Find the general solution of differential equations which are given below:

  1. y2 dx = x2 dy

  2. dx/dy = (1 + x2)/(1 + y2)

  3. dx/dy = 1 + x +y + xy , at x= 1 and y =0

Q.3 :

  1. In a city, the Covid-19 virus initially infected 5,000 citizens. Numbers have increased by 5 % in 2 days. In how many days will the count reach 30,000, if the rate of growth of virus is proportional to the number present?

  2. Show that every polynomial function is a continuous function.

  3. Find the equation of a curve which is passing through point ( 1, 3) and slope of the tangent at any point (x, y) is x 2/ 2xy .

Vector And Three-Dimensional Geometry

A quantity that has magnitude as well as direction is a vector. A scalar quantity has only magnitude.

Some important concepts are described below.

Position Vector

It is given with respect to the origin O (0,0,0), where the coordinates of point A are x, y, and z.

Magnitude of vector = | OA | = √(x2 +y2+z2)

  • Sum of two vectors,

R = P + Q

1672745642815

Product Of Vectors

There are two types of product

  1. Scalar product (Dot product) : If P and Q are two vectors then

P ⋅ Q = |P |⋅|Q|cos(θ) , where θ is the angle between P and Q .

  1. Vector product (Cross product):

Let P = ai +bj+ ck and Q =xi +yj+ zk

P × Q = (bz - cy)i + (cx - az)j+ (ay - bx)k

Magnitude of cross product

|P × Q| = |P |⋅|Q|sin(θ) , where θ is the angle between P and Q .

  • Vector equation of a line that passes through a vector a and parallel to the vector b is

r = a + λb

  • The position vector of a point R dividing a line segment joining the points P and Q whose position vectors a and b respectively in the ratio m: n is given by:

(i) Internal dividation = (na + mb) / (m + n)

(ii) External dividation = (na - mb) /( m - n )

  • The distance between parallel lines r = a1 + λb1 and r = a2 + μ b2

|(b1⨯b2).( a2 - a1) | / |(b1⨯b2)|

  • The Direction cosines of a line joining two points P (x1, y1, z1) and

Q (x2, y2, z2 ) are,

L = (x2- x1) / PQ, m= (y2 - y1) / PQ , n= (z2- z1) / PQ

Where, PQ = √{(x2-x1)2+(y2-y1)2+(z2-z1)2}

If l, m, and n are the direction cosines of a line, then l2 + m2 + n2 = 1.

  • Equation of a line which is passing through a point (x1 , y1 , z1 ), having direction cosines l, m, n is:

(x - x1) / l = (y - y1) / m = (z - z1) / n

  • Equation of a plane which is passing through a given point (x1 , y1 , z1 ) and perpendicular to a given line with direction ratios P, Q, R and is,

P (x - x1) + Q (y - y1)+ R (z - z1)= 0

  • Cartesian equation of a line which is passing through two points (x1,y1,z1) and (x2,y2,z2) is,

(x-x1)/(x2 - x1) = (y-y1)/(y2 - y1) = (z-z1)/(z2 - z1)

Some important questions are given below

Q.1 : Find the solution of given questions:

a. Find the cross product and dot products of given vectors

  1. P = 2i + 3j+4k and Q = i - 2j+3k

  2. P = i - 5j+2k and Q = 3i - 2j+2k

b. Find the unit vector of given vector P and Q in the direction of sum of vector P and Q .

  1. P = 1i + 2j-4k and Q = 2i - 3j+3k

  2. P = 2i - 3j+2k and Q = 3i - 2j+3k

c. Identify the linear, equal, and co-initial vectors in the given figure

1672745642558

Q. 2: Solve the following questions

  1. Find the value of x if the vectors P = 3i + xj+k and Q = 9i - j+3k are parallel.

  2. Find the value of angle between P = 3i + 2j+k and Q = 1i -2 j+3k

  3. Find the area of triangle ABC with vertices A( 2,2,4), B( 2,5,3), C( 3, 1, 2).

  4. Find the area of parallelogram whose diagonals are represented by vectors 4i -6 j+8k and 4i - 2j+4k .

  5. If P = i + 2j-3k and Q = 3i - j+2k are two vectors then show that (P + Q ) and (P - Q ) are perpendicular to each other.

  6. Show that (p - q)×(p + q) = 2(p × q).

Q.3: Solve following questions

  1. Find the equation of line which is passing through the point A ( 1, 4, 5) and parallel to the vector 3i + 2j+k .

  2. If a line makes angles 45º , 60º and 90º with the x, y and z-axis respectively, find its direction cosines.

  3. If a line makes angles 90º , 60º and θ with x, y, and z-axis respectively, where θ is an acute angle then find the value of θ .

  4. Find the cartesian equation of line which passes through the point ( -4, 8 , -10 ) and is parallel to the line (x+3)/6 = (4-y)/10 = (z+8)/12 .

Algebra

This unit includes matrices and determinants. These have various applications in science, engineering, economics and other fields. Some important properties and concepts are given below.

  • If two columns or rows are interchanged then the sign of determinant changes

  • If the rows and columns of a determinant are interchanged then the value of the determinant will be unchanged.

  • Value of a determinant will be zero (0), if two identical rows or columns are present in the given determinant.

  • If each element of a column or row is multiplied by constant k then the value of the determinant obtained is multiplied with k.

  • A square matrix A = [aij] is said to be symmetric if the transpose of A is equal to A, that is, [aij] = [aji] for all possible values of i and j.

  • A square matrix A = [aij] is said to be a skew-symmetric matrix if A’ = -A, that is aij=-aij for all possible values of i and j.

  • For a skew-symmetric, if we substitute i=j, we have aii=-aii and thus 2aii = 0 or aii=0 for all i. There, all the diagonal elements of a skew-symmetric matrix are zero.

  • Suppose A and B are two matrices with the same order m, such that AB = BA = I, then B is called the inverse matrix of A, and it is denoted by A-1. and A is known as invertible matrix.

Also check - CBSE Class 12 Biology: 10 Most Asked Questions On Genetics And Evolution In Boards

Some important questions are given below.

Q.1: Estimate the following

  1. If 2A - B = 1672745641416 And B = 1672745641815 Find the value of matrix A,


  1. If A is the Square matrix such that A2 = I then find the value of ( A + I )2 + ( A -I )2 + 5A .


  1. Find the value of ( a + b) if 1672745642164


  1. Find the value of x if 1672745641572


  1. Simplify 1672745643188


Q.2:

  1. Show that

  2. det 1672745643406= ( x - y )( y - z )( z - a )


  1. det 1672745643858= ( x -1 )3

b.using properties of determinant

  1. Find the value of det 1672745643619


  1. Find the value of f(2) - f’(1) if

f(x) = det 1672745644079, where a = 5


Probability

There are many applications of probability in real life. Every CBSE Class 12 board exam Maths paper has included questions related to dice, coins, playing cards and real life situations. There are several important concepts and formulas in probability theory that are used to calculate the likelihood of different events occurring. Some of the most important ones include:

  • The probability of an event occurring is given by the formula:

P(A) = n(A) / n(S),

Where, P(A) is the probability of the event,

n(A) is the number of favourable events that occur,

n(S) is the total number of possible outcomes.

  • The probability of two independent events:

P(A ∩ B) = P(A) * P(B),

Where P(A) and P(B) are the probabilities of both events occurring.

  • The probability of an event occurring given that another event has already occurred is called conditional probability and is given by the formula:

P(A | B) = P(A ∩ B) / P(B),

Where, P(A ∩ B) is the probability of both events occurring

P(B) is the probability of the second event occurring.

  • P(A ∪ B) = P(A) + P(B) - P(A∩B)

  • The probability of an event not occurring is given by the formula:

P(not A) = 1 - P(A),

Where P(A) is the probability of the event occurring.

  • Theorem of total probability:

Let S be a sample space and let A1 , A2, ...,An be a partition of S,

i.e. the events A1 , A2, ..., An are disjoint and their union is S. Let B be an event in S. Then the probability of B occurring is given by:

P(B) = P(B | A1 ) * P(A1 ) + P(B | A2) * P(A2) + ... + P(B | An) * P(An)

The questions given below – or ones similar to them – are likely to be asked.

Q.1: Calculate the probability for given situation

a. If a dice is thrown twice then estimate following:

  1. Probability that the sum of numbers obtained is more than 8.

  2. Probability that the number obtained is the prime number on both dice

  3. Probability of not getting a one on either throw .

  4. Probability of not getting the same number both times

b. A coin is tossed thrice. Calculate the probability of getting

  1. At least two heads

  2. Only one head

  3. Same face

  4. Only two tails

Q.2: Calculate the probability for a given situation

  1. Evaluate P(A ∪ B) if 2 P(A) = P(B) = 5/8 and P(A/B) = 3/5
  2. P and Q take turns to throw a pair of dice. P wins the game if he gets the total of 8 and Q wins the game if he gets the total of 9. If A starts the game then find the probability of B winning the game.
  3. A family has two children. What is the conditional probability that both are girls if it is known that
  1. One of them is a boy
  2. The youngest is a girl
  3. The older child is a boy

Also check - Class 12 Board Exams - Top Maths Formulae You Should Know

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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. 

5 Jobs Available
Vlogger

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. 

3 Jobs Available
Publisher

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.

3 Jobs Available
Journalist

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.

3 Jobs Available
Editor

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.

3 Jobs Available
Reporter

Individuals who opt for a career as a reporter may often be at work on national holidays and festivities. He or she pitches various story ideas and covers news stories in risky situations. Students can pursue a BMC (Bachelor of Mass Communication), B.M.M. (Bachelor of Mass Media), or MAJMC (MA in Journalism and Mass Communication) to become a reporter. While we sit at home reporters travel to locations to collect information that carries a news value.  

2 Jobs Available
Corporate Executive

Are you searching for a Corporate Executive job description? A Corporate Executive role comes with administrative duties. He or she provides support to the leadership of the organisation. A Corporate Executive fulfils the business purpose and ensures its financial stability. In this article, we are going to discuss how to become corporate executive.

2 Jobs Available
Multimedia Specialist

A multimedia specialist is a media professional who creates, audio, videos, graphic image files, computer animations for multimedia applications. He or she is responsible for planning, producing, and maintaining websites and applications. 

2 Jobs Available
Welding Engineer

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. 

5 Jobs Available
QA Manager
4 Jobs Available
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.

3 Jobs Available
Production Manager
3 Jobs Available
Product Manager

A Product Manager is a professional responsible for product planning and marketing. He or she manages the product throughout the Product Life Cycle, gathering and prioritising the product. A product manager job description includes defining the product vision and working closely with team members of other departments to deliver winning products.  

3 Jobs Available
QA Lead

A QA Lead is in charge of the QA Team. The role of QA Lead comes with the responsibility of assessing services and products in order to determine that he or she meets the quality standards. He or she develops, implements and manages test plans. 

2 Jobs Available
Structural Engineer

A Structural Engineer designs buildings, bridges, and other related structures. He or she analyzes the structures and makes sure the structures are strong enough to be used by the people. A career as a Structural Engineer requires working in the construction process. It comes under the civil engineering discipline. A Structure Engineer creates structural models with the help of computer-aided design software. 

2 Jobs Available
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.

2 Jobs Available
QA Manager
4 Jobs Available
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. 

4 Jobs Available
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. 

4 Jobs Available
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.

3 Jobs Available
Product Manager

A Product Manager is a professional responsible for product planning and marketing. He or she manages the product throughout the Product Life Cycle, gathering and prioritising the product. A product manager job description includes defining the product vision and working closely with team members of other departments to deliver winning products.  

3 Jobs Available
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 

3 Jobs Available
ITSM Manager
3 Jobs Available
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. 

2 Jobs Available
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