Let \(r_{1}\) and \(r_{2}\) be the radii of the largest and smallest circles, respectively, which pass through the point \((-4,1)\) and having their centres on the circumference of the circle \(x^{2}+y^{2}+2 x+4 y-4= 0.\) If \(\frac{r_{1}}{r_{2}}=a+b \sqrt{2}\), then \(a+b\) is equal to:

The differential equation of the family of curves, \(x^{2}=4 b(y+b), b \in R,\) is

Let \(x = x ( y )\) be the solution of the differential equation \(2( y +2) \log _e( y +2) dx +\left( x +4-2 \log _e( y +2)\right) dy =0\), \(y > -1\) with \(x\left(e^4-2\right)=1\). Then \(x\left(e^9-2\right)\) is equal to

In a survey of \(220\) students of a higher secondary school, it was found that at least \(125\) and at most \(130\) students studied Mathematics; at least \(85\) and at most \(95\) studied Physics; at least \(75\) and at most \(90\) studied Chemistry; \(30\) studied both Physics and Chemistry; \(50\) studied both Chemistry and Mathematics; \(40\) studied both Mathematics and Physics and \(10\) studied none of these subjects. Let \(\mathrm{m}\) and \(\mathrm{n}\) respectively be the least and the most number of students who studied all the three subjects. Then \(\mathrm{m}+\mathrm{n}\) is equal to ...........

Let \(A = \{2, 3, 4, 5, 6\}\). Let R be a relation on the set \(A \times A\) given by \((x, y) R (z, w)\) if and only if \(x\) divides \(z\) and \(y \leq w\). Then the number of elements in R is _______.

If \(\int_{0}^{\sqrt{3}} \frac{15 x^{3}}{\sqrt{1+x^{2}+\sqrt{\left(1+x^{2}\right)^{3}}}} d x=\alpha \sqrt{2}+\beta \sqrt{3}\), where \(\alpha, \beta\) are integers, then \(\alpha+\beta\) is equal to.

If the probability that the random variable \(X\) takes values \(x\) is given by \(P ( X = x )= k ( x +1) 3^{- x }, x =0\), \(1,2,3 \ldots\), where \(k\) is a constant, then \(P ( X \geq 2)\) is equal to

A natural number has prime factorization given by \(n =2^{ x } 3^{ y } 5^{ z },\) where \(y\) and \(z\) are such that \(y+z=5\) and \(y^{-1}+z^{-1}=\frac{5}{6}, y > z\). Then the number of odd divisors of \(n\), including \(1,\) is ..... .

For two non-zero complex number \(z_1\) and \(z_2\), if \(\operatorname{Re}\left(z_1 z_2\right)=0\) and \(\operatorname{Re}\left(z_1+z_2\right)=0\), then which of the following are possible ? \((A)\) \(\operatorname{Im}\left(z_1\right) > 0\) and \(\operatorname{Im}\left(z_2\right) > 0\) \((B)\) \(\operatorname{Im}\left(z_1\right) < 0\) and \(\operatorname{Im}\left(z_2\right) > 0\) \((C)\) \(\operatorname{Im}\left(z_1\right) > 0\) and \(\operatorname{Im}\left(z_2\right) < 0\) \((D)\) \(\operatorname{Im}\left( z _1\right) < 0\) and \(\operatorname{Im}\left( z _2\right) < 0\) Choose the correct answer from the options given below :

The number of three-digit even numbers, formed by the digits \(0,1,3,4,6,7\) if the repetition of digits is not allowed, is .... .

Let \(P _1\) be the plane \(3 x - y -7 z =11\) and \(P _2\) be the plane passing through the points \((2,-1,0)\), \((2,0,-1)\), and \((5,1,1)\). If the foot of the perpendicular drawn from the point \((7,4,-1)\) on the line of intersection of the planes \(P_1\) and \(P_2\) is \((\alpha, \beta\), \(\gamma\) ), then \(\alpha+\beta+\gamma\) is equal to \(............\).

A die is thrown two times and the sum of the scores appearing on the die is observed to be a multiple of \(4\). Then the conditional probability that the score \(4\) has appeared at least once is

Let \(\alpha\) and \(\beta\) be the roots of the equation \(5 x^{2}+6 x-2=0 .\) If \(S_{n}=\alpha^{n}+\beta^{n}, n=1,2,3 \ldots\) then :