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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 TRIGONOMETREY

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

$I{E}_1$ and $I{E}_2$ of Mg are 178 and 348 Kcal ${mol}^{-1}$ respectively. The energy required for the reaction

$M_{\left(g\right)}$ $\to$ ${Mg}^{2+}$_(g) + 2$e^{-}$ is

If the vertices of triangle are (0,2), (1,0) and (3,1), then the triangle is

$f\left(x\right)=\sqrt{x+1}\&g\left(x\right)=2x-3$

Find the domain of $\frac{f}{g}$

$\text{If}{\left(\frac{x}{a}\right)}^n+{\left(\frac{y}{b}\right)}^n=2then\frac{dy}{dx}at(a,b)is$

A die is rolled and the outcomes are observed. Event A is an even number turns up and event B is a prime number turns up. Match the events on left with the outcomes on right.

$\begin{array}{cc}1.\text{Event A or B}& P.\{4,6\}\\ 2.\text{Event A and B}& Q.\left\{2\right\}\\ 3.\text{Event A but not B}& R.\{2,3,4,5,6,\}\\ 4.\text{Complementary event of B}& S.\{1,4,6\}\end{array}$

Find the value of $(0.04{)}^{{\log }_5(0.1+0.01+0.001+....)}$

Three children are selected at random from a group of 6 boys and 4 girls. It is known that in this group exactly one girl and one boy belong to same parents. The probability that the selected group of children have no blood relations, is equal to :

If $|z|=1$ and $z\ne \pm 1$, then all the values of $\frac{z}{1-z^2}$ lies on

Two equations of two S.H.M. are $y=asin(\omega t-\alpha )$ and $y=bcos(\omega t-\alpha )$. The phase difference between the two is

If $T_0,T_1,T_2,..................T_n$ represent the terms in the expansion of $(x+a{)}^n$, then the value of $(T_0-T_2+T_4-T_6+.................{)}^2$ + $(T_1-T_3+T_5-.................{)}^2$ is

If a,b,c,d,e are prime integers, then the number of divisors of $a{b}^2{c}^{2}de$ excluding 1 as a factor, is

The conjugate of the complex number $\frac{2+5i}{4-3i}$ is

What are the principal solutions of the equation
$2si{n}^{2}x+(2-\sqrt{3})sinx-\sqrt{3}=0?$

Prove that $cot7\frac{1}{2}=tan82\frac{1}{2}=(\sqrt{3}+\sqrt{2})(\sqrt{2}+1)$

Or

Find the general solution of each of the equations.

(i) $2co{s}^{2}x=1$

(ii) $co{t}^{2}x=3$

If $3+\frac{1}{4}(3+P)+\frac{1}{4^2}(3+2P)+\frac{1}{4^3}(3+3P)+\dots =8,$ then the value of $P$ is

The distance of the point (1,2) from the line $3x+4y–5=0$ is units.

If the graph of $y=3x^2+2\sqrt{b}x+5$ does not touch x-axis , which of the following is true?

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

Using binomial theorem, prove that $(101{)}^{50}>({100}^{50}+{99}^{50}).$

Prove 1.2 + 2.3 + 3.4 + $\cdots +n(n+1)=\left[\frac{n(n+1)(n+2)}{3}\right].$

The area of the triangle formed by the coordinate axes and a line is $6$ square units and the length of its hypotenuse is $5$ units. Find the equation of the line.

$(3\sqrt{3}+5{)}^n$ = p + f, where p is an integer and f is a proper fraction(fractional part of $(3\sqrt{3}+5{)}^n$). Find the value of $(3\sqrt{3}-5{)}^n$ in terms of p and f, if n is an even integer.

The negation of the statement $(p\wedge q)\to (\sim p\vee r)$ is

If the equation 2 cos x + cos 2kx = 3 has only one solution then k is

If 9 HM's are inserted between 2 & 3, the general term of i^th HM is

If the centre of the circle $2x^2+pxy+q{y}^2+2gx+2fy+3$ = 0 is (1,-3) then the radius of the circle is

If $7^{103}$ is divided by 25 then the remainder is ___ .

Though Moseley's equation $\sqrt{v}$ taken y - axis and Z on x-axis,if a straight line is obtained at angle of ${45}^{\circ }$ and y intercept equal to 1 is obtained when the frequency is 25 ${sec}^{-1}$ .Then ,atomic number of the element is

Calculate the velocity of the centre of mass of the system of particles shown in figure.

Express the following in the form a + ib where a, b Є R.

A(0,6), B(8,12), C(8,0) are the Co-ordinate of vertices of triangle ABC. Then......
1. Co-ordinate of centroid P.(20,6)
2. Co-ordinate of In centre q.(0,16)
3. Co-ordinate of Ex centre r.($\frac{16}{3}$,6)
s.(0,-4)
t.(5,6)

The value of $\sum _{k=1}^{13}\frac{1}{\sin (\frac{\pi }{4}+\frac{(k-1)\pi }{6})\sin (\frac{\pi }{4}+\frac{k\pi }{6})}$ is equal to

If A = $\{\frac{1}{x}:xϵN\text{and}x<8\}\text{and}B=\{\frac{1}{2x}:xϵ\text{and}x\le 4\}$, find B- A

The number of integral terms in the expansion of ${(\sqrt{3}-\sqrt[8]{85})}^{256}$ is

Vectors $\overrightarrow{C}$ and $\overrightarrow{D}$ have magnitudes of 3 and 4 units respectively. What is the angle between the directions of $\overrightarrow{C}$ and $\overrightarrow{D}$ if the magnitude of vector product $\overrightarrow{C}\times \overrightarrow{D}$ is?

(x) 0 (y) 12 units

(i) 0 (ii)$\frac{\pi }{2}$ (iii)$\frac{\pi }{3}$ (iv) $\frac{\pi }{4}$

Solve the given inequality for real x:
$\frac{x}{4}<\frac{(5x-2)}{3}-\frac{(7x-3)}{5}$

Solve $|x+1|>4,xϵR$

Find the sum of the first n terms of the series:

3 + 7 + 13 + 21 + 31 + ...

If T=${}^{2n+1}{C}_1{+}^{2n+1}{C}_2+......{+}^{2n+1}{C}_n=255$. Find the value of n.

$\frac{1}{1!(n-1)!}$ + $\frac{1}{3!(n-3)!}$ + $\frac{1}{(n-5)!}$ + ....... =

The statement $(p\to q)\to \left[\right(\sim p\to q)\to q]$ is

Let $S$ be the standard deviation of $n$ observations. Each of the $n$ observations is multiplied by a constant $C$. Then the standard deviation of the resulting number is

Area of the triangle formed by the lines $y^2-9xy+18x^2=0$ and y=9 is

a) A consumer, Mr Aman is in state of equilibrium consuming two goods X and Y, with given prices Px and Py . What will happen if $\frac{MUx}{Px}>\frac{MUy}{Py}$ ?

b) Identify which of the following is not true for the Indifference Curves theory. Give valid reasons for choice of your answer:

a. Lower indifference curve represents lower level of satisfaction.
b. Two indifference curves can intersect each other.
c. Indifference curve must be convex to origin at the point of tangency with the budget line at the consumer’s equilibrium.
d. Indifference curves are drawn under the ordinal approach to consumer equilibrium.

OR

A consumer has total money income of Rs 500 to be spent on two goods X and Y with prices of Rs 50 and Rs 10 per unit respectively. On the basis of the given information, answer the following questions:
a. Give the equation of the budget line for the consumer.
b. What is the value of slope of the budget line?
c. How many units can the consumer buy if he is to spend all his money income on good X?
d. How does the budget line change if there is a 50% fall in price of good Y?

If $-7\le x^2+8x+5\le 14$, then the interval(s) in which $x$ lies is/are

If ${\log }_{a}b=2,{\log }_{b}c=2$ and ${\log }_{3}c=3+{\log }_{3}a,$ then the value of $a+b+c$ is