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Using properties of sets, show that

(i) $A\cup (A\cap B)=A$ (ii) $A\cap (A\cup B)=A.$

Let $z_1,z_2,z_3$ are the vertices of an equilateral triangle inscribed in the circle $\left|z\right|$ = 2. If $z_1,z_2,z_3$ are in the clockwise sense and $z_1$ = $1+i\sqrt{3}$, then:

If Z is a complex number such that $\left|z\right|$ greater than or equal to 2, then the minimum value of $|z+\frac{1}{2}|$.

Let $\alpha ,\beta$ be real and z be a complex number. If $z^2+\alpha z+\beta =0$ has two distinct roots on the line Re(z) = 1, then it is necessary that

The 5th term of an A.P. of n terms whose sum is $n^2$ - 2n, is:

1 . The sum of the series 1 + 3x + 6$x^2$ + 10 $x^3$ + ....... ∞ will be

Consider the lines represented by the equation $(x^2+xy-x)(x-y)=0$, forming a triangle, then
$\begin{array}{cc}Column1& Column2\\ \left(a\right)Orthocentreoftraingle& (\frac{1}{6},\frac{1}{2})\\ \left(b\right)Circumcenter& \left(\frac{1}{2+2\sqrt{2}}\right)\\ \left(c\right)Centroid& (0,\frac{1}{2})\\ \left(d\right)Incenter& (\frac{1}{2},\frac{1}{2})\end{array}$

Find the locus of the curve represented by $x=sec\theta +1andy=tan\theta -1$, where $\theta$ is a variable.

(De Morgan's laws) For any two sets A and B, prove that:
$I.(A\cup B{)}^{\prime }=(A^{\prime }\cap B^{\prime })$

$II.(A\cap B{)}^{\prime }=(A^{\prime }\cup B^{\prime })$

If sum of the perpendicular distances of a variable point P(x,y) form the lines $x+y-5=0$ and $3x-2y+7=0$ is always 10.
Show that P must move on a line.

Show that the effect of following transaction on the accounting equation

$\begin{array}{cc}& \\ & \left(Rs\right)\\ \text{(a)Manoj started business with}& \\ \text{(i) Cash}& 2,30,000\\ \text{(ii) Goods}& 1,00,000\\ \text{(iii) Building}& 2,00,000\\ \text{(b) He purchased goods for cash}& 50,000\\ \text{(c) He sold goods ( costing Rs 20,000)}& 35,000\\ \text{(d)He purchased goods from Rahul}& 55,000\\ \text{(e) He sold goods to Varun (costing Rs 52,000)}& 60,000\\ \text{(f)He paid cash to Rahul in full settlement}& 53,000\\ \text{(g) Salary paid by him}& 20,000\\ \text{(h) Received cash from Varun in full settlement}& 59,000\\ \text{(i)Rent outstanding}& 3,000\\ \text{(j)Prepaid Insurance}& 2,000\\ \text{(k)Commission received by him}& 13,000\\ \text{(l) Amount withdrawn by him for personal use}& 20,000\\ \text{(m) Depreciation charged on building}& 10,000\\ \text{(n) Fresh capital invested}& 50,000\\ \text{(0) Purchased goods from Rakhi}& 6,000\end{array}$

The numbers of diagonals in n sided polygon______.

In how many ways two students can exchange greeting cards to each other?

Value of the numerically greatest term in the expansion of $\sqrt{3}(1+\frac{1}{\sqrt{3}}{)}^{20}$ is

The sum of the following series,$\frac{1}{3}{}^n{c}_1+\frac{(1^2+2^2)}{5}{}^n{c}_2+\frac{(1^2+2^2+3^2)}{7}{}^n{c}_3$ + ...... upto n terms is

The solutions of the system of equations $x+y=\frac{2\pi }{3}$ and
$cosx+cosy=\frac{3}{2}$ where x and y are real, are

The negation of $q\vee \sim (p\wedge r)$ is ___.

The set of values of $\theta$ satisfying the inequation $2si{n}^2\theta -5sin\theta +2>0$, where 0<$\theta$<2$\pi$,is

A team of 4 students is to be sent for a competition. 12 students offered their services for the same . But from past experience, it was observed that 5 students who had offered the services were not true to their work and they did mischief one time or the other are not to be selected. In how many ways, can the 4 students be selected for a competition?

In the above Q(52) near the point B, Compressibility factor Z is about

If $sin\theta =3sin(\theta +2\alpha )$,then the value of tan $(\theta +\alpha )+2tan\alpha$ is

Find the domain and range of the real function f defined by $f\left(x\right)=\sqrt{x-1}$.

Find the value of tan $\frac{\pi }{8}$.

Find the range of each of the following functions:

(i) $f\left(x\right)=2-3x,xϵR,x>0$

(ii) $f\left(x\right)=x^2+2,x$ is a real number

(iii) $f\left(x\right)=x,x$ is a real number.

The negation of $\sim s\vee (\sim r\wedge s)$ is equivalent to

Which type of number is root negative 2

The interval that gives all possible values of the expression $\sqrt{x^2-4x+20}$ is

The second derivative of a single valued function parametrically represented by $x=\varphi \left(t\right)andy=\psi \left(t\right)$, ( where $\varphi \left(t\right)and\psi \left(t\right)$ are different functions and ${\varphi }^{\prime }\left(t\right)\ne 0$) is given by

If $lo{g}_{3}y=x$ and $lo{g}_{2}z=x$, find ${72}^x$ in terms of y and z

(Distributive laws) For any three sets A, B, C prove that:

I. $A\cup (B\cap C)=(A\cup B)\cap (A\cup C)$
[Distirbutive law of union over intersection]

II. $A\cap \left[\right(B\cup C\left)\right]=(A\cap B)\cup (A\cap C)$
[Distributive law of intersection over union]

If the sum of a certain number of terms of the A.P. 25, 22, 19, .... is 116, Find the last term.

If a and b are real numbers between 0 and 1 such that the points $z_1$=a+i,$z_2$=1+bi and $z_3$=0 form an equilateral triangle, then

Find the sum of the following series up to n terms:

(i) 5 + 55 + 555 + .....

(ii) .6 + .66 + .666 + .....

Two lines passing through the point (2, 3) intersects each other at an angle of ${60}^{\circ }$. If slope of one line is 2; Find equation of the other line.

If a point P(x,y) moves along the ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1$ and if C is the centre of the ellipse, then, 4 max{CP} + 5 min{CP}=___ .

Using principle of mathematical induction, prove that ${41}^n-{14}^n$ is a multiple of 27.

Or
Prove by the principle of mathematical induction $n<2^n$ for all $nϵN$.

Find the numbers of chords that can be drawn through 16 points on circle.

If $z=\sqrt{2}-i\sqrt{2}$ is rotated through an angle 45° in the anti-clockwise direction about the origin, then the coordinates of its new position are

Which of the following functions is depicted below?

The solution set of $x^2+4x+9\ge 0$ is

If $g\left\{f\right(x\left)\right\}=|sinx|$ and $f\left\{g\right(x\left)\right\}=(sin\sqrt{x}{)}^2$, then

Range of the function $f\left(x\right)=\frac{x^2+x+2}{x^2+x+1};xϵR$ is

The middle term in the expansion of $(x^2+\frac{1}{x^2}+2{)}^n$ is:

Express $(cos2\theta +isin2\theta {)}^{-5}$ $(cos3\theta -isin3\theta {)}^6$ $(sin\theta -icos\theta {)}^3$ in the form of A+iB is

Solution set of the inequality
$\frac{1}{2^x-1}>\frac{1}{1-2^{x-1}}is$

In a triangle ABC, tan A + tan B +tan C = 6 and tan A $\times$ tan B = 2, then the value of tan A, tan B and tan C are