and 
nuclei have the same mass number. Do they have the same binding energy?Despite the fact that $H e_{2}^{3}$ and $H e_{1}^{3}$ have the same mass number, their binding energies are different. Because the number of protons and neutrons in both nuclei differs, the binding...
Solution: (i) Provided, $\mathrm{g}=\mathrm{n},(1 / \mathrm{n}) \mid \mathrm{n}$ is a positive integer As a result, the element n is a positive integer and the corresponding 1/n will be a unique and...
Solution: (i) Provided, h = {(4, 6), (3, 9), (– 11, 6), (3, 11)} Hence, element 3 has two images, namely, 9 and 11. If every element of one set has one and only one image in other set then a...
and 
are the disintegration constants of A and B respectively. In which of the following cases, the two samples can simultaneously have the same decay rate at any time?
and 
The correct options are: (b) Initial rate of decay of A is twice the initial rate of decay of B and $\lambda _A>\lambda_B$ (d) Initial rate of decay of B is the same as the rate of decay of A at...
The correct option is: (b) elastic collision of neutrons with heavy nuclei will not slow them down
. If one of the neutrons in Triton decays, it would transform into He3 nucleus. This does not happen. This is because
nucleus nucleusThe correct option is: (a) Triton energy is less than that of a $\mathrm{He}^{3}$ nucleus
and 
denote the atomic masses of the parent and the daughter nuclei respectively in a radioactive decay. The Q-value for a 
– decay is 
and that for a 
decay is 
. If me denotes the mass of an electron, then which of the following statements is correct?
and 
and 
and 
and The correct option is: (a) $Q_ 1=(M _x-M_ y) c^{2}$ and $Q_ 2=(M _x-M_ y-2 m_ e) c^{2}$
= {(x, |x| ) |x is a real number} is a relation. Then find domain and range of .Solution: Provided, $\mathrm{R}_3=\{(\mathrm{x},|\mathrm{x}|) \mid \mathrm{x}$ is a real number $\}$ is a relation All the first elements of all the ordered pairs of R3, i.e., x, are included in...
The correct option is: (b) change for α and β radioactivity but not for γ-radioactivity
= {(x, y) | x and y are integers and x2 + y2 = 64} is a relation. Then find .Solution: Given, $\mathrm{R}_2=\{(\mathrm{x}, \mathrm{y})$ x and y are integers and $\left.x^{2}+y^{2}-64\right\}$ As a result, we have, $x^{2} = 0$ and $y^{2} = 64$ or $x^{2} = 64$ and $y^{2} = 0$...
The correct option is: (c) the containers will, in general, have different numbers of the atoms of the material but their average will be close to 5000
= {(x, y) | y = 2x + 7, where x ∈ R and – 5 ≤ x ≤ 5} is a relation. Then find the domain and Range of .Solution: Provided, $R_1$ $=\{(x, y) \mid y=2 x+7$, in which x ∈R and ${-} 5\leq x\leq 5}$ is a relation All the first elements of all the ordered pairs of $R_1$, i.e., x, are included in the domain...
Solution: Provided, R $=\left\{(x, y): x, y \in W,x^{2}+y^{2}=25\right\}$ R $=\{(0,5),(3,4),(4,3),(5,0)\}$ R's domain consists of all the first elements of all the ordered pairs of R. Domain of R...
x + y > 8 Finding the ordered pair such that x + y>8, where x and y belong to the given set A: 1 + 1 = 2<8 1 + 2 = 3<8 1 + 3 = 4<8 1 + 4 = 5<8 1 + 5 = 6<8 2 + 1 = 3<8 2 + 2 =...
are thirty sets each having 5 elements and 
are 
sets each with 3 elements, let 
and each element of 
belongs to exactly 10 of the 
‘s and exactly 9 of the B,’S. then is equal to A. 15 B. 3 C. 45 D. 35Solution: As per the question, $\bigcup_{i=1}^{30} A_{i}=\bigcup_{j=1}^{n} B_{j}=S$ As the elements are not repeating, no. of elements in $\mathrm{A}_{1} \cup \mathrm{A}_{2} \cup \mathrm{A}_{3} \cup...
Given, A = {1, 2, 3, 4, 5}, S = {(x, y) : x ∈A, y ∈A} (i) x + y = 5 Finding the ordered pair such that x + y = 5, where x and y belong to the given set A: 1 + 1 = 2≠5 1 + 2 = 3≠5 1 + 3 = 4≠5 1 + 4 =...
Solution: Given, $50 =$ Total number of students $13 =$ Number of students studying English $17 =$ Number of students studying French $15 =$ Number of students studying Sanskrit $9 =$ Number of...
Solution: Given, $50 =$ Total number of students $13 =$ Number of students studying English $17 =$ Number of students studying French $15 =$ Number of students studying Sanskrit $9 =$ Number of...
Solution: Given, $50 =$ Total number of students $13 =$ Number of students studying English $17 =$ Number of students studying French $15 =$ Number of students studying Sanskrit $9 =$ Number of...
Solution: Given, $50 =$ Total number of students $13 =$ Number of students studying English $17 =$ Number of students studying French $15 =$ Number of students studying Sanskrit $9 =$ Number of...
Solution: As per the question, $=10,000$ = Total number of families $\mathrm{A}=\mathrm{n}(\mathrm{A})=40 \%$ = Number of families buying newspaper $B=n(B)=20 \%$ = Number of families buying...
(i) Given, (2a + b, a – b) = (8, 3) The corresponding elements will be equal because the ordered pairs are equal. Hence, 2a + b = 8 and a–b = 3 Now a–b = 3 ⇒a = 3 + b On putting the value of a in...
Given, P = {x: x < 3, x ∈N}, Q = {x : x ≤ 2, x ∈W} where W is the set of whole numbers P = {1, 2} Q = {0, 1, 2} Now (P∪Q) = {1, 2}∪{0, 1, 2} = {0, 1, 2} And, (P∩Q) = {1, 2}∩{0, 1, 2} = {1, 2}...
Given, A = {–1, 2, 3} and B = {1, 3} (i) A × B {–1, 2, 3} × {1, 3} So, A × B = {(–1, 1), (–1, 3), (2, 1), (2, 3), (3, 1), (3, 3)} As a result, the Cartesian product is {(–1, 1), (–1, 3), (2, 1), (2,...
Solution: As per the question, n(U) = 200 = Total number of students n(M) = 120 = Number of students who study Mathematics n(P) = 90 = Number of students who study Physics n(C) = 70 = Number of...
Solution: As per the question, 60 = Total number of students 25 = Students who play cricket 20 = Students who play tennis 10 = Students who play both the games We need to find: the number of...
Solution: As per the question, 100 = Total number of students 15 = Number of students who passed in English 12 = Number of students who passed in Mathematics 8 = Number of students who passed in...
Solution: As per the question, 100 = Total number of students 15 = Number of students who passed in English 12 = Number of students who passed in Mathematics 8 = Number of students who passed in...
Solution: As per the question The three given sets are A, B and C We need to prove: $A \cap(B \cup C)=(A \cap B) \cup(A \cap C)$ Let's say $x \in A \cap(B \cup C)$ $\Rightarrow x \in A$ and $x \in(B...
Is T an empty set? Justify your answer.Solution: As per the question T$=\left\{x \mid \frac{x+5}{x-7}-5=\frac{4 x-40}{13-x}\right\}$ We need to check whether T is an empty set or not, Now solve, $\frac{x+5}{x-7}-5=\frac{4 x-40}{13-x}$...
Solution: As per the question, Two sets are A and B We need to prove: $(A \cup B)-B=A-B$ Left Hand Side $=(A \cup B)-B$ As, $A-B=A \cap B^{\prime}$, we have, $=(A \cup B) \cap B^{\prime}$ As,...
Solution: As per the question, Two sets A and B We need to prove: $A-(A \cap B)=A-B$ Left Hand Side =$\mathrm{A}-(\mathrm{A} \cap \mathrm{B})$ As, $A-B=A \cap B^{\prime}$, we have, $=A \cap(A \cap...
Solution: As per the question, Two sets are $A$ and $B$ We need to prove: $A-(A-B)=A \cap B$ Left Hand Side =$\mathrm{A}-(\mathrm{A}-\mathrm{B})$ As, $A-B=A \cap B^{\prime}$, we have,...
Solution: As per the question, Two sets A and B We need to prove: $A \cup(B-A)=A \cup B$ Left Hand Side $=A \cup(B-A)$ As, $A-B=A \cap B^{\prime}$, we have, $=A \cup\left(B \cap A^{\prime}\right)$...
Solution: The statement is true As per the question, Three sets are A, B and C We need to check: if $A \subset C$ and $B \subset C$, therefore $A \cup B \subset C$ is true or false Let's say $x \in...
Solution: The statement is true As per the question, Three sets are A, B and C We need to check: if $A \subset B$, therefore $A \cup C \subset B \cup C$ is true or false Let's say $x \in A \cup C$...
Solution: The statement is true As per the question, Three sets are A, B and C We need to check: if $A \subset B$, therefore $A \cap C \subset B \cap C$ is true or false Let's say $x \in A \cap C$...
Solution: The statement is false As per the question, Three sets are A, B and C We need to check: $A-(B-C)=(A-B)-C$ is true or false. Step 1: $B-C$ Step 2: $A-(B-C)$ Step 3: $A-B$ Step 4:...
and 
Show that
i) The energy density due to electric field $E$ is given as $\mathrm{uE}=1 / 2 \varepsilon_{0} \mathrm{E}^{2}$ The energy density due to magnetic field $B$ is givena as $\mathrm{uB}=1 / 2...
Solution: The statement is true As per the question, Two sets A and B We need to check: $(A-B) \cup(A \cap B)=A$ is true or false Left Hand Side $=(A-B) \cup(A \cap B)$ As, $A-B=A \cap B'$ , We...
to prove 
, prove that 
a) Substituting the above equations in the following equation we get ${c} \oint E . d l=-\frac{d \phi_{B}}{d t}=-\frac{d}{d t} \oint B \cdot d s$ So, $E_{0} / B_{0}=0$ b) We get $c=1 /...
Solution: As per the question, A, B and C are three sets We need to show: $(\mathrm{A}-\mathrm{B}) \cap(\mathrm{A}-\mathrm{C})=\mathrm{A}-(\mathrm{B} \cup \mathrm{C})$ $\text { Let's say }...
is placed symmetrically along the z-axis and has radius 
. The cable consists of a thin wire and a co-axial conducting tube. An alternating current 
sin 
flows down the central thin wire and returns along the co-axial conducting tube. The induced electric field at a distance 
from the wire inside the cable is 
. In 
, 
The displacement will be, $I_{0}^{\mathrm{d}} / \mathrm{I}_{0}=(\mathrm{am} / \lambda)^{2}$
SOLUTION: \[\begin{array}{*{35}{l}} x\text{ }+\text{ }2y\text{ }\le \text{ }3 \\ {} \\ \end{array}\] \[Line:\text{ }x\text{ }+\text{ }2y\text{ }=\text{ }3\] x 3 1 y 0 1 Also, (0, 0) satisfies...
has permittivity 
, permeability 
and resistivity 
. Imagine a parallel plate capacitor immersed in seawater and driven by an alternating voltage source 
What fraction of the conduction current density is the displacement current density?The separation between the plates of the capacitor is given as $V(t)=V_{0} \sin (2 \pi v t)$ Ohm's law for the conduction of current density is given as $\mathrm{J}_{0}{...
Solution: As per the question, Four sets given here are U, G, B, S So here, A universal set containing set of all boys and girls in a school = U Set of all girls in the school = G Set of all boys in...
is placed along the z-axis. The wire is set into motion along its length with a uniform velocity 
Calculate the pointing vectors 
The electric field in an infinitely long thin wire is given by the expression, $\vec{E}=\frac{\lambda \hat{e}_{s}}{2 \pi \epsilon_{0} a} \hat{j}$ Magnetic field due to the wire is given by the...
Solution: As per the question, $A=\{2,4,6,8,12,20\}, B=\{3,6,9,12,15\}, C=\{5,10,15,20\}$ U is a universal set here Hence, $\begin{array}{l} \Rightarrow \mathrm{A} \cap \mathrm{B}=\{2,4,6,8,12,20\}...
SOLUTION: Considering \[x\text{ }+\text{ }y\text{ }=\text{ }8,\] The shaded region and the origin both are on the same side of the graph of the line and (0, 0) satisfy the constraint \[x\text{...
SOLUTION: According to the question, Considering \[3x\text{ }+\text{ }2y\text{ }=\text{ }48,\] The shaded region and the origin both are on the same side of the graph of the line and (0, 0) satisfy...
Solution: For above fraction be greater than 0, either both denominator and numerator should be greater than 0 or both should be less than 0. \[\begin{array}{*{35}{l}} \Rightarrow ~6\text{ }\text{...
\[T\text{ }=\text{ }30\text{ }+\text{ }25\left( x\text{ }\text{ }3 \right),\text{ }3\text{ }\le \text{ }x\text{ }\le \text{ }15;\] where, T = temperature and x = depth inside the earth The...
Let the length of shortest side = ‘x’ cm According to the question, The longest side of a triangle is twice the shortest side ⇒ Length of largest side = 2x Also, the third side is 2 cm longer than...
Let temperature in Celsius be C Let temperature in Fahrenheit be F According to the question, Solution should be kept between 40° C and 45°C ⇒ 40 < C < 45 Multiplying each term by 9/5, we get...
Despite the fact that an electric field E imposes a force qE on a charged particle, the electric field of an EM wave has no effect on radiation pressure because radiation pressure is the product of...
According to the question, Let x litres of 3% solution is to be added to 460 liters of the 9% of solution Then, we get, Total solution = \[\left( 460\text{ }+\text{ }x \right)\text{ }litres\] Total...
Energy received by the surface per second is given as $E = IA$ Number of photons received by the surface per second is given as $N$ The perfect absorbing can be expressed as $h/ \lambda$ Hence,...
According to the question, First reading = 8.48 Second reading = 8.35 Now, let the third reading be ‘x’ Average pH should be between 8.2 and 8.5 Average \[pH~=\text{ }\left( 8.48\text{ }+\text{...
parallel plate capacitor. How would you establish an instantaneous displacement current of 
in the space between its plates?The capacitance of the capacitor is given by $\mathrm{C}=2 \mu \mathrm{F}$ Displacement current is given as $I_{d}=1 \mathrm{~mA}$ Hence, Charge in capacitor will be, $q=C V$ $\mathrm{dV} /...
Since, Profit = Revenue – cost Requirement is, profit > 0 \[C\left( x \right)\text{ }=\text{ }26,000\text{ }+\text{ }30\text{ }x;\] where x is number of cassettes \[\begin{array}{*{35}{l}}...
is given by 
Radiant flux density is given as $\vec{S}=\frac{1}{\mu_{0}}\left(\vec{E} \times \vec B\right)=c^{2} \epsilon_{0}\left({\vec{E}} \times \vec B\right)$ $\mathrm{E}=\mathrm{E}_{0} \cos...
\[\begin{array}{*{35}{l}} 4x\text{ }+\text{ }3\text{ }\ge \text{ }2x\text{ }+\text{ }17 \\ \Rightarrow ~4x\text{ }\text{ }-2x\text{ }\ge \text{ }17\text{ }\text{ }-3 \\ \Rightarrow ~2x\text{ }\ge...
Solution: Multiplying each term by 4, we get \[\Rightarrow ~-20\text{ }\le \text{ }2\text{ }\text{ }-3x\text{ }\le \text{ }36\] Adding -2 each term, we get \[\Rightarrow ~-22\text{ }\le \text{...
\[\left| x\text{ }\text{ }-1 \right|\le \text{ }5\] There are two cases, 1:-\[x\text{ }\text{ }-1\text{ }\le \text{ }5\] Adding 1 to LHS and RHS \[\begin{array}{*{35}{l}} \Rightarrow ~x\text{ }\le...
Solution: \[\begin{array}{*{35}{l}} \Rightarrow ~5-\text{ }\text{ }\left| x \right|\text{ }\le \text{ }0\text{ }and\text{ }\left| x \right|\text{ }\text{ }-3\text{ }>\text{ }0\text{ }or\text{...
Solution: Hence, \[\begin{array}{*{35}{l}} 1\text{ }\le \text{ }y\text{ }<\text{ }2 \\ \Rightarrow ~1\text{ }\le \text{ }\left| x-\text{ }\text{ }2 \right|\text{ }<\text{ }2 \\ \end{array}\]...
SOLUTION: Multiplying each term by \[\begin{array}{*{35}{l}} \left( x\text{ }+\text{ }1 \right) \\ \Rightarrow ~4\text{ }\le \text{ }3\left( x\text{ }+\text{ }1 \right)\text{ }\le \text{ }6 \\...
Let z = \[\left| z \right|\text{ }(cos\text{ }{{\theta }_{1}}~+\text{ }I\text{ }sin\text{ }{{\theta }_{1}})\text{ }and\text{ }w\text{ }=\text{ }\left| w \right|\text{ }(cos\text{ }{{\theta...
at a point in between the plates of a parallel plate capacitor during charging is 
Let $I_{d}$ be the displacement current in the magnetic field region between two parallel plate capacitor plates. The magnetic field induction at a point between two capacitor plates at a...
SOLUTION:
\[z\text{ }+\text{ }\surd 2\text{ }\left| \left( z\text{ }+\text{ }1 \right) \right|\text{ }+\text{ }i\text{ }=\text{ }0\text{ }\ldots \text{ }\left( 1 \right)\] Substituting\[z\text{ }=\text{...
\[\begin{array}{*{35}{l}} Re\text{ }({{z}^{2}})\text{ }=\text{ }0,\text{ }\left| z \right|\text{ }=\text{ }2 \\ Let\text{ }z\text{ }=\text{ }x\text{ }+\text{ }iy. \\ Then,\text{ }\left| z...
is defined as a vector whose magnitude is equal to the wave intensity and whose direction is along the direction of wave propagation. Mathematically, it is given by 
. Show that nature of versus 
graph.Capacitive reaction is given as, $\mathrm{Xc}=1 / 2 \mathrm{mf} \mathrm{C}$ As the frequency decreases, $X_ c$ rises, and the conduction current becomes inversely proportional to $X_ c$.
Let \[{{z}_{1}}~=\text{ }\left| {{z}_{1}} \right|\text{ }\left( cos\text{ }{{\theta }_{1}}~+\text{ }I\text{ }sin\text{ }{{\theta }_{1}} \right)\text{ }and\text{ }{{z}_{2}}~=\text{ }\left| {{z}_{2}}...
. The plates are very large and close together. Neglecting the edge effects, find the displacement current through the capacitor?Displacement current through the capacitor is given as, $\mathrm{Id}=\mathrm{Ic}=\mathrm{dq} / \mathrm{dt}$ Given, $q=q_{0} \cos 2 \pi v t$ On putting the values, we get $\mathrm{Id}=\mathrm{Ic}=-2...
Because the microwave's frequency and the resonant frequency of the water molecules are the same, the microwave oven cooks a food item containing water molecules most efficiently.
As electromagnetic waves are plane polarised, the antenna must be parallel to the vibration of the fields of the wave. The orientation of the portable radio with respect to the transmitting station...
on it. Which of the following are true?
if the wave is totally absorbed
if the wave is totally reflected
if the wave is totally reflected
for real surfaceThe correct options are: a) radiation pressure is $\mathrm{l} / \mathrm{c}$ if the wave is totally absorbed c) radiation pressure is $2 \mathrm{l} / \mathrm{c}$ if the wave is totally reflected d)...
. The electromagnetic waves produced:
The correct options are: a) will have a frequency of $10^{9} \mathrm{~Hz}$ c) will have a wavelength of $0.3 \mathrm{~m}$ d) fall in the region of radiowaves
\[\begin{array}{*{35}{l}} \left| {{z}_{1}} \right|\text{ }=\text{ }\left| {{z}_{2}} \right|\text{ }=\text{ }\ldots \text{ }=\text{ }\left| {{z}_{n}} \right|\text{ }=\text{ }1 \\ \Rightarrow...
z1 and z2 are conjugate complex numbers. The negative side of the real axis \[\begin{array}{*{35}{l}} =\text{ }{{r}_{1}}~\left( cos\text{ }{{\theta }_{1}}~\text{ }-i\text{ }sin\text{ }{{\theta...
Let z1 = x + iy Therefore, the real part of z2 is zero.
According to the question, Let \[{{z}_{1}}~=\text{ }|{{z}_{1}}|\text{ }(cos\text{ }{{\theta }_{1}}~+\text{ }I\text{ }sin\text{ }{{\theta }_{1}})\text{ }and\text{ }{{z}_{2}}~=\text{...
Solution: (i) As per the question, $Y=\{1,2,3, \ldots, 10\}$ in which a represents any element of $Y$ $Y=\{1,2,3, \ldots, 10\}$ $a$ is less than $6 \Rightarrow 1,2,3,4,5$ $1,2,3,4,5$ satisfy the...
Solution: (i) As per the question, $Y=\{1,2,3, \ldots, 10\}$ in which a represents any element of $Y$ $\begin{array}{l} Y=\{1,2,3, \ldots, 10\} \\ 1^{2}=1,2^{2}=4,3^{2}=9 \end{array}$ $1,4,9 \in Y...
\[Let\text{ }z\text{ }=\text{ }x\text{ }+\text{ }iy\] Now, \[\begin{array}{*{35}{l}} \Rightarrow ~{{x}^{2}}~\text{ }-1\text{ }+\text{ }{{y}^{2}}~=\text{ }0 \\ \Rightarrow ~{{x}^{2}}~+\text{...
\[\begin{array}{*{35}{l}} arg(\left| \left( z\text{ }\text{ }-2 \right)\text{ }/\text{ }\left( z-\text{ }\text{ }3 \right) \right|\text{ })=\text{ }2 \\ Substituting\text{ }z\text{ }=\text{...
Solution: (i) As per the question, $X=\{1,2,3\}$ in which $n$ represents any member of $X$ $X=\{1,2,3\}$ (ii) As per the question, $X=\{1,2,3\}$ in which $n$ represents any member of $X$...
Let \[z\text{ }=\text{ }x\text{ }+\text{ }iy\] Given that, \[arg\text{ }\left( z-\text{ }\text{ }1 \right)\text{ }=\text{ }arg\text{ }\left( z\text{ }+\text{ }3i \right)\] \[\Rightarrow ~arg\text{...
Solution: (i) As per the question, $X=\{1,2,3\}$ in which $n$ represents any member of $X$ $\begin{array}{l} X=\{1,2,3\} \\ \{4 n \mid n \in x\} \\ =\{4 \times 1,4 \times 2,4 \times 3\} \\...
\[\left| z\text{ }+\text{ }1 \right|\text{ }=\text{ }z\text{ }+\text{ }2\text{ }\left( 1\text{ }+\text{ }i \right)\] Substituting z = x + iy, we get, \[\Rightarrow ~\left| x\text{ }+\text{ }iy\text{...
Solution: It is known that, A set 'A' is a subset of a set 'B', if 'A' is "contained" inside 'B'. As a result, all elements of 'A' are also elements of 'B'. (i) As per the question, N$=\{1,2,3,...
As per the inquiry, We have, \[\left| z \right|\text{ }=\text{ }z\text{ }+\text{ }1\text{ }+\text{ }2i\] Subbing z = x + iy, we get, \[\begin{array}{*{35}{l}} \Rightarrow \left| x\text{ }+\text{ }iy...
Solution: (i) As per the question, Two subsets are A and B We need to prove: $(A \cap B) \subset A$ Proof: Let's say $x \in A \cap B$ $\Rightarrow x \in A$ and $x \in B$ $\Rightarrow \mathrm{X} \in...
Solution: (i) As per the question, $A$ and $B$ are two subsets We need to prove: $A \subset A \cup B$ Proof: Let's say $x \in A$ $\Rightarrow \mathrm{x} \in \mathrm{A}$ or $\mathrm{x} \in...
Let \[z\text{ }=\text{ }x\text{ }+\text{ }iy\] arg \[\left( \left( z-1 \right)/\left( z+1 \right) \right)\text{ }=\text{ }\pi /4\] \[\Rightarrow ~arg\text{ }\left( z\text{ }\text{ }1 \right)\text{...
Let z = x + iy Now, \[\Rightarrow ~{{x}^{2}}~+\text{ }x\text{ }\text{ }-2\text{ }+\text{ }{{y}^{2}}~=\text{ }4\text{ }\left( {{x}^{2}}~\text{ }-2x\text{ }+\text{ }1\text{ }+\text{ }{{y}^{2}}...
\[z\text{ }=\text{ }x\text{ }+\text{ }iy\] ⇒ z̅ = x – iy Now, we also have, z z̅ + 2 (z + z̅) + b = 0 \[\Rightarrow ~\left( x\text{ }+\text{ }iy \right)\text{ }\left( x\text{ }\text{ }iy...
SOLUTION: = -iz̅ Hence proved.
SOLUTION: a = cos θ + i sin θ
then find (a, b). Solution: = (i4)25 = 1 Hence, (a, b) = (1, 0)
FIG 1: SOLUTION: We have,
fig 1: SOLUTION: We have,
where n ϵ N. SOLUTION: WE have,
A/Q = (1 – i)n (1 + i)n = (1 – i2)n = 2n Therefore, (1 – i)n (1 – 1/i)n = 2n
Let P(n):\[1\text{ }+\text{ }2\text{ }+\text{ }{{2}^{2}}~+\text{ }\ldots \text{ }+\text{ }{{2}^{n}}~=\text{ }{{2}^{n}}{{~}^{+1}}~\text{ }1\] , for all natural numbers n \[P\left( 1 \right):\text{...
As per the inquiry, \[\Rightarrow P\left( k+1 \right)\] is valid when P(k) is valid. Along these lines, by Mathematical Induction, \[\surd n\text{ }<\text{ }1/\surd 1\text{ }+\text{ }1/\surd...
Solution: As per the question, $\mathrm{L}=\{1,2,3,4\}, \mathrm{M}=\{3,4,5,6\}$ and $\mathrm{N}=\{1,3,5\}$ We need to verify: $\begin{array}{l} L-(M \cup N)=(L-M) \cap(L-N) \\ M=\{3,4,5,6\},...
As indicated by the inquiry, \[P\left( n \right)\text{ }is\text{ }n\hat{\ }^2\text{ }<\text{ }2n\text{ }for\text{ }n\ge 5\] Let \[P\left( k \right)\text{ }=\text{ }k^2\text{ }<\text{ }2k\] be...
Solution: (i) The statement is true As per the question, $\begin{array}{l} 3 \notin\left\{x \mid x^{4}-5 x^{3}+2 x^{2}-112 x+6=0\right\} \\ x^{4}-5 x^{3}+2 x^{2}-112 x+6=0 \end{array}$ When we put...
As per the inquiry, \[P\left( n \right)\text{ }=\text{ }n\left( n^2\text{ }+\text{ }5 \right)\] is distinguishable by 6. Along these lines, subbing various qualities for n, we get, \[P\left( 0...
As indicated by the inquiry, \[P\left( n \right)\text{ }=\text{ }xn\text{ }\text{ }yn\] is detachable by \[x\text{ }\text{ }y,\text{ }x\] whole numbers with\[x\text{ }\ne \text{ }y\] . In this way,...
As indicated by the inquiry, \[P\left( n \right)\text{ }=\text{ }32n\text{ }\text{ }1\] is distinct by 8. Along these lines, subbing various qualities for n, we get, \[P\left( 0 \right)\text{...
-direction can have the following pairs of 
and 
The correct options are: b) $E_y, \mathbf{B_ z}$ d) $E_z, B_y$
As indicated by the inquiry, \[P\left( n \right)\text{ }=\text{ }23n\text{ }\text{ }1\] is detachable by 7. In this way, subbing various qualities for n, we get, \[P\left( 0 \right)\text{ }=\text{...
-axis is given as: 
. Choose the correct options from the following a) the associated magnetic field is given as 
a) the associated magnetic field is given as $B=\frac{1}{c} \hat{k} \times E=\frac{1}{\omega}(\hat{k} \times E)$ b) the electromagnetic field can be written in terms of the associated magnetic field...
Solution: (i) As per the question, $F=\left\{x \mid x^{4}-5 x^{2}+6=0, x \in R\right\}$ Roster form, $\begin{array}{l} x^{4}-5 x^{2}+6=0 \\ \Rightarrow x^{4}-3 x^{2}-2 x^{2}+6=0 \\ \Rightarrow...
(ii) 
Solution: (i) As per the question, $D=\left\{t \mid t^{3}=t, t \in R\right\}$ Roster form, $\begin{array}{l} t^{3}=t \\ \Rightarrow t^{3}-t=0 \\ \Rightarrow t\left(t^{2}-1\right)=0 \\ \Rightarrow...
Solution: (i) As per the question, $C=\{x \mid x$ is a positive factor of a prime number $p\}$ Roster form, $p=1$ and $p$ itself are the only possible positive factors of a prime number. Hence,...
Solution: (i) As per the question, $A=\{x: x \in R, 2 x+11=15\}$ Roster form, $\begin{array}{l} 2 x+11=15 \\ \Rightarrow 2 x=15-11 \\ \Rightarrow 2 x=4 \\ \Rightarrow x=2 \end{array}$ As a result,...
c) $1: 1$
bulb at a 
distance is E. The electric field intensity produced by the radiations coming from 
bulb at the same distance is
c) $\mathrm{E} / \sqrt{2}$
falls on a non-reflecting surface at normal incidence. If the surface has an area of 
, the total momentum delivered during 30 minutes is
b) $36 \times 10^{-4} \mathrm{~kg} \mathrm{~m} / \mathrm{s}$
is incident normally on a perfectly reflecting infinite wall at 
.
b)$E_{r}=E_{0} \hat{i} \cos (k z+\omega t)$
, the voltage source stops and R is short-circuited. Now write down how much energy is stored in each of L and C.i) The equation for charge motion variation with respect to time is as follows: $\mathrm{L} d^{2} q(\mathrm{t}) / \mathrm{dt}+\mathrm{R} \mathrm{dq}(\mathrm{t}) / \mathrm{dt}+\mathrm{q}(\mathrm{t})...
As indicated by the inquiry, P(n) which is valid for all n. Let P(n) be, ⇒ P(k) is valid for all k. In this manner, P(n) is valid for all n.
As indicated by the inquiry, P(n) which is valid for all n ≥ 4 however P(1), P(2) and P(3) are false Let \[P\left( n \right)\text{ }be\text{ }2n\text{ }<\text{ }n!\] Thus, the instances of the...
Given equation is, $L d i / d t+R i+q / C=v i=v m \sin \omega t$ i) On multiplying the above equation with I, we get, $\mathrm{d}\left(1 / 2 \mathrm{Li}^{2}\right) / \mathrm{dt}+1 / 2 \mathrm{C}...
i) It is given that the power station is $10 \mathrm{~km}$ away from the town Length of the $\mathrm{Cu}$ wire is given as $\mathrm{L}=20 \mathrm{~km}=20000 \mathrm{~m}$ Resistance of $\mathrm{Cu}$...
The reactance offered by an inductor increases with the increasing frequency of an alternating voltage because the induced emf is proportional to the rate of change of current.
$P_{s}$ is given as $60W$ $I_{s}$ is given as $0.54A$ Primary voltage is given as $220V$ $V_{s}=60/0.54=110V$ The transformer is a step-down transformer because the secondary voltage is lower than...
Solution: X could be an inductor, a capacitor, or a hybrid of the two.
$tan \phi = \frac{X_{l}-X_{c}}{R}$ $X_{l}<X_{c}$ At resonate frequency we have, $X_{l}=X_{c}$ $tan \phi = 0$
Frequency 1 is 0.8 rad/s Frequency 2 is 1.2 rad/s The bandwidth is 0.4 rad/s
(a) Under which conditions would the rms currents in the two circuits be the same? (b) Can the rms current in the circuit (b) be larger than that in (a)? Solution: a) $(I_{rms})a=(I_{rms})b$...
The magnetic energy of the LC circuit is equivalent to kinetic energy, while electrostatic energy due to a capacitor's charge change is analogous to potential energy.
The correct options are: (c) the charge on the plates is in phase with the applied voltage. (d) power delivered to the capacitor is zero.
The correct options are: (a) For a given power level, there is a lower current. (b) Lower current implies less power loss. (d) It is easy to reduce the voltage at the receiving end using step-down...
The correct options are: (a) Inductor and capacitor. (d) Resistor, inductor and capacitor.
(c) 14.4 W.
(b) another capacitor should be added in parallel to the first
and is calibrated to read 
.(c) the meter reads not v but $<v_{2}>$ and is calibrated to read $\sqrt <v_{2}>$
and an internal reactance 
. It is used to supply power to a passive load consisting of a resistance and a reactance 
. For maximum power to be delivered from the generator to the load, the value of is equal to.
..(c) $-X_{g}$.
Answer: It is advantageous to use a null-point in the Wheatstone bridge because it prevents any effect on the resistance of the galvanometer caused by the balance point. Kirchhoff's rule is used to...
Answer: The relaxation time (average time between successive collisions) decreases as the drift velocity increases, increasing the...
Answer: When a charge traverses a junction in an electric circuit, the momentum of the charge does not conserve. This is due to the fact that the drift velocity is directly proportional to the...
Solution: Velocity of the earth at perigee is given as $v_{p}$ Velocity of the earth at apogee is given as $v_{a}$ Angular velocity of the earth at perihelion is given as $\omega_{p}$ Angular...
$AE = AG + EG$ $AG + AG = 2AG$ $= 2l cos 30^{o}$ $AE = AC = \sqrt3l$ $AD = 2l$ As a result, $\frac{Gm^{2}}{l^{2}}$ is the force on A due to B along B to A $\frac{Gm^{2}}{3l^{2}}$ is the force on A...
When a body moves around a star in equilibrium, the gravitational attraction produces a centripetal force. Consider a body of mass $m$ revolving in a circular path of radius $r$ around the star S of...
Solution: Let the radius of the ring be r Let the mass of the ring be m When small element dM is considered as the mass, the gravitation force becomes, $dF=\frac{G(dM)m}{x^{2}}$ where...
Solution: Amongst the given figures, (c) show the focus of trajectory.
a) $\mathrm{K}=1 / 2 \mathrm{mv}^{2}=(1 / 2 \mathrm{~m})(\mathrm{GM} / \mathrm{R})$ b) PE of satellite is $U = -GMm/R = -2K$
The polar axis of the earth and its movement are E and E’ respectively. Translational motion is P’ After every 24 hours, earth's orbit is approximately advanced by $1^{o}$ As a result, time taken...
According to Kepler’s second law, a real velocity is constant and is given as: $r_{p}\times v_{p}=r_{A}\times v_{A}$ $\frac{r_{A}}{r_{p}}=\frac{v_{p}}{v_{A}}$ $r_{A}>r_{p}$ and...
R is the spherical shell's radius, while r is the distance between m and M. The point's mass is m, while the hollow spherical shell's mass is M. Now, $F=\frac{GMm}{r^{2}}$ When F = 0,...
The astronaut will experience variation when the size of the space station orbiting the earth is large, and this is due to acceleration due to gravity.
Yes. It is possible for a body to have inertia but no weight as inertia is associated with the mass of the body. Satellite revolving around the earth is an example of a body with inertia and has no...
The direction of the earth's areal velocity around the sun is determined by the product of r and v.
Example of Central force: Electrostatic force acting on the point charge Example of Non-central force: Nuclear force between the atoms
The gravitational force of the earth attracts molecules in the atmosphere, yet they do not descend into the earth since they are in random motion, whereas an apple moves downhill.
The correct options are c) we will be going around but not strictly in closed orbits d) after a sufficiently long time we will leave the solar system
The correct options are a) walking on ground would become more difficult c) raindrops will fall much faster d) aeroplanes will have to travel much faster
a) m will remain at rest b) m will move towards M c) m will move towards 2M d) m will have oscillatory motion Solution: The correct option is c) m will move towards 2M
The correct option is d) will move in orbits like planets and obey Kepler’s laws
The correct option is c) of viscous forces causing the speed of the satellite and hence height to gradually decrease
The correct option is c) not be true because the major gravitational force on mercury is due to sun
The correct option is d) cannot be zero at any point
Answer: Option i) & ii) Lithium exhibits a high ionization enthalpy in addition to a high degree of hydration. This is owing to the fact that it is so little.
Answer: Option i) & ii) Be mimics Al (diagonal relationship), and together they create a protective film of oxide that is resistant to acid assault. Because of the high hydration enthalpy of...
Answer: option i) & iii) The tendency to generate halide hydrates decreases gradually as one moves down the chemical group. The hydrates are MgCl2.6H2O, CaCl2.6H2O, SrCl2.6H2O, and BaCl2.2H2O.
ions (ii) 
ions (iii) 
ions (iv) 
ionsAnswer: Option ii) & iii) Because of this, when zeolite, which is sodium aluminium silicate, reacts with hard water, the sodium ion of zeolite is swapped for calcium and magnesium ions.
Answer: Option i) & ii) Because their hydration enthalpies are higher than their lattice enthalpies, BeSO4 and MgSO4 are extremely soluble in water. CaSO4, SrSO4, and BaSO4 have low hydration...
Answer: Option i) & iii) Both the chemicals are widely used in industries for preparation of other compounds in large scale.
Answer: Option ii) & iv) Periods begin with alkali metals. For their period, alkali metals have the biggest atomic radius. They have low density due to their huge size and low bulk. Alkali...
Answer: Option iv) Because they are hygroscopic in nature, the calcium, barium, and strontium halides operate as a dehydrating agent in the body. They are capable of absorbing moisture....
Answer: option iii) Ca(OH)2 is the chemical A that is used in the manufacturing of washing soda Na2CO3 in order to recover ammonia. It is a reaction that occurs during the Solvay process, which is...
Answer: Option ii) The alkali metal compounds and alkaline earth metal compounds give the flame its colour. In a flame, calcium turns brick red, strontium turns crimson red, barium turns apple...
Answer: Option iv) The formula for soda ash is $Na_2CO_3$ that is sodium carbonate.
Answer: Option i) When heated, all of the elements, with the exception of beryllium, mix with hydrogen to create their hydrides.
Answer: Option iii) Calcium hydroxide dissociates in an aqueous solution, releasing calcium cations and hydroxide anions. Calcium oxide is the chemical compound known as quicklime. Milk of lime is a...
Answer: Option i) When plaster of Paris is heated to 200°C, it transforms into anhydrous calcium sulphate, often known as dead plaster, which lacks setting properties since it absorbs water at a...
Answer: Option ii) The addition of gypsum serves only to slow down the setting process of the cement, allowing it to solidify to a sufficiently hard state before use.
Answer: option i) The concentration of these solvated ions in the sodium in liquid ammonia solution changes the color to copper. So the ammoniated electron is responsible for the solution's color.
Answer: Option i) Sodium carbonate is synthesised by Solvary ammonia soda process.
Answer: option i) The solubility of hydroxides of alkaline earth metals in water increases as the alkaline earth metals move from Be to Ba. Both Be(OH)2 and Mg(OH)2 are nearly insoluble in water....
Answer: Option ii) Lattice and hydration enthalpies encourage the dissolving of alkali metal halides in water. Fluorides are soluble in this order: LiF, NaF, KF, RbF, CsF. The high lattice energy of...
Answer: Option i) Beryllium haldies are essentially covalent compounds that are soluble in organic solvents such as ethanol and acetic acid.
Answer: Option i) As the ionisation enthalpy increases from Mg to Ba, the M – O bond weakens and weakens down the group, and as a result, basicity increases down the group as a function of time. As...
Answer: Option i) Because of the smaller cation and larger anion sizes (smaller cation stabilises smaller anion through crystal lattice energy), Beryllium Carbonate can only be preserved in CO2...
Answer: Option iv) The thermal stability of metal carbonates rises as the metal's electropositive character or basicity increases from Be(OH)2 to Ba(OH)2. The most stable is BaCO3.
Answer: option iii) Li's hydration enthalpy is likewise high (highly exothermic). Li atom has the highest hydration enthalpy, making it the strongest reducing agent in aqueous media. - Li atom's...
Answer: Option i) The answer is Li has a high hydrogen enthalpy. So its water reaction produces a lot of energy, which is used up infusion, vaporization, and ionization. So its water interaction is...
Solution: The correct answer is option (iv). The melting point of alkali metals drops as the strength of metallic bonding diminishes with increasing the size of the atom, as shown in the graph. As a...
Answer: Option (i) is correct. Beryllium carbonate is stable and decomposes to give beryllium oxide and carbon dioxide. The concentration of carbon dioxide grows in the right side, causing the...
Answer: Option i) is correct Lithium carbonate readily decomposes into lithium oxide and carbon dioxide. Others do not decompose. $Li _{2} CO _{3} \stackrel{\Delta}{\longrightarrow} Li _{2} O + CO...
Column I(i) Cs (ii) Na (iii) K (iv) Ca (v) Sr (vi) Ba Column II(a) Apple green ((b) Violet (c) Brick red (d) Yellow (e) Crimson red (f) Blue Answer: (i) is f (ii) is d (iii) is b (iv) is c...
Column I(i) CaCO3 (ii) Ca(OH)2 (iii) Cao (iv) CaSO4 Column II(a) Dentistry, ornamental work (b) Manufacture of sodium carbonate from caustic soda (c) Manufacture of high-quality paper (d) Used in...
Answer: (i) is c (ii) is b (iii) is d (iv) is a,e Explanation: Li: E is the most negative because of the extremely high hydration enthalpy. Na: NaOH is produced from the letter Na (strong base). One...
Answer: The gaseous/vapour state differs from the solid state in several ways. In the solid state, BeCl2 has a polymeric chain structure, which is similar to that of water. In the presence of...