Find the intervals in which the following functions are strictly increasing or decreasing:
$(a) x^2 + 2x − 5 $
$(b) 10 − 6x − 2x^2 $
$(c) −2x^3 − 9x^2 − 12x + 1 $
$(d) 6 − 9x − x^2 $
$(e) (x + 1)^3 (x − 3)^3$

A balloon, which always remains spherical has a variable radius. Find the rate at which its volume is increasing with the radius when the later is 10cm.

If $\sin(sin^{-1}\frac{1}{5}+\cos^{-1}x)=1$ , then find the value of x.

If the system of linear equations $ 8 x+y+4 z=-2 $, $ x+y+z=0 $, $ \lambda x-3 y=\mu$ has infinitely many solutions, then the distance of the point $\left(\lambda, \mu,-\frac{1}{2}\right)$ from the plane $8 x + y +4 z +2=0$ is :

The integrating factor of the differential equation.
$(1-y^{2})\frac{dx}{dy}+yx=ay$$(-1<y<1)$$is$

When a = 0, b = – 1, find the value of the given expressions: (i) 2a+2b (ii) 2a²+b²+1 (iii) 2a²b+2ab²+ab (iv) a²+ab+2

Prove that the perpendicular at the point of contact to the tangent to a circle passes through the centre.

There are 4 letters and 4 directed envelopes. The number of ways in which all the letters can be put in wrong envelope is

How many numbers of 6 digits can be formed from the digits of the number 112233 ?

DL and BM are the heights on sides AB and AD respectively of parallelogram ABCD (Fig 9.15). If the area of the parallelogram is $1470$ $cm^2$ , AB = $35$ $cm$ and AD = $49$ $cm$, find the length of BM and DL.

A diagonal is a line segment that joins any two vertices of the polygon and is not a side of the polygon. Draw a rough sketch of a pentagon and draw its diagonals.

For the function f(x)= e^cosx , Rolle’s theorem is

x + \frac{1}{x} = 2 \cos θ,

then for any integer

n, x^{n} + \frac{1}{x^{n}} =

For the real parameter

t

, the locus of the complex number

z = (1 - t^{2}) + i \sqrt{1 + t^{2}}

The equation of the locus of the point of intersection of the straight lines x sin θ + (1 – cos θ) y = a sin θ and x sin θ – (1 + cos θ) y + a sin θ = 0 is

If x^{2} + y^{2} = 4, then y \frac{d y}{d x} + x =

The diagonals of a rhombus are 7.5 cm and 12 cm. Find its area.

ABC is a triangle in which altitudes BE and CF to sides AC and AB are equal (see Fig). Show that

(i) ∆ ABE ≅ ∆ ACF

(ii) AB = AC, i.e., ABC is an isosceles triangle.