Let \(\vec{a}=\hat{i}+\hat{j}, \vec{b}=\hat{i}-\hat{j}\) and \(\vec{c}=\hat{i}+\hat{j}+\hat{k}\). If \(\hat{m}\) is a unit vector perpendicular to both \(\vec{a}\) and \(\vec{b}\), then \(|\vec{c} \cdot \hat{m}|\) is equal to :

A particle moves along the curve \(6 y=x^3+2\). The point(s) on the curve at which the \(y\)-coordinate is changing 8 times as fast as the \(x\)-coordinate are:

If \(A=\left[\begin{array}{cc}1 & 2 \\ 4 & -3\end{array}\right]\) and \(f(x)=2 x^2-4 x+5\), then \(f(A)\) is equal to

Determine the value of k for which the function \(f(x)=\left\{\begin{array}{ll}\frac{x^2-9}{x-3}, & x \neq 3 \\ k, & x=3\end{array}\right.\) is continuous at x = 3

If \(A=\left[\begin{array}{cc}\sin \alpha & -\cos \alpha \\ \cos \alpha & \sin \alpha\end{array}\right]\), and \(A+A^{\prime}=I\), then the value of \({ }^{\prime} \alpha^{\prime}\) is :

If the objective function \(z=p x+q y\) has its maximum value at the points \((2,1)\) and \((0,6)\), then the relationship between \(p\) and \(q\) is :

If we reject the null hypothesis when it is true, we might be making

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 :

The total EMF of a voltaic cell is greater than zero. Which of the following statements is true?

For strong electrolytes, \(\Lambda_m\) increases with dilution and is represented by the equation \(\Lambda_m=\Lambda_m^0-A c^{1 / 2}\). Here, the
(A) Nature of solvent.
(B) Temperature.
(C) Charges on cation and anions produced.
(D) Type of electrolyte.
Choose the correct answer from the options given below :

The frequency of an input signal is 50 Hz. The frequencies of an output signal for a half wave rectifier and a full wave rectifier will be respectively :

Match List-I with List-II

List - I	List - II
A. Activation Energy	I. Difference in the energy of activated complex and reactants
B. Threshold Energy	II. Minimum Energy possessed by reactant molecules to form effective collisions
C. Heat of Reaction	III. Negative for exothermic reaction
D. Role of a catalyst	IV. To give new path of reaction

Choose the correct answer from the options given below :

The molality of solution obtained by dissolving 1.5 g of ethanoic acid \(CH _3 COOH\) in \(2 5 ~ g\) of benzene is :