Write the intercept cut off by the plane 2x + y – z = 5 on x-axis.
Prove the following:
cot-1 1 + sin x + 1 - sin x 1 + sin x - 1 - sin x = x2, x ∈ 0, π4
Find the value of tan-1 xy - tan-1 x - yx + y
Using properties of determinants, prove that
- a2 ab ac ba -b2 bc ca cb - c2 = 4 a2b2c2
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.
Differentiate X x cos x + x2 + 1x2 - 1 w.r.t. x
If x = a θ - sin θ , y = 1 + cos θ , find d2ydx2
x = a θ - sin θ , y = a 1 + cos θ Differentiating x and y w.r.t. θ,dxdθ = a 1 - cos θ .........(i)dydθ = - a sin θ ..........(ii)Dividing ( 2 ) by ( 1 ),dydθdxdθ = - a sin θ a 1 - cos θ
⇒ dydx = - sinθ1 - cos θ⇒ dydx = - 2 sin θ2 cos θ22 sin2 θ2⇒ dydx =- cos θ2sin θ2⇒ dydx = - cot θ2
Differentiating w.r.t. x,
ddx dydx = ddθ dydx x dθdx⇒ d2ydx2 = ddθ dydx x dθdx⇒ d2ydx2 = ddθ - cot θ2 x dθdx ....[ From equation (iii) ]d2ydx2 = - - cosec2 θ2 x 12 x dθdx = 12 cosec2 θ2 x 1 dxdθ
= 12 cosec2 θ2 x 1a 1 - cos θ ........[From equation (i) ]= cosec2 θ22 a 1 - cos θ = cosec2 θ22 a 2 sin2 θ2 = 14a x cosec4 θ2
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?
Find the points on the curve x2 + y2 – 2x – 3= 0 at whichthe tangents are parallel to x-axis.
Using matrix method, solve the following system of equations:
2x + 3y + 10z = 4, 4x - 6y + 5z, 6x + 9y - 20z; x, y, z ≠ 0