The orthocentre of triangle formed by the lines x + 3y = 10 and 6

Subject

Mathematics

Class

JEE Class 12

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 Multiple Choice QuestionsMultiple Choice Questions

1.

In a competition A, B, C are participating the probability that A wins is twice that of B, the probability that B wins is twice that of C, then probability that A loses is

  • 17

  • 27

  • 47

  • 37


2.

The probability that a number selected at random from the set of numbers (1, 2, 3, ... , 100) is a cube, is

  • 125

  • 225

  • 325

  • 425


3.

Two dice are rolled simultaneously. The probability that the sum of the two numbers on the dice is a prime number, is

  • 512

  • 712

  • 912

  • 0.25


4.

The events A andB have probabilities 0.25 and 0.50, respectively. The probability that both A and B occur simultaneously is 0.14, then the probability that neither A nor B occurs, is

  • 0.39

  • 0.29

  • 0.11

  • 0.25


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5.

For all values of a and b the line (a + 2b)x + (a - by + (a + 5b) = 0 passes through the point.

  • (- 1, 2)

  • (2, - 1)

  • (- 2, 1)

  • (1, - 2)


6.

The lines 2x + 3y = 6 , 2x + 3y = 8 cut the X-axis at A and B, respectively. A line L drawn through the point (2, 2) meets the X-axis as C in such away that abscissae of A, B and C are in arithmetic progression. Then, the equation of the line L is

  • 2x + 3y = 10

  • 8x + 2y = 10

  • 2x - 3y = 10

  • 8x - 2y = 10


7.

The number of circles that touch all the straight lines x + y = 4,x - y = - 2 and y = 2 is

  • 1

  • 2

  • 3

  • 4


8.

The incentre of triangle formed by the lines x + y = 1, x =1, y = 1 is

  • 1 - 12, 1 - 12

  • 1 - 12, 12

  • 12, 12

  • 12, 1 - 12


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9.

The orthocentre of triangle formed by the lines x + 3y = 10 and 6x2 + xy - y2 = 0 is

  • (1, 3)

  • (3, 1)

  • (- 1, 3)

  • (1, - 3)


A.

(1, 3)

The given lines are

                x + 3y = 10                ...(i)

and 6x2 + xy - y2 = 0

or    6x2 + 3xy - 2xy - y2 = 0

or 3x(2x + y) - y(2x + y) = 0

                            3x - y = 0       ...(ii)

                           2x + y = 0      ...(iii)

On solving Eqs. (i) and (ii), we get

       x + 33x = 10             10x = 10                 x = 1and 3 - 1 - y = 0                 y = 3 Coordinates of B are (1, 3).On solving Eqs. (ii) and (iii), we get          x = 0, y = 0 Coordinates of A are (0, 0).

A line perpendicular to BC is3x - y = λIt passes through (0, 0), then 0 - 0 = λ     λ= 0The line AD is 3x - y = 0           ...(iv)A line perpendicular to AC isx - 2y = λIt passes through (1, 3), then 1 - 6 = λ         λ = - 5The line BE is x - 2y = - 5      ...(v)On solving Eqs. (iv) and (v), we getx = 1, y = 3Thus, the coordinates of required orthocentre is (1, 3).


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10.

If one of the lines of pair of straight lines ax2 + 2hxy + by2  = 0 bisects the angle between the coordinate axes, then

  • a2 + b2 = h2

  • (a + b)2 = 4h2

  • a2 + b2 = 4h2

  • (a + b)2 = h2


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