Differential of $$\log [\log (\log {x^5})]$$ w.r.t. x is

The number of all possible matrices of order 2 $$\times$$ 3 with each entry 1 or 2 is

A function f : R $$\to$$ R is defined as f(x) = x³ + 1. Then the function has

If $$\sin y = x\cos (a + y)$$, then $${{dx} \over {dy}}$$ is

The points on the curve $${{{x^2}} \over 9} + {{{y^2}} \over {25}} = 1$$, where tangent is parallel to x-axis are

Three points P(2x, x + 3), Q(0, x) and R(x + 3, x + 6) are collinear, then x is equal to

The principal value of $${\cos ^{ - 1}}\left( {{1 \over 2}} \right) + {\sin ^{ - 1}}\left( { - {1 \over {\sqrt 2 }}} \right)$$ is

If $${({x^2} + {y^2})^2} = xy$$, then $${{dy} \over {dx}}$$ is

If a matrix A is both symmetric and skew symmetric, then A is necessarily a

Let set X = {1, 2, 3} and a relation R is defined in X as : R = {(1, 3), (2, 2), (3, 2)}, then minimum ordered pairs which should be added in relation R to make it reflexive and symmetric are

A Linear Programming Problem is as follows:

Minimize z = 2x + y

subject to the constraints

x $$\ge$$ 3, x $$\le$$ 9, y $$\ge$$ 0

x $$-$$ y $$\ge$$ 0, x + y $$\le$$ 14

The feasible region has

The function $$f(x) = \left\{ {\matrix{ {{{{e^{3x}} - {e^{ - 5x}}} \over x},} & {if\,x \ne 0} \cr {k,} & {if\,x = 0} \cr } } \right.$$ is continuous at x = 0 for the value of k, as

If C_ij denotes the cofactor of element p_ij of the matrix $$P = \left[ {\matrix{ 1 & { - 1} & 2 \cr 0 & 2 & { - 3} \cr 3 & 2 & 4 \cr } } \right]$$, then the value of $${C_{31}}\,.\,{C_{23}}$$ is

The function $$y = {x^2}{e^{ - x}}$$ is decreasing in the interval

If R = {(x, y); x, y $$\in$$ Z, x² + y² $$\le$$ 4} is a relation in set Z, then domain of R is

The system of linear equations

5x + ky = 5,

3x + 3y = 5; will be consistent if

The equation of the tangent to the curve y (1 + x²) = 2 $$-$$ x, where it crosses the x-axis is

If $$\left[ {\matrix{ {3c + 6} & {a - d} \cr {a + d} & {2 - 3b} \cr } } \right] = \left[ {\matrix{ {12} & 2 \cr { - 8} & { - 4} \cr } } \right]$$ are equal, then value of ab $-$$ cd is

The principal value of $${\tan ^{ - 1}}\left( {\tan {{9\pi } \over 8}} \right)$$ is

For two matrices $$P = \left[ {\matrix{ 3 & 4 \cr { - 1} & 2 \cr 0 & 1 \cr } } \right]$$ and $${Q^T} = \left[ {\matrix{ { - 1} & 2 & 1 \cr 1 & 2 & 3 \cr } } \right]$$ P $$-$$ Q is

The function $$f(x) = 2{x^3} - 15{x^2} + 36x + 6$$ is increasing in the interval

If $$x = 2\cos \theta - \cos 2\theta $$ and $$y = 2\sin \theta - \sin 2\theta $$, then $${{dy} \over {dx}}$$ is

What is the domain of the function $${\cos ^{ - 1}}(2x - 3)$$ ?

A matrix $$A = {[{a_{ij}}]_{3 \times 3}}$$ is defined by $${a_{ij}} = \left\{ {\matrix{ {2i + 3j} & , & {i < j} \cr 5 & , & {i = j} \cr {3i - 2j} & , & {i > j} \cr } } \right.$$

The number of elements in A which are more than 5, is

If a function f defined by $$f(x) = \left\{ {\matrix{ {{{k\cos x} \over {\pi - 2x}}} & , & {if\,x \ne {\pi \over 2}} \cr 3 & , & {if\,x = {\pi \over 2}} \cr } } \right.$$ is continuous at $$x = {\pi \over 2}$$, then the value of k is

For the matrix $$X = \left[ {\matrix{ 0 & 1 & 1 \cr 1 & 0 & 1 \cr 1 & 1 & 0 \cr } } \right]$$, (X² $$-$$ X) is

Let X = {x² : x $$\in$$ N} and the function f : N $$\to$$ X is defined by f(x) = x², x $$\in$$ N. Then this function is

The corner points of the feasible region for a Linear Programming problem are P(0, 5), Q(1, 5), R(4, 2) and S(12, 0). The minimum value of the objective function Z = 2x + 5y is at the point

The equation of the normal to the curve ay² = x³ at the point (am², am³) is

If A is a square matrix of order 3 and |A| = $$-$$ 5, then |adj A| is

The simplest form of $${\tan ^{ - 1}}\left[ {{{\sqrt {1 + x} - \sqrt {1 - x} } \over {\sqrt {1 + x} + \sqrt {1 - x} }}} \right]$$ is

If for the matrix $$A = \left[ {\matrix{ \alpha & { - 2} \cr { - 2} & \alpha \cr } } \right]$$, | A³ | = 125, then the value of $$\alpha$$ is

If $$y = \sin (m{\sin ^{ - 1}}x)$$, then which one of the following equations is true?

The principal value of $$[{\tan ^{ - 1}}\sqrt 3 - {\cot ^{ - 1}}( - \sqrt 3 )]$$ is

The maximum value of $${\left( {{1 \over x}} \right)^{x}}$$ is

Let matrix $$X = [{x_{ij}}]$$ is given by $$X = \left[ {\matrix{ 1 & { - 1} & 2 \cr 3 & 4 & { - 5} \cr 2 & { - 1} & 3 \cr } } \right]$$. Then the matrix $$Y = [{m_{ij}}]$$, where m_ij = Minor of x_ij, is

A function f : R $$\to$$ R defined by f(x) = 2 + x² is

A Linear Programming Problem is as follows:

Maximize/Minimize objective function Z = 2x $$-$$ y + 5

Subject to the constraints

3x + 4y $$\le$$ 60

x + 3y $$\le$$ 30

x $$\ge$$ 0, y $$\ge$$ 0

If the corner points of the feasible region are A (0, 10), B (12, 6), C (20, 0) and O (0, 0) then which of the following is true?

If x = $$-$$4 is a root of $$\left| {\matrix{ x & 2 & 3 \cr 1 & x & 1 \cr 3 & 2 & x \cr } } \right| = 0$$, then the sum of the other two roots is

The absolute maximum value of the function $$f(x) = 4x - {1 \over 2}{x^2}$$ in the interval $$\left[ { - 2,{9 \over 2}} \right]$$ is

In a sphere of radius r, a right circular cone of height h having maximum curved surface area is inscribed. The expression for the square of curved surface of cone is

The corner points of the feasible region determined by a set of constraints (linear inequalities) are P(0, 5), Q(3, 5), R(5, 0) and S(4, 1) and the objective function is Z = ax + 2by where a, b > 0. The condition on a and b such that the maximum Z occurs at Q and S is

If curves y² = 4x and xy = c cut at right angles, then the value of c is

The inverse of the matrix $$X = \left[ {\matrix{ 2 & 0 & 0 \cr 0 & 3 & 0 \cr 0 & 0 & 4 \cr } } \right]$$ is

For an L.P.P. the objective function is Z = 4x + 3y, and the feasible region determined by a set of constraints (linear inequations) is shown in the graph.

Image

Which one of the following statements is true?

In a residential society comprising of 100 houses, there were 60 children between the ages of 10-15 years. They were inspired by their teachers to start composting to ensure that biodegradable waste is recycled. For this purpose, instead of each child doing it for only his/her house, children convinced the Residents Welfare Association to do it as a society initiative. For this they identified a square area in the local park. Local authorities charged amount of Rs.50 per square metre for space so that there is no misuse of the space and Resident Welfare Association takes it seriously. Association hired a labourer for digging out 250 m³ and he charged Rs.400 $$\times$$ (depth)². Association will like to have minimum cost.

Let side of square plot is x m and its depth is h metres, then cost c for the pit is

In a residential society comprising of 100 houses, there were 60 children between the ages of 10-15 years. They were inspired by their teachers to start composting to ensure that biodegradable waste is recycled. For this purpose, instead of each child doing it for only his/her house, children convinced the Residents Welfare Association to do it as a society initiative. For this they identified a square area in the local park. Local authorities charged amount of Rs.50 per square metre for space so that there is no misuse of the space and Resident Welfare Association takes it seriously. Association hired a labourer for digging out 250 m³ and he charged Rs.400 $$\times$$ (depth)². Association will like to have minimum cost.

Value of h (in m) for which $${{dc} \over {dh}} = 0$$ is

In a residential society comprising of 100 houses, there were 60 children between the ages of 10-15 years. They were inspired by their teachers to start composting to ensure that biodegradable waste is recycled. For this purpose, instead of each child doing it for only his/her house, children convinced the Residents Welfare Association to do it as a society initiative. For this they identified a square area in the local park. Local authorities charged amount of Rs.50 per square metre for space so that there is no misuse of the space and Resident Welfare Association takes it seriously. Association hired a labourer for digging out 250 m³ and he charged Rs.400 $$\times$$ (depth)². Association will like to have minimum cost.

$${{{d^2}c} \over {d{h^2}}}$$ is given by

In a residential society comprising of 100 houses, there were 60 children between the ages of 10-15 years. They were inspired by their teachers to start composting to ensure that biodegradable waste is recycled. For this purpose, instead of each child doing it for only his/her house, children convinced the Residents Welfare Association to do it as a society initiative. For this they identified a square area in the local park. Local authorities charged amount of Rs.50 per square metre for space so that there is no misuse of the space and Resident Welfare Association takes it seriously. Association hired a labourer for digging out 250 m³ and he charged Rs.400 $$\times$$ (depth)². Association will like to have minimum cost.

Value of x (in m) for minimum cost is

In a residential society comprising of 100 houses, there were 60 children between the ages of 10-15 years. They were inspired by their teachers to start composting to ensure that biodegradable waste is recycled. For this purpose, instead of each child doing it for only his/her house, children convinced the Residents Welfare Association to do it as a society initiative. For this they identified a square area in the local park. Local authorities charged amount of Rs.50 per square metre for space so that there is no misuse of the space and Resident Welfare Association takes it seriously. Association hired a labourer for digging out 250 m³ and he charged Rs.400 $$\times$$ (depth)². Association will like to have minimum cost.

Total minimum cost of digging the pit (in Rs.) is