If a line makes angles \(\alpha, \beta\) and \(\gamma\) with the positive directions of coordinate axes respectively, then \(\cos 2 \alpha+\cos 2 \beta+\cos 2 \gamma\) is equal to

The equation of the normal to the curve \(y=4 x^3+2 \sin x\) at \((0,3)\) is :

Consider the following L.P.P.
Max. \(z=5 x+2 y\), subject to \(-2 x-3 y \leq-6, x-2 y \leq 2,3 x+2 y \leq 12,-3 x+2 y \leq 3\) and \(x, y \geq 0\) then

Value of \(\tan ^{-1} 2+\tan ^{-1} 3+\tan ^{-1} 4\) is equal to:

If \(a, b, c\) are in A.P., then the value of determinant \(\left|\begin{array}{lll}x+2 & x+3 & x+2 a \\ x+3 & x+4 & x+2 b \\ x+4 & x+5 & x+2 c\end{array}\right|\) is:

The maximum number of equivalence relations on the set \(A=\{3,5,7\}\) is :

If the objective function \(z = 4x + 3y\) has maximum value on a line joining points (3, a) and (b, 2) where \(a > 0, b > 0\) such that \(a - b = 2\) then the maximum value of z is:

Match List-I with List-II

List-I (Contraceptive methods)	List-II (Examples)
(A) Non-medicated IUD	(I) LNG-20
(B) Hormone releasing IUD	(II) Multiload 375
(C) Copper releasing IUD	(III) Vaults
(D) Barrier method	(IV) Lippes loop

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Identify the incorrect statements from the following.
(A) A charged particle of charge e is moving through a magnetic field, work will be done by the magnetic field on the charge.
(B) A conductor with free electrons is placed in a magnetic field. No current is flowing through the conductor. No magnetic force acts on the conductor.
(C) A charged particle is moving perpendicular to the magnetic field. The magnetic force does not change the speed of the particle but changes its direction.
(D) A charged particle is moving in a straight direction without any change of direction. It shows that no magnetic field is there.
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Which of the following statements are correct?
(A) All mirrors follow the laws of reflection.
(B) The angle between the ray of incidence and the plane surface of the mirror is equal to the angle between the plane surface of mirror and the ray of reflection for plane mirror
(C) the rays coming parallel to the principal axis will go after reflection through the focus of the curved mirror
(D) the rays coming to the pole of a curved mirror making an angle with axis will be reflected making the equal angle with the axis on the other side of the
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If the two particles of the same mass and same charge are accelerated with a potential difference of 120 V and 480 V respectively, the ratio of de Broglie's wavelengths associated with the two particles respectively is

Identify the least basic amine in gaseous phase?

"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.
Arrange the following acids in order of their increasing \(\text{pKa}\) value:
(A) Benzoic acid
(B) 2-nitrobenzoic acid
(C) 4-nitrobenzoic acid
(D) 4-methoxybenzoic acid
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