Combination Based on Geometry

Q.

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

1. 

2. 

3. 

4. None of these

Q.

Let U be the set of all triangles in a plane. If A is the set of all triangles with at least one angle different from ${60}^{\circ }$ what is $A^{\prime }$?

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. 25
2. 16
3. 36
4. 9

Q.

Draw a rough sketch of a hexagon $ABCDEF$. How many diago

does it have? Draw them. How many of them pass through $D$? Write them.

Q.

There are $10$ points in a plane, out of these $6$ are collinear. If $\mathrm{N}$ is the number of triangles formed by joining these points, then

1. $\mathrm{N}>190$

2. $\mathrm{N}\le 100$

3. $100<\mathrm{N}\le 140$

4. $140\le \mathrm{N}\le 190$

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

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

Q. In a plane there are $37$ straight lines of which $13$ pass through point $A$ and $11$ pass through the point $B$. Besides, no three lines pass through one point, no line passes through both $A$ and $B$, and no two lines are parallel.
Then the total number of intersection points of the lines are

1. $426$
2. $535$
3. $601$
4. $728$

Q.

A polygon has prime number of sides. Its number of sides is equal to the sum of the two least consecutive primes. The number of diagonals of the polygon is

1. $4$

2. $5$

3. $6$

4. $7$

Q. Convert ${25}^{\circ }$ into radian.

Q. The number of lines drawn through $6$ points lying on a circle, is

Q.

The cartesian product $A\times A$ has 9 elements among which are found (-1, 0) and (0, 1). Find the set A and the remaining elements of $A\times A$.

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. 8
3. 5
4. 10

Q. $ABCD$ is a convex quadrilateral with $3,4,5$ and $6$ points marked on sides $AB,BC,CD$ and $DA$ respectively. Number of triangles with vertices on different sides is :

1. $243$
2. $342$
3. $432$
4. $534$

Q.

The sum of the squares of perpendicular on any tangent of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}$ = 1 from two Points on the minor axis each one at a distance of units from the centre is

1. a²

2. b²

3. 2a²

4. 2b²

Q. Number of ways in which three numbers in $A.P.$ can be selected from $\{1,2,3,.....,n\}$ is

1. ${\left(\frac{n-1}{2}\right)}^2$ if $n$ is even
   
2. $\frac{n(n-2)}{4}$ if $n$ is even
   
3. $\frac{(n-1{)}^2}{4}$ if $n$ is odd
   
4. $\frac{n(n-2)}{4}$ if $n$ is odd

Q.

Let $T_n$ denote the number of triangles which can be formed using the vertices of a

regular polygon of n sides. If $T_{n+1}-T_n=21$ , then n equals

__

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. There are 10 points in a plane, out of these 6 are collinear, if n is the number of triangles formed by joining these points. then:

1. $n>190$
2. $100<n\le 140$
3. $n\le 100$
4. $140<n\le 190$

Q. The area of a parallelogram formed by the lines $ax\pm by\pm c=0$ is

1. $\frac{c^2}{ab}$
2. $\frac{2c^2}{ab}$
3. $\frac{c^2}{2ab}$
4. None of these

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.

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

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

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.

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

1. ${(}^m{C}_2{)}^2$

2. None of these

3. ${(}^{m+1}{C}_2{)}^2$

4. ${(}^{m+2}{C}_2{)}^2$