iitjee 2011 practice test 1

1

The TestThe TestThe TestThe Test----1 is a multiple choice examination. In each of the following questions, at least one of the choices is/are correct. 1 is a multiple choice examination. In each of the following questions, at least one of the choices is/are correct. 1 is a multiple choice examination. In each of the following questions, at least one of the choices is/are correct. 1 is a multiple choice examination. In each of the following questions, at least one of the choices is/are correct. Please do tick the correct answer(s). You get Please do tick the correct answer(s). You get Please do tick the correct answer(s). You get Please do tick the correct answer(s). You get sixsixsixsix marks for each correct answer, marks for each correct answer, marks for each correct answer, marks for each correct answer, oneoneoneone mark for each (attempted) unanswered mark for each (attempted) unanswered mark for each (attempted) unanswered mark for each (attempted) unanswered question,question,question,question, and ( and ( and ( and (----3) marks for each incorrect answer3) marks for each incorrect answer3) marks for each incorrect answer3) marks for each incorrect answer. *** A few questions are of Integer type. You are allotted . *** A few questions are of Integer type. You are allotted . *** A few questions are of Integer type. You are allotted . *** A few questions are of Integer type. You are allotted twotwotwotwo hours to hours to hours to hours to answer these questions.answer these questions.answer these questions.answer these questions. [ PLEASE DO SHOW A BRIEF OF YOUR SOLUTION ON THE ALLOCATED ROUGH SPACE] [ PLEASE DO SHOW A BRIEF OF YOUR SOLUTION ON THE ALLOCATED ROUGH SPACE] [ PLEASE DO SHOW A BRIEF OF YOUR SOLUTION ON THE ALLOCATED ROUGH SPACE] [ PLEASE DO SHOW A BRIEF OF YOUR SOLUTION ON THE ALLOCATED ROUGH SPACE]

1>>1>>1>>1>> Let f(x) and g(x) be bijective fun Let f(x) and g(x) be bijective fun Let f(x) and g(x) be bijective fun Let f(x) and g(x) be bijective functions where f: {a,b,c,d}ctions where f: {a,b,c,d}ctions where f: {a,b,c,d}ctions where f: {a,b,c,d}→ {1,2,3,4} and g: {3,4,5,6}{1,2,3,4} and g: {3,4,5,6}{1,2,3,4} and g: {3,4,5,6}{1,2,3,4} and g: {3,4,5,6}→ {w,x,y,z} respectively. {w,x,y,z} respectively. {w,x,y,z} respectively. {w,x,y,z} respectively. The number of elements in the range set of g(f(x)) are The number of elements in the range set of g(f(x)) are The number of elements in the range set of g(f(x)) are The number of elements in the range set of g(f(x)) are (A) 1(A) 1(A) 1(A) 1 (B) 2(B) 2(B) 2(B) 2 (C) 3(C) 3(C) 3(C) 3 (D) 4(D) 4(D) 4(D) 4 SPACE FOR ROUGH WORKSPACE FOR ROUGH WORKSPACE FOR ROUGH WORKSPACE FOR ROUGH WORK

2>> D2>> D2>> D2>> Domain (D) and Range (R) of f(x) = omain (D) and Range (R) of f(x) = omain (D) and Range (R) of f(x) = omain (D) and Range (R) of f(x) = [ ]( )− −1 1sin cos x is respectively, where is respectively, where is respectively, where is respectively, where [ ]. denotes the greatest int.denotes the greatest int.denotes the greatest int.denotes the greatest int. function. function. function. function.

(A)(A)(A)(A) { }≡ ∈ ≡[1,2), 0D x R (B)(B)(B)(B) [ ] { }≡ ∈ ≡ −0,1 , 1,0,1D x R

(C)(C)(C)(C) [ ] ( )π

π− − ≡ ∈ − ≡

1 11,1 , 0,sin ,sin2

D x R (D)(D)(D)(D) [ ] { }π π≡ ∈ − ≡ −1,1 , ,0,

2 2D x R

SPACE FOR ROUGH WORKSPACE FOR ROUGH WORKSPACE FOR ROUGH WORKSPACE FOR ROUGH WORK

3>> 3>> 3>> 3>> The number of solutions of The number of solutions of The number of solutions of The number of solutions of { }− =2 1x nx , where , where , where , where ( )∈ ∞2,n and and and and {}. represents fractional part of x isrepresents fractional part of x isrepresents fractional part of x isrepresents fractional part of x is

(A) 1(A) 1(A) 1(A) 1 (B) 0(B) 0(B) 0(B) 0 (C) 2(C) 2(C) 2(C) 2 (D) none of these(D) none of these(D) none of these(D) none of these SPACE FOR ROUGH WORKSPACE FOR ROUGH WORKSPACE FOR ROUGH WORKSPACE FOR ROUGH WORK

4>> The polynomial p4>> The polynomial p4>> The polynomial p4>> The polynomial p(x) is such that for any polynomial(x) is such that for any polynomial(x) is such that for any polynomial(x) is such that for any polynomial q(x) we have q(x) we have q(x) we have q(x) we have ( )( ) ( )( )= .p q x q p x Then p(x) isThen p(x) isThen p(x) isThen p(x) is

(A) Even(A) Even(A) Even(A) Even (B) ODD(B) ODD(B) ODD(B) ODD (C) of even degree(C) of even degree(C) of even degree(C) of even degree (D) of odd degree(D) of odd degree(D) of odd degree(D) of odd degree SPACE FOR ROUGH WORKSPACE FOR ROUGH WORKSPACE FOR ROUGH WORKSPACE FOR ROUGH WORK

2

5>> 5>> 5>> 5>> The range of The range of The range of The range of α for which the points for which the points for which the points for which the points ( )α α≡ +, 2 and and and and α

α ≡

23,

2 lie on the opposite sides of the line lie on the opposite sides of the line lie on the opposite sides of the line lie on the opposite sides of the line

+ =2 3 6x y isisisis

(A) (A) (A) (A) ( )−2,1 (B)(B)(B)(B) ( ) ( )−∞ − ∪, 2 0,1 (C)(C)(C)(C) ( ) ( )− ∪ ∞2,0 1, (D)(D)(D)(D) ( ) ( )− ∪ ∞1,0 2,


6>>6>>6>>6>> The graph The graph The graph The graph + + =2 2 40 400y xy x divides the plane into regions. The area of the bounded region is……………divides the plane into regions. The area of the bounded region is……………divides the plane into regions. The area of the bounded region is……………divides the plane into regions. The area of the bounded region is……………;;;;


7>>7>>7>>7>> In a triangle ABC,In a triangle ABC,In a triangle ABC,In a triangle ABC, ( ) ( ) ( )α β≡ ≡ ≡, , 2,3 & 1,3A B C and point A lies on the lineand point A lies on the lineand point A lies on the lineand point A lies on the line = +2 3y x wherewherewherewhereα∈I . Area . Area . Area . Area

ofofofof ∆ABC ,,,, ∆ is such thatis such thatis such thatis such that[ ] 5∆ = . Possible co. Possible co. Possible co. Possible co----ordinates of ordinates of ordinates of ordinates of ∆ iiiis/ares/ares/ares/are

(A)(A)(A)(A) ( )2,3 (B)(B)(B)(B) ( )5,13 (C)(C)(C)(C) ( )− −5, 7 (D)(D)(D)(D) ( )− −3, 5


8>>8>>8>>8>> If If If If +

= ++

3 4

1

ix iy

i, then the point, then the point, then the point, then the point ( ),x y lies lies lies lies on the same side of the line on the same side of the line on the same side of the line on the same side of the line + − =2 6 0x y as the point as the point as the point as the point

(A) (3,2)(A) (3,2)(A) (3,2)(A) (3,2) (B) (1,3)(B) (1,3)(B) (1,3)(B) (1,3) (C) (1,5)(C) (1,5)(C) (1,5)(C) (1,5) (D) (0,0)(D) (0,0)(D) (0,0)(D) (0,0) SPACE FOR ROUGH WORKSPACE FOR ROUGH WORKSPACE FOR ROUGH WORKSPACE FOR ROUGH WORK

3

9>> 9>> 9>> 9>> The equationThe equationThe equationThe equation ( ) ++ − + =2 12 1 2 0x xa a has roots of opposite signs then the exhaustive set of values has roots of opposite signs then the exhaustive set of values has roots of opposite signs then the exhaustive set of values has roots of opposite signs then the exhaustive set of values of ‘a’ is of ‘a’ is of ‘a’ is of ‘a’ is

(A) (A) (A) (A) <0a (B)(B)(B)(B) ( )∈ −1,0a (C)(C)(C)(C) ( )∈ −∞,1/3a (D)(D)(D)(D) ( )∈ 0,1/3a


10>>10>>10>>10>> If the equation If the equation If the equation If the equation + + =2x bx c k has four real roots, thenhas four real roots, thenhas four real roots, thenhas four real roots, then

(A) (A) (A) (A) −

− > ∈

22 4

4 0 & 0,4

c bb c k (B)(B)(B)(B)

−− < ∈

22 4

4 0 & 0,4

c bb c k

(C)(C)(C)(C) −

− > >2

2 44 0 &

4

c bb c k (D)(D)(D)(D) NONE OF THESE NONE OF THESE NONE OF THESE NONE OF THESE


11>>11>>11>>11>> If If If If + − = − + +2 2 2 2 2sin 17 16 2sin cosx x x x x , then x lies in, then x lies in, then x lies in, then x lies in

(A) (A) (A) (A) [ ]−8,8 (B) (B) (B) (B) [ ]−4,4 (C)(C)(C)(C) − 17 , 17 (D) None of these(D) None of these(D) None of these(D) None of these


12>> 12>> 12>> 12>> The equationThe equationThe equationThe equation + + =8 4cos cos 1 0x b x will have a solution if “b” belongs to will have a solution if “b” belongs to will have a solution if “b” belongs to will have a solution if “b” belongs to

(A)(A)(A)(A) ( ]−∞,2 (B)(B)(B)(B) ( ]−∞ −, 2 (C)(C)(C)(C) [ )∞2, (D)(D)(D)(D) None of these None of these None of these None of these


4

13>>13>>13>>13>> If If If If α β β γ γ α, ; , ; , are the roots of the equationare the roots of the equationare the roots of the equationare the roots of the equation + + =2 0i i ia x b x c , for , for , for , for iiii = 1, 2, 3, (given that = 1, 2, 3, (given that = 1, 2, 3, (given that = 1, 2, 3, (given thatα β γ >, , 0 ). ). ). ).

IfIfIfIf α αβ αβγ=

− ++ + = +

∑ ∑ ∏

1/23

1

i i i

ii

a b ck

a, , , , then k =then k =then k =then k = --------------------------------------------------------------------------------------------;;;;


14>> 14>> 14>> 14>> If x, m satisfy the inequationsIf x, m satisfy the inequationsIf x, m satisfy the inequationsIf x, m satisfy the inequations ( )≥ + − ≤2 2 41/2 1/2log log 2 & 49 4 0x x x m , then , then , then , then

(A) (A) (A) (A) ( ]∈ −∞ −, 1m (B) (B) (B) (B) ( ]∈ −∞,2m (C)(C)(C)(C) ( ) ∈ −∞ − ∪ ∞ , 7 7 ,m (D)(D)(D)(D) None None None None

SPACE SPACE SPACE SPACE FOR ROUGH WORKFOR ROUGH WORKFOR ROUGH WORKFOR ROUGH WORK

15>>15>>15>>15>> If If If If − + =2 0ax bx c has two distinct roots lying in the intervalhas two distinct roots lying in the intervalhas two distinct roots lying in the intervalhas two distinct roots lying in the interval ( )0,1 , , , , ∈, ,a b c N , then , then , then , then

(A)(A)(A)(A) =5log 1abc (B)(B)(B)(B) =5log 3abc (C)(C)(C)(C) =6log 2abc (D)(D)(D)(D) =6log 4abc

SPACE FOR ROUGH WORKSPACE FOR ROUGH WORKSPACE FOR ROUGH WORKSPACE FOR ROUGH WORK 16>>16>>16>>16>> Find the positive integral values of “n” such thatFind the positive integral values of “n” such thatFind the positive integral values of “n” such thatFind the positive integral values of “n” such that

++ + + + + = +1 2 3 4 101.2 2.2 3.2 4.2 ......... .2 2 2.n nn (___________________);(___________________);(___________________);(___________________);


5

17>>17>>17>>17>> Find the minimum value of Find the minimum value of Find the minimum value of Find the minimum value of ( )( )( )+ + + + + +2 2 23 1 3 1 3 1a a b b c c

abc, where , where , where , where

+∈, ,a b c R , is (, is (, is (, is (------------------------------------------------).).).).


18>> 18>> 18>> 18>> Let Let Let Let 1 2 3, , ,........a a a be in A.P. and be in A.P. and be in A.P. and be in A.P. and 1 2 3, , ,........b b b be in G.P. if be in G.P. if be in G.P. if be in G.P. if = =1 1 2a b and and and and = =10 10 3a b thenthenthenthen

(A) (A) (A) (A) 7 19a b is not an integeris not an integeris not an integeris not an integer (B) (B) (B) (B) 19 7a b is an integeris an integeris an integeris an integer (C)(C)(C)(C) =7 19 19 10a b a b (D) none of these(D) none of these(D) none of these(D) none of these


19>> 19>> 19>> 19>> If If If If

=

+ =

+ + ∑ 4 3 2

1

2 1

2

n

nr

rS

r r rthen then then then 20S is equal to (____________).is equal to (____________).is equal to (____________).is equal to (____________).


20>> 20>> 20>> 20>> Number of ordered Number of ordered Number of ordered Number of ordered tripletstripletstripletstriplets ( )∈, ,p q r N , where , where , where , where ≤ ≤1 , , 10p q r , , , , such that such that such that such that + +2 3 5p q ris a multiple of 4 is is a multiple of 4 is is a multiple of 4 is is a multiple of 4 is

(A)(A)(A)(A) 1000100010001000 (B) 500(B) 500(B) 500(B) 500 (C) 250(C) 250(C) 250(C) 250 (D) 125(D) 125(D) 125(D) 125 SPACE FOR ROUGH WORKSPACE FOR ROUGH WORKSPACE FOR ROUGH WORKSPACE FOR ROUGH WORK

6

21>>21>>21>>21>> A seven digit number made up of all distinct digits 8, 7, 6, 4, 2, x and y is divisible by 3. Then the number of possible A seven digit number made up of all distinct digits 8, 7, 6, 4, 2, x and y is divisible by 3. Then the number of possible A seven digit number made up of all distinct digits 8, 7, 6, 4, 2, x and y is divisible by 3. Then the number of possible A seven digit number made up of all distinct digits 8, 7, 6, 4, 2, x and y is divisible by 3. Then the number of possible ordered pair (x, y) isordered pair (x, y) isordered pair (x, y) isordered pair (x, y) is (A) 4(A) 4(A) 4(A) 4 (B) 8(B) 8(B) 8(B) 8 (C) 10(C) 10(C) 10(C) 10 (D) 13(D) 13(D) 13(D) 13 SPACE FOR ROUGH WORKSPACE FOR ROUGH WORKSPACE FOR ROUGH WORKSPACE FOR ROUGH WORK

22>> 22>> 22>> 22>> TotTotTotTotal number of divisors of al number of divisors of al number of divisors of al number of divisors of = 5 7 93 .5 .7n that are of the formthat are of the formthat are of the formthat are of the form λ +4 1 , , , , λ ≥0 , is equal to, is equal to, is equal to, is equal to (A) 240(A) 240(A) 240(A) 240 (B) 30(B) 30(B) 30(B) 30 (C) 120(C) 120(C) 120(C) 120 (D) 15(D) 15(D) 15(D) 15

SPACE FOR ROUGH WORKSPACE FOR ROUGH WORKSPACE FOR ROUGH WORKSPACE FOR ROUGH WORK 23>> 23>> 23>> 23>> The number of ways in which we can choose The number of ways in which we can choose The number of ways in which we can choose The number of ways in which we can choose 2 distinct integers from 1 to 100 such that the difference between them is 2 distinct integers from 1 to 100 such that the difference between them is 2 distinct integers from 1 to 100 such that the difference between them is 2 distinct integers from 1 to 100 such that the difference between them is at most 10 isat most 10 isat most 10 isat most 10 is

(A) (A) (A) (A) −100 902 2C C (B) (B) (B) (B) −100 90

98 88C C (C)(C)(C)(C) −100 902 88C C (D)(D)(D)(D) None of these None of these None of these None of these


24>> 24>> 24>> 24>> 1 2 3 100, , ,.......,α α α α are all the are all the are all the are all the .100throots of unity.roots of unity.roots of unity.roots of unity. The numerical value of The numerical value of The numerical value of The numerical value of ( )

5

1 100i j

i j

α α≤ < ≤

∑ ∑ isisisis

(A) 20(A) 20(A) 20(A) 20 (B) 0(B) 0(B) 0(B) 0 (C) (C) (C) (C) ( )1/2020 (D) 100(D) 100(D) 100(D) 100


7

25>> 25>> 25>> 25>> Locus of z if Locus of z if Locus of z if Locus of z if ( )

3, 2

4arg 1

, 24

z zz i

z z

π

π

≤ −

− + = − > −

isisisis

(A) Straight lines passing through (2,0)(A) Straight lines passing through (2,0)(A) Straight lines passing through (2,0)(A) Straight lines passing through (2,0) (B) straight lines passing through (2, 0), (1,1)(B) straight lines passing through (2, 0), (1,1)(B) straight lines passing through (2, 0), (1,1)(B) straight lines passing through (2, 0), (1,1) (C) a line segment(C) a line segment(C) a line segment(C) a line segment (D) a set of two rays(D) a set of two rays(D) a set of two rays(D) a set of two rays


26>> Let 26>> Let 26>> Let 26>> Let ( ){ } ( ){ }2 : arg B :arg 3 3 .

4 3z C and if A z z and z z i

π π∈ = = = − − = Then Then Then Then ( )n A B∩ is equal tois equal tois equal tois equal to

(A) 1(A) 1(A) 1(A) 1 (B) 2(B) 2(B) 2(B) 2 (C) 3(C) 3(C) 3(C) 3 (D) 0(D) 0(D) 0(D) 0 SPACE FOR ROUGH WORKSPACE FOR ROUGH WORKSPACE FOR ROUGH WORKSPACE FOR ROUGH WORK

27>> Dividing 27>> Dividing 27>> Dividing 27>> Dividing ( )f z bybybyby ( )z i− , we obtain the remainder , we obtain the remainder , we obtain the remainder , we obtain the remainder ( )i and dividing it byand dividing it byand dividing it byand dividing it by ( )z i+ , we get the remaind, we get the remaind, we get the remaind, we get the remainderererer ( )1 i+ . . . .

The remainder upon the division of The remainder upon the division of The remainder upon the division of The remainder upon the division of ( )f z by by by by ( )2 1z + isisisis

(A) (A) (A) (A) ( )1

12

z i+ + (B)(B)(B)(B) ( )1

12

iz i+ + (C)(C)(C)(C) ( )1

12

iz i− + (D)(D)(D)(D) ( )1

12

z i− +


8

28>>28>>28>>28>> The number of solutions of the equation The number of solutions of the equation The number of solutions of the equation The number of solutions of the equation ( )2

1 1sin sec 1

2 2

xx

x

π− += −

is/areis/areis/areis/are

(A) 1(A) 1(A) 1(A) 1 (B) 2(B) 2(B) 2(B) 2 (C) 3(C) 3(C) 3(C) 3 (D) Infinite(D) Infinite(D) Infinite(D) Infinite SPACE FOR ROUGH WORKSPACE FOR ROUGH WORKSPACE FOR ROUGH WORKSPACE FOR ROUGH WORK

29>> 29>> 29>> 29>> The maximum value of the expressionThe maximum value of the expressionThe maximum value of the expressionThe maximum value of the expression2 2 2 2sin 2 2 1 cosx a a x+ − − − ,,,, where a and x are real numbers, is where a and x are real numbers, is where a and x are real numbers, is where a and x are real numbers, is

(A) (A) (A) (A) 2 (B) 1(B) 1(B) 1(B) 1 (C) (C) (C) (C) 3 (D) None of these(D) None of these(D) None of these(D) None of these


30>> 30>> 30>> 30>> If If If If 4nπ α= then the value of then the value of then the value of then the value of ( )tan .tan2 .tan3 .........tan 2 1nα α α α− is equal to is equal to is equal to is equal to

(A) 1(A) 1(A) 1(A) 1 (B) 0(B) 0(B) 0(B) 0 (C) (C) (C) (C) ----1111 (D) n(D) n(D) n(D) n SPACE FOR ROUGH WORKSPACE FOR ROUGH WORKSPACE FOR ROUGH WORKSPACE FOR ROUGH WORK

iitjee 2011 practice test 1

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