Let C be the circle with centre at(1, 1) and radius = 1. If T is the circle centered at (0, y) passing through the origin and touching the circle C externally, then the radius of T is equal to

(a)

\(\frac { \sqrt { 3 } }{ \sqrt { 2 } } \)

(b)

\(\frac { \sqrt { 3 } }{ 2 } \)

(c)

\(\frac { 1 }{ 2 } \)

(d)

\(\frac { 1 }{ 4 } \)

If \(\vec { a } ,\vec { b } ,\vec { c } \) are three unit vectors such that \(\vec { a } \) is perpendicular to \(\vec { b } \) and is parallel to \(\vec { c } \) then \(\vec { a } \times (\vec { b } \times \vec { c } )\) is equal to

(a)

\(\vec { a } \)

(b)

\(\vec { b} \)

(c)

\(\vec { c } \)

(d)

\(\vec { 0 } \)

If the distance of the point (1, 1, 1) from the origin is half of its distance from the plane x + y + z + k = 0, then the values of k are

(a)

\(\pm 3\)

(b)

\(\pm 6\)

(c)

-3, 9

(d)

3, -9

One of the closest points on the curve x² - y² = 4 to the point (6, 0) is

(a)

(2,0)

(b)

\(\left( \sqrt { 5 } ,1 \right) \)

(c)

\(\left( 3,\sqrt { 5 } \right) \)

(d)

\(\left( \sqrt { 13 } ,-\sqrt { 3 } \right) \)

If u(x, y) = x²+ 3xy + y - 2019, then \(\left.\frac{\partial u}{\partial x}\right|_{(4,-5)}\) is equal to

(a)

-4

(b)

-3

(c)

-7

(d)

13

If \(\int _{ 0 }^{ a }{ \frac { 1 }{ 4+{ x }^{ 2 } } dx=\frac { \pi }{ 8 } } \) then a is

(a)

4

(b)

1

(c)

3

(d)

2

The differential equation representing the family of curves y = Acos(x + B), where A and B are parameters,is

(a)

\(\frac { { d }^{ 2 }y }{ { dx }^{ 2 } } -y=0\)

(b)

\(\frac { { d }^{ 2 }y }{ { dx }^{ 2 } }+y=0\)

(c)

\(\frac { { d }^{ 2 }y }{ { dx }^{ 2 } }=0\)

(d)

\(\frac { { d }^{ 2 }x }{ { dy }^{ 2 } }=0\)

Integrating factor of the differential equation \(\frac{d y}{d x}=\frac{x+y+1}{x+1}\) is 

(a)

\(\frac{1}{x+1}\)

(b)

x+1

(c)

\(\frac { 1 }{ \sqrt { x+1 } } \)

(d)

\({ \sqrt { x+1 } } \)

Consider a game where the player tosses a six-sided fair die. If the face that comes up is 6, the player wins Rs. 36, otherwise he loses Rs. k², where k is the face that comes up k = {1, 2, 3, 4, 5}.
The expected amount to win at this game in Rs. is

(a)

\(\cfrac { 19 }{ 6 } \)

(b)

\(-\cfrac { 19 }{ 6 } \)

(c)

\(\cfrac { 3 }{ 2 } \)

(d)

\(-\cfrac { 3 }{ 2 } \)

If \(f(x)=\left\{\begin{array}{ll} 2 x & 0 \leq x \leq a \\ 0 & \text { otherwise } \end{array}\right.\) is a probability density function of a random variable, then the value of a is

(a)

1

(b)

2

(c)

3

(d)

4