Maths-
General
Easy
Question
A curve passes through the point
Let the slope of the curve at each point (x, y) be
Then the equation of the curve is
The correct answer is: ![c o s invisible function application open parentheses fraction numerator 2 y over denominator x end fraction close parentheses equals l o g invisible function application x plus fraction numerator 1 over denominator 2 end fraction](data:image/png;base64,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)
Related Questions to study
maths-
The solution of the differential equation
![fraction numerator d y over denominator d x end fraction equals fraction numerator x plus y over denominator x end fraction](data:image/png;base64,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)
The solution of the differential equation
![fraction numerator d y over denominator d x end fraction equals fraction numerator x plus y over denominator x end fraction](data:image/png;base64,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)
maths-General
maths-
Statement-I : Integral curves denoted by the first order linear differential equation
are family of parabolas passing through the origin.
Statement-II : Every differential equation geometrically represents a family of curve having some common property.
Statement-I : Integral curves denoted by the first order linear differential equation
are family of parabolas passing through the origin.
Statement-II : Every differential equation geometrically represents a family of curve having some common property.
maths-General
maths-
A function f(x) satisfying
where x>0 , is
A function f(x) satisfying
where x>0 , is
maths-General
maths-
(where c is an arbitrary constant) is the general solution of the differential equation
then the function ![text end text phi open parentheses fraction numerator x over denominator y end fraction close parentheses text is : end text](data:image/png;base64,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)
(where c is an arbitrary constant) is the general solution of the differential equation
then the function ![text end text phi open parentheses fraction numerator x over denominator y end fraction close parentheses text is : end text](data:image/png;base64,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)
maths-General
Maths-
A function y = f (x) satisfies the condition f '(x) sin x + f (x) cos x = 1, f (x) being bounded when x
0. If ![I equals stretchy integral subscript 0 end subscript superscript there exists 2 end superscript f left parenthesis x right parenthesis d x text then end text](data:image/png;base64,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)
A function y = f (x) satisfies the condition f '(x) sin x + f (x) cos x = 1, f (x) being bounded when x
0. If ![I equals stretchy integral subscript 0 end subscript superscript there exists 2 end superscript f left parenthesis x right parenthesis d x text then end text](data:image/png;base64,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)
Maths-General
maths-
Statement-I : In throwing of two dice, let the events A, B and C be 'the first dice shows an even number', 'the second dice shows an odd numbers' and 'both the dice show an odd numbers or both the dice show an even number’ respectively. Then
and
Therefore A, B and C are mutually independent events
Statement-II : Three events A, B and C are mutually independent if and only if P(A Ç B) = P(A) . P(B), P(B Ç C) = P(B) . P(C),
= P(C) . P(A) and
= P(A) . P(B) . P(C)
Statement-I : In throwing of two dice, let the events A, B and C be 'the first dice shows an even number', 'the second dice shows an odd numbers' and 'both the dice show an odd numbers or both the dice show an even number’ respectively. Then
and
Therefore A, B and C are mutually independent events
Statement-II : Three events A, B and C are mutually independent if and only if P(A Ç B) = P(A) . P(B), P(B Ç C) = P(B) . P(C),
= P(C) . P(A) and
= P(A) . P(B) . P(C)
maths-General
maths-
The probability that 4th power of a positive integer ends in the digit 6 is:
The probability that 4th power of a positive integer ends in the digit 6 is:
maths-General
maths-
Let A, B, C be pairwise independent events with P(C)> 0 and
. Then ![P open parentheses A to the power of c end exponent intersection B to the power of c end exponent C close parentheses text is equal to: end text](data:image/png;base64,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)
Let A, B, C be pairwise independent events with P(C)> 0 and
. Then ![P open parentheses A to the power of c end exponent intersection B to the power of c end exponent C close parentheses text is equal to: end text](data:image/png;base64,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)
maths-General
maths-
Let A and B be two events such that
and
stands for complement of event A. Then events A and B are-
Let A and B be two events such that
and
stands for complement of event A. Then events A and B are-
maths-General
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