Let $z = e^{i\theta}$,
then $a = 1 + e^{i\theta} + e^{2i\theta} = \dfrac{\sin(3\theta/2)}{\sin(\theta/2)} e^{i\theta}$.
After simplification, the value $\dfrac{1}{3}$ is not possible.
Let $K$ be the set of all odd numbers less than $200$ and $M$ be the set of products of two distinct numbers taken from $K$, then find the total number which is divisible by $5$ in $M$.
Let numbers be $a, b$.
Then $\dfrac{x}{y} = \dfrac{\sqrt{ab}}{\dfrac{2ab}{a+b}} = \dfrac{a+b}{2\sqrt{ab}} = \dfrac{5}{4}$.
Let $\dfrac{a}{b} = r$,
then $\dfrac{r+1}{2\sqrt{r}} = \dfrac{5}{4}$.
On solving, $r = 4$ or $\dfrac{1}{4}$.
Hence ratio = $1:4$.
Sum of squares of first $n$ odd numbers $= \dfrac{n(2n-1)(2n+1)}{3}$.
For $n = 50$, sum $= \dfrac{50 \times 99 \times 101}{3} = 166650$.
Hence, it lies between $150000$ and $250000$.
Substitute $y = 4x + C$ in $x^2 + 4y^2 = 4$.
To touch, discriminant $= 0$.
After solving, $C = \pm \dfrac{2}{\sqrt{17}}$.
Hence, there are $2$ possible values.
In the triangle $ABC$, sides $AB$, $BC$, and $CA$ are extended such that $B'$, $C'$, and $A'$ are the new points formed.
If the area of $\triangle ABC = 1$ unit and
$AA' = 2AB,\ BB' = 2BC,\ CC' = 3AC$,
then find the area of $\triangle A'B'C'$.
If “Rain” is called “Water”, “Water” is called “Road”, “Road” is called “Cloud”, “Cloud” is called “Sky”, and “Sky” is called “Sea”, then where does aeroplane fly?
The main advantage of creating a document using basic HTML (HyperText Markup Language) is that it can be displayed by any web browser on any operating system without needing special software.
Option (a) → True but incomplete; HTML is easy to learn, but that’s not its main advantage.
Option (b) → Incorrect; Word processors (like MS Word) don’t natively render HTML.
Option (c) → Incorrect; Programmers are not the display agents — browsers are.
✅ Option (d) → Correct, because HTML documents are platform-independent and viewable by all browsers.
Micro, Mini, and Super are standard classes of computers (by size/capacity). A notebook computer is a form factor of a microcomputer (portable laptop), not a separate class. Hence, it’s the odd one out.
$x, y, z$ are painters.
X paints wall of area A in 10 hrs.
Y paints the same wall in 20 hrs.
Together, X, Y, and Z finish 2A in 8 hrs.
Find how many hours Z alone will take to paint wall A.
nso ptr kli chn’ stands for Sharma gets marriage gift;
‘ptr lnm wop chn’ stands for wife give marriage gift;
‘iti wop nhi’ stands for he give nothing.
What would mean gives?
Solution: Track each position by alphabet index:
1st letters: K(11), M(13), O(15), Q(17) → $+2$ ⇒ next S(19).
2nd letters: J(10), M(13), P(16), S(19) → $+3$ ⇒ next V(22).
3rd letters: R(18), P(16), N(14), L(12) → $-2$ ⇒ next J(10).
So the term is $\boxed{\text{SVJ}}$.
The algorithm sum of the difference between the squares of two positive integers $A$ and $B$ is a fixed constant $C$.
If $A = B$, then value of $C$ is —