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q-When an elastic material with Young's modulus Y is subjected to a streching stress S, the elastic energy stored per unit volume of the material is; [ans- S^2/2Y

please give detail answer

q-If a wire is streched by hanging a weight from its end,the elastic potential energy per unit volume in terms of the longitudinal strain

and modulus of elasticity Y is? [ans-(Y

^2)/2

please give detail answer

1. The hybridization state of C atom in butendioic acid is :
2 3
(1) sp   (2) sp   (3) both two (4) sp

2. Which of the following is not a isomer of pentane :
(1) n-pentane
(2) 2, 2-dimethy 1 propane
(3) 2, 3-dimethy 1 butane
(4) 2-methy 1 butane

3. The oxidation number of C atom in Ch CI and CCI are respectiv
2 2 4
(1) -2 and ? 4 (2) 0 and ? 4 (3) 0 and 4 (4) 2 and 4

4. Which of the following dissolves in lonic solvents :
(1) C H (2) CH OH (3) CCI (4) C H
6 5 3 4 5 12

5. The conjugate acid of HS is :
-2
(1) S (2) H S (3) both two (4) none
2 2

6. Phenolphthalein of pH range [8-10] is used in which of the followin
titration as a suitable indicator :
OH and HCI
(1) NH
4
(2) NH
OH and HCOOH
4
(3) NH OH and C H O
4 2 4 2
(4) NaOH and C
O H
2 4 2

7. Which of the following is iron are :
(1) Malachite (2) Hernatite (3) Siderite (4) Limonite

8. The molar concentration of chloride ions in the resulting solution o
will be :
ml.of 3.0 M NaCI and 200 ml. of 4.0 M BaCl
2
(1) 1.7 M (2) 1.8 M (3) 5.0 M (4) 3.5 M

9. Which of the following has least bond energy :
-2 - +
(2) N   (3) N   (4) N
(1) N
2 2 2 2

10. Which of the following species has highest bond energy :
-2 + -
(1) O (2) O   (3) O (4) O
2 2 2 2

11. Which of the following compound is not aromatic :
(1) 1, 3-cyclobutene
(2) pyridine
(3) furane
(4) thiophene

12. Which of the following compound is used as refrigerant :
(1) CCI F
2 2
(2) CCI

4
(3) CF
4
(4) Acetone

13. Which of the following is weak acid :
(1) C H (2) CH -C=CH (3) CH =CH    (4) CH -C=C-CH
6 6 3 2 2 3 3

14. L.P.G. mainly consist of the following :
(1) Methane (2) Hydrogen (3) Acetylene (4) Butane

-9
15. The solubility product of CaCo is 5 x 10 . The solubility will be :
3
-5 -5 -4 -9
(1) 2.5 x 10   (2) 7 x 10 (3) 2.5 x 10   (4) 2.2 x 10

16. The outer electronic configuration of alkali earth metals is :
10 1 6
(1) nd
(2) ns   (3) np   (4) ns
2

17. The nature of 2, 4, 6-trinitrophenol is :
(1) Neutral (2) Basic (3) Acidic (4) Weak basic

18. Which of the following group is sharp ortho and para directive :
(1) ?C H (2)-OH (3) ?CH (4) ?CI
6 5 3

19. By which of the following process hydrocarbons are found from petroleum :
(1) combustion
(2) fractional distillation
(3) addition
(4) all above

20. A sample of petroleum contains 30% n-heptane, 10% 2-methyl hexane and
60% 2, 2, 4-trimethyl pentane, the octane no. of this sample will be :
(1) 30% (2) 60% (3) 10% (4) 70%
21. In which of the following halogens p-electrons does not take part in
resonance :
(1) CH
=CH-CH Cl (2) BrC H
2 2 6 5
H Cl   (4) CH =CHCl
(3) C
6 5 2
22. Which of the following statement is false :
(1) 40% solution HCHO is known as formalin
(2) HCHO is least reactive in its homologous series
(3) The B.P. of isovarelaldehyde is less than n-varelaldehyde
(4) The boiling point of ketones are higher than that of aldehydes
23. If n + ?? = 8 then the expected no. of orbitals will be :
??
(1) 4 (2) 9 (3) 16 (4) 25

Alc. KOH        2Cl             Ca(OH)

2 2
24. A              B             C                   here the compound C will be :
(1) Lewsite (2) Westron (3) Acetylene tetra chloride (4) Both 2 and 3

25. Which of the following is least hydrolysed :
(1) BeCl (2) MgCl (3) CaCl (3) BaCl
2 2 2 2

26. The laughing gas is :
O (2) NO (3) N O (4) N O
(1) N
2 4 2 2 5

-6
27. The hydrogen ion concentration of a solution is 3.98 x 10 mole per liter. The
pH value of this solution will be :
(1) 6.0 (2) 5.8 (3) 5.4 (4) 5.9

28. The reaction of sodium acetate and sodalime gives :
(1) Butane (2) Ethane (3) Methane (4) Propane

29. Which of the following acids does not contain ? COOH group :
(1) Carbamic acid (2) Barbituric acid
(3) Lactic acid (4) succinnic acid

30. Which of the following compound of xenone does not exists :
(1) XeF (2) XeF (4) XeF (4) XeF
6 4 5 2

31. FeSO
, 7H O is :
4 2
(1) Mohr?s salt     (2) Blue vitriol (3) Green vitriol (4) White vitriol
32. The solution of BiCl3 in dil. HCI when diluted with water white preci
formed which is :
(1) Bismith oxychloride (2) Bismith oxide
(3) Bismith hydroxide (3) none of these
33. The strongest acid is :
(1) acetic acid
(2) trichloroacetic acid
(3) dichloracetic acid
(4) monochloroacetic acid
34. The false statement regarding alkane is :
(1) This does not perform polymerization reaction
(2) This does not gives elimination reaction
(3) It does not disappear the colour of dilute KMnO
solution

(4) It does not decolourise bromine water
35. Which of the following is strongest base :
(1) C H NH   (2) CH NH

36. Which of the following aromatic compound gives sulphonation reaction very
easily :
(1) Chlorobenzene (2) Nitrobenzene (3) Toluene (4) benzene

37. The geometry of I3- is :
(1) Triangular (2) Linear (3) Tetrahedral (4) T-shape

38. The half life of a radio active element is 140 days. 1 gm. of this element after
560 days will become :
(1)   1
gm       (2)    1 gm (3)     1gm.   (4)    1 gm.
       16                       4                       8                                2

39. The volume concentration of hydrogen peroxide 6.8% concentration will be :
(1) 5 (2) 11.2 (3) 22.4 (4) 20

40. Which of the following on combustion give maximum energy :
(1) Ethane (2) Propane (3) Methane (4) Butane

                                                         Anhy. AICI
3
41. C6H6 + CH3CI                 C6H5CH3 + HCI The name of above reaction is :
(1) Gattermann (2) Reimer-tiemann
(3) Friedel-Craft (4) Cannizaro

42. The oxidation state of Cr in K Cr O is :
2 2 7
(1) + 4 (2) + 3 (3) + 6 (4) + 5

43. The natural rubber is the polymer of :
(1) 1, 3- butadiene (2) polyamide (3) isoprene (4) none of these

44. Nylone-66 is a :
(1) polyester (2) polyamide (3) polyacrylate (4) none of these

ANSWER SHEET
1.(2) 2.(3) 3.(3) 4.(2) 5.(2) 6.(4) 7.(1) 8.(3) 9.(1) 10.(4) 11.(1)
12.(1) 13.(2) 14.(4) 15.(2) 16.(4)17.(3)18.(2)19.(2)20.(2) 21.(1) 22.(2)
23.(3) 24.(4) 25.(4) 26.(3) 27.(3)28.(3)29.(2)30.(3)31.(3) 32.(1) 33.(2)
34.(3) 35.(2) 36.(3) 37.(2) 38.(1)39.(4)40.(4)41.(3)42(3) 43.(3) 44.(2)

Friend hope youu will find them useful please do comment i will post some more such questions cheers

1. The hybridization state of C atom in butendioic acid is :
2 3
(1) sp   (2) sp   (3) both two (4) sp

2. Which of the following is not a isomer of pentane :
(1) n-pentane
(2) 2, 2-dimethy 1 propane
(3) 2, 3-dimethy 1 butane
(4) 2-methy 1 butane

3. The oxidation number of C atom in Ch CI and CCI are respectiv
2 2 4
(1) -2 and ? 4 (2) 0 and ? 4 (3) 0 and 4 (4) 2 and 4

4. Which of the following dissolves in lonic solvents :
(1) C H (2) CH OH (3) CCI (4) C H
6 5 3 4 5 12

5. The conjugate acid of HS is :
-2
(1) S (2) H S (3) both two (4) none
2 2

6. Phenolphthalein of pH range [8-10] is used in which of the followin
titration as a suitable indicator :
OH and HCI
(1) NH
4
(2) NH
OH and HCOOH
4
(3) NH OH and C H O
4 2 4 2
(4) NaOH and C
O H
2 4 2

7. Which of the following is iron are :
(1) Malachite (2) Hernatite (3) Siderite (4) Limonite

8. The molar concentration of chloride ions in the resulting solution o
will be :
ml.of 3.0 M NaCI and 200 ml. of 4.0 M BaCl
2
(1) 1.7 M (2) 1.8 M (3) 5.0 M (4) 3.5 M

9. Which of the following has least bond energy :
-2 - +
(2) N   (3) N   (4) N
(1) N
2 2 2 2

10. Which of the following species has highest bond energy :
-2 + -
(1) O (2) O   (3) O (4) O
2 2 2 2

11. Which of the following compound is not aromatic :
(1) 1, 3-cyclobutene
(2) pyridine
(3) furane
(4) thiophene

12. Which of the following compound is used as refrigerant :
(1) CCI F
2 2
(2) CCI

4
(3) CF
4
(4) Acetone

13. Which of the following is weak acid :
(1) C H (2) CH -C=CH (3) CH =CH    (4) CH -C=C-CH
6 6 3 2 2 3 3

14. L.P.G. mainly consist of the following :
(1) Methane (2) Hydrogen (3) Acetylene (4) Butane

-9
15. The solubility product of CaCo is 5 x 10 . The solubility will be :
3
-5 -5 -4 -9
(1) 2.5 x 10   (2) 7 x 10 (3) 2.5 x 10   (4) 2.2 x 10

16. The outer electronic configuration of alkali earth metals is :
10 1 6
(1) nd
(2) ns   (3) np   (4) ns
2

17. The nature of 2, 4, 6-trinitrophenol is :
(1) Neutral (2) Basic (3) Acidic (4) Weak basic

18. Which of the following group is sharp ortho and para directive :
(1) ?C H (2)-OH (3) ?CH (4) ?CI
6 5 3

19. By which of the following process hydrocarbons are found from petroleum :
(1) combustion
(2) fractional distillation
(3) addition
(4) all above

20. A sample of petroleum contains 30% n-heptane, 10% 2-methyl hexane and
60% 2, 2, 4-trimethyl pentane, the octane no. of this sample will be :
(1) 30% (2) 60% (3) 10% (4) 70%
21. In which of the following halogens p-electrons does not take part in
resonance :
(1) CH
=CH-CH Cl (2) BrC H
2 2 6 5
H Cl   (4) CH =CHCl
(3) C
6 5 2
22. Which of the following statement is false :
(1) 40% solution HCHO is known as formalin
(2) HCHO is least reactive in its homologous series
(3) The B.P. of isovarelaldehyde is less than n-varelaldehyde
(4) The boiling point of ketones are higher than that of aldehydes
23. If n + ?? = 8 then the expected no. of orbitals will be :
??
(1) 4 (2) 9 (3) 16 (4) 25

Alc. KOH        2Cl             Ca(OH)

2 2
24. A              B             C                   here the compound C will be :
(1) Lewsite (2) Westron (3) Acetylene tetra chloride (4) Both 2 and 3

25. Which of the following is least hydrolysed :
(1) BeCl (2) MgCl (3) CaCl (3) BaCl
2 2 2 2

26. The laughing gas is :
O (2) NO (3) N O (4) N O
(1) N
2 4 2 2 5

-6
27. The hydrogen ion concentration of a solution is 3.98 x 10 mole per liter. The
pH value of this solution will be :
(1) 6.0 (2) 5.8 (3) 5.4 (4) 5.9

28. The reaction of sodium acetate and sodalime gives :
(1) Butane (2) Ethane (3) Methane (4) Propane

29. Which of the following acids does not contain ? COOH group :
(1) Carbamic acid (2) Barbituric acid
(3) Lactic acid (4) succinnic acid

30. Which of the following compound of xenone does not exists :
(1) XeF (2) XeF (4) XeF (4) XeF
6 4 5 2

31. FeSO
, 7H O is :
4 2
(1) Mohr?s salt     (2) Blue vitriol (3) Green vitriol (4) White vitriol
32. The solution of BiCl3 in dil. HCI when diluted with water white preci
formed which is :
(1) Bismith oxychloride (2) Bismith oxide
(3) Bismith hydroxide (3) none of these
33. The strongest acid is :
(1) acetic acid
(2) trichloroacetic acid
(3) dichloracetic acid
(4) monochloroacetic acid
34. The false statement regarding alkane is :
(1) This does not perform polymerization reaction
(2) This does not gives elimination reaction
(3) It does not disappear the colour of dilute KMnO
solution

(4) It does not decolourise bromine water
35. Which of the following is strongest base :
(1) C H NH   (2) CH NH

36. Which of the following aromatic compound gives sulphonation reaction very
easily :
(1) Chlorobenzene (2) Nitrobenzene (3) Toluene (4) benzene

37. The geometry of I3- is :
(1) Triangular (2) Linear (3) Tetrahedral (4) T-shape

38. The half life of a radio active element is 140 days. 1 gm. of this element after
560 days will become :
(1)   1
gm       (2)    1 gm (3)     1gm.   (4)    1 gm.
       16                       4                       8                                2

39. The volume concentration of hydrogen peroxide 6.8% concentration will be :
(1) 5 (2) 11.2 (3) 22.4 (4) 20

40. Which of the following on combustion give maximum energy :
(1) Ethane (2) Propane (3) Methane (4) Butane

                                                         Anhy. AICI
3
41. C6H6 + CH3CI                 C6H5CH3 + HCI The name of above reaction is :
(1) Gattermann (2) Reimer-tiemann
(3) Friedel-Craft (4) Cannizaro

42. The oxidation state of Cr in K Cr O is :
2 2 7
(1) + 4 (2) + 3 (3) + 6 (4) + 5

43. The natural rubber is the polymer of :
(1) 1, 3- butadiene (2) polyamide (3) isoprene (4) none of these

44. Nylone-66 is a :
(1) polyester (2) polyamide (3) polyacrylate (4) none of these

ANSWER SHEET
1.(2) 2.(3) 3.(3) 4.(2) 5.(2) 6.(4) 7.(1) 8.(3) 9.(1) 10.(4) 11.(1)
12.(1) 13.(2) 14.(4) 15.(2) 16.(4)17.(3)18.(2)19.(2)20.(2) 21.(1) 22.(2)
23.(3) 24.(4) 25.(4) 26.(3) 27.(3)28.(3)29.(2)30.(3)31.(3) 32.(1) 33.(2)
34.(3) 35.(2) 36.(3) 37.(2) 38.(1)39.(4)40.(4)41.(3)42(3) 43.(3) 44.(2)

Friend hope youu will find them useful please do comment i will post some more such questions cheers

can anyone tell me the sign convention used

Case 1 when a body is dropped from tower height H wit velocity v , gravity g

Case 2 when body is thrown upward from tower height H wit velocity v , gravity g

Pleas ehelp me i promise to rate for this concept

A uniform pole 50 kg is placed vertically on a rough surface of

=0.3.Lenght of pole is 8 m.A 500 N horizontal force is given to the pole suddenly at a height of 2m above the ground.Determine the acceleration of the topmost point of the pole..(g=9.8)....

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A stick length L and mass M lies on a frictionless horizontal surface on which it is free to move in any way a ball of mass m moving with speed v collides elastically with the stick as shown in the figure . it after collison the ball comes to rest then mass of ball should be

Ans M/4

Please give me a detailed ans for which i promise to rate every one

To get a magnified image click on the image

A rigid body each of length L fastened together to form letter H

The body is free to rotate about a horizontal axis that runs along the length of one of the legs of H The body is allowed to fall from rest from a position in which plane of H is horizontal

What is angular speed of the body when plane of body is vertical

Ans 1.5

(g/l)

Please give me a detailed ans i promise to rate everyone You may click on image to magnify it

^[

Very good book i am also using it & altough bit tough good for exams like AIIMS & IIT whic is everyones dream so it is a dream book

STUDY TIPS
Combination of Subjects
Study a combination of subjects during a day i. e. after studying 2Ò3 hrs of mathematics shift to
any theoretical subject for 2 horrs. When we study a subject like math, a particular part of the
brain is working more than rest of the brain. When we shift to a theoretical subject, practically the
other part of the brain would become active and the part studying maths will go for rest.

Revision
Always refresh your memory by revising the matter learned. At the end of the day you must revise
whatever youÌve learnt during that day (or revise the previous days work before starting studies
the next day). On an average brain is able to retain the newly learned information 80% only for 12
hours, after that the forgetting cycle begins. After this revision, now the brain is able to hold the
matter for 7 days. So next revision should be after 7 days (sundays could be kept for just
revision). This ways you will get rid of the problem of forgetting what you study and save a lot of
time in restudying that topic.

Use All Your Senses
Whatever you read, try to convert that into picture and visualize it. Our eye memory is many times
stronger than our ear memory since the nerves connecting brain to eye are many times stronger
than nerves connecting brain to ear. So instead of trying to mug up by repeating it loudly try to
see it while reapeating (loudly or in your mind). This is applicable in theoritical subjects. Try to use
all your senses while learning a subject matter. On an average we remember 25% of what we
read, 35% of what we hear, 50% of what we say, 75% of what we see, 95% of what we read,
hear, say and see.

Breathing and Relaxation
Take special care of your breathing. Deep breaths are very important for relaxing your mind and
hence in your concentration. Pranayam can do wonders to your concentration, relaxation and
sharpening your mined (by supplying oxygen to it). Aerobic exercises like skipping, jogging,
swimming and cycling are also very helpful.

The most powerful weapon on earth is
human soul on fire!

Hope you will find them useful please comment

Question 1

Let f and g be defined by f(x) = |x|, xÎ R, and g(x) = x, xÎ R. Find f + g, f - g, fg and f/g.

Ans.

( f + g )( x )	= 2x	x ³ 0
	0	x < 0

( f - g )( x )	= 0	x ³ 0
	-2x	x < 0

(fg)(x)	= x²	x ³ 0
	-x²	x < 0

(f/g)(x)	= 1	x > 0
	-1	x < 0

f/g is not defined at x = 0.

Question 2
Find the inverse of the function F(x) = x + 1, x ¹ 1 x - 1
Ans. Let us see whether the given function is invertable Let x₁ , x_₂ ¹ 1 x₁ ¹ x_₂ Then x₁ + 1 ¹ x_₂ + 1 And x₁ - 1 ¹ x_₂ - 1 \ x₁ + 1 ¹ x_₂ + 1 x1 - 1 ¹ x_₂ - 1 Þ f(x₁) ¹ f(x₂) \ f is invertable function Now y = x + 1 or xy - y = x + 1/X - 1 Or x(y - 1) = y + 1 or X = y +1/y - 1 we are given f = {(x,y) : y = x + 1/x - 1} \ f-1 = {(y,x) : y ¹ 1 , x = y +1/y - 1 }

Question 3
Find fof^-1 and f^{^-1}of the function F(x) = 1/x , x ¹ 0. Also prove that fof^-1 = f^-1of
Ans. f(x) = 1/x \ f^-1 = { ( 1/x = y,x)} = { (y, 1/y) } \ f^-1 (x) = 1/x \ fof ^-1 (x) = f(1/x) = 1 = x 1/x Also f ^-1of (x) = f^-1 (1/x) = 1 = x. 1/x \ fof ^-1(x) = f ^-1of (x) Hence f ^-1 of = f ^-1 of

Question 4
Evaluate lim 2x² + 3x x®¥ x² +1
Ans. Put x = 1/y, so that when x®¥, y®0. \ lim 2x^² + 3x = lim 2/y^² + 3/y x®¥ x^² +1 y®0 1/y² +1 = lim 2 + 3y = 2 + 0 = 2. y®0 1 +y² 1 + 0

Question 5
The profit function of a manufacturing firm is given by P(x) = -x³/3 + 729x - 2500. Find x so that the firm attains maximum profit, where x is the number of items manufactured.
Ans. The profit function is P(x) = -x³/3 + 729x - 2500 P¹(x) = -x² + 729; where p ₁ is the d(p(x))/dx \ P¹ (x) = 0 give us -x² + 729 = 0 or x = 27. [ -27 is rejected as x is number of items] Now P¹¹ (x) = -2x Which is -ve at x = 27 \ P(x) attain the maximum value at x = 27 \ Number of items (x) manufactured by firm = 27.

Question 6
_______________________ If y = Öcosx + Ö cosx +Ö cosx+ ?..¥ then prove that (2y -1) dy/dx + sinx = 0.
Ans. we are given that _______________________ y = Öcosx + Ö cosx +Ö cosx+ ?..¥ _______ = Öcosx + y or y² = cosx + y Differentiating both sides w.r.t. x1 we get 2y dy = -sinx + dy dx dx or 2y dy - dy sinx = 0 dx dx \ (2y - 1) dy/dx + sinx = 0.

Question 7
Show that x/sinx is an increasing function in the intercal 0< x < p/2.
Ans. Let f(x) = x/sinx Then f^¹(x) = sinx - xcosx sin²x Now sin²x >0 for 0< x < p/2 [ In fact sin² x> 0 < " x] we know that tanx > x for 0 < x < p/2 \ sinx > x for 0 < x < p/2 cos \ sinx - x cosx > 0 for 0 < x < p/2 \ f¹(x) = sinx - x cosx > 0 for 0 < x < p/2 sin²x \ f (x) is increasing in the interval { 0, p/2}

Question 8
Find the deviative of cot( 2x +1) w.r.t. x using the first principles.
Ans. Let y = cot [ 2 x +1] \ y +dy = cot {2(x + dx) + 1 } or dy = cot {2 (x +dx) +1} - cot (2x +1) = cos (2x+1+ 2dx ) - cos(2x+1) sin (2x+1+2dx) sin (2x+1) = cos (2x+1+2ds) sin (2x+1) - sin(2x+1+2dx) cos (2x+1) sin (2x+1+2fx) sin ( 2x+1 ) = sin (2x +1 - 2x -1 -2dx) sin(2x +1 +2dx) sin(2x+1) \ dy = -sin (2dx) sx sin(2x+1 + 2fx) sin(2x +1)dx \ lim dy = lim -sin(2dx) = lim-1 dx ®0 dx dx ®0 sin(2x+1+2dx) sin(2x+1)dx \ dy = -1 x 1 x 2 dx sin(2x+1) sin(2x +1) = -2 sin2 (2x +1) = -2 cosec² (2x +1)

Question 9
____ Use Lagrauge's mean value theorm to determine a point P on the curve y = Ö x - 2 defined in the interval [2,3] where the tangent is parallel to the chord joining the end points on the curve. Unit - 2 (Integral Calculus)
Ans. _____ Here f(x) = Ö x - 2 _____ Now Ö x - 2 is defined for all values of x ³ 2 \ f(x) is defined in the interval [2,3] again f¹(x) = 1 2Ö x - 2 1 is defined for all values of x > 2. 2Ö x - 2 \ f(x) is derivable in the interval ]2,3[. Thus both the conditions of LMV therom are satisfied. \ by LMV theorem there must exists at least one C in the interval ]2,3[ at which the tangent is parallel to the chord joining the end points on the curve, such that. F¹ ( C ) = f(3) - f(2) 3 - 2 or 1 = Ö3 -2 - Ö2 - 2 2Ö c - 2 1 \ 1 = 1 2Ö c - 2 or 2Ö c - 2 = 1 \ 4(c - 2) = 1 or c - 2 = ¼ \ c = ¼ + 2 = 9/4 E ]2,3[ \ Now f (9/4) = Ö9/4 - 2 = Ö¼ = ½ Hence the tangent is parallel to the chord at the point (9/4, ½).

Question 10
Evaluate : ò₁⁴ \|x - 2\| dx
Ans. ½x - 2½ = x - 2 if x ³ 2. And ½x - 2½ = -x - 2 if x < 2. \ ò^₁4½x - 2½dx = ò^₁2 - (x - 2) dx + ò^₂4 (x - 2) dx = ( -x²/2 + 2x)₁² + (x²/2 + 2x)₂⁴ = [ (-4/2 + 4 ) - ( -1/2 + 2)] + [(16/2 -8) (4/2 -4) = 2-3/2 + 0 +2 = 5/2.

Question 11
Evaluate òxe^x/(x +1)² dx
Ans. I = ò x e ^x dx (x + 1)2 = òe^x [ 1/ (x +1)- 1/(x + 1)² ] dx = òe^x [f(x) + f^¹(x) ], where f(x) = 1/ (x+1) = e^x f(x) + c = e^x /(x+1) + C.

Question 12
Evaluate ______ ò Öx / Öa³ - x³ dx
Ans. Let x ^3/2 = t. then 3/2 Öx dx = dt \ Öx dx = 2/3 dt \ I = òÖ x /Ö a³ - x³ dx = ò Ö2/3 dt/Öa³ - t² = 2/3 ò dt/ Ö( a ^3/2) ^² - t^² = 2/3 - sin^-1 t/a^3/2 + c {\ ò dx/Öa²-x² = sin^-1 x/a = c} = 2/3 sin^-1 (x/a)^3/2 + c.

Question 13
Evaluate ò_-1¹ e^x dx as the limit of a sum.
Ans. Let f (x) = e^x . Let nh = 1- ( -1) = 2. ò _-1¹ e^xd^x = lim h [ f (-1)+f(-1+h) +f(-1+2h)+ ------+f(-1+n-1h) ] h ® o = lim h[ e^-1+e^-1+h+e^-1+2 h+-------+e^-1^{+^(n-1)h} ] h ® o = lim h [e^-1{ 1+eh+e2h+---------+e ^{+^(n+)h} } ] h ®o = lim h [e^-1{ 1-e^nh/1-e^h } ] h ® o = lim h/1-e^h . lim e^-1 ( 1-e^nh) h ® o h ®o = (-1) [ e-1(1-e2)] lim eh-1/h h ® o = e^-1(e^² -1)}e² 1 = e-1 e

Question 14
Find the aver of the region bounded by the two parabolas y = x² and x = y²
Ans. Sloving the equation y = x2 and x = y2 , the points of the intersection of the two curves are 0(0,) and P(1,1). The requried area is PROS in fig. (1.1) Y \ A = ò y dx y = x2 \ Requried area A = area OSPQ - area ORPQ = ò₀¹ Öx dx - ò₀¹ x² dx =[2/3 x ^3/2]₀¹ - [x³/3]₀¹ = 2/3 - 1/3 = 1/3 sq units.

Question 15
Prove that ò₀^p/4 log(1 + tanq) dq = ^p/8 log 2.
Ans. Let I = ò₀ ^p/4 log(1 + tanq) dQ = ò_₀ ^p/4 log [ 1 + tan(p/4 -Q) dQ [\ ò₀^a f(x) dx = ò₀^a f ( a - x) dx] = ò₀ ^p/4 log [1 - tanQ/1 + tanQ] dQ = ò₀ ^p/4 log [ 2/1 + tanQ ] dQ = ò₀ ^p/4 logdQ - ò₀ ₀ ^p/4 log [1 - tanQ] dQ \ I = log 2 [0] ₀ ^p/4 - I or 2 I = log 2 [ ₀ p/4] or I = p/8 log 2.

Question 16
Evaluate: òx² + 1/x⁴ + 1 dx
Ans. ò x² + 1/x⁴ + 1 =dx = ò(1+1/x²)/(x² + 1/x2) dx = ò 1 + 1/x² dx x² + 1/ x² - 2 +2 Let x - 1/x = z, so that ( 1+1/x²) dx = dz \ òx² + 1/x⁴ + 1 dx = ò dz/ z² + 2 = ò dz / z²+(Ö2)² = (1/Ö2)( tan^-1 z/Ö2 )+ c [\ ò 1/x^² + a^² dx = 1/a tan ^-1 x/a] = 1/Ö2 tan^-1 x - 1/x/Ö2 + c = 1/Ö2 tan^-1 x² - 1/Ö2x + c

Question 17
Evaluate : òx⁴/x⁴ - 1 dx
Ans. Let x⁴ º x⁴ x⁴ - 1 (x-1)(x+1)(x^²+1) = 1 + A + B + (x + D) ?..(1) x-1 x+1 x+1 Multiplying by x⁴ - 1 we get º (x⁴ - 1) + A (x+1) (x²+1) +B (x - 1) (x² +1) + (x + D) (x + 1) ( x - 1 ) ?. (2) Putting x = 1, -1 in (II) , we get 1= A (2) (2) And 1 = B(-2)(2) or A = ¼ and B = -1/4 Equating ¥ efficients of x³ in (2) , we get A + B +C = 0 But a = ¼ and b = -1/4 \ c = 0 Puuting x = 0 in (2) , we get 0 = -1 + A -B - D or D = -1 +A -B = -1 + ¼ + ¼ = -1/2 \from (1) x⁴ = 1 + 1 - 1 - 1 . x⁴ -1 4(x-1) 4(x+1) 2(x²+1) x4/x4 -1 dx = 1 + 1/4(x-1) - 1/4/x+1 - 1/2(x^²+1).dx = x + ¼ log (x -1 ) -1/4 log (x + 1) -1/2 tan^-1 x + c = x + ¼ log x - 1 -1/2 tan ^-1 x + c x+1

Question 18
Find the area of the region bounded by the curve c:y = tanx, tangent drawn to C at x =p/4 and the x - axis.
Ans. Requried area = shaded area on the figure = Area (OPR) - area (DPQR) = ò_o ^p/4 tanx dx - DPQR ??..(1) To find DPQR, we proceed as follows : Now Point P is ( ^p/4 , 1) Y \ y = tan x \ dx/dy = sec²x \Slope of tangent at P = sec²p/4 = 2 i.e., tanQ = 2 \ QR = PR cot Q = 1 x ½ = ½ \ Area of DPQR = ½ x ½ x 1 = ¼ sq.unit. q R \ from (1) Required area = (log sec x )^p/4 - ¼ = log 02 - ¼ =[1/2 log2 - ¼ ] sr.units.

Question 19
Evaluate : ò_{_o}^p/4 (sinx + cosx)/(9+16 sin2x) dx
Ans. Let I = ò_o ^p/4 sinx + cosx dx 9+16 sin2x = ò_o ^p/4 cosx + sinx dx 9+16 { 1-(sinx - cosx)2} Let sinx - cosx = t , then ( cosx +sinx) dx = dt When x = 0, t = -1 and when x = p/4, t = 0 \ I = ò_-1^o dt 9 + 16 ( 1 - t^²) = ò_-1^o dt 25 - 16t^² = ò_-1^o dt 5² - (4t)(4E)^² = 1/2x5 x ¼ log [ 5 + 4t/5 - 4t]^o_-1 = 1/40 [log1 - log(1/9)] = 1/40 (log9) = 1/40 log 3² = 1/20 log3.

Question 20
_______ Evaluate ò(3x - 2) Öx² +x+ 1 dx.
Ans. Let I = ò(3x - 2) Öx² +x+ 1 dx. Here d/dx (x² +x+ 1) = 2x +1 \ we express 3x - 2 in terms of 2x +1 Now 3x - 2 = 3/2 (2x+1) -7/2 \ I = ò{ 3/2 (2x + 1) - 7/2 } Öx² +x+ 1 dx. = 3/2 ò (2x +1) Öx² +x+ 1 dx - 7/2 òÖx² +x+ 1 dx. = 3/2 I₁ - 7/2 I₂ + c Now I₁ = ò(2x + 2) Öx² +x+ 1 dx. Let x² +x+ 1 = t , then (2x +1) dx = dt. \ I₁ = ò Öt dt = 2/3 t ^3/2 = 2/3 (x2 +x+ 1)^3/2 and I₂ = ò Ö x² +x+ 1 dx = ò Ö ( x + ½)² + (Ö3/2)² dx. = (x + ½) Öx2 +x+ 1 + ¾ log [(x + ½) Ö x2 +x+ 1] +c 2 2 Ö3/2 = (2x + 1) Ö x2 +x+ 1 + 3 log [(2x + 1) Ö x2 +x+ 1] +c 4 8 Ö3 \ from (1) I = 3/2 . 2/3 (x² +x+ 1)^3/2 - 7/2[(2x+1) Ö x2 +x+ 1 + 4 3/8 log [{2x +1 +2 Öx² +x+ 1}] +c = (x2 +x+ 1)^3/2 - 7/8 (2x +1) Öx2 +x+ 1 - 21/16 log [ 2x + 1 + 2 Ö x² +x+ 1] +c Ö3

Question 21
Prove that ò_o^p/2 Öcotx/(ÖCotx + Ötanx) dx = p/4
Ans. Let I = ò_o^p/2 Öcotx / (ÖCotx + Ötanx) dx = ò_o^p/2 Öcotx (p/2 -x) _____ dx ÖCotx (p/2 -x) + Ötanx (p/2 -x) [\ò_{_o^a} f(x) dx = _{_òo^a} f(a - x)dx] = ò_o^p/2 Ötanx dx / Ötanx + Öcotx \ I + I = ò_o^p/2 Öcotx/ Ötanx + Öcotx dx + ò_o^p/2 Ötanx/Öcotx + Ötanx dx = ò_o^p/2 Öcotx + Ötanx/Öcotx + Ötanx dx = ò_o^p/2 1.dx = [x]_o^p/2 = ^p/2 \2I = ^p/2 or I = ^p/4.

Question 22
_____ Solve the differential equation (Ö a + x ) dy/dx + x = 0.
Ans. ____ The given equation is( Öa + x) dy/dx + x = 0 Þ dy = -x /Öa + x dx Þ òdy = - ò x dx / Öa + x Þ òdy = - ò a + x - a / Öa + x dx ____ Þ y = - ò Öa + x dx + òa (a+ x) ^-1/2 dx Þ y = - (a + x)^3/2 + a . (a + x)^½+ C 3/2 1/2 Þ y = -2/3 (a +x) ^3/2 + 2a Öa + x + C. which is the required solution.

Question 23
Find the differential equaion which corresponds to the equaion Y = ke ^sin^-1^x
Ans. We are given Y = k . e ^sin^-1^x???.(1) Defferentiating w.r.t. x we get dy/dx = k.e^sin-1x 1/Ö1- x2 ???(2) Putting k = y from (1) in (2) , we get Sin-1 x dy/dx = y . e^sin-1x e^sin-1x Ö1-x² or dy/dx = y/Ö1-x² Which is the required differential equation.