Let \(f: R \rightarrow R\) be defined as \(f(x)=3 x+4\). Choose the correct answer.

The value of \(\int_1^2 \frac{\sqrt{x}}{\sqrt{3-x}+\sqrt{x}} d x\) is:

Consider the curves, \(x^2+y^2=1\) and \((x-1)^2+y^2=1\).
Points of intersection of two curves are :

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

The random variable \(X\) can take values 0, 1, 2 . If \(P(X=0)=P(X=1)=\alpha\), and \(E\left(X^2\right)=E(X)\), then which of the following are correct?
\(E(X)=2-3 \alpha\)
\(E\left(X^2\right)=4+7 \alpha\)
\(\alpha=\frac{1}{2}\)
\(\alpha=\frac{1}{5}\)
Choose the correct answer from the options given below :

If \(A=\left[\begin{array}{cc}1 & 0 \\ -1 & 7\end{array}\right]\) and \(A^2-8 A+K I=O\), then the value of \(K\) is:

The following system of equations:
x + y - z = 7
\(4 x+\lambda y-\lambda z=3\)
3x + 2y - 4x = 5
does not possess a solution if the value of \(\lambda\) is:

In a large building, there are 20 bulbs of 40 W each, 5 bulbs of 100 W each, 5 fans of 80 W each, and 1 heater of 1 kW. The voltage supply is 220 V. The minimum capacity of the main fuse of the building should be

Depending upon the immediate urgency of the need to conserve biodiversity, arrange the options given below in a descending order.
(A) Hot spots
(B) Sacred groves
(C) Ex situ method
(D) Bioreserves
Choose the most appropriate answer from the options given below:

If \(f(x) = \begin{vmatrix} 0 & x-a & x-b \\ x+a & 0 & x-c \\ x+b & x+c & 0 \end{vmatrix}\), then \(f(0)\) is:

Two particles x and y having equal charge, after being accelerated through the same potential difference, enter a region of a uniform magnetic field. They describe circular paths of radii \( R_1 \) and \( R_2 \), respectively. The ratio of the mass of x to that of y 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 weakest among the following acid is :"

The equation of a plane passing through the points \((1,2,3),(4,2,3)\) and \((-1,0,6)\) is given by: