If P and Q are the points of intersection of the circles x2+ y2+

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21.

If P and Q are the points of intersection of the circles x2+ y2+ 3x + 7y + 2p – 5 = 0 and x2+ y2+ 2x + 2y – p2 = 0, then there is a circle passing through P, Q and (1, 1) for

  • all values of p

  • all except one value of p

  • all except two values of p

  • all except two values of p


B.

all except one value of p

Radical axis is x + 5y + p2 + 2p – 5 =0
Equation of circle is
x2 + y2 + 3x + 7y + 2p – 5 + λ [x + 5y + p2 + 2p – 5 ] = 0 …. (i)
(i) passes through (1, 1)

rightwards double arrow space straight lambda space equals negative space fraction numerator left parenthesis 2 straight p space plus 7 right parenthesis over denominator left parenthesis straight p plus 1 right parenthesis squared end fraction space space space space space space space space space space space space space space left parenthesis straight p space not equal to negative 1 right parenthesis

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22.

The shortest distance between the line y – x = 1and the curve x = y2 is 

  • fraction numerator 3 square root of 2 over denominator 8 end fraction
  • fraction numerator 2 square root of 3 over denominator 8 end fraction
  • fraction numerator 3 square root of 2 over denominator 5 end fraction
  • fraction numerator 3 square root of 2 over denominator 5 end fraction
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23.

The line passing through the points (5, 1, a) and (3, b, 1) crosses the yz−plane at the point open parentheses 0 space comma 17 over 2 comma space fraction numerator negative 13 over denominator 2 end fraction close parentheses. Then

  • a = 2, b = 8

  • a = 4, b = 6

  • a = 6, b = 4

  • a = 6, b = 4

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24.

If the straight lines fraction numerator straight x minus 1 over denominator straight k end fraction space equals fraction numerator straight y minus 2 over denominator 2 end fraction space equals space fraction numerator straight z minus 3 over denominator 3 end fraction space and space fraction numerator straight x minus 2 over denominator 3 end fraction space equals space fraction numerator straight y minus 3 over denominator straight k end fraction space equals fraction numerator straight z minus 1 over denominator 2 end fraction intersect at a point, then the integer k is equal to

  • -5

  • 2

  • 5

  • 5

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25.

Let L be the line of intersection of the planes 2x + 3y + z = 1 and x + 3y + 2z = 2. If L makes an angle α with the positive x-axis, then cos α equals-

  • fraction numerator 1 over denominator square root of 3 end fraction
  • 1/2

  • 1

  • 1

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26.

The resultant of two forces P N and 3 N is a force of 7 N. If the direction of the 3 N force were reversed, the resultant would be square root of 19N. The value of P is

  • 5N

  • 6N

  • 4N

  • 4N

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27.

If a line makes an angle of π/4 with the positive directions of each of x-axis and y-axis, then the angle that the line makes with the positive direction of the z-axis is

  • π/6

  • π/3

  • π/4

  • π/4

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28.

If (2, 3, 5) is one end of a diameter of the sphere x2+ y2+ z2 − 6x − 12y − 2z + 20 = 0, then the coordinates of the other end of the diameter are

  • (4, 9, –3)

  • (4, –3, 3)

  • (4, 3, 5)

  • (4, 3, 5)

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29.

Let A(h, k), B(1, 1) and C(2, 1) be the vertices of a right-angled triangle with AC as its hypotenuse. If the area of the triangle is 1, then the set of values which ‘k’ can take is given by  

  • {1, 3}

  • {0, 2}

  • {–1, 3}

  • {–1, 3}

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30.

Let P = (−1, 0), Q = (0, 0) and R = ( 3, 3 √3) be three points. The equation of the bisector of the angle PQR

  • square root of 3 straight x space plus straight y space equals space 0
  • straight x space plus space fraction numerator square root of 3 over denominator 2 end fraction straight y space equals space 0
  • fraction numerator square root of 3 over denominator 2 end fraction space straight x space plus straight y space equals 0
  • fraction numerator square root of 3 over denominator 2 end fraction space straight x space plus straight y space equals 0
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