Maths-
General
Easy

Question

How to define Wavy Curve Method f(x)?

  1. open parentheses x minus a subscript 1 close parentheses to the power of n minus 1 end exponent divided by open parentheses x minus a subscript 2 close parentheses to the power of n 2 end exponent divided by open parentheses x minus a subscript 3 close parentheses to the power of n 3 end exponent......divided by open parentheses x minus a subscript k close parentheses to the power of n k asterisk times end exponent open parentheses x minus b subscript 1 close parentheses to the power of m minus 1 end exponent divided by open parentheses x minus b subscript 2 close parentheses to the power of m 2 end exponent divided by open parentheses x minus b subscript 3 close parentheses to the power of m 3 end exponent......divided by open parentheses x minus b subscript 2 close parentheses to the power of m p end exponent
  2. open parentheses x minus a subscript 1 close parentheses to the power of n plus 1 end exponent plus open parentheses x minus a subscript 2 close parentheses to the power of n 2 end exponent plus open parentheses x minus a subscript 3 close parentheses to the power of n 3 end exponent.......plus open parentheses x minus a subscript k close parentheses to the power of n k end exponent divided by open parentheses x minus b subscript 1 close parentheses to the power of m 1 end exponent plus open parentheses x minus b subscript 2 close parentheses to the power of m 2 end exponent plus open parentheses x minus b subscript 3 close parentheses to the power of n 3 end exponent......plus open parentheses x minus b subscript p close parentheses to the power of m p end exponent
  3. open parentheses x minus a subscript 1 close parentheses to the power of n 1 end exponent open parentheses x minus a subscript 2 close parentheses to the power of n 2 end exponent open parentheses x minus a subscript 3 close parentheses to the power of n 3 end exponent......open parentheses x minus a subscript 1 close parentheses to the power of n k end exponent divided by open parentheses x minus b subscript 1 close parentheses to the power of m 1 end exponent open parentheses x minus b subscript 2 close parentheses to the power of m 2 end exponent open parentheses x minus b subscript 3 close parentheses to the power of m 3 end exponent......open parentheses x minus b subscript p close parentheses to the power of m p end exponent
  4. open parentheses x minus a subscript 1 close parentheses to the power of n 1 end exponent minus open parentheses x minus a subscript 2 close parentheses to the power of n 2 end exponent minus open parentheses x minus a subscript 3 close parentheses to the power of n minus end exponent.....negative open parentheses x minus a subscript k close parentheses to the power of n k divided by 2 end exponent divided by open parentheses x minus b subscript 1 close parentheses to the power of m 1 end exponent minus open parentheses x minus b subscript 2 close parentheses to the power of m 2 end exponent minus open parentheses x minus b subscript 3 close parentheses to the power of m 3 end exponent.....negative open parentheses x minus b subscript p close parentheses to the power of m p end exponent

The correct answer is: open parentheses x minus a subscript 1 close parentheses to the power of n 1 end exponent open parentheses x minus a subscript 2 close parentheses to the power of n 2 end exponent open parentheses x minus a subscript 3 close parentheses to the power of n 3 end exponent......open parentheses x minus a subscript 1 close parentheses to the power of n k end exponent divided by open parentheses x minus b subscript 1 close parentheses to the power of m 1 end exponent open parentheses x minus b subscript 2 close parentheses to the power of m 2 end exponent open parentheses x minus b subscript 3 close parentheses to the power of m 3 end exponent......open parentheses x minus b subscript p close parentheses to the power of m p end exponent

Related Questions to study

General
physics-

Two metal cubes A and B of same size are arranged as shown in the figure. The extreme ends of the combination are maintained at the indicated temperatures. The arrangement is thermally insulated. The coefficients of thermal conductivity of A and B are 300 W divided by m ℃ and 200 W divided by m ℃, respectively. After steady state is reached, the temperature of the interface will be

Temperature of interface T equals fraction numerator K subscript 1 end subscript theta subscript 1 end subscript plus K subscript 2 end subscript theta subscript 2 end subscript over denominator K subscript 1 end subscript plus K subscript 2 end subscript end fraction
equals fraction numerator 300 cross times 100 plus 200 cross times 0 over denominator 300 plus 200 end fraction equals 60 ℃

Two metal cubes A and B of same size are arranged as shown in the figure. The extreme ends of the combination are maintained at the indicated temperatures. The arrangement is thermally insulated. The coefficients of thermal conductivity of A and B are 300 W divided by m ℃ and 200 W divided by m ℃, respectively. After steady state is reached, the temperature of the interface will be

physics-General
Temperature of interface T equals fraction numerator K subscript 1 end subscript theta subscript 1 end subscript plus K subscript 2 end subscript theta subscript 2 end subscript over denominator K subscript 1 end subscript plus K subscript 2 end subscript end fraction
equals fraction numerator 300 cross times 100 plus 200 cross times 0 over denominator 300 plus 200 end fraction equals 60 ℃
General
physics-

The coefficient of thermal conductivity of copper is 9 times that of steel. In the composite cylindrical bar shown in the figure, what will be the temperature at the junction of copper and steel?

Temperature of interface
theta equals fraction numerator K subscript 1 end subscript theta subscript 1 end subscript l subscript 2 end subscript plus K subscript 2 end subscript theta subscript 2 end subscript l subscript 1 end subscript over denominator K subscript 1 end subscript l subscript 2 end subscript plus K subscript 2 end subscript l subscript 1 end subscript end fraction
It is given that K subscript C u end subscript equals 9 K subscript s end subscript. So, if K subscript s end subscript equals K subscript 1 end subscript equals K, then
K subscript C u end subscript equals K subscript 2 end subscript equals 9 K
rightwards double arrow theta equals fraction numerator 9 K cross times 100 cross times 6 plus K cross times 0 cross times 18 over denominator 9 K cross times 6 plus K cross times 18 end fraction
equals fraction numerator 5400 K over denominator 72 K end fraction equals 75 ℃

The coefficient of thermal conductivity of copper is 9 times that of steel. In the composite cylindrical bar shown in the figure, what will be the temperature at the junction of copper and steel?

physics-General
Temperature of interface
theta equals fraction numerator K subscript 1 end subscript theta subscript 1 end subscript l subscript 2 end subscript plus K subscript 2 end subscript theta subscript 2 end subscript l subscript 1 end subscript over denominator K subscript 1 end subscript l subscript 2 end subscript plus K subscript 2 end subscript l subscript 1 end subscript end fraction
It is given that K subscript C u end subscript equals 9 K subscript s end subscript. So, if K subscript s end subscript equals K subscript 1 end subscript equals K, then
K subscript C u end subscript equals K subscript 2 end subscript equals 9 K
rightwards double arrow theta equals fraction numerator 9 K cross times 100 cross times 6 plus K cross times 0 cross times 18 over denominator 9 K cross times 6 plus K cross times 18 end fraction
equals fraction numerator 5400 K over denominator 72 K end fraction equals 75 ℃
General
physics-

A student takes 50 g m wax (specific heat equals 0.6 blank k c a l divided by k g ℃) and heats it till it boils. The graph between temperature and time is as follows. Heat supplied to the wax per minute and boiling point are respectively

Since specific heat equals 0.6 k c a l divided by g cross times ℃ equals 0.6 blank c a l divided by g cross times ℃
From graph it is clear that in a minute, the temperature is raised from 0 ℃ to 50 ℃.
rightwards double arrow Heat required for a minute equals 50 cross times 0.6 cross times 50 equals 1500 blank c a l
Also from graph, Boiling point of wax is 200 ℃

A student takes 50 g m wax (specific heat equals 0.6 blank k c a l divided by k g ℃) and heats it till it boils. The graph between temperature and time is as follows. Heat supplied to the wax per minute and boiling point are respectively

physics-General
Since specific heat equals 0.6 k c a l divided by g cross times ℃ equals 0.6 blank c a l divided by g cross times ℃
From graph it is clear that in a minute, the temperature is raised from 0 ℃ to 50 ℃.
rightwards double arrow Heat required for a minute equals 50 cross times 0.6 cross times 50 equals 1500 blank c a l
Also from graph, Boiling point of wax is 200 ℃
parallel
General
physics-

A composite metal bar of uniform section is made up of length 25 blank c m of copper, 10 blank c m of nickel and 15 blank c m of aluminium. Each part being in perfect thermal contact with the adjoining part. The copper end of the composite rod is maintained at 100 ℃ and the aluminium end at 0 ℃. The whole rod is covered with belt so that no heat loss occurs at the sides. If K subscript C u end subscript equals 2 K subscript A l end subscript and K subscript A l end subscript equals 3 K subscript N i end subscript, then what will be the temperatures of C u minus N i and N i minus A l junctions respectively

If suppose K subscript N i end subscript equals K rightwards double arrow K subscript A l end subscript equals 3 K and K subscript C u end subscript equals 6 K
Since all metal bars are connected in series
So open parentheses fraction numerator Q over denominator t end fraction close parentheses subscript C o m b i n a t i o n end subscript equals open parentheses fraction numerator Q over denominator t end fraction close parentheses subscript C u end subscript equals open parentheses fraction numerator Q over denominator t end fraction close parentheses subscript A l end subscript equals open parentheses fraction numerator Q over denominator t end fraction close parentheses subscript N i end subscript
a n d blank fraction numerator 3 over denominator K subscript e q end subscript end fraction equals fraction numerator 1 over denominator K subscript C u end subscript end fraction plus fraction numerator 1 over denominator K subscript A l end subscript end fraction plus fraction numerator 1 over denominator K subscript N i end subscript end fraction equals fraction numerator 1 over denominator 6 K end fraction plus fraction numerator 1 over denominator 3 K end fraction plus fraction numerator 1 over denominator K end fraction equals fraction numerator 9 over denominator 6 K end fraction
rightwards double arrow K subscript e q end subscript equals 2 K

Hence, it open parentheses fraction numerator Q over denominator t end fraction close parentheses subscript C o m b i n a t i o n end subscript equals open parentheses fraction numerator Q over denominator t end fraction close parentheses subscript C u end subscript
rightwards double arrow fraction numerator K subscript e q end subscript A left parenthesis 100 minus 0 right parenthesis over denominator l subscript C o m b i n a t i o n end subscript end fraction equals fraction numerator K subscript C u end subscript A left parenthesis 100 minus theta subscript 1 end subscript right parenthesis over denominator l subscript C u end subscript end fraction
rightwards double arrow blank fraction numerator 2 K blank A left parenthesis 100 minus 0 right parenthesis over denominator left parenthesis 25 plus 10 plus 15 right parenthesis end fraction equals fraction numerator 6 K blank A left parenthesis 100 minus theta subscript 1 end subscript right parenthesis over denominator 25 end fraction rightwards double arrow theta subscript 1 end subscript equals 83.33 ℃
Similar if open parentheses fraction numerator Q over denominator t end fraction close parentheses subscript C o m b i n a t i o n end subscript equals open parentheses fraction numerator Q over denominator t end fraction close parentheses subscript A l end subscript
rightwards double arrow fraction numerator 2 K blank A left parenthesis 100 minus 0 right parenthesis over denominator 50 end fraction equals fraction numerator 3 K blank A left parenthesis theta subscript 2 end subscript minus 0 right parenthesis over denominator 15 end fraction rightwards double arrow theta subscript 2 end subscript equals 20 ℃

A composite metal bar of uniform section is made up of length 25 blank c m of copper, 10 blank c m of nickel and 15 blank c m of aluminium. Each part being in perfect thermal contact with the adjoining part. The copper end of the composite rod is maintained at 100 ℃ and the aluminium end at 0 ℃. The whole rod is covered with belt so that no heat loss occurs at the sides. If K subscript C u end subscript equals 2 K subscript A l end subscript and K subscript A l end subscript equals 3 K subscript N i end subscript, then what will be the temperatures of C u minus N i and N i minus A l junctions respectively

physics-General
If suppose K subscript N i end subscript equals K rightwards double arrow K subscript A l end subscript equals 3 K and K subscript C u end subscript equals 6 K
Since all metal bars are connected in series
So open parentheses fraction numerator Q over denominator t end fraction close parentheses subscript C o m b i n a t i o n end subscript equals open parentheses fraction numerator Q over denominator t end fraction close parentheses subscript C u end subscript equals open parentheses fraction numerator Q over denominator t end fraction close parentheses subscript A l end subscript equals open parentheses fraction numerator Q over denominator t end fraction close parentheses subscript N i end subscript
a n d blank fraction numerator 3 over denominator K subscript e q end subscript end fraction equals fraction numerator 1 over denominator K subscript C u end subscript end fraction plus fraction numerator 1 over denominator K subscript A l end subscript end fraction plus fraction numerator 1 over denominator K subscript N i end subscript end fraction equals fraction numerator 1 over denominator 6 K end fraction plus fraction numerator 1 over denominator 3 K end fraction plus fraction numerator 1 over denominator K end fraction equals fraction numerator 9 over denominator 6 K end fraction
rightwards double arrow K subscript e q end subscript equals 2 K

Hence, it open parentheses fraction numerator Q over denominator t end fraction close parentheses subscript C o m b i n a t i o n end subscript equals open parentheses fraction numerator Q over denominator t end fraction close parentheses subscript C u end subscript
rightwards double arrow fraction numerator K subscript e q end subscript A left parenthesis 100 minus 0 right parenthesis over denominator l subscript C o m b i n a t i o n end subscript end fraction equals fraction numerator K subscript C u end subscript A left parenthesis 100 minus theta subscript 1 end subscript right parenthesis over denominator l subscript C u end subscript end fraction
rightwards double arrow blank fraction numerator 2 K blank A left parenthesis 100 minus 0 right parenthesis over denominator left parenthesis 25 plus 10 plus 15 right parenthesis end fraction equals fraction numerator 6 K blank A left parenthesis 100 minus theta subscript 1 end subscript right parenthesis over denominator 25 end fraction rightwards double arrow theta subscript 1 end subscript equals 83.33 ℃
Similar if open parentheses fraction numerator Q over denominator t end fraction close parentheses subscript C o m b i n a t i o n end subscript equals open parentheses fraction numerator Q over denominator t end fraction close parentheses subscript A l end subscript
rightwards double arrow fraction numerator 2 K blank A left parenthesis 100 minus 0 right parenthesis over denominator 50 end fraction equals fraction numerator 3 K blank A left parenthesis theta subscript 2 end subscript minus 0 right parenthesis over denominator 15 end fraction rightwards double arrow theta subscript 2 end subscript equals 20 ℃
General
physics-

The plots of intensity of radiation v e r s u s blankwavelength of three black bodies at temperatures T subscript 1 end subscript comma blank T subscript 2 end subscript a n d T subscript 3 end subscript are shown. Then,

According to Wien’s law
lambda subscript m end subscript proportional to fraction numerator 1 over denominator T end fraction
And from the figure
open parentheses lambda subscript m end subscript close parentheses subscript 1 end subscript less than open parentheses lambda subscript m end subscript close parentheses subscript 3 end subscript less than open parentheses lambda subscript m end subscript close parentheses subscript 2 end subscript
Therefore, T subscript 1 end subscript greater than T subscript 3 end subscript greater than T subscript 2 end subscript

The plots of intensity of radiation v e r s u s blankwavelength of three black bodies at temperatures T subscript 1 end subscript comma blank T subscript 2 end subscript a n d T subscript 3 end subscript are shown. Then,

physics-General
According to Wien’s law
lambda subscript m end subscript proportional to fraction numerator 1 over denominator T end fraction
And from the figure
open parentheses lambda subscript m end subscript close parentheses subscript 1 end subscript less than open parentheses lambda subscript m end subscript close parentheses subscript 3 end subscript less than open parentheses lambda subscript m end subscript close parentheses subscript 2 end subscript
Therefore, T subscript 1 end subscript greater than T subscript 3 end subscript greater than T subscript 2 end subscript
General
maths-

Consider the system of linear equations: x subscript 1 end subscript plus 2 x subscript 2 end subscript plus x subscript 3 end subscript equals 3 semicolon 2 x subscript 1 end subscript plus 3 x subscript 2 end subscript plus x subscript 3 end subscript equals 3x subscript 1 end subscript plus 5 x subscript 2 end subscript plus 2 x subscript 3 end subscript equals 1 The system has

Consider the system of linear equations: x subscript 1 end subscript plus 2 x subscript 2 end subscript plus x subscript 3 end subscript equals 3 semicolon 2 x subscript 1 end subscript plus 3 x subscript 2 end subscript plus x subscript 3 end subscript equals 3x subscript 1 end subscript plus 5 x subscript 2 end subscript plus 2 x subscript 3 end subscript equals 1 The system has

maths-General
parallel
General
Maths-

If the system of equations x plus y plus z equals 6 comma x plus 2 y plus lambda z equals 0 comma x plus 2 y plus 3 z equals 10 has no solution, then Error converting from MathML to accessible text.

If the system of equations x plus y plus z equals 6 comma x plus 2 y plus lambda z equals 0 comma x plus 2 y plus 3 z equals 10 has no solution, then Error converting from MathML to accessible text.

Maths-General
General
physics-

Two bars of thermal conductivities K and 3 K and lengths 1 c m and 2 c m respectively have equal cross-sectional area, they are joined lengths wise as shown in the figure. If the temperature at the ends of this composite bar is 0 ℃ and 100 ℃ respectively (see figure), then the temperature ϕ of the interface is

Temperature of interface
theta equals fraction numerator K subscript 1 end subscript theta subscript 1 end subscript l subscript 2 end subscript plus K subscript 2 end subscript theta subscript 2 end subscript l subscript 1 end subscript over denominator K subscript 1 end subscript l subscript 2 end subscript plus K subscript 2 end subscript l subscript 1 end subscript end fraction equals fraction numerator K cross times 0 cross times 2 plus 3 K cross times 100 cross times 1 over denominator K cross times 2 plus 3 K cross times 1 end fraction
equals fraction numerator 300 K over denominator 5 K end fraction equals 60 ℃

Two bars of thermal conductivities K and 3 K and lengths 1 c m and 2 c m respectively have equal cross-sectional area, they are joined lengths wise as shown in the figure. If the temperature at the ends of this composite bar is 0 ℃ and 100 ℃ respectively (see figure), then the temperature ϕ of the interface is

physics-General
Temperature of interface
theta equals fraction numerator K subscript 1 end subscript theta subscript 1 end subscript l subscript 2 end subscript plus K subscript 2 end subscript theta subscript 2 end subscript l subscript 1 end subscript over denominator K subscript 1 end subscript l subscript 2 end subscript plus K subscript 2 end subscript l subscript 1 end subscript end fraction equals fraction numerator K cross times 0 cross times 2 plus 3 K cross times 100 cross times 1 over denominator K cross times 2 plus 3 K cross times 1 end fraction
equals fraction numerator 300 K over denominator 5 K end fraction equals 60 ℃
General
Maths-

If A equals open curly brackets table attributes columnalign left end attributes row i 0 row 0 cell negative i end cell end table close curly brackets then A2 equals

If A equals open curly brackets table attributes columnalign left end attributes row i 0 row 0 cell negative i end cell end table close curly brackets then A2 equals

Maths-General
parallel
General
maths-

The perimeter of a straight triangleis 6 times the A.M of the sines of its angles if the side text  'a'  end textis , then the angle A is

The perimeter of a straight triangleis 6 times the A.M of the sines of its angles if the side text  'a'  end textis , then the angle A is

maths-General
General
Maths-

If the angles of triangle are in the ratio 1:1:4 then ratio of the perimeter of triangle to its largest side

R a t i o space o f space a n g l e s space angle A colon angle B colon angle C space equals space 1 colon 1 colon 4
W e space k n o w space t h a t space s u m space o f space a n g l e s space o f space t h e space t r i a n g l e space a r e space 180 degree.
S o comma space angle A plus angle B plus angle C space equals 180
rightwards double arrow space x plus x plus 4 x equals 180
rightwards double arrow 6 x equals space 180
rightwards double arrow x equals 30
angle A equals 30 degree comma space angle B equals 30 degree comma space angle C equals 120 degree space
l e t space a equals b equals y
C o sin e space f o r m u l a comma space
space space space space c squared equals a squared plus b squared minus 2 a b space cos space C
rightwards double arrow space c squared space equals y squared plus y squared minus 2 y squared cos space C
rightwards double arrow space c squared equals 2 y squared minus 2 y squared cos space C
rightwards double arrow space c squared equals 2 y squared left parenthesis 1 minus cos space C right parenthesis
rightwards double arrow space c squared equals 2 y squared. space 2 sin squared C over 2
rightwards double arrow space c squared equals open parentheses 2 y space sin space C over 2 close parentheses squared
rightwards double arrow c equals 2 y space sin space C over 2
rightwards double arrow c equals 2 y space sin space 60 degree
rightwards double arrow c equals 2 y fraction numerator square root of 3 over denominator 2 end fraction
rightwards double arrow c equals square root of 3
T h e space r a t i o equals fraction numerator a plus b plus c over denominator c end fraction equals fraction numerator y plus y plus square root of 3 y over denominator square root of 3 y end fraction equals fraction numerator 2 plus square root of 3 over denominator square root of 3 end fraction equals 2 plus square root of 3 colon square root of 3

If the angles of triangle are in the ratio 1:1:4 then ratio of the perimeter of triangle to its largest side

Maths-General
R a t i o space o f space a n g l e s space angle A colon angle B colon angle C space equals space 1 colon 1 colon 4
W e space k n o w space t h a t space s u m space o f space a n g l e s space o f space t h e space t r i a n g l e space a r e space 180 degree.
S o comma space angle A plus angle B plus angle C space equals 180
rightwards double arrow space x plus x plus 4 x equals 180
rightwards double arrow 6 x equals space 180
rightwards double arrow x equals 30
angle A equals 30 degree comma space angle B equals 30 degree comma space angle C equals 120 degree space
l e t space a equals b equals y
C o sin e space f o r m u l a comma space
space space space space c squared equals a squared plus b squared minus 2 a b space cos space C
rightwards double arrow space c squared space equals y squared plus y squared minus 2 y squared cos space C
rightwards double arrow space c squared equals 2 y squared minus 2 y squared cos space C
rightwards double arrow space c squared equals 2 y squared left parenthesis 1 minus cos space C right parenthesis
rightwards double arrow space c squared equals 2 y squared. space 2 sin squared C over 2
rightwards double arrow space c squared equals open parentheses 2 y space sin space C over 2 close parentheses squared
rightwards double arrow c equals 2 y space sin space C over 2
rightwards double arrow c equals 2 y space sin space 60 degree
rightwards double arrow c equals 2 y fraction numerator square root of 3 over denominator 2 end fraction
rightwards double arrow c equals square root of 3
T h e space r a t i o equals fraction numerator a plus b plus c over denominator c end fraction equals fraction numerator y plus y plus square root of 3 y over denominator square root of 3 y end fraction equals fraction numerator 2 plus square root of 3 over denominator square root of 3 end fraction equals 2 plus square root of 3 colon square root of 3
General
Maths-

The angles of a triangle are in the ratio 3:5:10.Then ratio of smallest to greatest side is

I n space triangle A B C space comma r a t i o space o f space angle A colon angle B colon angle C equals space 3 colon 5 colon 10
S o comma space angle A equals space 3 x comma space angle B equals 5 x comma space angle C equals space 10 x
w e space k n o w space t h a t space angle A plus angle B plus angle C equals 180 degree
rightwards double arrow 3 x plus 5 x plus 10 x equals 180 degree
rightwards double arrow 18 x equals 180 degree
rightwards double arrow x equals 10 degree
S o comma space angle A equals 30 degree comma space angle B equals 50 degree comma space angle C equals 100 degree
A s space angle C space i s space t h e space g r e a t e s t space a n g l e comma space s o space t h e space s i d e space o p p o s i t e space t o space i t space i. e comma space c space i s space t h e space l a r g e s t space s i d e.
A s space angle A space i s space t h e space s m a l l e s t space a n g l e comma space s o space t h e space s i d e space o p p o s i t e space t o space i t space i. e comma space a space i s space t h e space s m a l l e s t space s i d e.
I n space triangle A B C comma space
u sin g space sin e space f o r m u l a comma
fraction numerator a over denominator S i n A end fraction equals fraction numerator b over denominator S i n B end fraction equals fraction numerator c over denominator S i n C end fraction
rightwards double arrow fraction numerator a over denominator sin space A end fraction equals fraction numerator c over denominator sin space C end fraction
rightwards double arrow a over c equals fraction numerator sin space A over denominator sin space C end fraction
space space space space space space space space space space equals fraction numerator sin space 30 degree over denominator sin 100 degree end fraction
space space space space space space space space space equals fraction numerator sin space 30 degree over denominator sin space left parenthesis 90 space plus 10 right parenthesis degree end fraction
space space space space space space space space space equals fraction numerator begin display style 1 half end style over denominator cos space 10 degree end fraction
space space space space space space space space equals fraction numerator 1 over denominator 2 cos space 10 degree end fraction
a colon c equals 1 colon 2 cos space 10 degree

The angles of a triangle are in the ratio 3:5:10.Then ratio of smallest to greatest side is

Maths-General
I n space triangle A B C space comma r a t i o space o f space angle A colon angle B colon angle C equals space 3 colon 5 colon 10
S o comma space angle A equals space 3 x comma space angle B equals 5 x comma space angle C equals space 10 x
w e space k n o w space t h a t space angle A plus angle B plus angle C equals 180 degree
rightwards double arrow 3 x plus 5 x plus 10 x equals 180 degree
rightwards double arrow 18 x equals 180 degree
rightwards double arrow x equals 10 degree
S o comma space angle A equals 30 degree comma space angle B equals 50 degree comma space angle C equals 100 degree
A s space angle C space i s space t h e space g r e a t e s t space a n g l e comma space s o space t h e space s i d e space o p p o s i t e space t o space i t space i. e comma space c space i s space t h e space l a r g e s t space s i d e.
A s space angle A space i s space t h e space s m a l l e s t space a n g l e comma space s o space t h e space s i d e space o p p o s i t e space t o space i t space i. e comma space a space i s space t h e space s m a l l e s t space s i d e.
I n space triangle A B C comma space
u sin g space sin e space f o r m u l a comma
fraction numerator a over denominator S i n A end fraction equals fraction numerator b over denominator S i n B end fraction equals fraction numerator c over denominator S i n C end fraction
rightwards double arrow fraction numerator a over denominator sin space A end fraction equals fraction numerator c over denominator sin space C end fraction
rightwards double arrow a over c equals fraction numerator sin space A over denominator sin space C end fraction
space space space space space space space space space space equals fraction numerator sin space 30 degree over denominator sin 100 degree end fraction
space space space space space space space space space equals fraction numerator sin space 30 degree over denominator sin space left parenthesis 90 space plus 10 right parenthesis degree end fraction
space space space space space space space space space equals fraction numerator begin display style 1 half end style over denominator cos space 10 degree end fraction
space space space space space space space space equals fraction numerator 1 over denominator 2 cos space 10 degree end fraction
a colon c equals 1 colon 2 cos space 10 degree
parallel
General
Maths-

text  If  end text f left parenthesis 9 right parenthesis equals 9 comma f to the power of apostrophe left parenthesis 9 right parenthesis equals 4 comma text  then  end text fraction numerator L t over denominator x not stretchy rightwards arrow t end fraction fraction numerator square root of f left parenthesis x right parenthesis end root minus 3 over denominator square root of x minus 3 end fraction text  equals  end text

f left parenthesis 9 right parenthesis equals 9 space comma space f apostrophe left parenthesis 9 right parenthesis equals 4
limit as x rightwards arrow t of fraction numerator square root of f left parenthesis x right parenthesis end root minus 3 over denominator square root of x minus 3 end fraction
equals limit as x rightwards arrow 9 of fraction numerator square root of f left parenthesis x right parenthesis end root minus 3 over denominator square root of x minus 3 end fraction
u sin g space L space h o s p i t a l space t h e o r e m comma
equals limit as x rightwards arrow 9 of fraction numerator begin display style fraction numerator f apostrophe left parenthesis x right parenthesis over denominator 2 square root of f left parenthesis x right parenthesis end root end fraction end style minus 0 over denominator begin display style fraction numerator 1 over denominator 2 square root of x end fraction end style minus 0 end fraction
equals fraction numerator begin display style fraction numerator f apostrophe left parenthesis 9 right parenthesis over denominator 2 square root of f left parenthesis 9 right parenthesis end root end fraction end style over denominator begin display style fraction numerator 1 over denominator 2 square root of 9 end fraction end style end fraction
equals fraction numerator begin display style fraction numerator 4 over denominator 2 square root of 9 end fraction end style over denominator begin display style fraction numerator 1 over denominator 2 square root of 9 end fraction end style end fraction
equals 4

text  If  end text f left parenthesis 9 right parenthesis equals 9 comma f to the power of apostrophe left parenthesis 9 right parenthesis equals 4 comma text  then  end text fraction numerator L t over denominator x not stretchy rightwards arrow t end fraction fraction numerator square root of f left parenthesis x right parenthesis end root minus 3 over denominator square root of x minus 3 end fraction text  equals  end text

Maths-General
f left parenthesis 9 right parenthesis equals 9 space comma space f apostrophe left parenthesis 9 right parenthesis equals 4
limit as x rightwards arrow t of fraction numerator square root of f left parenthesis x right parenthesis end root minus 3 over denominator square root of x minus 3 end fraction
equals limit as x rightwards arrow 9 of fraction numerator square root of f left parenthesis x right parenthesis end root minus 3 over denominator square root of x minus 3 end fraction
u sin g space L space h o s p i t a l space t h e o r e m comma
equals limit as x rightwards arrow 9 of fraction numerator begin display style fraction numerator f apostrophe left parenthesis x right parenthesis over denominator 2 square root of f left parenthesis x right parenthesis end root end fraction end style minus 0 over denominator begin display style fraction numerator 1 over denominator 2 square root of x end fraction end style minus 0 end fraction
equals fraction numerator begin display style fraction numerator f apostrophe left parenthesis 9 right parenthesis over denominator 2 square root of f left parenthesis 9 right parenthesis end root end fraction end style over denominator begin display style fraction numerator 1 over denominator 2 square root of 9 end fraction end style end fraction
equals fraction numerator begin display style fraction numerator 4 over denominator 2 square root of 9 end fraction end style over denominator begin display style fraction numerator 1 over denominator 2 square root of 9 end fraction end style end fraction
equals 4
General
Maths-

The sides of a triangle aresin invisible function application alpha comma cos invisible function application alpha text  and  end text square root of 1 plus sin invisible function application alpha cos invisible function application alpha end root text  for some  end text 0 less than alpha less than pi over 2 Then the greatest angle of the triangle is  [AIEEE 2014]

L e t space a equals sin space alpha comma b equals cos space alpha space a n d space c equals square root of 1 plus sin space alpha cos space alpha end root space comma space 0 less than alpha less than pi over 2
A s space c space i s space t h e space l a r g e s t space s i d e comma space s o space angle C space i s space t h e space g r e a t e s t space a n g l e. space
cos space C space equals fraction numerator a squared plus b squared minus c squared over denominator 2 a b end fraction
space space space space space space space space space space space equals fraction numerator sin squared space alpha plus cos squared space alpha space minus open parentheses square root of 1 plus sin space alpha space cos space alpha end root close parentheses squared space over denominator 2 space sin space alpha space cos space alpha end fraction
space space space space space space space space space space space equals fraction numerator sin squared space alpha space plus cos squared space alpha space minus 1 minus sin space alpha. cos space alpha over denominator 2 space sin alpha space cos alpha end fraction
space space space space space space space space space space space equals fraction numerator 1 minus 1 minus sin space alpha. cos space alpha over denominator 2 sin space alpha. cos space alpha end fraction
space space space space space space space space space space equals fraction numerator negative sin space alpha. cos space alpha over denominator 2 sin space alpha. cos space alpha end fraction
space space space space space space space space space equals fraction numerator negative 1 over denominator 2 end fraction
rightwards double arrow cos space C space equals fraction numerator negative 1 over denominator 2 end fraction
rightwards double arrow cos space C equals cos space 120 degree
rightwards double arrow C equals 120 degree

The sides of a triangle aresin invisible function application alpha comma cos invisible function application alpha text  and  end text square root of 1 plus sin invisible function application alpha cos invisible function application alpha end root text  for some  end text 0 less than alpha less than pi over 2 Then the greatest angle of the triangle is  [AIEEE 2014]

Maths-General
L e t space a equals sin space alpha comma b equals cos space alpha space a n d space c equals square root of 1 plus sin space alpha cos space alpha end root space comma space 0 less than alpha less than pi over 2
A s space c space i s space t h e space l a r g e s t space s i d e comma space s o space angle C space i s space t h e space g r e a t e s t space a n g l e. space
cos space C space equals fraction numerator a squared plus b squared minus c squared over denominator 2 a b end fraction
space space space space space space space space space space space equals fraction numerator sin squared space alpha plus cos squared space alpha space minus open parentheses square root of 1 plus sin space alpha space cos space alpha end root close parentheses squared space over denominator 2 space sin space alpha space cos space alpha end fraction
space space space space space space space space space space space equals fraction numerator sin squared space alpha space plus cos squared space alpha space minus 1 minus sin space alpha. cos space alpha over denominator 2 space sin alpha space cos alpha end fraction
space space space space space space space space space space space equals fraction numerator 1 minus 1 minus sin space alpha. cos space alpha over denominator 2 sin space alpha. cos space alpha end fraction
space space space space space space space space space space equals fraction numerator negative sin space alpha. cos space alpha over denominator 2 sin space alpha. cos space alpha end fraction
space space space space space space space space space equals fraction numerator negative 1 over denominator 2 end fraction
rightwards double arrow cos space C space equals fraction numerator negative 1 over denominator 2 end fraction
rightwards double arrow cos space C equals cos space 120 degree
rightwards double arrow C equals 120 degree
General
Maths-

L t subscript x not stretchy rightwards arrow negative straight infinity end subscript open square brackets fraction numerator x to the power of 4 sin invisible function application open parentheses 1 over x close parentheses plus x squared over denominator open parentheses 1 plus vertical line x vertical line cubed close parentheses end fraction close square brackets equals

space space space space L t subscript x not stretchy rightwards arrow negative straight infinity end subscript space open square brackets fraction numerator x to the power of 4 sin invisible function application open parentheses 1 over x close parentheses plus x squared over denominator open parentheses 1 plus vertical line x vertical line cubed close parentheses end fraction close square brackets
A s space 1 plus vertical line x vertical line cubed equals open curly brackets table attributes columnalign left columnspacing 1.4ex end attributes row cell 1 plus x cubed end cell cell i f space x greater or equal than 0 end cell row cell 1 minus x cubed end cell cell i f space x less than 0 end cell end table close
equals L t subscript x not stretchy rightwards arrow negative straight infinity end subscript space space open square brackets fraction numerator x to the power of 4 sin invisible function application open parentheses 1 over x close parentheses plus x squared over denominator open parentheses 1 minus x cubed close parentheses end fraction close square brackets
D i v i d i n g space b y space x cubed comma space w e space g e t colon
equals L t subscript x not stretchy rightwards arrow negative straight infinity end subscript space open square brackets fraction numerator x sin space begin display style 1 over x end style plus begin display style 1 over x end style over denominator begin display style 1 over x cubed end style minus 1 end fraction close square brackets
equals L t subscript x not stretchy rightwards arrow negative straight infinity end subscript space open square brackets fraction numerator begin display style fraction numerator sin space begin display style 1 over x end style over denominator begin display style 1 over x end style end fraction end style plus begin display style 1 over x end style over denominator begin display style 1 over x cubed minus 1 end style end fraction close square brackets
equals fraction numerator 1 plus 0 over denominator 0 minus 1 end fraction
equals negative 1.

L t subscript x not stretchy rightwards arrow negative straight infinity end subscript open square brackets fraction numerator x to the power of 4 sin invisible function application open parentheses 1 over x close parentheses plus x squared over denominator open parentheses 1 plus vertical line x vertical line cubed close parentheses end fraction close square brackets equals

Maths-General
space space space space L t subscript x not stretchy rightwards arrow negative straight infinity end subscript space open square brackets fraction numerator x to the power of 4 sin invisible function application open parentheses 1 over x close parentheses plus x squared over denominator open parentheses 1 plus vertical line x vertical line cubed close parentheses end fraction close square brackets
A s space 1 plus vertical line x vertical line cubed equals open curly brackets table attributes columnalign left columnspacing 1.4ex end attributes row cell 1 plus x cubed end cell cell i f space x greater or equal than 0 end cell row cell 1 minus x cubed end cell cell i f space x less than 0 end cell end table close
equals L t subscript x not stretchy rightwards arrow negative straight infinity end subscript space space open square brackets fraction numerator x to the power of 4 sin invisible function application open parentheses 1 over x close parentheses plus x squared over denominator open parentheses 1 minus x cubed close parentheses end fraction close square brackets
D i v i d i n g space b y space x cubed comma space w e space g e t colon
equals L t subscript x not stretchy rightwards arrow negative straight infinity end subscript space open square brackets fraction numerator x sin space begin display style 1 over x end style plus begin display style 1 over x end style over denominator begin display style 1 over x cubed end style minus 1 end fraction close square brackets
equals L t subscript x not stretchy rightwards arrow negative straight infinity end subscript space open square brackets fraction numerator begin display style fraction numerator sin space begin display style 1 over x end style over denominator begin display style 1 over x end style end fraction end style plus begin display style 1 over x end style over denominator begin display style 1 over x cubed minus 1 end style end fraction close square brackets
equals fraction numerator 1 plus 0 over denominator 0 minus 1 end fraction
equals negative 1.
parallel

card img

With Turito Academy.

card img

With Turito Foundation.

card img

Get an Expert Advice From Turito.

Turito Academy

card img

With Turito Academy.

Test Prep

card img

With Turito Foundation.