Maths-
General
Easy
Question
![log square root of 12](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAD0AAAATCAYAAAAjxAWvAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAQ3ZOC+gAAAepJREFUeNpjYBhcgAeI/9MBDyoQAcT7GUYY2A7EKSPJwyJA/B5K0w0MdFrPAOL1eOTVgLgWiC/gkLcG4jVA/AmIf0HVRQ92T4Pycgge+cVAnIbHnQeBOBJaGIKAFhAfhYoNSk8rAPFrIOagcoqUB+JLpBrmA8TXoMkFRPvh0GsJxGeA+AcQ3wXiRBIdVwHEs2mUDX+QYpg51AM6UL4OlG+Jpk4DKm4L5SsB8VISHXceiD1o4GlLaBIn2rA10JhGj/kNaGLzoXmNGMfxYBEDBeZzIGahsqdBWeUktIAj2jBQKciGJsYGFUcGoLwoSMA8FWjggMRs0NS1A3E/lWsZQWjkuJFqGC7DfxGpDlm8BuppUD18Ck3dfSA2oaKnlaAeViHHMGJj+hsQMxHpuCwg/gfE4lC+BRDfJiFpE/K0BrRA5CLXsDVYSusAIF6HJrYDS+EmiMdxoEBaBmVPBuIGKjWiQAG5isQAxDDMFFoqa0H5elA+esHgBcSHoZYyQUvhk3iqijlA/B3quNfQ2KGGp7eQYRbOevoGNB9fgcY0NpAMxE+h6k5CA+wTDrUCQPwXiNcC8XES3YevizjgXUlQrO/FI38Y6piCodxD2gSNXRCQhPIT8ag3hXpaZih7OhTaxoU1Q4mJwZCBciwAubKfYrqMNnIAAABldEVYdE1hdGhNTAA8bWF0aCB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMTk5OC9NYXRoL01hdGhNTCI+PG1pPmxvZzwvbWk+PG1zcXJ0Pjxtbj4xMjwvbW4+PC9tc3FydD48L21hdGg+Tczd1AAAAABJRU5ErkJggg==)
![log square root of 11](data:image/png;base64,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)
- log 12
- log 11
Hint:
In this question, we have given
. We have to find the sum of this series, for that open the bracket and find the sum.
The correct answer is: ![log square root of 12](data:image/png;base64,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)
Here we have to find the sum of the series.
Firstly, the given series is
![open square brackets 1 plus open parentheses 1 half plus 1 third close parentheses times 1 fourth plus open parentheses 1 fourth plus 1 fifth close parentheses times 1 over 4 squared plus open parentheses 1 over 6 plus 1 over 7 close parentheses times 1 over 4 cubed plus horizontal ellipsis horizontal ellipsis straight infinity close square brackets](data:image/png;base64,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)
S =
…..
We can write
S = ![1 half x 1 fourth plus 1 fourth x 1 over 4 squared plus 1 over 6 x 1 over 4 cubed...... plus 1 plus 1 third x 1 fourth plus 1 fifth x 1 over 4 squared.....](data:image/png;base64,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)
We know that,
Log ( 1 + x ) = ![x minus x squared over 2 space plus space x cubed over 3 minus x to the power of 4 over 4 plus..](data:image/png;base64,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)
Log (1 - x) =
- …
Log ( 1 – x2) = -2 [
.] ------(1)
Now ,
Log (
) = 2 [
.]
=2 x[
.] ----(2)
Replacing eq1 and eq2 in S, here x = ½
S = -1/2 log ( 1 – ¼ ) + 1/( 2 x ½) log ( 1 + ½ / 1 – ½ )
S = -½ log ( ¾ ) + log ( 3 )
S = -½ log ( 3 ) + ½ log ( 4) + log ( 3 )
S = ½ log ( 3 ) + ½ log ( 4)
S = ½ log ( 12 )
S = log √12
Therefore, the correct answer is log log√12
In this question, we have to find the sum of the series. Separate the series in even and odd series, and replace with Log ( 1 – x2) and Log ( 1 + x / 1-x ) .
Related Questions to study
maths-
If
and principal value be considered then value of z is-
If
and principal value be considered then value of z is-
maths-General
maths-
maths-General
maths-
The conditional
is
The conditional
is
maths-General
maths-
If p and q are simple propositions, then
is true when
If p and q are simple propositions, then
is true when
maths-General
maths-
Using Simpson’s
rule, the value of
for the following data, is
![](data:image/png;base64,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)
Using Simpson’s
rule, the value of
for the following data, is
![](data:image/png;base64,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)
maths-General
maths-
A curve passes through the points given by the following table :
![](https://mycourses.turito.com/tokenpluginfile.php/c161933dbfaab094c54655ab71e9b8f0/1/question/questiontext/506990/1/977896/Picture4.png)
By Simpson’s rule, the area bounded by the curve, the x-axis and the lines x=1, x=4, is
A curve passes through the points given by the following table :
![](https://mycourses.turito.com/tokenpluginfile.php/c161933dbfaab094c54655ab71e9b8f0/1/question/questiontext/506990/1/977896/Picture4.png)
By Simpson’s rule, the area bounded by the curve, the x-axis and the lines x=1, x=4, is
maths-General
maths-
With the help of trapezoidal rule for numerical integration and the following table
![](https://mycourses.turito.com/tokenpluginfile.php/c161933dbfaab094c54655ab71e9b8f0/1/question/questiontext/508625/1/977890/Picture3.png)
the value of
is
With the help of trapezoidal rule for numerical integration and the following table
![](https://mycourses.turito.com/tokenpluginfile.php/c161933dbfaab094c54655ab71e9b8f0/1/question/questiontext/508625/1/977890/Picture3.png)
the value of
is
maths-General
maths-
A river is 80 metre wide. Its depth d metre and corresponding distance x metre from one bank is given below in table
![](https://mycourses.turito.com/tokenpluginfile.php/c161933dbfaab094c54655ab71e9b8f0/1/question/questiontext/508624/1/977888/Picture2.png)
Then approximate area of cross-section of river by Trapezoidal rule, is
A river is 80 metre wide. Its depth d metre and corresponding distance x metre from one bank is given below in table
![](https://mycourses.turito.com/tokenpluginfile.php/c161933dbfaab094c54655ab71e9b8f0/1/question/questiontext/508624/1/977888/Picture2.png)
Then approximate area of cross-section of river by Trapezoidal rule, is
maths-General
maths-
A curve passes through
![](https://mycourses.turito.com/tokenpluginfile.php/c161933dbfaab094c54655ab71e9b8f0/1/question/questiontext/508623/1/977887/Picture244.png)
Then area bounded by x=1 and x=4 by Trapezoidal rule, is
A curve passes through
![](https://mycourses.turito.com/tokenpluginfile.php/c161933dbfaab094c54655ab71e9b8f0/1/question/questiontext/508623/1/977887/Picture244.png)
Then area bounded by x=1 and x=4 by Trapezoidal rule, is
maths-General
maths-
A curve passes through the points given by the following table :
![](https://mycourses.turito.com/tokenpluginfile.php/c161933dbfaab094c54655ab71e9b8f0/1/question/questiontext/508622/1/977885/Picture31.png)
By Trapezoidal rule, the area bounded by the curve, the x-axis and the lines x=1, x=5, is
A curve passes through the points given by the following table :
![](https://mycourses.turito.com/tokenpluginfile.php/c161933dbfaab094c54655ab71e9b8f0/1/question/questiontext/508622/1/977885/Picture31.png)
By Trapezoidal rule, the area bounded by the curve, the x-axis and the lines x=1, x=5, is
maths-General
maths-
From the following table, using Trapezoidal rule, the area bounded by the curve, the x-axis and the lines x=7.47, x=7.52, is
![](data:image/png;base64,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)
From the following table, using Trapezoidal rule, the area bounded by the curve, the x-axis and the lines x=7.47, x=7.52, is
![](data:image/png;base64,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)
maths-General
maths-
In the group (G={0,1,2,3,4},+5) the inverse of
is
In the group (G={0,1,2,3,4},+5) the inverse of
is
maths-General
Maths-
If (G,×) is a group such that
for all
then G is
If (G,×) is a group such that
for all
then G is
Maths-General
maths-
In any group the number of improper subgroups is
In any group the number of improper subgroups is
maths-General
maths-
Shaded region is represented by
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)
maths-General