hyperbola passes through the point P(√2, √3) and has foci at

Subject

Mathematics

Class

JEE Class 12

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 Multiple Choice QuestionsMultiple Choice Questions

1.

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

  • 9

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2.

The function straight f colon space straight R space rightwards arrow with space on top space open square brackets negative 1 half comma 1 half close square brackets space defined space as space straight f left parenthesis straight x right parenthesis space equals space fraction numerator straight x over denominator 1 plus space straight x squared end fraction comma space is

  • neither injective nor surjective.

  • invertible

  • injective but not surjective.

  • injective but not surjective.

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3.

The following statement
(p → q ) → [(~p → q) → q] is

  • a fallacy

  • a tautology

  • equivalent to ~ p → q

  • equivalent to ~ p → q

397 Views

4.

For any three positive real numbers a, b and c, (25a2 + b2) + 25(c2 – 3ac) = 15b(3a + c), Then

  • b , c and a are in G.P

  • b, c and a are in A.P

  • a, b and c are in A.P

  • a, b and c are in A.P

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5.

hyperbola passes through the point P(√2, √3) and has foci at (± 2, 0). Then the tangent to this hyperbola at P also passes through the point

  • left parenthesis negative square root of 2 comma space minus square root of 3 right parenthesis
  • left parenthesis 3 square root of 2 space comma space 2 square root of 3 right parenthesis
  • left parenthesis 2 square root of 2 space comma 3 space square root of 3 right parenthesis
  • left parenthesis 2 square root of 2 space comma 3 space square root of 3 right parenthesis


C.

left parenthesis 2 square root of 2 space comma 3 space square root of 3 right parenthesis

Equation of hyperbola is straight x squared over straight a squared minus straight y squared over straight b squared space equals space 1
foci is (±2, 0) hence ae = 2, ⇒ a2e2 = 4

b2 = a2(e2 – 1)
∴ a2 + b2 = 4
Hyperbola passes through √2,√3
therefore space 2 over straight a squared space minus 3 over straight b squared space equals space 1 space... space left parenthesis 2 right parenthesis
On space sol space left parenthesis 1 right parenthesis space and space left parenthesis 2 right parenthesis
straight a squared space equals 8 space left parenthesis is space rejected right parenthesis space and space straight a squared space equals space 1 space and space straight b squared space equals 3
therefore space straight x squared over 1 minus straight y squared over 3 space equals 1
Equation space of space tangent space is space fraction numerator square root of 2 straight x over denominator 1 end fraction minus fraction numerator square root of 3 straight y over denominator 3 end fraction space equals 1
Hence space left parenthesis 2 square root of 2 comma 3 space square root of 3 right parenthesis space satisfy space it.

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6.

Let a, b, c ∈ R. If f(x) = ax2 + bx + c is such
that a + b + c = 3 and f(x + y) = f(x) + f(y) + xy, ∀x,y ∈ R,then sum from straight n space equals 1 to 10 of space straight f space left parenthesis straight n right parenthesis is equal to

  • 225

  • 330

  • 165

  • 165

330 Views

7.

Twenty meters of wire is available for fencing off a flower-bed in the form of a circular sector.Then the maximum area (in sq. m) of the flower-bed, is

  • 30

  • 12.5

  • 10

  • 10

332 Views

8.

The value of
(21C110C1) + (21C210C2) + (21C310C3) + (21C410C4) + .... +
(21C1010C10) is

  • 220 – 210

  • 221 – 211

  • 221 – 210

  • 221 – 210

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9. stack lim space with straight x space rightwards arrow straight pi over 2 below space fraction numerator cot space straight x space minus cos space straight x over denominator left parenthesis straight pi minus 2 straight x right parenthesis cubed end fraction space equals
  • 1/4

  • 1/24

  • 1/16

  • 1/16

396 Views

10.

If 5(tan2x – cos2x) = 2cos 2x + 9, then the value of cos4x is

  • -7/9

  • -3/5

  • 1/3

  • 1/3

542 Views

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