# R D Sharma Class 11 Solutions Education is the most important stage in one’s life. Education not only makes you literate but also helps you to become a responsible citizen wherever you live. The biggest advantage of education is the development of mannerisms in a person. It helps a person to be a well-mannered and well-cultured human being. Learning Maths is more beneficial as it can lead to a good job with a good salary.

R D Sharma Solutions for Class 11 will help ambitious students to complete their dream by studying well in Maths. As we know, Maths is the base for Science, getting high scores in both subjects can make the future path easier. If the students refer to these solutions along with the regular textbooks they can be rest assured to acquire higher marks than usual.

## Mathematics

Chapter wise summary

Chapter 1 – Sets

This chapter speaks about the concept of sets. The chapters cover the topics types of sets, empty sets, singleton sets, finite & infinite sets, equal sets, subsets, power sets, universal sets, a union of sets, the intersection of sets, the difference of sets, and disjoint sets.

Chapter 2 – Relations

In this chapter, the students will learn about the Cartesian products of sets along with relations. The students will know about relations of which the definition is ‘A relation R is the subset of a Cartesian product of X x Y where X and Y are two non-empty elements, it is derived by stating the relationship between the first element and second element of the ordered pair of X x Y. The set of all primary elements of the ordered pairs is called a domain of R and the set of all second elements of the ordered pairs is called a range of R. The different types of relations are empty relation, universal relation, identity relation, inverse relation, reflexive relation, symmetric relation, transitive relation, and equivalence relation.

Chapter 3 – Functions

In this chapter, the students will learn about the functions of the Cartesian products. A relation ‘f’ is said to be a function, if every element of a non-empty set X, has only one image or range to a non-empty set Y. It can be also described as; if ‘f’ is the function from X to Y and (x,y) ϵ f, then f(x)=y, where y is the image of x, under the function f and x is the preimage of y, under f. The different types of functions are injective function or one-to-one function, surjective function or onto function, polynomial function, and an inverse function.

Chapter 4 – Measurement of Angles

Measurement of angles comes under trigonometry which is a branch of mathematics that deals with angles and sides of a triangle. The students will know the concepts of relation between degrees, radians, and real numbers. The students will learn

Angle: An angle can be defined as the rotation from the initial point to an endpoint of a ray.

Angle measurement: Angle measurement is the amount of rotation from the initial to n endpoint of a ray. The angle is a positive angle if the rotation is clockwise and a negative if the rotation is anticlockwise.

Systems of measurement of angles: There are three systems for measuring angles; sexagesimal system, centesimal system, and circular system.

Chapter 5 – Trigonometric Functions

In this chapter, the students will learn about the introduction to trigonometry, angles – degree and radians, trigonometric functions, sum & difference of two angles, various trigonometric identities like reciprocal, Pythagorean, ratio, opposite angles, complementary angles, supplementary angles, sum & difference of angles, double angle, half angle, product sum, and products trigonometric identities, and periodic function – a function returning to the same value at regular intervals.

Chapter 6 – Graphs of Trigonometric Functions

From this chapter, the students will learn to plot graphs of different functions. This chapter talks about graphs of trigonometric functions, sin graph, cos graph, tan graph, amplitude, period, phase, and a lot of practice graphs.

Chapter 7 – Trigonometric Ratios of Compound Angles

From this chapter, the students will know the formula which expresses the values of trigonometric functions at the sum or difference of two real numbers. The formulae for trigonometric ratios of compounded angles are as follows –

Sin (A+B) = sinA cosB + cosA sinB

Sin (A-B) = sin A cos B – cos sin B

Cos (A+B) = cos A cos B – sin A sin B

Cos (A-B) = cos A cos B + sin A sin B

Tan (A+B) = [tan AA + tan B] / [1- tan A tan B]

Tan (A-B) = [tan A –tan B] / [1+ tan A tan B]

Sin (A+B) sin (A-B) = sin2A –sin2B = cos2B – cos2A

Cos (A+B) cos (A-B) = cos2A – sin2A – sin2B = cos2B – sin2A

Chapter 8 – Transformation Formulae

In this chapter, two sets of transformation formulae are explained that are of fundamental importance. The products into sum or difference formulae are as follows –

Product to sum formulae

Cos(a) cos(b) = ½ [cos (a+b) + cos (a-b)]

Sin(a) sin(b) = ½ [cos (a-b) – cos (a+b)]

Sin(a) cos(b) = ½ [sin (a+b) + sin (a-b)]

Cos(a) sin(b) = ½ [sin (a+b) – sin (a-b)]

Sum to product formulae

Sin A + sin B = 2 sin (A+B)/ 2 cos (A-B)/2

Sin A – sin B = 2 cos (A+B)/ 2 sin (A-B)/2

Cos A + cos B = 2 cos (A+B)/2 cos (A-B)/2

Cos A –cos B = 2 sin (A+B)/ 2 sin (A-B)/2

Chapter 9 – Trigonometric Ratios of Multiple and Sub Multiple Angles

The formula expressing the values of trigonometric functions at multiples and submultiples of ‘x’ is the main concept. The values of trigonometric functions at some important points are the other concepts that students will learn from this chapter.

Values of trigonometric functions at ‘2x’ in terms of values at ‘x’:

Sin(2x) = 2sin(x).cos(x) = 2[tan x/(1+tan2x)]

Cos(2x) = cos2(x)-sin2(x) = [(1-tan2x)/(1+tan2x)]

Cos(2x) = 2cos2(x)-1 = 1-2sin2(x)

Tan(2x) = [2tan(x)]/[1-tan2(x)]

Sec(2x) = sec2x/(2-sec2x)

Csc(2x) = (sec x.csc x)/2

Values of trigonometric functions at ‘3x’ in terms of values at ‘x’:

Sin3 x = 3sin x – 4sin3x

Cos 3 x = 4cos3x – 3cos x

Tan 3 x = [3tanx-tan3x]/[1-3tan2x]

Chapter 10 – Sine and Cosine Formulae and their Applications

From this chapter, the students will learn about the different applications of sine and cosine formulae. The students can know –

Law of sines or sine rule: The law of sines is generally used to find the unknown angle or side of a triangle. This law can be used if certain combinations of measurement of a triangle are given.

ASA criteria: Given two angles and included side to find an unknown side

AAS criteria: Given two angles and a non-included side to find the unknown side.

Law of cosines formula: As per the cosines law formula, to find the length of sides of a triangle ABC, we can write

A2 = b2+c2-2abc cos α

B2 = a2+c2-2ac cos β

C2 = b2+a22-2ba cos γ

Chapter 11 – Trigonometric Equations

The equations that involve the trigonometric functions of a variable are called trigonometric equations. These equations have one or more trigonometric ratios of unknown angles. Sin x and cos x repeats themselves after an interval of pie. In this chapter, the students will learn solutions for trigonometric equations, proof of solutions of trigonometric equations, and trigonometric equations solutions.

Chapter 12 – Mathematical Induction

Mathematical induction is used to prove certain things. The following statements are important in mathematical induction; all the numbers lying on the real number line are known as real numbers, all the real numbers greater than zero are positive real numbers, and 25 is a real number. The students will learn what is mathematical induction, mathematical induction steps, mathematical induction examples, and mathematical induction problems.

Chapter 13 – Complex Numbers

From this chapter, the students will learn that a complex number is a number that can be represented in the form p+iq. The following topics are covered in this chapter; introduction to complex numbers, algebra of complex numbers, addition of two complex numbers, difference of two complex numbers, multiplication of two complex numbers, division of two complex numbers, power of i, the square root of a negative real number, the modulus and the conjugate of complex numbers, argand plane & polar representation, and polar representation of complex numbers.

From this chapter, the students will learn the following important things; a polynomial equation has at least one root, a polynomial equation of degree ‘n’ has ‘n’ roots, and the values of a variable that satisfy the given equation are called roots of a quadratic equation, the solution to quadratic equations can also be calculated using factorization method, if α & β are the roots of a quadratic equation, then the equation is x2-(α+β)x +αβ = 0, and the nature of roots depends on the discriminant (D) of the quadratic equation.

Chapter 15 – Linear Inequalities

In this chapter, the students will know the concept of linear inequalities in one variable and two variables with their algebraic solution and graphical solution. Linear inequalities are used to solve problems in different fields like engineering, science, and mathematics. The students will learn the introduction to linear inequalities, algebraic solutions for linear inequalities in one variable and its graphical representation, graphical solution of linear inequalities in two variables, and practice problems.

Chapter 16 – Permutation and Combination

If an event can occur in ‘m’ different ways, following which an event can occur in ‘n’ different ways, then the total number of occurrences of events in the order is ‘m*n’. The students will learn this principle for any finite number of events. They can also know about permutation definition, permutation when all the objects are distinct, permutation when all the objects are not distinct, combination definition, the relationship between permutation & combination, and solve problems.

Chapter 17 – Binomial Theorem

The binomial theorem states a formula for the expression of the powers of sums. The students will get detailed knowledge about binomial theorem and binomial theorem examples.

Chapter 18 – Arithmetic Progressions

In this chapter, the students will learn about sequences, series, and progressions. A sequence is a finite or infinite list of numbers following a specific pattern. A series is the sum of the elements in the corresponding sequence. A progression is a sequence in which the general term can be expressed using a mathematical formula. The students can also learn common differences in arithmetic progression, finite & infinite AP, general term of AP, the ‘n’th term of AP, the sum of terms in an AP, arithmetic mean, basic adding patterns, and the sum of first ‘n’ natural numbers.

Chapter 19 – Geometric Progressions

A sequence in which the ratio of the term and the preceding term is always a constant is called geometric progression. The students will learn that the geometric progression is represented by a, ar, ar2, ar3, and so on. Here, ‘a’ is the first term and ‘r’ is the common ratio. The students will also learn the general term or nth term of GP, and the sum of n terms of GP.

Chapter 20 – Some Special Series

The main concept of this chapter is the sum of n terms of some other special series. The students will learn –

Sum of first n natural numbers: The sum of first n natural numbers is sum = n(n+1)/2.

Sum of squares of first n natural numbers: The formula for the addition of squares of first n natural numbers is  Ʃn2 = [n(n+1)(2n+1)]/6

Chapter 21 – Brief Review of Cartesian System of Rectangular Coordinates

This chapter deals with the topics like Cartesian coordinate system, the distance between two points, the area of a triangle, section formula, locus, and shifting of origin. A Cartesian coordinate system is used to locate the position of any point and that point is plotted as an ordered pair (x,y) known as coordinates.

Chapter 22 – The Straight Lines

A straight line is a curve where every point on the line segment joining any two points lies on it. From this chapter, the students will understand the concept of straight lines with several examples.

Slope formula: If P(x1,y1) and Q(x2,y2) are the two points on a straight line, then the slope formula is m = change in y-coordinates/change in x-coordinates m = (y2-y1)/(x2-x1)

The slope of a line equation: The equation for the slope of a line and the points also called the point-slope form of the equation of a straight line is given by y-y1 = m(x-x1).

Chapter 23 – The Circle

This chapter speaks about the circle and the equation of any circle whose center and radius are given. The students will learn the following concepts –

Standard form of circle equation: (x-a)2 + (y-b)2 = r2, where, (a,b) are coordinates of center and r is radius.

General form of circle equation: x2 + y2 + Ax + By + C =0

Chapter 24 – Parabola

In this chapter, the students will learn about parabola and find the general equation of a conic section when its directrix, focus, and eccentricity are given. The following concepts are discussed in this chapter –

Parabola: The section of a right circular cone by a plane parallel to a generator of a cone is called a parabola.

Standard equation of a parabola: If the directrix is parallel to the y-axis, the standard equation for a parabola is y2 = 4ax.

If the directrix of a parabola is parallel to the x-axis, then the standard equation becomes x2 = 4ay.

Chapter 25 – Ellipse

This chapter deals with the concept of an ellipse and finding out the equation of the ellipse in standard and other forms. The explanation is in simple language for the students to understand easily.

Ellipse: The ellipse is one of the conic sections that is produced when a plane cuts the cone at an angle to the base.

Ellipse equation: When the center of the ellipse is at the origin (0,0) and the foci are on the x-axis and y-axis then we can easily derive the ellipse equation.

Standard equation of ellipse: The equation of the ellipse with center at origin and major axis along the y-axis is x2/b2 + y2/a2 = 1.

Chapter 26 – Hyperbola

Hyperbola is a particular case of conic. This chapter details the information on the hyperbola. A hyperbola is the locus of all those points in a plane such that the difference in their distances from two fixed points in a plane is constant. The students will learn hyperbola eccentricity, standard equation of hyperbola, latus rectum of hyperbola, hyperbolic sine function, hyperbolic cosine function, hyperbolic tangent function, properties of hyperbolic functions, hyperbolic function identities, and inverse hyperbolic functions.

Chapter 27 – Introduction to 3D Coordinate Geometry

The coordinate of a point in space, signs of coordinates of a point, distance formula, and section formula are the topics covered in this chapter. The students will understand the simple explanation of –

Distance formula: The formula to find the distance between two points PQ is given by:

PQ = √[(x2-x1)2 + [(y2-y1)2

Chapter 28 – Limits and Derivates

In mathematics, a limit is defined as a value that a function approaches the output for the given input values. the derivatives are the varying rate of change of a function with respect to an independent variable. From this chapter, the students will learn limits definition, properties of limits of a given function, derivatives definition, properties of derivatives of a given function, and general derivative formulas.

Chapter 29 – Mathematical Reasoning

In this chapter, the student will learn what is mathematical reasoning, mathematically acceptable statements, types of reasoning in maths, inductive reasoning, deductive reasoning, types of reasoning statements (simple, compound, and if-then statements), how to deduce mathematical statements, a negation of the given statement, contradiction method, counter statements, and examples of mathematical reasoning.

Chapter 30 – Statistics

In this chapter, the study of the measures of dispersion and the methods of calculating grouped and ungrouped data are elaborated on. Statistics is a branch that deals with the collection, organization, and interpretation of data. It is like a study of the probability of events occurring based on the collection of data or known quantities of data. The students will learn measures of dispersion, range, mean deviation, mean deviation of ungrouped data, the mean deviation for grouped data, discrete frequency distribution, mean deviation about mean & median, continuous frequency distribution, and standard deviation.

Chapter 31 – Probability

In this chapter, the students will learn probability & its concepts, events in probability, the axiomatic approach to probability, and solved examples.

Significance of Maths in life: –

Most students opt to leave the Science stream in 11th standard for the fear of Maths. However, to run away from Maths, one can only choose the Arts stream where one needs to read, write, and memorize excessively. Opting for Science, for those who are comfortable, will always be advisable because of the wide range of job availability.

Unknowingly, we all use Math in our daily life. We do calculations while purchasing, estimate the approximate costs, and so on but we never think of it as a subject to study. R D Sharma Solutions for Class 11 will help the students at this crucial stage of education as 11th std is considered the prelim of the ever-important year of 12th std.

## Conclusion

Class 11 is a very important year in a student’s life that have opted for the Science or Commerce stream. To pursue a good career, one must exhibit exceptional dedication to build a successful career and grow in life. For ambitious and studious students a lot of study material is available nowadays. Choosing the right study material and utilizing the resources fully is the student’s responsibility.

R D Sharma Solutions for Class 11 is one of such exclusive study materials to perform well in Math. R D Sharma solutions deal only with Math study material and hence they can provide the best in the relevant field. Their experienced and professional tutors have designed it to match the requirements of class 11 Math.

## FAQs

##### Can a student rely on R D Sharma study material?

R D Sharma Solutions for Class 11 is exclusively for Math and those who wish to score high marks can refer to it along with regular textbooks.

##### R D Sharma solutions are relevant to ICSE or CBSE?

R D Sharma solutions are relevant to the CBSE course of study.

##### Where can the students get R D Sharma solutions?

R D Sharma solutions are available offline as well as online. The students can also download them from the official website.

##### Why Math is rated as the toughest subject?

Any syllabus of all subjects relates to the capacity of the students of that level so s such no subject is tougher or toughest. It is the presumption of most students that Math is a tough subject.

##### Will the study of Math be beneficial for learning computers?

Though at basic levels there is no direct connection between Math and computer, at later stages of engineering or science it will definitely help the students.

##### Which subjects are more important to building career in Science?

Math and Science are the most important subjects if one wishes to build a career in Science, Math, or engineering. #### Robin Singh

Robin Singh is education professional tutor of NCERT. I have good knowledge of CBSE all subjects. Expert in maths, physics and chemistry. if students have any doubt about NCERT Solutions so contact us info@cbsenotes.in.

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