List of top Mathematics Questions asked in NEST

Evaluate \[ \int_{0}^{\sqrt{\pi}} x\sin^{2}(x^{2})\,dx \]

For \(x \in \mathbb{R}\), define \[ \langle x \rangle = x-[x] \] where \([x]\) denotes the greatest integer less than or equal to \(x\). Then for any arbitrary pair of real numbers \(x,y \in \mathbb{R}\), which of the following is always true?

Let \(S\) be the set of all rational numbers \(r\) such that if \[ r=\frac{p}{q} \] with \(p,q\in\mathbb{Z}\), \(q\neq0\), then the two roots of the quadratic equation \[ x^{2}+2px+q^{2}=0 \] are equal. Then the number of elements in \(S\) is:

Let \(\theta \neq \dfrac{(2n+1)\pi}{2}\), for \(n\in\mathbb{Z}\). Consider the system of equations \[ x\cos\theta+y\sec\theta=0 \] \[ x\sin\theta+y\tan\theta=0 \] This system does not have a unique solution if and only if \(\theta\) belongs to: