Maths-
General
Easy
Question
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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)
The correct answer is: ![fraction numerator pi over denominator 2 end fraction less than I less than fraction numerator pi to the power of 2 end exponent over denominator 4 end fraction](data:image/png;base64,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)
Given:-A function y=f(x) satisfies the condition f′(x)sinx+f(x)cosx=1,f(x) being bounded when x→0
we need to find I if I=
.
![s space i space n space x fraction numerator d y over denominator d x end fraction space plus space y space c space o space s space x space equals space 1
fraction numerator d y over denominator d x end fraction space plus space y space c space o space t space x space equals space c space o space s space e space c space x
I space F space equals space e to the power of integral space c space o space t space x space d space x space end exponent equals space e to the power of ln left parenthesis sin x right parenthesis end exponent equals space s space i space n space x
y space space space s space i space n space x space equals space integral space c space o space s space e space c space x space space space s space i space n space x space space space d space x space equals space x space plus space c
space I space f space x space equals space 0 space comma space y space space space i space s space space space f space i space n space i space t space e
space therefore space space space c space equals space 0
y space equals space x space left parenthesis space c space o space s space e space c space x space right parenthesis space equals fraction numerator space x over denominator space s space i space n space x end fraction
N o w comma space I less than pi squared over 4 space a n d space I greater than fraction numerator pi space over denominator 2 end fraction
H e n c e space fraction numerator pi space over denominator 2 end fraction less than space I space less than space pi squared over 4](data:image/png;base64,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8bW8+JiN4QTA7PC9tbz48bWk+ZTwvbWk+PG1vPiYjeEEwOzwvbW8+PG1pPmM8L21pPjxtbz4mI3hBMDs8L21vPjxtaT54PC9taT48bW8+JiN4QTA7PC9tbz48bW8+JiN4QTA7PC9tbz48bW8+JiN4QTA7PC9tbz48bWk+czwvbWk+PG1vPiYjeEEwOzwvbW8+PG1pPmk8L21pPjxtbz4mI3hBMDs8L21vPjxtaT5uPC9taT48bW8+JiN4QTA7PC9tbz48bWk+eDwvbWk+PG1vPiYjeEEwOzwvbW8+PG1vPiYjeEEwOzwvbW8+PG1vPiYjeEEwOzwvbW8+PG1pPmQ8L21pPjxtbz4mI3hBMDs8L21vPjxtaT54PC9taT48bW8+JiN4QTA7PC9tbz48bW8+PTwvbW8+PG1vPiYjeEEwOzwvbW8+PG1pPng8L21pPjxtbz4mI3hBMDs8L21vPjxtbz4rPC9tbz48bW8+JiN4QTA7PC9tbz48bWk+YzwvbWk+PG1zcGFjZSBsaW5lYnJlYWs9Im5ld2xpbmUiLz48bW8+JiN4QTA7PC9tbz48bWk+STwvbWk+PG1vPiYjeEEwOzwvbW8+PG1pPmY8L21pPjxtbz4mI3hBMDs8L21vPjxtaT54PC9taT48bW8+JiN4QTA7PC9tbz48bW8+PTwvbW8+PG1vPiYjeEEwOzwvbW8+PG1uPjA8L21uPjxtbz4mI3hBMDs8L21vPjxtbz4sPC9tbz48bW8+JiN4QTA7PC9tbz48bWk+eTwvbWk+PG1vPiYjeEEwOzwvbW8+PG1vPiYjeEEwOzwvbW8+PG1vPiYjeEEwOzwvbW8+PG1pPmk8L21pPjxtbz4mI3hBMDs8L21vPjxtaT5zPC9taT48bW8+JiN4QTA7PC9tbz48bW8+JiN4QTA7PC9tbz48bW8+JiN4QTA7PC9tbz48bWk+ZjwvbWk+PG1vPiYjeEEwOzwvbW8+PG1pPmk8L21pPjxtbz4mI3hBMDs8L21vPjxtaT5uPC9taT48bW8+JiN4QTA7PC9tbz48bWk+aTwvbWk+PG1vPiYjeEEwOzwvbW8+PG1pPnQ8L21pPjxtbz4mI3hBMDs8L21vPjxtaT5lPC9taT48bXNwYWNlIGxpbmVicmVhaz0ibmV3bGluZSIvPjxtbz4mI3hBMDs8L21vPjxtbz4mI3gyMjM0OzwvbW8+PG1vPiYjeEEwOzwvbW8+PG1vPiYjeEEwOzwvbW8+PG1vPiYjeEEwOzwvbW8+PG1pPmM8L21pPjxtbz4mI3hBMDs8L21vPjxtbz49PC9tbz48bW8+JiN4QTA7PC9tbz48bW4+MDwvbW4+PG1zcGFjZSBsaW5lYnJlYWs9Im5ld2xpbmUiLz48bWk+eTwvbWk+PG1vPiYjeEEwOzwvbW8+PG1vPj08L21vPjxtbz4mI3hBMDs8L21vPjxtaT54PC9taT48bW8+JiN4QTA7PC9tbz48bW8+KDwvbW8+PG1vPiYjeEEwOzwvbW8+PG1pPmM8L21pPjxtbz4mI3hBMDs8L21vPjxtaT5vPC9taT48bW8+JiN4QTA7PC9tbz48bWk+czwvbWk+PG1vPiYjeEEwOzwvbW8+PG1pPmU8L21pPjxtbz4mI3hBMDs8L21vPjxtaT5jPC9taT48bW8+JiN4QTA7PC9tbz48bWk+eDwvbWk+PG1vPiYjeEEwOzwvbW8+PG1vPik8L21vPjxtbz4mI3hBMDs8L21vPjxtbz49PC9tbz48bWZyYWM+PG1yb3c+PG1vPiYjeEEwOzwvbW8+PG1pPng8L21pPjwvbXJvdz48bXJvdz48bW8+JiN4QTA7PC9tbz48bWk+czwvbWk+PG1vPiYjeEEwOzwvbW8+PG1pPmk8L21pPjxtbz4mI3hBMDs8L21vPjxtaT5uPC9taT48bW8+JiN4QTA7PC9tbz48bWk+eDwvbWk+PC9tcm93PjwvbWZyYWM+PG1zcGFjZSBsaW5lYnJlYWs9Im5ld2xpbmUiLz48bWk+TjwvbWk+PG1pPm88L21pPjxtaT53PC9taT48bW8+LDwvbW8+PG1vPiYjeEEwOzwvbW8+PG1pPkk8L21pPjxtbz4mbHQ7PC9tbz48bWZyYWM+PG1zdXA+PG1pPiYjeDNDMDs8L21pPjxtbj4yPC9tbj48L21zdXA+PG1uPjQ8L21uPjwvbWZyYWM+PG1vPiYjeEEwOzwvbW8+PG1pPmE8L21pPjxtaT5uPC9taT48bWk+ZDwvbWk+PG1vPiYjeEEwOzwvbW8+PG1pPkk8L21pPjxtbz4mZ3Q7PC9tbz48bWZyYWM+PG1yb3c+PG1pPiYjeDNDMDs8L21pPjxtbz4mI3hBMDs8L21vPjwvbXJvdz48bW4+MjwvbW4+PC9tZnJhYz48bXNwYWNlIGxpbmVicmVhaz0ibmV3bGluZSIvPjxtaT5IPC9taT48bWk+ZTwvbWk+PG1pPm48L21pPjxtaT5jPC9taT48bWk+ZTwvbWk+PG1vPiYjeEEwOzwvbW8+PG1mcmFjPjxtcm93PjxtaT4mI3gzQzA7PC9taT48bW8+JiN4QTA7PC9tbz48L21yb3c+PG1uPjI8L21uPjwvbWZyYWM+PG1vPiZsdDs8L21vPjxtbz4mI3hBMDs8L21vPjxtaT5JPC9taT48bW8+JiN4QTA7PC9tbz48bW8+Jmx0OzwvbW8+PG1vPiYjeEEwOzwvbW8+PG1mcmFjPjxtc3VwPjxtaT4mI3gzQzA7PC9taT48bW4+MjwvbW4+PC9tc3VwPjxtbj40PC9tbj48L21mcmFjPjxtc3BhY2UgbGluZWJyZWFrPSJuZXdsaW5lIi8+PC9tYXRoPpNHuwIAAAAASUVORK5CYII=)
Related Questions to study
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
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