NEET Five-Year Analysis: Solving Vectors Algebra, Logarithmic, And Exponential Questions

Edited By Ramraj Saini | Updated on Jun 14, 2022 09:05 AM IST | #NEET

Vector algebra is one of the fundamental topics of algebra. There are two types of physical quantities – scalar quantity and vector quantity. In vector algebra, we study the vector quantities. Vector quantity is a quantity that has both magnitude and direction and follows vector addition rules whereas scalar quantity only has magnitude.

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This Story also Contains

Vectors Addition Rules
Subtraction And Addition Of Vectors
Multiplication Of Vectors
Other Important Formulas
Logarithmic
Exponential
Exponential Function Rules
NEET Previous Five Years’ Analysis
Previous Years Questions To Understand The Concepts

NEET Five-Year Analysis: Solving Vectors Algebra, Logarithmic, And Exponential Questions

Vector: A vector is a measurement or quantity that includes both magnitude and direction. Generally, it can be represented physically by an arrow, and mathematically as A⃗ can be read as “A vector”. Arrow (➙) indicates the direction of the vector and its length represents the magnitude of the vector that is given in the following figure. |A⃗| represents the magnitude of the vector A. If we say that two vectors are equal, then both must have the same direction as well as magnitude.

1654753334271

Unit Vector: It is a vector that has a unit magnitude. Generally, it is used to show the direction of the vector. Conventionally three-unit vectors namely $\hat{i}, \hat{j}, and \ \hat{k}$ are used in the direction of the x-axis, y-axis, and z-axis.

The unit vector can be calculated by dividing the vector by its magnitude and represented as mentioned below.

$\hat{a} = \frac{\vec{a}}{|\vec{a}|}$

Vectors Addition Rules

Triangle Rule

1654753331095

Let us consider two vectors and represented by two sides of a triangle, the head of is connected with the tail of as shown above in the image, then the third side in the opposite direction represents the resultant sum of these two vectors. Mathematically the resultant of two vectors can be calculated by the given formula.

$|\vec{A} + \vec{B}| = \sqrt{A^2+B^2+2ABcos \theta}$

$tan \alpha = \frac{Bsin\theta}{A+Bcos\theta}$

Where angle ⍺ is the angle between the resultant R vector and . ? is the angle between and . Remember that angle between two vectors can only be calculated when both vectors are connected head to head or tail to tail.

Parallelogram Rules

1654753331575

Let's consider two adjacent sides of a parallelogram represent two vectors and . As mentioned above in the diagram, the diagonal joining the intersecting point to the other corner of the parallelogram represents the resultant sum of vectors as shown in the image.

Subtraction And Addition Of Vectors

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Subtraction of the vector is nothing but the addition of the negative vector to the other vector. Keeping the same magnitude and reversing the direction of a vector gives a negative vector. In a Parallelogram, one diagonal shows the addition of vectors, and the other one shows the subtraction of vectors as shown above in the figure.

Mathematical formula of subtraction of two vectors and

$|\vec{A} - \vec{B}| = \sqrt{A^2+B^2-2ABcos \theta}$

$tan \alpha = \frac{Bsin\theta}{A-Bcos\theta}$

Where ? is the angle between and and angle ? is between resultant vector $\vec{A} - \vec{B}$ and

Multiplication Of Vectors

Multiplication of vectors can be of two types: scalar vector multiplication and vector-vector multiplication. If we multiply a vector with a scalar quantity “k”, when “k” is positive, the direction remains the same and magnitude becomes “k” times, and when “k” is negative, magnitude becomes “k” time, but the direction gets reversed.

Vector multiplication is two types i.e dot product or scalar product and cross product or vector product which are explained below.

Dot Product

It is also called a scalar product, represented as (.) between two vectors. Dot product physically represents the multiplication of the magnitude of one vector with a component of another vector in the direction of the vector.

$\\ \vec{P}.\vec{Q} = |\vec{P}| |\vec{Q}| cos \theta \\ \text{If P vector and Q vector are in the same direction, i.e. } \theta = 0 {\degree} \\ then; \vec{P}.\vec{Q} = |\vec{P}| |\vec{Q}| \\ \text{If P and Q are both orthogonal, i.e. } \theta = 90 \degree , \\ then; \vec{P}.\vec{Q} = \vec{0} \ \text{since } cos 90 \degree = 0 \\$

$\\ \vec{A} = a_{1} \ \hat{i}+a_{2} \ \hat{j}+a_{3} \ \hat{k} \\ \vec{B} = b_{1} \ \hat{i}+b_{2} \ \hat{j}+b_{3} \ \hat{k} \\ \vec{A}.\vec{B} = a_{1}b_{1}+a_{2}b_{2}+a_{3}b_{3}$

Cross Product

Cross product is also known as the vector product is represented by the “X” sign between two vectors. If and are two vectors.

$\\ \vec{A} = a_{1} \ \hat{i}+a_{2} \ \hat{j}+a_{3} \ \hat{k} \\ \vec{B} = b_{1} \ \hat{i}+b_{2} \ \hat{j}+b_{3} \ \hat{k} \\ \vec{A} \times \vec{B} = \begin{vmatrix} i & j & k\\ a_{1} & a_{2} & a_{3}\\ b_{1}& b_{2} & b_{3} \end{vmatrix}$

Other Important Formulas

$\hat{a} = \frac{\vec{a}}{|\vec{a}|}$

Logarithmic

Logarithmic Function is an inverse function of exponential function which is defined as

x = log_ba, for b > 0, a > 0, and b ≠ 1, if and only if a = b^x. Generally, we come across two types of bases while solving NEET problems. That is 10 and the mathematical constant “e”. When the base is “e” it is called a natural logarithmic function and when the base is 10 then it is called a common logarithmic function.

Natural logarithmic function f(x) = log_e x

Common logarithmic function f(x) = log₁₀ x

Logarithmic Function

1654753331731

Logarithmic Table

x	Base = 10 Log₁₀ x	Base = e ≈ 2.71; Log_e x
1	0	0
2	0.3010	0.693147
3	0.4771	1.098612
4	0.6020	1.386294
5	0.6989	1.609438
6	0.7781	1.791759
7	0.8450	1.94591
8	0.9030	2.079442
9	0.9542	2.197225
10	1	2.302585

Graph

1654753330769

Properties

1654753331425

Exponential

It is a mathematical function that is used to find exponential decay and exponential growth of many real-life events like population models, radioactive decay, wounds healing models, and many more. Mathematically defined by a function f (x) = a.b^x, where “x” is a variable that can be any real number, “a” is constant but can not be zero, and “b” is an also constant know as base of exponential function and it should be greater than zero and not equal to 1. And the most commonly used exponential bases are 10 and “e” which is approximately equal to 2.718. This article includes rules, formulas, graphs, etc of the exponential function.

Exponential Growth

Initially, the quantity increases very slowly, and then rapidly. The rate of change increases over time. A mathematical expression that defines exponential growth is y = a ( 1+ r )^x where “ r ” is the growth percentage.

Exponential Decay

Initially, quantity decreases very rapidly, and then slowly. The mathematical expression to define exponential decay is y = a ( 1- r )^x where “ r ” is the decay percentage.

Exponential Growth And Decay Graph

1654753330406

Exponential And Logarithmic Function

1654753330586

Exponential Function Rules

Following are some important rules that are repeatedly used. Here x and y are real numbers whereas “a” and “b” are positive numbers.

If a>0, and b>0, the following hold true for all the real numbers x and y:

1654753329739

NEET Previous Five Years’ Analysis

Since the NEET exam is highly competitive, smart study can help students to succeed in the exam. Careers360 came up with an analysis to help them. The following list shows the number of questions in which vector algebra, exponential, and logarithmic concepts are used.

Numbers Of Questions Asked In Previous Five Years NEET Exam

	Vector Algebra	Exponential	Logarithmic
2021	4	2	1
2020	1	2	2
2019	5	2	1
2018	4	1	1
2017	1	3	1

Previous Years Questions To Understand The Concepts

Here is a list of previous year's questions with step-by-step explanations to understand vector algebra, exponential, and logarithmic concepts. NEET aspirants can develop an understanding of the level of questions along with the spread and depth of the concepts.

Q.1 NEET - 2021

In the product

What will be the complete expression for

Solution:

$\bar{F}=q\left ( \bar{v}\times \bar{B} \right )=q\bar{v}\times \left ( B\hat{i}+B\hat{j}+B_{0}\hat{k} \right )$ $\left [ 4\hat{i}-20\hat{j}+12\hat{k} \right ]=\left [ 2\hat{i}+4\hat{j}+6\hat{k} \right ]\times \left [ B\hat{i}+B\hat{j}+B_{0}\hat{k} \right ]$

$=\hat{i}\left ( 4B_{0}-6B \right )-\hat{j}\left ( 2B_{0}-6B \right )$ $+\hat{k}\left ( -2B \right )$

$4B_{0}-6B=4$

$2B_{0}-6B=20$

$2B_{0}=-16,B_{0}=-8$

$\bar{B}=B\hat{i}+B\hat{j}+B_{0}\hat{K}$

$\bar{B}=-6\hat{i}-6\hat{j}-8\hat{k}$

Concepts used:

Cross product formula:

Q.2. NEET - 2020

Find the torque about the origin when a force of $3\hat j N$ acts on a particle whose position vector is $2\hat k m$

Solution:

$\\ \vec{F}=3 \hat{j}\\ \vec{r}=2 \hat{k}\\ \tau =\vec{r}\times \vec{F}=2 \hat{k}\times 3 \hat{j}=-6 \hat{i}$

Concepts used:

Cross product formula:

Q.3. NEET - 2021

If force [F], acceleration [A], and time [T] are chosen as the fundamental physical quantities. find the dimensions of energy.

Solution:

$[Energy ]=[F]^{x}[A]^{y}[T]^{Z}$

$\left[M^{1} L^{2} T^{-2}\right]=\left[M^{1} L^{1} T^{-2}\right]^{x}[L^{1}T^{-2}]^{y}[T^{1}]^{z}$

$\left[M^{1} L^{2} T^{-2}\right]=M^{x} L^{x+y} T^{-2 y-2 x+z}$

Concepts Used:

1654753329618

Q.4. NEET - 2020

The rate constant for a first-order reaction is $4.606 \times 10^{-3}s^{-1}$ The time required to reduce 2 g of the reactant to 0.2 g is:

Solution:

We have given $K=4.606 \times 10^{-3}s^{-1}$ Thus, the reaction is of the first order. $\mathrm{K\: t\: =\: ln\frac{A_{o}}{A}}$

$\mathrm{4.606\: x\: 10^{-3}\: t\: =\: 2.303log\left (\frac{2}{0.2} \right )}$

Thus,

$\text {t}=500\; \text {sec}$

Concepts Used:

Log₁₀ 10 = 1

Q.5 NEET - 2020

The slope of the Arrhenius Plot $\left(\ln \mathrm{k\ v} / \mathrm{s}\ \frac{1}{\mathrm{T}}\right)$ of a first-order reaction is $-5 \times 10^{3} K$ . The value of of the reaction is. Choose the correct option for your answer.

Solution:

We know the Arrhenius equation

$\mathrm{k}=\mathrm{Ae}^{-\mathrm{E}_{\mathrm{a}} / \mathrm{RT}}$

$\ln \mathrm{k}=\ln \mathrm{A}+\ln \mathrm{e}^{-\mathrm{E}_{\mathrm{a}} / \mathrm{RT}}$

$\mathrm{\ln k=\ln A-\frac{E_{a}}{R}\left(\frac{1}{T}\right)}$

The slope of $\left(\ln \mathrm{k\ v} / \mathrm{s}\ \frac{1}{\mathrm{T}}\right)$ of the above equation -

$\mathrm{m=-\frac{E_{a}}{R}}$

Given, m = $-5 \times 10^{3} \mathrm{~K}$ and $\mathrm{R}=8.314 \mathrm{JK}^{-1} \mathrm{~mol}^{-1}$

$\mathrm{-5 \times 10^{3}=-\frac{E_{a}}{8.314}}$

$\mathrm{E_{a}=5 \times 10^{3} \times 8.314 \mathrm{~J} / \mathrm{mol}}$

$\mathrm{E_{a} =41.57 \times 10^{3} \mathrm{~J} / \mathrm{mol}}$

$\mathrm{E_{a}\simeq 41.5 \mathrm{~kJ} / \mathrm{mol}}$

Concepts Used:

Using property ln x^p = p ln x, simplify the equation and ln e = 1.

After going through this article it is clear that the NEET exam does not go deeper and asks just formula-based and simple concept-based questions. Therefore, even without mastering complex formulas and concepts of vector algebra students can score well in the exam. With continuous practice, aspirants can get command of the concepts.

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Questions related to NEET

Have a question related to NEET ?

1st time my 12th % is 66.5 when i am regular student and 2 time % in 12th is 80 when i am private students of 12th Then Mukhyamantri Medhavi yojana applicable or not for me for neet for semi govt. college . please

The Madhya Pradesh Medhavi Chhatra Yojana (also known as the Medhavi Yojana) provides scholarships to meritorious students based on their performance in the 12th board examinations.the percentage should be more than 75 percent for mp board students and for CBSE and icse it is 85 percent.

You can apply for scholarship and If you come into the meritorious students then the amount is even more than 25000.

If you belong to any school then they will collect your bank information So all in all you do not have to do anything, all the work is to be done by the school. You just enjoy.

Hope it helps

Read Complete Answer

sadafreen356 18 November,2024

Is admission taken through neet

From your profile it is clear that you are asking about pharmacy admission through neet so,yes there are some colleges in India which considers NEET score for admission to pharmacy programs.

Other than NEET some state colleges use their own entrance exam like MHT CET, CUET, AP EAMCET, TS EAMCET, KCET, KEAM, and Graduate Pharmacy Aptitude Test (GPAT) for pharmacy.

If you are not satisfied with answer ask again i will try to help you better..

Here are top pharmacy colleges and their admission criteria you can consider

https://pharmacy.careers360.com/colleges/list-of-pharmacy-colleges-in-india

Read Complete Answer

sadafreen356 18 November,2024

My daughter is born in India but for last few years she has been studying abroad as we have shifted to Qatar for job. She has cleared 10th and willbe appearing for 12th from CBSE board affiliated school in Qatar and will be appearing for NEET in Qatar. Will she qualify for NRI seat or normal seat.

If your daughter is currently studying in a CBSE affiliated school in Qatar and has been living abroad for the past few years, she will likely qualify for an NRI seat when applying for NEET, as she meets the criteria of being an Indian citizen residing abroad and completing her 10th and 12th education from a foreign country.

NRI students can apply NEET UG under NRI category. Your daughter will be applying for seats under AIQ quota i.e 15% seats only .some states dont give reservation to NRI like Karnataka but some states like gujarat has NRI reservation.while considering colleges check specific eligibility criteria of the medical colleges you are applying to, as there may be minor variations in their NRI admission policies.

Have a look on details of NRI category NEET admission

https://medicine.careers360.com/articles/nri-quota-for-neet

Read Complete Answer

sadafreen356 18 November,2024

Hlo sir i want to confirm. can you give admission without clearing neet exam

Hello,

Yes, it is possible to get admission into medical-related courses without clearing the NEET exam. Here’s how:

Alternative Medical Courses :
- B.Sc. Nursing : Colleges like AIIMS (for nursing), Christian Medical College (CMC), Vellore , and St. John's Medical College, Bengaluru offer nursing programs.
- B.Sc. Biotechnology/Biomedical Science : Jamia Millia Islamia, Delhi and Manipal Academy of Higher Education provide such programs.
Allied Health Sciences :
- Programs such as B.Sc. Radiology , B.Sc. Medical Laboratory Technology (MLT) , and B.Pharm are offered at institutions like Amity University , Lovely Professional University (LPU) , and SRM University, Chennai .
Paramedical Courses :
- Courses like Physiotherapy (BPT) , Occupational Therapy , and Optometry are available at Manipal College of Health Professions and Apollo Institute of Hospital Administration .
Overseas Admissions :
- Countries like the Philippines , Russia , and China offer MBBS without NEET but require other eligibility criteria.

These colleges and courses offer reputable options outside of the NEET pathway.

To know more, visit : https://medicine.careers360.com/articles/medical-courses-without-neet

Hope it helps !

Read Complete Answer

Prachi Kumari 17 November,2024

sir 2025 neet exam will be held soon ...if I failed in that , what other government exams incan give which has the same syllabus as neet???

There is no other major government examination in India that overlaps with the NEET syllabus to provide similar careers in the medical field. NEET is, however the gateway to medical and dental colleges in the country.

In case you wish to pursue other fields of science, here are your options:

1. Engineering:

JEE Main: This examination offers entry to engineering colleges, providing a vast number of courses in engineering.

2. Other Related Health Sciences:

B.Sc Nursing: Eligibility criteria differ for this course, such as Physics, Chemistry, Biology, and English

B.Pharm: This could be pursued after 12th with PCB

B.Sc Biotechnology, Microbiology, Biochemistry, etc: These are courses one takes if a person interested in the life sciences and other related studies.

Note:

Eligibility: Remember that eligibility criteria differ for courses and universities.

Entrance Exams: Prepare for entrance exams that are related to NEET, JEE Main, or whichever the university-specific tests may be.

Guidance Counseling: Participate in the process of counseling to get a seat in the preferred college and course.

If you hold an ardent passion for a career in the healthcare sector, keep researching and consider the following options like research or pharmaceutical industry or public health.

Read Complete Answer

Samprikta Mondal 16 November,2024

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Popular NEET Questions

Column I ( Salivary gland)		Column II ( Their location)
Parotids	I	Below tongue
Sub-maxillary / sub-mandibular	Ii	Lower jaw
Sub-linguals	Iii	Cheek

Option: 1

a(i), b(ii) , c(iii)

Option: 2

a(ii), b(i), c(iii)

Option: 3

a(i), b(iii), c(ii)

Option: 4

a(iii), b(ii), c(i)

$Ethyl \; ester \xrightarrow[(excess)]{CH_{3}MgBr} P$

the product 'P' will be ,

Option: 1

Option: 2

Option: 3

$\left ( C_{2}H_{5} \right )_{3} - C- OH$

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	Valve name		Function
I	Aortic valve	A	Prevents blood from going backward from the pulmonary artery to the right ventricle.
II	Mitral valve	B	Prevent blood from flowing backward from the right ventricle to the right atrium.
III	Pulmonic valve	C	Prevents backward flow from the aorta into the left ventricle.
IV	Tricuspid valve	D	Prevent backward flow from the left ventricle to the left atrium.

Option: 1

I – A , II – B, III – C, IV – D

Option: 2

I – B , II – C , III – A , IV – D

Option: 3

I – C , II – D , III – A , IV – B

Option: 4

I – D , II – A , III – B , IV – C

Column A	Column B
A	a) Organisation of cellular contents and further cell growth.
B	b) Leads to formation of two daughter cells.
C	c) Cell grows physically and increase volume proteins,organells.
D	d) synthesis and replication of DNA.

Match the correct option as per the process shown in the diagram.

Option: 1

1-b,2-a,3-d,4-c

Option: 2

1-c,2-b,3-a,4-d

Option: 3

1-a,2-d,3-c,4-b

Option: 4

1-c,2-d,3-a,4-b

0.014 Kg of N₂ gas at 27 ⁰C is kept in a closed vessel. How much heat is required to double the rms speed of the N₂ molecules?

Option: 1

3000 cal

Option: 2

2250 cal

Option: 3

2500 cal

Option: 4

3500 cal

0.16 g of dibasic acid required 25 ml of decinormal $NaOH$ solution for complete neutralisation. The modecular weight of the acid will be

Option: 1

Option: 2

Option: 3

128

Option: 4

256

0.5 F of electricity is passed through 500 mL of copper sulphate solution. The amount of copper (in g) which can be deposited will be:

Option: 1

31.75

Option: 2

15.8

Option: 3

47.4

Option: 4

63.5

0.5 g of an organic substance was kjeldahlised and the ammonia released was neutralised by 100 ml 0.1 M HCl. Percentage of nitrogen in the compound is

Option: 1

Option: 2

Option: 3

Option: 4

0xone is

Option: 1

$\mathrm{KO}_{2}$

Option: 2

$\mathrm{Na}_{2} \mathrm{O}_{2}$

Option: 3

$\mathrm{Li}_{2} \mathrm{O}$

Option: 4

$\mathrm{CaO}$

(1) A substance known as "Smack"

(2) Diacetylmorphine

(3) Possessing a white color

(4) Devoid of any odor

(5) Crystal compound with a bitter taste

(6) Obtained by extracting from the latex of the poppy plant

The above statements/information are correct for:

Option: 1

Morphine

Option: 2

Heroin

Option: 3

Cocaine

Option: 4

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Narcotics Officer

A Narcotics Officer is an officer employed by the state to investigate the usage of drugs and their trafficking. A narcotics officer conducts undercover operations, investigates suspected drug dealers, executes raids and other appropriate actions for arresting these traffickers to reduce the circulation of drugs in the country.

A narcotics officer works in collaboration with other government agencies to stop drug trafficking at borders. He or she engages with various NGOs and public organisations to teach people about the dangerous effects of drug usage. A narcotics officer plays an important role in reducing the illegal activities of drug dealers and the circulation of drugs in the nation.

4 Jobs Available

Research Associate

If we talk about a career as a research associate, it all comes down to one thing - curiosity towards nature and the passion to find answers. A career as a research associate is full of thrill and excitement. However, a research associate also faces a lot of challenges and failures while working on a project. A job of a research associate includes a spectrum of Science as a subject in detail.

2 Jobs Available

Drug Inspector

A career as a Drug Inspector is regarded as one of the most diverse in the field of healthcare and pharmacy. Candidates must undergo a screening process administered by the UPSC and or SPSCs in order to become drug inspectors. Those who manage it through the selection process will have a rewarding career with a high salary.

2 Jobs Available

Biotechnologist

A Biotechnologist is a professional who possesses strong knowledge and techniques that are utilised in creating and developing innovative products that improve the quality of human life standards. A biochemist uses biological organisms to create and improve goods and procedures for agriculture, medicine, and sustainability. He or she researches the genetic, chemical, and physical characteristics of cells, tissues, and organisms to determine how they can be used industrially.

2 Jobs Available

R&D Personnel

A career as R&D Personnel requires researching, planning, and implementing new programs and protocols into their organization and overseeing new products’ development. He or she uses his or her creative abilities to improve the existing products as per the requirements of the target market.

2 Jobs Available