If \(\sin x+\sin ^{2} x=1\), then \(\cos ^{8} x+2 \cos ^{6} x+\cos ^{4} x\) is

If the normal to the curve \(y=f(x)\) at the point \((3,4)\) makes and angle \(\left(\frac{3 \pi}{4}\right)^{\mathrm{C}}\) with positive \(\mathrm{X}\)-axis, then \(f^{\prime}(3)\) is equal to

If \(\overline{\mathrm{a}}, \overline{\mathrm{b}}, \overline{\mathrm{c}}\) are non-coplanar vectors and \(\overline{\mathrm{p}}=\frac{\overline{\mathrm{b}} \times \overline{\mathrm{c}}}{[\overline{\mathrm{a}} \overline{\mathrm{b}} \overline{\mathrm{c}}]}, \overline{\mathrm{q}}=\frac{\overline{\mathrm{c}} \times \overline{\mathrm{a}}}{[\overline{\mathrm{a}} \overline{\mathrm{b}} \overline{\mathrm{c}}]}, \overline{\mathrm{r}}=\frac{\overline{\mathrm{a}} \times \overline{\mathrm{b}}}{[\overline{\mathrm{a}} \overline{\mathrm{b}} \overline{\mathrm{c}}]}, \quad\) then \(2 \overline{\mathrm{a}} \cdot \overline{\mathrm{p}}+\overline{\mathrm{b}} \cdot \overline{\mathrm{q}}+\overline{\mathrm{c}} \cdot \overline{\mathrm{r}}=\)

If \(\bar{a}+2 \bar{b}+3 \bar{c}=\overline{0}\) and \((\bar{b} \times \bar{b})+(\bar{b} \times \bar{c})+(\bar{c} \times \bar{a})=\lambda(\bar{b} \times \bar{c})\), then \(\lambda\) has the value

In a triangle \(A B C\), with usual notations, if \(a=5, b=4, \cos (A-B)=\frac{31}{32}\), then \(c=\)

If the variance of the numbers \(2,3,11\) and \(x\) is \(\frac{49}{4}\), then the values of \(x\) are

If the slope of the tangent of the curve at any point is equal to \(-y+\mathrm{e}^{-x}\), then the equation of the curve passing through origin is

In an adiabatic process for an ideal gas, the relation between the universal gas constant ' \(R\) ' and specific heat at constant volume ' \(\mathrm{C}_{\mathrm{v}}\) ' is \(R=0 \cdot 4 C_v\). The pressure ' \(P\) ' of the gas is proportional to the temperature ' T ', of the gas as \(\mathrm{T}^k\). The value of constant ' K ' is

A vertical spring oscillates with period 6 second with mass \(m\) is suspended from it. When the mass is at rest, the spring is stretched through a distance of
(Take, acceleration due to gravity, \(\mathrm{g}=\pi^2=10 \mathrm{~m} / \mathrm{s}^2\) )

Which one of the following is NOT the main postulate of Darwinism?

A parallel combination of three condensers of capacities ' \(2 \mathrm{C}\), ' \(\mathrm{C}\) ' and ' \(\mathrm{C} / 2\) ' is connected
Across \(10 \mathrm{~V}\) battery. All the condensers are fully charged to charges ' \(\mathrm{Q}_1\) ', ' \(\mathrm{Q}_2\) ' and ' \(\mathrm{Q}\) 3' respectively. The ratio \(\mathrm{Q}_1: \mathrm{Q}_2: \mathrm{Q}_3\) is

Match hormones given in Column I with their source gland in Column II.

	Column I		Column II
i	Oxytocin	a.	Adrenal medulla
ii	Melatonin	b.	Thyroid gland
iii	Catecholamines	c.	Pineal gland
Iv	Calcitonin	d.	Hypothalamus

Choose the correct option from below

A single slit diffraction pattern is formed with light of wavelength \(6195 Å\). The second secondary maximum for this wavelength coincides with the third secondary maximum in the pattern for light of wavelength ' \(\lambda_0\) '. The value of ' \(\lambda_0\) ' is