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Maths Handbook

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463 views222 pages

Maths Handbook

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nilabh singh
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IMD vos so toner cde We rk Ma strc a Mathematics HandBook (a) (b) © (d) @ ) @ @ (k) a (m) (n) (°) (p) (a) «) (s) () (u) ) LOGARITHM — LOGARITHM OF A NUMBER : The logarithm of the number N to the base ‘a’ is the exponent indicating the power to which the base ‘a’ must be raised to obtain the number N. This number is designated as log, N . log,N = x, read as log of N to the base a <> a° = N If a = 10 then we write log N or log,,N and if a =e we write In N or log,N (Natural log) Necessary conditions: N>0;a>O;a#1 log,1 = 0 log,a = 1 logy a=-1 log,(x.y) = log.x + logy; x, y > 0 \ tog,{*|=1oa, x10, 9 xy >0 y log, x” = plog, x; x > 0 log, x = Fog, x x>0 log. x= Feat x>0,x41 log,x = logx/loga ; x > 0,a,b>0,b# 1a log, b.log, c.log. d=log,d; a, b, c, d> 0, 4 1 a>O,a¥l1 ch": abc > 0;b41 xy. if a>, x>y if 0 x=y;x,y>0;a>0,a41 em =a log,,2 = 0.3010 ; log,,3 = 0.4771; fn2 = 0.693, fn10 = 2.303 Ifa > 1 then log,x 1 then log,x>p=>x>a? If 0 x>a? If 0 p=0 + tan Stan + tanS tan = 1 A B Cc AB (A) cot 5 + cot 5 + cot 5 = cot > cot5 cot 5 (e) sin 2A + sin 2B + sin 2C = 4 sinA sinB sinC (f) cos 2A + cos 2B + cos 2C =-1-4 cosA cosB cosC A B (9) sin A+ sinB+ sinC = 4.cos “cos 2 cos€ Z 2 2 (h) cos A + cos B + cos C=1+4 sin sinBsin& 2 2 2 DOMAINS, RANGES AND PERIODICITY OF TRIGONOMETRIC FUNCTIONS : T-RatioDomain Range Period sinx 1,11 2e cos x R (41,1) Qn tanx R(2n+)x/2;neh = RR z cot x R-{nz: nel] R 1 secx R-((2n+1)n/2:nel} (nll) ne cosecx R-{nn:nel} (oll) oe Mathematics HandBook ALLE ES -_ V3t1 ot (d) cos15°= oe = sin75% cos 5 Set on te 2 V3 t—— oe “Yee1 12 ¥B+1 X () tant = 24 V3 = Wea ie (q) tan(225°) = V2 - 1= cot(67.5°) = cot SE = tan (h) tan(67.5°) = V2 + 1= cot(225°) 7. MAXIMUM & MINIMUM VALUES OF TRIGONOMETRIC EXPRESSIONS : (a) a cos 0 + b sin 0 will always lie in the interval [-Va? +b? va? +b*], i.e. the maximum and minimum values are Va2+b?, Va? +b* respectively (b) Minimum value of a? tan?@ + b? cot? @ = 2ab, where a, b >0 (c) Minimum value of a*cos’0 + b* sec*0 (or a’sin?0 + b’cosec’®) is either 2ab (when |al > |b!) ora’ + b’(when lal < Ibl!). 8. IMPORTANT RESULTS : (a) sin 0 sin (60° - 0) sin (60° + 0) = Fsin3o (b) cos 6. cos(60°~6) cos(60° + 6) = e0830 (©) tan 0 tan (60° - 6) tan (60° + 6) = tan 30 (d) cot 0 cot (60° 0) cot (60° + 6) = cot 30 (©) (sin? 0 + sin? (60° + 6) + sin? (60° 6) = = (ii) cos’ 6 + cos? (60° + @) + cos* (60° — 0) 6 ALLEN Mathematics HandBook (xxviii) sin (A +B +C) = sinAcosBeosC + sinBeosAcosC + sinCcosAcosB ~sinAsinBsinC = ZsinA cosB cosC - Hsin A = cosA cosB cosC [tanA + tanB + tanC -tanA tanB tanC] (xxix) cos (A + B+C) =cosA cosBcosC —sinA sinB cosC -sinA cosB sinC —cosA sinB sinC =Tlcos A -Zsin A sin B cos C = cosA cosB cosC [1 -tanA tanB - tanB tanC - tanC tanA] (ax) tan (A +B +O) __tanA + tanB+tanC-tanAtanBtanC _ S$, - ~ 1-tanAtanB-tanBtanC-tanCtanA 1-S, (xxxi) sin a + sin (a+) + sin (a+28) +... sin (2 +n—-1f) : sofa (2s ho) (5) cos a + cos (+P) + cos (a + 2B) + .... +cos(a +n— 1p) teat) = i se 2 6. VALUES OF SOME T-RATIOS FOR ANGLES 18°, 36°, 15°, 22.5°, 67.5° etc. (a) sin18°=——— Wen 1. =cos72 = sn (xxii) (b) 0536 = Be : x =sin54° = CO a fe) snt5?= “22 -cos75" = sin 22 Mathematics HandBook C+D c-D (xiii) sin C + sin D = 2 sin [—3—] cos | 5 c+D (xiv) sin C-sin D = 2 cos C+D (v) cos C + cosD = 2 cos [5] cos (—5- C+D D-C (xvi) cos C- cos D = 2 sin | 3 | sin| oe ee (sil) sin 20 = 2 sin 0 cos 0= 77, (xviii) cos20 = cos’) sin’0 = 2oos" - 1 = 1-2 sin’ jeune £0520 = cost ~ sin’0 = 2cos'- 1 = 1-2 sin’ 1+tan?0 (xix) 1 +. cos 20 = 2 cos? 0 of Icos0 |= yee T= cos20 (xx) 1-cos20 = 2 sin? @ or Isin ol = J— cos20 sin20 i) tang =——COeh __ Sine _ Ct hs capcom renzo . 2tand (a) mans (xxiii) sin 30 = 3 sin 0 ~4 sin’ 6. (xxiv) cos 30 = 4 cos* 0 - 3 cos0. (aw) tan30 = 3tan0-tan*0 1-3tan? 0 (xvi) sin? A — sin? B = sin (A+B). sin (A-B) = cos? B ~ cos? A. (xxvii) cos* A - sin? B = cos (A+B). cos (A -B). 4 cc Efrat Se Vr Mas AL! EN Mathematics HandBook 4. TRIGONOMETRIC FUNCTIONS OF ALLIED ANGLES : (a) sin (2nz + 0) = sin 0, cos (2nz + 6) = cos 0, wheren eI (b) sin (0) =-sin 0 cos (-0) = cos 0 sin(90°- 6) = cos0 cos(90°- 0) = sind sin(90° + 6) = cos0 cos(90° + 8) =-sind sin(180°- 0) = sind cos(180°- 0) = -cos0 sin(180° + 6) = sind cos(180° + 6) = —cos0 sin(270°- 6) = -cos® cos(270°- 6) = -sin@ sin(270° + 0) = -cos0 cos(270° + 0) = sind Note : @) sin nm = 0; cos ne =(-1);; tan nx = 0, where n € I (ii) sin(2n+1)5 =(1); cost2n+1) 5 =0, where nel IMPORTANT TRIGONOMETRIC FORMULAE : () — sin(A+B)=sin A cosB +cos Asin B. (i) sin(A-B) = sin A cos B~ cos A sin. (iii) cos (A +B) = cos A cos B- sin A sin B (iv) cos(A-B) = cos A cos B + sin A sin B () — tan(A +B) = eA+tenB 1~tanAtanB (i) tan(A—B) = 20A=tenB 1+ tanA tanB i A+B cotBcotA -1 (wil) cot(A+ B)= "CoB + cotA cotBcotA +1 (itl) cot (A-B) = Ea (i) 2sin Acos B= sin (A+ B) + sin (A-B). (%) — 2cosA sin B = sin (A + B)~sin (A-B). (xi) 2.cos Acos B = cos (A + B) + cos (AB) (xii) 2 sin Asin B = cos (A-B) -cos (A + B) Mathematics HandBook ALLE IS TRIGONOMETRIC RATIOS & IDENTITIES ———— 1. RELATION BETWEEN SYSTEM OF MEASUREMENT OF ANGLES : DS 2c 90 100 x L Radian = *©° degree ~ 57°17'15" (approximately) I degree = 735 radian x 0.0175 radian 2. BASIC TRIGONOMETRIC IDENTITIES : (a) sin? 6 + cos? @ = 1 orsin? @ = 1 cos? @ or cos? 0 = 1 sin? @ (b) sec’ 0 - tan’? 0 = 1 or sec? @ = 1 + tan’ 0 or tan’ 0 = sec? 0-1 1 (©) If seo + tand=k => secd-tan® = = 2secO-k+ (A) cosec20 cot’ = 1 or cosec’d = 1 + cot?0 or cot = cosec’- 1 1 ©) Icosecd + cot0=k = cosecd-cot= ; => 2eosecd=k ot 3. SIGNS OF TRIGONOMETRIC FUNCTIONS IN DIFFERENT QUADRANTS : 90°, x/2 © Uquadrant | I quadrant only sine All + ve & cosec + ve 180°, 0°, 360°, 2x only tan & cot | only cos +ve & sec + ve @® I quadrant — | IV quadrant © 270°, 32/2 2 Mathematics HandBook 11. GRAPH OF TRIGONOMETRIC FUNCTIONS : (a) y= sinx (b) y= cosx (c) y=tanx (d) y=cotx i i i j i i i 3 j E Mathematics HandBook AL! ES 3. (ii) If & =p + Jq isone root in this case, ( where pis rational & Jq isa surd) then other root will be p- J/q - COMMON ROOTS OF TWO QUADRATIC EQUATIONS (a) Atleast one common root Let «be the common root of ax? + bx + c= 0 & ax? + bx += 0 thenaa?+ba +c=0 & aa?+ba+c =0. By Cramer's a 1 “aie—ac’ ab a'b ca'-c'a_be'— bre ab'= a'b- a'c—ac’ So the condition for a common root is (ca' - ca)’ = (ab' - a’b)(bc- b'c) Therefore, 4 = bic abc’ ROOTS UNDER PARTICULAR CASES Let the quadratic equation ax? + bx + c = 0 has real roots and (a) If b = 0 = roots are of equal magnitude but of opposite sign (b) If both roots are same then (b) If c= 0 = one roots is zero other is —b/a (©) Ifa=c = roots are reciprocal to each other (@) Ifac<0 = roots are of opposite signs If : : A : a ee a at = both roots are negative. () 1°? of . ie a of = both roots are positive. a<0,b>0,c< (g) If sign of a = sign of b # sign of ¢ => Greater root in magnitude is negative. (h) If sign of b = sign of c + sign of a = Greater root in magnitude is positive. () Ifa+b+c=0 = one rootis 1 and second root is c/a. 14 oma 3 i j } | i 1. ——————= QUADRATIC EQUATION EE SOLUTION OF QUADRATIC EQUATION & RELATION BETWEEN ROOTS & CO-EFFICIENTS : (a) The solutions of the quadratic equation, ax? + bx + c = 0 is b+ Vb? -dac 2a (b) The expression b? - dac = D is called the discriminant of the quadratic equation. given by x = (Q) If a &B are the roots of the quadratic equation ax’ + bx + c = 0, then ; @) a+B=-b/a (ii) aP=c/a_ iii) |a-B[=VD/lal (d) Quadratic equation whose roots are « & B is (x- a)(x— )=0 ie. x2-(0 +B) x + oP =Oice. x2—(sum of roots) x + product of roots = 0. @) Ifa, are roots of equation ax?+ bx+ c = 0, we have identity in x as ax + bx+ c= a (x- a) (xB) NATURE OF ROOTS: (a) Consider the quadratic equation ax’ + bx + c = 0 where a, b, c €R &a # Othen; (i) D> 0 < roots are real & distinct (unequal). (ii) D = 0 = roots are real & coincident (equal) (iii) D < 0 <= roots are imaginary. (iv) If p + iq is one root of a quadratic equation, then the other root must be the conjugate p — i q & vice versa (ppqeR & i= J). (b) Consider the quadratic equation ax? + bx + = 0 where a, b,c € Q&a # O then; (i) If Dis a perfect square, then roots are rational. Mathematics HandBook Mathematics HandBook ALLE ES (€) Check that denominator is not zero at any stage while solving equations. () ()_Iftan 6 or sec 0 is involved in the equations, 0 should not be odd multiple of 3 (ii) If cot 0 or cosec 0 is involved in the equation, should not be integral multiple of x or 0. (g) If two different trigonometric ratios such as tan @ and sec 6 are involved then after solving we cannot apply the usual formulae for general solution because periodicity of the functions are not same. (h) IfL.H.S. of the given trigonometric equation is always less than or equal to k and RHS is always greater than k, then no solution exists. If both the sides are equal to k for same value of 0, then solution exists and if they are equal for different value of @, then solution does not exist. 12 3 j é j 3 | 3 él Mathematics HandBook @) sin (nx + 0) = -1)’sin 0,n eI cos (nx + 6) = (-1)"cos @,n eI] GENERAL SOLUTION OF EQUATION asin @ +b cos 0 =c: Consider, a sin 0 + b cos 0 = C ......escses (i) ; ac c . Vere eee va? +b? equation (i) has the solution only if |e! < Va? +b? 2 ~ 0054, P= sing g tant Va? +b Va? +b? a by introducing this auxiliary argument $, equation (i) reduces to cos@ = let sin (0+) = Now this equation can be solved easily. GENERAL SOLUTION OF EQUATION OF FORM : a,sin’x + a,sin™'x cosx + a,sin™? xcos’x + .......... + a,cos"’x = 0 yy Aj,.-..++.€, are real numbers Such an equation is solved by dividing equation both sides by cos’x. IMPORTANT TIPS : (a) For equations of the type sin 0 = k or cos 0 = k, one must check that 1k 1 <1. (b) Avoid squaring the equations, if possible, because it may lead to extraneous solutions. () Do not cancel the common variable factor from the two sides of the equations which are in a product because we may loose some solutions. (d) The answer should not contain such values of 0, which make any of the terms undefined or infinite. Mathematics HandBook ALLE ES 1. TRIGONOMETRIC EQUATION ———— TRIGONOMETRIC EQUATION : An equation involving one or more trigonometrical ratios of unknown angles is called a trigonometric equation. SOLUTION OF TRIGONOMETRIC EQUATION : A value of the unknown angle which satisfies the given equations is called a solution of the trigonometric equation. (a) Principal solution :- The solution of the trigonometric equation lying in the interval [0, 27]. (b) General solution :- Since all the trigonometric functions are many one & periodic, hence there are infinite values of 0 for which trigonometric functions have the same value. All such possible values of 0 for which the given trigonometric function is satisfied is given by a general formula. Such a general formula is called general solutions of trigonometric equation: GENERAL SOLUTIONS OF SOME TRIGONOMETRIC EQUATIONS (TO BE REMEMBERED) : (a) If sin 0 = 0, then 0 = nz, n € I (set of integers) (b) Ifcos @=0, then @ = (2n+1)5, nel (0) Iftan0 =0, then =nz,nel (d) Ifsin0 =sina, then 0 =n +(-Ia,nel ) Ifcos = cosa, then @ = 2nrza,nel (f) Iftan@ = tana, then®=nr+a,nel @) sin =1, then 0 = 2ne+5 =(4n+1)5,nel (h) If cos 6 = 1 then 0 = 2nz,n 1 (i) If sin? 0 = sin? a. or cos” 6 = cos? a or tan* 6 = tan? a, then @=nt+a,nel 10 AL! EN Mathematics HandBook () y=secx (f) y = cosecx 12. IMPORTANT NOTE: (a) The sum of interior angles of a polygon of n-sides = (n= 2) x 180° = (n = 2). (6) Each interior angle of a regular polygon of n sides = £=2).1g9°- 8-9), n n (c) Sum of exterior angles of a polygon of any number of sides i j & 3 = 360° = 2z. 4 j | j ; AL! EN Mathematics HandBook 5. MAXIMUM & MINIMUM VALUES OF QUADRATIC EXPRESSION : Maximum or Minimum Values of expression y = ax” + bx + c is ae which occurs at x =— (b/2a) according as a < Oora>0. ye [2] fa>0 & ye(~2| ifa<0. 6. LOCATION OF ROOTS: Let f(x) = ax? + bx +c, where a,b,c ¢ R,a¥0 (a) Conditions for both the roots of f(x) = 0 to be greater than a specified number ‘d’ are D> O and a. aa > 0 & (-b/2a) > d. D20 XY 220 (6) Condition for the both roots of f(x) = 0 to lie on either side of the number ‘d’ in other words the number ‘d’ lies between the roots of f(x) = is a.f(d) <0. ALT (c) Condition for exactly one root of f(x) = 0 to lie in the interval (d,e) ie. d 0 & af(e) > 0 and d<(b/2a) G2H () A=G=Hea=b (il) Let a,, a,,...... ,a, be n positive real numbers, then we define their arithmetic mean (A), geometric mean (G) and harmonic ay tay +... +a, mean (H) asA = 12 G=(aa f Agess crv a) and H= ay It can be shown that A > G > H. Moreover equality holds at either place if and only if a, ARITHMETICO - GEOMETRIC SERIES : Sum of First n terms of an Arithmetico-Geometric Series : Let S, =a+(atd)r+(at2d)r? +.........¢la+(n—1d}e" [a+(n—1)d] 1-r , rel Sum to infinity : If f]<1 & ne then Limr"=0 = S, SIGMA NOTATIONS Theorems : @) Neth )= da, edb, &) Dka, =kYa, rel rel r=1 rel rol (©) Sk=nk ; where kis a constant. ral Mathematics HandBook 4. MEANS. (a) () © Arithmetic mean (AM) : If three terms are in AP then the middle term is called the AM between the other two, so if a, b, c are in AP, bis AM ofa&c. n-arithmetic means between two numbers : If a,b are any two given numbers & a, A,, Ay, v.04, A,, bare in AP then A,, A,,...A, are the n AM’s between a & b, then -a b A=atd,A,=a+2d,......, A= a+nd, whered =—— z n+1 Note : Sum of n AMS; inserted between a & bis equal to n times the single AM between a & bie. YA, =nA where A is the si b single AM between a &b ie. 95> Geometric mean (GM) : Ifa, b, care in GP, then bis the GM between a & c i.e. b? = ac, therefore b= Jac n-geometric means between two numbers : If a, b are two given positive numbers & a, G,, Gyy sss, Gis b arein GP then G,,G,,G,,......G, aren GMs between a & b. G,= ar, G, = ar’, ....... G,= ar", where r= (b/a)!/**! Note : The product of n GMs between a & b is equal to nth power of the single GM between a & bie. tT G, =(G)" where re Gis the single GM between a & bie. Jab Harmonic mean (HM) : Ifa, b, care in HP, then b is HM between a & c, then b = 22. a +¢ 20 cc Efrat Se Vr Mas él Mathematics HandBook Note : () InaG.P. product of k term from beginning and kterm from the last is always constant which equal to product of first term and last term. (i) Three numbers in G.P. a/r, a, ar Five numbers in G.P. a/r’, a/r, a, ar, ar? FournumbersinG.P. : a/r’, a/r, ar, ar® SixnumbersinG.P. : — a/r°, a/r’, a/r, ar, ar’, ar® (iii) If each term of a G.P. be raised to the same power, then resulting series is also a G.P. (iv) If each term of a G.P. be multiplied or divided by the same non-zero quantity, then the resulting sequence is also a G.P. @ Ifa, a,, a, ..... and b,, b,, by, ....... be two G.P’s of common ratio r, and r, respectively, then a,b,, a,b, ..... and will also form a G.P. common ratio will be r, r, 5 : and + respectively. b wi Ina positive G.P. every term (except first) is equal to square root of product of its two terms which are equidistant from it. ie. T= VT aban k b= 28% o 2-2-2 are progression is — c b-c Mathematics HandBook ALLE ES (vi) (a) If each term of an A.P. is increased or decreased by the same number, then the resulting sequence is also an A.P. having the same common difference. ) If each term of an A.P. is multiplied or divided by the same non zero number (k), then the resulting sequence is also an A.P. whose common difference is kd & d/k respectively, where dis common difference of original A.P. (vii) Any term of an AP (except the first & last) is equal to half the sum of terms which are equidistant from it. 7, =the, ker GEOMETRIC PROGRESSION (GP) : GP is a sequence of numbers whose first term is non-zero & each of the succeeding terms is equal to the preceding terms multiplied by a constant. Thus in a GP the ratio of successive terms is constant. This constant factor is called the common ratio of the series & is obtained by dividing any term by the immediately previous term. Therefore a, ar, ar’, ar’, ar’, .......... is a GP with ‘a’ as the first term & 'r' as common ratio. (a) n® term |T,= ar"? (b) Sum of the first n terms |S,, , ifs] _a(r" -1) ae () Sum of infinite GP when |r[<1 (n+, m0) (d) Ifa, b, care in GP > b? = ac = loga, logb, loge, are in A.P. 18 i j j } j | ; Mathematics HandBook ———— SEQUENCE & SERIES ————_ ARITHMETIC PROGRESSION (AP) : AP is sequence whose terms increase or decrease by a fixed number. This fixed number is called the common difference. If ‘a’ is the first term & ‘d’ is the common difference, then AP can be written as a,a+d, a+2d,...,a+(n-1)d,... (a) n® term of this AP [T,= a + (n-1)d], where d=T,-T,, (b) The sum of the first n terms : |S, = F12a+(n-1)d]= Fla+ where ( is the last term. (0) Alson'* term [T,=S, = Sa Note: (i) Sum of first n terms of an A.P. is of the form An’ + Bn i.e. a quadratic expression in n, in such case the common difference is twice the coefficient of n’. i.e. 2A (ii) n" term of an A.P. is of the form An + Bie. alinear expression inn, in such case the coefficient of n is the common difference of the A.P. i.e. A (iii) Three numbers in AP can be taken as a -d, a, a + d; four numbers in AP can be taken as a - 3d,a-d,a+d,a+ 3d five numbers in AP are a~ 2d,a—d,a,a+d,a+2d & six terms in AP are a-5d,a-3d,a-d,a+d,a+3d,a+ 5d etc. (iv) If a, b, care in A.P., then b= a (W) Ifa, a,, a, a,+b,a,+ and b,, b,, by ........ are two APs, then one ate also in A.P. Mathematics HandBook ALLE IS 7. GENERAL QUADRATIC EXPRESSION IN TWO VARIABLES : f(x, y) = ax” + 2 hxy + by? + 2gx + 2 fy + ¢ may be resolved into two linear factors if ; A = abc + 2fgh - af?-bg’-ch?=0 OR os. aos ome i ° THEORY OF EQUATIONS : If Gy 01p, 0tgy--s0.240n are the roots of the equation ; f(x) = a, x” ta, x! +a, x24 tax ta,=0 where ay , a,,......a, are constants a, # 0 then, : aS ay La, =-22, Yaya, =+22, Layayay ag ao in Ay sey 101g eens Oy = (I)? 2B. a Note: () Every odd degree equation has at least one real root whose sign is opposite to that of its constant term, when coefficient of highest degree term is (+)ve {If not then make it (+) ve} Ex. x8-x?+x-1=0 (ii) Even degree polynomial whose constant term is (-)ve & coefficient of highest degree term is (+)ve has atleast two real roots, one (+)ve & one (ve. (iii) If equation contains only even power of x & all coefficient are (+)ve, then all roots are imaginary P (iv) Rational root theorem : If a rational number q (p,qeZ,)isa root of polynomial equation with integral coefficient ax" + a,x"! +......4 a) = 0, then p divides a, and q divides a, 16 IM vss 01 s0\cn\etacracanb Ma Sheerach a start as Mathematics HandBook (@) Number of ways in which n distinct things can be distributed to P persons if there is no restriction to the number of things received by them is p" (e) Number of ways in which n identical things may be distributed among p persons if each person may receive none, one or more things is™*?-1C,, DERANGEMENT : Number of ways in which n letters can be placed in n directed envelopes so that no letter goes into its own envelope is : aff ddd nooner ra] na 3a al IMPORTANT RESULT : (a) Number of rectangles of any size in a square of size n x n is Dr & number of squares of any size is Ut” 5 S (6) Number of rectangles of any size in a rectangle of size n x p (n0 & + fe A} for b < 0 where Iz|= Va? +b’. ROTATION : cl, Zo = 29 _ 21% 29 T2_21 124-29! Take 0 in anticlockwise direction Famke IM vss 01 s0\cn\etacracanb Ma Sheerach a start as él Mathematics HandBook IMPORTANT PROPERTIES OF CONJUGATE : @ (o) % +z, = 342% Z-% @ 42% =%.% 240 (® Iffisa polynomial with real coefficient such that f(a+ip) =x + iy, then f(a —ip) = x-iy. IMPORTANT PROPERTIES OF MODULUS : (@) 220 () > Rel) — (&) (zi Ime) @ e=lZl=}2=+2| @2zz=kf — & 2, 2)=!2,2, 24 E . 2,40 (h) je" = 2" , ee Il? z @ Iz, +2,1%= l2l?+ 12,12 + 2Re(z,zZ,) or [z,+2,/2= 12,1? + 2,12 +21z,1 12,1 cos(®, - 0) ® |e +2, + fy 2° =2 [lal + leek | |+|z,| {Triangular Inequality] () 2 ,|+|z,| [Triangular Inequality] (m) If & min Iz I 5(Ve +4 -a) IMPORTANT PROPERTIES OF AMPLITUDE : (a) () amp (,. z,) = amp z, +ampz, + 2kn; kel =a (a> 0), then max!z! = 1 z+ — Zz \ (i) amp (24) = amp z,—ampz,+2kn; kel 22 (ii) ample") =n amp) + 2kr, where proper value of k must be chosen so that RHS lies in Gx, x]. (b) log(2) = log(re") = logr + i = log! z! +i ample) Mathematics HandBook (iv) If zis purely real then z-Z = 0 (v) If zis purely imaginary then z +2 =0 REPRESENTATION OF A COMPLEX NUMBER IN VARIOUS FORMS : (a) Cartesian Form (Geometrical Representation) : Every complexnumber z=x-+iy can ‘S Imaginary py axis (x, y) be represented by a point on the cartesian plane known as complex plane (Argand diagram) by the ordered pair (x , y) Length OP is called modulus of the complex number denoted by |z| & Beal ands 9 is called the principal argument or amplitude, (0 ¢ (-r, 7). eg.|z|= x+y? & 0 = tant % (angle made by OP with positive x-axis), x > 0 Geometrically |z| represents the distance of point P from origin. (|z| 20) (b) Trigonometric / Polar Representation : z=r(cos 0+ isin @) where |z|=r ;arg z=0 ; Z =r (cos0-isin®) Note : cos 6 + isin 0 is also written as CiS 0. Euler's formula : The formula e* = cosx + i sin x is called Euler's formula. cee ee Also cos x = 7 & sinx = a ae known as i Euler's identities. Exponential Representation : Let z be a complex number such that |z|=r & argz then z = re® rats gh a IM xenon toe We Sh rch Mathematics HandBook ———— COMPLEX NUMBER DEFINITION : Complex numbers are defined as expressions of the form a + ib where a,beR & i= 1. It is denoted by z ie. z=a+ib. ‘a’ is called real part of z (a= Re 2) and‘b’is called imaginary part of z (b=Imz). Every Complex Number Can Be Regarded As ete et Purely real Imaginary ifb=0 ifb40 Purely imaginary ifa=0 Note : () The set R of real numbers is a proper subset of the Complex Numbers. Hence the Complex Number system is Ne WelcQcReC. (ii) Zero is both purely real as well as purely imaginary but not imaginary. (iii)i = J-1_ is called the imaginary unit. Also i? it = Lete. (iv) Ja Vb = Jab only if atleast one of aor b is non-negative. CONJUGATE COMPLEX : If z=a+ib then its conjugate complex is obtained by changing the sign of its imaginary part & is denoted byZ.ie. Z =a-ib. Note that: @ z+ Z = 2Ret) (i) 2-Z = 2ilme) (iii) 2Z = 02 +b? which is real Mathematics HandBook ALL Mi ES x x xe at= 1+ /ma+ (n’a+— in’a+.......0, where a >0 1 2! 3! 6. LOGARITHMIC SERIES : | (@) mn{l +x) =x oo , where -1 Iz,-z,1 > Anellipse i) If 2k = |z,-z,1 = Aline segment (iii) If 2k < 1z,-z,1 = No solution (b) Equation | |z~z,1~ |z—z,1 |= 2k (a constant) represent @) If2k 1z,-z,| = No solution IMD vss 120 earch era a arto la 8s IM xenon toe We Sh rch Mathematics HandBook 13. (iii) Two lines with complex slope p, & y, are parallel or perpendicular if p,= pw, or H+ hy = 0. (iv) Length of perpendicular from point A(q) to line az +az +b =0 laa+aa+b! seial EQUATION OF CIRCLE : (a) Circle whose centre is z, & radius = r lz-z,l =r (b) General equation of circle is wZ+az+az+b=0 centre '-a' & radius = Vial’ —b (©) Diameter form (z~z,)(2 -2,) + (2-22 -2,) =0 ot al (@ Equation ifk=1. (e) Equation 12-2, 2+ Iz-z,l?=k Pe) Hee LE 2 represent circle if k > 3 lz, -z, 1 = B A CC) ona(? A\ea O 4264725 = 2,2) +2525 +252; 1 1 i + + =0 or MZ, %-%y yh (b) Isosceles triangle : A(z,) Be) a Cle.) then 4cos’a (z, - 2,)(z, -2,) = (2, - 2, rl (©) Area of triangle ABC given by modulus of 722 23 EQUATION OF LINE THROUGH POINTS z, & z, ] 1]=0 > 2(% -%,)+2(z, —z,) + 2,2 -Zz, =0 ] )=0 Z)i+ Bey —2,)i + ile,Z, Let (z,-z,)i= a, then equation of line is |az +aZ +b = 0| where ae C&beR. Note : (i) Complex slope of line joining points z, & z,is ( z note that slope of a line in Cartesian plane is different from complex slope of a line in Argand plane. (i) Complex slope of line az +az +b =0 is-2, beR Famke Mathematics HandBook 6. SYSTEM OF EQUATION : (a) System of equation involving two variable : ax+by+c,=0 ax + by +c,=0 Pee Consistent Inconsistent (System of equation has solution) (System of equation >| has no solution) i 7 ey Bie G unique solution Infinite solution ao ees ab alba ee aoe (Equations represents eat ti llel disjoint lines) ora,b,-a,b,#0 (Equations represents _ Parallel disjoint lines (Equations represents coincident lines) intersecting lines) (b) System of equations involving three variables : ax +by+cz=d, ax + by+¢,2=d, a,x + by + cz =d, To solve this system we first define following determinants a bh d; by cy, A=|a2 be %], a=|d by cy abs cy be a dy cy A,=|a d, ¢, a3 d, C3 IMD vss 120 earch era a arto la 8s Mathematics HandBook AL! ES ORTHOGONAL MATRIX A square matrix is said to be orthogonal matrix if A AT = I Note: (i) The determinant value of orthogonal matrix is either 1 or -1. Hence orthogonal matrix is always invertible (ii) AAT=1= ATA Hence A1= AT. SOME SPECIAL SQUARE MATRICES : (a) Idempotent Matrix : A square matrix is idempotent provided AP=A, For idempotent matrix note the following : @) Av*=AvneN. (ii) determinant value of idempotent matrix is either 0 or 1 ( ff idempotent matrix is invertible then it will be an identity matrix i.e. 1 (b) Periodic Matrix : A square matrix which satisfies the relation A"! =A, for some positive integer K, is a periodic matrix. The period of the matrix is the least value of K for which this holds true Note that period of an idempotent matrix is 1. (c) Nilpotent Matrix : A square matrix is said to be nilpotent matrix of order m, m € N, if A"=O, A™1 #0. Note that a nilpotent matrix will not be invertible. (d) Involutary Matrix : If A? = I, the matrix is said to be an involutary matrix. Note that A = A“ for an involutary matrix. (2) If A and B are square matrices of same order and AB = BA then (A + BY =°C,A" + °C,A™B + °C,A™Be + rats gh a IM xenon toe We Sh rch él Mathematics HandBook (©) Distributivity : A(B+C)=AB+AC (A+B)C = AC+BC for respective products Provided A,B & C are conformable (V) Positive Integral Powers of A square matrix : (a) Aran = Am (b) (Arp = Ame (A) () I'=1, mneN CHARACTERISTIC EQUATION : Let A be a square matrix. Then the polynomial in x, 1A — xl! is called as characteristic polynomial of A & the equation |A—xI| = 0 is called characteristic equation of A CAYLEY - HAMILTON THEOREM : Every square matrix A satisfy its characteristic equation ie. ax" + ax! + ........ + a,x + a, = 0 is the characteristic equation of matrix A, then a,A"+a,A™! + +a, A+al=0 TRANSPOSE OF A MATRIX : (Changing rows & columns) Let A be any matrix of order m x n. Then transpose of A is AT or A’ of order n x mand (A"), = (A),. Properties of transpose : If AT & BT denote the transpose of A and B (a) (A+B)' = A™4B" ; note that A & B have the same order, (b) (AB)' = BTAT (Reversal law) A & B are conformable for matrix product AB (©) (A =A (@) (kA) = kA", where k is a scalar. General : (Ay. Ay. AJ? = AT. AT. AT (reversal law for transpose) Mathematics HandBook AL! ES 5. ALGEBRA OF MATRICES : @ Addition : A + B = [a, + b,] where A & B are of the same order. (a) Addition of matrices is commutative : A +B=B+A (b) Matrix addition is associative : (A + B)+ C=A+(B+C) (©) A+ 0 =0 +A (Additive identity) (@) A+(-A)=(-A) + A = 0 (Additive inverse) () Multiplication of A Matrix By A Scalar : abe ka kb ke If A=|b c a|,then kA=|kb ke ka cab kc ka kb (Ii) Multiplication of matrices (Row by Column) : Let A be a matrix of order m xn and B be a matrix of order p x q then the matrix multiplication AB is possible if and only if n=p. Let A, = [a] and B, = [bJ, then order of AB ism xp xp, & (AB), = Y.a,b, 7 (V)Properties of Matrix Multiplication : (a) AB =O 4 A=O orB =O (ingeneral) Note: If A and B are two non-zero matrices such that AB = O, then A and Bare called the divisors of zero. If A and B are two matrices such that () AB =BA then A and Bare said to commute (i) AB =-BA then A and B are said to anticommute (b) Matrix Multiplication Is Associative : If A, B & Care conformable for the product AB & BC, then (AB)C = A(BC) IPM xeon tocando Sh rk hs ar as IM xenon toe We Sh rch Mathematics HandBook SQUARE MATRICES : SQUARE MATRICES v Triangular Matrix 13 -2 10 0 A=|0 2 4 ;B=|2 -3 0 005 43 3 Upper Triangular Lower Triangular a= 0 Vi>j a= 0 Vi m. (e) Vertical Matrix : A matrix of order m x nis vertical matrix ifm>n. (f) Square Matrix : (Order n) If number of rows = number of column, then matrix is a square matrix. Note: (The pair of elements a, & a, are called Conjugate Elements. (il) The elements a,;, aj, ayy. a,,, are called Diagonal Elements. The line along which the diagonal elements lie is called "Principal or Leading diagonal. "The quantity Xa, = trace of the matrix written as, t, (A) IMD vss 120 earch era a arto la 8s

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