Combination Based on Geometry

Q.

How many triangles can be formed by joining four points on a circle?

1. $4$

2. $6$

3. $8$

4. $10$

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

Q. From $25$ points on a plane, $8$ are on a straight line, $7$ are on another straight line and $10$ are on a third straight line.
Then the number of triangles that can be drawn by connecting some three points from these $25$ is

1. ${}^{25}{C}_3$
2. ${}^{25}{C}_3-({}^8{C}_3+{}^7{C}_3+{}^{10}{C}_3)$
3. ${}^{25}{C}_3+({}^8{C}_3+{}^7{C}_3+{}^{10}{C}_3)$
4. ${}^8{C}_3$+${}^7{C}_3+{}^{10}{C}_3$

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

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

Q. A polygon has 44 diagonals. Then the number of sides is

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

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. 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. $791$
3. $842$
4. None of these

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

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

Q. The number of triangles whose vertices are at the vertices of an octagon but none of whose sides happen to come from the octagon, is

1. $16$
2. $28$
3. $56$
4. $70$

Q.

The number of parallelograms that can be formed from a set of four parallel lines intersecting another set of three parallel lines is

1. 6
2. 18
3. 12
4. 9

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

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. 35
2. 175
3. 210
4. 420

Q. The maximum number of points of intersection of five lines and four circles in a plane is

1. $60$
2. $62$
3. $70$
4. $72$

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

Q. A polygon has 44 diagonals. Then the number of sides is

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

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. 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. $791$
3. $842$
4. None of these

Q. If a convex polygon has $35$ diagonals, then the number of points of intersection of diagonals which lies inside the polygon is

1. $45$
2. $120$
3. $210$
4. $235$

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

Q. The sides $AB,BC$ and $CA$ of $△ABC$ are marked with $3,4$ and $5$ interior points respectively. Number of triangles that can be constructed using these interior points as vertices is

1. $205$
2. $150$
3. $200$
4. $210$

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. A parallelogram is cut by two sets of $m$ lines parallel to its sides. The number of parallelograms thus formed is

1. ${\left({}^m{C}_2\right)}^2$
2. ${\left({}^{m+1}{C}_2\right)}^2$
3. ${\left({}^{m+2}{C}_2\right)}^2$
4. ${}^{m+2}{C}_4$

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. The number of triangles whose vertices are at the vertices of an octagon but none of whose sides happen to come from the octagon, is

1. $16$
2. $28$
3. $56$
4. $70$

Q. Consider the fourteen lines in a plane given by y = x + r, y = -x + r where r= 0, 1, 2, 3, 4, 5, 6 The number of squares formed by these lines whose diagonals are of length 2 is

1. 9
2. 16
3. 25
4. 36

Q. Maximum number of points of intersection of $n$ straight lines if $n$ satisfies ${}^{n+5}{P}_{n+1}=\frac{11(n-1)}{2}\times {}^{n+3}{P}_n$ is

1. $15$
2. $28$
3. $21$
4. $10$

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. Let $T_n$ be the number of all possible triangles formed by joining vertices of an n-sided reqular polygon. If $T_{n+1}-T_n=10,$ then the value of $n$ is;

1. 7
2. 5
3. 10
4. 8

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. If a convex polygon has $35$ diagonals, then the number of triangles formed by joining the vertices of the polygon is :

1. $93$
2. $105$
3. $120$
4. $84$