For all real values of x, \(\frac{3x^2-6x+12}{x^2+2x+4}\) lies between 1 and k, and does not take any value above k. Then k equals............

A shopkeeper reduces the price of a pen by 25% as a result of which the sales quantity increased by 20%. If the revenue made by the shopkeeper decreases by x%, then x is...

The maximum distance between the point (-5, 0) and a point on the circle \(X^2 + Y^2 = 4\) is......

Three friends divided some apples in the ratio 3 : 5 : 7 among themselves. After consuming 16 apples they found that the remaining number of apples with them was equal to largest number of apples received by one of them at the beginning. Total number of apples these friends initi…

If x, y, z are positive real numbers such that \(x^{12} = y^{16} = z^{24}\), and the three quantities \(3\log_y x\), \(4\log_z y\), \(n\log_x z\) are in arithmetic progression, then the value of n is

If A is a \(3\times 3\) non-zero matrix such that \(A^2 = 0\) then determinant of \([(1+A)^2 - 50A]\) is equal to

The number of pairs (x, y) satisfying the equation \(\sin x+\sin y=\sin(x+y)\) and \(|x|+|y|=1\) is

Let A, B, C be three \(4\times 4\) matrices such that det A = 5, det B = -3, and det C = \(\frac{1}{2}\). Then the det \((2AB^{-1}C^{3}B^{T})\) is .....

The circle \(x^2+y^2-6x-10y+k=0\) does not touch or intersect the coordinate axes. If the point (1, 4) does not lie outside the circle, and the range of k is [a, b] then a + b is____