Given: \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1\]…(i) Since, \[25>9\] So, above equation is of the form, \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1\]…(ii)...

### Find the(v) length of the latus rectum of each of the following ellipses.

### Express each of the following angles in radians – 36°

Answer: Formula: Angle in radians = $ Angle\,in\,\deg \times \frac{\pi }{180} $ Therefore, Angle in radians = $ 36\times \frac{\pi }{180}=\frac{\pi }{5} $

### If A ( – 1, 6), B( – 3, – 9) and C(5, – 8) are the vertices of a ΔABC, find the equations of its medians.

### If A(0, 0), b(2, 4) and C(6, 4) are the vertices of a ΔABC, find the equations of its sides.

### Prove that the points A(1, 4), B(3, – 2) and C(4, – 5) are collinear. Also, find the equation of the line on which these points lie.

### Find the equation of a line passing through the origin and making an angle of 1200 with the positive direction of the x – axis.

### Find the equation of a line parallel to the y – axis at a distance of (i) 6 units to its right

(ii) 3 units to its left

(ii) 3 units to its left

Answer : (i) Equation of line parallel to y - axis is given by x = constant, as the x - coordinate of every point on the line parallel to y - axis is 6 i.e. constant. Now the point lies to the right...

### Using slopes. Prove that the points A(-2, -1), B(1,0), C(4, 3) and D(1, 2) are the vertices of a parallelogram.

### Find the value of x so that the line through (3, x) and (2, 7) is parallel to the line through (-1, 4) and (0, 6).

### Find the slope of a line which passes through the points

(i) (0, 0) and (4, -2)

(ii) (0, -3) and (2, 1)

(iii) (2, 5) and (-4, -4)

(iv) (-2, 3) and (4, -6)

### Arrange the bonds in order of increasing ionic character in the molecules: LiF, , and .

Solution: The difference in electronegativity between constituent atoms determines the ionic character of a molecule. As a result, the greater the difference, the greater the ionic character of a...

### Explain with the help of suitable example polar covalent bond.

Solution: The bond pair of electrons are not shared equally when two unique atoms with different electronegativities join to form a covalent bond. The bond pair is attracted to the nucleus of an...

### Define electronegativity. How does it differ from electron gain enthalpy?

Solution: "Electronegativity refers to an atom's ability to attract a bond pair of electrons towards itself in a chemical compound." Sr. No Electronegativity Electron affinity 1 A tendency to...

### Write the significance/applications of dipole moment.

Solution: There is a difference in electro-negativities of constituents of the atom in a heteronuclear molecule, which causes polarisation. As a result, one end gains a positive charge, while the...

### Although both and are triatomic molecules, the shape of the molecule is bent while that of is linear. Explain this on the basis of dipole moment.

Solution: $CO_2$ has a dipole moment of 0 according to experimental results. And it's only possible if the molecule's shape is linear, because the dipole moments of the C-O bond are equal and...

### Use Lewis symbols to show electron transfer between the following atoms to form cations and anions :(iii) Al and N.

Solution: Below is a list of Lewis symbols. To form a cation, a metal atom loses one or more electrons, while a nonmetal atom gains one or more electrons. Ionic bonds are formed between cations and...

### Use Lewis symbols to show electron transfer between the following atoms to form cations and anions : (i) K and S (ii) Ca and O

Solution: Below is a list of Lewis symbols. To form a cation, a metal atom loses one or more electrons, while a nonmetal atom gains one or more electrons. Ionic bonds are formed between cations and...

### Temperature dependence of resistivity ρ(T) of semiconductors, insulators, and metals is significantly based on the following factors:

a) number of charge carriers can change with temperature T

b) time interval between two successive collisions can depend on T

c) length of material can be a function of T

d) mass of carriers is a function of T

The correct answer is a) number of charge carriers can change with temperature T b) time interval between two successive collisions can depend on T

### Write the resonance structures for , and

Solution: Resonance is the phenomenon that allows a molecule to be expressed in multiple ways, none of which fully explain the molecule's properties. The molecule's structure is called a resonance...

### can be represented by structures 1 and 2 shown below. Can these two structures be taken as the canonical forms of the resonance hybrid representing ? If not, give reasons for the same.

Solution: The positions of the atoms remain constant in canonical forms, but the positions of the electrons change. The positions of atoms change in the given canonical forms. As a result, they...

### Explain the important aspects of resonance with reference to the ion.

Solution: However, while the carbonate ion cannot be represented by a single structure, the properties of the ion can be described by two or more different resonance structures. The actual structure...

### Define Bond length.

Solution: Bond length is defined as the equilibrium distance between the nuclei of two bonded atoms in a molecule.

### How do you express the bond strength in terms of bond order?

Solution: During the formation of a molecule, the extent of bonding that occurs between two atoms is represented by the bond strength of the molecule. As the bond strength increases, the bond...

### Although geometries of and molecules are distorted tetrahedral, bond angle in water is less than that of Ammonia. Discuss.

Solution: Ammonia's central atom (N) has one lone pair and three bond pairs. In water, the central atom (O) has two lone pairs and two bond pairs. As a result, the two bond pairs repel the two lone...

### Discuss the shape of the following molecules using the VSEPR model:

Solution: $BeCl_2$ The central atom does not have a lone pair, but it does have two bond pairs. As a result, its shape is AB2, or linear. $BCl_3$ The central atom has three bond pairs but no lone...

### Write the favourable factors for the formation of an ionic bond.

Solution: Ionic bonds are formed when one or more electrons are transferred from one atom to another. As a result, the ability of neutral atoms to lose or gain electrons is required for the...

### Define the octet rule. Write its significance and limitations

Solution: “Atoms can combine either by transferring valence electrons from one atom to another or by sharing their valence electrons in order to achieve the closest inert gas configuration by having...

### Draw the Lewis structures for the following molecules and ions :

Solution: The lewis dot structures are:

### Write Lewis symbols for the following atoms and ions: Sand and and

Solution: For S and S2- A sulphur atom has only 6 valence electrons, which is a very small number. As a result, the Lewis dot symbol for the letter S is The presence of a...

### Write Lewis dot symbols for atoms of the following elements :e) N f) Br

Solution: Nitrogen atoms have only five valence electrons in total. As a result, the Lewis dot symbol for N is Bromine, because the atom has only seven valence electrons. As a result,...

### Write Lewis dot symbols for atoms of the following elements :c) B d) O

Solution: Boron atoms have only three valence electrons, which is a very small number. As a result, the Lewis dot symbols for B are as follows: The oxygen atom has only six valence...

### Write Lewis dot symbols for atoms of the following elements :

a) Mg

b) Na

Solution: Only two valence electrons exist in the magnesium atom. As a result, the Lewis dot symbols for Mg are as follows: Only one valence electron exists in the sodium atom. As a...

### Explain the formation of a chemical bond.

Answer: "A chemical bond is an attractive force that holds a chemical species' constituents together." For chemical bond formation, many theories have been proposed, including valence shell electron...

### Differentiate the following functions with respect to x:

As per the given question,

### In how many ways can the letters of the word ASSASSINATION be arranged so that all the S’s are together?

In the given word $ASSASSINATION,$ there are $4\;‘S’.$ Since all the $4\;‘S’$ have to be arranged together so let as take them as one unit. The remaining letters are \[=\text{ }3\text{ }A,\text{...

### From a class of 25 students, 10 are to be chosen for an excursion party. There are 3 students who decide that either all of them will join or none of them will join. In how many ways can the excursion party be chosen?

In this question we get $2$ options that is (i) Either all $3$ will go Then remaining students in class are: \[25\text{ }-\text{ }3\text{ }=\text{ }22\] Number of students remained to be chosen for...

### It is required to seat 5 men and 4 women in a row so that the women occupy the even places. How many such arrangements are possible?

Given there are total $9\;people$ Women occupies even places that means they will be sitting on \[{{2}^{nd}},\text{ }{{4}^{th}},\text{ }{{6}^{th}}and\text{ }{{8}^{th}}\] place where as men will be...

### Determine the number of 5-card combinations out of a deck of 52 cards if each selection of 5 cards has exactly one king.

We have a deck of cards has $4\;kings.$ The numbers of remaining cards are $52.$ Ways of selecting a king from the deck \[\Rightarrow {{~}^{4}}{{C}_{1}}=\] Ways of selecting the remaining $4\;cards$...

### In an examination, a question paper consists of 12 questions divided into two parts i.e., Part I and Part II, containing 5 and 7 questions, respectively. A student is required to attempt 8 questions in all, selecting at least 3 from each part. In how many ways can a student select the questions?

The student can choose $3$ questions from $part\;I$ and $5$ from $part\;II$ Or $4\;questions$ from $part\;I$ and $4$ from $part \;II$ $5$ questions from $part\;I$ and $3$ from $part \;II$

### The English alphabet has 5 vowels and 21 consonants. How many words with two different vowels and 2 different consonants can be formed from the alphabet?

We know that there are $5$ vowels and $21$ consonants in English alphabets. Choosing two vowels out of $5$ would be done in \[^{5}{{C}_{2}}\] ways Choosing $2$ consonants out of $21$ can be done in...

### How many 6-digit numbers can be formed from the digits 0, 1, 3, 5, 7 and 9 which are divisible by 10 and no digit is repeated?

The number is divisible by $10$ if the unit place has $0$ in it. The $6-digit$ number is to be formed out of which unit place is fixed as $0$ The remaining $5\;places$ can be filled by $1, \;3,\;...

### If the different permutations of all the letter of the word EXAMINATION are listed as in a dictionary, how many words are there in this list before the first word starting with E?

In dictionary words are listed alphabetically, so to find the words Listed before $E$ should start with letter either $A,\;B, \;C \;or \;D$ But the word $EXAMINATION$ doesn`t have $B,\;C \;or \;D$...

### A committee of 7 has to be formed from 9 boys and 4 girls. In how many ways can this be done when the committee consists of: At most 3 girls?

Given at most $3\;girls$ In this case the numbers of possibilities are $0\;girl\;and\;7\;boys$ $1\;girl\;and\;6\;boys$ $2\;girl\;and\;5\;boys$ $3\;girl\;and\;4\;boys$ Number of ways to choose...

### A committee of 7 has to be formed from 9 boys and 4 girls. In how many ways can this be done when the committee consists of: (i) Exactly 3 girls? (ii) At least 3 girls?

(i) Given exactly $3$ girls Total numbers of girls are $4$ Out of which $3$ are to be chosen ∴ Number of ways in which choice would be made \[\Rightarrow {{~}^{4}}{{C}_{3}}=\] Numbers of boys are...

### How many words, with or without meaning, can be formed using all the letters of the word EQUATION at a time so that the vowels and consonants occur together?

In the word $EQUATION$ there are $5$ vowels $(A,\;E,\;I,\;O,\;U)$ and $3$ consonants $(Q,\;T,\;N)$ The numbers of ways in which $5$ vowels can be arranged are \[^{5}{{C}_{5}}\] …………… (i) Similarly,...

### How many words, with or without meaning, each of 2 vowels and 3 consonants can be formed from the letters of the word DAUGHTER?

The word DAUGHTER has $3$ vowels $A,$ $E,$ $U$ and $5$ consonants $D,$ $G,$ $H,$ $T$ and $R.$ The three vowels can be chosen in \[^{3}{{C}_{2}}\] as only two vowels are to be chosen. Similarly, the...

### In how many ways can a student choose a programme of 5 courses if 9 courses are available and 2 specific courses are compulsory for every student?

Given $9$ courses are available and $2$ specific courses are compulsory for every student Here $2$ courses are compulsory out of $9$ courses, so a student need to select \[5\text{ }-\text{ }2\text{...

### A bag contains 5 black and 6 red balls. Determine the number of ways in which 2 black and 3 red balls can be selected.

Given a bag contains $5$ black and $6$ red balls Number of ways we can select $2$ black balls from $5$ black balls are \[\Rightarrow {{~}^{5}}{{C}_{2}}=\] Number of ways we can select $3$ red balls...

### In how many ways can one select a cricket team of eleven from 17 players in which only 5 players can bowl if each cricket team of 11 must include exactly 4 bowlers?

Given $17\;players$ in which only $5\;players$ can bowl if each cricket team of $11$ must include exactly $4\;bowlers$ There are $5\;players$ how bowl, and we can require $4\;bowlers$ in a team of...

### Determine the number of 5 card combinations out of a deck of 52 cards if there is exactly one ace in each combination.

Given a deck of $52$ cards There are $4\;\;Ace$ cards in a deck of $52\;\;cards.$ According to question, we need to select $1\;\;Ace$ card out the $4\;\;Ace\;\;cards$ ∴ Number of ways to select...

### Find the number of ways of selecting 9 balls from 6 red balls, 5 white balls and 5 blue balls if each selection consists of 3 balls of each colour.

Given $6$ red balls, $5$ white balls and $5$ blue balls We can select $3$ red balls from $6$ red balls in \[\Rightarrow {{~}^{6}}{{C}_{3}}=\] ways Similarly, we can select $3$ white balls from $5$...

### In how many ways can a team of 3 boys and 3 girls be selected from 5 boys and 4 girls?

Given $5$ boys and $4$ girls are in total We can select $3$ boys from $5$ boys in \[\Rightarrow {{~}^{5}}{{C}_{3}}=\] ways Similarly, we can select $3$ boys from $54$ girls in \[\Rightarrow...

### How many chords can be drawn through 21 points on a circle?

Given $21$ points on a circle We know that we require two points on the circle to draw a chord ∴ Number of chords is are \[\Rightarrow {{~}^{21}}{{C}_{2}}=\] ∴ Total number of chords can be drawn...

### Determine n if (i) 2nC^3:nC^3 = 12: 1 (ii) 2nC^3: nC^3 = 11: 1

Simplifying and computing \[\Rightarrow ~4\text{ }\times \text{ }\left( 2n\text{ }-\text{ }1 \right)\text{ }=\text{ }12\text{ }\times \text{ }\left( n\text{ }-\text{ }2 \right)\] \[\Rightarrow...

### If nC^8 = nC^2, find nC^2.

As per the given question,

### In how many ways can the letters of the word PERMUTATIONS be arranged if the.There are always 4 letters between P and S?

Number of places are as \[1~2~3~4~5~6~7~8~9~10~11~12\] There should always be $4$ letters between $P$ and $S.$ Possible places of $P$ and $S$ are $1$ and $6, 2$ and $7, 3$ and $8, 4$ and $9, 5$ and...

### In how many ways can the letters of the word PERMUTATIONS be arranged if the (i) Words start with P and end with S, (ii) Vowels are all together

(i) Total number of letters in PERMUTATIONS \[=\text{ }12\] Only repeated letter is $T;$ $2times$ First and last letter of the word are fixed as $P$ and $S$ respectively. Number of letters remaining...

### In how many of the distinct permutations of the letters in MISSISSIPPI do the four I’s not come together?

Total number of letters in MISSISSIPPI \[=\text{11}\] Letter Number of occurrence ⇒ Number of permutations = We take that $4 I’s$ come together, and they are treated as $1$ letter, ∴ Total number of...

### How many words, with or without meaning can be made from the letters of the word MONDAY, assuming that no letter is repeated, if. all letters are used but first letter is a vowel?

Number of vowels in MONDAY \[=\text{ }2\text{ }\left( O\text{ }and\text{ }A \right)\] ⇒ Number of permutations in vowel = Now, remaining places \[=\text{ }5\] Remaining letters to be used \[=\text{...

### How many words, with or without meaning can be made from the letters of the word MONDAY, assuming that no letter is repeated, if. (i) 4 letters are used at a time, (ii) All letters are used at a time

(i) Number of letters to be used \[=\text{ }4\] ⇒ Number of permutations = (ii) Number of letters to be used \[=\text{ }6\] ⇒ Number of permutations =

### How many words, with or without meaning, can be formed using all the letters of the word EQUATION, using each letter exactly once?

Total number of different letters in EQUATION \[=\text{ }8\] Number of letters to be used to form a word \[=\text{ }8\] ⇒ Number of permutations =

### Find r if (i)5Pr = 26Pr-1 (ii) 5Pr = 6Pr-1

As per the given question,

### Find n if n-1P^3: nP^3 = 1: 9.

As per the given question,

### From a committee of 8 persons, in how many ways can we choose a chairman and a vice chairman assuming one person cannot hold more than one position?

Total number of people in committee \[=\text{ }8\] Number of positions to be filled \[=\text{ }2\] ⇒ Number of permutations =

### Find the number of 4-digit numbers that can be formed using the digits 1, 2, 3, 4, 5 if no digit is repeated. How many of these will be even?

Total number of digits possible for choosing \[=\text{ }5\] Number of places for which a digit has to be taken \[=\text{ }5=4\] As there is no repetition allowed, ⇒ Number of permutations = The...

### How many 3-digit even numbers can be made using the digits 1, 2, 3, 4, 6, 7, if no digit is repeated?

Even number means that last digit should be even, Number of possible digits at one’s place \[=\text{ }3\text{ }\left( 2,\text{ }4\text{ }and\text{ }6 \right)\] ⇒ Number of permutations= One of digit...

### How many 4-digit numbers are there with no digit repeated?

To find four digit number (digits does not repeat) Now we will have $4$ places where $4$ digits are to be put. So, at thousand’s place = There are $9$ ways as $0$ cannot be at thousand’s place = $9...

### How many 3-digit numbers can be formed by using the digits 1 to 9 if no digit is repeated?

As per the given question,

### Evaluate: When (i) n = 6, r = 2 (ii) n = 9, r = 5

, Solution: As per the given question,

### Find x. If

Solution: As per the given question,

### Compute

Solution: As per the given question,

### Is 3! + 4! = 7!?

Consider LHS 3! +4! Computing left hand side, we get $ \begin{array}{l} 3 !+4 !=(3 \times 2 \times 1)+(4 \times 3 \times 2 \times 1) \\ =6+24 \\ =30 \end{array} $ Again consider RHS and computing we...

### Evaluate (i) 8! (ii) 4! – 3!

(i) Consider $8 !$ We know that $8 !=8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1$ $=40320$ (ii) Consider 4!-3! $ 4 !-3 !=(4 \times 3 !)-3 ! $ Above equation can be written as $...

### Given 5 flags of different colours, how many different signals can be generated if each signal requires the use of 2 flags, one below the other?

Given $5$ flags of different colours We know the signal requires $2$ flags. The number of flags possible for upper flag is $5.$ Now as one of the flag is taken, the number of flags remaining for...

### A coin is tossed 3 times and the outcomes are recorded. How many possible outcomes are there?

Given $A$ coin is tossed $3$ times and the outcomes are recorded The possible outcomes after a coin toss are head and tail. The number of possible outcomes at each coin toss is $2.$ ∴The total...

### How many 5-digit telephone numbers can be constructed using the digits 0 to 9 if each number starts with 67 and no digit appears more than once?

Let the five-digit number be $ABCDE.$ Given that first $2$ digits of each number is $ 67.$ Therefore, the number is $67CDE.$ As the repetition is not allowed and $6$ and $7$ are already taken, the...

### How many 4-letter code can be formed using the first 10 letters of the English alphabet, if no letter can be repeated?

Suppose the $4$ digit code be $1234.$ Hence, the number of letters possible is $10.$ Let’s suppose any $1$ of the ten occupies place $1.$ So, as the repetition is not allowed, the number of letters...

### How many 3-digits even numbers can be formed from the digits 1, 2, 3, 4, 5, 6 if the digits can be repeated?

Let the $3-digit$ number be $ABC,$ where $C$ is at the unit’s place, $B$ at the tens place and $A$ at the hundreds place. As the number has to even, the digits possible at $C$ are $2$ or $4$ or $6.$...

### How many 3-digit numbers can be formed from the digits 1, 2, 3, 4 and 5 assuming that (i) Repetition of the digits is allowed? (ii) Repetition of the digits is not allowed?

(i) Let the $3-digit$ number be $ABC,$ where $C$ is at the units place, $B$ at the tens place and $A$ at the hundreds place. Now when repetition is allowed, The number of digits possible at $C$ is...

### Construct a 2 × 2 matrix A = [ai j] whose elements ai j are given by: ai j = e2ix sin x j

Given \[{{a}_{i\text{ }j}}~=\text{ }{{e}^{2ix}}~sin\text{ }x\text{ }j\] Let \[A\text{ }=\text{ }{{[{{a}_{i\text{ }j}}]}_{2\times 2}}\] So, the elements in a \[2\text{ }\times \text{ }2\] matrix are...

### Using binomial theorem, prove that 23n – 7n – 1 is divisible by 49, where n ∈ N.

Answer: Given, 23n – 7n – 1 23n – 7n – 1 = 8n – 7n – 1 Using binomial theorem, 8n = 7n + 1 8n = (1 + 7) n 8n = nC0 + nC1 (7)1 + nC2 (7)2 + nC3 (7)3 + nC4 (7)2 + nC5 (7)1 + … + nCn (7) n 8n = 1 + 7n...

### The planes: 2x – y + 4z = 5 and 5x – 2.5y + 10z = 6 are

A. Perpendicular

B. Parallel

C. intersect y–axis

D. passes through

Solution: Given: First Plane: $2 x-y+4 z=5$ [On multiply both the sides by $2.5]$ We obtain, $5 x-2.5 y+10 z=12.5 \ldots$ Second Plane: $5 x-2.5 y+10 z=6 \ldots$ Therefore, $\begin{array}{l}...

### Distance between the two planes: 2x + 3y + 4z = 4 and 4x + 6y + 8z = 12 is

A. 2 units

B. 4 units

C. 8 units

D. 2/√29 units

Solution: It is known to us that the distance between two parallel planes $A x+B y+C z=d_{1}$ and $A x+B y+C z=d_{2}$ is given as...

### Prove that if a plane has the intercepts a, b, c and is at a distance of p units from the origin, then

Solution: It is known to us that the distance of the point $\left(\mathrm{x}_{1}, \mathrm{y}_{1}, \mathrm{z}_{1}\right)$ from the plane $\mathrm{Ax}+\mathrm{By}+\mathrm{Cz}$ $=\mathrm{D}$ is given...

### Find the vector equation of the line passing through the point (1,2,-4) and perpendicular to the two lines: and

Solution: The vector eq. of a line passing through a point with position vector $\vec{a}$ and parallel to a vector $\overrightarrow{\mathrm{b}}$ is...

### Find the vector equation of the line passing through and parallel to the planes and

Solution: The vector eq. of a line passing through a point with position vector $\vec{a}$ and parallel to a vector $\vec{b}$ is $\vec{r}=\vec{a}+\lambda \vec{b}$ It is given that the line passes...

### Find the distance of the point from the point of intersection of the line and the plane

Solution: It is given that, The eq. of line is $\overrightarrow{\mathrm{r}}=(2 \hat{\mathrm{i}}-\hat{\mathrm{j}}+2 \hat{\mathrm{k}})+\lambda(3 \hat{\mathrm{i}}+4 \hat{\mathrm{j}}+2 \hat{\mathrm{k}})...

### Find the equation of the plane which contains the line of intersection of the planes and And which is perpendicular to the plane

Solution: It is known that, The eq. of any plane through the line of intersection of the planes $\overrightarrow{\mathrm{r}} \cdot \overrightarrow{\mathrm{n}_{1}}=\mathrm{d}_{1}$ and...

### If O be the origin and the coordinates of P be (1, 2, –3), then find the equation of the plane passing through P and perpendicular to OP.

Solution: It is known to us that the eq. of a plane passing through $\left(\mathrm{x}_{1}, \mathrm{y}_{1}, \mathrm{z}_{1}\right)$ and perpendicular to a line with direction ratios $A, B, C$ is given...

### Find the equation of the plane passing through the line of intersection of the planes and and parallel to -axis.

Solution: It is known to us that, The eq. of any plane through the line of intersection of the planes $\vec{r} \cdot \overrightarrow{n_{1}}=d_{1}$ and $\vec{r} \cdot \overrightarrow{n_{2}}=d_{2}$ is...

### If the points (1, 1, p) and (–3, 0, 1) be equidistant from the plane , then find the value of .

Solution: It is known to us that the distance of a point with position vector $\vec{a}$ from the plane $\vec{r} \cdot \vec{n}=d$ is given as $\left|\frac{\vec{a} \cdot \vec{n}-d}{|\vec{n}|}\right|$...

### Find the equation of the plane passing through the point (–1, 3, 2) and perpendicular to each of the planes x + 2y + 3z = 5 and 3x + 3y + z = 0.

Solution: It is known to us that the eq. of a plane passing through $\left(x_{1}, y_{1}, z_{1}\right)$ is given by $A\left(x-x_{1}\right)+B\left(y-y_{1}\right)+C\left(z-z_{1}\right)=0$ Where, A, B,...

### Find the coordinates of the point where the line through (3, –4, –5) and (2, –3, 1) crosses the plane 2x + y + z = 7.

Solution: It is known to us that the eq. of a line passing through two points $A\left(x_{1}, y_{1}, z_{1}\right)$ and $B\left(x_{2}, y_{2}, z_{2}\right)$ is given as...

### Find the coordinates of the point where the line through (5, 1, 6) and (3, 4, 1) crosses the ZX – plane.

Solution: It is known to us that the vector eq. of a line passing through two points with position vectors $\vec{a}$ and $b$ is given as...

### Find the coordinates of the point where the line through (5, 1, 6) and (3, 4,1) crosses the YZ – plane.

Solution: It is known to us that the vector eq. of a line passing through two points with position vectors $\vec{a}$ and $\vec{b}$ is given as...

### Find the shortest distance between lines

Solution: It is known to us that the shortest distance between lines with vector equations $\vec{r}=\overrightarrow{a_{1}}+\lambda \overrightarrow{b_{1}}$ and $\vec{r}=\overrightarrow{a_{2}}+\lambda...

### Find the equation of the plane passing through (a, b, c) and parallel to the plane

Solution: The eq. of a plane passing through $\left(x_{1}, y_{1}, z_{1}\right)$ and perpendicular to a line with direction ratios $A, B, C$ is given as...

### Find the vector equation of the line passing through (1, 2, 3) and perpendicular to the plane

Solution: The vector eq. of a line passing through a point with position vector $\vec{a}$ and parallel to vector $\vec{b}$ is given as...

### If the lines are perpendicular, find the value of .

Solution: It is known to us that the two lines $\frac{x-1}{3 k}=\frac{y-2}{1}=\frac{z-3}{-5} \text { and }$ $\frac{\mathrm{x}-1}{3 \mathrm{k}}=\frac{\mathrm{y}-2}{1}=\frac{\mathrm{z}-3}{-5}$ are...

### If the coordinates of the points A, B, C, D be (1, 2, 3), (4, 5, 7), (–4, 3, –6) and (2, 9, 2) respectively, then find the angle between the lines AB and CD.

Solution: It is known to us that the angle between the lines with direction ratios $a_{1}, b_{1}, c_{1}$ and $a_{2}, b_{2}, c_{2}$ is given by $\cos \theta=\left|\frac{a_{1} a_{2}+b_{1} b_{2}+c_{1}...

### Find the equation of a line parallel to x – axis and passing through the origin.

Solution: It is known to us that, eq. of a line passing through $\left(x_{1}, y_{1}, z_{1}\right)$ and parallel to a line with direction ratios $a, b, c$ is...

### Find the angle between the lines whose direction ratios are a, b, c and b – c, c – a, a – b.

Solution: The angle between the lines with direction ratios $a_{1}, b_{1}, c_{1}$ and $a_{2}, b_{2}, c_{2}$ is given by $\cos \theta=\left|\frac{a_{1} a_{2}+b_{1} b_{2}+c_{1}...

### If and are the direction cosines of two mutually perpendicular lines, show that the direction cosines of the line perpendicular to both of these are

Solution: Let's consider $l, m, n$ be the direction cosines of the line perpendicular to each of the given lines. Therefore, $ll_{1}+m m_{1}+n n_{1}=0 \ldots(1)$ And...

### Show that the line joining the origin to the point (2, 1, 1) is perpendicular to the line determined by the points (3, 5, –1), (4, 3, –1).

Solution: Let's consider $O A$ be the line joining the origin $(0,0,0)$ and the point $A(2,1,1)$. And let $B C$ be the line joining the points $B(3,5,-1)$ and $C(4,3,-1)$ Therefore the direction...

### In the following cases, find the distance of each of the given points from the corresponding given plane. Point Plane

(a) (2, 3, -5) x + 2y – 2z = 9

(b) (-6, 0, 0) 2x – 3y + 6z – 2 = 0

Solution: (a) The length of perpendicular from the point $(2,3,-5)$ on the plane $x+2 y-2 z=9 \Rightarrow x+2 y-2 z-9=0$ is $\frac{\left|a x_{1}+b y_{1}+c z_{1}+d\right|}{\sqrt{a^{2}+b^{2}+c^{2}}}$...

### In the following cases, find the distance of each of the given points from the corresponding given plane. Point Plane

(a) (0, 0, 0) 3x – 4y + 12 z = 3

(b) (3, -2, 1) 2x – y + 2z + 3 = 0

Solution: (a) The distance of the point $(0,0,0)$ from the plane $3 x-4 y+12=3 \Rightarrow$ $3 x-4 y+12 z-3=0$ is $\begin{array}{l} \frac{\left|a x_{1}+b y_{1}+c...

### In the following cases, determine whether the given planes are parallel or perpendicular, and in case they are neither, find the angles between them.

(a) 4x + 8y + z – 8 = 0 and y + z – 4 = 0

Solution: (a) $4 x+8 y+z-8=0$ and $y+z-4=0$ It is given that The eq. of the given planes are $4 x+8 y+z-8=0 \text { and } y+z-4=0$ It is known to us that, two planes are $\perp$ if the direction...

### In the following cases, determine whether the given planes are parallel or perpendicular, and in case they are neither, find the angles between them.

(a) 2x – 2y + 4z + 5 = 0 and 3x – 3y + 6z – 1 = 0

(b) 2x – 2y + 4z + 5 = 0 and 3x – 3y + 6z – 1 = 0

Solution: (a) $2 x-2 y+4 z+5=0$ and $3 x-3 y+6 z-1=0$ It is given that The eq. of the given planes are $2 x-2 y+4 z+5=0$ and $x-2 y+5=0$ It is known to us that, two planes are $\perp$ if the...

### In the following cases, determine whether the given planes are parallel or perpendicular, and in case they are neither, find the angles between them.

(a) 7x + 5y + 6z + 30 = 0 and 3x – y – 10z + 4 = 0

(b) 2x + y + 3z – 2 = 0 and x – 2y + 5 = 0

Solution: (a) $7 x+5 y+6 z+30=0$ and $3 x-y-10 z+4=0$ It is given that The eq. of the given planes are $7 x+5 y+6 z+30=0$ and $3 x-y-10 z+4=0$ Two planes are $\perp$ if the direction ratio of the...

### Find the angle between the planes whose vector equations are

Solution: It is given that The eq. of the given planes are $\vec{r}(2 \hat{i}+2 \hat{j}-3 \hat{k})=5 \text { and } \vec{r}(3 \hat{i}-3 \hat{j}+5 \hat{k})=5$ If $\mathrm{n}_{1}$ and $\mathrm{n}_{2}$...

### Find the equation of the plane through the line of intersection of the planes x + y + z = 1 and 2x + 3y + 4z = 5 which is perpendicular to the plane x – y + z = 0.

Solution: Let's say that the eq. of the plane that passes through the two-given planes $x+y+z=1$ and $2 x+3 y+4 z=5$ is $\begin{array}{l} (x+y+z-1)+\lambda(2 x+3 y+4 z-5)=0 \\ (2 \lambda+1) x+(3...

### Find the vector equation of the plane passing through the intersection of the planes and through the point

Solution: Let's consider the vector eq. of the plane passing through the intersection of the planes are $\overrightarrow{\mathrm{r}} \cdot(2 \hat{\mathrm{i}}+2 \hat{\mathrm{j}}-3 \hat{\mathrm{k}})=7...

### Find the equation of the plane through the intersection of the planes 3x – y + 2z – 4 = 0 and x + y + z – 2 = 0 and the point (2, 2, 1).

Solution: It is given that Eq. of the plane passes through the intersection of the plane is given by $(3 x-y+2 z-4)+\lambda(x+y+z-2)=0$ and the plane passes through the points $(2,2,1)$ Therefore,...

### Find the equation of the plane with intercept 3 on the -axis and parallel to ZOX plane.

Solution: It is known to us that the equation of the plane $\mathrm{ZOX}$ is $\mathrm{y}=0$ Therefore, the equation of plane parallel to $\mathrm{ZOX}$ is of the form, $\mathrm{y}=\mathrm{a}$ As the...

### Find the intercepts cut off by the plane 2x + y – z = 5.

Solution: It is given that The plane $2 x+y-z=5$ Let us express the equation of the plane in intercept form $x / a+y / b+z / c=1$ Where $a, b, c$ are the intercepts cut-off by the plane at $x, y$...

### Find the equations of the planes that passes through three points.

(a) (1, 1, –1), (6, 4, –5), (–4, –2, 3)

(b) (1, 1, 0), (1, 2, 1), (–2, 2, –1)

Solution: (a) It is given that, The points are $(1,1,-1),(6,4,-5),(-4,-2,3)$. Let, $\begin{array}{l} =\left|\begin{array}{ccc} 1 & 1 & -1 \\ 6 & 4 & -5 \\ -4 & -2 & 3...

### Find the vector and Cartesian equations of the planes

(a) that passes through the point and the normal to the plane is

(b) that passes through the point and the normal vector to the plane is

Solution: (a) That passes through the point $(1,0,-2)$ and the normal to the plane is $\hat{\mathrm{i}}+\hat{\mathrm{j}}-\hat{\mathrm{k}}$ Let's say that the position vector of the point $(1,0,-2)$...

### In the following cases, find the coordinates of the foot of the perpendicular drawn from the origin.

(a) x + y + z = 1

(b) 5y + 8 = 0

Solution: (a) $x+y+z=1$ Let the coordinate of the foot of $\perp \mathrm{P}$ from the origin to the given plane be $P(x, y, z)$ $x+y+z=1$ The direction ratio are $(1,1,1)$ $\begin{array}{l}...

### Find the Cartesian equation of the following planes:

(a)

Solution: Let $\overrightarrow{\mathrm{r}}$ be the position vector of $\mathrm{P}(\mathrm{x}, \mathrm{y}, \mathrm{z})$ is given by $\overrightarrow{\mathrm{r}}=\mathrm{x} \hat{\mathrm{i}}+\mathrm{y}...

### Find the Cartesian equation of the following planes:

(a)

(b)

Solution: (a) It is given that, The equation of the plane. Let $\overrightarrow{\mathrm{r}}$ be the position vector of $\mathrm{P}(\mathrm{x}, \mathrm{y}, \mathrm{z})$ is given by $\vec{r}=x...

### Find the vector equation of a plane which is at a distance of 7 units from the origin and normal to the vector

Solution: It is given that, The vector $3 \hat{\mathrm{i}}+5 \hat{\mathrm{j}}-6 \hat{\mathrm{k}}$ Vector equation of the plane with position vector $\overrightarrow{\mathrm{r}}$ is $\vec{r} \cdot...

### In each of the following cases, determine the direction cosines of the normal to the plane and the distance from the origin.

(a) 2x + 3y – z = 5

(b) 5y + 8 = 0

Solution: (a) $2 x+3 y-z=5$ It is given that The eq. of the plane, $2 x+3 y-z=5 \ldots$. (1) The direction ratio of the normal $(2,3,-1)$ Using the formula,...

### In each of the following cases, determine the direction cosines of the normal to the plane and the distance from the origin.

(a) z = 2

(b) x + y + z = 1

Solution: (a) $z=2$ It is given that The eq. of the plane, $z=2$ or $0 x+0 y+z=2 \ldots (1) .$ The direction ratio of the normal $(0,0,1)$ Using the formula, $\begin{array}{l}...

### Find the shortest distance between the lines whose vector equations are

Solution: Consider the given equations $\begin{array}{l} \Rightarrow \vec{r}=(1-t) \hat{i}+(t-2) \hat{j}+(3-2 t) \hat{k} \\ \vec{r}=\hat{i}-t \hat{i}+t \hat{j}-2 \hat{j}+3 \hat{k}-2 t \hat{k} \\...

### Find the shortest distance between the lines whose vector equations are and

Solution: It is known to us that shortest distance between two lines $\vec{r}=\overrightarrow{a_{1}}+\lambda \overrightarrow{b_{1}}$ and $\vec{r}=\overrightarrow{a_{2}}+\mu \overrightarrow{b_{2}}$...

### Find the shortest distance between the lines and

Solution: It is known to us that the shortest distance between two lines $\frac{x+1}{7}=\frac{y+1}{-6}=\frac{z+1}{1}$ and $\frac{x-3}{1}=\frac{y-5}{-2}=\frac{z-7}{1}$ is given as:...

### Find the shortest distance between the lines

Solution: It is known to us that the shortest distance between two lines $\overrightarrow{\mathrm{r}}=\overrightarrow{\mathrm{a}_{1}}+\lambda \overrightarrow{\mathrm{b}_{1}}$ and...

### Show that the lines and are perpendicular to each other.

Solution: The equations of the given lines are $\frac{\mathrm{x}-5}{7}=\frac{\mathrm{y}+2}{-5}=\frac{\mathrm{z}}{1}$ and $\frac{\mathrm{x}}{1}=\frac{\mathrm{y}}{2}=\frac{\mathrm{z}}{3}$ Two lines...

### Find the values of p so that the lines and are at right angles.

Solution: The standard form of a pair of Cartesian lines is:...

### Find the angle between the following pairs of lines:

(i) and

(ii) and

Solution: Let's consider $\theta$ be the angle between the given lines. If $\theta$ is the acute angle between $\vec{r}=\overrightarrow{a_{1}}+\lambda \overrightarrow{b_{1}}$ and...

### Find the vector and the Cartesian equations of the line that passes through the points (3, –2, –5), (3, –2, 6).

Solution: It is given that Let's calculate the vector form: The vector eq. of as line which passes through two points whose position vectors are $\vec{a}$ and $\vec{b}$ is...

### Find the vector and the Cartesian equations of the lines that passes through the origin and (5, –2, 3).

Solution: Given: The origin $(0,0,0)$ and the point $(5,-2,3)$ It is known to us that The vector eq. of as line which passes through two points whose position vectors are $\vec{a}$ and $\vec{b}$ is...

### The Cartesian equation of a line is Write its vector form.

Solution: It is given that The Cartesian equation is $\frac{x-5}{3}=\frac{y+4}{7}=\frac{z-6}{2} \ldots \text { (1) }$ It is known to us that The Cartesian eq. of a line passing through a point...

### Find the Cartesian equation of the line which passes through the point and parallel to the line given by

Solution: It is given that The points $(-2,4,-5)$ It is known that Now, the Cartesian equation of a line through a point $\left(\mathrm{x}_{1}, \mathrm{y}_{1}, \mathrm{z}_{1}\right)$ and having...

### Find the equation of the line in vector and in Cartesian form that passes through the point with position vector and . is in the direction

Solution: Given: Vector equation of a line that passes through a given point whose position vector is $\vec{a}$ and parallel to a given vector $\vec{b}$ is $\vec{r}=\vec{a}+\lambda \vec{b}$ Let,...

### Find the equation of the line which passes through the point (1, 2, 3) and is parallel to the vector

Solution: Given that, Line passes through the point $(1,2,3)$ and is parallel to the vector. It is known to us that Vector eq. of a line that passes through a given point whose position vector is...

### Show that the line through the points (4, 7, 8), (2, 3, 4) is parallel to the line through the points (–1, –2, 1), (1, 2, 5).

Solution: The points $(4,7,8),(2,3,4)$ and $(-1,-2,1),(1,2,5)$. Consider $A B$ be the line joining the points, $(4,7,8),(2,3,4)$ and $C D$ be the line through the points $(-1,-2$, 1), $(1,2,5)$. So...

### Show that the line through the points (1, –1, 2), (3, 4, –2) is perpendicular to the line through the points (0, 3, 2) and (3, 5, 6).

Solution: Given that The points $(1,-1,2),(3,4,-2)$ and $(0,3,2),(3,5,6)$. Let's consider $A B$ be the line joining the points, $(1,-1,2)$ and $(3,4,-2)$, and $C D$ be the line through the points...

### Show that the three lines with direction cosines Are mutually perpendicular.

Solution: Consider the direction cosines of $L_{1}, L_{2}$ and $L_{3}$ be $l_{1}, m_{1}, n_{1} ; l_{2}, m_{2}, n_{2}$ and $l_{3}, m_{3}, n_{3}$. It is known that If $\mathrm{f}_{1}, \mathrm{~m}_{1},...

### Find the direction cosines of the sides of the triangle whose vertices are (3, 5, –4), (-1, 1, 2) and (–5, –5, –2).

Solution: Given that, The vertices are $(3,5,-4),(-1,1,2)$ and $(-5,-5,-2)$. Firstly find the direction ratios of $\mathrm{AB}$ Where, $A=(3,5,-4)$ and $B=(-1,1,2)$ Ratio of $A...

### Show that the points (2, 3, 4), (–1, –2, 1), (5, 8, 7) are collinear.

Solution: If the direction ratios of two lines segments are proportional, then the lines are collinear. It is given that $\mathrm{A}(2,3,4), \mathrm{B}(-1,-2,1), \mathrm{C}(5,8,7)$ The direction...

### If a line has the direction ratios –18, 12, –4, then what are its direction cosines?

Solution: Given that, The direction ratios are $-18,12,-4$ Where, $a=-18, b=12, c=-4$ Consider the direction ratios of the line as $\mathrm{a}, \mathrm{b}$ and $\mathrm{c}$ Direction cosines are...

### Find the direction cosines of a line which makes equal angles with the coordinate axes.

Solution: Given that, Angles are equal. Let the angles be $\alpha, \beta, \mathrm{Y}$ The direction cosines of the line be I, $\mathrm{m}$ and $\mathrm{n}$ $I=\cos \alpha, m=\cos \beta \text { and }...

### If a line makes angles 90°, 135°, 45° with the x, y and z-axes respectively, find its direction cosines.

Solution: Let's consider the direction cosines of the line be $I, m$ and $n$. Let $\alpha=90^{\circ}, \beta=135^{\circ}$ and $\mathrm{y}=45^{\circ}$ Therefore, $I=\cos \alpha, m=\cos \beta \text {...

### Integrate the function in

As per the given question, Let I = = = = = = = = =

### If the starting material for the manufacture of silicones is RSiCl3, write the structure of the product formed.

Solution:

### Elements of group 14 (a) exhibit oxidation state of only (b) exhibit oxidation state of and (c) form and ion (d) form and ions

Solution: (b) Group 14 components have 4 valence electrons. Thus, bunch oxidation status is $+4$. In any case, the lower oxidation state turns out to be progressively steady because of the inactive...

### An aqueous solution of borax is

Solution: (b) Borax is a strong base salt (NaOH) and a feeble corrosive $\left(\mathrm{H}_{3} \mathrm{BO}_{3}\right)$. In this way, it is essential thing in nature.

### Give one method for industrial preparation and one for laboratory preparation of CO and each.

Solution: Carbon dioxide CO2 can be ready in the lab through the activity of weaken hydrochloric corrosive on calcium carbonate. Their response is as per the following: CO2 is industrially ready by...

### Write balanced equations for: (ii) (iii) (iv) (v) (vi)

Solution: The balanced equations are as follow:

### A certain salt gives the following results. (i) Its aqueous solution is alkaline to litmus. (ii) It swells up to a glassy material on strong heating. (iii) When conc. is added to a hot solution of , a white crystal of an acid separates out Write equations for all the above reactions and identify X, , and .

Solution: The salt given to litmus is antacid. $X$ is, subsequently, a salt with a solid base, and a feeble corrosive. When $X$ is warmed unnecessarily, it additionally enlarges to frame material...

### When metal is treated with sodium hydroxide, a white precipitate (A) is obtained, which is soluble in excess of to give soluble complex (B). Compound (A) is soluble in dilute HCI to form compound (C). The compound (A) when heated strongly gives (D), which is used to extract the metal. Identify (X), (A), (B), (C) and (D). Write suitable equations to support their identities.

Solution: The given metal $X$ gives sodium hydroxide to a white accelerate, and the encourage breaks up surpassing sodium hydroxide. $X$ must, consequently, be made of aluminum. The acquired white...

### (a) Classify the following oxides as neutral, acidic, basic or amphoteric. (B) Write suitable equations to show their nature.

Solution: $\rightarrow$ CO $=$ Neutral $\rightarrow \mathrm{B}_{2} \mathrm{O}_{3}=$ Acidic Being acidic, it responds with bases to frame salts. It responds with $\mathrm{NaOH}$ to frame sodium...

### What are allotropes? Sketch the structure of two allotropes of carbon namely diamond and graphite. What is the impact of structure on the physical properties of two allotropes?

Solution: Allotropy is the presence of a component in more than one structure, having diverse actual properties however similar substance properties. Diamond's solid 3-D construction makes it a...

### How would you explain the lower atomic radius of Ga as compared to Al?

Solution: A tomic sweep (in pm) Aluminum Gallium In spite of the fact that Ga has more than one shell than Al, it is more modest in size than Al. This is on the grounds that the $3 \mathrm{~d}$...

### Explain why is there a phenomenal decrease in ionization enthalpy from carbon to silicon?

Solution: Carbon ionizing enthalpy (the primary component in bunch 14 ) is exceptionally high $(1086 \mathrm{~kJ}/\mathrm{mol})$. That is normal on account of its little size. Nonetheless, there is...

### Explain the following reactions (a) Silicon is heated with methyl chloride at high temperature in the presence of copper; (b) Silicon dioxide is treated with hydrogen fluoride; (c) CO is heated with ZnO; (d) Hydrated alumina is treated with aqueous solution.

Solution: (a) Silicon is warmed with methyl chloride at high temperature within the sight of copper A class of organosilicon polymers called methyl-subbed chlorosilane $\mathrm{MeSiCl}_{3},...

### What happens when (a) Borax is heated strongly (b) Boric acid is added to water (c) Aluminum is treated with dilute NaOH (d) is reacted with ammonia?

Solution: (a) Borax is warmed firmly Borax goes through different changes when warmed. It is losing atoms and expands of water right away. Then, at that point, it turns into a clear fluid, which...

### Explain the structures of diborane and boric acid.

Solution: (a) Diborane $\mathrm{B}_{2} \mathrm{H}_{6}$ is a compound that does not have an electron. $\mathrm{B}_{2} \mathrm{H}_{6}$ just has 12 electrons $-6 \mathrm{e}^{-}$of $6 \mathrm{H}$...

### Aluminium trifluoride is insoluble in anhydrous HF but dissolves when NaF is added. It precipitates out of the resulting solution when gaseous (boron trifluoride) is bubbled through. Give reasons.

Solution: Hydrogen fluoride is a covalent compound with an exceptionally solid intermolecular clinging to hydrogen. Along these lines, it doesn't give particles and doesn't break up aluminum...

### If B-CI bond has a dipole moment, explain why molecule has zero dipole moment.

Solution: The $B-C l$ bond is normally polar as a result of the distinction in the electronegativities of $\mathrm{Cl}$ and $B$. However the particle of $\mathrm{BCl}_{3}$ is non-polar. That is on...

### Suggest reasons why the B-F bond lengths in and differ.

Solution; In $\mathrm{BF}_{3}$, the length of the $\mathrm{B}-\mathrm{F}$ bond is more limited than that of the $\mathrm{B}-\mathrm{F}$ bond in $\mathrm{BF}_{4}^{-} \cdot \mathrm{BF}_{3}$ is an...

### Rationalize the given statements and give chemical reactions: – Lead (II) chloride reacts with to give . – Lead (IV) chloride is highly unstable towards heat. – Lead is known not to form an iodide, .

Solution: - Lead is an individual from bunch 14 of the occasional table. The two oxidation situations with bunch shows are $+2$ and $+4$. The $+2$ oxidation state turns out to be more steady when...

### Explain the difference in properties of diamond and graphite on the basis of their structures.

Solution: Diamond: 1. It has a glasslike grid 2. In precious stone, every carbon molecule is sp3 hybridized and attached to four other carbon particles through a sigma bond. 3. It has an unbending...

### What is the state of hybridisation of carbon in (a) diamond (c) graphite?

Solution: The condition of hybridization of carbon in: (a) $C O_{3}^{2-}$ c in $\mathrm{CO}_{3}^{2-}$ is sp $^{2}$ hybridized and is attached to 3 oxygen iotas. (b) Diamond Every precious stone...

### Write the resonance structures of and .

Solution: For $C O_{3}^{2-}$ There are just 2 resounding designs for the bicarbonate particle. For $\mathrm{HCO}_{3}^{-}$

### Write reactions to justify amphoteric nature of aluminium

Solution: Amphoteric substances will be substances that display both acidic and essential characteristics. Since aluminum breaks up in the two acids and bases, it is said to have an amphoteric...

### Describe the shapes of and . Assign the hybridisation of boron in these species

Solution: (I) $\mathbf{B} \boldsymbol{F}_{3}$ Boron will in general shape monomeric covalent halides in view of its little size and high electronegativity. These halides of boron generally have a...

### Explain what happens when boric acid is heated .

Solution: After warming orthoboric corrosive at a temperature of $370 \mathrm{~K}$ or above, it is changed over into metaboric corrosive and, upon additional warming, yields boric oxide...

### Is boric acid a protic acid? Explain.

Solution: Boric corrosive is a powerless monobasic corrosive which acts as a Lewis corrosive. In this way, it's anything but a protic corrosive. $\mathrm{B}(\mathrm{OH})_{3}+2 \mathrm{HOH}...

### Consider the compounds, and . How will they behave with water? Justify.

Solution: Since it is a Lewis corrosive, $\mathrm{BCl}_{3}$ promptly goes through hydrolysis to frame boric corrosive. $\mathrm{BCl}_{3}+3 \mathrm{H}_{2} \mathrm{O} \rightarrow 3...

### Why does boron trifluoride behave as a Lewis acid?

Solution: The electronic arrangement of boron is $n s^{2} n p^{1}$. It contains 3 electrons in its valence shell. In this way, it can frame just 3 covalent bonds which imply that there are just 6...

### How can you explain the higher stability of as compared to ?

Solution: Thallium and boron have a place with bunch 13 of the intermittent table and $+1$ oxidation state turns out to be more steady as we drop down the gathering. Boron is more steady than...

### Discuss the pattern of variation in the oxidation states of (i) to (ii) to

Solution: (I) $B$ to $\mathrm{TI}$ Gathering 13 components have their electronic design of $\mathrm{ns}^{2} \mathrm{np}^{1}$ and the oxidation state displayed by these components ought to be 3 ....

### List all events associated with the random experiment of tossing of two coins. How many of them are elementary events?

According to the question, two coins are tossed once. We know, when two coins are tossed then the total number of possible outcomes are will be $2^2=4$ So, the Sample space is {HH, HT, TT, TH} ∴...

### Depict the galvanic cell in which the reaction is: Further show: (i) which of the electrode is negatively charged? (ii) the carriers of the current in the cell. (iii) individual reaction at each electrode.

Solution: The galvanic cell relating to the given redox response can be displayed as: $\mathrm{Zn}\left|Z n_{(a q)}^{2+} \| A g_{(a q)}^{+}\right| \mathrm{Ag}$ (I) Zn anode is contrarily charged on...

### Given the standard electrode potentials, Arrange these metals in their increasing order of reducing power.

Solution: The diminishing specialist is more grounded as the terminal potential declines. Subsequently, the expanding request of the lessening force of the given metals is as given underneath: Ag...

### Arrange the given metals in the order in which they displace each other from the solution of their salts. Al, Fe, Cu, Zn, Mg

Solution: A metal with more grounded diminishing force uproots one more metal with more vulnerable lessening power from its answer of salt. The request for the expanding diminishing force of the...

### Predict the products of electrolysis in each of the following: (i) An aqueous solution of with silver electrodes (ii) An aqueous solution with platinum electrodes (iii) A dilute solution of with platinum electrodes (iv) An aqueous solution of with platinum electrodes.

Solution: (I) In fluid arrangement, AgNO3 ionizes to give Ag+(aq) and NO3–(aq) particles. \[AgN03\left( aq \right)\text{ }\to \text{ }Ag+\left( aq \right)\text{ }+\text{ }NO3\left( aq...

### Using the standard electrode potentials given in Table 8.1, predict if the reaction between the following is feasible: (a) and (b) and (c) and (d) and (e) and

Solution: (a) $F e_{(a q)}^{3+}$ and $I_{(a q)}^{-}$ $2 F e_{(a q)}^{3+}+2 I_{(a q)}^{-} \rightarrow 2 F e_{(a q)}^{2+}+I_{2(s)}$ Oxidation half response: $2 I_{(a q)}^{-} \rightarrow I_{2}(s)+2...

### In Ostwald’s process for the manufacture of nitric acid, the first step involves the oxidation of ammonia gas by oxygen gas to give nitric oxide gas and steam. What is the maximum weight of nitric oxide that can be obtained starting only with 10.00 g. of ammonia and 20.00 g of oxygen?

Solution: The reasonable response is as given underneath: $4 \mathrm{NH}_{3(g)}+5 \mathrm{O}_{2}(g) \rightarrow 4 \mathrm{NO}_{(g)}+6 \mathrm{H}_{2} \mathrm{O}_{(g)}$ $4 N H_{3}=4 \times 17...

### Refer to the periodic table given in your book and now answer the following questions: (a) Select the possible non – metals that can show disproportionation reaction? (b) Select three metals that show disproportionation reaction?

Solution: One of the responding components consistently has a component that can exist in somewhere around 3 oxidation numbers. (I) The non - metals which can show disproportionation responses are...

### Chlorine is used to purify drinking water. Excess of chlorine is harmful. The excess of chlorine is removed by treating with sulphur dioxide. Present a balanced equation for this redox change taking place in water.

Solution: The redox response is as given beneath: Cl2(s)+SO2(aq)+H2O(l)→Cl(aq)-+SO4(aq)2- The oxidation half response: SO2(aq)→SO4(aq)2- Add 2 electrons to adjust the oxidation no. :...

### The ion is unstable in solution and undergoes disproportionation to give , and ion. Write a balanced ionic equation for the reaction.

Solution: The response is as given beneath: $\mathrm{Mn}_{(a q)}^{3+} \rightarrow \mathrm{Mn}_{(a q)}^{2+}+\mathrm{MnO}_{2(s)}+\mathrm{H}_{(a q)}^{+}$ The oxidation half response: $\mathrm{Mn}_{(a...

### What sorts of informations can you draw from the following reaction?

Solution: The oxidation no. of $\mathrm{C}$ in $(C N)_{2}, C N^{-}$and $C N O^{-}$are $+3,+2$ and $+4$ separately. Let the oxidation no. of $\mathrm{C}$ be $\mathrm{y}$. $(C N)_{2}$ $2(y-3)=0$ Along...

### Balance the following equations in basic medium by ion-electron method and oxidation number methods and identify the oxidising agent and the reducing agent. (a) (s) (b) (c)

Solution:

### Balance the following redox reactions by ion – electron method : (a) (Basic medium) (b) (Acidic medium) (c) (Acidic medium) (d) (Acidic medium)

Solution: (a) $M n O_{4}^{-}(a q)+I_{(a q)}^{-} \rightarrow \operatorname{MnO}_{2}(s)+I_{2}(s)$ Stage 1 The two half responses are given beneath: Oxidation half response: $I_{(a q)} \rightarrow...

### Consider the reactions: (a) (b) (c) (d) No change is observed What inference do you draw about the behavior of and from these reactions?

Solution: $\mathrm{Ag}^{+}$and $C u^{2+}$ acts as oxidizing specialist in responses (I) and (ii) individually. In response (iii), $\mathrm{Ag}^{+}$oxidizes $\mathrm{C}_{6} \mathrm{H}_{5}...

### Why does the following reaction occur? What conclusion about the compound (of which is a part) can be drawn from the reaction?

Solution: $X e O_{6(a q)}^{4-}+2 F_{(a q)}^{-}+6 H_{(a q)}^{+} \rightarrow X e O_{3(g)}+F_{2(g)}+3 H_{2} O_{(l)}$ The oxidation no. of Xe decreases from $+8$ in $\mathrm{XeO}_{6}^{4-}$ to $+6$ in...

### Justify giving reactions that among halogens, fluorine is the best oxidant and among hydrohalic compounds, hydroiodic acid is the best reductant.

Solution: $F_{2}$ can oxidize $C l^{-}$to $C l_{2}, B r^{-}$to $B r_{2}$, and $I^{-}$to $I_{2}$ as: $F_{2(a q)}+2 C l_{(s)}^{-} \rightarrow 2 F_{(a q)}^{-}+C l_{2(g)}$ $F_{2}(a q)+2 B r_{(a q)}^{-}...

### Consider the reactions : Why does the same reductant, thiosulphate react differently with iodine and bromine?

Solution: The normal oxidation no. of $\mathrm{S}$ in $\mathrm{S}_{2} \mathrm{O}_{3}^{2-}$ is $+2$. The normal oxidation no. of $\mathrm{S}$ in $S_{4} \mathrm{O}_{6}^{2-}$ is $+2.5$. The oxidation...

### Identify the substance oxidised, reduced, oxidising agent and reducing agent for each of the following reactions: (a) (b) (c) (d) (e)

Solution: (a) $2 \mathrm{AgBr}_{(s)}+C_{6} H_{6} O_{2}(a q) \rightarrow 2 \mathrm{Ag}_{(s)}+2 \mathrm{HBr}_{(a q)}+C_{6} \mathrm{H}_{4} O_{2}(a q)$ $\mathrm{C}_{6} \mathrm{H}_{6}...

### How do you count for the following observations? (a) Though alkaline potassium permanganate and acidic potassium permanganate both are used as oxidants, yet in the manufacture of benzoic acid from toluene we use alcoholic potassium permanganate as an oxidant. Why? Write a balanced redox equation for the reaction. (b) When concentrated sulphuric acid is added to an inorganic mixture containing chloride, we get colourless pungent smelling gas HCl, but if the mixture contains bromide then we get red vapour of bromine. Why?

Solution: (a) While producing benzoic corrosive from toluene, alcoholic potassium permanganate is utilized as an oxidant because of the given reasons. (I) In an impartial medium, $O H^{-}$ions are...

### Whenever a reaction between an oxidisina adent and a reducina aqent is carried out, a compound of lower oxidation state is formed if the reducing agent is in excess and a compound of higher oxidation state is formed if the oxidising agent is in excess. J ustify this statement giving three illustrations. Justify the above statement with three examples.

Solution: When there is a response between lessening specialist and oxidizing specialist, a compound is framed which has lower oxidation number if the diminishing specialist is in abundance and a...

### The compound is unstable compound. However, if formed, the compound acts as a very strong oxidising agent. Why?

Solution: The oxidation no. of $A g$ in $A g F_{2}$ is $+2$. Be that as it may, $+2$ is entirely unsound oxidation no. of Ag. Consequently, when $A g F_{2}$ is framed, silver acknowledges an...

### Consider the reactions: (a) (b) Why it is more appropriate to write these reactions as : (a) aq (b) Also suggest a technique to investigate the path of the above (a) and (b) redox reactions

Solution: (a) Stage 1: $\mathrm{H}_{2} \mathrm{O}$ breaks to give $\mathrm{H}_{2}$ and $\mathrm{O}_{2}$. $2 \mathrm{H}_{2} \mathrm{O}_{(\mathrm{l})} \rightarrow 2...

### While sulphur dioxide and hydrogen peroxide can act as oxidising as well as reducing agents in their reactions, ozone and nitric acid act only as oxidants. Why?

Solution: In sulfur dioxide $\left(S O_{2}\right)$ the oxidation no. of $\mathrm{S}$ is $+4$ and the scope of oxidation no. of sulfur is from $+6$ to $-2$. Consequently, $S O_{2}$ can go about as a...

### Suggest a list of substances where carbon can exhibit oxidation states from -4 to +4 and nitrogen from -3 to +5.

Solution: The compound where carbon has oxidation number from -4 to +4 are given below in the table:

### Write formulas for the following compounds: (a) Mercury (II) chloride (b) Nickel (II) sulphate (c) Tin (IV) oxide (d) Thallium (I) sulphate (e) Iron (III) sulphate (f) Chromium (III) oxide

Solution: Formulas are: (a) Mercury (II) chloride $H g C l_{2}$ (b) Nickel (II) sulphate $\mathrm{NiSO}_{4}$ (c) Tin (IV) oxide $\mathrm{SnO}_{2}$ (d) Thallium (I) sulphate $\mathrm{Tl}_{2}...

### Calculate the oxidation number of sulphur, chromium and nitrogen in H2SO5, Cr2O2 and NOT. Suggest structure of these compounds. Count for the fallacy. nitrogen in H2SO5, Cr2O2 and NOT. Suggest structure of these compounds. Count for the fallacy.

Solution: O.N. of S in H2SO5. By traditional strategy, the O.N. of S in H2SO5 is 2 (+1) + x + 5 (- 2) = 0 or x = +8 This is outlandish on the grounds that the most extreme O.N. of S can't be more...