Let `f(n)` be the number of regions in which `n` coplanar circles can divide the plane. If it is known that each pair of circles intersect in two different points and no three of them have common points of intersection, then `(i)` `f(20) = 382` `(ii)` `f(n)` is always an even number `(iii)` `f^(-1)(92) = 10` `(iv)` `f(n)` can be odd

Total numbers of regions in which |'n:)' coplanar lines can divide the plane,if it is known that no two lines are parallel and no three of them are concurrent,is equal to

Let N denotes,the greatest number of points in which m straight lines and n circles intersect. Then,

Let f(n) denote the number of ways in which n letters go into n envelops so that no letter is in the correct envelope, (where n>= 5 ), then f(5)/11 equals

Let N be the set of natural numbers and f : N->N be a function given by f(x)=x+1 for x in N.Which one of the following is correct? a. f is one-one and onto b. f is one-one but not onto c. f is only onto d. f is neither one-one nor onto

Fill in the blanks (i) A line intersecting a circle in two distinct points is caled a…. (ii) A circle can have… parallel tangents at the most. (iii) The common point of a tangent to a circle and the circle is called the…. (iv) A circle can have.... tangents.

Let f(x)=[n+p sin x],x in(0,pi),n in Z,p a prime number and [x]= the greatest integer less than or equal to x.The number of points at which f(x) is not not differentiable is:

State which of the following statements are true (T) and which false (F): Point has a size because we can see it as a thick dot on paper. By lines in geometry, we mean only straight lines. Two lines in a place always intersect in a point. Any plane through a vertical line is vertical. Any plane through a horizontal line is horizontal. There cannot be a horizontal line is a vertical plane. All lines in a horizontal plane are horizontal. Two lines in a plane always intersect in a point. If two lines intersect at a point P , then P is called the point of concurrence of the two lines. If two lines intersect at a point p , then p is called the point of intersection of the two lines. If A , B , C a n d D are collinear points D , P a n d Q are collinear, then points A , B , C , D , P a n d Q and always collinear. Two different lines can be drawn passing through two given points. Through a given point only one line can be drawn. Four points are collinear if any three of them lie on the same line. The maximum number of points of intersection of three lines in three. The minimum matching of the statements of column A and Column B .

Let f be real valued function from N to N satisfying. The relation f(m+n)=f(m)+f(n) for all m,n in N . If domain of f is first 3m natural numbers and if the number of elements common in domain and range is m, then the value of f(1) is

Let f(x)=ax^(4)+bx^(2)+3x+7 and f(-4)=2286 and f=(4)=N . The number of ways in which N can be resolved as a product of two divisors which are relatively prime: