If \(\theta\) is the angle between the pair of tangents drawn from the point \(A\left(0, \frac{7}{2}\right)\) to the circle \(x^2 + y^2 - 14x + 16y + 88 = 0\), then \(\tan\theta\) equals

The difference between the maximum real root and the minimum real root of the equation \((x^2-5)^4 + (x^2-7)^4 = 16\) is

The numbers \(2^{2024}\) and \(5^{2024}\) are expanded and their digits are written out consecutively on one page. The total number of digits written on the page is

For some non-zero real values of a, b and c, it is given that \(\left|\frac{c}{a}\right| = 4\), \(\left|\frac{a}{b}\right| = \frac{1}{3}\) and \(\frac{b}{c} = -\frac{3}{4}\). If ac > 0, then \(\left(\frac{b+c}{a}\right)\) equals

A boat goes 96 km upstream in 8 hours and covers the same distance moving downstream in 6 hours. On the next day boat starts from point A, goes downstream for 1 hour, then upstream for 1 hour and repeats this four more time that is, 5 upstream and 5 downstream journeys. Then the…

Let \(a = \frac{(\log_7 4)(\log_7 5 - \log_7 2)}{(\log_7 25)(\log_7 8 - \log_7 4)}\). Then the value of \(5^a\) is

If the shortest distance of a given point to a given circle is 4 cm and the longest distance is 9 cm, then the radius of the circle is

The side AB of a triangle ABC is c. The median BD is of length k. If \(\angle BDA = \theta < 90^\circ\), then the area of triangle ABC is

In a survey of 500 people, it was found that 250 owned a 4-wheeler but not a 2-wheeler, 100 owned a 2-wheeler but not a 4-wheeler, and 100 owned neither a 4-wheeler nor a 2-wheeler. Then the number of people who owned both is