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Multiple choice statistics information processing fundamental principle of addition fundamental principles of counting principles of counting

The greatest possible number of points of intersection of 8 straight lines and $4$ circles is $104$.

  1. True

  2. False
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

According to question,

There are 8 lines, for two lines meet in a point,

=>  $^8C _2\times 1= \dfrac{8.7}{1.2}=28$

Line and circle meet in two points,

=>$(^8C _1\times ^4C _1) \times 2 =64$

Two circles meet in two points,

=>  $(^4C _2)\times2 =\dfrac{4.3}{1.2}.2= 104$

Multiple choice statistics information processing fundamental principle of addition fundamental principles of counting principles of counting

Solve:$\dfrac{2}{2}+\dfrac{3}{3}+\dfrac{4}{4}+$...... + upto $1000$ terms= ?

  1. $1000$
  2. twice of $500$
  3. four times of $250$

  4. All of these
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

All the terms simplify to $1$ which is being added $1000$ times.
Therefore, the final answer is $1000$ and it can be seen that all the options are correct.

Multiple choice statistics information processing fundamental principle of addition fundamental principles of counting principles of counting

There are 6 equally spaced points A, B, C, D, E and F marked on a circle with radius R. How many convex pentagons of distinctly different areas can be drawn using these points advertises?

  1. $^6P _5$
  2. $1$
  3. $55$
  4. $42$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

For any 5 points chosen from 6 equally spaced points on a circle, the area of the resulting convex pentagon is determined by the relative positions of the points. Due to the symmetry of the points on the circle, all convex pentagons formed by choosing 5 out of 6 points are congruent and thus have the same area.

Multiple choice statistics information processing fundamental principle of addition fundamental principles of counting principles of counting

A point $(a, b)$ is called a good point if both $a$ and $b$ are integers. Number of good points on the curve $xy$ $=$ $225$ are

  1. 20

  2. 18

  3. 16

  4. 14
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The order pair $(x, y)$ satisfying $xy=225$ are $(1, 225), (3, 75) (5, 45), (9, 25), (15, 15)$. Order can be changed in the first four pairs and both $x$ and $y$ can be negative also, so the no. of pairs $=2(2\times 4+1)=18$

Multiple choice statistics information processing fundamental principle of addition fundamental principles of counting principles of counting

A graph may be defined as a set of points connected by lines called edges. Every edge connects a pair of points. Thus, a triangle is a graph with 3 edges and 3 points. The degree of a point is the number of edges connected to it. For example, a triangle is agraph with three points of degree 2 each. Consider a graph with 12 points. It is possible to reach any point from any other point through a sequence of edges. The number of edges "e" in the graph must satisfy the condition

  1. $11 \leq e \leq 66$
  2. $10 \leq e \leq 66$
  3. $11 \leq e \leq 65$
  4. $0 \leq e \leq 11$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

(A) Since every edge connects a pair of points, the given 12 points have to be joined using lines. We may have minimum number of edges if all the 12 points are collinear.
No. of edges in this particular case 
$=12-1=11$
Maximum number of edges are possible when all the 12 points are non-collinear. In this particular case number of different straight lines that can be formed using 12 points which is equal to $^12C _{2}$
$=\frac{12\times 11}{2}=66$
Therefore, following inequality holds for "e"
$11 \leq e  \leq 66$

Multiple choice statistics information processing fundamental principle of addition fundamental principles of counting principles of counting

Total number of ways of selecting two numbers from the set ${1,2,3,...90}$ so that their sum is divisible by $3$ is

  1. $885$
  2. $1335$
  3. $1770$
  4. $3670$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Numbers 1 to 90 can be classified by remainder when divided by 3: 30 numbers each in categories congruent to 0, 1, and 2 mod 3. For sum of two numbers to be divisible by 3: either both are ≡ 0 (mod 3), or one is ≡ 1 and other ≡ 2 (mod 3). Ways from first category: C(30,2) = 30×29/2 = 435. Ways from second category: 30 × 30 = 900. Total ways = 435 + 900 = 1335.

Multiple choice statistics information processing fundamental principle of addition fundamental principles of counting principles of counting

There are $6$ boxes numbered $1, 2 ....... 6$. Each box is to be filled up either with a red or a green ball in such a way that at least $1$ box contains a green ball and the boxes containing green balls are consecutively numbered. The total number of ways in which this can be done is:

  1. $5$
  2. $21$
  3. $33$
  4. $60$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

(B) The number of ways in which 1 green ball can be put $=6$ . The number of ways in which two green balls can be put such that the boxes are consecutive 
$=5$ $(i.e., (1, 2),(2, 3),(3, 4),(4, 5),(5, 6))$

Similarly, the number of ways in which three green balls can be put 
$=4( i.e. (1, 2, 3),(2, 3, 4),(3, 4, 5),(4, 5, 6))$
$\cdots \cdots \cdots \cdots \cdots $ and so on.
$\therefore $ Total number of ways of doing this
$=6+5+4+3+2+1=21$

Multiple choice statistics information processing fundamental principle of addition fundamental principles of counting principles of counting

The sides of a quadrilateral are all positive integers and three of them are $5, 10, 20.$ How many possible value are there for the fourth side?

  1. $29$
  2. $31$
  3. $32$
  4. $34$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

For any quadrilateral with sides a, b, c, and d, the triangle inequality requires that the longest side is less than the sum of the other three. Here, three sides are 5, 10, and 20. If x is the fourth side, cases depend on whether x is the longest side or not, yielding possible integer values calculated via the inequality bounds.