RD Sharma Solutions for Class 12 Maths

RD Sharma Solutions for Class 12 Maths – Chapter-wise Free PDF Download Updated for 2023-24

RD Sharma Solutions for Class 12 Maths PDF can be easily accessed by students to start adequate preparation for their upcoming exams. Students can now solve any problem from the RD Sharma textbooks by referring to RD Sharma Solutions. BYJU’S experts formulate these questions in an easy and understandable manner that helps students solve problems in the most efficient ways. We hope these solutions will help CBSE Class 12 students to build a strong foundation of the basics and secure excellent marks in their final exam.

As Mathematics is a complex subject for Class 12 students, these solutions will change every student’s approach towards Mathematics and will definitely make them recognise how interesting and easy the subject is. RD Sharma Solutions are focused on learning various Mathematics tricks and shortcuts for quick and easy calculations. Download the RD Sharma Solutions for Class 12 Maths now and practise all the CBSE Textbook questions. Solving these questions will ensure that students have good practice with all the types of questions that can be framed in the examination.

RD Sharma Chapter-wise Solutions for Class 12 Maths

Students are also given access to additional online study materials and resources, available at BYJU’S, such as notes, books, question papers, exemplar problems, worksheets, etc. Students are also advised to practise Class 12 Sample papers to get an idea of the question pattern in the final exam. Practising RD Sharma Class 12 Solutions on a regular basis helps students improve the skills that are important to score optimum marks in the board exams.

Download Class 12 Maths RD Sharma Solutions to answer the chapter-wise questions effortlessly.

RD Sharma Solutions for Class 12 Maths – Exercise-wise Solutions

Students can refer to these solutions of RD Sharma Class 12, which helps in gaining knowledge. For better guidance, it is best to solve these solutions. Solving the exercises in each chapter will ensure that students score good marks in the board exams.

RD Sharma Solutions for Class 12 Maths Chapter 1 – Relations

Chapter 1 of the RD Sharma textbook deals with relations and their properties, types of relations, inverse of a relation, equivalence relation, some useful results on relations, reflexive relation, symmetric relation, and transitive relation.

Relation: It defines the relationship between two sets of different information. If we consider two sets, the relation between them will be established if there is a connection between the elements of two or more non-empty sets.

Types of Relations: The different types of relations are empty, universal, identity, inverse, reflexive, symmetric, transitive, and equivalence relation.

Representation of Types of Relations

Here, you can find the exercises solution links for the topics covered in this chapter.

Also access the following resources for Class 12 Chapter 1 Relations at BYJU’S:

• NCERT Solutions for Class 12 Maths Chapter 1 Relations and Functions
• NCERT Exemplar Solutions for Class 12 Maths Chapter 1 Relations and Functions

RD Sharma Solutions for Class 12 Maths Chapter 2 – Functions

Chapter 2, Functions of RD Sharma Class 12 Maths, gives an account of various topics such as the definition of functions, function as a correspondence, function as a set of ordered pairs, the graph of a function, vertical line test, constant function, identity function, modulus function, greatest integer function, properties of greatest integer function, smallest integer functions and its properties, fractional part function, signum function, exponential function, logarithmic function, reciprocal and square root function, square function, square root function, cube function, reciprocal squared function, operations on real function, kinds of functions such as one-one, many-one and onto function, bijection, the composition of functions and its properties and composition of real functions. Meanwhile, in this chapter, students get to learn how to relate graphs of a function and its inverse.

Function: A function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output.

Types of Functions: The types of functions are constant, identity, modulus, integer, exponential, logarithmic, reciprocal, and square root functions.

Inverse Function: An inverse function or an anti-function is a function that can reverse into another function. In other words, if any function “f” takes x to y, then the inverse of “f” will take y to x.

Also, access the following resources for Class 12 Chapter 2 Functions at BYJU’S:

• NCERT Solutions for Class 12 Maths Chapter 1 Relations and Functions
• NCERT Exemplar Solutions for Class 12 Maths Chapter 1 Relations and Functions

RD Sharma Solutions for Class 12 Maths Chapter 3 – Binary Operations

In Chapter 3 of the RD Sharma textbook, we shall see the definition of a binary operation, the number of binary operations, types of binary operations such as commutativity, associativity and distributivity, an identity element, inverse of an element, composition table, addition modulo ‘n’ and multiplication modulo ‘n’.

Binary Operations: It is defined as an operation * which is performed on a set A. The function is given by *: A * A → A. So, the operation * performed on operands a and b is denoted by a * b.

Types of Binary Operations: The types of Binary Operations are Distributivity, Associativity, and Commutativity.

Here, you can find the exercises solution links for the topics covered in this chapter.

RD Sharma Solutions for Class 12 Maths Chapter 4 – Inverse Trigonometric Functions

Chapter 4 of RD Sharma 12 Maths Solutions discusses the topic of the inverse of a function. Students will get to learn about the definition and meaning of inverse trigonometric functions, the inverse of the sine function, inverse of the cosine function, inverse of the tangent function, inverse of secant function, inverse of cosecant function, inverse of cotangent function, and properties of inverse trigonometric functions.

Inverse Trigonometric Functions: They are the inverse functions of the basic trigonometric functions such as sine, cosine, tangent, cotangent, secant, and cosecant functions. We can also term it arcus functions, antitrigonometric functions or cyclometric functions.

Properties of Inverse Trigonometric Function

Property Set 1:

• Sin⁻¹(x) = cosec⁻¹(1/x), x∈ [−1,1] − {0}
• Cos⁻¹(x) = sec⁻¹(1/x), x ∈ [−1,1] − {0}
• Tan⁻¹(x) = cot⁻¹(1/x), if x > 0 (or) cot⁻¹(1/x) −π, if x < 0
• Cot⁻¹(x) = tan⁻¹(1/x), if x > 0 (or) tan⁻¹(1/x) + π, if x < 0

Property Set 2:

• Sin⁻¹(−x) = −Sin⁻¹(x)
• Tan⁻¹(−x) = −Tan⁻¹(x)
• Cos⁻¹(−x) = π − Cos⁻¹(x)
• Cosec⁻¹(−x) = − Cosec⁻¹(x)
• Sec⁻¹(−x) = π − Sec⁻¹(x)
• Cot⁻¹(−x) = π − Cot⁻¹(x)

Below we have links provided to each exercise solution covered in this chapter.

Also, access the following resources for Class 12 Chapter 4, Inverse Trigonometric Functions at BYJU’S:

• NCERT Solutions for Class 12 Maths Chapter 2 Inverse Trigonometric Functions
• NCERT Exemplar Solutions for Class 12 Maths Chapter 2 Inverse Trigonometric Functions

RD Sharma Solutions for Class 12 Maths Chapter 5 – Algebra of Matrices

Chapter 5 of the RD Sharma textbook starts with the definitions of Matrices. Students go on to learn about types of matrices, equality of matrices, the addition of matrices, properties of matrix addition, multiplication of a matrix by a scalar, properties of scalar multiplication, subtraction of matrices, multiplication of matrices, properties of matrix multiplication, positive integral powers of a square matrix, transpose of a matrix, properties of transpose, symmetric and skew-symmetric matrices via examples.

Matrices: A rectangular array of m × n numbers (real or complex) in the form of m horizontal lines (called rows) and n vertical lines (called columns), is called a matrix of order m by n, written as m × n matrix.

Types of Matrices: The types of matrices are symmetric matrix, skew-symmetric matrix, Hermitian and skew-Hermitian matrix, orthogonal matrix, idempotent matrix, involuntary matrix, and nilpotent matrix.

Transpose of Matrix: The matrix which is obtained from a given matrix A by changing its rows into columns or columns into rows is known as the transpose of matrix A and is denoted by A^T or A’.

Here, students can see the exercises explaining these concepts properly with solutions.

Also, access the following resources for Class 12 Chapter 5 Algebra of Matrices at BYJU’S:

• NCERT Solutions for Class 12 Maths Chapter 3 Matrices
• NCERT Exemplar Solutions for Class 12 Maths Chapter 3 Matrices

RD Sharma Solutions for Class 12 Maths Chapter 6 – Determinants

Chapter 6 of the RD Sharma Mathematics Class 12 textbook, provides a definition of determinants, determinant of a square matrix of order 1, 2, and 3, determinant of a square matrix of order 3 by using Sarrus diagram, singular matrix, minors, and cofactors, properties of determinants, evaluation of determinants, applications of determinants to coordinate geometry and applications of determinants in solving a system of linear equations and condition for consistency.

Determinants: We can define a determinant by expressing in terms of columns of the matrix as n x n matrix in terms of column vectors.

Properties of Determinants –

• Reflection Property
• All-zero property
• Proportionality property or Repetition Property
• Switching Property
• Sum Property
• Scalar multiple Property
• Factor Property
• Triangle Property
• Invariance Property
• The determinant of Cofactor matrix

Here, students can find exercises explaining these concepts properly.

Also access the following resources for Class 12 Chapter 6 Determinants at BYJU’S:

• NCERT Solutions for Class 12 Maths Chapter 4 Determinants
• NCERT Exemplar Solutions for Class 12 Maths Chapter 4 Determinants

RD Sharma Solutions for Class 12 Maths Chapter 7 – Adjoint and Inverse of a Matrix

In Chapter 7, Adjoint and Inverse of a Matrix, we shall see the definition of the adjoint of a square matrix, the inverse of a matrix, some useful results on invertible matrices, elementary transformations of elementary operations of a matrix via examples and verbal problems related to it.

Adjoint of a Matrix: The adjoint of a square matrix A = [a_ij]_nxn is the transpose of matrix [A_ij]_nxn, where A_ij is the cofactor of the element a_ij. In other words, the transpose of a cofactor matrix of the square matrix is called the adjoint of the matrix.

Inverse of a Matrix: or every m × n square matrix, there exists an inverse matrix. If A is the square matrix, then A^-1 is the inverse of matrix A and satisfies the property:

AA^-1 = A^-1A = I, where I is the Identity matrix.

Here, we have given exercises with solutions based on these topics from the chapter.

Also access the following resources for Class 12 Chapter 7 Adjoint and Inverse of a Matrix at BYJU’S:

• NCERT Solutions for Class 12 Maths Chapter 4 Determinants
• NCERT Exemplar Solutions for Class 12 Maths Chapter 4 Determinants

RD Sharma Solutions for Class 12 Maths Chapter 8 – Solutions of Simultaneous Linear Equations

In Chapter 8 of the RD Sharma textbook, we continue the discussions about equations starting with definition, consistent system, homogeneous and non-homogeneous systems, matrix method for the solution of a non-homogeneous system, and final solution of a homogeneous system of linear equations.

Homogeneous and Non-homogeneous equations: There are two types of linear equations, homogeneous and non-homogeneous. A homogeneous equation does not have zero on the right-hand side of the equality sign, while a non-homogeneous equation has a function of the independent variable on the right-hand side of the equal sign.

From the below-given links, students can access the exercises with solutions explaining the concepts from this chapter.

RD Sharma Solutions for Class 12 Maths Chapter 9 – Continuity

In this chapter, students concentrate on the definition of continuity, continuity at a point, algebra of continuous function, continuity on an interval, continuity on an open interval, continuity on a closed interval, continuous function, everywhere continuous function, and properties of continuous functions.

Continuity: First, a function f with variable x is continuous at the point “a” on the real line if the limit of f(x), when x approaches the point “a”, is equal to the value of f(x) at “a”, i.e., f(a). Second, the function (as a whole) is continuous if it is continuous at every point in its domain.

Continuous function: A continuous function is a function that does not have discontinuities which means any unexpected changes in value.

Open Intervals and Closed Intervals: An interval is called open if it doesn’t include the endpoints. It is denoted by ( ). A closed interval is one that includes all the limit points. It is denoted by [ ].

Solutions to the exercises from this chapter can be accessed here.

Also, access the following resources for Class 12 Chapter 9 Continuity at BYJU’S:

• NCERT Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability
• NCERT Exemplar Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability

RD Sharma Solutions for Class 12 Maths Chapter 10 – Differentiability

This chapter deals with differentiability. By reviewing the basic facts and formulae studied earlier in Class 11, we shall implement those formulas again in this chapter along with the advanced concepts mentioned, i.e., differentiability at a point, differentiability in a set, and some useful results on differentiability. We have compiled here the solutions to exercises from this chapter, which have explained the concept thoroughly.

Differentiability: f(x) is said to be differentiable at the point x = a if the derivative f ‘(a) exists at every point in its domain. It is given by

f'(a)=

Differentiability formula: Assume that if f is a real function and c is a point in its domain. The derivative of f at c is defined by 0

The derivative of a function f at c is defined by-

Also, access the following resources for Class 12 Chapter 10 Differentiability at BYJU’S:

• NCERT Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability
• NCERT Exemplar Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability

RD Sharma Solutions for Class 12 Maths Chapter 11 – Differentiation

Chapter 11 of  Maths RD Sharma Class 12 constitutes topics related to differentiation, differentiation of inverse trigonometric functions from first principles, differentiation of a function, differentiation of inverse trigonometric function by chain rule, differentiation by using trigonometric substitutions, differentiation of implicit functions, logarithmic differentiation, differentiation of infinite series, differentiation of parametric functions and differentiation of a function with respect to another function via illustrations.

Differentiation: Differentiation can be defined as a derivative of a function with respect to an independent variable. Differentiation, in calculus, can be applied to measure the function per unit change in the independent variable.

Let y = f(x) be a function of x. Then, the rate of change of “y” per unit change in “x” is given by:

dy / dx

If the function f(x) undergoes an infinitesimal change of ‘h’ near to any point ‘x’, then the derivative of the function is defined as

Product Rule: As per the product rule, if the function f(x) is the product of two functions u(x) and v(x), the derivative of the function is,

If f(x) = u(x) × v(x)

then f’(x) = u’(x) × v(x) + u(x) × v’(x)

Quotient Rule: If the function f(x) is in the form of two functions [u(x)]/[v(x)], the derivative of the function is

If f(x) = u(x)/v(x)

Then, f’(x) = [u’(x) × v(x) – u(x) × v’(x)]/[v(x)]²

Below we have listed links to the solutions for the exercises from this chapter.

Also, access the following resources for Class 12 Chapter 11 Differentiation at BYJU’S:

• NCERT Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability
• NCERT Exemplar Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability

RD Sharma Solutions for Class 12 Maths Chapter 12 – Higher Order Derivatives

In this chapter, students are introduced to higher-order derivatives. The concepts covered in this chapter are proving relations involving various order derivatives of cartesian functions, proving relations involving various order derivatives of parametric functions, and proving relations involving various order derivatives via illustrations.

Higher order derivative in Parametric Form: The derivative of the first order in parametric equations is given by:

dy/dx = dy/dt × dt/dx = y’(t)/x’(t)

Second-order derivative: It is the derivative of the first derivative of the given function.

If y = f(x)

Then dy/dx = f’(x)

If f’(x) is differentiable, then differentiating dy/dx again w.r.t x the 2^nd order derivative is

d/dx (dy/dx) = d²y/dx² = f’’(x)

RD Sharma Solutions for Class 12 Maths Chapter 13 – Derivative as a Rate Measurer

In this chapter, students are introduced to the derivative as a rate measurer. The concepts in this chapter cover how to find the rate measurer of derivative and related rates in which the rate of change of one of the quantities involved is required, corresponding to the given rate of change of another quantity. This chapter contains more word problems that help the students to learn sentence formation in an effective way.

Also, access the following resources for Class 12 Chapter 13 Derivative as a Rate Measurer at BYJU’S:

• NCERT Solutions for Class 12 Maths Chapter 6 Application of Derivatives
• NCERT Exemplar Solutions for Class 12 Maths Chapter 6 Application of Derivatives

RD Sharma Solutions for Class 12 Maths Chapter 14 – Differentials, Errors, and Approximations

Chapter 14 of RD Sharma Class 12 Maths deals with the topic of differentials and errors. The topics that are discussed include the definition of differentials, absolute error, relative error, percentage error, the geometrical meaning of differentials with algorithms, and finding the approximate value using differentials.

Differentials: They deal with the rate of change of one quantity with another.

If f(x) is a function, then f'(x) = dy/dx is the differential equation, where f’(x) is the derivative of the function, y is a dependent variable and x is an independent variable.

f’(x)=dy/dx; x ≠0

Absolute Error: If x is the actual value of a quantity and x₀ is the measured value of the quantity, then the absolute error value can be calculated using the formula

Δx = x₀-x.

Here, Δx is called an absolute error.

Relative Error: The relative error is defined as the ratio of the absolute error of the measurement to the actual measurement. Using this method, we can determine the magnitude of the absolute error in terms of the actual size of the measurement.

If x is the actual value of a quantity, x₀ is the measured value of the quantity, and Δx is the absolute error, then the relative error can be measured using the below formula.

Relative error = (x₀-x)/x = (Δx)/x

Percentage Error: Percent error is the difference between the estimated value and the actual value in comparison to the actual value and is expressed as a percentage.

The formula for percent error is:

PE = (|Estimated value – Actual value|/ Actual value) × 100

RD Sharma Solutions for Class 12 Maths Chapter 15 – Mean Value Theorems

In this chapter, we shall discuss theorems related to mean values. It also deals with Rolle’s theorem, geometrical interpretation of Rolle’s theorem, algebraic interpretation of Rolle’s theorem, the applicability of Rolle’s theorem, Lagrange’s mean value theorem, geometrical interpretation of Lagrange’s mean value theorem, verification of Lagrange’s mean value theorem, applications of Lagrange’s mean value theorem and proving inequalities by using Lagrange’s mean value theorem.

Rolle’s Theorem: A special case of Lagrange’s mean value theorem is Rolle’s Theorem which states that:

If a function f is defined in the closed interval [a, b] in such a way that it satisfies the following conditions.

i) The function f is continuous on the closed interval [a, b]

ii)The function f is differentiable on the open interval (a, b)

iii) Now, if f (a) = f (b), then there exists at least one value of x, let us assume this value to be c, which lies between a and b, i.e. (a < c < b) in such a way that f ‘(c) = 0.

Precisely, if a function is continuous on the closed interval [a, b] and differentiable on the open interval (a, b), then there exists a point x = c in (a, b) such that f'(c) = 0

Lagrange’s Mean Value Theorem: If a function f is defined on the closed interval [a, b] satisfying the following conditions –

i) The function f is continuous on the closed interval [a, b]

ii)The function f is differentiable on the open interval (a, b)

Then there exists a value x = c in such a way that

f'(c) = [f(b) – f(a)]/(b-a)

This theorem is also known as the first mean value theorem or Lagrange’s mean value theorem.

From the below-given links, students can access the exercises with solutions explaining the concepts from this chapter.

RD Sharma Solutions for Class 12 Maths Chapter 16 – Tangents and Normals

Students can learn the concepts related to tangents and normals in this chapter. It also deals with the slope of a line, slopes of tangent and normal, finding slopes of tangent and normal at a given point, finding the point on a given curve at which tangent is parallel or perpendicular to a given line, equations of tangent, and normal with an algorithm, finding the equation of tangent and normal to a curve at a point, finding tangent and normal parallel or perpendicular to a given line, finding tangent or normal passing through a given point, angle of intersection of two curves and orthogonal curves.

Slope of a Line: The slope of a line gives the measure of its steepness and direction. The slope of a straight line between two points says (x₁, y₁) and (x₂, y₂) can be easily determined by finding the difference between the coordinates of the points. The slope is usually represented by the letter ‘m’.

If P (x₁, y₁) and Q (x₂, y₂) are the two points on a straight line, then the slope formula is given by:

Slope, m = Change in y-coordinates/Change in x-coordinates

m = (y₂ – y₁)/ (x₂ – x₁)

Equation of Tangent: The equation of the tangent (x₀, y₀) to the curve y=f(x) is

y – y₀ = f ′(x₀)(x – x₀)

Equation of Normal: The equation of the normal to the curve y=f(x) at the point (x₀, y₀) is given as:

(y-y₀) f’(x₀) + (x-x₀) = 0

Solution links are provided below to each exercise, which is covered in this chapter.

RD Sharma Solutions for Class 12 Maths Chapter 17 – Increasing and Decreasing Functions

Chapter 17 of RD Sharma textbook deals with the topic of increasing and decreasing functions, solution of rational algebraic inequations with algorithms, strictly increasing functions, strictly decreasing functions, monotonic functions, monotonic increasing, and monotonic decreasing functions, necessary and sufficient conditions for monotonicity, finding the intervals in which a function is increasing or decreasing and proving the monotonicity of a function on a given interval.

Increasing Function: For a function, y = f(x) to be increasing dy/dx ≥ 0 for all such values of interval (a, b) and equality may hold for discrete values.

Decreasing Function: For a function, y = f(x) to be monotonically decreasing dy/dx ≤ 0 for all such values of interval (a, b) and equality may hold for discrete values.

Monotonic function: A monotonic function is defined as any function which follows one of the four cases mentioned above.

Students can access the exercises solution links provided below, which cover all the topics discussed in this chapter.

RD Sharma Solutions for Class 12 Maths Chapter 18 – Maxima and Minima

In Chapter 18, Maxima and Minima of RD Sharma textbook deal with the maximum and minimum values of a function in its domain, definition of maximum, local maxima and local minima, definition and meaning of local maximum, first derivative test for local maxima and minima along with algorithm, higher-order derivative test, point of inflection, properties of maxima and minima, maximum and minimum values in the closed interval and applied problems on maxima and minima.

Maxima and Minima:  Wherever the double derivative is positive it is the point of minima. Wherever the double derivative is negative it is the point of maxima on the curve.

Local Maxima and Minima: The local maxima and minima are defined as:

1. If f(a) ≤ f(x) for all x in P’s neighbourhood (within the distance nearby P, where x = a), f is said to have a local minimum at x = a.
2. If f(a) ≥ f(x) for all x in P’s neighbourhood (within the distance nearby P, where x = a), f is said to have a local maximum at x = a.

Here, students can see the exercises explaining these concepts properly with solutions.

RD Sharma Solutions for Class 12 Maths Chapter 19 – Indefinite Integrals

In this chapter, students concentrate on indefinite integral, primitive and antiderivative, fundamental integration formulae, some standard results on integration, integration of trigonometric functions, integration of exponential functions, geometrical interpretation of indefinite integral, comparison between differentiation and integration, methods of integration, integration by substitution, integration by parts, integration of rational algebraic functions by using partial fractions and integration of some special irrational algebraic functions. These concepts are explained very well through examples.

Indefinite Integrals: An integral which does not have any upper and lower limit is known as an indefinite integral.

Mathematically, if F(x) is any anti-derivative of f(x) then the most general antiderivative of f(x) is called an indefinite integral and denoted,

∫f(x) dx = F(x) + C

Indefinite Integrals Formula:

• ∫ 1 dx = x + C
• ∫ a dx = ax + C
• ∫ xⁿdx = ((xⁿ⁺¹)/(n+1)) + C; n ≠ 1
• ∫ sin x dx = – cos x + C
• ∫ cos x dx = sin x + C
• ∫ sec²x dx = tan x + C
• ∫ cosec²x dx = -cot x + C
• ∫ sec x tan x dx = sec x + C
• ∫ cosec x cot x dx = -cosec x + C
• ∫ (1/x) dx = ln |x| + C
• ∫ e^xdx = e^x + C
• ∫ a^xdx = (a^x/ln a) + C; a > 0, a ≠ 1

Solutions to the exercises from this chapter can be accessed here.

Also, access the following resources for Class 12 Chapter 19 Indefinite Integrals at BYJU’S:

• NCERT Solutions for Class 12 Maths Chapter 7 Integrals
• NCERT Exemplar Solutions for Class 12 Maths Chapter 7 Integrals

We realise the importance of practising problems for a better understanding of concepts and their applications. For this reason, we, at BYJU’S, bring to students an all-inclusive RD Sharma Solutions for Class 12 Maths. These solutions will help students to practise a variety of questions from different chapters to develop their analytical and reasoning skills. The solutions given here are in a well-structured format with different shortcut methods to ensure a proper understanding of the concept.

Benefits of RD Sharma Solutions for Class 12 Maths

• RD Sharma Solutions provide an algorithmic approach to solving each problem.
• These solutions give more than one way to solve, with detailed illustrations and brief summaries, which consist of concepts and formulae.
• RD Sharma 12 Solutions are prepared by our BYJU’S expert team, focusing completely on accuracy.
• Students get detailed study material on topics from CBSE Class 12 Maths.
• To make learning more interesting, each solution is provided with a pictorial representation to improve analysing skills among students.

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