On the ellipse 4x2 + 9y2 = 1, the points at which the tangen

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 Multiple Choice QuestionsMultiple Choice Questions

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1.

On the ellipse 4x2 + 9y2 = 1, the points at which the tangents are parallel to the line 8x = 9y, are

  • 25, 15

  • - 25, 15

  • - 25, - 15

  • 25, - 15


B.

- 25, 15

D.

25, - 15

Given equation of ellipse is,

4x2 + 9y2 = 1          ...(i)

On differentiating, we get

 8x + 18yy' = 0 y' = - 8x18y = mAlso, 8x = 9y            equation of lineOn differentiating, we get 8 = 9y' y' = 89 = m       tangents are parallel to this line

So, by Eqs. (ii) and (iii), we get

- 8x18y = 89 - x = 2y x = - 2y

On substituting x = - 2y in Eq. (i), we get

4(- 2y)2 + 9y2 = 1

 16y2 + 9y2 = 1 y = ± 15 x = ± 25So, required points are 25, - 15 and - 25, 15


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2.

A container s the shape of an inverted cone. Its height is 6 m and radius is 4m at the top. If it is filled with water at the rate of 3m/min then the rate of change of height of water(in mt/min) when the water level is 3 m is

  • 34π

  • 29π

  • 16π

  • 2π


3.

 If α, β, γ are the lengths of the tangents from the vertices of a triangle to its incircle. Then

  • α + β + γ = 1r2αβγ

  • α + β + γ = 1rαβγ

  • 1α + 1β + 1γ = rαβγ

  • α2 + β2 + γ2 = 2rαβγ


4.

If a cylindrical vessel of given volume V with no lid on the top is to be made from a sheet of metal, then the radius (r) and height(h) of the vessel so that the metal sheet used is minimum is

  • r = πV3, h =  πV3

  • r = πV, h = πV

  • r = Vπ3, h = Vπ3

  • r = Vπ, h = Vπ


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5.

A wire of length 2 units is cut into two parts which are bent respectively to form a square of side=x units and a circle of radius=r units. If the sum of the areas of the square and the circle so formed is minimum, then:

  • 2x=(π+4)r

  • (4−π)x=πr

  • x=2r

  • x=2r

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6.

Let f (x) be a polynomial of degree four having extreme values at x =1 an x =2. If limit as straight x rightwards arrow 0 of open square brackets 1 plus fraction numerator straight f left parenthesis straight x right parenthesis over denominator straight x squared end fraction close square brackets space equals space 3 comma then f(2) is equal to 

  • -8

  • -4

  • 0

  • 0

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7.

A spherical balloon is filled with 4500π cubic meters of helium gas. If a leak in the balloon causes the gas to escape at the rate of 72π cubic meters per minute, then the rate (in meters per minute) at which the radius of the balloon decreases 49 minutes after the leakage began is

  • 9/7

  • 7/9

  • 2/9

  • 2/9

1221 Views

8.

Let a, b ∈ R be such that the function f given by f(x) = ln |x| + bx
2+ ax, x ≠ 0 has extreme values at x = –1 and x = 2.
Statement 1: f has local maximum at x = –1 and at x = 2.
Statement 2: straight a space equals space 1 half space and space straight b space equals space fraction numerator negative 1 over denominator 4 end fraction

  • Statement 1 is false, statement 2 is true

  • Statement 1 is true, statement 2 is true; statement 2 is a correct explanation for statement 1

  • Statement 1 is true, statement 2 is true; statement 2 is not a correct explanation for statement 1

  • Statement 1 is true, statement 2 is true; statement 2 is not a correct explanation for statement 1

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9.

If dy/dx = y + 3 > 0 and y(0) = 2, then y(ln2) is equal to:

  • 7

  • 5

  • 13

  • 13

158 Views

10.

Equation of the ellipse whose axes are the axes of coordinates and which passes through the point (-3, 1) and has eccentricity square root of 2 over 5 end root is

  • 3x2 + 5y2 -32 = 0

  • 5x2 + 3y2 - 48 = 0

  • 3x2 + 5y2 - 15 = 0 

  • 3x2 + 5y2 - 15 = 0 

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