If \(A=\left[\begin{array}{ccc}1 & 2 & 3 \\ -4 & -5 & -6\end{array}\right], \quad B=\left[\begin{array}{cc}2 & -3 \\ 4 & -5 \\ 2 & -1\end{array}\right]\) and \(B A=\left[b_{i j}\right]\), then \(\left(b_{23}-b_{31}\right)\) is equal to

If is equal to \(A=\left[\begin{array}{cc}3 & 1 \\ -1 & 2\end{array}\right]\), then \(A^2-5 A\) (where I is identity matrix of order 2)

For \(\ x \in R -\{-1,0,1\} \), \(\int \frac{1}{x-x^5} d x\) is equal to:

Match List-l with List-II

List - l	List - lI
(A) The chance of observing a specific outcome in an experiment	(I) Parameter
(B) A method used to estimate the population parameters	(II) Estimation
(C) A value that describes an entire population	(III) Statistic
(D) A measurable characteristic of a sample	(IV) Probability distribution

Choose the correct option:

The feasible region of a LPP is bounded. The corresponding objective function is \(Z=6 x-7 y\). Then the objective function attains:

consider the linear programming problem: Maximise \(z=3 x+y\) subject to the constraints \(x+y \leq 6, x \geq 0, y \geq 0\). Which of the following statements are true?
(A) \(x \geq 0, y \geq 0\) are non-negative restrictions.
(B) The point \((1,1)\) does not lie in the half plane \(x+y \leq 6\).
(C) There are 2 corner points of the feasible region of given LPP.
(D) Maximum value of \(z\) is 18.
Choose the correct answer from the options given below:

The acute angle between the lines \(\vec{r}=(4 \hat{i}-\hat{j})+\lambda(2 \hat{i}+\hat{j}-3 \hat{k})\) and \(\frac{x-1}{1}=\frac{y+1}{-3}=\frac{z-2}{2}\) is

Germanium and silicon behave as semiconductors because

At any instant t the values of voltage applied and the current through a series LCR circuit are given as : \( V = \frac{1}{\sqrt{2}} \sin(100\pi t) \text{ V}, I = \frac{1}{\sqrt{2}} \sin(100\pi t + \frac{\pi}{3}) \text{ A} \). The average power consumed in the circuit is given by :

Chromosome number of in human meiocyte is :

Read the passage carefully and answer the questions
Sodium dichromate \(Na _2 Cr _2 O _7\) and Potassium dichromate \(K _2 Cr _2 O _7\) are strong oxidizing agents; in acidic solution, its oxidising action can be represented as follows :
\(Cr _2 O _7^{2-}+14 H ^{+}+6 e^{-} \rightarrow 2 Cr ^{3+}+7 H _2 O \left( E ^{\circ}=1.33 V\right)\)
The chromates and dichromates are interconvertible in aqueous solution depending upon pH of the solution.
The chromate ion is tetrahedral whereas the dichromate ion consists of two tetrahedra sharing one corner with \(Cr - O - Cr\) bond angle of \(126^{\circ}\)
Potassium permanganate is prepared by fusion of \(MnO _2\) with an alkali metal hydroxide and an oxidising agent like \(KNO _3\).
This produces the dark green \(K _2 MnO _4\) which disproportionates in a neutral or acidic solution to give permanganate. It is a strong oxidizing agent. Hydrogen ion concentration of the solution plays 
an important part in the reduction of permanganate to manganate, manganese dioxide and manganese(II) salt.
Potassium permanganate forms dark purple (almost black) crystals which are isostructural with those of \(KClO _4\).
The salt is not very soluble in water \((6.4 g / 100 g\) of water at 293 K ), but when heated it decomposes at 513 K.
The number of moles of KMnO₄ required to react with one mole of sulphide ion in acidic solution is

In a meeting, 70% of the members favour and 30% oppose a certain proposal.
A member is selected at random and we take X = 0 if he opposed, and X = 1 if he is in favour.
Then, E(X) is:

"Answer the question on basis of passage given below :
Carboxylic acids are weaker than mineral acids, but they are stronger acids than alcohols and many simple phenols (\(\text{pKa}\) is ~ 16 for ethanol and 10 for phenol). The higher acidity of carboxylic acids as compared to phenols can be understood similarly. The conjugate base of carboxylic acid, a carboxylate ion, is stabilised by two equivalent resonance structures in which the negative charge is at the more electronegative oxygen atom. The conjugate base of phenol, a phenoxide ion, has non-equivalent resonance structures in which the negative charge is at the less electronegative carbon atom. Further, the negative charge is delocalised over two electronegative oxygen atoms in carboxylate ion whereas it is less effectively delocalised over one oxygen atom and less electronegative carbon atoms in phenoxide ion. Thus, the carboxylate ion is more estabilised than phenoxide ion, so carboxylic acids are more acidic than phenols.
Effect of substituents on the acidity of carboxylic acids: substituents may affect the stabliity of the conjugate base and thus, also affect the acidity of the carboxylic acids. Electron withdrawing group increase the acidity of carboxylic acids by stabilising the conjugate base through delocalisation of the negative charge by inductive and/or resonance effects. Conversely, electron donating groups decrease the acidity by destabilising the conjugate base.
The correct order of decreasing acid strength of:
Acetic acid (A), Formic acid (B), Benzoic acid (C) is:"