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)...
Answer: Formula: Angle in radians = $ Angle\,in\,\deg \times \frac{\pi }{180} $ Therefore, Angle in radians = $ 36\times \frac{\pi }{180}=\frac{\pi }{5} $
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...
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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{...
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...
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...
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 $4\;cards$...
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 $part \;II$
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...
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,\;...
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 $B,\;C \;or \;D$...
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...
(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...
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,...
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...
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{...
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...
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...
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...
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 $5$...
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...
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...
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...
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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...
(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...
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...
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{...
(i) Number of letters to be used \[=\text{ }4\] ⇒ Number of permutations = (ii) Number of letters to be used \[=\text{ }6\] ⇒ Number of permutations =
Total number of different letters in EQUATION \[=\text{ }8\] Number of letters to be used to form a word \[=\text{ }8\] ⇒ Number of permutations =
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Total number of people in committee \[=\text{ }8\] Number of positions to be filled \[=\text{ }2\] ⇒ Number of permutations =
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...
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...
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...
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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...
(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 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...
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...
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...
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...
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 $6.$...
(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...
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...
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...
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}...
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...
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...
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...
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...
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}})...
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...
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...
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...
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 $\left|\frac{\vec{a} \cdot \vec{n}-d}{|\vec{n}|}\right|$...
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,...
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...
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...
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...
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...
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...
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...
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...
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}...
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...
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}...
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...
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...
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 $\frac{\left|a x_{1}+b y_{1}+c z_{1}+d\right|}{\sqrt{a^{2}+b^{2}+c^{2}}}$...
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...
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...
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...
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...
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 $\mathrm{n}_{2}$...
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...
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...
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,...
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...
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 $x, y$...
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...
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 $(1,0,-2)$...
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}...
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}...
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...
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...
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,...
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}...
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} \\...
Solution: It is known to us that shortest distance between two lines $\vec{r}=\overrightarrow{a_{1}}+\lambda \overrightarrow{b_{1}}$ and $\vec{r}=\overrightarrow{a_{2}}+\mu \overrightarrow{b_{2}}$...
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:...
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...
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...
Solution: The standard form of a pair of Cartesian lines is:...
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...
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...
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...
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...
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...
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,...
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...
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...
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...
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},...
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...
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...
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...
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 }...
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 {...
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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} ∴...
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,...
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) =...
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...
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...
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...
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
Answers: (i) Consider, p ∈ B p ∈ B ∪ A ∴ B ⊂ A ∪ B (ii) Consider, p ∈ A ∩ B p ∈ A and p ∈ B ∴ A ∩ B ⊂ A
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,...
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,...
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) =...
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}....
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}
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...
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...
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}...
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,...
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,...
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:...
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 =...
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...
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,...
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 -...
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.
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.
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.
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.
Answers: (i) This statement is True. Reason = {6, 7, 8} ∈ A. (ii) This statement is True Reason = {{4, 5}} is a subset of 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
Answers: (i) This statement is False Reason - ϕ ⊂ A (ii) This statement is False Reason - ϕ ⊂ 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
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
Answers: (i) This statement is True Reason - {c, d} is a subset of A. (ii) This statement is True Reason - a belongs to 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
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
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}}
Answers: (iii) This statement is incorrect Correct Form = {a} ∈ {{a}, b} (iv) This statement is incorrect Correct Form = {a} ∈ {{a}, b}
Answers: (i) This statement is incorrect Correct Form = a ∈{a,b,c} (ii) This statement is incorrect Correct Form = {a} ⊂ {a, b, c}
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,...
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...
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...
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
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...
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.
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...
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...
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...
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...
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
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...
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}...
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...
(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 =...
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...
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,...
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...
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...
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....
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...
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...
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 -...
(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...
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}.
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,……}
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...
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{...
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{...
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{...
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{...
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,...
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,...
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...
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 =...
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)...
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,...
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,...
Answers: (i) The repetition of letter is not allowed in a set Then, Better = b, e, t, r ∴ Roster form = {b, e, t, r}
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...