MPC BRIDGE COURSE CONTENTS MATHEMATICS : 2 TO 43 PHYSICS : 44 TO 69 CHEMISTRY : 70 TO 101 KEY : 102 TO 108

Transcription

1 CLASS - VIII CONTENTS MATHEMATICS : TO 4 PHYSICS : 44 TO 69 CHEMISTRY : 7 TO KEY : TO 8 NARAYANA GROUP OF SCHOOLS

2 MATHEMATICS DAY- : SYNOPSIS Vrible: A letter symbol which c tke vrious umericl vlues clled vrible or literl. Exmples: x, y, z etc. Costt : Qutities which hve oly oe fixed vlue re clled costts. Term: Numericls or literls or their combitios by opertio of multiplictio re clled terms. Costt Term : A term of expressio hvig o literl clled costt term. TYPES OF ALGEBRAIC EXPRESSIONS: * A expressio cotiig oly oe term clled moomil. * A expressio cotiig two terms clled biomil. * A expressio cotiig three terms clled triomil. * A expressio cotiig two or more terms clled multiomil. * A expressio cotiig oe or more terms with positive itegrl idices (powers) clled polyomil. Note: Every o-zero umber cosidered moomil with degree zero. Degree of polyomil : The highest power of terms i polyomil clled the degree of polyomil. Zero polyomil : If ll the coefficiets i polyomil re zeroes, the it clled zero polyomil. Zero of the polyomil : The umber for which the vlue of polyomil zero, clled zero of the polyomil. Note: Degree of zero polyomil ot defied. Substitutios : The method of replcig umericl vlues i the plce of literl umbers clled substitutio. Like d Ulike terms : 8 Ex: 8b, b, b ; m, 4m, m. * Terms which do ot hve the sme literl fctors re clled Ulike terms. Exmple :,,b; xy,x y Additio of lgebric expressios : Additio of lgebric expressios mes ddig the like terms of the expressios. * Combiig the coefficiets of like terms of expressio through dditio or subtrctio clled simplifictio of lgebric expressio. There re two methods of ddig lgebric expressios. They re i) Horizotl method ii) Verticl method. Horizotl method : I th method, like terms should be dded d ulike terms should be writte seprtely Subtrctio of Algebric expressios: Additive Iverse of expressio : * Terms which coti the sme literl fctors re clled like terms or similr terms. I like terms, the umericl coefficiet my be differet. fid their product s give below. NARAYANA GROUP OF SCHOOLS * The dditive iverse or the egtive of expressio obtied by replcig ech term of the expressio by its dditive iverse. Exmple: Additive iverse of -9x 9x * To subtrct st expressio from the d expressio, dditive iverse of the st expressio should be dded to the d expressio. If P d Q re two lgebric expressios the P Q = P + ( Q). Exmple: Subtrct 6b from 7 + 4b Solutio: (7 + 4b) ( 6b) = 7 + 4b + 6b = 4 +b Subtrctio c lso be doe i two wys. * Horizotl method * Verticl method Multiplictio of Polyomils: Multiply ech term of the first Polyomil with ech term of the secod d dd the like terms i the product. Suppose (+b) d (c + d) re two Polyomils. By usig the dtributive lw of multiplictio over dditio, we c

3 bc d c d b c d c d b c b d c d bc bd Colum Method of Multiplictio: I th method we write multiplicd d the multiplier i descedig powers of, rrge oe uder other, d multiply the multiplicd by every term of the multiplier d dd. DAY- : WORKSHEET Coceptul Uderstdig Questios :. The degree of the polyomil 4 7 x x + x ) ) 4 ) 7 4). The zeroes of the polyomil x ) ) ) 4). The simplified form of. x.7 x +.x + x ).7 x ) x ) x 4) x 4. The dditio of 7x 4x + d x + x ) 4x x + 4 ) 4x + x + 4 ) 4x x 4 4) 4x + x 4. Subtrct x 4x + x + from 4x + x + x + 6, the the resultet vlue ) 6x + x x + ) x + x x+ ) x x x + 4) x x + x 6. Additive iverse of x + bx + c ) x bx c ) x + bx c ) x + bx + c 4) x + bx + c 7.The product of 6 b, bc d bc 6 9 ) ) 4 b c b c ) 4) 4 b c b c 8. Divide 4x x + x by x, the the resultet vlue ) x x ) 4x x ) 4x x 4) 4x x Sigle Correct Choice Type : 9. Two djcet sides of rectgle re b d 6b the its perimeter ) + b ) 4 + b ) b 4) 4 b.the perimeter of trigle whose sides re y + z, z y, 4y z ) y + z ) y + 4z ) y z 4) y + z.the perimeter of rectgle, 6x 6x + x + 4. If oe of its sides 8x + x, the the other side ) 6x 4x + x + 4 ) 8x x + x + ) 6x + 4x + x 4 4) 8x + x + x. Subtrct x xy + x y y from y 6x y xy + x ) y 8x y + xy x ) x 8xy + x y y ) x y 4) x y. Wht must be dded to x xy + y to get x xy y ) x + xy 6y ) x + xy 6y ) x xy + 6y 4) x xy 6y 4. The vlue of 4 8 = ) 4 ) 4 ) 4 4) 4. Wht must be dded to x + x 8 to get x + x + 6? ) x + x x + 4 ) x + x + 4 ) x + x 6x 4 4) x + x 4 NARAYANA GROUP OF SCHOOLS

4 6.If A 6x 8 4 d B x, the A B = ) 8x ) 9x ) 7x 4) 6x 7.The product of 4x ) x 7xy 99y ) x 7xy 99y ) x 7xy 99y 4) x 7xy 99y 9y x y 8.The product of.x(x y xy ) ) x y x y ) x y + x y ) x y x y 4) x y + x y 9. If x y 6x y 4x y A, 4xy 4 6x y A B B 4x y, the 4x y ) 8x 7 y + 4x y + x ) 8x 7 y 4x y xy ) 8x 7 + 4x y xy 4) 8x 7 4x + xy DAY- : SYNOPSIS MULTIPLICATION BY USING FORMULAE: x x b x b x b * * x x b x b x b * x x b x b x b * x x b x b x b b b b * b b b * * b b b. * (+b) = + b+b +b or +b +b(+b) * (-b) = - b+b -b or -b -b(-b) * +b =(+b)( -b+b ) or (+b) -b(+b) * -b =(-b)( +b+b ) or (-b) +b(-b) * + b + c bc = ( + b + c)( + b + c b bc c) * ( + b + c) = + b + c + b + bc + c DAY- : WORKSHEET Coceptul Uderstdig Questios :. The product of (x + ) d (x + 4) ) x + 9x + ) x 9x + ) x 9x 4) x + 9x. (x y) = ) x 4 y xy ) x 4 + y xy ) x 4 y + x y 4) x 4 + y x y. (x + y) (x y) = ) 4x + 9y ) x y ) x + y 4) 4x 9y 4. b b = ) 4 9 ) 4 b 9 b ) 4) b b. (x + y) = ) 8x + 7y + 8xy (x +y) ) 8x + 7y + 6x y + 4xy ) 8x + 7y + 8xy (x + y) 4) Both & 6. (x + y) (x xy + 4y ) = ) x + y ) (x + y) ) x + 8y 4) (x + y) 7. If x =, y = d z =, the x + y + z = ) 9 ) 9 ) 4) 9xyz Sigle Correct Choice Type : 8. Usig the idetity the vlue of 497 ) 476 ) 479 ) 76 4) The vlue of ).4 ).4 ).4 4).4.If P 6 6, Q4 4 d R =, the the vlue of P Q R ) ) ) 4 4) NARAYANA GROUP OF SCHOOLS 4

5 . If p b b d q 4b, the p + q ) b ) b ) b.the expsio squre ) ). The product of ) 4. If ) 99 = 4 4 ) ) b 9 4 s perfect ) 4) 7 7 d 6 6 ) 8 ) 6 6 4) 8 ) 99 4) = () (b), the the vlues of d b re ), ), 48 ) 4, 4) 4, 6.The vlue of b b b 4 b 4 whe =, b = ) ) ) 4) 7.If x, the x = x x ) ) ) 4 4) 4. If x + y + z = the x y z ) xyz ) xyz ) 4). x y y z z x. If ) (x y) (y z) (z x) ) ) 4) xyz b c the b c ) ) ) bc 4) 7bc DAY- : SYNOPSIS Literl fctor (Divor): If two or more lgebric expressios re multiplied, their products re obtied. The lgebric expressios which multiplied to form the product, re clled the fctors of the product. Exmple: xy = x 4y x d 4y re fctors of xy Gretest/Highest commo fctor (G.C.F./ H.C.F): G.C.F./H.C.F of two or more moomils the highest moomil which divides ech of the give moomils completely. L.C.M of Moomils: The L.C.M of two or more moomils moomil hvig the lest powers of costts d vribles such tht ech of the give moomils fctor of it. The sig of the coefficiet of the L.C.M of the moomils the sme s the sig of the coefficiet of the product of the moomils. Fctoriztio: The process of resolvig the give expressio ito fctors clled fctoriztio. Differet types of Fctoriztio: If x + y = 8; xy = the x y. Tkig out commo fctor: Steps : * Fid the H.C.F. of ll the terms of the ) ) ) 4) give expressio. 9. If z 4 the z * Divide ech term by th H.C.F. d z z eclose the quotiet withi the brckets, ) 8 ) 6 ) 4 4) 8 keepig the commo fctor outside the brckets. NARAYANA GROUP OF SCHOOLS

6 Fidig fctors of multiomils : To fctorize multiomil, i geerl we hve to express the multiomil s product of two or more expressios. These two or more expressios whose product equl to the give multiomil, re clled the fctors of the multiomil. Th the reverse process of multiplictio. Prime multiomil: A multiomil sid to be prime if it divible by oe d itself oly. Commo fctor: A umber or umber letter combitio which divides ll the terms of multiomil clled commo fctor of the terms of the multiomil. Commo Biomil fctor: The gretest commo fctor of the terms of multiomil eed ot be moomil lwys. It c lso be biomil. Fctortio d rerrgemet of terms: If we observe the terms of lgebric expressio i the order they re give, they my ot hve commo fctor. But by rerrgig the terms i such wy tht ech group of terms hs commo fctor, some lgebric expressios c be fctorized. While rerrgig the sig d vlue of ech term should ot be ltered. Exmple: x by y bx = x( b) y( b) = ( b)(x y) The fctors re ( + b) d (x y) To fid the fctors of differece of two squres: The differece of two squres lwys equl to the product of the sum d differece of the squre roots of the squre terms i the expressio. Exmple: b ( b)( b) Fctoriztio of ( + b), ( b) forms : * b b b b or b b b b * b b b b or b b b b Fctoriztio of + b, b forms * b b b b (or) b b b b * b b b b (or) b b b b DAY- : WORKSHEET Coceptul Uderstdig Questios :. The HCF of x d x ) x ) x ) x 4) x. The HCF of x d yx ) x ) yx ) x 4) yx. The HCF of umericl coefficiet of the give moomils 6x b c, 8x b c d b c ) ) 8 ) 6 4) 4. The fctoriztio of 4x 9y ) (x y) (x + y)) (x y) (x y) ) (x y) (x y) 4) (x y) (x y). The fctoriztio of x 4 y 4 ) (x + y) (x y) ) (x + y) (x y) (x + y ) ) (x + y) (x y) (x y ) 4) (x + y ) (x + y) (y x) 6. The fctoriztio of x + 8y ) (x + y) (x + xy + 4y ) ) (x + y) (x xy + 4y ) ) (x y) (x + xy + 4y ) 4) (x y) (x xy + 4y ) 7. The fctoriztio of 8 7 ) ( + ) ( ) ) ( + ) ( ) ) ( ) ( ) 4) ( ) ( ) Sigle Correct Choice Type : 8. If A = 84 x 4 y z d B = 6 x y z, the their G.C.D 7 7 ) x y z ) x y z ) x y z 4) x y z 9. The fctors of 9x 7y ) 9 (x + y) ) 9 (x + y) ) 9 (x y) 4) 9 (x y) NARAYANA GROUP OF SCHOOLS 6

7 .The H.C.F of pq, qr, rp ) pqr ) pqr ) 4).The fctors of b + b b ) b (b + b ) ) b (b + b ) ) b (b + + b ) 4) b (b + + b ).The fctors of (x + x) (x +x) y(x + x) + (x + x) re ) (x + x)(x + x y) ) (x + x )(x + x + y) ) (x + x + )(x + x + y) 4) (x + x )(x + x y).the fctors of 6b b re ) b 9b ) b 9b ) b 9b 4) b 9b 4.If you fctore 4p (q + r) the fctors re ) p q r ) p q r ) p q r 4) Both d.the fctors of x 6 y 6 /re ) x y, x y ) x y, x 4 x y y 4 ) x y, x 4 x y y 4 4) Both d 6.The fctor of x 8y re ) (x + 4y)(x 4y) ) (x 4y)(x 4y) ) (x + 4y)(x + 4y) 4) (4y )(x + y) 7. The fctors of x + x y -xy ) x x 4yx y ) x x 4yx y ) x x 4yx y 4) x x 4yx y 8. p 8q = ) p q pq p q ) p q pq q p ) p q p pq 4q 4) both () d () 9. 8 = ) 4 ) 4 ) 4 4) 4 DAY-4 : SYNOPSIS Itriductio: A secod-degree equtio polyomil equtio i which the highest degree of the vrible. I prticulr, secod-degree equtio i oe ukow clled qudrtic equtio. We defie the stdrd form of qudrtic equtio s Ax + Bx + C = A The zero-product rule : If. b =, the = or b = Solvig Qudrtic Equtio by usig the Qudrtic Formul: Cosider the qudrtic equtio x bx c. By solvig th equtio with completio of squre method we get x b b 4c Let the roots re deoted by sy b b 4c, b b 4c We use theseformuls to fid the roots of y other qudrtic equtio. NARAYANA GROUP OF SCHOOLS 7

8 NATURE OF ROOTS: Dcrimit :b 4c deoted by or D clled the dcrimit of qudrtic equtio x bx c where, b, c R d. Thus b 4c Nture of the Roots of Qudrtic Equtio x bx c where, b, c R d : ) If, d,b,c R, the the roots re complex d cojugtes. I th cse the grph of the curve y = f(x) does ot itersect x - x. ) If, d,b,c R, the the roots re rel d ech of the root clled double or repeted root d equl to b. I th cse the curve y = f(x) touch b x - x i oe poit,. Also the qudrtic expressio will be perfect squre expressio whe ) If, d,b,c R, the the roots re rel d dtict. I th cse the curve y = f(x) itersect x - x i two dtict poits 4) If d,b,c Q d perfect squre, the the roots re rtiol d dtict. ) If d,b,c Q d ot perfect squre, the the roots re irrtiol d cojugtes. Nture of the Roots of Qudrtic Equtio x bx c where, b, c C d rel prt of ot equl to : I th cse roots re complex d my or my ot be cojugtes. DAY-4 : WORKSHEET Coceptul Uderstdig Questios :. The solutio set of x bcx d ) ) b d, c b d, c ),c 4) b, d. The roots of x x 6 re ), ), ), 4),. The roots of x 4x re ), 6 ),6 ),6 4), 6 4. The ture of the roots of the equtio x 4x 4 re ) Rel d rtiol ) Rel d irrtiol ) Rel d equl 4) ll of these Sigle Correct Choice Type :. x x b the the vlue of x ),b ), b 6. ), b 4), b b x x, the x ) ) 4 ) 4 4) 7. If b the roots of the equtio x x b b re ) rel d dtict ) rel d equl ) rel 4) imgiry 8. The umber of rel roots of the equtio x x x ) ) ) 4) 9. The vlue of m for which the equtio m x m x 8m h s equl roots, ) ) ) 4) NARAYANA GROUP OF SCHOOLS 8

9 . The roots of x x re ) Rtiol d equl ) Rtiol d ot equl ) Irrtiol 4) Imgiry. If the roots of the equtio x - - m(x-8)= re equl the m = ), - ), ) -, 4) -, -. Oly oe of the roots of x bx c,, zero if ) c ) c, b ) b, c 4) b, c ) ) ) 4) 4 4. The roots of the equtio c b x cx b c re ), c b b c ), b c c b b c ), c 4), c c b. If b cx bc x c b perfect squre, the, b, c re i ) A.P. ) G.P. ) H.P. 4) A.G.P. 6. If the roots of p q x q p r x q r be rel d equl, the p, q, r will be i ) A.P ) G.P ) H.P 4) Noe DAY- : SYNOPSIS Complex Numbers: Euler ws he first Mthemtici. Who itoduced the symbol i ( red s iot ) for with property i i. e. i. He lso clled th symbol s imgiry uit.. i R For exmple : ( i ) 6 6 i 6 i4 ( i4) 6 ( ii ) 6 6 i 66 i6 ( i6) 6 Imgiry umber: Squre root of egtive umber clled imgiry umber. For exmle :,, 6, etc. re ll imgiry umbers. Itegrl powers of iot ( i ) : ( i ) Positve Itegrl Power of i : we kow tht, i = we c write higher powers of i s follows : i, i i, i, i i i i i i 4, ( i ) ( ), 4 i i i i i, 6 4 i ( i) i ( ), 7 4 i i i i i ( ), i ( i ) (), i i i i i d so o. I order to compute i p for p> 4, we divide p by 4 d obti the remider r. Let q be the quotiet, whe p divided by 4. The, p = 4q + r. were r 4 i i ( i ) i. r i p 4qr 4 q r r r Thus, the vlue of p i for p 4 i r, where r the remider whe p s divided by 4. ( ii ) Negtive itegrl Powers of i : Complex Numbers: A umber of the form +ib clled comples umber, wher d b re rel umbers d i. Complex umber geerlly deoted by z i.e. z ib For exmple : +i, +i,-i etc. re complex umbers. NARAYANA GROUP OF SCHOOLS 9

10 Rel d imgiry prts of complex umber : Let +ib be complex umber the, clled rel d b the imgiry prt of z d my be deoted by Re(z) d Im(z) repectively. For exmple : If z=+i, the Re(z) =, im(z) =. Purely rel d purey imgiry complex umbers: A complex z ib clled purely rel, if b = i.e Im(z)= d clled purely imgiry, if =. i.e., Re(z) =. For exmple : z= purely rel d z=i purely imgiry Set of complex umbers: The Product set R R costig of the ordered pirs of rel umbers clled the set of complex umbers. The set of complex umber deoted by C i.e. C ib :, b R Note: We observe tht the system of complex umbers icludes the system of rel umbers i.e. R C. Equlity of complex umbers: Two complex umbers z ib d z c id re sid to be equl, if = c d b=d. Proof: ib c id c i( d b) ( c) ( d b) ( c) ( d b) Here, sum of two positive umbers zero. Th oly possible, if ech umber zero. ( c) c d ( d b) b d. For exmple : If + ib =+i, the =,b= Zero complex umber: A complex umber z sid to be zero, if its both rel d imgiry prts re zero. I other words, z = +ib =, if d oly if = d b =. Note: Order reltios greter th d less th re ot defied for complex umbers. The iequlities like i>, +i < etc. re meigless. Additio of two complex umbers: If z ( x, y) C d z (, b) C the zz ( x, y) (, b) ( x yb, xb y). If z, z C, the z z C d z z clled the sum of two complex umbers z d z. Multiplictio of two complex umbers: The postulte (iii) defies the biry opertio of multiplictio of two complex umbers. If z ( x, y) C d z (, b) C, the zz ( x, y) (, b) ( x yb, xb y) e.g. (,) (, 9) (..9,.9.) ( 9, ) If z, z C, the z z C d z z clled the product of two complex umbers z d z. Divio of complex umbers: The divio of complex umber z by o zero complex umber z defied s the multiplictive iverse of z by the multiplictive iverse of z d deoted by Therefore, b b b b = b b NARAYANA GROUP OF SCHOOLS z z. z z = z, z z.. z Let z ib d z ib The = z ib ( ib ) ( ib ) z ib ( ib )( ib ) b b i ( b b b

11 For exmple : If z i d z 4 i, the z i ( i)( 4 i) 8i i z 4 i ( 4 i) ( 4 i) 6 7i 7 i DAY- : WORKSHEET Coceptul Uderstdig Questios : i.. ) i ) -i ) 4) -. i i. ) ) - ) 4). If z =-i the Rel prt of z. ) i ) - ) 4) 4. If z = +i the imgiry prt of z. ) ) ) i 4). If p+iq= the p+q. ) ) - ) - 4) 6. If z i, z i the z z. ) ) ) 4) i 7. If z i, z i the z z. ) i ) ) 4 4) If z i, z i, the z z. ) ) ) - 4) 9. If z z i z i the z,,. ) -i ) ) 4) i Sigle Correct Choice Type :.. b ) b ) b ) b 4) b i. i ) i ) ) -i 4) - 7.Whe simplified the vlue of i (/ i ) ) ) i ) -i 4).If i, the i i i 4 i 6... to ( + ) terms equl to ) ) -i ) 4) The vlue of ( i) ( i) equl to ) 8 ) ) -8 4) -4.If N, the vlue of i i 4 4 ) i ) ) -i 4) - 6.The vlue of 4 6 i i i ) ) - ) 4) 7.Which of the followig true? 8. 7 ) ( i) 6 i8 ) i ) 6 i9 4) 48 7 i6 4 i 9. i i 9 i7 ) -4 ) -i ) 4 4) i i 7 4 ) -i ) - ) i 4). (-i) (+i) (-i) = ) -6i ) 8-6i ) 6-8i 4) 6+8i. Express ( +i) (-4i) i the form of +ib ) 6+i ) 7+i ) 8+i 4) Noe. Express (4 i) i the form of (+ib). If ) (-44-77i) ) 44-7i) ) -44+7i) 4) Noe 6 (y ) i ( x) i, the ) x = / ) y = / ) Both () & () 4) Noe NARAYANA GROUP OF SCHOOLS

12 4. If i y, the y = ) ) ) 4). Mtrix mtch type: Colum-I Colum-II ) If (y-) +i (7-x) =, the x = p) /4 b) If (y-) +i (7-x) =, the y = q) /4 c) If 4x+i (x-y) =-6i, the x = r) / d) If 4x+i (x-y) =-6i, the y = s) 7/ 6. If x+i4y=i, the x 7. If x+4iy = i x-y+, the y = 8. Which of the followig true? ) ( ) ( ) 4 ( 44 7 i) ) ( )( ) 7 i ) ( i) ( i) 4) ( 7 i) ( ) i DAY-6 : SYNOPSIS Ordered Pir: A pir of umbers d b lted i specific order with t the first plce d b t the secod plce clled ordered pir (, b). Note tht, b b,. Thus, (, ) oe ordered pir d (, ) other ordered pir. CO-ORDINATE SYSTEM We represet ech poit i ple by mes of ordered pir of rel umbers, clled the coordites of tht poit. The positio of poit i ple determied with referece to two fixed mutully perpediculr lies, clled the coordite xes. O grph pper, let us drw two mutully perpediculr stright lies X'OX d YOY', itersectig ech other t the poit O. These lies re kow s the coordite xes or xes of referece. The horizotl lie X 'OX clled the x-x. The verticl lie YOY ' clled the y-x. NARAYANA GROUP OF SCHOOLS 4 X' Y O Y' 4 The poit clled the origi. The cofigurtio so formed clled the coordite system or coordite ple. Coordites of poit i ple: Let P be poit i ple. Let the dtce of P from the y-x = uits. Ad, the dtce of P from the x-x = b uits. The, we sy tht the coordites of P re (, b). clled the x-coordites or bscs of P. b clled the y-coordites or ordite of P. X' Y O Y' b P(, b) Cosider the poit P show o the djoiig grph pper. Drw PM OX. Qudrts: Let X'OX d YOY' be the coordites xes. These xes divide the ple of the grph pper ito four regios, clled qudrts. The regio XOY clled the First Qudrt. The regio YOX ' clled the Secod Qudrt. The regio X 'OY ' clled the Third Qudrt. The regio Y ' OX clled the Fourth Qudrt. X X

13 Usig the covetio of sigs, we hve the sigs of the coordites i vrious qudrts s give below: Regio XOY YOX' X'OY ' Y'OX Qudrt I II III IV Nture of x d y x >, y > x <, y > x <, y < x >, y < Sigs of coordites (+, +) (, +) (, ) (+, ) Ay poit o x-x: If we cosider y poit o x-x, the its dtce from x-x. So, its ordite zero. Thus, the coordites of y poit o x-x (x, ). Ay poit o y-x: If we cosider y poit o y-x, the its dtce from y-x. So, its bscs zero. Thus, the coordites of y poit o y-x (, y). Slope of x-x ; slope of y-x ot defied. Dtce betwee two poits Let A,,, x y B x y be y two poits o lie ot prllel to the xes. From the djcet figure we hve the right gle trigle ABC. X O Y AB AC BC x x, y A x x y x B But AC x x,bc y y AB x x y y AB x x y y NOTE : The dtce to the poit Ax, y from origi x y C y DAY-6 : WORKSHEET Coceptul Uderstdig Questios :. If the x co-ordite of poit d its y co-ordite, the it represeted s ), ), ) (, ) 4) (, ). If the bscs & ordite of poit re d respectively the the poit represeted s ), ), ) (, ) 4) (, ). If poit t dtce of uits from Y x d uits from X x the the poit represeted s ), ), ) (, ) 4) (, ) 4. A Poit (4, ) lies o ) X x ) Y x ) Origi 4) X d Y xes. A Poit (, ) lies o ) X x ) Y x ) Origi 4) X d Y xes 6. The dtce betwee two poits (,) d (,) ) uits ) 9 uits ) 7 uits 4) 7 uits 7. The dtce betwee two poits (,) d (,4) ) uits ) uits ) 7 uits 4) 7 uits 8. The dtce betwee two poits (,) d ( 4,) ) uits ) uits ) uits 4) uits Sigle Correct Choice Type : 9. I which of the followig qudrt does the give poit (, 8) lie? ) I qudrt ) II qudrt ) III qudrt 4) IV qudrt.i which of the followig qudrt does the give poit (, ) lie? ) I qudrt ) II qudrt ) III qudrt 4) IV qudrt NARAYANA GROUP OF SCHOOLS

14 .I which of the followig qudrt does the give poit ( 6, 8) lie? ) I qudrt ) II qudrt ) III qudrt 4) IV qudrt.the Horiztl x clled. ) X x ) Y xix ) Origi 4) I Qurdrt.The erest poit from the origi ) (, ) ) (, ) ) (, ) 4) (, ) 4.If Q(x,y) lies i the Fourth Qudrt the x ) Positive ) Negetive ) Both & 4) Noe.The trigle formed by (, ), (, ) d (, ) (through grph) ) Right gle oceles trigle ) Sclee trigle ) Equilterl trigle 4) Cot form trigle 6.The X co ordite o Y x ) ) ) Udifie 4) Noe 7.The dtce betwee (4, ) ( 4,) ) uits ) uits )4uits 4)uits 8. If the dtce betwee the poits (, ) (,), the the vlue of ) uits ) uits )4uits 4) uit 9. The poit o x-x which equidtt from the poits (, 4), (, ) ) (,) ) (4,) )(,) 4) (,). The dtce betwee A(7,) d B o the x-x whose bscs ) uits ) uits ) uits 4) uits MATRIX MATCH TYPE.Colum-I Colum-II ) Dtce betwee (, ), (8, 7) ) - b) Dtce betwee (, ), (-4, ) ) c) Dtce betwee,,, ) 4) If (, x) t uits from(, ) the x = 4) ) 4 DAY-7 : SYNOPSIS Dividig lie segmet i give rtio (sectio formule) : A (P divides AB P m P i the rtio m: iterlly.) A (P divides AB i the rtio m: exterlly.) Sectio formule : The poit P which divides the lie segmet joiig the poits,,, i) iterlly A x y B x y i the rtio m: ii) exterlly B mx x my y, ; m m m mx x my y, ; m m m Mid poit of lie segmet : The mid poit of lie segmet joiig of x, y d x y, x x, y y NOTE :. The poit P(x,y) divides the lie segmet joiig A x, y d, B x y i x x : x x (or) y y : y y i.e AP = PB = x x x x. x-x divides the lie segmet joiig x y d,, x y i the rtio y : y. y-x divides the lie segmet joiig x y d,, x y i the rtio x : x Secod - order determit : The expressio c b d clled the secodorder determit. NARAYANA GROUP OF SCHOOLS 4

15 b It defied s c d = d-bc 4 Exmple : Are of trigle :. The re of the trigle formed by the poits A x, y, B x, y d C x, y (or) = x y y x x x x (or) y y y y x x x x y y y y sq.uits. The re of the trigle formed by the poits O,, A x, y, B x, y = x y x y sq.uits. NOTE :. Three poits A,B,C re collier if the re of ABC zero.. If D,E,F re the mid poits of the sides of the ABC the the re of ABC = 4 (re of DEF ).. If G the cetroid of the ABC the re of ABC = (re of GAB ) DAY-7 : WORKSHEET Coceptul Uderstdig Questios :. The vertices of trigle re A(, 4), B(4,) d C(,), so ABC ) Right gled trigle ) Isosceles trigle ) Right gled, Isoceles trigle 4) Equilterl trigle. The mid poit of (,) d (,4) ) (,) ) (,) ) (,4) 4) (,). The rtio i which (,) divides the lie segmet joiig (4,8), (, 7) ) : exterlly ) : iterlly ) 4 : exterlly 4) : iterlly. 4. x - x divides the lie segmet joiig (, ), (,7) i the rtio ) : ) : 7 ) 4 : 4) : 4. The re of the trigle formed by the poits (,), (,), (,) )4 sq. uits ) sq.uits ) sq.uits 4) Sigle Correct Choice Type : 6. If the poits, 8, 4, d, k re collier the, the vlue of k ) 4 ) -8 ) 4 4) 7. The trigle formed by (,), (,) d (,) ) Right gle osceles trigle ) Sclee trigle ) Equilterl trigle 4) Cot form trigle 8. The mid poit of the lie joiig the poits,4 d x, y, the x y ) ) ) 7 4) - 9. If the poit p, divides the lie joiig the poits,6 d 8,9,the the rtio NARAYANA GROUP OF SCHOOLS ) : iterlly ) : exterlly ) : iterlly 4) : exterlly. The coordites of the poit which divides the lie segmet joiig poits A (,) d B(9, ) i the rtio : re ) (, 4) ) (, 4) ) (, 4) 4) Noe of these. The poit which divides the lie joiig the poits b, b d b, b i the rtio : b ) ) b b, b b ) b b b, b b b b b, b b 4) Noe of these.the rtio i which the lie segmet joiig the poits (, 4) d (, 6) divided by the x x ) : ) : ) : 4) Noe

16 .Let P d Q be the poits o the lie segmet joiig A(, ) d B(, ) such tht AP = PQ = QB. The the midpoit of PQ ), ),4 ), 4),4 4. The coordites of poits A, B, C re x, y, x, y d x, y d poit D divides AB i the rtio l : k. If P divides lie DC i the rtio m : k, the the coordites of P re ) ) ) kx lx mx ky ly my, k l m k l m lx mx kx ly my ky, l m k l m k mx kx lx my ky ly, m k l m k l 4) Noe of these. P = (,4) d Q = (, ). If PQ produced to R such tht P divides QR exterlly i the rtio :, the R ), ), ), 4), DAY-8 : SYNOPSIS Vrible: A symbol which c tke vrious umericl vlues clled vrible. Exmple : x, y, z,, b, c etc. Note: Vribles re lso kow s literls. Costt: A symbol hvig fixed vlue clled costt. Exmple:,,, 7 etc. Term: Numericls or literls or their combitio formed by opertio of multiplictio re clled terms. Exmple:, etc. x, x,7 m l, y,,x, 6 Expoetil form: The product of umber x with itself times ( turl umber) give by x x x x... x ( times) d writte s x which clled the expoetil form. Here x clled bse d clled the expoet (or) idex of x. x c be red s th power of x (or) x red to the power. Note: Expoetil form lso clled s power ottio. Exmple: Here clled the product form (or) expded form d clled the expoetil form. Note: * The first power of umber the umber itself. i.e., =. * The secod power clled squre. Exmple: Squre of * The third power clled cube. Exmple: Cube of x x * red to y itegrl power gives Exmple: = * ( ) odd turl umber = Exmple: ( ) 7 = Lws of Expoets (or) Idices:. The product of the two powers of the sme bse power of the sme bse with the idex equl to the sum of the idices. i.e., If be y rtiol umber d m, be positive itegers, the m. m m NARAYANA GROUP OF SCHOOLS 6 x m x itegers, x,where m, re positive. Power of Product b b where, b, d positive iteger 4. Quotiet of powers of the sme bse. m m if m if m m

17 . Power of quotiet m m i.e., m where, b, d m b b positive iteger. 6. Powers with expoet zero: If we pply the bove lws of idices of evlute m m where m =, d m DAY-8: WORKSHEET ; the. If x the fourth power of d y the third power of 4, the (x + y) + (y x) + (y + x) A) 68 B) 66 C) 64 D) 6. If = 4, x =, the x x A) 6 B) 64 C) D) 4. If = 4, b = 6 d c =, the b 4 c A) 47 B) 48 C) 4 D) If x 4 = 6 d y = 4, the x y A) 6 B) 8 C) 4 D) 9. If P = , where p = x, the (p) A) 764 B) 768 C) 76 D) If x x + =, y y + x+y x+y = 79, the x y A) (6) B) () C) () D) 7. If (4 ) =, the x x A) 4 B) 6 C) 8 D) 8. If ( ) multiplied by the the resultt -, 4. If p = where p m = 4 p m A) B) 4 C) D) 7. If = 6 9 = 6, the A) B) C) D) 9. If (x y ) (4 b ) = p where x = y, y =, = 4b d b =, the the vlue of p Note: * The smllest whole umber A) 6 (48) 6 (4) 4 (8) 6 () 9 d the lrgest whole umber cot B) 6(48) (4) 4 () 9 be determied C) 6(48) 6 (4) 4 (8) () 9 * All turl umbers re whole umbers. D) 6(48) 6 (74) (8) () 8 NARAYANA GROUP OF SCHOOLS 7 y x d, the (x + y) A) 4 B) 6 C) D) 4 4. If x b x b c x c = p, the the vlue of p A) B) C) D). If = 9, b = d c = 4, the ( b c 4 ) A) ( 4 ) B) ( 4 ) C) ( 4 ) 4 D) ( 4 ) 4. If 8 4 b = () 8 (4), p q =, the p b q q q A) 6 B) 8 C) D) 64 DAY-9 : SYNOPSIS Nturl Numbers: Coutig umbers,,, 4,... re clled Nturl umbers, deoted by N. N = {,,, 4,...}. Note: * The smllest Nturl umbers d the lrgest umber c t be determied. * The umber of turl umbers betwee d b where < b b. * The umber of turl umbers from to b where < b b +. Whole Numbers : The turl umbers log with zero re clled whole umbers, deoted by W. W = {,,,, 4,...}.

18 * The differece of y two cosecutive whole umbers. Frctios: * Tej hs tke crd bord which i the shpe of squre. * He hs cut the crd bord s show. He hs observed tht he got two pieces of crd bord d ech prt of the whole. Hece, we c sy tht ech prt frctio. * Therefore from the bove illustrtios we sy tht A prt of the whole clled Frctio. Comp ro of frctio s: By c ross multiplictio: If two frctios b d c re to be compred,we cross multiply d c (i) If d b c,the b d c (ii) If d b c,the b d c (iii) If d b c,the b d Exmple: Compre the d 6 Solutio: O cross multiplictio we get 6 d d < 6 By tkig the L.C.M: Tkig the L.C.M of the deomitor of the give frctio. Covert ech of the frctio ito equivlet frctio with deomitor equl to the L.C.M. Compre their umertors. The higher the vlue of the umertor, the greter the frctio Exmple: Arrge,,, 4 order. i scedig Solutio: The L.C.M of,4,, = 4 8 Now,, , Now compre the umertors of like 8 8 frctios,,, Arrgig them i scedig order, we 8 8 get So, 9 4 Fudmetl opertios o frctios: Additio: While ddig like terms, dd the umertors d reti the commo deomitor. I, geerl c c Exmple: Add b b b Subtrctio: While subtrctig like terms, subtrctig the umertors d reti the commo deomitor. I, geerl c c b b b Multiplictio of frctios: If b d c re two frctios, the the d product of these frctios = c product of their umertors b d = product of their deomi tors Exmple: Multiply d 7. Solutio: Divio of frctios: If b d c d frctios, the c b d = d. b c Exmple: Divide 9 Solutio: 9 9 re two NARAYANA GROUP OF SCHOOLS 8

19 DAY-9: WORKSHEET Coceptul Uderstdig Questios :. Compre d 4 ) ) ) 4) Both & A Frctio greter th ) ) ) 4) 4. pe s frctio of Rs. ) ) ) 4) 4. Ascedig order of,, d 6 ),, ),, 6 6 ),, 4),, 6 6. Descedig order of, d 7 ),, 7 ),, 7 6. = ) ) 7. = ) 6 7 = 7 ) 7 = ) 7 6 ) ) ) 7 6 ) ),, 7 4),, 7 ) 6 6 ) 84 ) 4) All the bove 4) 6 4) ) 6 7. Product of 8 d 6 7 ) ) ) 4) The vlue of of ) ) ) 4) 6 Sigle Correct Choice Type :. Equivlet frctio of 4 ) 6 ) ) 8. Simplest form of ) ) ) 4) 8 8 4) 7. If m - =,the the vlue of m ) ) 7 ) 4) 7 4. The sum of three sides of trigle 6 cm. If two of its sides mesure 7 cm d 6 4 cm respectively, the the legth of the third side ) ) 4 cm ) 7 cm 4) 7 cm 4 cm DAY- : SYNOPSIS Defiitio: If N d re y two positive rel umbers d for some rel x, such tht x N, the x sid to be logrithm of N to the bse. It writte s log N x. Thus, x N log N x Note: * It should be oted tht log bbrevtio of the word logrithm. * Logrithms re defied oly for positive rel umbers. NARAYANA GROUP OF SCHOOLS 9

20 * There exts uique x which stfies the equtio x N. So log N lso uique. Logrithmic Fuctio: Fuctios defied by such equtios re clled logrithmic fuctios. We c express expoetil forms s logrithmic forms. Expoetil form Logrithmic form (i) 4 = 6 4 log 6 (ii) log (iii) Note: log 9 * For y positive rel umber we hve. Therefore log. i.e., The logrithm of y o - zero positive umber to the sme bse uity. x * If N x log N x Replcig x by log N i log N N 9 N * The logrithm of uity to y o - zero bse zero. Recll tht, if o, log LAWS OF LOGARITHMS:. If m, re positive rtiol umbers, the log m log m log. i.e., The log of the product of two umbers equl to the sum of their logs. Proof: Let log m x d log y x y m... d... from () & () x y x y m. m Apply log to the bse o both sides log m x y log m log m log x log m d y log GENERALISATION: If x,x,...x re positive rtiol umbers, the NARAYANA GROUP OF SCHOOLS log x,x,..x log x log x log x... log x. If m d re two positive rtiol m umbers, the log log m log log m. log m. 4. log m log m m m. log log b b 6. If m, b re positive rel umbers the log b log log m m b 7. If, b re two positive rel umber the log b log b. DAY-: WORKSHEET Coceptul Uderstdig Questios :. If 4 6, the which of the followig true? ) log 4 6 ) log 6 ) log ) log 6 4. log.. = ). ). ) 4). log 7 = ) ) ) 7 4) 7 4. log ) ) ) 4)

21 . log 4 = ) log 4 log 6 ) log log ) log log 8 6. log 4 log = ) log 4 ) ) 4 log log 4) log log 4 ) 4 (log ) ) 4) All of these log 4 log log log 4 log 4 4 ) 4 log 4) 4 log Sigle Correct Choice Type :. log 4 x, the x = ) 4 ) 6 ) 4). If, the which of the followig 9 true? ) log ) log 9 9 ) log 4) log 9 9. The vlue of log 9 8 = ) ) ) 9 4) 8 4. The vlue of log. ) 4 ) 4 ) 4). The vlue of log 4 ) ) 4 ) 4) 6. The vlue of x if log 6 x 6 ) ) 6 ) 4 4) 7. Logrithm of y o-zero umber to the sme bse ) Sme o-zero umber ) ) 4) Negtive of the o-zero umber 8. Logrithm of uity to y o-zero bse ) No-zero umber ) Uity ) 4) 9. log. = ) log ) log log ) 4) Both &. log log ) log x ) ) oly () 4) Both &. log log ) log log log ) ) log ( ) 4) log. The vlue of log 8 log log log ) ) 4 ) 4) 6.The vlue of log ) ) 4. log b x log b = 7 ) 4) 7 ) log x ) log x b ) log x 4) log b x. log m = ) log m log ) log m + log m ) log 4) log m log 6. If log., the logrithm of with bse ) 9.7 ) 7. ) 7.7 4) log log 4 = ) 4 ) ) 4) 8. Express log log log s sigle logrithm ) log ) log ) 4) log NARAYANA GROUP OF SCHOOLS

22 DAY- : SYNOPSIS Sequece: It rrgemet of umbers i defiite order ccordig to sme rule Exmple:,,,,... etc. d 4, 44, 444,... i) T = + represets sequece 7,, 7... ii) T = - 4 represets sequece -,,... iii) T = represets sequece, 7, 9,... Series: A expressio costig of the terms of sequee, ltertig with the symbol + clled Series For exmple, ssocited with the sequece 7,,..., we hve series 7,, Arithmetic me (A.M.): The rithmetic me betwee two umbers the umber which whe plced betwee them forms with them rithmetic progressio. Thus the rithmetic me betwee d b b A.M = where d b re y two positive umbers. If there re A.M s betwee, b the commo differece, b d Sum of first turl umbers, th term: The umber occurig t the th plce of sequece clled its th term. S t t... t t = t ; t s s Deoted by T. DAY-: WORKSHEET Progressio: The sequece which obey the defiite rule d its geerl term Coceptul Uderstdig Questios : lwys expressible i terms of clled. Choose sequece from the give progressio. ),,,7,9...),,4,,7,9,,... Exmple: ), 4, 6, 8,,... ) -,,7,,...4),7,,,,... ),, 9, Choose series from the give ),,,... ) ) Arithmetic progressio (A.P):A Sequece ) clled rithmetic progressio if its terms cotiully icrese or decrese 4) or decrese by the sme umber..,4,8,,6 forms / Exmple:, + d, + d, + d,... i A.P ) Arithmetic sequece th term of A.P t = + (-)d ) Geometric sequece Sum of terms of A. P ) Arithmetic progressio 4) Geometric progressio S d d S where l the lst term. NARAYANA GROUP OF SCHOOLS S Sum of the squres of first turl umbers, S Sum of the cubes of first turl umbers, 6 4 S

23 4. The first term of Ap d its commo differece, the first five terms of give AP re ), +, + 4, + 6, + 8 ) +, + 4, + 6, + 8, + ), -, - 4, - 6, - 8 4), -, 4 -, 6 -, 8-4. If the first term of AP d its commo differece, the its th term ) + ) - ) - 4) -. If, 6,, -4, -9 forms AP, the its commo differece ) - ) 4 ) 4) -4 Sigle Correct Choice Type : 6. The ext term i the series 7,, 9... ) 9 ) 8 ) 6 4) 4 7., 7,,, 6, (...) ) 9 ) ) 7 4) 8. If x +, 4x -, x + re i A.P, the the first term ) 8 ) 7 ) 9 4)6 9. If the th term of AP 4 - the the 8 th term ) 7 ) 7 ) 74 4) 7. If 8k + 4, 6k -, k + re i AP the the vlue of k ) - ) - ) 9 4). The th term of the A.P., 6,, -, -6, -,... ) - 86 ) 6 ) 96 4) 46. If first term d 8 th term 4 eh the 68 th term ) 4 ) ) 4) 8. If the rd d 7 th terms of A.P. re 7 d 7 respectively, the the f i r s t term of the A.P ) 9 ) ) 4 4) 6 4. If, x, x, x...x b re i A.P the x b b b ) ) ) 4) b.if the th term of the A.P,, 7, 9... d -7, -, -, 4... re equl, the the vlue of = ) = ) = 8 ) = 9 4) = DAY- : SYNOPSIS Geometricl Progressio: A sequece (fiite or ifiite) of o-zero umber i which every term, except the first oe, bers costt rti with its precedig term, clled geometric progressio, bbrevited s G.P. Illustrtio: The sequeces give below: i), 4, 8, 6,,... ii), -6,, -4, 48,... iii),,,,, Note: I G.P. y term my be obtied by multiplyig the precedig term by the commo rtio of the G.P. Therefore, if y oe term d the commo rtio of G.P. be kow, y term c be writte out, i.e., the G.P. the completely kow. I prticulr, if the first term d the commo rtio re kow, the G.P. completely kow. The first term d the commo rtio of G.P. re geerlly deoted by d r respectively. Geerl term of G.P.: Let be the first term d r be the commo rtio of G.P. Let t, t, t,..., t deote st, d, rd,..., th terms, respectively. The, we hve t = tr, t = t r, t 4 = tr,..., t = t-r. O multiplyig these, we get t t t... t = t t t... t r t = t r, but t = Geerl term = t = r -. Thus, if the first term d r the commo rtio of G.P. the the G.P., r, r,... r - or, r, r... ccordig s it fiite or ifiite. If the lst term of G.P. costig of terms deoted by l, the l = r -. Note: * If the first term d r the commo rtio of fiite G.P. costig of m terms, the the th term from the ed give by r m -. NARAYANA GROUP OF SCHOOLS

24 * The th term from the ed of G.P. with the lst term l d commo rtio r i s l/r -. * Three umbers i G.P. c be tke s /r,, r, four umbers i G.P. c b e tke s /r, /r, r, r, five umbers i G.P. c be tke s /r, /r,, r, r, etc... * Three umbers, b, c re i G.P. if d oly if b/ = c/b, i.e., if d oly if b = c. DAY-: WORKSHEET Coceptul Uderstdig Questios :.,,,, 4,... forms / ) Arithmetic sequece ) Geometric sequece ) Arithmetic progressio 4) Geometric progressio. If the first term of GP d its commo rtio, the first five terms of the GP re ),6,9,, ),,7,9, ),9,7,8,4 4),7,,,9. if t 7 of GP 8 d its commo rtio, the its first term ) ) ) 4) 4 4. The first term of GP, d commo rtio. The the sum of its terms ) ) ) 7 4). Geometric Me of 7 d ) 4 ) 6 ) 7 4) 9 Sigle Correct Choice Type: 6. If x, x +, x +,... re i geometric progressio, the fourth term ) -7 ) ) 4). The first term of the progressio G.P. d the 4 th term, the the th term ) 8 ) ) 4 4) 6.Sum of three cosecutive terms i G.P. 4 d their product. F i d the lrgest of these umbers ) 8 ) 6 ) 4).The sum of the 8 terms of, 6,, 4,... ) 76 ) 46 ) 6 4) 64. Number of terms of G.P.,,,... re eeded to give the sum ) 4 ) ) 8 4) 4. Sum to terms of the series equls to 7. If the th term of G.P 9 d 4 th term 4, the the 7 th term ) ) 6 ) 4 4) 8 8. If th, 8 th, d th term of G.P. re p, q d s respectively, the ) p = qs ) q = ps ) s = pq 4) s = pq 9. If x + 9, x - 6, 4 re i G.P the the vlue of x ) 8 ) ) 6 4) followig. NARAYANA GROUP OF SCHOOLS 4 ) 9 9. ) 9 9. ) 4) The sum to terms of the progressio, -,, -,,...(-) + = ) if eve ) - if eve ) if odd 4) - if odd DAY- : SYNOPSIS. Itroductio : I everydy life, we hve to del with the collectios of objects of oe kid or the other. For Exmple : i) The collectio of eve turl umbers less th i.e., the umbers, 4, 6, 8 d. ii) The collectio of vowels i the Eglh lphbet i.e., the letters, e, i, o, u. I ech of the bove collectios, it defiitely kow whether give object to be icluded i the collectio or ot to be icluded. Ech oe of the bove collectios exmple of set. However, the collectio of ll itelliget studets i clss of give school ot set. Here it difficult to decide who itelliget d who ot. The sme studet my be itelliget i the eyes of oe techer d my ot be itelliget i the eyes of other. We sy tht such collectio ot well defied. With th bsic otio, we hve the

25 Defiitio : Ay well defied collectio of objects clled set. By well-defied collectio we me tht give set d object, it must be possible to decide whether or ot the object belogs to the set. The objects re clled the members or the elemets of the set. Sets re usully deoted by cpitl letters d their members re deoted by smll letters. We write the elemets of set with i the brcess { }. If x member of the set S, we write x S (red s x belogs to S) d if x ot member of the set S, we write x S (red s x does ot belog to S). If x d y both belogs to set S, we write x, ys. Represettio of Sets : There re two wys to represet give set. ) Roster or Tbulr Form or lt form. I th form, lt ll the members of the set, seprte these by comms d eclose these withi brces (curly brckets) For exmple : i) The set S of eve turl umbers less th i the tbulr form writte s S = {, 4, 6, 8, }. Note tht 8S while 7S. ii) The set S of prime turl umbers less th i the tbulr form writte s S ={,,, 7,,, 7, 9 } iii)the set N of turl umbers i the tbulr form writte s N = {,,,... }, the dots idictig ifiitely my msig positive itegers.. Set Builder or rule form : I th form, write oe or more (if ecessry) vribles (sy x, y etc.) represetig rbitrry member of the set, th followed by sttemet or property which must be stfied by ech member of the set. For exmple : i) The set S of eve turl umbers less th i the set builder form writte s S = { x/x eve turl umber less th }. ii) The set of prime turl umber less th i the set builder form writte s {x/x prime turl umber less th }. The symbol stds for the words such tht or where. Sometimes we use the symbol ; or : i plce of the symbol. iii)the set N of turl umbers i the set builder form writte s N = { x :x turl umber}. Some Stdrd Sets : We elt below some sets of umbers which re most commoly used i the study of sets : i) The set of turl umbers (or positive itegers). It usully deoted by N. i.e. N = {,,, 4,... } ii) The set of whole umbers. It usully deoted by W. i.e. W = {,,,,... }. iii) The set of itegers. It usully deoted by Z. i.e. Z = {..., -, -, -,,,,,...} iv) The set of rtiol umbers. It usully deoted by Q i.e. Q = { x : x rtiol umber} or m Q x : x, where m d re it egers d v) The set of rel umbers. It usully deoted by R. i.e. R = { x : x rel umber } or R = { x : x either rtiol umber or irrtiol umber } Note :- i Types of sets : The Empty set : A set cotiig o elemet clled the empty set. It lso clled the ull set or void set. There oly oe such set.it deoted by or by { }. For exmple : i) The collectio of ll itegers whose squre less th the empty set. ( Squre of iteger cot be egtive) ii) The collectio of ll girl studets i boys college the empty set. iii) The collectio of ll the rel roots of the equtio x + = the empty set. ( The equtio x + = ot stfied by y rel umber, for if rel root of x + =, the + = =, which ot possible s squre of rel umber cot be egtive ). The order of the empty set zero. NARAYANA GROUP OF SCHOOLS

26 Sigleto set : A set sid to be sigleto set if it cot oly oe elemet. The set { 7 }, { } re sigleto sets. { x : x + 4 =, x Z } sigleto set, becuse the set cot oly oe iteger mely, 4. For exmple : i) The sets { } d {,, } re equivlet. ii) The sets {, 4 } d { x : x = 4 } re equivlet sets. iii) The sets {, b, c, d, e }, {,,, 4, } d {, e, i, o, u } re equivlet sets s ech of these sets cot dtict elemets. Equl sets :Two sets A d B re sid to be equl, writte s A = B, iff every member of A member of B d every member B member of A.Remember tht equl sets re lwys equivlet but equivlet sets my ot be equl. For exmple :i) The sets {, + } d { x : x = } re equl. ii) The sets {, } d { } re ot equl, but they re equivlet. The set { x : x + 4 =, x N } ull set, becuse there o turl umber which my stfy the equtio x + 4 =. A set whose order clled sigleto set. Thus, sigleto set set which cot oly oe dtict elemet. For exmple : i) If A = { x : x positive divor of }, the (A) = 6 s A = {,, 4,,, } ii) If B = { x : x positive eve prime }, the (B) = s B = { }. DAY-: WORKSHEET Note tht B sigleto set. Coceptul Uderstdig Questios : iii) If C = { x : x iteger either. Which of the followig ot well defied positive or egtive }, the (C) = s collectio of objects C = {}. C sigleto set. ) The collectio of dys i week Fiite d Ifiite sets : A set clled fiite ) The Colloectio of ll eve itegers if the process of coutig of its differet ) The Vowels i the Eglh Alphbet elemets comes to ed; otherwe, it clled ifiite. The empty set tke s fiite. 4) The collectio of te most tleted i your Clss For exmple : i) The set S = {, 4, 6, 8 } fiite set.. Write the set {x : x positive iteger ii) The set of ll studets studyig i d x 4 } i the roster form give school fiite set. ) {,,,4,,6} ) {,,,4,,6,7} iii) The set N of ll turl umbers ) {,,4,,6,7} 4) {,,,,,6} ifiite set.. Roster form of set of S odd turl umbers less th iv) The set of divors of give turl umber fiite set. ) {,,,7,9,} ){,,,7,9,,} v) The set of ll prime umbers ){x/x odd turl umbers } ifiite set. 4){,, } Order of fiite set : The umber of 4. Set builder or roster form of set S of odd differet elemets i fiite set S turl umbers lessth clled order of S, it deoted by O(S) or ) {,,,7,9,} ){,,,7,9,,} (S). ){x/x odd turl umbers } Note : The order of ifiite set ot 4){,, } defied. Equivlet sets : Two fiite sets A d B Sigle Correct Choice Type: re sid to be equivlet writte A ~ B.Which of the followig collectios re sets. (or A B), iff they coti the sme i) The collectio of ll prime umbers umber of dtict elemets i.e., iff (A) betwee 7 d 9. = (B). ii) The collectio of ll rich persos i NARAYANA GROUP OF SCHOOLS Idi. 6

27 iii) Collectio of ll fctors of which re greter th 6. ) (i) d (ii) ) (i), (ii) d (iii) ) (ii) d (iii) 4) (i) d (iii). Which of the followig collectios re sets. i) The collectio of ll moths of yer, begiig with letter J. ii) The collectio of most tleted writers of Idi. iii) The collectio of ll turl umbers less th. iv) The collectio of most dgerous imls of the world. ) Oly (i) ) Oly (ii) ) Both (i) d (iii) 4) Both (ii) d (iv). The solutio set of the equtio x x i roster form ), ), ), 4) x : x R, x 4. If A 6, the set builder form ) A x : x Rd 6 x ) A x : x Rd 6 x ) A x : x R d 6 x 4) A={6,7,8,9,,,}. Which of the followig sets re sigleto sets? i) A = { x : x =, xq } ii) B = { x : x =, xr } iii) C = { x : x 9 =, xn } iv) D = { x : x =, xz } ) (i) d (iii) ) Oly (iii) ) (i) d (ii) 4) Oly (iv) 6. Which of the followig sets re ull sets? i) A = { x : x < d x > } ii) B = { x : x = 9 d x = 7 } iii) C = { x : x =, xr } iv) D = { x : x eve prime umber } ) (i) d (iii) ) (i) d (ii) ) (ii) d (iv) 4) (iii) d (iv) 7. Which of the followig sttemets re ot true? i) { x : x + <, xn } = {,,, 4 } ii) { x : x <, xn } = {,,, 4, } ) Oly (ii) ) Both (i) & (ii) ) Oly (i) 4) oe of these 8. Which of the followig empty set? ) { x : x R d x = } ) { x : x R d x + = } ) { x : x R d x 9 = } 4) { x : x R d x = x + } 9. If Q = x : x =, where x, y N y, the ) Q ) Q ) Q 4) Q. Which of the followig sets re fiite. i) Prime umbers set ii) Hum beigs set i the world iii) A rel umber set betwee d. iv) The set of multiples of. ) Oly (ii) ) Oly (iii) ) (i) d (ii) 4) (iii) d (iv).which of the followig ifiite set. i) Nturl umber set less th oe millio. ii) The set which c be writte by usig the digit, repeted y umber of times. iii) The set of elephts i the world. iv) Spheres set pssig through give poit. ) Oly (ii) ) (ii) d (iv) ) Oly (iii) 4) (i) d (iv) DAY-4 : SYNOPSIS Crdil umber of set : The umber of dtrict elemets cotied i fiite set clled its crdil umber d deoted by (A). Exmples : If A = {,,, 4, } the (A) =, if B = {,, } the (B) =. Subsets: Let A, B be two sets such tht every member of A member of B, the A clled subset of B, it writte s A B. Thus, A B iff (red s if d oly if ) xa xb. NARAYANA GROUP OF SCHOOLS 7

28 If (red s there exts ) tlest oe elemet i A which ot member of B, the A ot subset of B d we write it s A B. For exmple: i) Let A = {,, } d B = {,,, 7, }, the A B. Note tht B A ii) The set of ll eve turl umbers subset of the set of turl umbers. Some properties of subsets : i) The ull set subset of every set. Let A be y set. A, s there o elemet i which ot i A. ii) Every set subset of itself. Let A be y set. x A x A A A. iii) If A B d B C, the A C. Let x A. x B A B x C B C A C. iv) A = B iff A x A x B A B d B A. Let A = B. A B B. Similrly, x B x A A B B A. Coversely, let A x A x B x B x A B d B A. A B B A d A = B Note:-. Two sets A d B re equl iff A B d B A.. Sice every elemet of set A belogs to A, it follows tht every set subset of itself. Proper subset: Let A be subset of B. We sy tht A proper subset of B if A B i.e., if there exts tlest oe elemet i B which does ot belog to A. A subset, which ot proper, clled improper subset. Observe tht every set improper subset of itself. If set A o-empty, the the ull set proper subset of A. Ex : If A = {,, }, the proper subsets of A re, { },{ },{ }, {, }, {, }, {, } Ex : A = {,,, 4, }, B = {,, 4 } Every elemet of B i.e.,, d 4 lso elemet of A. B A Further we ote tht there re two more elemets tht re i A d ot i B. They re d. The A B. I such circumstces we sy tht B proper subset of A. Ex : N W Z Q R Remrk : If A B the every elemet of A i B d there chce tht A my be equl to B i.e., every elemet of B A, but if A B, the every elemet of A i B d there o chce tht A my be equl to B i.e., there will ext t lest oe elemet i B which ot i A. A B A B, A B i.e., A B, B A. Remrk : If A B, we my hve B A,. but if A B, we cot hve B A Power Set: The set formed by ll the subsets of give set A clled the power set of A, it usully deoted by P(A). For exmple: i) Let A = {}, the P(A) = {, {}}. Note tht (P(A)) = =. ii) Let A = {, b}, the P(A) = {, {}, {b}, {, b}}. Note tht (P(A)) = 4 = iii) Let A = {,, }, the P(A) = {, {}, {}, {}, {, }, {, }, {, }, A}. Note tht O(P (A)) = 8 =. I ll these exmples, we hve observed tht (P(A)) = (A). Rule to write dow the power set of fiite set A: First of ll write. Next, write dow sigleto subsets ech cotiig oly oe elemet of A. I the ext step write ll the subsets which coti two elemets from the set A. Cotiue th wy d i the ed write A itself s A lso subset of A. Eclose ll these subsets i brces to get the power set of A. NARAYANA GROUP OF SCHOOLS 8

29 Comprble Sets : Two sets A d B re sid to be comprble iff either A B or B A. For exmple : i) The sets A = {, } d B = {,, 4, } re comprble s A B. ii) The sets A = {,, } d B = {, } re comprble s B A. iii) The sets A = {, } d B = {x : x = } re comprble s A B d lso B A. Clerly, equl sets re lwys comprble. However, comprble sets my ot be equl Uiversl Set: I y pplictio of the theory of sets, ll sets uder ivestigtio re regrded s subsets of fixed set. We cll th set the uiversl set, it usully deoted by X or U or. OPERATIONS OF SETS UNION OF SETS The uio of two sets A d B the set of ll those elemets, which re either i A or i B (icludig those which re i both) I symbolic form, uio of two sets A d B deoted s, A B. It red s A uio B. Clerly, x A B x A or x B. Ad, x A B x A d x B. It evidet from defiitio tht A A B; B A B SOLVED EXAMPLES (i) A,e,i,o,u, B,b,c (ii) A,,, B,, Solutio:- (i) We hve, A B,e,i,o,u,b,c A B,b,c,e,i,o,u Here, the commo elemet hs bee tke oly oce, while writig A B. (ii) We hve A B,,,, A B,,, Here, the commo elemets d hve bee tke oly oce, while writig A B. UNION OF THREE OR MORE SETS The uio of for sets A, A..., A defied s the set of ll those elemets which re i A i i for tlest oe vlue of i. The uio of A, A, A,...A deoted A I symbols, we write Ai { x : x A i i i for t lest oe vlue of i, i } SOLVED EXAMPLE :- If A = {,,, 4} B = {, 4,, 6}, C = {, 6, 7, 8} d D = {7, 8, 9, }, fid (i) A B (ii) A B C (iii) B C D Solutio :- (i) We hve, A B,,,4,4,,6 A B,,,4,,6 (ii) We hve, A B C,,,4,4,,6,6,7,8 A B C,,,4,,6,7,8 (iii) We hve, B C D,4,,6,6,7,8 7,8,9, B C D,4,,6,7,8,9, INTERSECTION OF SETS The itersectio of two sets A d B the set of ll those elemets which belog to both A d B. Symboliclly, we write A B = {x : x A d x B } d red s A itersectio B. Let x A B x A d x B d x A B x A or x B It evidet from the defiitio tht A B A, A B B SOLVED EXAMPLES Exmple : (i) If A = {,,7,9,}, B = {7,9,,}, fid A B. i NARAYANA GROUP OF SCHOOLS 9