Circle A circle is the set of all points in a plane, which are at a fixed distance from a fixed point in the plane. The fixed point is called the centre of the circle and the distance from centre to any point on the circle is called the radius of the circle. The equation […]

# CBSE Class 11th Maths, NCERT & R.D Sharma Solutions & Sample Papers

## CBSE Class 11 Maths Chapter 10 Straight Lines Notes

Distance Formula The distance between two points A(x1, y1) and B (x2, y2) is given by The distance of a point A(x, y) from the origin 0 (0, 0) is given by OA = Section Formula The coordinates of the point which divides the joint of (x1, y1) and (x2, y2) in the ratio m […]

## CBSE Class 11 Maths Chapter 9 Sequences and Series Notes

Sequence A succession of numbers arranged in a definite order according to a given certain rule is called sequence. A sequence is either finite or infinite depending upon the number of terms in a sequence. Series If a1, a2, a3,…… an is a sequence, then the expression a1 + a2 + a3 + a4 + … + an is called series. […]

## CBSE Class 11 Maths Chapter 8 Binomial Theorem Notes

Binomial Expression An expression consisting of two terms, connected by + or – sign is called binomial expression. Binomial Theorem If a and b are real numbers and n is a positive integer, then The general term of (r + 1)th term in the expression is given by Tr+1 = nCr an-r br Some Important Observations from the Binomial Theorem […]

## CBSE Class 11 Maths Chapter 7 Permutations and Combinations Notes

Fundamental Principles of Counting Multiplication Principle: Suppose an operation A can be performed in m ways and associated with each way of performing of A, another operation B can be performed in n ways, then total number of performance of two operations in the given order is mxn ways. This can be extended to any […]

## CBSE Class 11 Maths Chapter 6 Linear Inequalities Notes

Inequation A statement involving variables and the sign of inequality viz. >, <, ≥ or ≤ is called an inequation or an inequality. Numerical Inequalities Inequalities which do not contain any variable is called numerical inequalities, e.g. 3 < 7, 2 ≥ -1, etc. Literal Inequalities Inequalities which contains variables are called literal inequalities e.g. […]

## CBSE Class 11 Maths Chapter 5 Complex Numbers and Quadratic Equations Notes

Imaginary Numbers The square root of a negative real number is called an imaginary number, e.g. √-2, √-5 etc. The quantity √-1 is an imaginary unit and it is denoted by ‘i’ called Iota. Integral Power of IOTA (i) i = √-1, i2 = -1, i3 = -i, i4 = 1 So, i4n+1 = i, i4n+2 = -1, i4n+3 = -i, […]

## CBSE Class 11 Maths Chapter 4 Principle of Mathematical Induction Notes

Principle of Mathematical Induction Mathematical induction is one of the techniques, which can be used to prove a variety of mathematical statements which are formulated in terms of n, where n is a positive integer. Let P(n) be given statement involving the natural number n such that (i) The statement is true for n = […]

## CBSE Class 11 Maths Chapter 3 Trigonometric Functions Notes

Angle Angle is a measure of rotation of a given ray about its initial point. The original ray is called the initial side and the final position of ray after rotation is called terminal side of the angle. The point of rotation is called vertex. If the direction of rotation is anti-clockwise, the angle is […]

## CBSE Class 11 Maths Chapter 2 Relations and Functions Notes

Ordered Pair An ordered pair consists of two objects or elements in a given fixed order. Equality of Two Ordered Pairs Two ordered pairs (a, b) and (c, d) are equal if a = c and b = d. Cartesian Product of Two Sets For any two non-empty sets A and B, the set of […]

## CBSE Class 11 Maths Chapter 1 Sets Notes

Set A set is a well-defined collection of objects. Representation of Sets There are two methods of representing a set Roster or Tabular form In the roster form, we list all the members of the set within braces { } and separate by commas. Set-builder form In the set-builder form, we list the property or […]

## Class 11th Maths Formulas

Chapter 1 – Sets Chapter 2 – Relations and Functions Chapter 3 – Trigonometric Functions Chapter 4 – Principle of Mathematical Induction Chapter 5 – Complex Numbers and Quadratic Equations Chapter 6 – Linear Inequalities Chapter 7 – Permutations and Combinations Chapter 8 – Binomial Theorem Chapter 9 – Sequences and Series Chapter 10 – […]

## Mathematics Class 11 MCQs

Sets Relations And Functions Trigonometric Functions Principle Of Mathematical Induction Complex Numbers And Quadratic Equations Linear Inequalities Permutations and Combinations Binomial Theorem Sequence and Series Straight Lines Conic Sections Introduction To Three Dimensional Geometry Limits and Derivatives Mathematical Reasoning Statistics […]

## Limits and Derivatives, Class 11 Mathematics R.D Sharma Solutions

LIMITS Page 29.11 Ex 29.1 Q1. Answer : limx→0 xx Left hand limit: limx→0- xx Let x=0-h, where h→0.⇒limh→0 0-h0-h=limh→0 -hh=-1 Right hand limit: limx→0+ xxLet x=0+h, where h→0.limh→0 0+h0+h=limh→0 hh=1 Left hand limit ≠ Right hand limit Thus,lim x→0 xx does not exist. Q2. Answer : fx=2x+3,x≤2x+k,x>2Left hand limit:limx→2- fx=limx→2- 2x+3Let x=2-h, where h→0.limh→0 […]

## Introduction to Three Dimensional Geometry, Class 11 Mathematics R.D Sharma Solutions

Page 28.6 Ex 28.1 Q1. Answer : (i) The x-coordinate, the y-coordinate and the z-coordinate of the point (5, 2, 3) are all positive. Therefore, this point lies in XOYZ octant. (ii) The x-coordinate, the y-coordinate and the z-coordinate of the point (−5, 4, 3) are negative, positive and positive, respectively. Therefore, this point lies […]

## Conic Sections, Class 11 Mathematics R.D Sharma Solutions

CIRCLES Page 24.16 Ex 24.1 Q1. Answer : Let (h, k) be the centre of a circle with radius a. Thus, its equation will be x-h2+y-k2=a2. (i) Here, h = −2, k = 3 and a = 4 ∴ Required equation of the circle: x+22+y-32=42 ⇒x+22+y-32=16 (ii) Here, h = a, k = b and radius […]

## Binomial Theorem, Class 11 Mathematics R.D Sharma Solutions

Page 18.11 Ex 18.1 Q1. Answer : (i) (2x + 3y)5 =C05(2x)5(3y)0+C15(2x)4(3y)1+C25(2x)3(3y)2+C35(2x)2(3y)3+C45(2x)1(3y)4+C55(2x)0(3y)5 =32×5+5×16×4×3y+10×8×3×9y2+10×4×2×27y3+5×2x×81y4+243y5=32×5+240x4y+720x3y2+1080x2y3+810xy4+243y5 (ii) (2x − 3y)4 =C04(2x)4(3y)0-C14(2x)3(3y)1+C24(2x)2(3y)2-C34(2x)1(3y)3+C44(2x)0(3y)4=16×4-4×8×3×3y+6×4×2×9y2-4×2x×27y3+81y4=16×4-96x3y+216x2y2-216xy3+81y4 (iii) x-1×6=C06 x61x0-C16 x51x1+C26 x41x2-C36 x31x3+C46 x21x4-6C5 x11x5+C66 x01x6=x6-6 x5×1x+15 x4×1×2-20×3×1×3+15×2×1×4-6 x×1×5+1×6=x6-6×4+15×2-20+15×2-6×4+1×6 (iv) (1 − 3x)7 =C07(3x)0-C17(3x)1+C27(3x)2-C37(3x)3+C47(3x)4-C57(3x)5+C67(3x)6-C77(3x)7=1-7×3x+21×9×2-35×27×3+35×81×4-21×243×5+7×729×6-2187×7=1-21x+189×2-945×3+2835×4-5103×5+5103×6-2187×7 (v) (ax-bx)6=C06(ax)6(bx)0-C16(ax)5(bx)1+C26(ax)4(bx)2-C36(ax)3(bx)3+C46(ax)2(bx)4-C56(ax)1(bx)5+C66(ax)0(bx)6 =a6x6-6a5x5×bx+15a4x4×b2x2-20a3b3×b3x3+15a2x2×b4x4-6ax×b5x5+b6x6=a6x6-6a5x4b+15a4x2b2-20a3b3+15a2b4x2-6ab5x4+b6x6 (vi) xa-ax6=C06xa6ax0-C16xa5ax1+C26xa4ax2-C36xa3ax3+C46xa2ax4-C56xa1ax5+C66xa0ax6=x3a3-6x2a2+15xa-20+15ax-6a2x2+a3x3 (vii) x3-a36=C06(x3)6(a3)0-C16(x3)5(a3)1+C26(x3)4(a3)2-C36(x3)3(a3)3+C46(x3)2(a3)4-C56(x3)1(a3)5+C66(x3)0(a3)6=x2-6×5/3a1/3+15×4/3a2/3-20xa+15×2/3a4/3-6×1/3a5/3+a2 (viii) (1+2x-3×2)5Consider 1-2x and 3×2 as two separate entities and apply the binomial […]

## Probability, Class 11 Mathematics R.D Sharma Solutions

Probability Page No.33.6 Ex.33.1 Q1. Answer : If a coin is tossed once, the possible outcomes are a head (H) and tail (T). Therefore, the sample space of this experiment is S = { H, T}. Q2. Answer : If a coin is tossed twice, the possible outcomes are HH, HT, TH, TT. Therefore, the […]