The projection of the vector \(5 \hat{i}+\hat{j}-3 \hat{k}\) on the vector \(\hat{i}+2 \hat{j}-3 \hat{k}\) is

Which of the region shown in the given figures represents the feasible region bounded by the following constraint?
\(4 x+y \geq 80,2 x+y \geq 60, x+y \leq 80, x \geq 0, y \geq 0\)

The value of the determinant \(\left|\begin{array}{ccc}85 & 15 & 30 \\ 1 & 3 & 4 \\ 17 & 3 & 6\end{array}\right|\) is:

\(\int \frac{\sin 2 x d x}{\sqrt{9-\cos ^4 x}}\) equals

The rate of change of volume of a sphere with respect to its surface area, when the radius is 6 cm is:

The inverse of the matrix \(A=\left[\begin{array}{ll}2 & 3 \\ 1 & 2\end{array}\right]\) is:

The general solution of the differential equation \(\frac{d y}{d x}=e^{x-y}+x^2 e^{-y}\) is equal to:

An a.c. voltage source of fixed amplitude \( E_0 \) and variable angular frequency \( \omega \) is connected with an electric bulb of resistance R (inductance zero) and a capacitor of capacitance C in series. On increasing the angular frequency of the source, the total impedance of the circuit

Arrange the following compounds in increasing order of their acidity :
(A) Phenol
(B) 4 -methylphenol
(C) 3-nitrophenol
(D) 4-nitrophenol
Choose the correct answer from the options given below :

For the given values 8, 10, 12, 14, 16, the three-year moving averages are:

Choose the correct statement/statements:
(A) Denaturation of protein causes loss of secondary and tertiary structure
(B) Nucleotides are joined together by diester linkage
(C) DNA contains Adenine, Guanine, cytosine and Uracil
(D) Amino acids show amphoteric behaviour
Choose the correct answer from the options given below :

Which of the following is a hormonal method of contraception?

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