The value of the determinant \(\begin{vmatrix} 3 & -4 & 5 \\ 1 & 1 & -2 \\ 2 & 3 & 1 \end{vmatrix}\) is:

If \(\hat{i}, \hat{j}, \hat{k}\) are unit vectors along X, Y, Z axes then,
(A) \(\hat{i} \times \hat{i}=1\)
(B) \(\hat{j} \cdot \hat{j}=0\)
(C) \(\hat{i} \times \hat{j}=\hat{k}\)
(D) \(\hat{k} \times \hat{i}=\hat{j}\)
(E) \(\hat{i} \cdot \hat{j}=0\)
Choose the correct answer from the options given below:

For \(|x|<1\), if \(x=\cos \left(\frac{1}{a} \log y\right)\), then

The value of \(\left|\begin{array}{ccc}1 & 1+p & 1+p+q \\ 2 & 3+2 p & 1+3 p+2 q \\ 3 & 6+3 p & 1+6 p+3 q\end{array}\right|\) is:

Let L be the set of all lines in a plane and R be the relation in L defined as \(R = \{(L_1, L_2) : L_1 \text{ is perpendicular to } L_2\}\) then R is:A. Reflexive
B. Symmetric
C. neither reflexive nor transitive
D. Transitive
E. neither reflexive nor symmetric
Choose the correct answer from the options given below:

A spherical balloon has a variable radius. The rate at which its volume is increasing with respect to its radius when the radius is 5 cm is:

The non-negative remainder when \(7^{30}\) is divided by 5 is :

Read the following statements regarding energy levels, and identify the correct statements.
(A) The energy of an atom is least when its electron is revolving in an orbit closest to the nucleus
(B) the energy levels of the excited states come closer and closer together as principal quantum number increases
(C) ionisation energy of the hydrogen atom is the minimum energy required to free the electron from the ground state of the hydrogen atom
(D) as the excitation of hydrogen atom increases, the value of minimum energy required to free the electron from the excited atom decreases.
Choose the correct answer from the options given below:

Using monochromatic light for diffraction in a single slit of width 0.1 mm , with a central maximum of 5 mm width on a screen 50 cm away, the wavelength of light used is :

Osmotic pressure measurements are less accurate and difficult to carry out. Therefore, this method is not useful for measuring molecular weights. But it is used to find out molecular weights of polymers eg. plastics, proteins and starch having many monomer units. Knowing the value of osmotic pressure of a given solution, molecular weight of the solute can be calculated using Van't Hoff equation.
Determine the Molarity of a solution containing 4g of NaOH in 1L solution

For any events \(A\) and \(B\) of a sample space \(S\), which of the following statements are TRUE?
(A) \(P(S \mid B)=1\)
(B) \(P(A \cap B)=P(A)+P(B)+P(A \cup B)\)
(C) \(P(\bar{A} \mid B)=1-P(A \mid B)\)
(D) \(P(A \mid B)=\frac{P(A \cap B)}{P(B)}, P(B) \neq 0\)
Choose the correct answer from the options given below :

Which of the following animal shows adaptive radiation?

Read the passage carefully and answer the Questions
Molar conductivity \(\left(\Lambda_m\right)\) of a solution at a given concentration (c) is the conductance of volume, V of solution, containing one mole of electrolyte kept between the two electrodes with area of cross-section A and at a distance of unit length. It increases with the decrease in concentration and when the concentration approaches zero, the molar conductivity is called limiting molar
conductivity \(\left(\Lambda_m^0\right)\). For a strong electrolyte, \(\Lambda_m\) increases linearly with dilution and is given by \(\Lambda_m=\Lambda_m^0-A c^{1 / 2}\). The value of the constant A for a given solvent depends on the type of electrolyte along with temperature. According to Kohlrausch law, the value of \(\Lambda_m^0\) for an electrolyte is \(\Lambda_m^0=\nu_{+} \lambda_{+}^0+\nu_{-} \lambda_{-}^0\), where \(\nu_{+}\)and \(\nu_{-}\) are the number of cations and anions, respectively, per molecule of the electrolyte and \(\lambda_{+}^0\) and \(\lambda_{-}^0\) are limiting molar conductivities of cation and anion, respectively. Kohlrausch law finds many applications, like determining the solubility of a sparingly soluble salt, determining the degree of dissociation \(\left(\Lambda_m / \Lambda_m^0\right)\), and the dissociation constant of a weak electrolyte.
The plot between \(\Lambda_m\) and \(c^{1 / 2}\) is a straight line with