Kota Chandigarh Ahmedabad

Transcription

1 TARGT : J 0 SCOR J (Advanced) Home Assignment # 0 Kota Chandigah Ahmedabad

2 \\NOD6\_NOD6 ()\DATA\0\IIT-J\TARGT\MATHS\HOM ASSIGNMNT (Q.BANK)\NG\HOM ASSIGNMNT # 0.P65 HOM ASSIGNMNT # 0 STRAIGHT OBJCTIV TYP J-Mathematics. Length of each edge of a egula tetahedon is. The distance between the centoid of two faces is -. (A) If a,b,c,d,e,f ae position vectos of 6 points A, B, C, D, & F espectively such that a + b = 6c + d = e + f = x, then - uuu uuu (A) AB is paallel to CD line AB, CD and F ae concuent B uuu is pependicula to AF 7 x is position vecto of the point dividing CD in : 6. z is a complex numbe satisfying z and wz w = a (whee w is complex cube oot of unity), then - (A) 0 a 0 a 8 a 8 a. The coefficient of x 6 æ ö in the expansion of ç x + + must be - è x (A) Read following statements : (i) if two lines in space ae pependicula to a thid line then they will be paallel (ii) if two planes ae pependicula to a thid plane then they will be paallel (iii) if two lines ae paallel to a plane then they will be paallel (iv) a b is a vecto along the line of intesection of two nonpaallel planes.a+ 5= 0 and 5.b- 7= 0. Identify the tue and false statement (in ode of statement) - (A) TTTT TFFT FFTF FFFT 6. Acute angle between the lines L : x + y = 0 = y + z and L : x + y + z = 0, x + y + z = 0 is (A) cos cos sin sin 7. quation of a line lying in xy plane is x + 5y = 5 its vecto equation is - (A) = j ˆ+l (i ˆ+ 5j) ˆ, whee l Î R = 5i ˆ+l(5iˆ-j) ˆ, whee l Î R =m(5i ˆ j) ˆ, whee m Î R = (- ˆi + j) ˆ +l(5j ˆ-i) ˆ, whee l Î R 8. A vecto a = i ˆ+ j ˆ+ 7kˆ is thee in ight handed ectangula coodinate system. The coodinate system - is otated about z-axis fom positive x to positive y-axis though angle p/, then new components of a will be - (A) (,, 7) (,, 7) (,, 7) (,, 7) 5 FILL TH ANSWR HR. A B C D. A B C D. A B C D. A B C D 5. A B C D 6. A B C D 7. A B C D 8. A B C D

3 J-Mathematics 9. A plane x + y + 5z = has point P which is at minimum distance fom the line joining A(, 0, ) and B(, 5, 7), then distance AP is equal to - (A) none of these 0. Sum of coefficients of tems which contains integal powes of x in the expansion 0 (+ x) is - (A) æ. If Ag z - ö p ç =, then coect statement is - èz-i (A) minimum value of z is 0 minimum value of z is - maximum value of z is maximum value of z is +. log (a + c), log(a + b), log(b + c) ae in A.P. and a, c, b ae in H.P., whee a, b, c > 0. If a + b = kc, then the value of k is - (A) 6 8. a,b,c ae thee unit vectos and evey two ae inclined to each othe at an angle cos (/5) If a b = pa + qb + c, whee p, q, ae scalas, then 55q is equal to - (A) If a, a, a... a n ae in A.P. with common diffeence d & (n )d = p, then sin a sin a + sin a sin a sin a (n ) sin a n is equal to - æ d ö sin a - æ d ö -cos a (A) -sin a ç sin è ç a + è 5. If 6 th tem of a G.P. is 96 & n th tem is lying between 500 & 780, whee fist tem is a & ([a] ¹ 0, whee [.] denotes geatest intege function) & common atio is positive intege ( ¹ ), then numbe of tems in the seies ae - (A) In a G.P. consisting of positive numbes the poduct of the fist fou tems is and the second tem is the ecipocal of the fouth tem. Then sum of the G.P. up to infinite tem is - (A) 8 8 ( + ) ( ) 7. The sum of the infinitely deceasing geometic pogession is equal to the geatest value of the function ƒ (x) = x x 76 on the inteval [0, ] ; the fist tem of the pogession is equal to the squae of the common atio. The common atio of the G.P. is - (A) A B C D 0. A B C D. A B C D. A B C D. A B C D. A B C D 5. A B C D 6. A B C D 7. A B C D + \\NOD6\_NOD6 ()\DATA\0\IIT-J\TARGT\MATHS\HOM ASSIGNMNT (Q.BANK)\NG\HOM ASSIGNMNT # 0.P65

4 8. If a and b ae vectos such that a =, b =, p a^b= the value of (a c).b is - (A) 6 J-Mathematics and c satisfies (a+ b) + c= b c, then x- y- z- x- y- z- 9. Two lines whose equations ae L: = = and L : = = lie in the same l plane. If L intesects a plane x + y + z = 5 at P, then distance of P fom (,, ) is - (A) 6 0. If the shotest distance between the line = (i ˆ + j ˆ + k) ˆ +l ˆ ˆ ˆ (i + j+ k) and = (i ˆ+ j ˆ+ 5k) ˆ +l (i ˆ+ j ˆ+ 5k) ˆ - is x, then cos (cos 6x) is equal to - (A). If a,b and c ae non-coplana, then 0 [a + b b + c c + a] is equal to - [a b c] (A) If 't' is the peiod of ƒ (x) satisfying ƒ(x + 5) + ƒ (x) = 0, " x Î R and th tem in the expansion of æ x ö ç - è 5 t has the geatest numeical value, then x belongs to - \\NOD6\_NOD6 ()\DATA\0\IIT-J\TARGT\MATHS\HOM ASSIGNMNT (Q.BANK)\NG\HOM ASSIGNMNT # 0.P65 (A) é5 0ù ê, 6 7 ú ë û é ê ë 0 5, 7 ù ú û é ê ë 5,9 ù ú û é5 5ù ê, 6 ú ë û æp ö æp ö. If z - + i = z sin ç + ag z + cosç -ag z è è, whee z = + i, then locus of z is - (A) a pai of staight lines cicle paabola ellipse. Vectos a,b and c with magnitude, & espectively ae coplana. A unit vecto d is pependicula to all of them. If (a b) (c d) = ˆ i ˆ j k - + ˆ and the angle between a and b is 0, then ˆ ˆ c.i + c.j + c.kˆ 6 is equal to - (A) If the value of éa + b + c b + c + a c + a + bù = k éa b cù ë û ë û, then k equals - (A) A B C D 9. A B C D 0. A B C D. A B C D. A B C D. A B C D. A B C D 5. A B C D

5 J-Mathematics 6. If the coefficient of x in the expansion of æ x - è ax ö ç 0 is 5, then the value of 'a' is equal to - (A) 5 7. A plane x + y z = has point P which is at minimum distance fom line joining A(, 0, ) and B(,, ), then (AP) is equal to - (A) ö 8. The tem independent of x in the poduct (+ x+ 7x ) ç x- is - è x (A) 7. C 6 6. C 6 5. C Let a = and b =. If a +l b and a -l b ae mutually pependicula vectos, then l is - æ (A) 9 0. If angle between line = ˆi + k ˆ +l(j ˆ-k) ˆ and xy-plane is a and angle between the planes x + y = 0 and x + y = 0 is b, then cos sin a b is equal to - (A) is - 7. Numbe of ational tems in ( ) (A) 5. If T m & T n denotes the m th & n th tems of an A.P. espectively, such that T = m,tn n = m, then which of the following is necessaily a oot of the equation (a b)x + (b + 5a)x + (b 6a) = 0 - (A) T mn T m+n T m + T n T m. T n. Cente of the tetahedon OABC with O being oigin and A(a,, ), B(, b, 6), C(,, c) is (,, ). The value of a b + c is - (A) If the sum to n tems of a seies is given by n(n + )(n + ) 6, then the n th tem of the seies is - (A) å n ( å n) å n n n 5. If a+ b+ 6c= 0, then (a b) -(b c) + (c a) is equal to - (A) (b c) (b c) 5(b c) 6(b c) 6. A B C D 7. A B C D 8. A B C D 9. A B C D 0. A B C D. A B C D. A B C D. A B C D. A B C D 5. A B C D +å \\NOD6\_NOD6 ()\DATA\0\IIT-J\TARGT\MATHS\HOM ASSIGNMNT (Q.BANK)\NG\HOM ASSIGNMNT # 0.P65

6 \\NOD6\_NOD6 ()\DATA\0\IIT-J\TARGT\MATHS\HOM ASSIGNMNT (Q.BANK)\NG\HOM ASSIGNMNT # 0.P65 5 J-Mathematics 6. The line L : x - y + z - = = makes an isosceles tiangle with the planes x + y + z = and k x y + z = 5, then the point of intesection of L with plane x y + z + = 0 is at a distance 'p' fom the oigin, whee 'p' is equal to - (A) 7. If a, b, c ae thee distinct numbes in A.P., then e a + e b, e b, e b + e c ae in - (A) A.P. G.P. H.P. none of these 8. The numbe of solution(s) of the fom (x, y, z) satisfying x 8 + y 8 + z 8 = 8xyz 5 is (ae) - (A) 8 9. If a,b and c ae thee unit vectos, no two of which ae collinea, a + b is collinea with c and b + c collinea with a, then a+ b+ 8c is equal to - (A) 0 0. The value of æ n n k lim åå - C n. C k. n ç = k = 5 is equal to - è ö (A). a $ and c $ ae unit vectos and b =. If the angle between a $ and c $ is cos (/), then b- c$ =la$, whee l can be equal to (A) /. If a + b + c = 6 whee a, b, c Î R + then the maximum value of a b c is - (A) a, b, c ae thee distinct positive numbes and ab c has the geatest value (A) a =, b =,c= a =,b=,c= 6. If squae oot of, (A) a =,b =,c= 6 a =,b =,c= 6, then - a a a 8a a.(a).(a).(8a)... is 8, then the value of 'a' is If a, b, c ae non-zeo numbes in A.P., then -bc, -ca, - ab ae in - a b c (A) A.P. G.P. H.P. none of these 6. A B C D 7. A B C D 8. A B C D 9. A B C D 0. A B C D. A B C D. A B C D. A B C D. A B C D 5. A B C D

7 J-Mathematics 6. The sum of thee numbes in G.P. is and the sum of thei squaes is 89. If the sum to infinite tems fo the given G.P. is defined, then the common atio is - (A) none of these 7. If a, b, c ae in G.P., x and y be the aithmetic mean between a, b and b, c espectively, then æa c öæb bö ç + ç + is equal to - èx yèx y (A) 6 8. If a & b ae eal numbes such that a + b = 5, then minimum value of a + b is equal to - (A) 5 5 not defined 9. If a, a, a, a, a 5 ae oots of the equation z 5 + z + z + z + z + = 0, then Õ (-ai) is equal to - (A) If b and c ae two non-collinea vectos such that a.(b+ c) = = (x - x+ 6)b + (siny)c, then the point (x, y) always lies on - (A) x = y = y = p x + y = i= and a (b c) 5. The equation of a plane is x y z = 5 and A(,,), B(,, ), C(,, ) and D(,, ) ae fou points. The line segment which does not intesect the plane is - (A) AC AB BC BD 5. If pincipal agument of z satisfying inequalities z- and z-6- i is q, then tanq is equal to - (A) If the imaginay pat of the expession constant such that q ¹ ag(z )) - (A) a staight line paallel to x-axis a cicle of adius and cente (, 0) iq e z- + z- e iq 7 5 is zeo, then the locus of z is (whee q is a a paabola a staight line paallel to y-axis 5. If z satisfies iz = z + z, then ag z is equal to (z is non zeo complex numbe) - (A) p p p p 55. Let z and z be the non eal oots of the equation z + 6z + b = 0. If the oigin and the points epesented by z and z in agand plane foms an equilateal tiangle then value of b is - (A) A B C D 7. A B C D 8. A B C D 9. A B C D 50. A B C D 5. A B C D 5. A B C D 5. A B C D 5. A B C D 55. A B C D \\NOD6\_NOD6 ()\DATA\0\IIT-J\TARGT\MATHS\HOM ASSIGNMNT (Q.BANK)\NG\HOM ASSIGNMNT # 0.P65

8 J-Mathematics 56. If = 0 a a -b c c -b and a, b, c ae not in A.P., then - (A) a, b, c ae in G.P. a, b,cae in A.P. b a,,c ae in H.P. a, b, c ae in H.P The sum to n tems of the seies is n 6n 9n (A) n n+ n+ n+ n+ 58. The maximum value of the sum of the A.P. 50, 8, 6,,... is - (A) The numbe of common tems to the two sequences 7,, 5,...7 and 6,, 6,..., 66 is - (A) The value of s n n s åå Cs C is - = 0 s= s (A) n n + n ( n ) 6. If the numeically geatest tem in the expansion of ( 5x), whee x = 5 is 85l, then l is equal \\NOD6\_NOD6 ()\DATA\0\IIT-J\TARGT\MATHS\HOM ASSIGNMNT (Q.BANK)\NG\HOM ASSIGNMNT # 0.P65 to - (A) 8 79 none of these 6. If A( + i), B( + i) and C(z) ae the vetices of a DABC in which ÐBAC = p and AC = AB. Then z is - (A) + i + i ( + i) (+ i) + i( + i) (+ i) + i (+ i) ( + i) + i (+ i) 6. If the shaded potion epesents the set of complex numbes, then which of the following set of complex numbes satisfy the inequality tan (log z ) > tan (log z + ) - (A) Im(z) z-z 6. If -zz Re(z) Im(z) Re(z) = and z ¹, then z is - (A) 56. A B C D 57. A B C D 58. A B C D 59. A B C D 60. A B C D 6. A B C D 6. A B C D 6. A B C D 6. A B C D 7 Im(z) Re(z) Im(z) Re(z)

9 J-Mathematics [ ] - 5 n n+ n+ n+ 65. Agi + { i + i + i + i + i } is - (A) 0 p p p 66. If ( + x) 5 = a 0 + a x + a x + a x + a x + a 5 x 5, then the value of (a 0 a + a ) + (a a + a 5 ) is equal to- (A) If z =, then the points epesenting the complex numbes i + - 5i z 8 lies on the cicle - (A) whose cente is (0, 0) and adius = whose cente is (0, ) and adius = whose cente is (, 0) and adius = whose cente is (0, ) and adius = MULTIPL OBJCTIV TYP 68. If OABC is a tetahedon (whee O is oigin and points A, B, C lies on x, y, z axes espectively). Let A' is point on OA(a), B' is point on OB(b) & C' is point on OC(c) such that A'B'C' plane is paallel to ABC and volume of tetahedon OA'B'C' is half the volume of tetahedon OABC, then - (A) equation of plane though thee points A', B', C' is x y z + + = a b c equation of plane though thee points A', B', C' is x + y + z = / a b c aea of ( DABC) = aea of ( DA'B'C') aea of ( DABC) = aea of ( DA'B'C') / / 69. z = a + ib and z = c + id ae two complex numbes (a, b, c, d Î R) such that z = z = and Im (z z ) = 0. If w = a + ic and w = b + id, then - æ w ö æ w ö (A) Im(ww ) = 0 Im(ww ) = 0 Imç = 0 Reç = 0 èw èw 70. Fou positive numbes fom a G.P. The poduct of the fist numbe and the fouth one equals the + log0 x geate oot of the equation x = (0.00) - and the sum of the squae of the second and squae of the thid numbe is equal to 50. Then - (A) the fou numbes in G.P. will be ational. the common atio of the G.P. will be ational the fist tem of the G.P. will be ational the poduct of II nd and III d tem of G.P. will be ational 65. A B C D 66. A B C D 67. A B C D 68. A B C D 69. A B C D 70. A B C D \\NOD6\_NOD6 ()\DATA\0\IIT-J\TARGT\MATHS\HOM ASSIGNMNT (Q.BANK)\NG\HOM ASSIGNMNT # 0.P65

10 \\NOD6\_NOD6 ()\DATA\0\IIT-J\TARGT\MATHS\HOM ASSIGNMNT (Q.BANK)\NG\HOM ASSIGNMNT # 0.P65 9 J-Mathematics 7. The line x = y = z meets the plane x + y + z = at the point P and the sphee cented at oigin and adius equal to, at the points R and S, then - (A) PR + PS = 8 PR. PS = 7 PR = PS PR + PS = RS 7. A line passes though two points whose position vectos ae ˆi + ˆj - kˆ and ˆi- j ˆ+ kˆ. The position vecto of a point on it at a unit distance fom the fist point is - (A) (5i ˆ+ ˆj - 7k) ˆ (i ˆ+ 9j ˆ- k) ˆ (6i ˆ+ ˆj - 7k) ˆ (5i ˆ+ 9j ˆ-k) ˆ If ( + x + x + x ) 00 = a 0 + a x + a x a 00 x 00, then - (A) a 0 + a + a + a a 00 is divisible by 0 a 0 + a + a a 00 = a + a a 99 coefficients equidistant fom beginning and end ae equal a = Let P º x + y + z = 6 & P º x + y + z = be the two paallel planes. Let thee points A, B & C ae such that thei x, y & z co-odinates can take only positive integal values, then - (A) if point A & oigin lies on same side of plane P, then A can take 0 diffeent locations in space. if point B & oigin lies on same side of plane P, then B can take 0 diffeent locations in space. if point B & oigin lies on same side of plane P, then B can take0 diffeent locations in space. if point C lies between planes P & P, then it can take 00 diffeent locations in space. 75. The coect statement(s) is/ae - (A) The line of intesection of planes.n = q,.n = q n.n n.n = n.n n.n. if ( )( ) ( )( ) and.n = q,.n = q ae pependicula If thee distinct planes.n = q,.n = q,.n = q intesect in a line which is contained by the plane.n = q, then [ n n n ] n = [ n n n ] n If fou distinct planes.n = q,.n = q,.n = q and.n = q intesect in a line, then n n n n = n n n n [ ] [ ] If a plane contains line of intesection of planes.n = q,.n = q and is paallel to line of intesection of planes.n = q,.n = q then [ n n n ] n = [ n n n ] n. 76. If a, a, a, a, a 5 ae distinct positive tems in AP having common diffeence d, then - (A) 5a > d sum of all tems = 5a a + 5a 5, a, a + a ae in A.P. a a 5 < a a 7. A B C D 7. A B C D 7. A B C D 7. A B C D 75. A B C D 76. A B C D

11 J-Mathematics 77. The equation of plane passing though A(,, ) and paallel to vectos n ˆ ˆ ˆ = i -j-k n ˆ ˆ ˆ = i+ j+ k ae (whee a denotes the position vecto of A) - (A).(i ˆ- 9j ˆ+ 5k) ˆ = x- 9y+ 5z+ = 0 = (i ˆ+ j ˆ+ k) ˆ +l(i ˆ-j ˆ- k) ˆ + µ(iˆ+ ˆj + k) ˆ [nn] = [ann] and 78. Points that lie on the lines bisecting the angle between the lines x - y - z - = = 6 and 6 x- y- z-6 = = ae - 6 (A) (7,, ) (0,, ) (, 0, 0) (, 6, ) 79. The value fo which 0 5 C C C C C 0 C is maximum is/ae - (A) 80. If, w, w,... w n ae the n, n th oots of unity, then ( w)( w )... ( w n ) equal to - (A) n n C 0 + n C + n C n C n n+ n+ n+ 0 n C + C +... C - n (n + ) 8. In the expansion of (x + y + z) 9 - (A) evey tem is of the fom 9 C C k x 9 y k z k coefficient of x y 7 z is 0 the numbe of tems is 55 coefficient of x y z is 60 æ p ö 8. Let z = expç expi è, then - (A) l æ ö n(re z) = + n ç cos l è l æ ö n(im z) = + n ç sin l è 0 æ ö ln(re z) = + ln ç cos è æ ö ln(im z) = -ln ç sin è 8. If the complex numbes z, z, z & z taken in that ode ae the vetices of a hombus, then - (A) z + z = z + z z z = z z z z - z - z is puely imaginay z z = z z 77. A B C D 78. A B C D 79. A B C D 80. A B C D 8. A B C D 8. A B C D 8. A B C D \\NOD6\_NOD6 ()\DATA\0\IIT-J\TARGT\MATHS\HOM ASSIGNMNT (Q.BANK)\NG\HOM ASSIGNMNT # 0.P65

12 8. Fo the equation ( + i)z + ( i) z 6 = 0, which of the following is tue - J-Mathematics (A) The complex numbe with minimum modulus satisfying the given equation is 6-9 i. - The agument of any complex numbe satisfying the given equation is tan. Any z satisfying the given equation also satisfies z = ( + i) + l( + i) fo suitable l Î R. If z & z ae the two points satisfying given equation such that Re(z ) = 0 & Im(z ) = 0, then z z =. 85. Let w be a non-eal cube oot of unity & z = å ( -)( -w)( -w ), then - p (A) Ag(z) = 0 z = 6 7 z = 6-8 Ag( z) = 86. If z is the complex numbe satisfying z i 0 and a = sin (sin( z max ), z max z min cos æ - æ cos + ö b= ö ç ç - è è, then - (A) a b = p a b = -p sin a + cos b = sin a + cos b = 87. The sides of a tiangle ABC (ight angled at B) ae equimultiples of thee consecutive natual numbes 8 = such that peimete of the tiangle is units. If z, z & z epesent affixes of vetices A, B & C espectively and B(z ) is given by 6 + i.0 and AB coincide with eal axis, then - (A) inadius of the D is z max = cicumcicle of the D is z + z - = z min = 6 z 0 \\NOD6\_NOD6 ()\DATA\0\IIT-J\TARGT\MATHS\HOM ASSIGNMNT (Q.BANK)\NG\HOM ASSIGNMNT # 0.P Let x = ˆi + j ˆ-kˆ, y = iˆ+ j ˆ+ kˆ and a vecto z satisfying x z= x y and z.x = 0 geate than, (whee [.] denotes geatest intege function) (A) 89. If w is the imaginay cube oot of unity such that. Then [ z ] æ n n æ P- öö æ æ P-öö ç. ( w ) -55w = ç. w -55w ç ç è è å å å å, è = p= è = p= then n is equal to - (A) If z, z, z ae the vetices of an equilateal tiangle with centoid at oigin & length of cicum adius of tiangle is unit. Then which of the following may be the vetices of an equilateal tiangle - (A) z, z, z (z + z ), (z + z ), (z + z ) (z z),, + (z + z ) (z + z ) z z z,, z z z 8. A B C D 85. A B C D 86. A B C D 87. A B C D 88. A B C D 89. A B C D 90. A B C D is

13 J-Mathematics COMPRHNSION Paagaph fo Question 9 and 9 Let ( + x + x ) 0 = a 0 + a x + a x a 0 x 0. Futhe S = å a. Two coefficients ae chosen fom the coefficients a 0, a,..., a 0 and the pobability that they ae equal is p (consideing no thee coefficients ae equal). Units digit of S is equal to b and a = - p. 0p On the basis of above infomation, answe the following : 9. If ( a- b + b) 6 = I + F, whee 0 F < and I Î N, then the value of I is - (A) The coefficient a is equal to - 0 = 0 (A).0! 8!.0! 8! 0! 7!! 0! 8! Conside a plane having equation.n= d B( b ) ae lying on same side w..t. the plane. Paagaph fo Question 9 to 95 On the basis of above infomation, answe the following : (whee n should not be unit vecto) & two points A( a ) & 9. If foot of pependiculas fom A & B to the plane ae A' & B' espectively then distance A'B' is equal to (A) (b - a).n (b - a).n n 9. Reflection of the A( a ) w..t. the plane has the position vecto - (A) a + (d -a.n)n n (d - a.n) a+ n n (b- a) n (b- a) n n a + (d + a.n)n n none of these 95. If a plane is dawn fom the point a paallel to.n = d & anothe plane is dawn fom the point b paallel to.n = d & the distance between two planes is d then (A'B') + d is equal to (A'B' is mentioned in Q.) (A) (b- a) n b- a n (b - a) n 9. A B C D 9. A B C D 9. A B C D 9. A B C D 95. A B C D \\NOD6\_NOD6 ()\DATA\0\IIT-J\TARGT\MATHS\HOM ASSIGNMNT (Q.BANK)\NG\HOM ASSIGNMNT # 0.P65

14 J-Mathematics Paagaph fo Question 96 to 98 Let ƒ (x) = ae x + be x + cx satisfies the condition ƒ(0) =, ƒ '(ln) = and ln ò 0 9 (ƒ (x) - cx)dx =. Image of a line x - y - z - = = in the plane (a )x + (b + a)y + z + c = 0 be a line L = 0 whose 5 diection atios ae, p, q espectively. Let l = ai ˆ+ bj ˆ+ ckˆ and m = (p - )iˆ+ j ˆ+ (q + )kˆ. On the basis of above infomation, answe the following : 96. a + b + c is equal to - (A) p + q is equal to - (A) Component of vecto l which is pependicula to vecto m, is - 0iˆ+ 0j ˆ+ 0kˆ (A) 0i ˆ- j ˆ+ kˆ 5i ˆ- 6j ˆ+ kˆ 5i ˆ- 6j ˆ+ 7kˆ 9 9 Paagaph fo Question 99 to 0 The base of a pyamid is ectangula, thee of its vetices of the base ae A(,, ), B(,, ) and æ 6 0 ö C(,, ) (may o may not be in ode). Its vetex at the top is Pç, -, - and fouth vetex of è the base is D. On the basis of above infomation, answe the following : 99. Co-odinates of D ae - (A) (, 0, ) (,, 0) (, 0, ) (0,, ) 00. Co-odinates of foot of the nomal dawn fom P on the base of the pyamid ae - \\NOD6\_NOD6 ()\DATA\0\IIT-J\TARGT\MATHS\HOM ASSIGNMNT (Q.BANK)\NG\HOM ASSIGNMNT # 0.P65 æ ö (A),, æ ö æ ö æ ö ç,, è ç ç,, ç,, è è è 0. Volume of the pyamid is - (A) 0 cubic units 0 cubic units 0 cubic units 0 cubic units Paagaph fo Question 0 to 0 F' Thee foces ƒ,ƒ &ƒ of magnitude, and 6 units espectively act G F along thee face diagonals of a cube as shown in figue. Let P be a ƒ D' paallelopiped whose thee co-teminus edges be thee vectos D ƒ,ƒ &ƒ. Let the joning of mid-points of each pai of opposite edges ƒ O C ƒ of paallelopiped P meet in point X. On the basis of above infomation, answe the following questions : 0. The magnitude of the esultant of the thee foces is - (A) none of these 96. A B C D 97. A B C D 98. A B C D 99. A B C D 00. A B C D 0. A B C D 0. A B C D A B B'

15 J-Mathematics 0. The volume of the paallelopiped P is - (A) l(ox) is equal to - (A) Paagaph fo Question 05 to 08 Roots of z n = 0 ae, a, a, a,... a n p p whee a = cos + isin n n so z n = (z )(z a)... (z a n ) On the basis of above infomation, answe the following questions : 05. If n = 0, then + a 0 + a 0 + a a 90 is equal to - (A) none of these 06. n- p ål n sin is equal to (n Î I + ) n = (A) ln(n) -(n -) l n ln(n) + (n -) l n ln(n) -(n + ) l n ln(n) + (n + ) ln 07. If n = 7, then the equation whose oots ae a + a + a and a + a 5 + a 6 is - (A) x x + = 0 x + x = 0 x x = 0 x + x + = If n = 5, then ( + a)( + a )( + a )( + a ) is equal to - (A) zeo Paagaph fo Question 09 to Conside a cicle cented at oigin O and passing though points A(z ), B(z ) and C(z ). The tangents to the cicle at A, B and C intesect at D, and F; D,, F ae opposite to A, B, C espectively. æp ö Also ÐAOB = ÐBOC = q, whee qîç, p è ö 7sec x + 6ç tan x sec x - =- tan x( + sin x) è æ satisfying the equation On the basis of above infomation, answe the following : 09. Which of the following is not the equation of tangent to the cicle at A (z epesents any vaiable point on the tangent at A) - (A) z= z (+ i l), lî R0 z-z z-z + = 0 z z æz-z ö p ag ç =± è z z-z z-z - = 0 z z 0. The complex epesentation of intesection point F is given by - (A) z + z zz zz z + z 0. A B C D 0. A B C D 05. A B C D 06. A B C D 07. A B C D 08. A B C D 09. A B C D 0. A B C D z + z zz \\NOD6\_NOD6 ()\DATA\0\IIT-J\TARGT\MATHS\HOM ASSIGNMNT (Q.BANK)\NG\HOM ASSIGNMNT # 0.P65

16 J-Mathematics. z, z, z ae in G.P. with common atio, then [ iln ] is equal to (whee [.] denotes geatest intege function) - (A) 0 Paagaph fo Question to Two vectos a and b having unit modulus and angle between them is q. Now f( q ) = ƒ (x)dx and (a b) ò -(a.b) ƒ satisfies the condition ƒ(y) + ƒ(x) = x + y fo all x, y Î R {0} and h(q) = -f(q) + a b a.b, xy whee b = b. On the basis of above infomation, answe the following questions :. Fundamental peiod of f(q) is - (A) p. The volume of the paallelopiped fomed by a,b which h(q) is minimum is - p p 6p and a b, whee angle between a&b is taken fo (A) 8 \\NOD6\_NOD6 ()\DATA\0\IIT-J\TARGT\MATHS\HOM ASSIGNMNT (Q.BANK)\NG\HOM ASSIGNMNT # 0.P65. If a & b ae non collinea vectos, then numbe of solution of the equation f(q) + f'( q ) = 0 in [0, p], ae - (A) 6 8 Paagaph fo Question 5 to 7 Let the oigin O of the agand plane epesents the home town of a peson. Suppose the man tavels units in noth-east diection to each city A. Then he tavels unit in east diection to each town B. Fom town B he stats to move along a cicula path, in anticlockwise diection with cente O and adius OB, to each to city C such that Ð BOC is ight angle. Finally he etuns home by the shotest path fom city C. Let z, z and z be the complex numbes coesponding to the points A, B and C espectively. On the basis of above infomation answe the following : 5. The image of complex numbe z unde the locus z z = z z - (A) i + i i - + i Let z p & z q be the two complex numbes satisfying the equations z z = & z z = espectively, then maximum value of z p z q is - (A) none of these. A B C D. A B C D. A B C D. A B C D 5. A B C D 6. A B C D 5

17 J-Mathematics 7. The aea enclosed by the path taced by the peson duing his complete jouney is - (A) 5 p - 5 p Paagaph fo Question 8 to 0 5 p - 5 p- Conside an equation z + z 6 + = 0, whose oots ae a, a,...a and q, q,...q ae the aguments of the oots of the equation, whee 0 < q < q <... < q. On the basis of above infomation answe the following : 8. cos è æ ö æ ö ç q i + isin ç qi i= i= å å is equal to - è (A) 0 9. The value of sec(tan (tan(ag(a ))) + 6cos (cos(ag(a ))) is equal to - (A) sec0 0. Sum of eal pats of all oots of the equation whose imaginay pat is negative, is - (A) cos0 cot0 6 cos 0 - sin0 0 MATCH TH COLUMN. Column-I Column-II (A) Let a = ˆi+ ˆj b= i ˆ -k ˆ. If the point of intesection of the lines a = b a (P) 0 & b= a b is 'P', then l (OP) (whee O is the oigin) is If a = ˆi + j ˆ+ k, ˆ b = i ˆ- ˆj + kˆ and c= i ˆ+ j ˆ+ kˆ and a (b c) is equal to (Q) 5 xa + yb + zc, then x + y + z is equal to The numbe of values of x fo which the angle between the vectos (R) 7 ˆ ˆ ˆ 9 a = x i + (x - )j + k & b = (x - )i ˆ+ xj ˆ+ kˆ is obtuse Let P º x y + z = 7 & P º x + y + z =. If P be a point that lies on (S) P, P and XOY plane, Q be the point that lies on P, P and YOZ plane and R be the point that lies on P, P & XOZ plane, then [Aea of tiangle PQR] (whee [.] is geatest intege function) 7. A B C D 8. A B C D 9. A B C D 0. A B C D. (A) \\NOD6\_NOD6 ()\DATA\0\IIT-J\TARGT\MATHS\HOM ASSIGNMNT (Q.BANK)\NG\HOM ASSIGNMNT # 0.P65

18 \\NOD6\_NOD6 ()\DATA\0\IIT-J\TARGT\MATHS\HOM ASSIGNMNT (Q.BANK)\NG\HOM ASSIGNMNT # 0.P65 7 J-Mathematics. Conside thee planes P º x + y + z = P º x y + z = P º ax y + z = 5 The thee planes intesects each othe at point P on XOY plane and at point Q on YOZ plane. O is the oigin. Column-I Column-II (A) The value of a is (P) The length of pojection of PQ on x-axis is (Q) If the co-odinates of point R situated at a minimum distance fom (R) point 'O' on the line PQ ae (a, b, c), then value of 7a + b + c is (S) If the aea of DPOQ is a, then value of a b is b. Column-I Column-II (A) The diection cosines of a line satisfy the elations l(l + m) = n (P) 0 and mn + nl + lm = 0. The value of l fo which the two lines ae (Q) pependicula to each othe, is In the expansion of ( + x) 6, 7 th and 8 th tems ae equal and the (R) value of æ7 ö ç + 6 èx is l, then the value of l / is (S) 6 The numbe of complex numbes satisfying z + i + z i = 8 and z i + z + i = is. If z = x + iy, then on the basis of infomation given in column-i match the answes in column-ii Column-I Column-II (A) z + i z 5i = a epesents a hypebola then numbe of (P) integal values of 'a' ae Aea enclosed by the ays with equations ag (z + ) = ± p (Q) & ag(z ) = ± p is ƒ (z) = (z )i whee ƒ (z) is always eal & 5 ƒ (z) 5, then (R) numbe of common solution(s) of ƒ(z) & 5 z- = is/ae If oots of (z 5) 6 = 6 epesents a egula hexagon and (S) 6. (A) distance between its two paallel sides is l; value of. (A) l is. (A)

19 J-Mathematics 5. Let z, z, z ae vetices of a tiangle Column-I Column-II (A) If (z z ) + (i + ) (z z ) = 0, then tiangle is (P) ight angled æ z-z ö Reç = 0, then tiangle is (Q) obtuse angled èz-z æ z-z ö Reç < 0, then tiangle is (R) isosceles èz-z z z - z - z = i, then tiangle is (S) equilateal 6. Column-I Column-II (A) Let z = + i, z = + i and z be a complex numbe such that (P) 0 z - z lnre(z ) & lnim(z ) ae defined fo which z - z is puely eal, then tan(agz ) is (Q) i Fo a complex numbe z the minimum value of z + z - - is In the following figue, the value of Re(z ) Im(z ) is equal to (R) 5. (A) Im(z) z 6 z 5 z z 0 (5,) z z (, ) If z & z ae two complex numbes such that ag(z + z ) = 0 (S) & Im(z z ) = 0, then z- z is z Re(z) 8 6. (A) \\NOD6\_NOD6 ()\DATA\0\IIT-J\TARGT\MATHS\HOM ASSIGNMNT (Q.BANK)\NG\HOM ASSIGNMNT # 0.P65

20 J-Mathematics 7. Column-I Column-II (A) If the angle between the plane x y + z = and the (P) line x - y - z - = = is q, then the value of cosecq is - If a,b,c ae non coplana and éabcù ë û =, 7 (Q) then [a -b b -c c -a] is Let = (a b)sin x + (b c)cos y + (c a), whee a,b,c ae thee non-coplana vectos. If is pependicula (R) to (a+ b+ c ) and the minimum value of (x + y ) is then 'k' is k p, Locus of complex numbe 'z' satisfying æz i ö p ag ç = è z + - i (S) 5 is the ac of a cicle whose adius is equal to 0 p (p Î N), then 'p' is (T) an odd intege \\NOD6\_NOD6 ()\DATA\0\IIT-J\TARGT\MATHS\HOM ASSIGNMNT (Q.BANK)\NG\HOM ASSIGNMNT # 0.P65 7. (A) INTGR TYP / SUBJCTIV TYP 8. Fo any sequence of eal numbe A = {a, a, a...}, a sequence DA is defined such that DA = {a a, a a, a a,...}. Suppose that all of the tems of the sequence D(DA) ae and that a 9 = a 9 = 0. Find a. 9. Let [x], 6{x} and x be fist thee tems of a G.P. whee x Î R + and [.] and {.} epesent geatest intege and factional pat functions espectively. If thee is anothe G.P. whose tems ae squae of ecipocal of the tems of given G.P. and the sum of infinite tems of this G.P. is S then find the value of 00S. 0. Let ( + ix) 8 = ƒ(x) + ig(x) whee x Î R and i= -. If S be sum of all coefficient of ƒ(x), then the value of log (S) is

21 J-Mathematics. A squae ABCD of side length 50 is folded along diagonal AC so that planes ACB' and ACD ae pependicula to one anothe whee B' is the new position of B. If the shotest distance between AB' and CD is 0 n, then n is x-6 y-7 z- x y+ 9 z-. Let L: = = and L : = =. If L - - & L ae skew lines such that P & Q ae two points neaest to each othe lying on lines L & L espectively. If image of P with espect to plane x + y z = is R(a,b,c), then (a + b + c) equals. The volume of the tetahedon whose vetices ae the points with position vectos ˆi + ˆj + k, ˆ -ˆi - j ˆ+ 7k, ˆ ˆi + j ˆ- 7kˆ and i ˆ- j ˆ+l kˆ is, then the digit at unit place of l is p a. Given ƒ(a,b,c), whee - ƒ(a, b,c) = - + (b - c) + sin ( + ((a - ) + (b - ) )). Also P b denotes the plane though the line y z b + c =, x = 0 and paallel to the line x - z = a c, y = 0. If the plane P cuts the co-odinate axes at A, B, C, then volume of tetahedon OABC, O being the oigin, is é-ù éù 5. Let a ê ú =,b= ê ú ê ú ê ú & êë úû êëúû éù c= ê ú ê ú. If V is the volume of paallelopiped whose thee coteminous êë0 úû edges ae the vectos a+ b,b+ c,c+ a & V is the volume of tetahedon whose coteminous edges ae the vectos a b,b c,c a, then the value of (V + V ) is 6 Given a egula tetahdedon OABC with side length unit. Let D & ae mid points of AB & OC uuu uuu m espectively. If D.AC = (whee m & n ae copime), then (m + n) is n uuu 7. If P, Q, R ae thee points such that PQ = uuu & QR = 0 uuu uuu uuu, then the value of PR.(QR + QP) is \\NOD6\_NOD6 ()\DATA\0\IIT-J\TARGT\MATHS\HOM ASSIGNMNT (Q.BANK)\NG\HOM ASSIGNMNT # 0.P65

22 J-Mathematics 8. The pojection of the line x y - z - = = on a plane P is x y - z - = =. If plane P passes though - (a,, 0), then a is equal to 9. Let ˆ ˆ ˆ ˆ a b+ c= b+ aˆ cˆ, then aˆ + bˆ - c ˆ is equal to (whee â,b ˆ and ĉ ae unit vectos) 0. Two vaiable complex numbes z & z with thei aguments in ( p, p] ae such that z -(6+ i) and z -( + i ) k, whee maximum value of ag(z ) = minimum value of ag(z ), then k is equal to \\NOD6\_NOD6 ()\DATA\0\IIT-J\TARGT\MATHS\HOM ASSIGNMNT (Q.BANK)\NG\HOM ASSIGNMNT # 0.P

23 J-Mathematics ANSWR KY. C. B. B. B 5. D 6. B 7. B 8. D 9. B 0. D. D. D. D. A 5. B 6. A 7. B 8. B 9. D 0. C. D. A. B. D 5. D 6. C 7. D 8. B 9. D 0. C. C. A. B. C 5. C 6. D 7. C 8. C 9. C 0. A. A. C. B. B 5. C 6. A 7. C 8. C 9. A 50. A 5. C 5. B 5. C 5. B 55. A 56. D 57. B 58. C 59. B 60. A 6. C 6. A 6. B 6. A 65. C 66. B 67. D 68. B,C 69. A,B,C,D 70. B,D 7. A,B,D 7. A,D 7. A,B,C,D 7. A,C,D 75. A,B,C,D 76. A,B,D 77. B,C,D 78. A,B,C,D 79. B,C 80. A,B,C 8. A,B,C,D 8. A,C 8. A,B,C 8. A,C,D 85. A,C 86. B,C 87. A,B 88. A,B 89. A,B,C 90. A,B,C,D 9. C 9. B 9. C 9. A 95. C 96. B 97. D 98. D 99. B 00. A 0. A 0. B 0. C 0. A 05. C 06. A 07. D 08. C 09. D 0. C. A. A. B. B 5. D 6. C 7. A 8. C 9. B 0. D. (A) (S), (R), (P), (P). (A) (R); (P); (Q); (S). (A) (Q), (S), (P). (A) (Q), (Q), (P), (S) 5. (A) (S), (P), (Q), (P,R) 6. (A) (Q); (R); (S); (P) 7. (A) (P); (R); (S,T); (Q,T) \\NOD6\_NOD6 ()\DATA\0\IIT-J\TARGT\MATHS\HOM ASSIGNMNT (Q.BANK)\NG\HOM ASSIGNMNT # 0.P65

24 Impotant Notes J-Mathematics \\NOD6\_NOD6 ()\DATA\0\IIT-J\TARGT\MATHS\HOM ASSIGNMNT (Q.BANK)\NG\HOM ASSIGNMNT # 0.P65