Combination Based on Geometry

Q. The number of diagonals that can be drawn in a hexagon is

Q.

Write the number of diagonals of an n-sided polygon.

Q. A polygon has $35$ diagonals. The number of sides of polygon is

1. $9$
2. $10$
3. $11$
4. $8$

Q. There are 12 points in a plane of which 5 are in a line. Then the number of distinct quadrilaterals with vertices at these points is

1. 420
2. 35
3. 175
4. 210

Q. Let $P_1,P_2,...,P_{15}$ be $15$ points on a circle. The number of distinct triangles formed by points $P_i,P_j,P_k$ such that $i+j+k\ne 15,$ is

1. $455$
2. $12$
3. $419$
4. $443$

Q. There are 16 points in a plane, no three of which are in a straight line except 8 which are all in a straight line. The number of triangles that can be formed by joining them equals

1. 504
2. 552
3. 560
4. 1120

Q.

Consider the set of eight vectors
V={ai+bj+ck}:a, b, c $ϵ\{-1,1\}$. Three non - coplanar vectors can be chosen from V in $2^p$ ways, Then $p$ is ___

Q.

If three lines are non concurrent and no two of them are parallel, number of circles drawn touching all the three lines

1. 4

2. 3

3. 1

4. Infinite

Q. If three of the six vertices of a regular hexagon are chosen at random, then the probability that the triangle formed with these chosen vertices is equilateral is :

1. $\frac{1}{10}$
2. $\frac{1}{5}$
3. $\frac{3}{10}$
4. $\frac{3}{20}$

Q. The maximum number of points of intersection of $8$ straight lines is

Q. In a plane there are two families of lines $y=x+r,y=-x+r$, where $r\in \{0,1,2,3,4\}$.The number of squares that can be formed from these family of lines such that its diagonal is of length $2$ units is

1. $9$
2. $16$
3. $25$
4. None of the above

Q. Consider the $5$ points comprising of vertices of a square and the intersection point of its diagonals. Then the number of triangles that can be formed using these points are

Q. The shortest distance between the two opposite edges of a regular tetrahedron of edge $\sqrt{8}$ units is ___

Q.

Out of 18 points in a plane, no three are in the same straight line except five points which are collinear. How many (i) straight lines (ii) triangles can be formed by joining them ?

Q. If $4$ circles and $4$ straight lines intersect each other in a plane, then

1. Maximum number of intersection points made by lines $=38$
2. Maximum number of intersection points made by lines $=6$
3. Maximum number of intersection points made by circles $=44$
4. Maximum number of intersection points made by circles $=12$

Q. As shown in the diagram, points $P_1,P_2,P_3,..........P_{10}$, are either the vertices or midpoints of the edges of a tetrahedran respectively. If the number of groups of four points

$(P_1,P_i,P_j,P_k)(1<i<j<k\le 10)$ lying on the same plane is ${}^{\prime }{m}^{\prime }$ then the sum of digits of ${}^{\prime }{m}^{\prime }$ is.

Q. Let $A_i,i=1,2,3,......21$ be the vertices of a $21-$sided regular polygon inscribed in a circle with centre at $O$. If triangles are formed by joining the vertices of the $21-$sided polygon then

1. The number of equilateral triangles formed by joining the vertices are $7$
2. The number of isosceles triangles formed by joining the vertices are $196$
3. The number of equilateral triangles formed by joining the vertices are $6$
4. The number of isosceles triangles formed by joining the vertices are $186$

Q. If there are $12$ points in a plane out of which only $5$ are collinear, then the number of quadrilaterals that can formed using these points is

1. $210$
2. $280$
3. $350$
4. $420$

Q. Straight lines are drawn by joining $m$ points on a straight line to $n$ points on another line. Then excluding the given points, the number of point of intersections of the lines drawn is (no two lines drawn are parallel and no three lines are concurrent)

1. $\frac{1}{2}{m}^2{n}^2$
2. $\frac{1}{4}mn(m-1)(n-1)$
3. $\frac{1}{4}{m}^2{n}^2$
4. $\frac{1}{2}mn(m-1)(n-1)$

Q.

There are $\mathrm{n}$ points in a plane of which $\mathrm{p}$ points are collinear. How many lines can be formed from these points?

1. ${}^{\left(\mathrm{n}-\mathrm{p}\right)}\mathrm{C}_2$

2. ${}^{\mathrm{n}}\mathrm{C}_2-{}^{\mathrm{P}}\mathrm{C}_2$

3. ${}^{\mathrm{n}}\mathrm{C}_2-{}^{\mathrm{P}}\mathrm{C}_2+1$

4. ${}^{\mathrm{n}}\mathrm{C}_2-{}^{\mathrm{P}}\mathrm{C}_2-1$

Q.

The number of straight lines joining 8 points on a circle is______.

1. 8

2. 16

3. 24

4. 28

Q. The number of the rectangles excluding squares that can be formed from a rectangle of size $9\times 6$ if it is divided by set of parallel lines of unit length as shown in the diagram is

1. $391$
2. None of these
3. $842$
4. $791$

Q.

There are 10 points in a plane of which 4 are collinear. How many different straight lines can be drawn by joining these points.

Q.

How many diagonals are there in a hexagon?

Q. There are $10$ points in a plane of which no $3$ points are collinear and $4$ points are concyclic. Number of different circles that can be drawn through at least $3$ points is

1. $100$
2. $116$
3. $117$
4. $120$

Q.

How many triangles can be obtained by joining 12 points, five of which are collinear ?

Q.

Find the number of diagonals of (i) a hexagon (ii) a polygon of 16 sides.

Q. The number of the points in the cartesian plane with integral coordinates satisfying the inequalities $|x|\le k$ , $|y|\le k$ , $|x-y|\le k$ is $($where $k\in N)$

1. $(k+1{)}^3-k^3$
2. $(k+2{)}^3-(k+1{)}^3$
3. $(k^2+1)$
4. None of these

Q. Six points in a plane be joining in all possible way by staright lines, and if no two of them be coincident or parallel, and no three pass through the same point (with the exception of the original $6$ points). The number of distinct points of intersection is equal to

1. $105$
2. $65$
3. $51$
4. $45$

Q. Let $n\ge 2$ be an integer. Take $n$ distinct points on a circle and join each pair of points by a line segment. Colour the line segment joining every pair of adjacent points by blue and the rest by red. If the number of red and blue line segments are equal, then the value of $n$ is