The value of \(\left|\begin{array}{llr}\sin 15^{\circ} & \cos 15^{\circ} & 2 \sin 30^{\circ} \\ \sin 30^{\circ} & \cos 30^{\circ} & \sin 45^{\circ} \\ \cos 75^{\circ} & \sin 75^{\circ} & \sin 90^{\circ}\end{array}\right|\) is

The general solution of the differential equation \(\frac{d y}{d x}=1+x+y+x y\) is given by:

Which of the following matrices is not invertible?

Ram had invested Rs. 15,000 in a mutual fund and the value of the investment at the time of redemption was Rs. 25,000. If the compound annual growth rate (CAGR) is 8.88%, then the number of years for which Ram has invested the amount is: [Given: log 1.089 = 0.0370 and log 1.667 = 0.2220]

For what value of \(k\), the following system of equations has infinitely many solutions?
\(x+2 y=5\), \(3 x+k y=15\)

The plane \(x(1+2 \lambda)+y(2+\lambda)+z(3-\lambda)=0\) passes through the intersection of :

\(\int \frac{1}{(x+1)(x+2)} d x=\)

The slope of normal to the curve \(y=3 x^2-6 x\) at \(x=0\) is:

n-Hexane is obtained when glucose is

Benzene ring in phenol is expected to give the characteristic substitution or replacement reactions i.e. halogenation, nitration, sulphonation, alkylation etc. In these reactions, the attacking reagent is electrophile. The phenolic group activates the ortho and para positions in the ring due to conjugation. Disubstitution reactions are quite fast as the electron density in the ring has increased considerably. Phenols do not respond to nucleophilic substitution reactions as it is influenced by phenolic group.
Victor Meyer test gives following observations. Match List I with List II:

List-I	List-II
A. Blood red colouration	I. Tertiary alcohol
B. Blue colouration	II. Phenols
C. Colourless solution	III. Primary alcohol
D. Violet colouration with \(\text{FeCl}_3\)	IV. Secondary alcohol

Choose the correct answer from the options given below:

For a parallel plate capacitor having plate area \( 1.13 \times 10^3 \, m^2 \) and separation between plates 0.5 cm, the capacitance is

The interval in which \(y=x^2 e^{-x}\) is strictly increasing is:

"Answer the question on basis of passage given below:
\(G = \frac{\kappa A}{l}\) = k (both \(A\) and \(l\) are unity in their appropriate units in m or cm)
Molar conductivity of a solution at a given concentration is the conductance of the volume \(V\) of solution containing one mole of electrolyte kept between two electrodes with area of cross section \(A\) and distance of unit length (\(l\)). Therefore,
\(\Lambda_m = \frac{\kappa A}{l}\)
Since \(l = 1\) and \(A = V\) (volume containing 1 mole of electrolyte),
\(\Lambda_m = \kappa V\)
What does \(\kappa\) represent in the paragraph?"