The locus of the mid point of the line segment joining the point \((4,3)\) and the points on the ellipse \(x^{2}+2 y^{2}=4\) is an ellipse with eccentricity

Three circles of radii \(a, b, c\, ( a < b < c )\) touch each other externally. If they have \(x -\) axis as a common tangent, then

A candidate has to go to the examination centre to appear in an examination. The candidate uses only one means of transportation for the entire distance out of bus, scooter and car. The probabilities of the candidate going by bus, scooter and car, respectively, are \(\dfrac{2}{5}\), \(\dfrac{1}{5}\) and \(\dfrac{2}{5}\). The probabilities that the candidate reaches late at the examination centre are \(\dfrac{1}{5}\), \(\dfrac{1}{3}\) and \(\dfrac{1}{4}\) if the candidate uses bus, scooter and car, respectively. Given that the candidate reached late at the examination centre, the probability that the candidate travelled by bus is:

If the four distinct points \((4,6),(-1,5),(0,0)\) and \((\mathrm{k}, 3 \mathrm{k})\) lie on a circle of radius r , then \(10 \mathrm{k}+\mathrm{r}^2\) is equal to

Let the hyperbola \(H : \frac{ x ^{2}}{ a ^{2}}-\frac{ y ^{2}}{ b ^{2}}=1\) pass through the point \((2 \sqrt{2},-2 \sqrt{2})\). A parabola is drawn whose focus is same as the focus of \(H\) with positive abscissa and the directrix of the parabola passes through the other focus of \(H\). If the length of the latus rectum of the parabola is e times the length of the latus rectum of \(H\), where \(e\) is the eccentricity of \(H\), then which of the following points lies on the parabola?

The region represented by \(\left| {x - y} \right| \leq 2\) and \(\left| {x + y} \right| \leq 2\) is bounded by a

Let a circle \(C\) have its centre in the first quadrant, intersect the coordinate axes at exactly three points and cut off equal intercepts from the coordinate axes. If the length of the chord of \(C\) on the line \(x + y = 1\) is \(\sqrt{14}\), then the square of the radius of \(C\) is _______.

Let \(f:(1,\infty)\to\mathbb{R}\) be a function defined as \(f(x) = \dfrac{x-1}{x+1}\). Let \(f^{i+1}(x) = f(f^i(x))\), \(i=1, 2, \ldots, 25\), where \(f^1(x)=f(x)\). If \(g(x) + f^{26}(x) = 0\), \(x \in (1, \infty)\), then the area of the region bounded by the curves \(y=g(x)\), \(2y=2x-3\), \(y=0\) and \(x=4\) is:

For, \(\alpha, \beta, \gamma, \delta \in N\), if \(\int\left(\left(\frac{x}{e}\right)^{2 x}+\left(\frac{e}{x}\right)^{2 x}\right) \log _{ e } x d x=\frac{1}{\alpha}\left(\frac{ x }{ e }\right)^{\beta x}-\frac{1}{\gamma}\left(\frac{ e }{ x }\right)^{\delta x }+ C ,\) Where \(e =\sum \limits_{ n =0}^{\infty} \frac{1}{ n !}\) and \(C\) is constant of integration, then \(\alpha+2 \beta+3 \gamma-4 \delta\) is equal to:

Let the equation of the plane passing through the line \(x-2 y-z-5=0=x+y+3 z-5\) and parallel to the line \(x + y +2 z -7=0=2 x +3 y + z -2\) be \(a x+b y+c z=65\). Then the distance of the point \((a, b, c)\) from the plane \(2 x+2 y-z+16=0\) is \(..........\).

The area (in sq. units) of the largest rectangle \(ABCD\) whose vertices \(A\) and \(B\) lie on the \(x\)-axis and vertices \(C\) and \(D\) lie on the parabola, \(y = x ^{2}-1\) below the \(x\) -axis, is

Let \(x_1, x_2, \ldots, x_{10}\) be ten observations such that \(\sum_{i=1}^{10}\left(x_i-2\right)=30, \sum_{i=1}^{10}\left(x_i-\beta\right)^2=98, \beta\gt2\), and their variance is \(\frac{4}{5}\). If \(\mu\) and \(\sigma^2\) are respectively the mean and the variance of \(2\left(x_1-1\right)+4 \beta\), \(2\left(x_2-1\right)+4 \beta, \ldots ., 2\left(x_{10}-1\right)+4 \beta\), then \(\frac{\beta \mu}{\sigma^2}\) is equal to :

For the system of linear equations : \(x-2 y=1, x-y+k z=-2, k y+4 z=6, k \in R\) consider the following statements : \((A)\) The system has unique solution if \(k \neq 2\), \(k \neq-2\) \((B)\) The system has unique solution if \(k =-2\). \((C)\) The system has unique solution if \(k =2\). \((D)\) The system has no-solution if \(k =2\). \((E)\) The system has infinite number of solutions if \(k \neq-2\) Which of the following statements are correct?