Maths
General
Easy
Question
The adjoining figure shows the graph of Then –
 a > 0
 b > 0, c > 0
 c < 0

Hint:
So in this question, we have a graph given and we have to do it. As we can see the curve and it is of the parabola and by using the properties of the parabola and its equation, we can answer these questions easily.
The correct answer is: b > 0, c > 0
The adjoining figure shows the graph of
The given parabola is open downwards that means coefficient of
At x = 0, y > 0 ( From graph)
x coordinates of vertex = > 0 ( From the graph)
Thus, b > 0, c > 0 is the correct answer.
Here we can see that the graph was given to us and us to take out the conclusion from that since we have the options available so I would suggest you to always start to check from the options because by the use of options we can see how easily we concluded this question.
Related Questions to study
Maths
Graph of y = ax^{2} + bx + c = 0 is given adjacently. What conclusions can be drawn from this graph –
As we can see from the graph we have a parabola curve and since it is opening in an upward direction. So we can say that a > 0 and
Hence, the option (a) is correct.
Here, we can see that the vertex of the parabola is located in the fourth quadrant , therefore it will be =
On further solving this, we get
Therefore, the option (b) is also correct.
Since, at x=0 , the y intercept will be positive and from this, we can conclude that c < 0 and
Hence, the option (c) will also be correct
On checking all the options, and we can see all options are correct and
Therefore, we conclude that all the options available are correct.
Hence, the option (a) is correct.
Here, we can see that the vertex of the parabola is located in the fourth quadrant , therefore it will be =
On further solving this, we get
Therefore, the option (b) is also correct.
Since, at x=0 , the y intercept will be positive and from this, we can conclude that c < 0 and
Hence, the option (c) will also be correct
On checking all the options, and we can see all options are correct and
Therefore, we conclude that all the options available are correct.
Graph of y = ax^{2} + bx + c = 0 is given adjacently. What conclusions can be drawn from this graph –
MathsGeneral
As we can see from the graph we have a parabola curve and since it is opening in an upward direction. So we can say that a > 0 and
Hence, the option (a) is correct.
Here, we can see that the vertex of the parabola is located in the fourth quadrant , therefore it will be =
On further solving this, we get
Therefore, the option (b) is also correct.
Since, at x=0 , the y intercept will be positive and from this, we can conclude that c < 0 and
Hence, the option (c) will also be correct
On checking all the options, and we can see all options are correct and
Therefore, we conclude that all the options available are correct.
Hence, the option (a) is correct.
Here, we can see that the vertex of the parabola is located in the fourth quadrant , therefore it will be =
On further solving this, we get
Therefore, the option (b) is also correct.
Since, at x=0 , the y intercept will be positive and from this, we can conclude that c < 0 and
Hence, the option (c) will also be correct
On checking all the options, and we can see all options are correct and
Therefore, we conclude that all the options available are correct.
Maths
For the quadratic polynomial f (x) = 4x^{2} – 8kx + k, the statements which hold good are
For the quadratic polynomial f (x) = 4x^{2} – 8kx + k, the statements which hold good are
MathsGeneral
Maths
The graph of the quadratic polynomial y = ax^{2} + bx + c is as shown in the figure. Then :
Clearly, y = represent a parabola opening downwards. Therefore, a < 0
y = cuts negative y axis , Putting x = 0 in the given equation
y = c
y = c
c < 0
Thus, from the above graph c < 0.
The graph of the quadratic polynomial y = ax^{2} + bx + c is as shown in the figure. Then :
MathsGeneral
Clearly, y = represent a parabola opening downwards. Therefore, a < 0
y = cuts negative y axis , Putting x = 0 in the given equation
y = c
y = c
c < 0
Thus, from the above graph c < 0.
Maths
The greatest possible number of points of intersections of 8 straight line and 4 circles is :
Complete stepbystep answer:
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 = = 28
For calculating the points of intersection between two circles, we can use the formula which is mentioned in the hint as follows = = 12
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 = 64
Hence, the total number of points of intersection is = 28 + 12 + 64 = 104
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 = = 28
For calculating the points of intersection between two circles, we can use the formula which is mentioned in the hint as follows = = 12
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 = 64
Hence, the total number of points of intersection is = 28 + 12 + 64 = 104
The greatest possible number of points of intersections of 8 straight line and 4 circles is :
MathsGeneral
Complete stepbystep answer:
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 = = 28
For calculating the points of intersection between two circles, we can use the formula which is mentioned in the hint as follows = = 12
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 = 64
Hence, the total number of points of intersection is = 28 + 12 + 64 = 104
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 = = 28
For calculating the points of intersection between two circles, we can use the formula which is mentioned in the hint as follows = = 12
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 = 64
Hence, the total number of points of intersection is = 28 + 12 + 64 = 104
Maths
How many different nine digit numbers can be formed from the number 223355888 by rearranging its digits so that the odd digits occupy even position ?
Complete stepbystep answer:
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.
Number of ways to arrange the odd digits in 4 even places =
On finding the value of the factorials, we get
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 =
On finding the value of the factorials, we get
Number of ways to arrange the even digits in 5 odd places =
On further simplification, we get
Number of ways to arrange the even digits in 5 odd places =10
Total number of 9 digits number = 6×10 = 60
Hence, the required number of 9 digit numbers = 60
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.
X−X−X−X−X
Here, symbol ( − ) is for the even places and (X) 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
Number of odd digits = 4
We have to arrange the odd digits in even places.
On finding the value of the factorials, we get
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 =
On finding the value of the factorials, we get
Number of ways to arrange the even digits in 5 odd places =
On further simplification, we get
Number of ways to arrange the even digits in 5 odd places =10
Total number of 9 digits number = 6×10 = 60
Hence, the required number of 9 digit numbers = 60
How many different nine digit numbers can be formed from the number 223355888 by rearranging its digits so that the odd digits occupy even position ?
MathsGeneral
Complete stepbystep answer:
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.
Number of ways to arrange the odd digits in 4 even places =
On finding the value of the factorials, we get
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 =
On finding the value of the factorials, we get
Number of ways to arrange the even digits in 5 odd places =
On further simplification, we get
Number of ways to arrange the even digits in 5 odd places =10
Total number of 9 digits number = 6×10 = 60
Hence, the required number of 9 digit numbers = 60
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.
X−X−X−X−X
Here, symbol ( − ) is for the even places and (X) 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
Number of odd digits = 4
We have to arrange the odd digits in even places.
On finding the value of the factorials, we get
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 =
On finding the value of the factorials, we get
Number of ways to arrange the even digits in 5 odd places =
On further simplification, we get
Number of ways to arrange the even digits in 5 odd places =10
Total number of 9 digits number = 6×10 = 60
Hence, the required number of 9 digit numbers = 60
Maths
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 :
Matches whose prediction are correct can be selected in ways.
Since each match can result either in a win, loss or tie for the home team.
Now, each wrong prediction can be made in two ways (i.e. the correct result is win and the person predicts either lose or tie)
and there are 10 matches for which he predicted wrong.
Total number of ways in which he can make the predictions so that exactly 10 predictions are correct =
Thus, total number of ways in which he can make the predictions so that exactly 10 predictions are correct, is equal to
Since each match can result either in a win, loss or tie for the home team.
Now, each wrong prediction can be made in two ways (i.e. the correct result is win and the person predicts either lose or tie)
and there are 10 matches for which he predicted wrong.
Total number of ways in which he can make the predictions so that exactly 10 predictions are correct =
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 :
MathsGeneral
Matches whose prediction are correct can be selected in ways.
Since each match can result either in a win, loss or tie for the home team.
Now, each wrong prediction can be made in two ways (i.e. the correct result is win and the person predicts either lose or tie)
and there are 10 matches for which he predicted wrong.
Total number of ways in which he can make the predictions so that exactly 10 predictions are correct =
Thus, total number of ways in which he can make the predictions so that exactly 10 predictions are correct, is equal to
Since each match can result either in a win, loss or tie for the home team.
Now, each wrong prediction can be made in two ways (i.e. the correct result is win and the person predicts either lose or tie)
and there are 10 matches for which he predicted wrong.
Total number of ways in which he can make the predictions so that exactly 10 predictions are correct =
Thus, total number of ways in which he can make the predictions so that exactly 10 predictions are correct, is equal to
Maths
The foot of the perpendicular from the point on the line is
The foot of the perpendicular from the point on the line is
MathsGeneral
Maths
The point of intersection of the lines is
The point of intersection of the lines is
MathsGeneral
Maths
The line passing through and perpendicular to is
The line passing through and perpendicular to is
MathsGeneral
Maths
The equation of the line passing through is
The equation of the line passing through is
MathsGeneral
Maths
The length of the perpendicular from (1, π/6) to the line is
The length of the perpendicular from (1, π/6) to the line is
MathsGeneral
Maths
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) :
Complete stepbystep answer:
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 get sum as 2+3+4+5=14.
Now, we get a sum of digits in units place for all the numbers as 14×6=84.
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.
So, we get the sum of all the numbers that can be formed with the digits 2,3,4,5 taken all at a time is
(84×1000)+(84×100)+(84×10)+(84×1)
Sum = 84000+8400+840+84
Sum = 93324
We have found the sum of all the numbers that can be formed with the digits 2,3,4,5 taken all at a time as 93324.
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 get sum as 2+3+4+5=14.
Now, we get a sum of digits in units place for all the numbers as 14×6=84.
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.
So, we get the sum of all the numbers that can be formed with the digits 2,3,4,5 taken all at a time is
(84×1000)+(84×100)+(84×10)+(84×1)
Sum = 84000+8400+840+84
Sum = 93324
We have found the sum of all the numbers that can be formed with the digits 2,3,4,5 taken all at a time as 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) :
MathsGeneral
Complete stepbystep answer:
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 get sum as 2+3+4+5=14.
Now, we get a sum of digits in units place for all the numbers as 14×6=84.
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.
So, we get the sum of all the numbers that can be formed with the digits 2,3,4,5 taken all at a time is
(84×1000)+(84×100)+(84×10)+(84×1)
Sum = 84000+8400+840+84
Sum = 93324
We have found the sum of all the numbers that can be formed with the digits 2,3,4,5 taken all at a time as 93324.
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 get sum as 2+3+4+5=14.
Now, we get a sum of digits in units place for all the numbers as 14×6=84.
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.
So, we get the sum of all the numbers that can be formed with the digits 2,3,4,5 taken all at a time is
(84×1000)+(84×100)+(84×10)+(84×1)
Sum = 84000+8400+840+84
Sum = 93324
We have found the sum of all the numbers that can be formed with the digits 2,3,4,5 taken all at a time as 93324.
Maths
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
To solve this question, we should know that the total number of divisors of any number x of the form are prime numbers and is given by (m + 1) (n + 1) (p + 1)….. we know that 480 can be expressed as
So, according to the formula, the total number of divisors of 480 are (5 + 1) (1 + 1) (1 + 1) = 6×2×2=24
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. So, the total number of odd divisors that are possible are (1 + 1) (1 + 1) = 2×2=4, according to the property.
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,
So, the number of divisors that are multiples of 4 are (3 + 1) (1 + 1) (1 + 1) = 4×2×2 = 16.
Hence, we can say that there are 16 divisors of 480 which are multiple of 4.
So, the total number of divisors which are even but not divisible by 2 can be given by 20 – 16 = 4.
Hence, we can say that there are 4 divisors of 480 that are of 4n + 2 form, n≥0
.
To solve this question, we should know that the total number of divisors of any number x of the form are prime numbers and is given by (m + 1) (n + 1) (p + 1)….. we know that 480 can be expressed as
So, according to the formula, the total number of divisors of 480 are (5 + 1) (1 + 1) (1 + 1) = 6×2×2=24
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. So, the total number of odd divisors that are possible are (1 + 1) (1 + 1) = 2×2=4, according to the property.
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,
So, the number of divisors that are multiples of 4 are (3 + 1) (1 + 1) (1 + 1) = 4×2×2 = 16.
Hence, we can say that there are 16 divisors of 480 which are multiple of 4.
So, the total number of divisors which are even but not divisible by 2 can be given by 20 – 16 = 4.
Hence, we can say that there are 4 divisors of 480 that are of 4n + 2 form, n≥0
.
.
Total number of divisors of 480, that are of the form 4n + 2, n 0, is equal to :
MathsGeneral
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
To solve this question, we should know that the total number of divisors of any number x of the form are prime numbers and is given by (m + 1) (n + 1) (p + 1)….. we know that 480 can be expressed as
So, according to the formula, the total number of divisors of 480 are (5 + 1) (1 + 1) (1 + 1) = 6×2×2=24
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. So, the total number of odd divisors that are possible are (1 + 1) (1 + 1) = 2×2=4, according to the property.
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,
So, the number of divisors that are multiples of 4 are (3 + 1) (1 + 1) (1 + 1) = 4×2×2 = 16.
Hence, we can say that there are 16 divisors of 480 which are multiple of 4.
So, the total number of divisors which are even but not divisible by 2 can be given by 20 – 16 = 4.
Hence, we can say that there are 4 divisors of 480 that are of 4n + 2 form, n≥0
.
To solve this question, we should know that the total number of divisors of any number x of the form are prime numbers and is given by (m + 1) (n + 1) (p + 1)….. we know that 480 can be expressed as
So, according to the formula, the total number of divisors of 480 are (5 + 1) (1 + 1) (1 + 1) = 6×2×2=24
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. So, the total number of odd divisors that are possible are (1 + 1) (1 + 1) = 2×2=4, according to the property.
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,
So, the number of divisors that are multiples of 4 are (3 + 1) (1 + 1) (1 + 1) = 4×2×2 = 16.
Hence, we can say that there are 16 divisors of 480 which are multiple of 4.
So, the total number of divisors which are even but not divisible by 2 can be given by 20 – 16 = 4.
Hence, we can say that there are 4 divisors of 480 that are of 4n + 2 form, n≥0
.
.
Maths
If ^{9}P_{5} + 5 ^{9}P_{4} = ^{10}P_{r }, then r =
Given :
Using Formula :
Dividing both sides by 9!
Using Formula :
Dividing both sides by 9!
If ^{9}P_{5} + 5 ^{9}P_{4} = ^{10}P_{r }, then r =
MathsGeneral
Given :
Using Formula :
Dividing both sides by 9!
Using Formula :
Dividing both sides by 9!
Maths
The number of proper divisors of . . 15^{r} is
. .  We need to find proper divisors.
Suppose is a number then factors of = ( and a is proper
i.e. has total division = (n + 1)
Now, =
We know that
Thus, = =
Total factors = (p+q+1)(q+r+1)(r+1)
However, proper divisors exclude $1$ and the number itself.
Hence, the answer is $(p+q+1)(q+r+1)(r+1)−2.$
Suppose is a number then factors of = ( and a is proper
i.e. has total division = (n + 1)
Now, =
We know that
Thus, = =
Total factors = (p+q+1)(q+r+1)(r+1)
However, proper divisors exclude $1$ and the number itself.
Hence, the answer is $(p+q+1)(q+r+1)(r+1)−2.$
The number of proper divisors of . . 15^{r} is
MathsGeneral
. .  We need to find proper divisors.
Suppose is a number then factors of = ( and a is proper
i.e. has total division = (n + 1)
Now, =
We know that
Thus, = =
Total factors = (p+q+1)(q+r+1)(r+1)
However, proper divisors exclude $1$ and the number itself.
Hence, the answer is $(p+q+1)(q+r+1)(r+1)−2.$
Suppose is a number then factors of = ( and a is proper
i.e. has total division = (n + 1)
Now, =
We know that
Thus, = =
Total factors = (p+q+1)(q+r+1)(r+1)
However, proper divisors exclude $1$ and the number itself.
Hence, the answer is $(p+q+1)(q+r+1)(r+1)−2.$