The equation sin x(sin(x) + cos(x)) = k has real solutions, where

Subject

Mathematics

Class

JEE Class 12

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 Multiple Choice QuestionsMultiple Choice Questions

1.

The number of all numbers having 5 digits, with distinct digits is

  • 99999

  • 9 × P49

  • P510

  • P49


2.

The greatest integer which divides (p + 1) (p + 2) (p + 3) .... (p + q) for all p E N and fixed q  N is

  • p!

  • q!

  • p

  • q


3.

Let 1 +x + x29 = a0 +a1x +a2x2 +... + a18x18. Then,

  • a0 +a2 +... +a18 = a0 +a3 + ... + a17

  • a0 + a2 +... + a18 is even

  • a0 + a2 +... + a18 is divisible by 9

  • a0 + a2 +... + a18 is divisible by 3 but not by 9


4.

Let f : R ➔ R be such that f is injective and f(x)f(y) = f(x + y) for  x, y  R. If f(x), f(y), f(z) are in G.P., then x, y, z are in

  • AP always

  • GP always

  • AP depending on the value of x, y, z

  • GP depending on the value of x, y, z


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5.

The probability that a non-leap year selected at random will have 53 Sunday is,

  • 0

  • 1/7

  • 2/7

  • 3/7


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6.

The equation sin x(sin(x) + cos(x)) = k has real solutions, where k is a real number. Then,

  • 0  k  1 + 22

  • 2 - 3  k  2 + 3

  • 0  k  2 - 3

  • 1 - 22  k  1 + 22


D.

1 - 22  k  1 + 22

Let, f(x) = sin(x) (sin(x) + cos(x))

             = sin2x + sinxcosx

             = 1 - cos2x2 + 2sinxcosx2

             = 12 - 12cos2x + 12sin2x

             = 12 + 12sin2x - cos2x

Now, - 12 + 12  sin2x - cos2x  12 + 12

- a2 + b2  asinx + bcosx  a2 + b2

 - 2  sin2x - cos2x  2

 - 22  sin2x - cos2x2  22

 12 - 22  12 + sin2x - cos2x2  12 + 22

 1 - 22  f(x)  1 + 22

 1 - 22  k  1 + 22


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7.

Transforming to parallel axes through a point (p, q), the equation

2x2 + 3xy + 4y2 + x + 18y + 25 = 0 becomes 2x2 + 3xy + 4y= 1. Then,

  • p = - 2, q = 3

  • p = 2, q = - 3

  • p = 3, q = - 4

  • p = - 4, q = 3


8.

Let A(2,- 3) and B(- 2,1) be two angular points of AABC. If the centroid of the triangle moves on the line 2x + 3y = 1, then the locus of the angular point C is given by

  • 2x + 3y = 9

  • 2x - 3y = 9

  • 3x + 2y = 5

  • 3x + 2y = 3


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9.

The point P(3, 6) is first reflected on the line y =x and then the image point Q is again reflected on the line y = - x to get the image point Q'. Then, the circumcentre of the APQQ' is

  • (6, 3)

  • (6,- 3)

  • (3,- 6)

  • (0, 0)


10.

Let dand d2 be the lengths of the perpendiculars drawn from any point of the line 7x - 9y + 10 = 0 upon the lines 3x + 4y = 5 and 12x +5y = 7, respectively. Then,

  • d> d2

  • d= d2

  • d1 < d2

  • d= 2d2


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