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

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 sum of a given infinite geometric progression is 80 and the sum of its first two terms is 35. Then the value of n for which the sum of its first n terms is closest to 100, is

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…

The greatest number among \(2^{300}\), \(3^{200}\), \(4^{100}\), \(2^{100} + 3^{100}\) 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

The number of values of x for which \(C^{17-x}_{3x+1}\) is defined as an integer is

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 number of solutions of the equation \(x_1 + x_2 + x_3 + x_4 = 50\), where \(x_1, x_2, x_3, x_4\) are integers with \(x_1 \ge 1, x_2 \ge 2, x_3 \ge 0, x_4 \ge 0\) is