List of practice Questions

Which of the following statements are false?
Find the number of automorphisms of the cyclic group $\mathbb{Z}_n$ for $n = 30$.
Let $P$ be a $5 \times 5$ matrix such that $\det(P) = 2$. If $Q$ is the cofactor matrix of $P$, then find $\det(Q)$.
Determine whether the sequence \[ a_n = 1 - (-1)^n + \frac{1}{n} \] is convergent or divergent.
Let $G = P(N)$, where the operation is
\[ A \Delta B = A \cup B - A \cap B \] Which of the following is true?
Solve the system: \[ x + 2y + 2z = 1 \] \[ 2x + 3y + 2z = 2 \] \[ ax + 5y + bz = b \] Find $a + b$ for infinite solutions.
Find the radius of convergence of the series \[ \sum_{n=0}^{\infty} \frac{(n!)^2}{(2n)!}\, x^n. \]
Compare the M–C bond length in the following complexes:
$Na_2[Mn(CO)_4]$, $Na[Mn(CO)_5]$, $Na[Mn(CO)_6]$.
Given $C_p = a + bT$
$a = 19.5$, $b = 0.042$, $n = 1$ mole
Temperature changes from $27^\circ C$ to $327^\circ C$.
Find $\Delta H$.
T-shape geometry is shown by
Which of the following has tetrahedral coordination?
For the reaction: \[ N_2O_3(g) \rightarrow NO(g) + NO_2(g) \] If the degree of dissociation is $\alpha = 0.2$, find $\Delta n_g$ and total number of moles at equilibrium (starting with 1 mole).
The radioactive decay is given as: \[ {}^{238}_{92}U \rightarrow {}^{210}_{82}Pb + 7\alpha + x\beta^{-}. \] Find the value of $x$.
Among the following compounds of Xenon, identify those having exactly one lone pair on Xenon:
$XeF_2$, $XeO_2F_2$, $XeO_3$, $XeO_3F_2$, $XeOF_2$.
Isothermal compressibility of an ideal gas \[ -\frac{1}{V}\left(\frac{dV}{dP}\right)_T \] is equal to
Which of the following has the highest melting point?
Determine the shape of $ClF_3$ using VSEPR theory.
List the compounds of Xenon having one lone pair on the central atom.
Write the van der Waals equation and explain the van der Waals isotherm (P–V curve).
Langmuir isotherm
Find the interplanar spacing $d_{hkl}$ (in Å) using Bragg’s law.
Given: $n = 1$, $\lambda = 1.54\,\text{Å}$
Use Bragg’s law: $n\lambda = 2d\sin\theta$.
Find the number of P–O–P (pyrophosphate) bonds present in $P_4O_8$.

For the non-inverting amplifier shown in the figure, the input voltage is 1 V. The feedback network consists of 2 k$\Omega$ and 1 k$\Omega$ resistors as shown. 
If the switch is open, $V_o = x$. 
If the switch is closed, $V_o = ____ x$. 
 

Consider the system described by the difference equation 
\[ y(n) = \frac{5}{6}y(n-1) - \frac{1}{6}(4-n) + x(n). \] Determine whether the system is linear and time-invariant (LTI). 
 

Find the area of the square shown in the figure whose vertices are at $(0,0)$, $(1,1)$, $(2,0)$ and $(1,-1)$.