In a 500 m race, the ratio of speeds of two participants, A and B, is 4 : 5 respectively.
If A has a start of 180 m, then the distance by which A wins is :

The shortest distance between the lines \(\frac{x-1}{2}=\frac{y-2}{3}=\frac{x-3}{4}\) and \(\frac{x-2}{4}=\frac{y-4}{6}=\frac{x-5}{8}\) is equal to:

The corner points of the feasible region determined by the set of constraints (linear inequations) are P(0,5), Q(3, 5), R(5, 0) and S(4, 1) and the objective function z = ax + 2by where a, b > 0 The condition on a and b such that maximum z occurs at Q and S is :

If \(\vec{a}=2 \hat{i}-\hat{j}+\hat{k}, \vec{b}=\hat{i}+\hat{j}-2 \hat{k}, \vec{c}=\hat{i}+3 \hat{j}-\hat{k}\), such that \(\vec{a}\) is perpendicular to \((\lambda \vec{b}+\vec{c})\), then the value of \(\lambda\) is :

The derivative of \((\log x)^x\) with respect to log \(x\) is:

If \(x=a \sec ^3 \theta, y=a \tan ^3 \theta\), then \(\frac{d y}{d x}\) at \(\theta=\frac{\pi}{3}\) is

Area (in sq. units) of the region bounded by curves \(y^2=x\) and \(x=4\) is

Match List - I with List - II

List - I	List - II
(A) Denaturation	(I) Sequence of amino acids in a protein
(B) Primary structure of protein	(II) alpha-Helix and beta-pleated sheets
(C) Quaternary structure of protein	(III) \(2\) degree and \(3\) degree structures gets destroyed
(D) Secondary structure of protein	(IV) The spatial arrangement of subunits (containing \(2\) or more polypeptide chains) with respect to each other

Choose the correct answer from the options given below:

The value of \(\tan ^{-1}(1)+\cos ^{-1}\left(-\frac{\sqrt{3}}{2}\right)+\sin ^{-1}\left(\frac{1}{2}\right)\) is:

Four capacitors are connected in the circuit as shown. The equivalent capacitance between P and Q will be :

The equivalent capacitance between P and Q will be :

Match List-I with List-II

List-I (Electromagnetic Waves)	List-II (Production)
(A) Radio waves	(I) Vibration of atoms and molecules
(B) X-rays	(II) originates from an atomic nucleus
(C) \( \gamma \)-radiation	(III) produced by accelerating electrons in a circuit
(D) infrared waves	(IV) emitted from heavy atoms

Choose the correct answer from the options given below:

Eight cells each having emf 2 V and internal resistance \( r = 0.02 \Omega \) are connected in series with an external resistance 9 \( \Omega \). The current flowing through the external resistance (I) 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 .
Choose the correct statement from the options given below :