Mathematics Tes...

Consider the two sets :

\(A=\left\{m \in R\right.\) : both the roots of \(x^2-(m+1) x+m+4=0\) are real \(\}\) and \(B=(-3,5)\) Which of the following is not true?

The condition of being one root of the equations \(a x^{2}+b x+c=0\) and \(a^{\prime} x^{2}+b^{\prime} x+c^{\prime}=0\) common is:

For the given curve: \(y=2 x-x^{2}\), when \(x\) increases at the rate of 3 units/sec, then how the slope of curve changes?

Let \(\vec{\alpha}=3 \hat{i}+\hat{j}\) and \(\vec{\beta}=2 \hat{i}-\hat{j}+3 \hat{k}\). If \(\vec{\beta}=\vec{\beta}_1-\vec{\beta}_2\), where \(\vec{\beta}_1\) is parallel to \(\vec{\alpha}\) and \(\vec{\beta}_2\) is perpendicular to \(\vec{\alpha}\), then \(\vec{\beta}_1 \times \vec{\beta}_2\) is equal to:

Let \(\alpha\) and \(\beta\) be the roots of the quadratic equation \(x^2 \sin \theta-x(\sin \theta \cos \theta+1)+\cos \theta=0\left(0<\theta<45^{\circ}\right)\), and \(\alpha<\beta\). Then \(\sum_{n=0}^{\infty}\left(\alpha^n+\frac{(-1)^n}{\beta^n}\right)\) is equal to :

Let \(P Q\) be a focal chord of the parabola \(y^2=4 x\) such that it subtends an angle of \(\frac{\pi}{2}\) at the point \((3,0)\). Let the line segment PQ be also a focal chord of the ellipse \(E: \frac{x^2}{a^2}+\frac{y^2}{b^2}=1, a^2>b^2\). If \(e\) is the eccentricity of the ellipse \(E\), then the value of \(\frac{1}{e^2}\) is equal to:

The degree of the differential equation:

\(\frac{d^{2} y}{d x^{2}}+3\left(\frac{d y}{d x}\right)^{2}=x^{2} \log \left(\frac{d^{2} y}{d x^{2}}\right)\)

Find the 8^th term in the following sequence whose nth term is\(a_{n}=\frac{n^{2}}{2^{n}}\).

What is \(\int_{0}^{a} \frac{f(a-x)}{f(x)+f(a-x)} d x\) equal to?