Let S be a non-empty subset of R. Consider the following statement:P: There is a rational number $x\in S$such that $x>0$.Which of the following statements is the negation of the statement P ? There is no rational number $x\in S$ such that $x\le 0$(9) Every rational number $x\in S$ satisfies $x\le 0$(18) $x\in S$ and $x\le 0\Rightarrow x$(27) is not rational There is a rational number $x\in S$ such that $x\le 0$(36)

Solution of the differential equation $\cos xdy=y\left(\sin x-y\right)dx,0<x<\frac{\pi }{2}$ (A) $\sec x=\left(\tan x+c\right)y$ (B) $y\sec x=\tan x+c$ (C) $y\tan x=\sec x+c$ (D) $\tan x=\left(\sec x+c\right)y$

Let $f:R\to R$be defined by $f\left(x\right)=\left\{k-2x,\text{if}x\le -12x+3,fx\succ 1\right\}$. If f has a local minimum at $x=1$, then a possible value of k is

Let $\vec{a}=\hat{j}-\hat{k}$and $\vec{c}=\hat{i}-\hat{j}-\hat{k}$. Then vector $\vec{b}$satisfying $\vec{a}\times \vec{b}+\vec{c}=\vec{0}$and $\vec{a}\dot{\vec{b}}=3$is

The area bounded by the curves $y=\cos xandy=\text{sin}x$between the ordinates $x=0andx=\frac{3\pi }{2}$is

Let $\cos \left(\alpha +\beta \right)=\frac{4}{5}$and let $s\in \left(\alpha \beta \right)=\frac{5}{13}$where $0\le \alpha ,\beta \le \frac{\pi }{4}$, then $tan2\alpha =$

The equation of the tangent to the curve $y=x+\frac{4}{x^2}$, that is parallel to the x-axis, is

The number of 3 ´ 3 non-singular matrices, with four entries as 1 and all other entries as 0, is

Consider the following relations:R = {(x, y) | x, y are real numbers and x = wy for some rational number w};$S=\left\{\left(\frac{m}{n},\frac{p}{q}\right)\text{m , n , pandqa r ei n t e g e r ss u c ht h a tn ,q}\ne \text{0andq m = p n}\right\}$. Then

If two tangents drawn from a point P to the parabola $y^2=4x$ are at right angles, then the locus of P is $\left(a\right)$2x+1=0$\left(b\right)$x=-1$\left(c\right)$2x-1=0$\left(d\right)$x=1