Find the points on the curve  x2 + y2 – 2x – 3=

Subject

Mathematics

Class

CBSE Class 12

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Sample Papers

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 Multiple Choice QuestionsShort Answer Type

11.

Write the intercept cut off by the plane 2x + y – z = 5 on x-axis.


 Multiple Choice QuestionsLong Answer Type

12.

Prove the following:

cot-1   1 + sin x +  1 - sin x 1 + sin x -  1 - sin x  = x2,   x   0, π4 


13.

Find the value of  tan-1  xy  - tan-1  x - yx + y


14.

Using properties of determinants, prove that

  - a2      ab         ac     ba -b2      bc    ca  cb  - c2  = 4 a2b2c2


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15.

Find the value of ‘a’ for which the function f defined as

f ( x ) =  a sin π2 ( x + 1 ),       x  0tan x - sin x x3,            x > 0 

is continuous at x = 0.


16.

Differentiate  X x cos x +  x2 + 1x2 - 1  w.r.t. x


17.

If   x = a  θ - sin θ ,   y =  1 + cos θ ,    find d2ydx2


18.

Sand is pouring from a pipe at the rate of 12 cm3/s. The falling sand forms a cone on the ground in such a way that the height of the cone is always one-sixth of t heradius of the base. How fast is the sand cone increasing when the height is 4 cm?


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19.

Find the points on the curve  x2 + y2 – 2x – 3= 0  at  whichthe tangents are parallel to x-axis.


Let  P ( x, y )  be any point on the given curve  x2 + y2 - 2 x - 3 = 0.

Tangent to the curve at the point (x, y ) is given by dydx.

Differentiating the equation of the cueve w.r.t. x we get

2 x + 2 y dydx - 2 = 0dydx = 2 - 2 x2 y = 1 - xy

Let P ( x1, y1 ) be the point on the given curve at which the tangents are parallel to the x-axis.

 dydx  x1, y1  = 0 1 - x1y1 = 0 1 - x1 = 0 x1 = 1

To get the value of  y1  just substitute  x1 = 1  in the equation  x2 + y2 - 2 x - 3 = 0, we get

( 1 )2 + ( y1 )2 - 2 x 1 - 3 = 0

  y1 2 - 4 = 0  y1 2 = 4   y1 = ± 2 

So, the points on the given curve at which the tangents are parallel to the x-axis are  ( 1, 2 )  and  ( 1, - 2 ).


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20.

Using matrix method, solve the following system of equations:

2x + 3y + 10z = 4,       4x - 6y + 5z,       6x + 9y - 20z;    x, y, z  0


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