Let  S _k be the sum of an infinite GP series whose first term k is k and common  ratio is k/(k + 1) (k> 0). Then, the value of

1) log_e 4
2) log_e 2 – 1
3) 1 - log_e 2
4) 1 - log_e 4

The differential equation of the family of curves y = e^2x (a cos x + b sin x), where a and b are arbitrary constants, is given by

1) y₂ - 4y₁ + 5y = 0
2) 2y₂ - y₁ + 5y = 0
3) y₂ + 4y₁ - 5y = 0
4) y₂ - 2y₁ + 5y = 0

If a and b be the roots of the equation x² - 2x + 2 = 0, then the least value of n for which 
$$\left(\frac{\alpha}{\beta}\right)^{n}=1$$  is

1) 5
2) 2
3) 4
4) 3

For all values of  $$\Theta \in \left( 0 , \dfrac { \pi } { 2 } \right)$$ the determinants of the matrix $$\left[ \begin{array} { c c c } { - 2 } & { \tan \theta + \sec ^ { 2 } \theta } & { 3 } \\ { - \sin \theta } & { \cos \theta } & { \sin \theta } \\ { - 3 } & { 4 } & { 3 } \end{array} \right]$$  always lies in the interval:

1) (-4,3)
2) (1,2)
3) (4,3)
4) (4, 5)

If the mean of a binomial distribution is $$25$$, then its standard deviation lies in the interval given below:

1) $$[0,5)$$
2) $$(0,5]$$
3) $$[0,25)$$
4) $$(0,25]$$

If ax²  + 2hxy + by²   = 1, then d²y/ dx2  is equal to

If A=[a_ij] is a square matrix of even order such that [a_ij]=i²−j², then

1) A is a skew-symmetric matrix and |A| = 0
2) A is skew-symmetric matrix and |A| is a square
3) A is symmetric matrix and |A| = 0
4) none of these

If R is the set of all real numbers and f : R → R is given by f(x) = 3x²+ 1. Then, the set f ^-1 ([1, 6]) is

If   then the difference between the maximum and minimum values of  u² is given by

1) 2(a² + b²)
2) 2 Ö(a² + b²)
3) (a + b)²
4) (a - b)²

If  x = cos α  + cos β  – cos( α  + β ) and y =  , then (x – y) equals

1) 0
2) 1
3) – 1
4) – 2