13. (i) Value of 2. (iii) Value of P. P. P... = P. (iv) Value of ( ) [Where n no. of times P repeated].

Transcription

1 NUMER SYSTEM. Method to multiply -digit umber. (i) CD = C / D + C / D = / + 0 / 35 = / 4 / 35 = 645 C = / ( + C) / C = 7 / 7(4 + 6) / 4 6 = 49 / 70 / 4 = 4 9 / 7 0 / 4 = 564 (iii) CC = C / ( + )C / C = = 3 4 / (3 + 5) 4 / 5 4 = / 3 / 0 = / 3 / 0 = 540. Method to multiply 3-digit o. C DEF = D / E + D / F + E + CD / F + CE / CF = 4 / / / / 6 4 = 8 / + 0 / / / 4 = 8 / /4 3 / 3 8 / 4 = If i a series all umber cotais repeatig 7. To fid their sum, we start from the left multiply 7 by,, 3, 4, 5 & 6. Look at the example below =? = 7 / 7 / 7 3 / 7 4 / 7 5 / 7 6 = 7 / 4 / / 8 / 3 5 / 4 = =? To fid the sum of those umber i which oe umber is repeated after decimal, the first write the umber i either icreasig or decreasig order. The -fid the sum by usig the below method = 5 4 / 5 3 / 5 / 5 = 0 / 5 / 0 / 5 = Those umbers whose all digits are 3. (33) = 089 Those umber. i which all digits are umber is 3 two or more tha times repeated, to fid the square of these umber, we repeat ad 8 by ( ) time. Where Number of times 3 repeated. (333) = 0889 (3333) = Those umber whose all digits are 9. (99) = 980 (999) = (9999) = (99999) = Those umber whose all digits are. umber whose oe s, te s, hudred s digit is i.e.,,,,... I this we cout umber of digits. We write,, 3,... i their square the digit i the umber, the write i decreasig order up to. = = 3 = 343

2 S- 0 Shortcuts i Quatitative ptitude 8. Some properties of square ad square root: (i) Complete square of a o. is possible if its last digit is 0,, 4, 5, 6 & 9. If last digit of a o. is, 3, 7, 8 the complete square root of this o. is ot possible. If last digit of a o. is, the last digit of its complete square root is either or 9. (iii) If last digit of a o. is 4, the last digit of its complete square root is either or 8. (iv) If last digit of a o. is 5 or 0, the last digit of its complete square root is either 5 or 0. (v) If last digit of a o. is 6, the last digit of its complete square root is either 4 or 6. (vi) If last digit of a o. is 9, the last digit of its complete square root is either 3 or Prime Number : (i) Fid the approx square root of give o. Divide the give o. by the prime o. less tha approx square root of o. If give o. is ot divisible by ay of these prime o. the the o. is prime otherwise ot. For example : To check 359 is a prime umber or ot. Sol. pprox sq. root = 9 Prime o. < 9 are, 3, 5, 7,, 3, is ot divisible by ay of these prime os. So 359 is a prime o. For example: Is is prime or ot? Þ Remider = 0, + \ is ot prime. There are 5 prime o. from to 50. (iii) There are 5 prime o. from to 00. (iv) There are 68 prime o. from to If a o. is i the form of x + a, the it is divisible by (x + a); if is odd.. If x (x ), the remaider is always.. If x (x + ) (i) If is eve, the remaider is. If is odd, the remaider is x. 3. (i) Value of P + P + P +... = 4P+ + Value of P - P - P -... = 4P+ - (iii) Value of P. P. P.... = P (iv) Value of ( ) P P P P P = P - [Where o. of times P repeated]. Note: If factors of P are & ( + ) type the value of P + P + P +... = ( + ) ad P - P - P -... =. 4. Number of divisors : (i) If N is ay o. ad N = a b m c p... where a, b, c are prime o. No. of divisors of N = ( + ) (m + ) (p + )... e.g. Fid the o. of divisors of N = = = 3 5 ( 5) = So, the o. of divisors = (4 + ) ( + ) (4 + ) = 75 N = a b m c p, where a, b, c are prime The set of co-prime factors of N = [( + ) (m + ) (p + ) + m + mp + p + 3mp] (iii) If N = a b m c p..., where a, b & c are prime o. The sum of the divisors = ( a + m p )( b + )( c ) ( a-)( b-)( c-)

3 0 Shortcuts i Quatitative ptitude S-3 5. To fid the last digit or digit at the uit s place of a. (i) If the last digit or digit at the uit s place of a is, 5 or 6, whatever be the value of, it will have the same digit at uit s place, i.e., (...) = (...) (...5) = (...5) (...6) = (...6) If the last digit or digit at the uits place of a is, 3, 5, 7 or 8, the the last digit of a depeds upo the value of ad follows a repeatig patter i terms of 4 as give below : last digit of (...) last digit of (...3) last digit of (...7) last digit of (...8) 4x x x x 6 6 (iii) If the last digit or digit at the uit s place of a is either 4 or 9, the the last digit of a depeds upo the value of ad follows repeatig patter i terms of as give below. last digit of (...4) last digit of (...9) x 6 x (i) Sum of atural umber = () ( + ) Sum of eve umber = () ( + ) (iii) Sum of odd umber = 7. (i) Sum of sq. of first atural o. = ( + )( + ) ( 4 -) Sum of sq. of first odd atural o. = 6 3 (iii) Sum of sq. of first eve atural o. = ( + )( + ) 3 8. (i) Sum of cube of first atural o. = ( + ) é ( + ) ù = ê ú 4 ë Sum of cube of first eve atural o. = ( + ) (iii) Sum of cube of first odd atural o. = ( ) 9. (i) x y is divisible by (x + y) Whe is eve x y is divisible by (x y) Whe is either odd or eve. 0. For ay iteger, 3 is divisible by 3, 5 is divisible by 5, is divisible by, 3 is divisible by 3.. Some articles related to Divisibility : (i) o. of 3-digits which is formed by repeatig a digit 3-times, the this o. is divisible by 3 ad 37. e.g.,,, 333,... o. of 6-digit which is formed by repeatig a digit 6-times the this o. is divisible by 3, 7,, 3 ad 37. e.g.,,, , ,...

4 S-4 0 Shortcuts i Quatitative ptitude. Divisible by 7 : We use osculator ( ) for divisibility test : = : = : 98 0 = 98 Now 98 is divisible by 7, so is also divisible by Divisible by : I a umber, if differece of sum of digit at eve places ad sum of digit at odd places is either 0 or multiple of, the o. is divisible by. For example, 34 Sum of eve place digit = + 4 = 6 Sum of odd place digit = = 6 Differece = 6 6 = 0 \ 34 is divisible by. 4. Divisible by 3 : We use (+ 4) as osculator. e.g., : = = = = = 3 3 is divisible by 3. \ is also divisible by Divisible by 7 : We use ( 5) as osculator. e.g., 94678: = : = : = 55 55: = 0 \ is completely divisible by Divisible by 9 : We use (+ ) as osculator. e.g: 4964: = = = = = 9 9 is divisible by 9 \ 4964 is divisible by HCF (Highest Commo factor) There are two methods to fid the HCF (a) Factor method (b) Divisio method (i) For two o. a ad b if a < b, the HCF of a ad b is always less tha or equal to a. The greatest umber by which x, y ad z completely divisible is the HCF of x, y ad z. (iii) The greatest umber by which x, y, z divisible ad gives the remaider a, b ad c is the HCF of (x a), (y b) ad (z c). (iv) The greatest umber by which x, y ad z divisible ad gives same remaider i each case, that umber is HCF of (x y), (y z) ad (z x). a c (v) H.C.F. of, b d ad e H.C.M. of (a, c, e) f = L.C.M. of (b, d, f) 8. LCM (Least Commo Multiple) There are two methods to fid the LCM (a) Factor method (b) Divisio method (i) For two umbers a ad b if a < b, the L.C.M. of a ad b is more tha or equal to b. If ratio betwee two umbers is a : b ad their H.C.F. is x, the their L.C.M. = abx.

5 0 Shortcuts i Quatitative ptitude S-5 (iii) x If ratio betwee two umbers is a : b ad their L.C.M. is x, the their H.C.F. = ab (iv) The smallest umber which is divisible by x, y ad z is L.C.M. of x, y ad z. (v) The smallest umber which is divided by x, y ad z give remaider a, b ad c, but (x a) = (y b) = (z c) = k, the umber is (L.C.M. of (x, y ad z) k). (vi) The smallest umber which is divided by x, y ad z give remaider k i each case, the umber is (L.C.M. of x, y ad z) + k. (vii) a c L.C.M. of, b d ad e L.C.M. of (a, c, e) f = H.C.F. of (b, d, f) (viii) For two umbers a ad b LCM HCF = a b (ix) If a is the H.C.F. of each pair from umbers ad L is L.C.M., the product of umbers = a.l LGER 9. lgebra Idetities: (i) (a + b) + (a b) = (a + b ) (a + b) (a b) = 4ab (iii) a 3 + b 3 = (a + b) (a ab + b ) (iv) a 3 b 3 = (a b) (a + ab + b ) (v) a 4 + a + = (a + a + ) (a a + ) (vi) If a + b + c = 0, the a 3 + b 3 + c 3 =3abc (vii) = 4 (viii) ( a+ b) -( a-b) ab b e h k æ b e h kö = ç c f i l èc f i l ø If a + b + c = abc, the (ix) a d g j ( a d g j) (x) ( a+ b) + ( a-b) a + b = æ a ö æ b ö æ c ö ç + ç + ç è-a ø è-b ø è-c ø = æ a ö æ b ö æ c ö ç.. è ç ç -a ø è-b ø è-c ø ad æ è-3a ø è-3b ø è-3c ø = 3a -a öæ 3b -b öæ3c -c ö ç.. ç - ç è-3a øè -3b øè-3c ø æ a -a ö æ3b -b ö æ3c -c ö + + ç ç ç 30. If a x + b y = c ad a x + b y = c, the (i) If a b a b a b c ¹, oe solutio. If = =, Ifiite may solutios. a b c (iii) If a b c a b c = ¹, No solutio 3. If a ad b are roots of ax + bx + c = 0, the a ad b are roots of cx + bx + a = 0 3. If a ad b are roots of ax + bx + c = 0, the (i) Oe root is zero if c = 0. oth roots zero if b = 0 ad c = 0. (iii) Roots are reciprocal to each other, if c = a. (iv) If both roots a ad b are positive, the sig of a ad b are opposite ad sig of c ad a are same. (v) If both roots a ad b are egative, the sig of a, b ad c are same. a+b =- b c, a ab= a, the ( ) a-b= ( a+b) -4ab ( ) 4 4 a +b = a +b -a b = é ( a + b) - abù - ( ab) ë

6 S-6 0 Shortcuts i Quatitative ptitude 33. rithmetic Progressio: (i) If a, a + d, a + d,... are i.p., the, th term of.p. a = a + ( )d Sum of terms of this.p. = S = éëa + ( -d ) ù = [ l ].M. = a + b [Q.M. = rithmetic mea] 34. Geometric Progressio: (i) G.P. a, ar, ar,... The, th term of G.P. a = ar ( ) a r - S =,r > r- ( ) a(- r ) =,r < (- r) S = - r [where r = commo ratio, a = first term] G.M. = ab 35. If a, b, c are i H.P.,,, are i.p. a b c a + wherel = last term a = first term d = commo differece th term of H.M. = th term of.p. ab H.M. = a + b Note : Relatio betwee.m., G.M. ad H.M. (i).m. H.M. = G.M..M. > G.M. > H.M..M. rithmetic Mea G.M. Geometric Mea H.M. Harmoic Mea 36. (i) verage of first atural o. = + verage of first eve o. = ( + ) (iii) verage of first odd o. = VERGE ( )( ) 37. (i) verage of sum of square of first atural o. = + + verage of sum of square of first eve o. = 6 ( + )( + ) 3 (iii) verage of sum of square of first odd o. = æ 4 -ö ç 3 è ø

7 0 Shortcuts i Quatitative ptitude S-7 ( + ) 38. (i) verage of cube of first atural o. = 4 verage of cube of first eve atural o. = ( + ) (iii) verage of cube of first odd atural o. = ( ) ( + ) 39. m verage of first multiple of m = 40. (i) If average of some observatios is x ad a is added i each observatios, the ew average is (x + a). If average of some observatios is x ad a is subtracted i each observatios, the ew average is (x a). (iii) If average of some observatios is x ad each observatios multiply by a, the ew average is ax. (iv) If average of some observatios is x ad each observatios is divided by a, the ew average is x a. (v) If average of is, & average of is, the verage of ( + ) is + + ad verage of ( ) is Whe a perso is icluded or excluded the group, the age/weight of that perso = No. of persos i group (Icrease / Decrease) i average ± New average. For example : I a class average age of 5 studets is 8 yrs. Whe the age of teacher is icluded their average icreased by yrs, the fid the age of teacher. Sol. ge of teacher = 5 + (8 + ) = = 50 yrs. 4. Whe two or more tha two persos icluded or excluded the group, the average age of icluded or excluded perso is ( ) ( ) No. of perso Icrease / Decrease i average ± New average No. of perso icluded or excluded = No. of icluded or perso For example : verage weight of 3 studets is 44 kg. fter icludig two ew studets their average weight becomes 48 kg, the fid the average weight of two ew studets. Sol. verage weight of two ew studets ( ) = = = = 74 kg 43. If a perso travels two equal distaces at a speed of x km/h ad y km/h, the average speed = xy x+ y km/h 44. If a perso travels three equal distaces at a speed of x km/h, y km/h ad z km/h, the average speed = RTIO & PROPORTION 3xyz km/h. xy + yz + zx 45. (i) If = = =..., the a + b + c +... K+ K + K = c K a b c K K K3 3 For example: If P = Q = R, the fid P + Q + R R Sol. P = 3, Q = 4, R = 7 The P + Q + R = 7 = R 7 If a a a3 a4 a... K a a3 a4 a5 a + = = = = =, the a : a + = (K)

8 S-8 0 Shortcuts i Quatitative ptitude ad - bc 46. umber added or subtracted from a, b, c & d, so that they are i proportio = ( a+ d) -( b+ c) For example : Whe a umber should be subtracted from, 3, & 5 so that they are i proportio. Fid that umber = = Sol. Req. No. = ( ) ( ) 47. If X part of is equal to Y part of, the : = Y : X. For example: If 0% of = 30% of, the fid :. Sol. : = 30% = 3 = 3 : 0% 48. Whe X th part of P, Y th part of Q ad Z th part of R are equal, the fid : : C. The, : : C = yz : zx : xy TIME, DISTNCE ND WORK 49. ca do a/b part of work i t days ad c/d part of work i t days, the t t = a/b c/d 50. (i) If is K times efficiet tha, The T(K + ) = Kt If is K times efficiet tha ad takes t days less tha Kt The T = K - or t K-, t t = K- = kt 5. (i) If a cister takes X mi to be filled by a pipe but due to a leak, it takes Y extra miutes to be filled, the the time take by leak æ X + XY ö to empty the cister = mi ç Y è ø If a leak empty a cister i X hours. pipe which admits Y litres per hour water ito the cister ad ow cister is emptied æx+ Y+ Zö i Z hours, the capacity of cister is = ç è Z-X ø litres. (iii) If two pipes ad fill a cister i x hours ad y hours. pipe is also a outlet C. If all the three pipes are opeed together, é xyt ù the tak full i T hours. The the time take by C to empty the full tak is = ê ú ëyt + xt -xy 5. (i) If t ad t time take to travel from to ad to, with speed a km/h ad b km/h, the distace from to is ( ) d t t æ ab ö èa+ bø = + ç ( ) d t t æ ab ö èa-bø = - ç æ tt ö d = ( a-b) ç èt-t ø If Ist part of distace is covered at the speed of a i t time ad the secod part is covered at the speed of b i t time, the the average speed = æat + btö ç è t+ t ø

9 0 Shortcuts i Quatitative ptitude S-9 PERCENTGE 53. Simple Fractio Their Percetage 00% % 33.3% 5% 0% 6.67% Simple Fractio Their Percetage.5%.% 0% 9.09% 8.33% 7 4.8% æ a ö 54. (i) If is ç x% = è b ø more tha, the is æ a ö ç % èa+ b ø less tha. æ a ö If is ç x% = è b ø less tha, the is æ a ö ç % more tha èa-b ø if a > b, we take a b if b > a, we take b a. æb-a ö 55. If price of a article icrease from ` a to ` b, the its expeses decrease by ç 00 % so that expediture will be same. è b ø 56. Due to icrease/decrease the price x%, ma purchase a kg more i ` y, the æ xy ö Per kg icrease or decrease = ç è00 a ø xy 00 x a Per kg startig price = ` ( ± )

10 S-0 0 Shortcuts i Quatitative ptitude 57. For two articles, if price: Ist IId Overall Icrease (x%) Icrease (y%) æ xy ö Icrease ç x+ y + % è 00 ø Icrease (x%) Decrease (y%) æ xy ö ç x-y - % è 00 ø If +ve (Icrease) If ve (Decrease) Decrease (x%) Decrease (y%) æ xy ö Decrease ç x+ y - % è 00 ø Icrease (x%) Decrease (x%) æ x ö Decrease % ç 00 è ø 58. If the side of a square or radius of a circle is x% icrease/decrease, the its area icrease/decrease = 59. If the side of a square, x% icrease/decrease the x% its perimeter ad diagoal icrease/decrease. æ x ö x ± % ç 00 è ø t æ00 ± R ö 60. (i) If populatio P icrease/decrease at r% rate, the after t years populatio = Pç è 00 ø If populatio P icrease/decrease r % first year, r % icrease/decrease secod year ad r 3 % icrease/decrease third year, æ r the after 3 years populatio = öæ r öæ r P 3 ö ç ± 00 ç ± ± 00 ç 00 è øè øè ø If icrease we use (+), if decrease we use ( ) 6. If a ma sped x% of this icome o food, y% of remaiig o ret ad z% of remaiig o cloths. If he has ` P remaiig, the P total icome of ma is = ( 00 -x )( 00 -y )( 00 -z ) [Note: We ca use this table for area icrease/decrease i mesuratio for rectagle, triagle ad parallelogram]. 6. If CP of x thigs = SP of y thigs, the éx- y ù Profit/Loss = ê 00 ú % ë y If +ve, Profit; If ve, Loss 63. If after sellig x thigs P/L is equal to SP of y thigs, y x y 00 the P/L = ( ± ) éprofit =-ù ê Loss =+ ú ë PROFIT ND LOSS

11 0 Shortcuts i Quatitative ptitude S- 64. If CP of two articles are same, ad they sold at Ist IId Overall (x%) Profit (y%) Profit æx+ y ö ç % Profit è ø (x%) Profit (y%) Loss æx-y ö ìprofit,if x > y ç % í è ø î Loss,if x < y (x%) Loss (y%) Loss æx+ y ö ç % Loss è ø (x%) Profit (y%) Loss No profit, o loss 65. If SP of two articles are same ad they sold at Ist IId Overall Profit (x%) Loss(x%) æ x ö Loss % ç 00 è ø Profit (x%) Loss (y%) ( ) æ00 x -y -xy ö ç % è 00+ x-y or ø ( )( ) If ve,the Pr ofit% é 00 + x 00 -y ù ì + ê -00 ú% í ë 00 + x - y î If - ve,the Loss% é P+ D ù 66. fter D% discout, requires P% profit, the total icrease i C.P.= ê 00 % 00-D ú ë ( 00 + P ) 67. M.P. = C.P ( 00 - D) M.P. - C.P Profit % = ( ) C.P. é æ00+ r 69. (i) For discout r % ad r %, successive discout öæ00+ röæ00 + r3 ö ù = êç - 00 è 00 ç øè 00 øè ç 00 ø ú ë é æ00+ r For discout r %, r % ad r 3 %, successive discout öæ00+ röæ00 + r3 ö ù = êç - 00 è 00 ç øè 00 øè ç 00 ø ú ë SIMPLE ND COMPOUND INTEREST If P = Pricipal, R = Rate per aum, T = Time i years, SI = Simple iterest, = mout 70. (i) PRT SI = 00 RT P SI P é ù = + = ê + ë 00 ú

12 S- 0 Shortcuts i Quatitative ptitude 7. If P = Pricipal, = mout i years, R = rate of iterest per aum. R P é ù = ê + ë 00 ú, iterest payable aually 7. (i) 73. (i) é R ù = P + 00 ê ë ú R = R/, = R P é ù = ê + ë 400 ú 4, iterest payable half-yearly, iterest payable quarterly; é R ù ê + ë 400 ú is the yearly growth factor; é R ù ê ë 400 ú is the yearly decay factor or depreciatio factor. 74. Whe time is fractio of a year, say 3 4 4, years, the, é 3 ù 4 R é R ù ê mout P 4 ú = ê + ê + ú ë 00 ú ë 00 L F I O HG K J - NM QP R 75. CI = mout Pricipal = P Whe Rates are differet for differet years, say R, R, R 3 % for st, d & 3 rd years respectively, the, é R ùé R ùé R3 ù 77. mout = Pê+ + + ë 00 úê ë 00 úê ë 00 ú I geeral, iterest is cosidered to be SIMPLE uless otherwise stated. 78. (i) Sum of all the exterior agle of a polygo = 360 Each exterior agle of a regular polygo = 360 GEOMETRY (iii) Sum of all the iterior agles of a polygo = ( ) 80 ( - ) (iv) Each iterior agle of a regular polygo = (v) No. of diagoals of a polygo = ( - ) 3 80, o. of sides. (vi) The ratio of sides a polygo to the diagoals of a polygo is : ( 3) (vii) Ratio of iterior agle to exterior agle of a regular polygo is ( ) : 79. Properties of triagle: (i) Whe oe side is exteded i ay directio, a agle is formed with aother side. This is called the exterior agle. There are six exterior agles of a triagle. Iterior agle + correspodig exterior agle = 80.

13 0 Shortcuts i Quatitative ptitude S-3 (iii) exterior agle = Sum of the other two iterior opposite agles. (iv) Sum of the legths of ay two sides is greater tha the legth of third side. (v) Differece of ay two sides is less tha the third side. Side opposite to the greatest agle is greatest ad vice versa. (vi) triagle must have at least two acute agles. (vii) Triagles o equal bases ad betwee the same parallels have equal areas. (viii) If a, b, c deote the sides of a triagle the (i) if c < a + b, Triagle is acute agled. if c = a + b, Triagle is right agled. (iii) if c > a + b, Triagle is obtuse agled. (ix) If triagles are equiagular, their correspodig sides are proportioal. I triagles C ad XYZ, if Ð = ÐX, Ð = ÐY, ÐC = ÐZ, the C C = =.. XY XZ YZ (i) I DC, Ð = 90 D ^ C \ D C = C = + D C D (iii) D = D DC C (x) The perpediculars draw from vertices to opposite sides (called altitudes) meet at a poit called Orthocetre of the triagle. (xi) The lie draw from a vertex of a triagle to the opposite side such that it bisects the side is called the Media of the triagle. media bisects the area of the triagle. (xii) Whe a vertex of a triagle is joied to the midpoit of the opposite side, we get a media. The poit of itersectio of the medias is called the Cetroid of the triagle. The cetroid divides ay media i the ratio :. (xiii) gle isector Theorem I the figure if D is the agle bisector (iterior) of Ð C. The, D C. /C = D/DC.. x C D x DC = D. (xiv) Midpoit Theorem I a triagle, the lie joiig the mid poits of two sides is parallel to the third side ad half of it. (xv) asic Proportioality Theorem lie parallel to ay oe side of a triagle divides the other two sides proportioally. If DE is parallel to C, the D E C D E C D = =, = ad so o. D EC D E DE C

14 S-4 0 Shortcuts i Quatitative ptitude 80. Properties of circle (i) Oly oe circle ca pass through three give poits. There is oe ad oly oe taget to the circle passig through ay poit o the circle. (iii) From ay exterior poit of the circle, two tagets ca be draw o to the circle. (iv) The legths of two tagets segmet from the exterior poit to the circle, are equal. (v) The taget at ay poit of a circle ad the radius through the poit are perpedicular to each other. (vi) Whe two circles touch each other, their cetres & the poit of cotact are colliear. (vii) If two circles touch exterally, distace betwee cetres = sum of radii. (viii) If two circles touch iterally, distace betwee cetres = differece of radii (ix) Circles with same cetre ad differet radii are cocetric circles. (x) Poits lyig o the same circle are called cocyclic poits. (xi) Measure of a arc meas measure of cetral agle. m(mior arc) + m(major arc) = 360. (xii) gle i a semicircle is a right agle. (xiii) Oly oe circle ca pass through three give (xxv) If ON is ^ from the cetre O of a circle to a chord, the N = N. O N (^ from cetre bisects chord) (xv) If N is the midpoit of a chord of a circle with cetre O, the ÐON = 90. (Coverse, ^ from cetre bisects chord) (xvi) Two cogruet figures have equal areas but the coverse eed ot be true. (xvii) diagoal of a parallelogram divides it ito two triagles of equal area. (xviii) Parallelograms o the same base ad betwee the same parallels are equal i area. (xix) Triagles o the same bases ad betwee the same parallels are equal i area. (xx) If a triagle ad a parallelogram are o the same base ad betwee the same parallels, the the area of the triagle is equal to the half of the parallelogram. If PT is a taget to the circle, the OP = PT = OT T P O If PT is taget ad P is secat of a circle, the PT = P.P T O P

15 0 Shortcuts i Quatitative ptitude S-5 If P & PD are two secat of a circle, the P.P = PC.PD P C D If two circles touch exterally, the distace betwee their cetres = (r + r ) r r If two circles touch iterally, the distace betwee their cetres = r r where r > r. MENSURTION 8. (i) rea of triagle = base altitude rea of triagle usig hero s formula = S / S -a (S -b) (S - c), where S = a + b + c 8. I a equilateral triagle with side a, the where rea of triagle 4 4h P a 3 = 3 9 P Perimeter h Height 83. I a isosceles triagle PQR P ar D PQR = - b 4 a a Height = 4a - b Q b R

16 S-6 0 Shortcuts i Quatitative ptitude 84. (i) rea of D = bc SiP where ÐP = ÐQPR P rea of D = ac SiQ c b (iii) rea of D = ab SiR Q a R 85. b + c -a a + c -b CosP =, CosQ =, bc ac a + b -c CosR = ab 86. Sie Rule : a b c = = SiP SiQ SiR 87. Perimeter of square rea of square = 4 Diagoal of square = = side of square 4 Square 88. I a circle with radius r. D = where - rea of circle C 4 C - Circumferece of circle D - Diameter of circle O r 89. If q = 60, ar D O = 3 r 4 If q = 90, ar D O = If q, ar D O = r r æqö æqö siq = r si ç.cosç èø èø r q r 90. (i) circle with largest area iscribed i a right agle triagle, the area of DC r =. Perimeter of DC r C

17 0 Shortcuts i Quatitative ptitude S-7 If C is a equilateral triagle with side a, the rea of circle = pa (iii) If C is a equilateral triagle with side a, the area of circle = r C pa 3. r (iv) If DC is a equilateral triagle, ad two circles with radius r ad R, the r = ad R pr = pr 4 R r C r 3. (v) Three equal circle with radius r ad a equilateral triagle C, the area of shaded regio = ( -p) C 9. CD is a square placed iside a circle with side a ad radius of circle r, the area of square 7 = area of circle a a r a D a C

18 S-8 0 Shortcuts i Quatitative ptitude 9. Diagoal of a cube = 3 side 93. Diagoal of a cuboid = l + b + h ; where l Legth, b breadth, h height 94. For two cubes v 3 a d = = = v a d where, rea of cubes v, v Volume a, a Sides d, d Diagoals 95. Uits of Measuremet of rea ad Volume The iter-relatioships betwee various uits of measuremet of legth, area ad volume are listed below for ready referece: Legth Cetimetre (cm) = 0 milimetre (mm) Decimetre (dm) = 0 cetimetre Metre (m) = 0 dm = 00 cm = 000 mm Decametre (dam) = 0 m = 000 cm Hectometre (hm) = 0 dam = 00 m Kilometre (km) = 000 m = 00 dam = 0 hm Myriametre = 0 kilometre rea cm = cm cm = 0 mm 0 mm = 00 mm dm = dm dm = 0 cm 0 cm = 00 cm m = m m = 0 dm 0 dm = 00 dm dam or are = dam dam = 0 m 0 m = 00 m hm = hectare = hm hm = 00 m 0000m = 00 dm km = km km = 0 hm 0 hm = 00 hm or 00 hectare Volume cm 3 = ml = cm cm cm = 0 mm 0 mm 0 mm = 000 mm 3 litre = 000 ml = 000 cm 3 m 3 = m m m = 00 cm 00 cm 00 cm = 0 6 cm 3 = 000 litre = kilometre dm 3 = 000 cm 3, m 3 = 000 dm 3, km 3 = 0 9 m 3 If a, b, c are the edges of a cuboid, the 96. The logest diagoal = a + b + c (i) If the height of a cuboid is zero it becomes a rectagle. If a be the edge of a cube, the (iii) The logest diagoal = aö3 97. Volume of pyramid = ase rea height ( H) (i) If & deote the areas of two similar figures ad l & l deote their correspodig liear measures, the F l = H G I l K J If V & V deote the volumes of two similar solids ad l, l deote their correspodig liear measures, the V V F 3 l = H G I l K J (iii) The rise or fall of liquid level i a cotaier = Total volume of objects submerged or take out Cross sectioal area of cotaier

19 0 Shortcuts i Quatitative ptitude S If a largest possible cube is iscribed i a sphere of radius a cm, the (i) the edge of the cube = a. 3 If a largest possible sphere is iscribed i a cylider of radius a cm ad height h cm, the for h > a, the radius of the sphere = a ad the radius = h (for a > h) (iii) If a largest possible sphere is iscribed i a coe of radius a cm ad slat height equal to the diameter of the base, the the radius of the sphere = a 3. (iv) If a largest possible coe is iscribed i a cylider of radius a cm ad height h cm, the the radius of the coe = a ad height = h. (v) If a largest possible cube is iscribed i a hemisphere of radius a cm, the the edge of the cube = a 3.

20 S-0 0 Shortcuts i Quatitative ptitude 00. I ay quadrilateral (i) rea = oe diagoal (sum of perpediculars to it from opposite vertices) = d (d + d ) rea of a cyclic quadrilateral = bs - agbs - bgbs - cgbs - dg where a, b, c, d are sides of quadrilateral ad s = semi perimeter = a+ b + c + d 0. If legth, breadth & height of a three dimesioal figure icrease/decrease by x%, y% ad z%, the Chage i area = é æ00 ± x öæ00 ± y ö ù êç ç - 00% ú ëè øè ø é æ00 ± x öæ00 ± y öæ00 ± z ö ù Chage i Volume = êç ç ç - 00% ú ëè øè øè ø