Let a1, a2, ..........., a21 be an AP such that \sum\limits_{n = 1}^{20} {{1 \over {{a_n}{a_{n + 1}}}} = {4 \over 9}}. If the sum of this AP is 189, then a6a16 is equal to :

The function f(x) = {{4{x^3} - 3{x^2}} \over 6} - 2\sin x + \left( {2x - 1} \right)\cos x :

If the tangent to the curve y = x3 at the point P(t, t3) meets the curve again at Q, then the ordinate of the point which divides PQ internally in the ratio 1 : 2 is :

For which of the following curves, the line $$x+\sqrt{3}y=2\sqrt{3}$$ is the tangent at the point \left( {{{3\sqrt 3 } \over 2},{1 \over 2}} \right)?

Let $$f:R\to R$$ be defined as f(x) = \left\{ {\matrix{ { - 55x,} & {if\,x 4,} \cr } } \right. Let A = {x $$\in$$ R : f is increasing}. Then A is equal to :

If the curve y = ax2 + bx + c, x$$\in$$R, passes through the point (1, 2) and the tangent line to this curve at origin is y = x, then the possible values of a, b, c are :

If the curves, {{{x^2}} \over a} + {{{y^2}} \over b} = 1 and {{{x^2}} \over c} + {{{y^2}} \over d} = 1 intersect each other at an angle of 90$${}^{\circ }$$, then which of the following relations is TRUE?

If Rolle's theorem holds for the function $$f(x)=x^3-a{x}^2+bx-4$$, $$x\in [1,2]$$ with f'\left( {{4 \over 3}} \right) = 0, then ordered pair (a, b) is equal to :

The maximum slope of the curve y = {1 \over 2}{x^4} - 5{x^3} + 18{x^2} - 19x occurs at the point :

Let f be any function defined on R and let it satisfy the condition : $$|f(x)-f(y)|\le |(x-y{)}^2|,\forall (x,y)\in R$$ If f(0) = 1, then :

Let slope of the tangent line to a curve at any point P(x, y) be given by {{x{y^2} + y} \over x}. If the curve intersects the line x + 2y = 4 at x = $$-$$2, then the value of y, for which the point (3, y) lies on the curve, is :

Let f be a real valued function, defined on R $$-$$ {$$-$$1, 1} and given by f(x) = 3 loge \left| {{{x - 1} \over {x + 1}}} \right| - {2 \over {x - 1}}. Then in which of the following intervals, function f(x) is increasing?

The maximum value of f(x) = \left| {\matrix{ {{{\sin }^2}x} & {1 + {{\cos }^2}x} & {\cos 2x} \cr {1 + {{\sin }^2}x} & {{{\cos }^2}x} & {\cos 2x} \cr {{{\sin }^2}x} & {{{\cos }^2}x} & {\sin 2x} \cr } } \right|,x \in R is :

Consider the function f : R $$\to$$ R defined by f(x) = \left\{ \matrix{ \left( {2 - \sin \left( {{1 \over x}} \right)} \right)|x|,x \ne 0 \hfill \cr 0,\,\,x = 0 \hfill \cr} \right.. Then f is :

Let $$A=[a_{ij}]$$ be a 3 $$\times$$ 3 matrix, where {a_{ij}} = \left\{ {\matrix{ 1 & , & {if\,i = j} \cr { - x} & , & {if\,\left| {i - j} \right| = 1} \cr {2x + 1} & , & {otherwise.} \cr } } \right. Let a function f : R $$\to$$ R be defined as f(x) = det(A). Then the sum of maximum and minimum values of f on R is equal to:

Let 'a' be a real number such that the function f(x) = ax2 + 6x $$-$$ 15, x $$\in$$ R is increasing in \left( { - \infty ,{3 \over 4}} \right) and decreasing in \left( {{3 \over 4},\infty } \right). Then the function g(x) = ax2 $$-$$ 6x + 15, x$$\in$$R has a :

The sum of all the local minimum values of the twice differentiable function f : R $$\to$$ R defined by f(x) = {x^3} - 3{x^2} - {{3f''(2)} \over 2}x + f''(1) is :

Let f : R $$\to$$ R be defined as f(x) = \left\{ {\matrix{ { - {4 \over 3}{x^3} + 2{x^2} + 3x,} & {x > 0} \cr {3x{e^x},} & {x \le 0} \cr } } \right.. Then f is increasing function in the interval

Let $$f(x)=3{\sin }^{4}x+10{\sin }^{3}x+6{\sin }^{2}x-3$$, x \in \left[ { - {\pi \over 6},{\pi \over 2}} \right]. Then, f is :

The local maximum value of the function f(x) = {\left( {{2 \over x}} \right)^{{x^2}}}, x > 0, is