NDA Mathematics 2016 Question Paper

Suppose $\omega$ is a cube root of unity with $\omega \ne 1$. Suppose $P$ and $Q$ are the points on the complex plane defined by $\omega$ and ${\omega }^2$. If $O$ is the origin, then what is the angle between $OP$ and $OQ$?

Suppose there is a relation $\ast$ between the positive numbers $x$ and $y$ given by $x\ast y$ if and only if $x\le y^2$. Then which one of the following is correct?

If $x^2-px+4>0$ for all real values of $x$, then which one of the following is correct?

If $z=x+iy={\left(\frac{1}{\sqrt{2}}-\frac{i}{\sqrt{2}}\right)}^{-25}$, where $i=\sqrt{-1}$, then what is the fundamental amplitude of $\frac{z-\sqrt{2}}{z-i\sqrt{2}}$?

If $f(x_1)-f(x_2)=f\left(\frac{x_1-x_2}{1-x_1{x}_2}\right)$ for $x_1,x_2\in (-1,1)$, then what is $f(x)$ equal to?

What is the range of the function $y=\frac{x^2}{1+x^2}$ where $x\in \mathbb{R}$?

A straight line intersects $x$ and $y$ axes at $P$ and $Q$ respectively. If $(3,5)$ is the middle point of $PQ$, then what is the area of the triangle $OPQ$?

If a circle of radius $b$ units with centre at $(0,b)$ touches the line $y=x-\sqrt{2}$, then what is the value of $b$?

Directions : Consider the function $f(\theta )=4({\sin }^2\theta +{\cos }^4\theta )$

What is the maximum value of the function $f(\theta )$?

Directions : Consider the function $f(\theta )=4({\sin }^2\theta +{\cos }^4\theta )$

What is the minimum value of the function $f(\theta )$?