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2 ] Oswaal CBSE Mind Maps, Mathe matics, Class-XI To know about more useful books click here MIND MAP AN INTERACTIVE MAGICAL TOOL What? When? Why? How? Result anytime, as frequently as you like till it becomes a habit! presenting words and concepts as pictures!! Learning made simple ‘a winning combination’ with a blank sheet of paper coloured pens and your creative imagination! Mind Maps for each chapter show the breakthrough system of planning and note-taking which help make schoolwork fun and cut homework time in half !! OSWAAL BOOKS LEARNING MADE SIMPLE nlock the imagination To u and come up with ideas emember facts and To r gures easily ake clearer and To m better notes oncentrate and save To c time lan with ease and ace To p exams
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8 ] Oswaal CBSE Mind Maps, Mathematics, Class-XI To know about more useful books click here ,1*9=.;#ff,8695.A7>6+.;:>*-;*=2,.:>*=287< Let squaring both sides, we get (x+iy) 2= a+ib i.e. x2 −y2=a, 2xy=b solving these equations, we get square root of z. , x iy a ib + = + 22 22 , a b i ab ab − + + + z If z= a+ib is a complex number (i)Distance of z from origin is called as modulus of complex number z. It is denoted by (ii) Angle θ made by OP with +ve direction of X-axis is called argument of z. 2 2 r z a b = = + P (a,b) Re (z) b a O (0,0) 2 1, 1 i i = − = − In general, 4 r 1; 0 ; 1 1; 2 ; 3 k r i r i = r i r + = = − = − = A complex number z=a+ib can be represented by a unique point P(a,b) in the argand plane z=a+ib is represented by a point P (a,b) Im(z) θ P (a,b) Re (z) r O (0,0) Im(z) Let: z = a+ 1 ib and z = c+ 1 id be two complex numbers, where a,b,c,d R and 1. Addition: 2. Subtraction: 3. Multiplication: 4. Division: 1 i = − ∈ 1 2 ( ) ( ) ( ) (b d) z z a ib c id a c i + = + + + = + + + 1 2 ( ) ( ) ( ) (b d)i z z a ib c id a c − = + − + = − + − 1 2 2 ( ) ( ) ( )( ) ( ) ( ) ( 1) z z a ib c id a c id ib c i d ac bd ad bc i i . = + + = + + + = − + + = − ∵ 0 a ≠ 1 2 22 22 . z a ib a ib c id z c id c id c id ac bd bc ad i cd cd + + − = = + + − + − = + + + Powers of ‘i’ Argand P al ne Numbers Algebra of Complex mu N eb sr oiti nif e D o n Cf mo p el x So ul t oi n of Quadrat ci Equat oi n Polar Representation Complex Number & Quadratic Equations f o C mo p el x Number o M du ul s & Argument Let a = r cos b = r sin where, θ θ θ θ θ θ r z = and = arg (z) The argument ‘θ’ of complex number z = a+ib is called principal argument of z if (cos sin ) z a ib r i θ θ ∴ = + = + P (a,b) r sin r cos Real (z) Im (z) . π θ π − < ≤ r · General form of quadratic equation in x is ax2+bx+c=0, Where The solutions of given quadratic equation , , & 0 a b c ∈R a ≠ are given by b2–4ac 2 b x a − ± = Note: A monomial equation has one root. A polynomial equation of degree n has n roots. A number of the form a+ib, where a,b R and is called a complex number and denoted by ‘z’. z= a+ib ∈ 1 i = − Imaginary part Real part Conjugate of a complex number: For a given complex number z=a+ib, its conjugate is defined as z= a – ib For a non-zero complex number z=a+ib there exists a complex number + i denoted by – or z−l, called multiplicative inverse of z 22 22 a a ib i i −b ( ) 1 0 1 ab ab + + = + = + + Such that: ( 0, 0), a b ≠ ≠ Note: If a+ib = c+id ⇔ a = c & b = d 1 z a a2+b2 –b a2+b2 ( ( b2–4ac<0 ∴
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10 ] Oswaal CBSE Mind Maps, Mathematics, Class-XI To know about more useful books click here ,1*9=.;# 9.;6>=*=287<,86+27*=287< Permutations of alike Objects Permutations & Combinations 1. The no. of circular permutations of ‘n’ distinct objects is (n-1)! 2. If anti-clockwise and clockwise order of arrangements are not distinct then the no. of circular permutations of n 1 ( 1)! 2 n − e.g.: arrangements of beads in a necklace, arrangements of flower in a garland etc. The no. of permutations of n objects taken all at a time, where p1 objects are of first kind, p2 objects are of the second kind, ....pk objects are of the kth kind and rest, if any one are all different is 1 2 ! 3 ! ! ! ! k n p p p p … Each of the different arrangements which can be made by taking some or all of a number of distinct objects is called a permutation. The no. of permutations of n different things taken r at a time, where repetition is not allowed, is denoted by and is given by ! ( )! n n r = − F.P.C. of Multiplication: If an event can occur in m different ways, following which another event in n different ways, then the total number of occurance of the events in the given order is m×n F.P.C. of Addition: If there are two events such that they can performed independently in m and n ways respectively, then either of the two events can be performed in ( m +n) ways. r n 2 1 n + + = − … … Certain Conditions C mo i b an oit s n of Count ni g F( P. C. .) Fund ma ental Pr ni c pi el Permutation C ri uc al r Pe mr utat oi ns Relation between ! !( )! n r n r = − In particular, r =n ! 1 !0! n n = = = + = n Cr n Cn n+1 Cr n Cr n Cr n C n-r C r-1 i.e., Corresponding to each combination of we have r! permutations, because r objects in every combination can be rearranged in r! ways n Cn The no. of arrangements of n distinct objects taken r at a time, so that k particular objects are (i) Always included: (ii) Never included: The no. of selections from ‘n’ different objects, taking at least one= + + n C1 n C2 n C +..............C 3 =2 n n–1 n–k Cr–k.r! n–k C r. r! n Cr , The no. of all permutations of ‘n’ different objects taken r at a time: (i) When a particular object is to be always included in each arrangement is (ii) When a particular object is never taken in each arrangement in (iii) The no. of permutation of n different things taken r at a time, where repetition is allowed is (n)r n-1 P n-1 Pr n Pr n Pr Each of different selections made by taking source or all of a number of distinct objects or item, irrespective of their arrangements or order in which they are placed, is called a combination. The no. of combinations of n different things taken r at a time, denoted by is given by ! ; 0 r!( )! n n r = ≤ ≤ − n Cr n Cr r n r Since, n P= C ×r! n n n r r 0<r≤n distinct items is ∴
Oswaal CBSE Mind Maps, Mathematics, Class-XI [ 11 To know about more useful books click here ,1*9=.;#!+27862*5=1.8;.6 E U Q \ QQ² U DUH U U U U Q [ ² [ \ Q Q Q² Q&& & ² Q
12 ] Oswaal CBSE Mind Maps, Mathematics, Class-XI To know about more useful books click here ,1*9=.;#"<.:>.7,.<*7-<.;2.< Sequences and Series If a, b, c are in G.P., b is the GM between a & c, b2 = ac, therefore ; a > 0, c >0 If a, b are two given numbers & a, G1, G2, …, Gn, b are in G.P., then G1, G2 ,G3 … Gn are n GMs between a & b. b ac = ( ) 1 1 1 n b G a a + = ⋅ ( )2 1 2 n b G a a + = ⋅ ( ) 1 n n n b G a a + = ⋅ ...... or where 2 1 2 , , n n G ar G ar G ar = = … = ( ) 1 1 n b r a + = In a series,where each term is formed by multiplying the corresponding term of an AP & GP is called Arithmetico‐Geometric Series. Eg: 1+3x+5x2+7x3+… Here, 1, 3, 5, … are in AP and 1, x, x2, … are in GP. If A.M.: G.M. of two positive numbers a and b is m : n, then a : b = Sum of first ‘n’ natural numbers Sum of first ‘n’ odd natural numbers Sum of squares of first n natural numbers Sum of cubes of first n natural numbers If A and G be the AM & GM between two positive numbers, then the numbers are 2 2 A A G ± − m+ m2 – n2 ( ( m– m2 – n2 :( ( Let A and G be the AM and GM of two given positive real numbers a & b, respectively. Then, 2 a b A + = and G ab = Thus, we have , ( )2 2 2 2 0 2 ab ab ab A G ab a b + + − − = − = − = ≥ From (1), we obtain the relationship A ≥ G ( ) 1 1 1 2 3 2 n k n n k n = + = + + + + = ∑ … ( )( ) 2 2 2 2 2 1 1 2 1 1 2 3 6 n k n n n k n = + + = + + + + = ∑ … ( ) 2 2 3 3 3 3 3 1 1 1 1 2 3 2 n n k k n n k n k = = + = + + + + = = ∑ ∑ … 2 1 (2 1) 1 3 5 (2 1) n k k n n = − = + + + + − = ∑ … If three numbers are in A.P., then the middle term is called AM between the other two, so if a, b, c, are in A.P., b is AM of a and c. AM for any ‘n’ +ve numbers a1, a2, a3 , …, an is 1 2 3 n a a a +a AM n + + + = … n-Arithmetic Mean between Two Numbers: If a, b are any two given numbers & a, A1, A2, .., An, b are in A.P. then A1, A2, …, An are n AM’s between a & b. ( ) ( ) 1 2 2 , 1 1 1 n b a n b a b a A a A a A a n n n − − − =+ =+ = + + + + … or A = 1 a+ d, A2 = a+2d, …, An = a + nd, where 1 b a d n − = + An A.P. is a sequence in which terms increase or decrease regularly by the fixed number (same constant). This fixed number is called ‘common difference’ of the A.P. If ‘a’ is the first term & ‘d’ is the common difference and ‘l’ is the last term of A.P., then general term or the nth term of the A.P. is given by an = a+ (n−1)d from starting and a =n l–(n–1) d from the end. The sum Sn of the first n terms of an A.P. is given by [2 ( 1) ] [ ] 2 2 n n n S a n d a l = + − = + Sequence is a function whose domain is the set of N natural numbers. Real sequence: A sequence whose range is a subset of R is called a real sequence Series: If a1, a2, a3, ...an, is a sequence, then the expression a +a 1 +a 2 +…a 3 +.. is a series. A series is finite or infinite n according to the no. of terms in the corresponding sequence as finite or infinite. Progression: Those sequences whose terms follow certain patterns are called progression. A sequence is said to be a geometric progression or G.P., if the ratio of any term to its preceding term is same throughout.This constant factor is called ‘Common ratio‘. Usually we denote the first term of a G.P. by ‘a’ and its common ratio by ‘r’. The gereral or nth term of G.P. is given by a =ar n n–1 The sum Sn of the first n terms of G.P. is given by ( 1) 1 n n a r S r − = − (1 ) 1 n a r r − − or Sum of infinite terms of G.P. is given by ; 1 1 n a S r r = < − Def ni it oi ns Geometric Mean (GM) Arithmetic Mean (AM) Arithmetic Progression (A.P.) Special Sequences Sum upto terms of some Re al t oi n hs pi be wt e ne MA na d MG be wt een wt o quantit ei s P or pert ei s of AM & GM (1) ............... if r≠1 · · r P og er ss oi n G( P. .)
Oswaal CBSE Mind Maps, Mathematics, Class-XI [ 13 To know about more useful books click here ,1*9=.;#<=;*201=527. 1.The image of a point(x1, y1) about a line ax + by+ c =0 is: 2.Similarly, foot of perpendicular from a point on the line is: x − x1 a y − y1 b − (ax1 + by1 + c) a2 + b2 = = x − x1 a y − y1 b = = ax1 + by1 + c a2 + b2 − Straight Line po ni t about a l ni e Ref el ct oi n & of ot of perpendicu al r a The length of the perpendicular from P(x1, y1) on ax + by +c =0 is: 2 2 ax by c + + a b + L ne g ht of ht e pe pr end ci u al r rf mo a po ni t on a l ni e If m1 and m2 are the slopes of two intersecting lines (m m1 2 ≠−1) and θ be the acute angle between them 2 1 21 ta n m m 1 m m θ − = + Area of triangle whose vertices are (x1, y1), (x2, y2) and (x3, y3) is The distance between the points A(x1, y1) and B(x2, y2) is ni senil mret os ht f ri e ol s ep s wt eb el gnA nee o wt rts gi a t h A er a of Tr ai ng el Distance Fo mr ula S ol pe Fo mr u al Fo mr u al Sect oi n √ √ √ (x 1 –x 2 ) 2 +(y 1 –y2 ) 2 1. When two lines are parallel their slopes are equal. Thus, any line parallel to y = mx +c is of the type y = mx +d , where d is any parameter 2. Two lines ax + by +c=0 and are parallel if 3. The distance between two parallel lines with equations ax +by +c1 = 0 and ax +by +c2 = 0 is d= 2 1 2 2 c c a b a’x + b’x + c’ = 0 − + a b c = ≠ a’ b’ c’ 1.When two lines of the slope m1 & m2 are at right angles, the Product of their slope is −1, i.e. , m m1 2 = −1. Thus, any line perpendicular to y=mx + c is of the form , y = x+d where d is any parameter. 1 m − 2.Two lines ax + by +c =0 and are perpendicular if Thus, any line perpendicular to ax + by +c =0 is of the form bx – ay+ k =0, where k is any parameter. aa’ + bb’ =0. a’x + b’y + c’=0 1. POINT-SLOPE FORM: y – y1 = m(x – x1) is the equation of a straight line whose slope is ‘m’ and passes through the point (x1, y1). 2. SLOPE INTERCEPT FORM: y = mx +c is the equation of a straight line whose slope is ‘ m’ and makes an intercept c on the y-axis. 3.TWO POINT FORM: is the equation of a straight line which passes through (x1, y1) & (x2, y2). 4. INTERCEPT FORM: is the equation of a straight line which makes intercepts a & b on x and y axis respectively. 5. NORMAL / PERPENDICULAR FORM: xcos α + ysin α = p (where p > 0,≤0 α < 2π) is the equation of a straight line where the length of the perpendicular from origin O on the line is p and this perpendicular makes an angle α with +ve x-axis. 6. GENERAL FORM: ax +by+ c =0 is the equation of a straight line in general form. In this case, slope of line = 1 x y a b + = a b − y − y1 = y2 − y1 x2 − x1 (x −x1) 1 The P(x, y) divided the line joining A(x , y) and B(x , y) in the ratio m:n, then Note:>;fflfffflff;fflfflffffinal, but (–ve) signs tells, the division is external. >1 ff4fflff$flfffffflff8 A & B. x = mx2 + nx1 – – – – m + n y1 = my + ny 2 1 m + n ; 1 1 2 x1(y1 − y2) +x2(y3 − y1) + x3 (y1 − y2) >1θ is the angle at which a straight line is inclined to be+ve ffiff $ ffl? ≤θ < 180o , θ 90o, then the slope of the line, denoted by m, is defined by m = tan θ. If θ is 90o, m doesn’t exist, but the line is parallel to y$ ffl1 θ=0o, then m fffflflffiff $ ffl. ≠ >1-x1, y1) and B(x2, y2), x1 ≠ x2 are points on straight line, then the slope m of the line is given by m = Parallel lines 1 1 2 2 Perpendicular lines then (y1 – y2 ) (x1 – x2) d =
14 ] Oswaal CBSE Mind Maps, Mathematics, Class-XI To know about more useful books click here ,1*9=.;#,872,<.,=287< Conic Sections A hyperbola is the set of all points in a plane, the difference of whose distances from two fixed points in the plane is a constant. The two fixed points are called the ‘foci’ of the hyperbola. The mid-point of the line segment joining the foci is called the ‘centre’ of the hyperbola. The line through the foci is called ‘transverse axis’. Line through centre and perpendicular to transverse axis is called ‘conjugate axis’. Points at which hyperbola intersects transverse axis are called ‘vertices’. A parabola is the set of all points in a plane that are equidistant from a fixed line and a fixed point in the plane. Fixed line is called ‘directrix’ of parabola. Fixed point F is called the ‘focus’. A line through focus & perpendicular to directrix is called ‘axis’. Point of intersection of parabola with axis is called ‘vertex’. Main facts about the parabola Circles, ellipses, parabolas and hyperbolas are known as conic sections because they can be obtained as intersections of plane with a double napped right circular cone . From the given figure. (i) Section will represent circle, if =90° (ii) Section will represent an Ellipse, if < (iii) Section will represent a parabola if if = (iv) Section will represent a hyperbola if 0 es pill E Hyperbola Forms of Parabolas Axis Directix Vertex Focus Lenght of latus rectum Equations of latus rectum y2=4ax y =0 x a =– (0, 0) ( , 0) a a4 x =a y2=–4ax y =0 x a = (0, 0) (– , 0) a a4 x =–a x2=4ay x =0 y a =– (0, 0) (0, ) a a4 y =a x2=–4ay x =0 y a = (0, 0) (0,– ) a a4 y =–a An ellipse is the set of all points in a plane, the sum of whose distances from two fixed points in the plane is constant. The two fixed points are called the ‘foci’ of the ellipse. The midpoint of line segment joining foci is called the ‘centre’ of the ellipse. The line segment through the foci of the ellipse is called’ major axis’. The line segment through centre & perpendicular to major axis is called minor axis. Forms of ellipse Equation of major axis Length of major axis Equation of Minor axis Lenght of Minor axis Directrices Equation of latus rectum Length of latus rectum Centre a b > y=0 2a x=a 2b x2 a2 + =1 — y2 b2 — x =+ a —e – x =+ ae – (0, 0) 2b2 —–a a b > x=0 2a y=0 2b x2 b2 + =1— y2 a2 — y =+ a —e – y =+ ae – (0, 0) 2b2 —–a Forms of the hyperbola Equation of transverse axis Equation of conjugate axis Length of transverse axis Foci Equation of latus rectum Length of latus rectum Centre y=0 x=a 2a x2 a2 – =1— y2 b2 — x =+ ae – (0, 0) 2b2 —–a +ae, 0) –( x=0 y=a 2a y2 a2 – =1— x2 b2 — x =+ ae – (0, 0) 2b2 —–a +ae, 0) –( Y x = –a X' X F ( 0) a, = O y = ax 2 4 Y' = F ( 0) –a, X' X' O Y Y' y = ax 2 –4 x = +a (a) (b) Y F ( 0) a, = O y=–a X x = ay 2 4 F ( – ) 0, a y=a O = X x = ay 2 –4 (c) (d) % % % % % % % Y % % C P ( , ) x y B X F ( , 0) 2 ae F ( , 0) 1 –ae O A D x2 a2 — y2 b2 —+ =1 (0, ) a % % % % % % % (0, ) ea X (– 0) b, ( 0) b, O (0, – ) ae (0 – ) , a x2 b2 — y2 a2 —+ =1 Y • • (0, ) ae • (0, ) a X X' O O (0, – ) a (0, – ) ae Y' y2 a2 —– – =1 x2 b2 —– • • • • x2 a2 —– – =1 y2 b2 —– Y' X X' (– , 0) ae (– , 0) a ( , 0) a ( , 0) ae Y O A circle is a set of all points in a plane that are equidistant from a fixed point in the plane. The fixed point is called the ‘centre’ of the circle and the distance from the centre to a point on the circle is called the ‘radius’ of the circle. The equation of a circle with centre (h, k) and the radius r is The general equation of circle in x2 +y2+2 +2 + =0 gx fy c its centre is (– , – ) and radius = g f r g 2 +f 2– c • • Plane Generator Axis Upper nappe V • Lower nappe Here, > and = (1– ), <1 a b b a e e 2 2 2 Here, = ( ), >1 b a e e 2 2 2–1 ( – ) +( – ) = xh yk r 2 2 2
Oswaal CBSE Mind Maps, Mathematics, Class-XI [ 15 To know about more useful books click here ,1*9=.;#fl27=;8->,=287=8=1[..-26.7<287*50.86.=;B Introduction to Three Dimensional Geometry In three dimensions, the coordinate axes of a rectangular cartesian coordinate system are three mutually perpendicular lines. The axes are called x, y and z axes. The three planes determined by the pair of axes are the coordinate planes, called xy, yz and zx-planes. The three coordinate planes divide the space into eight parts known as octants. The coordinates of a point P in 3D Geometry is always written in the form of triplet like (x,y,z). Here, x, y and z are the distances from yz, zx and yx planes, respectively. Eg: Any point on x-axis is : (x, 0, 0) Any point on y-axis is : (0, y, 0) Any point on z-axis is : (0, 0, z) Distance between two points P(x1, y1, z1) and Q(x2, y2, z2) is given by ( ) ( ) ( ) 2 2 2 2 1 2 1 2 1 PQ x x y y z z = − + − + − Eg: Find the distance between the points P(1, –3, 4) and (–4, 1, 2). Sol: The distance PQ between the points P & Q is given by ( ) ( ) ( ) 2 2 2 4 1 1 3 2 4 PQ = − − + + + − 25 16 4 = + + 45 = 3 5 = units i D ts na ec eb wt nee o wT st ni oP Sect oi n Fo mr ula oo C dr ni t a se f o Ma di op ni t el gnairTaf o r oo C et ani d ne Ceht f os ort di Introduction The coordinates of the midpoint of the line segment joining two points P(x1, y1, z1) and Q(x2, y2, z2) are 1 2 1 2 1 2 , , 2 2 2 x x y y z z + + + Eg: Find the midpoint of the line joining two points P(1, –3, 4) and Q(–4, 1, 2). Sol: Coordinates of the midpoint of the line joining the points P & Q are 14 31 4 2 , , 2 2 2 − − + + 3 , 1,3 − 2 − i.e. . The coordinates of the point R which divides the line segment joining two points P(x1, y1, z1) and Q(x2, y2, z2) internally and externally in the ratio m : n are given by 2 12 1 2 1 , , mx nx my ny mz nz mn mn mn + + + + + + 2 12 1 2 1 , , mx nx my ny mz nz mn mn mn − − − − − − & respectively. Eg: Find the coordinates of the point which divides the line segment joining the points (1,–2, 3)and (3, 4, –5) in the ratio 2:3 internally. Sol : Let P(x, y, z) be the point which divides line segment joining A (1,–2, 3) and B (3, 4, –5) internally in the ratio 2:3. Therefore, ( ) ( ) 2 3 3 1 9 2 3 5 x + = = + ( ) ( ) 2 4 3 2 2 2 3 5 y + − = = + ( ) ( ) 2 5 3 3 1 2 3 5 z − + − = = + 1 5 − Thus, the required point is 9 5 2 5 , , ( . ( The coordinates of the centroid of the triangle, whose vertices are (x1, y1, z1), (x2, y2, z2) and (x3, y3, z3) are 1 2 3 1 2 3 1 2 3 , , 3 3 3 x x x y y y z z z + + + + + + Eg: The centroid of a triangle ABC is at the point (1, 1, 1). If the coordinates of A and B are (3, –5, 7) and (–1, 7, –6), respectively, find the coordinates of the point C. Sol: Let the coordinates of C be (x, y, z) and the coordinates of the centroid G be (1, 1, 1). Then 3 1 3 x + − =1, i.e., x=1; 5 7 3 y − + 7 6 3 z + − =1, i.e., z=2. So, C (x,y,z) = (1,1,2) = 1, i.e., y=1;
16 ] Oswaal CBSE Mind Maps, Mathematics, Class-XI To know about more useful books click here ,1*9=.;#ffi5262=<*7--.;2?*=2?.< Limits and Derivatives Let f(x) = anxn + an-1 xn-1 + – – +a1x+a0 be a polynomial function, where ai s are all real numbers and a ≠0. Then the derivative function is given by n ( ) ( ) − − − = + − + − − 1 2 1 1 n n n n d f x na x n a x dx 2 1 2a x a + + The derivative of a function f at a is defined by ( ) ( ) ( ) 0 limh f a h f a f a h → + − = ′ Eg: Find derivative of ( ) 1 f x x = Sol: We have ( ) ( ) ( ) 0 limh f x h f x f x h → + − ′ = ( ) 0 0 1 – 1 1 lim lim h h x h h x h h x x h → → + − = = + 1 2 − x = . For functions u and v the following holds: ( ) u v u v ′ ′ ′ ± = ± ( ) uv u v v u ′ ′ ′ = + 2 u u v uv v v ′ ′ ′ − = provided all are defined and v ≠ 0 du u dx ′ = Here, and dv v dx ′ = ( ) 1 n n d x nx dx − = ( ) sin cos d x x dx = ( ) cos sin d x x dx = − 1 lim n n n x a x a na x a − → − = − 0 sin lim 1 x x x → = 0 1 cos lim 0 x x x → − = For functions f and g the following holds: () () () ( ) lim lim lim x a xa xa f x g x f x g x → → → ± = ± () () () () lim . lim .lim x a xa xa f x g x f x g x → → → = ( ) ( ) ( ) ( ) lim lim x a lim x a x a f x f x gx gx → → → = We say is the expected value of f at x=a given that the values of f near x to the left of a. This value is called the left hand limit of f at ‘a’ We say is the expected value of f at x=a given that the values of f near x to the right of a. This value is called the right hand limit of f ‘at a’. If the right and left hand limits coincide, we call that common value as the limit of f(x) at x=a and denoted it by Eg: Find limit of function f(x) =(x–1)2 at x=1. Sol: For f(x) =(x–1)2 Left hand limit (LHL) (at x=1)= and Right hand limit (RHL) (at x=1)= ( )2 1 lim 1 0 x x − → − = ( )2 1 lim 1 0 x x + → − = L mi it uf nct oi n at a‘ ’ Der vi at vi e of a function Derivative of polynomial Der vi at vi es S mo e S at nda dr mi L sti e moS nat S ad dr Algebra of Limits of functions Algebra of derivatives ( ) limx a g x → ≠ 0 , ( )2 1 lim 1 0 x x → − = LHL = RHL .. . lim x a− → f(x) lim x a+ → f(x) lim x a → f(x). .. .
Oswaal CBSE Mind Maps, Mathematics, Class-XI [ 17 To know about more useful books click here ,1*9=.;#ffl6*=1.6*=2,*5;.*<87270 Mathematical Reasoning • The contrapositive of a statement p ⇒ q is the statement ∼q ⇒ ∼p • The converse of a statement p ⇒ q is the statement q ⇒ p e.g. If the physical environment changes, then biological environmental changes. • Contrapositive : If the biological environment does not change then the physical environment does not change. In this statement, the two important symbols are used. The symbol ‘ ∀’stand for “all values of”; The symbol ‘∃ ’ stand for “there exists” e.g. For every prime number P, P is an irrational number. This means that if S denotes the set of all prime numbers, then for all the members of P of the set S, P is an irrational number. Many mathematical statements are obtained by combining one or more statements using some connecting words like “and”, “or” etc. Each statement is called a “Compound Statement.” e.g. ‘The sky is blue’ and ‘the grass is green’ is a compound statement where connecting word is “and”; the components of Compound Statement are p : The sky is blue q : The grass is green A sentence is called a mathematically acceptable statement if it is either true or false but not both. e.g. The sum of two positive numbers is positive. All prime numbers are odd numbers. first is ‘true’ and second is ‘false’. A statement which is formed by changing the true value of a given statement by using the word like ‘no’, ‘not’ is called negation of given statement. If P is a statement, then negation is denoted by ∼ p e.g. New Delhi is a city. The negation of this statement: It is not the case that New Delhi is a city. It is false that New Delhi is a city. New Delhi is not a city. The following methods are used to check the validity of statements (i) Direct Method (ii) Contrapositive Method (iii) Method of Contradiction (iv) Using a Counter example Qu na tif ei rs S at et ment oC mpound Sentence mI plicat oi ns • Converse: If the biological environment changes then physical environment changes. These are statements with word “if then”, “only if”and “if and only if”. e.g. r: if a number is a multiple of 9, then it is multiple of 3. p: a number is a multiple of 9. q: a number is a multiple of 3. Then, if p then q is the same as following: p implies q is denoted by p ⇒ q, then symbol ⇒ stands for implies, p is a sufficient condition for q. then symbol ⇒ p only if q q is a necessary condition for p ∼q implies ∼p
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Oswaal CBSE Mind Maps, Mathematics, Class-XI [ 19 To know about more useful books click here ,1*9=.;#fi9;8+*+252=B Probability Events A & B are called mutually exclusive events if occurance of any one of them excludes occurrance of other event, i.e. they cannot occur simultaneously. Eg: A die is thrown. Event A= All even outcomes & event B = All odd outcomes. Then A & B are mutually exclusive events, they cannot occur simultaneously. Note: Simple events of a sample space are always mutually exclusive. Many events that together form sample space are called exhaustive events. Eg: A die is thrown. Event A = All even outcomes and event B = All odd outcomes. Event A & B together forms exhaustive events as it forms sample space. It is the set of favourable outcomes. Any subset E of a sample space S is called an event. Occurance of event: The event E of a sample space ‘S’ is said to have occurred if the outcome w of the experiment is such that w ∈E. If the outcome w is such that w ∉E, we say that event E has Eg: not occurred. Event of getting an even number (outcome) in a throw of a die. An Experiment is called random experiment if it satisfies the following two conditions: It has more than one possible outcome. It is not possible to predict the outcome in advance. Outcome: A possible result of a random experiment is called its outcome. Sample Space: Set of all possible outcomes of a random experiment is called sample space. It is denoted by symbol ‘S’. Eg: In a toss of a coin, sample space is Head & Tail. i.e., S= {H,T} Sample Point: Each element of the Sample Space is called a sample point. Eg: In a toss of a coin, head is a sample point Equally Likely Outcomes: All outcome with equal probability. If A and B are any two events, then P(AUB) = P(A) + P(B) – P(A B) P(A B) = P(A) + P(B) – P(AUB) If A and B are mutually exclusive, then P(AUB) = P(A) + P(B) (As) P (A B) = φ) Random Exper mi ent Events Mutually Exclusive Definition Events Exhaustive A gl ebra of Events A B S Event A or B or (A ∪B) A ∪B = {w: w∈A or w∈B} Event A and B or (A ∩B) A ∩B = {w: w∈A and w∈B} Event A but not B or (A–B) A–B = A ∩ B’ A B A B A B A A–B ∪ B A ∩ B Impossible and Sure Event: The empty set φis called an Impossible event, where as the whole sample space ‘S’ is called ‘Sure event’. Eg: In a rolling of a die, impossible event is that number more than 6 and event of getting number less than or equal to 6 is sure event. Simple Event: If an event has only one sample point of a sample space, it is called a ‘simple event’. Eg: In rolling of a die, simple event could be the event of getting number 4. Compound Event: If an event has more than one sample point, it is called a ‘compound event’. Eg: In rolling of a die, compound event could be event of getting an even number. Complementary Event: Complement event to A = ‘not A’ Eg: If an event A= Event of getting odd number in a throw of a die i.e., {1, 3, 5} then , complementary event to A = Event of getting an even number in a throw of a die , i.e. {2, 4, 6} A–B B–A Α∩Β U U U Probability of the event ‘not A’ P(A′) = P(not A) = 1–P(A) Probability is the measure of uncertainty of various phenomenon, numerically. It can have positive value from 0 to 1. No.of favourableoutcomes Probability = Total no.of outcomes Eg: Probability of getting an even no. in a throw of a die. Sol. Here, favourable outcomes = {2, 4, 6} Total no. of outcomes = {1, 2, 3, 4, 5, 6} Probability = 3 1 6 2 = ∩A B na d P n( o At ) r P bo ba ili yt of AUB, Event A′= {W: W∈ S and W∉A } = S − A (where S is the sample space)