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If (a-b) sin ($\theta +\varphi$) = (a+b) sin ($\theta +\varphi$) and a tan$\left(\frac{\theta }{2}\right)$ - btan$\left(\frac{\varphi }{2}\right)$ = C,the the value of sin$\varphi$ is equal

If a and b are two positive quantities whose sum is $\lambda$, then the minimum value of $\sqrt{(1+\frac{1}{a})(1+\frac{1}{b})}$ is

Prove that sin $({40}^{\circ }+\theta )$ cos$({10}^{\circ }+\theta )$ - cos $({40}^{\circ }+\theta )$ sin $({10}^{\circ }+\theta )$ = $\frac{1}{2}$

(i) If sin $x=\frac{1}{3}$, find the value of sin $3x$

(ii) If cos $x=\frac{1}{2}$, find the value of cos $3x$

Find the term independent of x in the expansion of $(1+x+2x^3){(\frac{3}{2}{x}^2-\frac{1}{3x})}^9$

The value of $\underset{x\to 1^{-}}{lim}\left[sinsi{n}^{-1}x\right]$ is

The two points A and B in a plane are such that for all points P lies on circle satisfied $\frac{PA}{PB}=k$, then k will not be

equal to

Find the coordinates of the foci, and the vertices, the eccentricity and the length of the latus rectum of the hyperbola,

$\frac{x^2}{16}-\frac{y^2}{9}=1$

Let sum of n, 2n, 3n terms of an A.P. be $S_1,S_2$ and $S_3$ respectively, show that $S_3=3(S_2-S_1)$

The value of |$\frac{(9+i)(5+i)}{(2+3i)(4+5i)}$| is ___________

A point is on the XZ-plane. What can you say about its y-coordinate?

Find the coordinatesof the foce, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse.
$16x^2+y^2=16$

If $|4x-3|=|x+5|$, then $x$ is/are

The solution set of $(x-1{)}^{99}(x+1{)}^{100}\le 0$ is

Let $x_1,x_2,\cdots ,x_n$ be n observations, and let $\overline{x}$ be their arithmetic mean and ${\sigma }^2$ be the variance.
Statement – 1: Variance of $2x_1,2x_2,\cdots ,2x_n\text{is}4{\sigma }^2$
Statement – 2: Arithmetic mean $2x_1,2x_2,\cdots ,2x_n\text{is}4\overline{x}$

Find the sum upto first 11 terms of the series 1.4.7 + 4.7.10 + 7.10.13 + . . . . . is __

Find the range of $\frac{1}{\sqrt{x^2+5x+6}}$

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

a. Lower indifference curve represents lower level of satisfaction.

b. Two regular convex to origin 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 250 to be spent on two goods X and Y with prices of Rs 25 and Rs 10 per unit respectively. On the basis of the information given, 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 fall in price of good Y?

If two of the three lines represented by the equation $a{x}^3+b{x}^{2}y+cx{y}^2+d{y}^3=0$ are perpendicular, then

If ${}^{n}Cr-1$ = 36,${}^{n}Cr$ = 84 and ${}^{n}Cr+1$ = 126, then the value of r is

Let U be the universal set and $A\cup B\cup C=U$. Then $\left\{\right(A-B)\cup (B-C)\cup (C-A){\}}^{\prime }$ is equal to

A visitor with sign board 'DO NOT LITTER' is moving on a circular path in an exhibition. During the movement, he stops at points represented by (3, -2) and (-2, 0). Also, centre of the circular path is on the line 2x - y = 3. What is the equation of the path? What message he wants to give in the public domain?

If the sum of the first $40$ term of the series $3+4+8+9+13+14+18+19+\cdots$ is $102\left(m\right)$, then the value of $m$ is

If $A$ is the solution set of the equation ${\log }_{x}2\cdot {\log }_{2x}2={\log }_{4x}2$ and $B$ is the solution set of the equation $x^{{\log }_x(3-x{)}^2}=25,$ then $n(A\cup B)$ is equal to

Let $\alpha$ and $\beta$ be the roots of $x^2-6x-2=0$. If $a_n={\alpha }^n-{\beta }^n$ for $n\ge 1$, then the value of $\frac{a_{10}-2a_8}{3a_9}$ is:

Two capillary tubes P and Q are dipped in water. The height of water level in capillary P is $\frac{2}{3}$ to the height in Q capillary. The ratio of their diameters is

${}^n{C}_0-{\frac{1}{2}}^n{C}_1+{\frac{1}{3}}^n{C}_2-...........+(-1{)}^n\frac{{}^n{C}_n}{n+1}$ =

Write the set $C=\{2,4,8,16,32\}$ in the set-builder form.

A man has 7 friends. In how many ways he can invite one or more of them for a tea party

Find the ${12}^{th}$ term of a G.P. whose $8^{th}$ term is $192$ and the common ratio is $2$.

While throwing a pair of dice, an event A is defined as 'sum of faces will be at least 10'. Find the total number of favorable outcomes.

(i) Find the value of r, if the coefficients of (2r + 2)th and (r-2)th terms in the expansion of (1+x)^{18} are equal.

(ii) Find the coefficient of $x^{-}17$ in the expansion of ${(x^4-\frac{1}{x^3})}^{15}$

(iii) Find the term independent of x in the expansion of ${(2x-\frac{1}{x})}^{10}$

Draw the graph of the greatest integer function:

$f:R\to R:f\left(x\right)=\left[x\right]$ for all $xϵR$

The sum to $n$ terms of the series $\frac{1}{1+1^2+1^4}+\frac{2}{1+2^2+2^4}+\frac{3}{1+3^2+3^4}+\dots \dots$ is $\frac{1}{2}-\frac{1}{a(n^b+n^c+d{n}^4+1)}$, then $a+b+c+d=$

If the coefficients of $p^{th}$, $(p+1{)}^{th}$ and $(p+2{)}^{th}$ terms in the expansion of $(1+x{)}^n$ are in A.P., then

$\underset{\overrightarrow{A}}{\overset{}{\to }}$ $\underset{\overrightarrow{B}}{\overset{}{\to }}$

$|\overrightarrow{A}|=2$units and $|\overrightarrow{B}|=5$units as shown above.

Find $\overrightarrow{A}-\overrightarrow{B}$

Which of the following depicts zero overlap?

If $y^2$ = $a{x}^2+bx+c$, then $y^3.\frac{d^{2}y}{d{x}^2}$ is

Three positive numbers form an increasing G.P. If the middle number is doubled, then the new numbers are in A.P. The common ratio of the G.P. is

The set of all x in $(-\pi ,\pi )$ satisfying $|4sinx-1|<\sqrt{5}$ is given by

If $x^p$ occurs in the expansion of ${(x^2+\frac{1}{x})}^{2n}$
Prove that its coefficient is
$\frac{\left(2n\right)!}{\{\left(\frac{4n-p}{3}\right)!\}\times \left\{\left(\frac{2n+p}{3}\right)\right\}!}$

The $(m+1{)}^{th}$ term of $(\frac{x}{y}+\frac{y}{x}{)}^{2m+1}$ depends on -

If ${\sin }^4\alpha +4{\cos }^4\beta +2=4\sqrt{2}\sin \alpha \cos \beta$;
$\alpha ,\beta \in [0,\pi ]$, then $\cos (\alpha +\beta )-\cos (\alpha -\beta )$ is equal to :

Let $f:R\to R$ be defined by $f\left(x\right)=2x+|x|$. Then $f\left(2x\right)+f(-x)-f\left(x\right)$ is

If a + b + c = 0, a,b,c $ϵ$ Q, then the roots of the equation $(b+c-a)x^2+(c+a-b)x+(a+b-c)=0$ are :

Let R be the relation on Z defined by R = {(a, b): a, b $\in$ Z a - b is an integer}. Find the domain and range of R.

Calculate mean deviation about median for following readings
$\begin{array}{ccccc}Class& 20-40& 40-60& 60-80& 80-100\\ F_i& 20& 44& 30& 40\end{array}$