PLEASURE TEST SERIES (XI) - 04 By O.P. Gupta (For stuffs on Math, click at theopgupta.com)

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1 wwwtheopguptacom wwwimathematiciacom For all the Math-Gya Buy books by OP Gupta A Compilatio By : OP Gupta ) For more stuffs o Maths, please visit : wwwtheopguptacom Time Allowed : 0 Miutes Max Marks : 00 SECTION A Questio umbers 0 to 0 carry mark each π Q0 Fid the value of si Q0 Write the derivative of x (3 6x ) with respect to x Fid f ( ), if f (x) cos x Q03 Write the real ad imagiary part of the zero complex umber Q0 If z 3 i, fid the value of zz SECTION B Questio umbers 05 to carry marks each Q05 If P(A) 3/5 ad, P(B) /5, fid P(A or B), if A ad B are mutually exclusive evets Q06 (i) Write the coverse of the followig statemet : If a umber is eve, the is eve (ii) Write the cotrapositive of the followig statemet : If you are bor i Idia, the you are a citize of Idia Q07 A committee of 5 is to be formed out of 6 males ad females I how may ways this ca be doe whe icluded females are i majority? Q0 Differetiate x w r t x x Q09 If ta x cot x, prove that ta x cot x, N Fid the value of ta(9π/) Q0 Fid the distace betwee the lies 5x y 3 ad 5x y 3 0 Check if the lies 3x y ad x 3y are perpedicular or ot Q State whether the sets {x : x is a eve iteger} ad {x : x is a odd iteger} are pair of disjoit sets or ot x 3 cos x e e Q Evaluate : Write the value of xπ/ (π x) x3 x 3 SECTION C Questio umbers 3 to 3 carry marks each Q3 Fid the term idepedet of x i the biomial expasio of (a) x 3x (b) x 3 3 x Q Fid the equatio of the circle passig through the poits (, 3) ad (, ) ad whose cetre is o the lie x 3y Fid the equatio of ellipse, with major axis alog the x-axis ad passig through the poits P(, 3) ad Q(, ) PLEASURE TEST SERIES (XI) - 0 By OP Gupta (For stuffs o Math, click at theopguptacom)

2 PTS-0 By OP Gupta (M ) Q5 Let A {x : x N, x 0} Defie a relatio R from A to A by R {(a,b) : a b 0; a,b N} Depict the relatio R usig roster form Write its domai ad rage also Q6 Fid the coordiates of a poit o y-axis which are at a distace of 5 from the poit (3,, 5) Q7 Determie the coordiates of the foot of perpedicular draw from the poit (, 3) from the lie 3x y 6 0 Fid the equatio of the circle which is circumscribed about the whose vertices are (, 3), (5, ) ad (6, ) Q If the sum of terms of a AP is 3 5 ad its m th term is 6, fid the value of m Q9 A girl has 3 library book passes ad books of her iterest are there i the library Of these books she does ot wat to borrow Mathematicia Vol uless Mathematicia Vol is also borrowed I how may ways ca she choose the three books to be borrowed? Q0 Fid the domai of f (x) [x] [x] 3 Fid the domai ad rage of the real valued fuctio f (x) x Q If o o x cos ycos 0 z cos 0, the fid the value of xy yz zx Q Three squares of a chess board are selected at radom, fid the probability of selectig two squares of oe colour ad the other of a differet colour Q3 Usig priciple of iductio, show that N ( )( 3) 3( 3) SECTION D Questio umbers to 9 carry 6 marks each Q The mea ad SD of a group of 00 observatios were foud to be 0 ad 3 respectively O recheckig, it was observed that three etries were icorrect, which were recorded as, ad, Fid the mea ad SD if the icorrect etries were omitted Q5 I a survey of 60 people, it was foud that 5 people read ewspaper H, 6 read ewspaper T ad 6 read ewspaper I, 9 read both Had I, read H ad T, read both T ad I ad 3 read all the three ewspapers Fid the umber of people who read (a) at least oe of the ewspapers (b) exactly oe ewspaper I a tow of 0000 families, it was foud that 0% families buy fruit A, 0% families buy fruit B, 0% families buy fruit C, 5% families buy A ad B, 3% buy B ad C ad, % buy A ad C If % families buy all the three kid of fruits, fid the umber of families which buy (a) fruit A oly (b) oe of A, B ad C Q6 (a) Fid the geeral solutio of the equatio ta 5x ta x 3 7 (b) Evaluate cos cos cos cos o o o o o o o o Prove that si0 si 30 si 50 si 70 Hece evaluate cos0 cos0 cos60 cos0 6 Q7 Let S to terms Differetiate S w r t (si x cos x) Evaluate : x si x Q Solve the system of iequatios graphically : x y, 3x y, x y, x 0, y 0 z Q9 If Im, the show that the locus of the poit represetig z i the Argad plae is a iz straight lie PLEASURE TEST SERIES (XI) - 0 By OP Gupta (For stuffs o Math, click at theopguptacom)

3 PTS-0 By OP Gupta (M ) 3 HINTS & ANSWERS for PTS XI 0 SECTION A Q0 π π π 3 si si π si Q0 Let y x (36x ) 5 y 3x 6x dy 5 O differetiatig w r t x, we get : 5x x dx We have f (x) cos x f (x) cos x si x six f ( ) si 0 Q03 Let z 0 0i Re(z) 0 ad Im(z) 0 Q0 7 SECTION B Q05 3 P(A or B) P(A) P(B) P(A ad B) [As A ad B are mutually exclusive evets so, P(A ad B) = 0 Q06 (i) If a umber is eve, the is eve (ii) If you are ot a citize of Idia, the you are ot bor i Idia Q07 36 x dy Q0 Let y x x dx (x ) Q09 As ta x cot x ta x (ta x ) 0 ta x ta x ta( /) x /, Z Now LHS: ta x cot x ta cot RHS cos si 9π π We have ta ta ta cos cos cos cos 9 ta Q0 Note that the lies 5x y 3 ad 5x y 3 0 are parallel lies as they have same slopes Writig them i slope-itercept form, we have : L : y x ad L : y x c c 65 Sice distace betwee two parallel lies uits m Slope of 3x y ad x 3y are respectively m ad m 3 3 Now m m so, these lies are perpedicular to each other 3 Q Let A {x : x is a eve iteger} ad B {x : x is a odd iteger} PLEASURE TEST SERIES (XI) - 0 By OP Gupta (For stuffs o Math, click at theopguptacom)

4 PTS-0 By OP Gupta (M ) So, A {,,0,,,} ad B {, 3,,,3,} Clearly A B A ad B are disjoit sets π π si x si x cos x () π π π x 0 π x0 π x0 π cos x Q xπ/ (π x) x x x x 3 3 x3 e e e (e ) 3 3 e e x3 x 3 x30 x 3 SECTION C Q3 (a) 7 th term ie, 9 C6 (b) 3 rd term ie, 0 C 6 0 Q6 (0,, 0), (0, 6, 0) Q 7 Q Let x cos k cos (i) k x So, k cos (ii) ad k cos (iii) 3 y 3 z Addig these equatios, we get : k cos k cos k cos x y z 3 3 yz zx zy k cos cos cos k cos cos xyz C C Q 6 C3 SECTION D Q5 (a) 5 (b) 30 Q6 (a) ta 5x cot x ta x 5x x x, Z (b) cos cos cos cos cos cos cos cos cos cos cos cos cos cos cos cos cos cos cos cos cos Q7 Sice S a a 3 3 S 3 3 ( )() ( ) S 3 3 S ds Now differetiatig w r t, we get : 0 d ( ) ( ) (si x cos x) si x x 5 (si x cos x) ( ) (si x cos x) ( ) si xcosx 5 5 xy yz zx 0 PLEASURE TEST SERIES (XI) - 0 By OP Gupta (For stuffs o Math, click at theopguptacom)

5 PTS-0 By OP Gupta (M ) (si x cos x) ( ) (si x cos x) ( ) si xcosx 5( ) 5 (si x cos x) ( ) (si x cos x) ( ) ( ) x iy x iy y ix Q9 Let z x iy Im Im ix i y y ix y ix ( x y) i(y y x x) y y x x Im ( y) x ( y) x y y x x y y x x y 0, which is a straight lie PLEASURE TEST SERIES (XI) - 0 By OP Gupta (For stuffs o Math, click at theopguptacom)