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

The value of 'c' of Lagrange's mean value theorem for $f space left parenthesis x right parenthesis equals x squared minus 3 x minus 2 text for end text x element of left square bracket negative 1 , 2 right square bracket$ is

The value of 'c' of Rolle's mean value theorem for $f space left parenthesis x right parenthesis equals vertical line x vertical line equals i n space left square bracket negative 1 , 1 right square bracket$ is

The value of 'c' of Rolle's theorem for $f space left parenthesis x right parenthesis equals l o g space open parentheses x squared plus 2 close parentheses$ – $l o g subscript 3 end subscript$ on [–1, 1] is

For $f space left parenthesis x right parenthesis equals 4 minus left parenthesis 6 minus x right parenthesis to the power of 2 divided by 3 end exponent$ in [5, 7]

The value of 'c' in Lagrange's mean value theorem for $f space left parenthesis x right parenthesis equals x cubed minus 2 x squared minus x plus 4$ in [0, 1] is

The value of 'c' in Lagrange's mean value theorem for $f space left parenthesis x right parenthesis equals x space left parenthesis x minus 2 right parenthesis squared$ in [0, 2] is

The equation $r equals a c o s space theta plus b s i n space theta$ represents

The polar equation of the circle whose end points of the diameter are $open parentheses square root of 2 comma fraction numerator pi over denominator 4 end fraction close parentheses$ and $open parentheses square root of 2 comma fraction numerator 3 pi over denominator 4 end fraction close parentheses$ is

The radius of the circle $r equals 8 s i n space theta plus 6 c o s space theta$ is

The adjoining figure shows the graph of $y equals a x to the power of 2 end exponent plus b x plus c$ Then –

Graph of y = ax² + bx + c = 0 is given adjacently. What conclusions can be drawn from this graph –

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.

Graph of y = ax² + bx + c = 0 is given adjacently. What conclusions can be drawn from this graph –

For the quadratic polynomial f (x) = 4x² – 8kx + k, the statements which hold good are

The graph of the quadratic polynomial y = ax² + bx + c is as shown in the figure. Then :

The roots (also called as zeros, y = 0) of the quadratic equation $a x squared space plus space b x space plus space c space equals space 0$ $a x^{2}$ are given by x = $fraction numerator negative b space plus-or-minus square root of b squared minus 4 a c end root over denominator 2 a end fraction$ $\frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}$ . The quantity $b squared minus space 4 a c$ is called the discriminant of the equation and determines the nature of its roots.
If $b squared minus space 4 a c$   ≥ 0, the roots are real.
If $b squared minus space 4 a c$   = 0, the roots are real and equal.
If $b squared minus space 4 a c$   < 0, the roots are complex and conjugates of each other.

The graph of the quadratic polynomial y = ax² + bx + c is as shown in the figure. Then :

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

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.

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