Differentiate  X x cos x + &nb

Subject

Mathematics

Class

CBSE Class 12

Pre Boards

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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.


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16.

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


Let  y = xx cos x  and  z = x2 + 1x2 - 1Consider  y = xx cos x 

Taking log on both sides,

log y = log ( xx cos x

log y = x cos x log x

Differentiating with respect to x,

1y dydx =  x cos x  1x + log x ddx  x cos x 1y dydx =  cos x + log x    cos x - x sin x dydx = y    cos x + log x    cos x - x sin x  dydx = xx cos x   cos x + log x    cos x - x sin x         .......(i)Consider  z = x2 + 1x2 - 1

Differentiating with respect to x,

dzdx =  x2 - 1  . ddx x2 + 1  -  x2 + 1  .  ddx x2 - 1   x2 + 1 2=  x2 - 1   2 x  -   x2 + 1   2 x   x2 + 1 2= 2 x3 - 2 x - 2 x3 - 2 x x2 + 1 2= - 4 x x2 + 1 2                                                   ............(ii)

Adding (i)  and  (ii)

ddx  xx cos x + x2 + 1x2 - 1  = dydx + dzdx=  xx cos x  cos x + log x  cos x - x sin x  - 4 x  x2 - 1 2


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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.


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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