If a line along a chord of the circle \(4 x^{2}+4 y^{2}+120 x+675=0\), passes through the point \((-30,0)\) and is tangent to the parabola \(\mathrm{y}^{2}=30 \mathrm{x}\), then the length of this chord is :

If the coefficient of x in the expansion of \( (ax^{2}+bx+c)(1-2x)^{26} \) is - 56 and the coefficients of \( x^{2} \) and \( x^{3} \) are both zero, then \( a+b+c \) is equal to:

Let \(A,B\) be points on the two half-lines \(x-\sqrt{3}|y|=\alpha\), \(\alpha>0\) at a distance of \(\alpha\) from their point of intersection \(P\). The line segment \(AB\) meets the angle bisector of the given half-lines at the point \(Q\). If \(PQ=\dfrac{9}{2}\) and \(R\) is the radius of the circumcircle of \(\triangle PAB\), then \(\dfrac{\alpha^2}{R}\) is equal to ______

The number of integral values of \(k\) for which the line, \(3 x+4 y=k\) intersects the circle, \(\mathrm{x}^{2}+\mathrm{y}^{2}-2 \mathrm{x}-4 \mathrm{y}+4=0\) at two distinct points is

Consider three vectors \(\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}}, \overrightarrow{\mathrm{c}}\). Let \(|\overrightarrow{\mathrm{a}}|=2,|\overrightarrow{\mathrm{b}}|=3\) and \(\overrightarrow{\mathrm{a}}=\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}}\). If \(\alpha \in\left[0, \frac{\pi}{3}\right]\) is the angle between the vectors \(\vec{b}\) and \(\vec{c}\), then the minimum value of \(27|\overrightarrow{c}-\overrightarrow{a}|^2\) is equal to :

Let \(\quad \overrightarrow{\mathrm{a}}=2 \hat{\mathrm{i}}+\alpha \hat{\mathrm{j}}+\hat{\mathrm{k}}, \quad \overrightarrow{\mathrm{b}}=-\hat{\mathrm{i}}+\hat{\mathrm{k}}, \quad \overrightarrow{\mathrm{c}}=\beta \hat{\mathrm{j}}-\hat{\mathrm{k}}\), where \(\alpha\) and \(\beta\) are integers and \(\alpha \beta=-6\). Let the values of the ordered pair \((\alpha, \beta)\) for which the area of the parallelogram of diagonals \(\vec{a}+\vec{b}\) and \(\vec{b}+\vec{c}\) is \(\frac{\sqrt{21}}{2}\), be \(\left(\alpha_1, \beta_1\right)\) and \(\left(\alpha_2, \beta_2\right)\). Then \(\alpha_1^2+\beta_1^2-\alpha_2 \beta_2\) is equal to

A water tank has the shape of a right circular cone with axis vertical and vertex downwards. Its semivertical angle is \(\tan ^{-1} \frac{3}{4}\). Water is poured in it at a constant rate of \(6\) cubic meter per hour. The rate (in square meter per hour), at which the wet curved surface area of the tank is increasing, when the depth of water in the tank is \(4\) meters, is.

The area (in sq. units) of the region described by \(\{(x,y):\)\({y^2} \le 2x \,and\,y \ge 4x - 1\)\(\}\) is

If a circle \(C\) passing through the point \((4, 0)\) touches the circle \(x^2 + y^2 + 4x -6y = 12\) externally at the point \((1, -1)\), then the radius of \(C\) is

The equation \(e^{4 x}+8 e^{3 x}+13 e^{2 x}-8 e^x+1=0, x \in R\) has:

Let \(f(x)=2 x^{2}-x-1\) and \(S =\{n \in Z :|f(n)| \leq 800\}\) . Then value of \(\sum_{n \in S} f(n)\) is . . . .  .

Let \(A=\{z\in\mathbb{C}:|z-2|\le4\}\) and
\(B=\{z\in\mathbb{C}:|z-2|+|z+2|=5\}\).
Then the max \(\left\{\left| z _1- z _2\right|: z _1 \in A\right.\) and \(\left.z _2 \in B\right\}\) is

If an unbiased dice is rolled thrice, then the probability of getting a greater number in the \(i^{\text {th }}\) roll than the number obtained in the \((i-1)^{\text {th }}\) roll, \(i=2,3\), is equal to :