The plane \(2 x-(1+\lambda) y+3 \lambda z=0, \lambda \in R\) passes through the intersection of planes :

If \(x=a \sec ^3 \theta, y=a \tan ^3 \theta\) then \(\frac{d^2 y}{d x^2}\) equals.

If \(\left|\begin{array}{ccc}x-4 & 2 x & 2 x \\ 2 x & x-4 & 2 x \\ 2 x & 2 x & x-4\end{array}\right|=(A+B x)(x-A)^2\), then the ordered pair (A, B) is (where A and B are real numbers):

The function \(f:[-1,1] \rightarrow R\) is given by \(f(x)=\frac{x}{x+2}\)

If \(A=\left[\begin{array}{ccc}1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 1\end{array}\right]\), then \(A^{-1}\) is equal to :

Three pipes A, B and C are installed to fill a tank. Pipes A and B opened together can fill the tank in the same time in which \(C\) can alone fill the tank. If pipe \(B\) can fill the tank 15 minutes faster than pipe \(A\) and 5 minutes slower than pipe \(C\), then the time required by pipe \(A\) to fill the tank alone is:

If \(\vec{a}\) and \(\vec{b}\) represent the diagonals of a square, then

If the points \(( a , b ),( c , d )\) and \(( a + c , b + d )\) are collinear, then

In a coil of self inductance 10 H, the rate of change of current is \( 4 \text{ A s}^{-1} \). The electromotive force (emf) induced in the coil will be

Read the passage given below and answer the question.
The transition elements which give the greatest number of oxidation states occur in or near the middle of the series. Manganese, for example, exhibits all the oxidation states from +2 to +7 . The lesser number of oxidation states at extreme ends stems from either too few electrons to lose or share (Sc, Ti) on too many d-electrons (hence fewer orbitals available in which to share electrons with others) for higher valence \(( Cu , Zn )\). Thus early in the series scandium (II) is virtually unknown and titanium (IV) is more stable than Ti (III) or Ti (II). At the other end, the only oxidation state of zinc is \(+ 2\) (no d electrons are involved). The maximum oxidation states of reasonable stability correspond in value to the sum of the s - and d-electrons upto manganese ( \(Ti ^{I V} O _2, V^V O _2^{+}, Cr ^{V I} O _4^{2-}\), \(Mn ^{\text {VII }} O _4\) ) followed by a rather abrupt decrease in stability of higher oxidation states, so that the typical species to follow are \(Fe ^{I I, I I I}, Co ^{I I, I I I}, Ni ^{I I}, Cu ^{I, I I}, Zn ^{I I}\).
The variability of oxidation states, a characteristic of transition elements, arises out of incomplete filling of d-orbitals in such a way that their oxidation states differ from each other by unity, e.g. \(V ^{I I}, V^{I I I}, V^{I V}, V^V\). This is in contrast with the variability of oxidation states of non transition elements where oxidation states normally differ by a unit of two.
An interesting feature in the variability of oxidation states of the d-block elements is noticed among the groups (groups 4 through 10). Although in the p-block, the lower oxidation states are favoured by the heavier members (due to inert pair effect), the opposite is true in the groups of dblock. For example, in group 6, \(Mo ( VI )\) and \(W ( VI )\) are found to be more stable than \(Cr ( VI )\). The \(Cr ( VI )\) in the form of dichromate in acidic medium is a strong oxidising agent, whereas \(MoO _3\) and \(WO _3\) are not.
Low oxidation states are found when a complex compound has ligands capable of \(\pi\)-acceptor character in addition to the \(\sigma\)-bonding. For example, in \(Ni ( CO )_4\) and \(Fe ( CO )_5\), the oxidation state of nickel and iron is zero.
Most acidic species among the following is :

A measure of an average yearly growth of an investment over a certain time period when returns are reinvested is

If the lines \(\frac{1-x}{3}=\frac{y-2}{2 \lambda}=\frac{z-3}{2}\) and \(\frac{x-1}{3 \lambda}=\frac{y-1}{1}=\frac{6-z}{5}\) are perpendicular, then the value of \(\lambda\) is:

Carbon, silicon and germanium have four valence electrons each. Their valence and conduction bands are separated by energy gap \( E_g \). If we represent \( (E_g)_C, (E_g)_{Si} \) and \( (E_g)_{Ge} \) as their respective energy gaps, then which of the following is true