Set Builder Form

Q.

If p and q are the lengths of perpendicular from the origin to the lines $xcos\theta -ysin\theta =kcos2\theta$ and $xsec\theta +ycosec\theta =k$, respectively.

Prove that $p^2+4q^2=k^2$

Q. Write the following sets in the set-builder form: (i) (3, 6, 9, 12) (ii) {2, 4, 8, 16, 32} (iii) {5, 25, 125, 625} (iv) {2, 4, 6 …} (v) {1, 4, 9 … 100}

Q. Match each of the set on the left in the roster form with the same set on the right described in set-builder form: (i) {1, 2, 3, 6} (a) { x : x is a prime number and a divisor of 6} (ii) {2, 3} (b) { x : x is an odd natural number less than 10} (iii) {M, A, T, H, E, I, C, S} (c) { x : x is natural number and divisor of 6} (iv) {1, 3, 5, 7, 9} (d) { x : x is a letter of the word MATHEMATICS}

Q. List all the elements of the following sets: (i) A = { x : x is an odd natural number} (ii) B = { x : x is an integer, } (iii) C = { x : x is an integer, } (iv) D = { x : x is a letter in the word “LOYAL”} (v) E = { x : x is a month of a year not having 31 days} (vi) F = { x : x is a consonant in the English alphabet which proceeds k }.

Q. 5. Find log 48 base 24 in terms of alpha if log 36 base 12 = alpha

Q. Prove by the method of induction that every even power of every odd integer greater than 1 , when divided by 8 leaves the remainder 1.

Q. Let $P=\{\theta :\sin \theta -\cos \theta =\sqrt{2}\cos \theta \}$ and $Q=\{\theta :\sin \theta +\cos \theta =\sqrt{2}\sin \theta \}$ be two sets. Then

1. $P\subset QandQ-P\ne \varphi$
2. $Q\subset ̸P$
3. P = Q
4. $P\subset ̸Q$

Q. In a village, there are 600 families. 100 have no children, 200 have only boys and 200 have only girlsHow many families have boys and girls ?

Q. The given figure shows a relationship between the sets P and Q. write this relation (i) in set-builder form (ii) in roster form. What is its domain and range?

Q. The set builder form of $\{\frac{1}{2},\frac{2}{3},\frac{3}{4},\frac{4}{5},\frac{5}{6}\}$ is

1. $\{x:x=\frac{n}{n+1}$, where $n$ is a natural number and $1\le n\le 6\}$
2. $\{x:x=\frac{n}{n+1}$, where $n$ is a natural number and $1\le n\le 5\}$
3. $\{x:x=\frac{n}{n+1}$, where $n$ is a natural number and $2\le n\le 6\}$
4. $\{x:x=\frac{n}{n-1}$, where $n$ is a natural number and $1\le n\le 5\}$

Q. The set builder form of $\{2,4,6,8,10\}$ is

1. $\{x:x$ is even natural number less than $10\}$
2. $\{x:x=2n$ where $n\in N$ such that $n<6\}$
3. $\{x:x=2n+1$ where $n\in \{1,2,3\}\}$
4. $\{x:x=2n$ where $n\in Z$ such that $n<6\}$

Q. 39. Find log48 to the base 24 in terms of α if log36 to the base 12=α.

Q.

Write the following sets in roster form:

(i) A = {x: x is an integer and –3 < x < 7}.

(ii) B = {x: x is a natural number less than 6}.

(iii) C = {x: x is a two-digit natural number such that the sum of its digits is 8}

(iv) D = {x: x is a prime number which is divisor of 60}.

(v) E = The set of all letters in the word TRIGONOMETRY.

(vi) F = The set of all letters in the word BETTER.

Q. A set $Y$ is set of all integers $k$ greater than $-10$ and less than $5$ can be represented as:

1. $Y=\{-10,-9,-8,...,3,4,5\}$
2. $Y=\{-9,-8,-7,-6,...,2,3,4\}$
3. $Y=\{k:k>-10\&k<5\&k\in I\}$
4. $Y=\{k:k<-10\&k>5\&k\in I\}$

Q. 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 is a natural number and divisor of 10}
(v)	{A, H, J, R, S, T, N}	(v)	x : x2 − 25 = 0
(vi)	{2, 5}	(vi)	{x : x is a letter of the word "APPLE"}

Q. Describe the following sets in Roster form:
(i) {x : x is a letter before e in the English alphabet};
(ii) {x ∈ N : x² < 25};
(iii) {x ∈ N : x is a prime number, 10 < x < 20};
(iv) {x ∈ N : x = 2n, n ∈ N};
(v) {x ∈ R : x > x}.
(vi) {x : x is a prime number which is a divisor of 60}
(vii) {x : x is a two digit number such that the sum of its digits is 8}
(viii) The set of all letters in the word 'Trigonometry'
(ix) The set of all letters in the word 'Better'.

Q. $\text{Let}A=\{x\in R:x^3=1\}$
$\text{and}B=\{x\in R:x^2=1\},\text{then}$

1. $A\subset B$
2. $A=B$
3. $B\subset A$
4. $A$ and $B$ are disjoint.

Q. The set builder representation of the set B = { 29, 31, 37, 41, 43, 47}, is

1. B = {x: x is an odd number more than 27 and less than 50}
2. B = {x: x is a prime number more than 27 and less than 50}
3. B = {x: x is an odd number more than 25 and less than 50}
4. B = {x: x is a prime number more than 25 and less than 50}

Q. The middle term in the expansion of ${\left(\frac{2x^2}{3}+\frac{3}{2x^2}\right)}^{10}$ is
(a) 251
(b) 252
(c) 250
(d) none of these

Q. Let A = (3, 5) and B = (7, 11). Let R = {(a, b) : a ∈ A, b ∈ B, a − b is odd}. Show that R is an empty relation from A into B.

Q. what is the rank of the word school in a dictionary which was formed by arranging its letters??

Q. Rewrite the following statement in five different ways conveying the same meaning.

Q. Which of the following alternatives is correctly matched?

(A)

Set builder form	Roster form
{x: x is a consonant in the word MATHEMATICS}	{M, T, H, S}

(B)

Set builder form	Roster form
{x: x is even and a prime number}	ϕ

(C)

Set builder form	Roster form
{x: x is odd and a multiple of 2}	ϕ

(D)

Set builder form	Roster form
{x: x is a vowel in the word ARITHMETIC}	{A, E, I, R}