Three real numbers are in Arithmetic Progression, the first term is 9. If 2 is added to the second term and 20 is added to the third term, the resulting three numbers are in Geometric Progression. What is the smallest possible value for the third term of the Geometric Progression?

Let $z=(1-t^2)+i\sqrt{1-t^2}$ be a complex number where $t$ is a real number such that $|t|<1$. Then the locus of $z$ in the complex plane is

The length of a side of a square is ‘a’ metres. A second square is formed by joining the middle points of the sides of this square. Then, a third square is formed by joining the middle points of the second square and so on. Calculate the sum of the area of all squares which are formed in the way as mentioned above.

Let ${\int }_1^2{e}^{x^2}dx=a$. Then the value of ${\int }_e^{e^4}\sqrt{{\log }_{e}x}dx$ is

Evaluate: $\int \frac{(\sqrt{x}{)}^5}{(\sqrt{x}{)}^7+x^6}dx$

There are 4 Red, 5 Black and 6 Blue balls in an urn. 4 balls are drawn at random without replacement from this urn. What is the probability that these 4 balls will contain at least one ball of each colour?

Consider four events $P,Q,R$ and $S$ such that if any of $P$ and $Q$ occurs, then either $R$ occurs or $S$ doesn’t occur. If exactly one of $R$ and $S$ always occurs, which of the following statements is necessarily true? (The notation $E^c$ denotes the complement of the event $E$)

Let ${\Delta }_a=\left|\begin{array}{ccc}a-1& n& 6\\ (a-1{)}^2& 2n^2& 4n-2\\ (a-1{)}^3& 3n^3& 3n^2-3n\end{array}\right|$, find the value of ${\sum }_{a=1}^n{\Delta }_a$.

Let $A=(1,-1),B=(-2,0),C=(1,2)$ and $D$ be the vertices of a parallelogram in the $X-Y$ plane listed clockwise. Then the point $D$ is

Solve the equation: $3^{\sin 2x+2{\cos }^{2}x}+3^{1-\sin 2x+2{\sin }^{2}x}=28$.

The distance of the point $(1,-2,3)$ from the plane $x-y+z=11$ measured along a line parallel to $\frac{x}{2}=\frac{y}{3}=\frac{z}{6}$ is