Maths-
General
Easy
Question
The value of 'c' in Lagrange's mean value theorem for 
in [0, 2] is
- 0
- 1
- 2/3
- 3/2
Hint:
Lagrange mean value theorem states that for any two points on the curve there exists a point on the curve such that the tangent drawn at this point is parallel to the secant through the two points on the curve.
The correct answer is: 2/3
Given : is a polynomial
We know that polynomials are continuous and differentiable in their range
Simplify f(x) for ease of calculation
Related Questions to study
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The equation 
represents
The equation represents
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The polar equation of the circle whose end points of the diameter are 
and 
is
The polar equation of the circle whose end points of the diameter are and is
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The radius of the circle 
is
The radius of the circle is
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The adjoining figure shows the graph of 
Then –
The adjoining figure shows the graph of Then –
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Graph of y = ax2 + 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 = ax2 + bx + c = 0 is given adjacently. What conclusions can be drawn from this graph –
Maths-General
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) = 4x2 – 8kx + k, the statements which hold good are
For the quadratic polynomial f (x) = 4x2 – 8kx + k, the statements which hold good are
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The graph of the quadratic polynomial y = ax2 + bx + c is as shown in the figure. Then :
Clearly, y =
represent a parabola opening downwards. Therefore, a < 0y =
cuts negative y- axis , Putting x = 0 in the given equation
-y = cy = -cc < 0Thus, from the above graph c < 0.
The graph of the quadratic polynomial y = ax2 + bx + c is as shown in the figure. Then :
Maths-General
Clearly, y =
represent a parabola opening downwards. Therefore, a < 0y =
cuts negative y- axis , Putting x = 0 in the given equation-y = cy = -cc < 0Thus, from the above graph c < 0.
Maths-
The greatest possible number of points of intersections of 8 straight line and 4 circles is :
Complete step-by-step 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 =
= 28For calculating the points of intersection between two circles, we can use the formula which is mentioned in the hint as follows =
= 12For calculating the points of intersection between one line and one circle, we can use the formula which is mentioned in the hint as follows
= 64Hence, 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 :
Maths-General
Complete step-by-step 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 =
= 28For calculating the points of intersection between two circles, we can use the formula which is mentioned in the hint as follows =
= 12For calculating the points of intersection between one line and one circle, we can use the formula which is mentioned in the hint as follows
= 64Hence, 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 step-by-step 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 ?
Maths-General
Complete step-by-step 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
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
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 :
Maths-General
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
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
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The foot of the perpendicular from the point 
on the line 
is
The foot of the perpendicular from the point on the line is
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The point of intersection of the lines 
is
The point of intersection of the lines is
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The line passing through 
and perpendicular to 
is
The line passing through and perpendicular to is
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The equation of the line passing through 
is
The equation of the line passing through is
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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
Maths-General
