The normal to the curve x = a(cosθ + θ sinθ), y = a( sinθ -

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 Multiple Choice QuestionsMultiple Choice Questions

31.

If one of the lines of my2+ (1 − m2)xy − mx2 = 0 is a bisector of the angle between the lines xy = 0, then m is

  • −1/2

  • -2

  • 1

  • 1

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32.

Angle between the tangents to the curve y = x2 − 5x + 6 at the points (2, 0) and (3, 0) is

  • π/2

  • π/4

  • π/6

  • π/6

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33.

The image of the point (−1, 3, 4) in the plane x − 2y = 0 is

  • open parentheses 9 over 5 comma negative 13 over 5 comma space 4 close parentheses
  • open parentheses fraction numerator negative 17 over denominator 3 end fraction comma fraction numerator negative 19 over denominator 3 end fraction comma 1 close parentheses
  • (15, 11, 4)

  • (15, 11, 4)

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34.

The normal to the curve x = a(cosθ + θ sinθ), y = a( sinθ - θ cosθ) at any point ‘θ’ is such that

  • it passes through the origin

  • it makes angle π/2 + θ with the x-axis

  • it passes through (aπ/2 ,-a)

  • it passes through (aπ/2 ,-a)


D.

it passes through (aπ/2 ,-a)

Clearly dy/dx = an θ
⇒ slope of normal = - cot θ
Equation of normal at ‘θ’ is
y – a(sin θ - θ cos θ) = - cot θ(x – a(cos θ + θ sin θ)
⇒ y sin θ - a sin2 θ + a θ cos θ sin θ = -x cos θ + a cos2 θ + a θ sin θ cos θ
⇒ x cos θ + y sin θ = a
Clearly this is an equation of straight line which is at a constant distance ‘a’ from origin.

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35.

The line parallel to the x−axis and passing through the intersection of the lines ax + 2by + 3b = 0 and bx − 2ay − 3a = 0, where (a, b) ≠ (0, 0) is

  • below the x−axis at a distance of 3/2 from it

  • below the x−axis at a distance of 2 /3 from it

  • above the x−axis at a distance of 3/ 2 from it

  • above the x−axis at a distance of 3/ 2 from it

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36.

If the angle θ between the line fraction numerator straight x plus 1 over denominator 1 end fraction space equals space fraction numerator straight y minus 1 over denominator 2 end fraction space equals space fraction numerator straight z minus 2 over denominator 2 end fractionand the plane space 2 straight x minus space straight y plus space square root of straight lambda space straight z space plus 4 space equals 0 is such of sin θ = 1/3 the value of λ is

  • 5/3

  • -3/5

  • 3/4

  • 3/4

105 Views

37.

The angle between the lines 2x = 3y = − z and 6x = − y = − 4z is

  • 0o

  • 90o

  • 45o

  • 45o

237 Views

38.

If the plane 2ax − 3ay + 4az + 6 = 0 passes through the midpoint of the line joining the centres of the spheres

x2 + y2 + z2 + 6x − 8y − 2z = 13 and x2 + y2 + z2 − 10x + 4y − 2z = 8, then a equals

  • -1

  • 1

  • -2

  • -2

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39.

The distance between the line straight r with rightwards arrow on top space equals space 2 straight i with hat on top space minus 2 straight j with hat on top space plus 3 straight k with hat on top space plus space straight lambda space left parenthesis straight i with hat on top minus straight j with hat on top space plus 4 straight k with hat on top right parenthesis and the plane straight r with rightwards arrow on top. left parenthesis straight i with hat on top space plus 5 straight j with hat on top space plus straight k with hat on top right parenthesis space equals space 5 space is

  • 10/9

  • fraction numerator 10 over denominator 3 square root of 3 end fraction
  • 3/10

  • 3/10

104 Views

40.

If a vertex of a triangle is (1, 1) and the mid-points of two sides through this vertex are (-1, 2) and (3, 2), then the centroid of the triangle is

  • (-1, 7/3)

  • (-1/3, 7/3)

  • (1, 7/3)

  • (1, 7/3)

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