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 Multiple Choice QuestionsMultiple Choice Questions

101.

The sum of the coefficients of all odd degree terms in the expansion of x + x3 -15 + x - x3-15, (x>1) is

  • 2

  • -1

  • 0

  • 1


A.

2

x  + x3 -15 + x  -x3 -15 =2 [C02x5 + C25 x3 (x3-1) + C45 x (x3-1)2] = 2 [x5 + 10 (x6 - x3) + 5x(x6-2x3 +1)] = 2[ x5 + 10x6 - 10x3 + 5x7 -10x4 + 5x]= 2[5x7 + 10x6 + x5 - 10x4-10x3 + 5x]

Sum of odd degree terms coefficients
= 2(5 + 1 – 10 + 5)
= 2


102.

If z  3, then the least value of z + 14

  • 112

  • 114

  • 3

  • 14


B.

114

z + 14       = z - - 14 z - - 14       = z - 14  3 - 14 = 114Hence, z + 14  114


103.

If α is an nth root of unity, then 1 + 2α + 3α2 + ... + n - 1equals

  • - n1 - α

  • - n1 + α2

  • n1 - α

  • None of these


A.

- n1 - α

Let S = 1 + 2α + 3α2 + ... + n - 1Then,      αS = α + 2α2 + 3α3 + ... + n - 1αn - 1 + n    S - αS = 1 + α + α2 + ... + αn - 1 - n S1 - α = αn - 1α - 1 - nαn S1 - α = 1 - 1α - 1 - n              αn = 1                  S = - n1 - α


104.

If x2 + x + 1 = 0, then the value of n = 16xn + 1xn2 is

  • 13

  • 12

  • 9

  • 14


B.

12

Given equation is x2 + x + 1 = 0.

On solving equation, we get

x = w and x = w2

Case - I When x = w

Then, n = 16xn + 1xn2 = n = 16wn + w2n2                  1w = w2= w + w22 + w2 + w42 + w3 + w62      + w4 + w82 + w5 + w102 + w6 + w122= - 12 + - 12 + 22 + - 12 + - 12 + 22

Case -  II When x = w2

Then, n = 16xn + 1xn2 = n = 16wn + w2n2                                                     1w = w2                                    = 12


105.

If α, β ∈ C are the distinct roots, of the equation x2 -x + 1 = 0, then α101 + β107 is equal to

  • 2

  • -1

  • 0

  • 1


D.

1

x2-x + 1 = 0

Roots are -ω, -ω2

Let α = -ω, β = -ω2

α101 + β107 = (-ω)101 + (-ω2)107

= -( ω101214)
= - (ω2 + ω)
= 1


106.

If the complex number z lies on a circle with centre at the origin and radius = 14, then the 4 complex number - 1 + 8z lies on a circle with radius

  • 4

  • 1

  • 3

  • 2


D.

2

Given,   z = 14Let,         z' = - 1 +8z           z = z' + 18         z = z' + 18          14 = z' + 18 z' + 1 = 2

Hence, z' lies on a circle with centre (-1, 0) and radius 2.


107.

The normal to the curve x2 + 2xy-3y2 =0 at (1,1)

  • does not meet the curve again

  • meets the curve again in the second quadrant

  • meets the curve again in the third quadrant

  • meets the curve again in the fourth quadrant


D.

meets the curve again in the fourth quadrant

Given equation of curve is
x2+ 2xy -3y2 = 0    .... (i)
On differentiating w.r.t. we get
2x + 2xy' + 2y-6yy' = 0

straight y apostrophe space equals space fraction numerator straight x plus straight y over denominator 3 straight y minus straight x end fraction
At, x = 1, y = 1, y'=1
i.e, open parentheses dy over dx close parentheses subscript left parenthesis 1 comma 1 right parenthesis end subscript
Equation of normal at (1,1) is 
y-1 = - 1 over 1 left parenthesis straight x minus 1 right parenthesis
⇒ y-1 = - (x-1)
⇒ x+y = 2 .... (ii)
On solving Eqs. (i) and (ii) sumultaneously we get
x2+ 2x(2-x)-3(2-x)2 = 0
⇒x2+4x-2x2-3(4+x2-4x)=0
⇒-x2 +4x-12-3x2+12x = 0
⇒-4x2 +16x-12 =0
⇒ 4x2-16x+12 = 0
⇒x2-4x+3 = 0
(x-1)(x-3) = 0x= 1,3
Now when x =1, then y=1
and when x=3 theny = -1
therefore, p = (1,1) an Q = (3,-1)
Hence, normal meets the curve again at (3, -1)in fourth quadrant.

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108.

The sum of all real values of x satisfying the equation
left parenthesis straight x squared minus 5 straight x space plus 5 right parenthesis to the power of straight x squared plus 4 straight x minus 60 space equals space 1 end exponent is:

  • 3

  • -4

  • 6

  • 5


A.

3

Given, 

left parenthesis straight x squared minus 5 straight x space plus 5 right parenthesis to the power of straight x squared plus 4 straight x minus 60 end exponent space equals space 1
clearly comma space this space is space possible space when
straight I. space straight x squared space plus space 4 straight x minus 60 space equals space 0 space and space straight x squared minus 5 straight x space plus 5 space not equal to space 0
II. space straight x squared minus 5 straight x plus 5 space equals space 1
III. space straight x squared minus 5 straight x space plus 5 space equals space minus 1 space and space straight x squared plus 4 straight x minus 60 space equals space Even

Case space straight I space when space straight x squared space plus space 4 space straight x minus 60 space equals space 0 comma space then
straight x squared space plus space 10 space straight x space minus 6 straight x space minus 60 equals 0
rightwards double arrow space straight x left parenthesis straight x plus 10 right parenthesis minus 6 left parenthesis straight x plus 10 right parenthesis space equals space 0
rightwards double arrow left parenthesis straight x plus 10 right parenthesis left parenthesis straight x plus 6 right parenthesis space equals space 0
rightwards double arrow straight x space equals space minus space 10 space or space straight x equals space 6
Note space that comma space for space these space two space vaues space of space straight x comma space straight x squared minus 5 straight x space plus space 5 space not equal to 0
Case space II space when space straight x squared space minus 5 straight x space plus 5 space equals space 1
straight x squared space minus 5 straight x space plus 4 space equals space 0
straight x squared minus 4 straight x minus straight x space plus 4 equals 0
straight x left parenthesis straight x minus 4 right parenthesis minus 1 left parenthesis straight x minus 4 right parenthesis space equals 0
left parenthesis straight x minus 4 right parenthesis left parenthesis straight x minus 1 right parenthesis space equals 0
straight x equals space 4 space or space straight x space equals 1

Case space III space when space straight x squared minus 5 straight x space plus 5 space equals space minus 1
straight x squared minus 5 straight x space plus 5 space equals space minus 1
straight x squared minus 5 straight x space plus space 6 space equals space 0
rightwards double arrow space straight x squared minus 2 straight x minus 3 straight x space plus 6 space equals space 0
straight x left parenthesis straight x minus 2 right parenthesis minus 3 left parenthesis straight x minus 2 right parenthesis space equals space 0
left parenthesis straight x minus 2 right parenthesis left parenthesis straight x minus 3 right parenthesis space equals 0
straight x equals 2 space or space straight x space equals space 3
Now comma space when space straight x space equals space 2 comma space straight x to the power of 2 space end exponent plus 4 straight x minus 60 space equals space 4 plus 8 minus 60
equals negative 48 comma space which space is space an space interger
when space straight x space equals space 3 comma space straight x squared plus space 4 straight x space minus 60 space equals space 9 space plus 12 minus 60 space equals space minus 39
which space is space not space an space even space interger.
Thus comma space in space this space case comma space we space get space straight x space equals space 2
Hence comma space the space sum space of space all space real space values space of space
straight x space equals space minus 10 plus 6 plus 4 plus 1 plus 2 space equals 3

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109.

If, for a positive integer n, the quadratic equation,
x(x + 1) + (x + 1) (x + 2) + .....
+ (x + n -1 ) (x + n) = 10n
has two consecutive integral solutions, then n is equal to :

  • 11

  • 12

  • 9

  • 10


A.

11

nx squared space plus space straight x left parenthesis 1 space plus space 3 plus 5 plus..... space plus space left parenthesis 2 straight n minus 1 right parenthesis right parenthesis space plus
space left parenthesis 1.2 space plus space 2.3 space plus space..... space plus space left parenthesis straight n minus 1 right parenthesis. straight n right parenthesis minus 10 straight n space equals space 0
rightwards double arrow space nx squared space plus space straight x space left parenthesis straight n squared right parenthesis space plus space fraction numerator straight n left parenthesis straight n squared minus 1 right parenthesis over denominator 3 end fraction space minus space 10 straight n space equals space 0
rightwards double arrow space straight x squared space plus space straight x space left parenthesis straight n right parenthesis space plus space fraction numerator left parenthesis straight n squared minus 1 right parenthesis over denominator 3 end fraction minus 10 space equals space 0 space less than subscript straight beta superscript straight alpha
left parenthesis straight alpha space minus space straight beta right parenthesis squared space equals space 1
rightwards double arrow space left parenthesis straight alpha space plus space straight beta right parenthesis squared space minus space 4 αβ space equals space 1
rightwards double arrow space straight n squared space minus space 4 space open parentheses fraction numerator straight n squared minus 1 over denominator 3 end fraction minus 10 close parentheses space equals space 1
rightwards double arrow space straight n space equals space 11
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