Question

# The greatest possible number of points of intersections of 8 straight line and 4 circles is :

- 32
- 64
- 76
- 104

Hint:

### The number point of intersection between two lines can be counted by finding the number of ways in which two lines can be selected out of the lot as two lines can intersect at most one point.

The number point of intersection between two circles can be counted by finding the number of ways in which two circles can be selected out of the lot multiplied by 2 as two circles can intersect at most two points.

## The correct answer is: 104

### Detailed Solution

The number point of intersection between two circles can be counted by finding the number of ways in which one circle and one line can be selected out of the lot multiplied by 2 as one circle and one line can intersect at most two points.

For selecting r objects from n objects can be done by using the formula as follows

As mentioned in the question, we have to find the total number of intersection points.

For calculating the points of intersection between two lines, we can use the formula which is mentioned in the hint as follows =

For calculating the points of intersection between two circles, we can use the formula which is mentioned in the hint as follows =

For calculating the points of intersection between one line and one circle, we can use the formula which is mentioned in the hint as follows =

Hence, the total number of points of intersection is = 28 + 64 + 12 = 104

The students can make an error if they don’t know about the formula for calculating the number of points as mentioned in the hint which is as follows

The number point of intersection between two lines can be counted by finding the number of ways in which two lines can be selected out of the lot as two lines can intersect at most one point.

The number point of intersection between two circles can be counted by finding the number of ways in which two circles can be selected out of the lot multiplied by 2 as two circles can intersect at most two points.

The number point of intersection between two circles can be counted by finding the number of ways in which one circle and one line can be selected out of the lot multiplied by 2 as one circle and one line can intersect at most two points.

### Related Questions to study

### Assertion (A):If ,then A

Reason (R) :

### Assertion (A):If ,then A

Reason (R) :

A force–time graph for a linear motion is shown in figure where the segments are circular. The linear momentum gained between zero and is

As the area above the time axis is numerically equal to area below the time axis therefore net momentum gained by body will be zero because momentum is a vector quantity

A force–time graph for a linear motion is shown in figure where the segments are circular. The linear momentum gained between zero and is

As the area above the time axis is numerically equal to area below the time axis therefore net momentum gained by body will be zero because momentum is a vector quantity

### How many different nine digit numbers can be formed from the number 2,2,3,3,5,5,8,8,8 by rearranging its digits so that the odd digits occupy even position ?

Here we need to find the total number of nine digit numbers that can be formed using the given digits i.e. 2, 2, 3, 3, 5, 5, 8, 8, 8.

Here, e is for the even places and O is for the odd places of the digit number.

The digits which are even are 2, 2, 8, 8 and 8.

Number of even digits = 5

The digits which are odd are 3, 3, 5 and 5.

Number of odd digits = 4

We have to arrange the odd digits in even places.

Number of ways to arrange the odd digits in 4 even places = = 6

Now, we have to arrange the even digits in odd places.

Number of ways to arrange the even digits in 5 odd places =

Total number of 9 digits number =

### How many different nine digit numbers can be formed from the number 2,2,3,3,5,5,8,8,8 by rearranging its digits so that the odd digits occupy even position ?

Here we need to find the total number of nine digit numbers that can be formed using the given digits i.e. 2, 2, 3, 3, 5, 5, 8, 8, 8.

Here, e is for the even places and O is for the odd places of the digit number.

The digits which are even are 2, 2, 8, 8 and 8.

Number of even digits = 5

The digits which are odd are 3, 3, 5 and 5.

Number of odd digits = 4

We have to arrange the odd digits in even places.

Number of ways to arrange the odd digits in 4 even places = = 6

Now, we have to arrange the even digits in odd places.

Number of ways to arrange the even digits in 5 odd places =

Total number of 9 digits number =

### A person predicts the outcome of 20 cricket matches of his home team. Each match can result either in a win, loss or tie for the home team. Total number of ways in which he can make the predictions so that exactly 10 predictions are correct, is equal to :

We have to find the number of ways in which exactly 10 predictions are correct which can be shown by ways in which his prediction is correct.

And in the remaining 10 matches, he makes wrong predictions i.e. out of 3 outcomes (win, lose, tie) he can pick 2 outcomes out of 3 , which can be done in ways.

Thus, total number of ways in which he can make the predictions so that exactly 10 predictions are correct, is equal to

### A person predicts the outcome of 20 cricket matches of his home team. Each match can result either in a win, loss or tie for the home team. Total number of ways in which he can make the predictions so that exactly 10 predictions are correct, is equal to :

We have to find the number of ways in which exactly 10 predictions are correct which can be shown by ways in which his prediction is correct.

And in the remaining 10 matches, he makes wrong predictions i.e. out of 3 outcomes (win, lose, tie) he can pick 2 outcomes out of 3 , which can be done in ways.

Thus, total number of ways in which he can make the predictions so that exactly 10 predictions are correct, is equal to

### The sum of all the numbers that can be formed with the digits 2, 3, 4, 5 taken all at a time is (repetition is not allowed) :

According to the problem, we need to find the sum of all the numbers that can be formed with the digits 2,3,4,5 taken all at a time.

Let us first fix a number in a unit place and find the total number of words possible due on fixing this number.

We need to arrange the remaining three places with three digits. We know that the number of ways of arranging n objects in n places is n! ways.

So, we get 3! = 6 numbers on fixing the unit place with a particular digit.

Now, let us find the sum of all digits.

We use the same digits in ten, hundred and thousand places also. So, the sum of those digits will also be 84 but with the multiplication of its place value.

i.e., We multiply the sum of the digits in tenth place with 10, hundredth place with 100 and so on. We then add these sums.

Thus, the sum of all the numbers that can be formed with the digits 2, 3, 4, 5 taken all at a time is (repetition is not allowed) : 93324

### The sum of all the numbers that can be formed with the digits 2, 3, 4, 5 taken all at a time is (repetition is not allowed) :

According to the problem, we need to find the sum of all the numbers that can be formed with the digits 2,3,4,5 taken all at a time.

Let us first fix a number in a unit place and find the total number of words possible due on fixing this number.

We need to arrange the remaining three places with three digits. We know that the number of ways of arranging n objects in n places is n! ways.

So, we get 3! = 6 numbers on fixing the unit place with a particular digit.

Now, let us find the sum of all digits.

We use the same digits in ten, hundred and thousand places also. So, the sum of those digits will also be 84 but with the multiplication of its place value.

i.e., We multiply the sum of the digits in tenth place with 10, hundredth place with 100 and so on. We then add these sums.

Thus, the sum of all the numbers that can be formed with the digits 2, 3, 4, 5 taken all at a time is (repetition is not allowed) : 93324

### Total number of divisors of 480, that are of the form 4n + 2, n 0, is equal to :

In this question, we have been asked to find the total number of divisors of 480 which are of the form 4n + 2, n 0

In order to solve this question, we should know that the number of the divisor of any number

where a, b, c are prime numbers and is given by (m + 1) (n + 1) (p + 1)…..

We know that 480 can be expressed as

Now, we have been asked to find the number of divisors which are of the form 4n + 2 = 2 (2n + 1), which means odd divisors cannot be a part of the solution.

Now, we can say the total number of even divisors are = all divisors – odd divisor = 24 - 4 = 20

Now, we have been given that the divisor should be of 4n + 2, which means they should not be a multiple of 4 but multiple of 2. For that, we will subtract the multiple of 4 which are divisor of 480 from the even divisors.

And, we know that,

Hence, we can say that there are 16 divisors of 480 which are multiple of 4.

Thus, total number of divisors of 480, that are of the form 4n + 2, n 0, is equal to 4.

### Total number of divisors of 480, that are of the form 4n + 2, n 0, is equal to :

In this question, we have been asked to find the total number of divisors of 480 which are of the form 4n + 2, n 0

In order to solve this question, we should know that the number of the divisor of any number

where a, b, c are prime numbers and is given by (m + 1) (n + 1) (p + 1)…..

We know that 480 can be expressed as

Now, we have been asked to find the number of divisors which are of the form 4n + 2 = 2 (2n + 1), which means odd divisors cannot be a part of the solution.

Now, we can say the total number of even divisors are = all divisors – odd divisor = 24 - 4 = 20

Now, we have been given that the divisor should be of 4n + 2, which means they should not be a multiple of 4 but multiple of 2. For that, we will subtract the multiple of 4 which are divisor of 480 from the even divisors.

And, we know that,

Hence, we can say that there are 16 divisors of 480 which are multiple of 4.

Thus, total number of divisors of 480, that are of the form 4n + 2, n 0, is equal to 4.

### If ^{9}P_{5} + 5 ^{9}P_{4} = , then r =

Here we need to find the value of the given variable.

The given expression is :

### If ^{9}P_{5} + 5 ^{9}P_{4} = , then r =

Here we need to find the value of the given variable.

The given expression is :

### Assertion (A) :If , then

Reason (R) :

### Assertion (A) :If , then

Reason (R) :

A particle is released from a height. At certain height its kinetic energy is three times its potential energy. The height and speed of the particle at that instant are respectively

Or (i)

Potential energy at (ii)

Kinetic energy potential energy

A particle is released from a height. At certain height its kinetic energy is three times its potential energy. The height and speed of the particle at that instant are respectively

Or (i)

Potential energy at (ii)

Kinetic energy potential energy