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)

### 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 = = = = = = = = =

### 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} ∴...

### For any two sets A and B, prove that: A‘ – B‘ = B – A

Answer: To show, A’ – B’ ⊆ B – A Consider, x ∈ A’ – B’ x ∈ A’ and x ∉ B’ [A ∩ A’ = ϕ] x ∉ A and x ∈ B x ∈ B – A x ∈ A’ – B’ ∴ A’ – B’ = B – A Thus,...

### For any two sets, prove that: (i) A ∪ (A ∩ B) = A (ii) A ∩ (A ∪ B) = A

Answers: (i) We know that, A ∪ (A ∩ B) [A ∪ A = A] (A ∪ A) ∩ (A ∪ B) ∴ A ∩ (A ∪ B) = A (ii) A ∩ (A ∪ B) = A We know that, (A ∩ A) ∪ (A ∩ B) [A ∩ A = A] ∴ A ∪ (A ∩ B) =...

### For three sets A, B, and C, show that (i) A ∩ B = A ∩ C need not imply B = C. (ii) A ⊂ B ⇒ C – B ⊂ C – A

Answers: (i) Consider, A = {1, 2} B = {2, 3} C = {2, 4} A ∩ B = {2} A ∩ C = {2} Thus, A ∩ B = A ∩ C and B is not equal to C. (ii) A ⊂ B C–B ⊂ C–A Consider, x ∈ C– B x ∈ C and x ∉ B x ∈ C and x ∉ A...

### For any two sets A and B, show that the following statements are equivalent: (i) A ∪ B = B (ii) A ∩ B = A

Answers: (i) A ∪ B = B Proving, (iii)=(iv) Let us take, A ∪ B = B A ∩ B = A. A ⊂ B and A ∩ B = A Thus, (iii)=(iv) is proved. (ii) A ∩ B = A Proving, (iv)=(i) Let us take, A ∩ B = A A ⊂ B A ∩ B = A...

### For any two sets A and B, show that the following statements are equivalent: (i) A ⊂ B (ii) A – B = ϕ

Answers: (i) A ⊂ B Proving, (i)=(ii) ( A ⊂ B) A–B = {x ∈ A: x ∉ B} All element of A is also an element of B ∴ A–B = ϕ Thus, (i)=(ii) Proved. (ii) A – B = ϕ Proving, (ii)=(iii) Let us take, A–B = ϕ...

### For any two sets A and B, prove that A ⊂ B ⇒ A ∩ B = A

Answer: A ⊂ B ⇒ A ∩ B = A Consider, p ∈ A ⊂ B x ∈ B Let, p ∈ A ∩ B x ∈ A and x ∈ B x ∈ A and x ∈ A ∴ (A ∩ B) = A

### For any two sets A and B, prove that (i) B ⊂ A ∪ B (ii) A ∩ B ⊂ A

Answers: (i) Consider, p ∈ B p ∈ B ∪ A ∴ B ⊂ A ∪ B (ii) Consider, p ∈ A ∩ B p ∈ A and p ∈ B ∴ A ∩ B ⊂ A

### If U = {2, 3, 5, 7, 9} is the universal set and A = {3, 7}, B = {2, 5, 7, 9}, then prove that: (i) (A ∪ B)’ = A’ ∩ B’ (ii) (A ∩ B)’ = A’ ∪ B’

Answers: (i) LHS, A ∪ B = {x: x ∈ A or x ∈ B} A ∪ B = {2, 3, 5, 7, 9} (A∪B)’ = Complement of (A∪B) with U. (A∪B)’ = U – (A∪B)’ U – (A∪B)’ = {x ∈ U: x ∉ (A∪B)’} U = {2, 3, 5, 7, 9} (A∪B)’ = {2, 3,...

### Let A = {1, 2, 4, 5} B = {2, 3, 5, 6} C = {4, 5, 6, 7}. Verify the following identities: (i) A – (B ∩ C) = (A – B) ∪ (A – C) (ii) A ∩ (B △ C) = (A ∩ B) △ (A ∩ C)

Answers: (i) LHS, (B ∩ C) = {x: x ∈ B and x ∈ C} (B ∩ C) = {5, 6} A – (B ∩ C) = {x ∈ A: x ∉ (B ∩ C)} A = {1, 2, 4, 5} (B ∩ C) = {5, 6} (A – (B ∩ C)) = {1, 2, 4} RHS, A – B = {x ∈ A: x ∉ B} A = {1,...

### Let A = {1, 2, 4, 5} B = {2, 3, 5, 6} C = {4, 5, 6, 7}. Verify the following identities: (i) A ∩ (B – C) = (A ∩ B) – (A ∩ C) (ii) A – (B ∪ C) = (A – B) ∩ (A – C)

Answers: (i) LHS, B–C = {x ∈ B: x ∉ C} B = {2, 3, 5, 6} C = {4, 5, 6, 7} B–C = {2, 3} (A ∩ (B – C)) = {x: x ∈ A and x ∈ (B – C)} (A ∩ (B – C)) = {2} RHS, (A ∩ B) = {x: x ∈ A and x ∈ B} (A ∩ B) =...

### Let A = {1, 2, 4, 5} B = {2, 3, 5, 6} C = {4, 5, 6, 7}. Verify the following identities: (i) A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) (ii) A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)

Answers: (i) LHS, (B ∩ C) = {x: x ∈ B and x ∈ C} (B ∩ C) = {5, 6} A ∪ (B ∩ C) = {x: x ∈ A or x ∈ (B ∩ C)} A ∪ (B ∩ C) = {1, 2, 4, 5, 6} RHS, (A ∪ B) = {x: x ∈ A or x ∈ B} (A ∪ B) = {1, 2, 4, 5, 6}....

### Find the smallest set A such that A ∪ {1, 2} = {1, 2, 3, 5, 9}.

Answer: A ∪ {1, 2} = {1, 2, 3, 5, 9} The smallest set of A, A = {1, 2, 3, 5, 9} – {1, 2} ∴ A = {3, 5, 9}

### If A and B are two sets such that A ⊂ B, then Find: (i) A ⋂ B (ii) A ⋃ B

Answers: (i) A ∩ B - A intersection B (Same elements of A and B). A ⊂ B denotes that both A and B have the same elements. ∴ A ∩ B = A (ii) A ∪ B - A union B (Elements of either A or B or in both A...

### If A = {1, 2, 3, 4, 5}, B = {4, 5, 6, 7, 8}, C = {7, 8, 9, 10, 11} and D = {10, 11, 12, 13, 14}. Find: (i) A ∪ B (ii) A ∪ C

Answers: (i) A = {1, 2, 3, 4, 5} B = {4, 5, 6, 7, 8} A ∪ B = {x: x ∈ A or x ∈ B} ∴ A ∪ B = {1, 2, 3, 4, 5, 6, 7, 8} (ii) A = {1, 2, 3, 4, 5} C = {7, 8, 9, 10, 11} A ∪ C = {x: x ∈ A or x ∈ C} ∴ A ∪ C...

### If A = {1, 2, 3, 4, 5}, B = {4, 5, 6, 7, 8}, C = {7, 8, 9, 10, 11} and D = {10, 11, 12, 13, 14}. Find: (i) B ∪ C (ii) B ∪ D

Answers: (i) B = {4, 5, 6, 7, 8} C = {7, 8, 9, 10, 11} B ∪ C = {x: x ∈ B or x ∈ C} ∴ B ∪ C = {4, 5, 6, 7, 8, 9, 10, 11} (ii) B = {4, 5, 6, 7, 8} D = {10, 11, 12, 13, 14} B ∪ D = {x: x ∈ B or x ∈ D}...

### If A = {1, 2, 3, 4, 5}, B = {4, 5, 6, 7, 8}, C = {7, 8, 9, 10, 11} and D = {10, 11, 12, 13, 14}. Find: (i) A ∪ B ∪ C (ii) A ∪ B ∪ D

Answers: (i) A = {1, 2, 3, 4, 5} B = {4, 5, 6, 7, 8} C = {7, 8, 9, 10, 11} A ∪ B = {x: x ∈ A or x ∈ B} A ∪ B = {1, 2, 3, 4, 5, 6, 7, 8} A ∪ B ∪ C = {x: x ∈ A ∪ B or x ∈ C} ∴ A ∪ B ∪ C = {1, 2, 3, 4,...

### If A = {1, 2, 3, 4, 5}, B = {4, 5, 6, 7, 8}, C = {7, 8, 9, 10, 11} and D = {10, 11, 12, 13, 14}. Find: (i) B ∪ C ∪ D (ii) A ∩ (B ∪ C)

Answers: (i) B = {4, 5, 6, 7, 8} C = {7, 8, 9, 10, 11} D = {10, 11, 12, 13, 14} B ∪ C = {x: x ∈ B or x ∈ C} B ∪ C = {4, 5, 6, 7, 8, 9, 10, 11} B ∪ C ∪ D = {x: x ∈ B ∪ C or x ∈ D} ∴ B ∪ C ∪ D = {4,...

### If A = {1, 2, 3, 4, 5}, B = {4, 5, 6, 7, 8}, C = {7, 8, 9, 10, 11} and D = {10, 11, 12, 13, 14}. Find: (i) (A ∩ B) ∩ (B ∩ C) (ii) (A ∪ D) ∩ (B ∪ C)

Answers: (i) A = {1, 2, 3, 4, 5} B = {4, 5, 6, 7, 8} C = {7, 8, 9, 10, 11} (A ∩ B) = {x: x ∈ A and x ∈ B} (A ∩ B) = {4, 5} (B ∩ C) = {x: x ∈ B and x ∈ C} (B ∩ C) = {7, 8} (A ∩ B) ∩ (B ∩ C) = {x:...

### Let A = {x: x ∈ N}, B = {x: x = 2n, n ∈ N), C = {x: x = 2n – 1, n ∈ N} and, D = {x: x is a prime natural number} Find: (i) A ∩ B (ii) A ∩ C

Answers: Given, A = {1, 2, 3…..} B = {2, 4, 6, 8…} C = {1, 3, 5, 7……} D = {1, 2, 3, 5, 7, 11, …} (i) A ∩ B B ⊂ A = {2, 4, 6, 8…} ∴ A ∩ B = B (ii) A ∩ C C ⊂ A = {1, 3, 5…} ∴ A ∩ C =...

### Let A = {x: x ∈ N}, B = {x: x = 2n, n ∈ N), C = {x: x = 2n – 1, n ∈ N} and, D = {x: x is a prime natural number} Find: (i) A ∩ D (ii) B ∩ C

Answers: Given, A = {1, 2, 3…..} B = {2, 4, 6, 8…} C = {1, 3, 5, 7……} D = {1, 2, 3, 5, 7, 11, …} (i) A ∩ D D ⊂ A = {2, 3, 5, 7..} ∴ A ∩ D = D (ii) B ∩ C ∴ B ∩ C = ϕ A natural number cannot be both...

### Let A = {x: x ∈ N}, B = {x: x = 2n, n ∈ N), C = {x: x = 2n – 1, n ∈ N} and, D = {x: x is a prime natural number} Find: (i) B ∩ D (ii) C ∩ D

Answers: Given, A = {1, 2, 3…..} B = {2, 4, 6, 8…} C = {1, 3, 5, 7……} D = {1, 2, 3, 5, 7, 11, …} (i) B ∩ D ∴ B ∩ D = 2 The number which is even and a prime number is 2. (ii) C ∩ D C ∩ D = {1, 3, 5,...

### Let A = {ϕ, {ϕ}, 1, {1, ϕ}, 2}. Which of the following are true? (i) {{2}, {1}} ⊄ A (ii) {ϕ, {ϕ}, {1, ϕ}} ⊂ A (iii) {{ϕ}} ⊂ A

Answers: (i) This statement is True Reason - Neither {2} and nor {1} is a subset of set A. (ii) This statement is True Reason - All three {ϕ, {ϕ}, {1, ϕ}} are subset of set A. (iii) True Reason -...

### Let A = {ϕ, {ϕ}, 1, {1, ϕ}, 2}. Which of the following are true? (i) 2 ⊂ A (ii) {2, {1}} ⊄A

Answers: (i) This statement is False Reason - 2 is not a subset of set A, it is an element of set A. (ii) This statement is True Reason - {2, {1}} is not a subset of set A.

### Let A = {ϕ, {ϕ}, 1, {1, ϕ}, 2}. Which of the following are true? (i) {1} ∈ A (ii) {2, ϕ} ⊂ A

Answers: (i) This statement is False Reason - 1 is not an element of A. (ii) This statement is True Reason - {2, Φ} is a subset of A.

### Let A = {ϕ, {ϕ}, 1, {1, ϕ}, 2}. Which of the following are true? (i) ϕ ∈ A (ii) {ϕ} ∈ A

Answers: (i) This statement is True Reason - Φ belongs to set A. (ii) This statement is True Reason - {Φ} is an element of set A.

### Let A = {{1, 2, 3}, {4, 5}, {6, 7, 8}}. Determine which of the following is true or false: (i) ϕ ∈ A (ii) ϕ ⊂ A

Answers: (i) This statement is False Reason - Φ is a subset of A, not an element of A. (ii) This statement is True Reason - Φ is a subset of every set, so it is a subset of A.

### Let A = {{1, 2, 3}, {4, 5}, {6, 7, 8}}. Determine which of the following is true or false: (i) {6, 7, 8} ∈ A (ii) {4, 5} ⊂ A

Answers: (i) This statement is True. Reason = {6, 7, 8} ∈ A. (ii) This statement is True Reason = {{4, 5}} is a subset of A.

### Let A = {{1, 2, 3}, {4, 5}, {6, 7, 8}}. Determine which of the following is true or false: (i) 1 ∈ A (ii) {1, 2, 3} ⊂ A

Answers: (i) This statement is False Reason - 1 is not an element of A. (ii) This statement is True Reason - Correct Form = {1,2,3} ∈ A

### Let A = {a, b,{c, d}, e}. Which of the following statements are false and why? (i) ϕ A (ii) {ϕ} ⊂ A

Answers: (i) This statement is False Reason - ϕ ⊂ A (ii) This statement is False Reason - ϕ ⊂ A

### Let A = {a, b,{c, d}, e}. Which of the following statements are false and why? (i) {a, b, e} A (ii) {a, b, c} ⊂ A

Answers: (i) This statement is False Reason - {a, b, e} does not belong to A. Correct Form = {a, b, e} ⊂ A (ii) This statement is False Reason - {a, b, c} is not a subset of A

### Let A = {a, b,{c, d}, e}. Which of the following statements are false and why? (i) a ⊂ A. (ii) {a, b, e} ⊂ A

Answers: (i) This statement is False Reason - a is not a subset of A and belongs to A. (ii) This statement is True Reason - {a, b, e} is a subset of A

### Let A = {a, b,{c, d}, e}. Which of the following statements are false and why? (i) {{c, d}} ⊂ A (ii) a A

Answers: (i) This statement is True Reason - {c, d} is a subset of A. (ii) This statement is True Reason - a belongs to A

### Let A = {a, b,{c, d}, e}. Which of the following statements are false and why? (i) {c, d} ⊂ A (ii) {c, d} A

Answers: (i) This statement is False Reason - {c, d} is not a subset of A but it belong to A. (ii) This statement is True Reason - {c, d} belongs to A

### Which of the following statements are correct? Write a correct form of each of the incorrect statements. (i) ϕ {a, b} (ii) ϕ ⊂ {a, b, c} (iii) {x: x + 3 = 3}= ϕ

Answers: (i) This statement is incorrect Correct Form = ϕ ⊂ {a, b} (ii) This statement is the correct form. (iii) This statement is incorrect Correct Form = {x: x + 3 = 3} ≠ ϕ

### Which of the following statements are correct? Write a correct form of each of the incorrect statements. (i) {b, c} ⊂ {a,{b, c}} (ii) {a, b} ⊂ {a,{b, c}}

Answers: (i) This statement is incorrect Correct Form = {b, c} ∈ {a,{b, c}} (ii) {a, b} is not a subset of given set. Correct Form = {a, b}⊄{a,{b, c}}

### Which of the following statements are correct? Write a correct form of each of the incorrect statements. (i) a {{a}, b} (ii) {a} ⊂ {{a}, b}

Answers: (iii) This statement is incorrect Correct Form = {a} ∈ {{a}, b} (iv) This statement is incorrect Correct Form = {a} ∈ {{a}, b}

### Which of the following statements are correct? Write a correct form of each of the incorrect statements. (i) a ⊂ {a, b, c} (ii) {a} {a, b, c}

Answers: (i) This statement is incorrect Correct Form = a ∈{a,b,c} (ii) This statement is incorrect Correct Form = {a} ⊂ {a, b, c}

### Write which of the following statements are true? Justify your answer. (i) The sets P = {a} and B = {{a}} are equal. (ii) The sets A={x: x is a letter of word “LITTLE”} AND, b = {x: x is a letter of the word “TITLE”} are equal.

Answers: (i) This statement is False P = {a} B = {{a}} But {a} = P B = {P} P and B are not equal (ii) This statement is True A = For “LITTLE” A = {L, I, T, E} = {E, I, L, T} B = For “TITLE” B = {T,...

### Write which of the following statements are true? Justify your answer. (i) The set of all integers is contained in the set of all rational numbers. (ii) The set of all crows is contained in the set of all birds.

Answers: (i) The statement is True Reason - A rational number is represented by the form p/q where p and q are integers and (q not equal to 0) keeping q = 1 we can place any number as p. Which then...

### Write which of the following statements are true? Justify your answer. (i) The set of all rectangles is contained in the set of all squares. (ii) The set of all rectangle is contained in the set of all squares.

Answers: (i) The statement is False Reason - Every square can be a rectangle, but every rectangle cannot be a square. (ii) The statement is False Reason - Every square can be a rectangle, but every...

### Decide among the following sets, which are subsets of which: A = {x: x satisfies x2 – 8x + 12=0}, B = {2,4,6}, C = {2,4,6,8,….}, D = {6}

Answer: A = x2 – 8x + 12=0 (x–6) (x–2) =0 x = 2 or 6 Then, A = {2, 6} B = {2, 4, 6} C = {2, 4, 6, 8} D = {6} Thus, D⊂A⊂B⊂C

### State whether the following statements are true or false: (i) {a} ∈ {a,b,c} (ii) {a, b} = {a, a, b, b, a} (iii) The set {x: x + 8 = 8} is the null set.

Answers: (iii) The statement is False Reason - a is a subset of the set. So, it cannot be an element. (iv) The statement is True Reason - Repetition of elements is not allowed in a set. (v) The...

### State whether the following statements are true or false: (i) 1 ∈ { 1,2,3} (ii) a ⊂ {b,c,a}

Answers: (i) The statement is True Reason - 1 is present in the given set. (ii) The statement is False Reason - a is an element of a set and not a subset.

### Which of the following statements are true? Give a reason to support your answer. (v) {a, b, a, b, a, b,….} is an infinite set. (i) {a, b, c} and {1, 2, 3} are equivalent sets. (ii) A set can have infinitely many subsets.

Answers: (v) The statement is False. Reason - Repetition is not allowed in the elements of a set. (vi) The statement is True. Reason - The equivalent sets have same number of elements. (vii) The...

### Which of the following statements are true? Give a reason to support your answer. (i) Every subset of a finite set is finite. (ii) Every set has a proper subset.

Answers: (iii) The statement is True Reason - Basically, the smaller part of something which is finite can never be infinite. (iv) The statement is False Reason - Null set or empty set does not have...

### Which of the following statements are true? Give a reason to support your answer. (i) For any two sets A and B either A B or B A. (ii) Every subset of an infinite set is infinite.

Answers: (i) The statement is False. Reason - It is not required for the two sets A and B to be A B or B A. (ii) The statement is False. Reason - It is finite subset of infinite set which belongs to...

### Show that the set of letters needed to spell “CATARACT” and the set of letters needed to spell “TRACT” are equal.

In the word “CATARACT” , the distinct letters = {C, A, T, R} = {A, C, R, T} In the word “TRACT”, the distinct letters = {T, R, A, C} = {A, C, R, T} Excluding the repetition of letters, both the sets...

### Which of the following sets are equal? A = {x: x ∈ N, x < 3} B = {1, 2}, C= {3, 1} D = {x: x ∈ N, x is odd, x < 5} E = {1, 2, 1, 1} F = {1, 1, 3}

Set A = {1, 2} Set B = {1, 2} Set C = {3, 1} Set D = {1, 3} Set E = {1, 2} Set F = {1, 3} Equal Sets: (i) A, B, E (II) C, D, F

### From the sets given below, select equal sets and equivalent sets. A = {0, a}, B = {1, 2, 3, 4}, C = {4, 8, 12}, D = {3, 1, 2, 4}, E = {1, 0}, F = {8, 4, 12}, G = {1, 5, 7, 11}, H = {a, b}

Equivalent sets: (i) Set A, Set E, Set H - Because of two identical elements (ii) Set B, Set D, Set G - Because of four identical elements (iii) Set C, Set F - Because of three identical elements...

### Are the following pairs of sets equal? Give reasons. (i) A = {2, 3}, B = {x: x is a solution of x2 + 5x + 6= 0} (ii) A={x: x is a letter of the word “WOLF”} B={x: x is letter of word “FOLLOW”}

Answers: (i) A = {2, 3} B = x2 + 5x + 6 = 0 x2 + 3x + 2x + 6 = 0 x(x+3) + 2(x+3) = 0 (x+3) (x+2) = 0 x = -2 and -3 x = {–2, –3} A and B are not equal. (ii) A = Every letter in WOLF A = {W, O, L, F}...

### From the sets given below, pair the equivalent sets: A= {1, 2, 3}, B = {t, p, q, r, s}, C = {α, β, γ}, D = {a, e, i, o, u}.

If the number of elements is same but the elements are different in a set, then it is said to be an equivalent sets. (i) A = {1, 2, 3} The number of elements = 3 (ii) B = {t, p, q, r, s} The number...

### Are the following sets equal? A={x: x is a letter in the word reap}, B={x: x is a letter in the word paper}, C={x: x is a letter in the word rope}.

(i) A - x is the letters in the word reap A ={R, E, A, P} = {A, E, P, R} (ii) B - x is the letters in the word paper B = {P, A, E, R} = {A, E, P, R} (iii) C - x is the letters in the word rope C =...

### Which of the following sets are finite and which are infinite? (i) Set of concentric circles in a plane. (ii) Set of letters of the English Alphabets.

Answers: (i) This set is an infinite set. Reason - Infinite set of concentric circles can be drawn in a plane. (ii) This set is a finite set. Reason - Only 26 letters in English Alphabets are...

### Which of the following sets are equal? (i) A = {1, 2, 3} (ii) B = {x ∈ R:x2–2x+1=0} (iii) C = (1, 2, 2, 3} (iv) D = {x ∈ R : x3 – 6×2+11x – 6 = 0}.

When all the elements of two sets are similar, then those two sets are considered to be the same. (i) A = {1, 2, 3} (ii) B ={x ∈ R: x2–2x+1=0} x2–2x+1 = 0 (x–1)2 = 0 ∴ x = 1. B = {1} (iii) C= {1, 2,...

### Which of the following sets are finite and which are infinite? (i) {x ∈ Z: x < 5} (ii) {x ∈ R: 0 < x < 1}.

Answers: (i) This set is an infinite set. Reason - The integers less than 5 can be infinity. (ii) This set is an infinite set. Reason - In between two real numbers, the real numbers are...

### Which of the following sets are finite and which are infinite? (i) {x ∈ N: x > 5} (ii) {x ∈ N: x < 200}

Answers: (i) This set is an infinite set. Reason - Natural numbers greater than 5 can go till infinity. (ii) This set is a finite set. Reason - The natural numbers start from 1 and there are 199...

### Which of the following are examples of empty set? (i) {x: x2–2=0 and x is rational}. (ii) {x: x is a natural number, x < 8 and simultaneously x > 12}. (iii) {x: x is a point common to any two parallel lines}.

Answers: (i) It is an empty set. Reason - There isn't any natural number whose square is 2. (ii) It is an empty set. Reason - There isn't any natural number which is less than 8 and greater than 12....

### Which of the following are examples of empty set? (i) Set of all even natural numbers divisible by 5. (ii) Set of all even prime numbers.

Answers: (i) It is not an empty set Reason - All the numbers ending with 0. Except 0 is divisible by 5 and is even natural number. (ii) It is not an empty set. Reason - Two is the only even prime...

### A letter is chosen at random from the word ‘’. Find the probability that letter is (i) a vowel (ii) a consonant

We are given the word ‘$ASSASSINATION$’. The total letters in the given word $ = 13$. Number of vowels in the given word $ = 6$. Number of consonants in the given word $ = 7$. Then, the sample space...

### If is the probability of an event, what is the probability of the event ‘not ’.

We are given that, $\frac{2}{{11}}$ is the probability of an event $A$, $P(A) = \frac{2}{11}$ Then, the probability of ‘not $A$’ is $P\left( {not{\text{ }}A} \right) = 1-P\left( A \right)$ $ = 1 -...

### Match each of the sets on the left in the roster form with the same set on the right described in the set-builder form:

(i) {A,P,L,E} (i) {x : x+5=5, x ∈ z} (ii) {5,-5} (ii) {x : x is a prime natural number and a divisor of 10} (iii) {0} (iii) {x : x is a letter of the word “RAJASTHAN”} (iv) {1, 2, 5, 10} (iv) {x : x...

### Write the set of all vowels in the English alphabet which precede q.

The set of all vowels which precede the letter q = A, E, I, O Then, the set of this can be written as, ∴ X = {A, E, I, O}.

### Write the set of all positive integers whose cube is odd.

Given, All odd number has an odd cube. Odd numbers = 2n+1. {2n+1: n ∈ Z, n>0} ∴ The set of all positive integers whose cube is odd = {1,3,5,7,……}

### Write the set {1/2, 2/5, 3/10, 4/17, 5/26, 6/37, 7/50} in the set-builder form.

Set Builder form = {n/(n2+1): n ∈ N, 1≤ n≤ 7} In this, 12 + 1 => 2 22 + 1 => 5 32 + 1 => 10 . . 72 + 1 => 50 The denominator is the square of the numerator +1. The set builder form can...

### Three coins are tossed once. Find the probability of getting (ix) at most two tails

When a coin is tossed the possible outcomes are either a Head $\left( H \right)$ or Tail $\left( T \right)$. Here, coin is tossed three times then the sample space contains, $S{\text{ }} = {\text{...

### Three coins are tossed once. Find the probability of getting (vii) Exactly two tails (viii) no tail

When a coin is tossed the possible outcomes are either a Head $\left( H \right)$ or Tail $\left( T \right)$. Here, coin is tossed three times then the sample space contains, $S{\text{ }} = {\text{...

### Three coins are tossed once. Find the probability of getting (v) no head (vi) 3 tails

When a coin is tossed the possible outcomes are either a Head $\left( H \right)$ or Tail $\left( T \right)$. Here, coin is tossed three times then the sample space contains, $S{\text{ }} = {\text{...

### Three coins are tossed once. Find the probability of getting (iii) at least 2 heads (iv) at most 2 heads

### List all the elements of the following sets: (i) E = {x : x is a month of a year not having 31 days} (ii) F={x : x is a letter of the word “MISSISSIPPI”}

Answers: (i) E = {x : x is a month of a year not having 31 days} The months may have 28, 29, 30, 31 days. Months having 31 days are February, April, June, September, November. ∴ E: {February, April,...

### List all the elements of the following sets: (i) C = {x : x is an integer, -1/2 < x < 9/2} (ii) D={x : x is a vowel in the word “EQUATION”}

Answers: (i) x - integer and is between -1/2 and 9/2 Then, x = 0, 1, 2, 3, 4 ∴ C = {0, 1, 2, 3, 4} (ii) The vowels in the word ‘EQUATION’ = E, U, A, I, O ∴ D = {A, E, I, O,...

### List all the elements of the following sets: (i) A={x : x2≤ 10, x ∈ Z} (ii) B = {x : x = 1/(2n-1), 1 ≤ n ≤ 5}

Answers: (i) x - integer (+ or -) x2 ≤ 10 (-3)2 = 9 < 10 (-2)2 = 4 < 10 (-1)2 = 1 < 10 02 = 0 < 10 12 = 1 < 10 22 = 4 < 10 32 = 9 < 10 The square root of next integers are...

### Describe the following sets in set-builder form: (i) A = {1, 2, 3, 4, 5, 6} (ii) B = {1, 1/2, 1/3, 1/4, 1/5, …..}

Answers: (i) Set-builder form of A = {x : x ∈ N, x<7} This set builder form can be explained as x is such that x belongs to the natural number also is less than 7. (ii) Set-builder form of B =...

### Describe the following sets in set-builder form: (i) C = {0, 3, 6, 9, 12,….} (ii) D = {10, 11, 12, 13, 14, 15}

Answers: (i) Set-builder form of C = {x : x = 3n, n ∈ Z+} Z+ - the set of positive integers This set builder form can be explained as x is such that C is the set of multiples of 3 including 0. (ii)...

### Describe the following sets in set-builder form: (i) E = {0} (ii) {1, 4, 9, 16,…,100}

Answers: (i) Set-builder form of E = {x : x = 0} This set builder form can be explained as x is such that E is an integer which is equal to 0. (ii) Set-builder form = {x2: x ∈ N, 1≤ x ≤10} In this,...

### Describe the following sets in set-builder form: (i) {2, 4, 6, 8,….} (ii) {5, 25, 125, 625}

Answers: (i) Set-builder form = {x: x = 2n, n ∈ N} This set builder form can be explained as x is such that the given set are multiples of 2. (ii) Set-builder form = {5n: n ∈ N, 1≤ n ≤ 4} In this,...

### Describe the following sets in Roster form: (i) The set of all letters in the word ‘Better.’

Answers: (i) The repetition of letter is not allowed in a set Then, Better = b, e, t, r ∴ Roster form = {b, e, t, r}

### Describe the following sets in Roster form: (i) {x : x is a two digit number such that the sum of its digits is 8} (ii) The set of all letters in the word ‘Trigonometry’

Answers: (i) x is a 2 digit number. The sum of the digits in x is 8 Then, x = 17, 26, 35, 44, 53, 62, 71, 80 ∴ Roster form = {17, 26, 35, 44, 53, 62, 71, 80}. (ii) The repetition of a letter is...