For the curve \(y=5 x-2 x^3\), if \(x\) increases at the rate of 2 units/sec, the rate of change in the slope of the curve at \(x=3\) is ......../sec

\(\frac{\tan A}{1-\cot A}+\frac{\cot A}{1-\tan A}=\)

\(\mathbf{a}, \mathbf{b}\) are non-collinear vectors, \(|\mathbf{a}|=2 \sqrt{2},|\mathbf{b}|=3\) and the angle between \(\mathbf{a}\) and \(\mathbf{b}\) is \(45^{\circ}\). Then, the lengths of the diagonals of the parallelogram whose adjacent sides are represented by the vectors \(5 \mathbf{a}+2 \mathbf{b}\) and \(\mathbf{a}-3 \mathbf{b}\) are

If one of the roots of \(\left|\begin{array}{lll}3 & 5 & x \\ 7 & x & 7 \\ x & 5 & 3\end{array}\right|=0\) is -10 , then the other roots are

A manufacturer can sell \(x\) items at a price of rupees \(\left(5-\frac{x}{100}\right)\) each. The cost price of \(x\) items is Rs. \(\left(\frac{x}{5}+500\right)\). The number of items that the manufacturer should sell to earn the maximum profit is

The equation of the tangent to the curve \(y=x^3-2 x+7\) at the point \((1,6)\) is

If \((h, k)\) is the image of the point \((2,-3)\) with respect to the line \(5 x-3 y=2\), then \(\mathrm{h}+\mathrm{k}=\)

Which of the following is / are ionic in nature?
(i) \(\mathrm{GeF}_4\) (ii) \(\mathrm{SnF}_4\) (iii) \(\mathrm{PbF}_4\)

A block of mass \(10 \mathrm{~kg}\) is placed on a horizontal frictionless surface and is attached to a cord which passes over two light frictionless pulleys as shown in the figure. The hanging block tied to the other end of the cord is initially at rest \(2 \mathrm{~m}\) above the horizontal floor.If the hanging block strikes the floor \(2 \mathrm{~s}\) after the system is released, then weight of the hanging block is ....... \(\left(g=10 \mathrm{~ms}^{-2}\right)\)

If \(b>a\), then \(\int_a^b \frac{d x}{\sqrt{(x-a)(b-x)}}\) is equal to

Observe the following statements :
a. The thermal stability of hydrides of group 16 elements follow the order :
\(\mathrm{H}_2 \mathrm{O}>\mathrm{H}_2 \mathrm{~S}>\mathrm{H}_2 \mathrm{Se}>\mathrm{H}_2 \mathrm{Te}\)
b. Acidic nature of hydrides of group 16 elements follow the order :
\(\mathrm{H}_2 \mathrm{O}>\mathrm{H}_2 \mathrm{~S}>\mathrm{H}_2 \mathrm{Se}>\mathrm{H}_2 \mathrm{Te}\)
c. The reducing nature of \(\mathrm{H}_2 \mathrm{~S}, \mathrm{H}_2 \mathrm{Se}\) and \(\mathrm{H}_2\) Te follow the order :
\(\mathrm{H}_2 \mathrm{~S} < \mathrm{H}_2 \mathrm{Se} < \mathrm{H}_2 \mathrm{Te}\)
The correct statement are:

If \(f(x)\)
\(\left|\begin{array}{ccc}1 & x & x+1 \\ 2 x & x(x-1) & x(x+1) \\ 3 x(x-1) & x(x-1)(x-2) & (x-1) x(x+1)\end{array}\right|\)
then \(f(2012)\) is equal to

If the tangent of the curve \(4 y^3=3 a x^2+x^3\) drawn at the point \((a, a)\) forms a triangle of area \(\frac{25}{24}\) sq.units with the coordinate axes then \(\mathrm{a}=\)