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

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

    #NEET
    Ramraj SainiUpdated on 14 Jun 2022, 09:05 AM IST

    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

    1. Vectors Addition Rules
    2. Subtraction And Addition Of Vectors
    3. Multiplication Of Vectors
    4. Other Important Formulas
    5. Logarithmic
    6. Exponential
    7. Exponential Function Rules
    8. NEET Previous Five Years’ Analysis
    9. Previous Years Questions To Understand The Concepts
    NEET Five-Year Analysis: Solving Vectors Algebra, Logarithmic, And Exponential Questions
    NEET Vector Algebra(Image: Shutterstock)

    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 1654753335271 and 1654753333496 represented by two sides of a triangle, the head of 1654753335896 is connected with the tail of 1654753333784 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 1654753335111. ? is the angle between 1654753336183 and 1654753333182. 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 1654753329310 and 1654753328841. 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.

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    Subtraction And Addition Of Vectors

    1654753329173

    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 1654753335442 and 1654753332886

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

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

    Where ? is the angle between 1654753336047 and 1654753332578 and angle ? is between resultant vector\vec{A} - \vec{B} and 1654753335607

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    Multiplication Of Vectors

    Multiplication of vectors can be of two types: scalar vector multiplication and vector-vector multiplication. If we multiply a vector 1654753335759 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 1654753328983and 1654753328566 are two vectors.

    1654753328714

    \\ \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}

    1654753334768

    Other Important Formulas

    1654753330230

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

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    Logarithmic

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

    x = logb a, for b > 0, a > 0, and b ≠ 1, if and only if a = bx. 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) = loge x

    Common logarithmic function f(x) = log10 x

    Logarithmic Function

    1654753331731

    Logarithmic Table

    x

    Base = 10

    Log10 x

    Base = e ≈ 2.71;

    Loge 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.bx, 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

    16547533274901654753328127

    1654753326354

    1654753326102

    What will be the complete expression for 1654753327010

    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 )

    -2B=12

    B=-6

    4B_{0}-6B=4

    2B_{0}-6B=20

    2B_{0}=-16,B_{0}=-8 B=-6

    \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:

    1654753334941

    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:

    1654753334597

    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}

    x=1

    x+y=2

    y=1

    -2x-2y+z=-2

    Z=-2+4=2

    1654753322652

    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 E_A of the reaction is. Choose the correct option for your answer.

    1654753321149

    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 xp = 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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