MULTIPLE CHOICE TEST
1)Let x,y,z be non-zero real nos.The set of all possible values of the expression
|x+y|/|x|+|y| +|z+y|/|z|+|y| +|x+z|/|x|+|z|
a)[0,5] b)[3,4]
c) [1,3] d)[-1,2]
2) The no N=12321 is aperfect square,viz.(111)^2 ,in base 10.In what other bases is it a perfect square?
a)exactly two bases b>=7
b)exactly three bases b>=4
c)for no base b other than 10
d)any base b>=4
3)Let ? =(45+29(2)^.5)^1/3 +(45-29(2)^.5)^1/3 Then which of the following statements is true?
a) ? is an irrational number and 5<= ?<=7
b) ? is an irrational number and 5< ?<7
c) ? is rational number and 5< ?<6
d) ? is an irrational number and 5<= ?<=6
4)The +ve soln(x,y) of the system of eqns a^2/x^2-b^2/y^2=8(y^4-x^4) , (a,b>0) is, (assume x?y)
a)({(a+b)^1/3 +(a-b)^1/3} /2 , {(a+b)^1/3 - (a-b)^1/3} /2 )
b) ({(a+b)^1/3 +(a-b)^1/3} /4 , {(a+b)^1/3 - (a-b)^1/3} /4 )
c) ({(a+b)^1/3 -(a-b)^1/3} /4 , {(a+b)^1/3 + (a-b)^1/3} /4 )
d) ({(a+b)^1/3 -(a-b)^1/3} /2 , {(a+b)^1/3 + (a-b)^1/3} /2 )
5)The remainder R(x) when the polynomial x^100 is divided by x^2-3x+2 is
a)(2^100 -1)x-2(2^99 -1)
b) 2^100x-2*(2^100-1)
c)(2^100-1)x+2(2^99-1)
d) (2^100 +1)x-2*(2^99+1)
6)Three str lines r drawn thr a pt P lying inside a triangle ABC ,parallel to its sides. The areas of the resulting triangles r 1,4,9 sqcm.Then the area of triangle ABC is(in sqcm.)
a)25 b)36 c)49 d)144
7)Let N=2^744-1.Then abt the divisors of N which of the following statement is true?
a) 2^248-2^124+1 is a divisor of N but 2^93+2^47+1 is not
b) 2^93+2^47+1 is a divisor of N but 2^248-2^124+1 is not
c) 2^248-2^124+1 and ) 2^93+2^47+1 r both divisors of N
d) Neither 2^248-2^124+1 nor 2^93+2^47+1 is a divisor of N
8)Let a1,a2,??.aN be a sequence of real nos such hat sum of every 5 conseq terms is +ve whereas sum of every 9 conseq term is ?ve.The seq can have atmost
a)5*9 terms b)14 terms c)13 terms d)12 terms
9)A polynomial f(x) with real coefficient satisfies the functional eqn f(x)f(1/x)=f(x)+f(1/x) If f(2)=33, then f(3)=?
a)244 b)1024 c)81 d)1023
10)ABCD is a square and P is a pt inside the square such that PA=6(2)^1/2 cm, PB=13cm and PD=5cm.Then the angle APD is
a)75 degree b)135 degree c)120 degree d)112 degree
11)Repunits are the nos that contain only the digit 1 in their writing , namely nos of the form 111?.1The 123-digit repunit 111?1(123times) when divided by 271 leaves a remainder of
a)123 b)110 c)101 d)111
12)In triangle ABC , the ratio of side BC to AC is 2+3^1/2 and angle C=60.The measures of angle A and B r
a)45 ,75 b)75,45 c)105,15 d)15,105
13)The min value of sec^4A/ tan^2B+sec^4B/tan^2A ,A,B?npi/2, n belongs to Z is
a)4 b)8 c)1 d)16
14)Let a and b be non-zero integers with |a|<=100, |b|<=100.Then about the bound of |a*2^1/2+b*3^1/2| which of the following is true?
a) |a*2^1/2+b*3^1/2|>=1/350 b) |a*2^1/2+b*3^1/2|<=1/450
c) |a*2^1/2+b*3^1/2|<1/450 d) |a*2^1/2+b*3^1/2|<1/550
15)If f(x)=e^2x-1/1+e^2x-1 then the value of
f(1/2006)+f(2/2006)+??f(2005/2006)
a)1002.5 b)1001.5 c)1003 d)1004
16)There r N boxes , each containing atmost r balls .If the no of boxes containing at least i balls is Ni for i=1,2?.r, then the total no of balls contained in these boxes is
a)exactly equal to N1+N2+?.+Nr
b)is strictly larger than N1+N2+?.+Nr
c) is strictly smaller than N1+N2+?.+Nr
d)cannot b determined frm the above info
17)Let n=2006!+1.Then the no of primes among n+1,n+1,?.n+2005 is
a)2 b)1 c)0 d)>5
18)Let P(x) b a polynomial of degree11 such that P(x)=1/1+x , for x=0,1,2,?11.The value of P(12) is
a)1/13 b) 1 c)0 d)cannot b determined
19) {(2)^1/3 -1}^1/3 is equivalent to
a)(1/9)^1/3+(2/9)^1/3-(4/9)^1/3 b) (1/9)^1/3-(2/9)^1/3+(4/9)^1/3
c) (1/9)^1/3+(2/9)^1/3+(4/9)^1/3 d) -(1/9)^1/3+(2/9)^1/3+(4/9)^1/3
20)If a,b,c r the roots of the eqn x^3+2x^2+3x+3=0, then the value of
(a/a+3)^3+(b/b+3)^3+(c/c +3)^3 is
a)14 b)144 c)45 d)15
21)If the seq {an}, satisfies the recurrence ,an+1=3an-2an-1 , n>=2, a0=2, a1=3, then a2007 is
a)2^2006+1 b) 2^2007-1 c) 2^2007+1 d)2^2006-1
22)If A,B and C r the angles of a triangle and e^iA, e^iB and e^iC r in AP , then the triangle is
a)rt angled but not isosceles
b)isosceles but not rt angled
c)equilateral
d)rt angled isosceles
23)Statistics show that 20% of smokers get lung cancer and 80% of lung cancer patients r smokers. If 30% of population smokes then the %of population having lung cancer
a)16 b)7.5 c)8 d)25
24)Let f(x) b a fuctn such that
f(x-1)+f(x+1)=(2)^.5 f(x)
Then the pd of f(x)
a)8 b)6 c)10 d)4
25)The soln in integers of the eqn x^2+xy=y^2+xz can b expressed as, (where n,a,b, r arbitrary integers)
a)x=na^2, y=nab, z=n(a^2+ab-b^2)
b) x=na^2, y= -nab, z=n(a^2+ab-b^2)
c} x=na^2, y=nab, z=n(a^2-ab+b^2)
d) x= -na^2, y=nab, z=n(a^2+ab+b^2)
26)The neighbouring sides AB and BC of a square ABCD a units r tangent to a circle. The vertex D of the square lies on the circumference of the circle. The radius of the circle
is
a)a(2*2^1/2 -1) b)2a*(2^1/2 -1)
c)a(2-(2)^1/2) d)a(2+(2)^1/2)
27)The sum of all distinct 4 digit nos that can b formed using the digits 1,2,3,4 and 5, each digits appearing at most once is
a)399960 b)396990 c)399600 d)369960
28)For what values of d is the product of 2 nos of the form x^2-d*y^2 and u^2-d*v^2 is also of the same form ?(d?a square)
a) d>=10 b)d=2 only c)d=2 and 5 d)for any d
29)The minimum value of the exp x^3(x^3+1)( x^3+2)( x^3+3) is, (x belongs to R)
a)1 b)-1 c)4 d)none
30)How many strings of 6 digits r there which use only the digits 0,1,2 and in which digit 2, whenever it appears , it always does so after 1?
SHORT ANSWER TYPE TEST
31)Let a1,a2,?.an b n nos such that each ai is either 1 or -1 .If a1a2a3a4+a2a3a4a5+?+ana1a2a3=0 then prove that 4 divides n
32)Let a, b be integers.Then show that the polynomial (x-a)^2(x-b)^2+1 is not the product of two polynomials with integral coefficients.
33)Let n be a +ve integer. Find all pairs(x,y) such that x^2(x+y)=y^n+1
34)Let f:[a,b]àR be a continuous+ve functn differentiable on (a,b).Prove that there exists a c in(a,b) such that f(b)/f(a)=e^{(b-a)*f?(c)/f(c)}
35)Prove that the no 1280000401 is composite
36)Solve in real nos the system for a,b,c and d a+b=8, ab+c+d=23, ad+bc=28 , cd=12
37)In the trapezoid PQRS PQ||RS, PQ=4cm ,RS=10cm.Also the lines PS and QS intersect at rt angles , and that lines PS and QR when extended to the pt N , form an angle of 45.Find the area of trapezoid PQRS
38)Find the minm value of | sinx+cosx+tanx+cotx+secx+cosecx| for real nos
39)figure based
40)Let A be any set of 19 distinct integers chosen frm the AP 1,4,7,?100.Prove that there must be two distinct integers in A ,whose sum is 104
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