Maths-
General
Easy
Question
Let A and B be two events such that
and
stands for complement of event A. Then events A and B are-
- mutually exclusive and independent
- independent but not equally likely
- equally likely but not independent
- equally likely and mutually exclusive
The correct answer is: independent but not equally likely
Related Questions to study
maths-
A random variable X has the probability distribution :
![](data:image/png;base64,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)
For the events E = {X is a prime number} and F = {X < 4}, the probability
is-
A random variable X has the probability distribution :
![](data:image/png;base64,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)
For the events E = {X is a prime number} and F = {X < 4}, the probability
is-
maths-General
maths-
A fair coin is tossed 3 times. Consider the events A : first toss is head ; B : second toss is head C : exactly two consecutive heads or exactly two consecutive tails
Statement - I A,B,C are independent events.
Statement - II A,B,C are pairwise independent.
A fair coin is tossed 3 times. Consider the events A : first toss is head ; B : second toss is head C : exactly two consecutive heads or exactly two consecutive tails
Statement - I A,B,C are independent events.
Statement - II A,B,C are pairwise independent.
maths-General
maths-
An experiment resulting in sample space as S = {a, b, c, d, e, f}
Let three events A, B and C are defined as A = {a, c, e,}, B = {c, d, e, f} and C = {b, c, f}.
then the correct order sequance is
An experiment resulting in sample space as S = {a, b, c, d, e, f}
Let three events A, B and C are defined as A = {a, c, e,}, B = {c, d, e, f} and C = {b, c, f}.
then the correct order sequance is
maths-General
maths-
An ellipse is drawn with major and minor axes of lengths 10 and 8 respectively. Using one focus as centre, a circle is drawn that is tangent to the ellipse, with no part of the circle being outside the ellipse. The radius of the circle is
An ellipse is drawn with major and minor axes of lengths 10 and 8 respectively. Using one focus as centre, a circle is drawn that is tangent to the ellipse, with no part of the circle being outside the ellipse. The radius of the circle is
maths-General
maths-
From a point P, perpendicular tangents PQ and PR are drawn to ellipse
Locus of circumcentre of triangle PQR is
From a point P, perpendicular tangents PQ and PR are drawn to ellipse
Locus of circumcentre of triangle PQR is
maths-General
maths-
The least integral value that contained is the range of the function f defined by
is ___
The least integral value that contained is the range of the function f defined by
is ___
maths-General
maths-
Let
then
Let
then
maths-General
maths-
If m is the slope of tangent to the curve
then
If m is the slope of tangent to the curve
then
maths-General
maths-
If
is equal to e7, then ![stack l i m with x greater-than or slanted equal to 0 below fraction numerator f open parentheses x close parentheses over denominator x to the power of 3 end exponent end fraction equals horizontal ellipsis horizontal ellipsis](data:image/png;base64,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)
If
is equal to e7, then ![stack l i m with x greater-than or slanted equal to 0 below fraction numerator f open parentheses x close parentheses over denominator x to the power of 3 end exponent end fraction equals horizontal ellipsis horizontal ellipsis](data:image/png;base64,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)
maths-General
maths-
Set of all values of x such that
is non-zero and finite number when n N
is
Set of all values of x such that
is non-zero and finite number when n N
is
maths-General
maths-
The period of the function
is
The period of the function
is
maths-General
maths-
Let
is a prime number and [x] = the greatest integer
The number of points at which f(x) is not differentiable is _______
Let
is a prime number and [x] = the greatest integer
The number of points at which f(x) is not differentiable is _______
maths-General
maths-
If the line
is one of the angle bisectors of the lines
and
then the value of k is
If the line
is one of the angle bisectors of the lines
and
then the value of k is
maths-General
physics-
A particle when projected in vertical plane moves along smooth surface with initial velocity 20m/s at an angle of 60°, so that its normal reaction on the surface remains zero throughout the motion. Then the slope of the tangent to the surface at height 5 m from the point of projection will be
![](data:image/png;base64,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)
A particle when projected in vertical plane moves along smooth surface with initial velocity 20m/s at an angle of 60°, so that its normal reaction on the surface remains zero throughout the motion. Then the slope of the tangent to the surface at height 5 m from the point of projection will be
![](data:image/png;base64,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)
physics-General
physics-
Consider a particle moving without friction on a rippled surface, as shown. Gravity acts down in the negative h direction. The elevation h(x) of the surface is given by h(x) = d cos(kx). If the particle starts at x = 0 with a speed v in the x direction, for what values of v will the particle stay on the surface at all times?
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)
Consider a particle moving without friction on a rippled surface, as shown. Gravity acts down in the negative h direction. The elevation h(x) of the surface is given by h(x) = d cos(kx). If the particle starts at x = 0 with a speed v in the x direction, for what values of v will the particle stay on the surface at all times?
![](data:image/png;base64,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)
physics-General