The line y = x + λ is tangent to the ellipse 2x2 + 3y

Subject

Mathematics

Class

JEE Class 12

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 Multiple Choice QuestionsMultiple Choice Questions

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1.

The line y = x + λ is tangent to the ellipse 2x2 + 3y2 = 1. Then, λ is

  • - 2

  • 1

  • 56

  • 23


C.

56

The equation of the line y = x + λ    ...(i)

On comparing it with y = mx + c, we get

m = 1 and c = λ

The equation of the ellipse is

2x2 + 3y2 = 1

or x2122 + y2132 = 1

On comparing it with x2a2 + y2b2 = 1, we get

a2 = 12 and b2 = 13

If the line touches the ellipse, then

c2 = a2m2 + b2

 λ2 = 12 . 1 + 13

 λ2 = 56

 λ = 56


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2.

Let S be the set of points, whose abscissae and ordinates are natural numbers. Let p e S, such that the sum of the distance of P from (8, 0) and (0, 12) is minimum among all elemants in S. Then, the number of such points P in S is

  • 1

  • 3

  • 5

  • 11


3.

The cosine of the angle between any two diagonals of a cube is

  • 13

  • 12

  • 23

  • 13


4.

If x is a positive real number different from 1 such that logax, logbx, logcx are in AP, then

  • b = a + c2

  • ac

  • c2 = aclogab

  • None of these


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5.

If a, x are real numbers and a < 1, x < 1, then 1 + (1+ a) x + (1+ a + a2)x2 + ... is equal to

  • 11 - a1 - ax

  • 11 - a1 - x

  • 11 - x1 - ax

  • 11 + ax1 - a


6.

If log0.3x - 1 < log0.09x - 1, then x lies in the interval

  • 2, 

  • (1, 2)

  • (- 2, - 1)

  • None of these


7.

The value of n = 113in + in + 1, i = - 1 is

  • i

  • i - 1

  • 1

  • 0


8.

If z1 , z2, z3 are imaginary numbers such that z1 = z2 = z3 = 1z1 + 1z2 + 1z3 = 1, then z1 + z2 + z3 is

  • equal to 1

  • less than

  • greater than 1

  • equal to 3


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9.

If p.q are the · roots of the equation x2 + px + q =0, then

  • p = 1, q = - 2

  • p = 0, q = 1

  • p = - 2, q = 0

  • p = - 2, q = 1


10.

The number of ways in which the letters of the word ARRANGE can be permuted such that the R's occur together, is

  • 7!2! 2!

  • 7!2!

  • 6!2!

  • 5! × 2!


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