Buds Public School , DubaiGrade 12 Science Mathematics Assignment Continuity and Differentiability, Applications of Derivatives:

1. Find the equation of tangent to the curve x=sin 3 t , y=cos2t at t=π4

2. Show that the rectangle of maximum area that can be inscribed in a circle is square.

3. Show that the height of the cylinder of maximum volume that can be inscribed in a cone of

height h is 13h .

4. If y=√x2+1−log( 1x+√1+ 1

x2 ) find dydx

5, If y=cot−1( √1+sin x+√1−sin x√1+sin x−√1−sin x ) find

dydx

6. Discuss the continuity of the following function at x=0

f ( x )= x4+2 x3+x2

tan−1x¿

,¿ x≠0 ,

¿0 x=0 7. Verify Lagrange’s mean value theorem for the following function: f ( x )=x2+2 x+3 , for [ 4 ,6 ]

8. If f ( x )=√ sec x−1sec x+1

, find f ' ( x ) . Also find f ‘( π2 )

9. If x √1+ y+ y √1+x=0 , find dydx

10. Prove that the curves x= y2 and xy=k intersect at right angles if 8k2=1.

11. Find dydx for the following functions:

a) ¿¿ b) xsin x+¿ c) y = ¿¿

d) xsin x+¿

12. Find the intervals in which the function f given by f ( x )=x3+ 1x3 , x ≠0is

a) increasing b) decreasing .

13. Find the equation of the tangent to the curve y = √3 x−2 which is parallel to the line 4 x−2 y+5=0.

Matrices and Determinants

1. Find the value of x∧ y if : 2[1 30 x]+[ y 0

1 2]=[5 61 8 ]

2. Evaluate | a+ib c+id−c+id a−ib|

3. Find the cofactor of a12 and ain the following : |2 −3 56 0 41 5 7|

4. Let A=|3 2 54 1 30 6 7|Express A as sum of two matrices such that one is symmetric and the

other is skew symmetric .

5. If A=|1 2 22 1 22 2 1|verify that A2−4 A−5 I=0

6. Using properties of determinants, prove the following :

i) | α β γα2 β2 γ2

β+γ γ+α α+β|=(α−β ) (β−γ ) (α+β+γ )

ii) | α β γα2 β2 γ2

β+γ γ+α α+β|=(α+ β+γ )|α β γα 2 β2 γ2

1 1 1|7. Using Properties of determinants, prove the following

|1+a2−b2 2ab −2b2ab 1−a2+b2 2a2b −2a 1−a2−b2|=(1+a2+b2 )3

8. Using Properties of determinants, prove the following :

|a+b+2c a bc b+c+2a bc a c+a+2b|=2 (a+b+c )3

9. If [ x 3 y y7 −x 4 ]=[4 −1

0 4 ] find the values of x∧ y .

10. If |x+2 3x+5 4|=3 , find the value of x .

11. If (123 4 )(3 1

25)=(7 11k23) find K .

12. Express A as sum of two matrices such that one is symmetric and the other is skew

symmetric . Verify your result for A = (3−2−43−2−5−11 2 )

13. Using matrices, solve the following system of linear equations. a¿2 x− y+z=3 , −x+2 y−z=−4 x− y+2 z=1 b¿3 x−2 y+3 z=8 , 2 x+ y−z=1 , 4 x−3 y+2 z=4 c) 2 x−3 y+5 z=11 ,3 x+2 y−4 z=−5 , x+ y−2 z=−3 d) x+ y+z = 6 , x+2z = 7 , 3 x+ y+ z=12

12. Using elementary transformations, find the inverse of the following matrix: [2 −1 44 0 23 −2 7 ]

13. Using properties of determinants, prove the following: | a a+b a+2ba+2b a a+ba+b a+2b a |=9b2 (a+b )

14. Using elementary transformations, find the inverse of the following matrix:

a) [2 5 33 4 11 6 2] b) |30−1

2 300 41 |

15. If a ,b, and c are all positive and distinct .show that ∆=|abcb cacab| has a negative value .

16 . By using Properties of determinants prove the following :

a) |x−42 x2 x2 x x−4 2 x2 x2 x x−4| = (5 x+4 )(4−x )2 b) | abc

a−bb−c c−ab+cc+aa+b | = a3+b3+c3−3abc

c) | 11+ p1+ p+q23+2 p1+3 p+2q3 6+3 p1+6 p+3q| = 1 d) | x+ y x x

5x+4 y 4 x2 x10 x+8 y8 x 3x|=x3

17. Use product [1−120 2−33−2 4 ][−2 0 1

9 2−36 1−2] and hence solve the system

of equations : x− y+2 z=1 ,2 y−3 z=1,3 x−2 y+4 z=2.

18. A diet is to contain 30 units of vitamin A, 40 units of vitamin B and 20 units of vitamin C. Thee types of foods F1, F2 and F3 are available . One unit of Food F1 contains 3 units of vitamin A, 2 units of vitamin B and 1 unit of vitamin C. One unit of Food F2 contains 1 units of vitamin A, 2 units of vitamin B and 1 unit of vitamin C. 2 units of Food F3 contains 5 units of vitamin A, 3 units of vitamin B and 2 unit of vitamin C. Represent the above situation algebraically and find the diet contains each types of food by using matrix method . Why a proper diet is required by us ? (Ans : 5,15,0)19. A merchant plans to sell three types of personal computers – a palmtop model , a portable model and desktop model that will cost Rs 8000, Rs 10500 and Rs 10000 respectively. He makes a the survey by two persons , one person estimate that the total monthly demand of computers will be 70 units and the other person survey that palmtop model type computers will be demanded 30 units and total units required is 273 units . If a dealer wants to invest Rs 7 lakh on it . Represent the above situation algebraically and find each type of unit sales. How we can use computer in student life and which is best computer model for students? ( Ans : 7,28,35) 20. A mixture is to be made of three foods A, B, C . The three foods A,B,C contain nutrients P, Q,R as shown below :

Food Grams per Kg of nutrientsP Q R

A 1 2 5B 3 1 1C 4 2 1

How to form a mixture which will have 8 grams of P , 5 grams Q and 7 grams of R ? (Ans: 1, 1, 1)21. Three schools A ,B and C organized a mela for collecting funds for helping the rehabilitation of flood victims . They sold hand – made fans , mats and plates from recycled material at a cost of Rs. 25 , Rs 100 and Rs 50 each . The number of articles sold are given below : Articles / School A B CHand – fans 40 25 35Mats 50 40 50Plate 20 30 40

Find the funds collected by each school separately by selling the above articles . Also find the

total funds collected for the purpose . Write one value generated by the above situation .

22. In a parliament election , a political party hired a public relations firm to promote its candidate in

three ways – telephone , house calls and letters . The cost in paise is given in matrix A as

A=[140200150] . The number of contacts of each type made in two cities X and Y is given in matrix

Telephone House calls Letters

B as [ 1000500500030001000 10000] . Find the total amount spent by the two cities . What

should one consider before casting his/her vote – party’s promotional activity or their social

activites ?

23. Two schools P and Q want to award their selected students on the values of Tolerance,

Kindness and Leadership . The school P wants to award Rs x each , Rs y each and Rs x each

for the three respective values to 3,2 and 1 students respectively with a total award money

Rs2200. School Q wants to spend Rs 3100 to award its 4 , 1 and 3 students on the

respective values (by giving the same award money to the three values as in school P) . If the

total amount of award for one prize on each value is Rs 1200 , Using matrices find the

award money for each value. Apart from this three values , suggest one more value which

should be considered for award .

24. Two schools A and B want to award their selected students on the values of sincerity ,

truthfulness and helpfulness . The school A wants to award Rs x each , Rs y each and Rs z

each for the three respective values 3,2 and 1 students respectively with a total award money

Rs 1600 . School B wants to spend Rs 2300 to award its 4,1 and 3 students on the respective

values (by giving the same award money to the three values as before ) . If the total amount of

award for one prize on each value is Rs 900, using matrices find the award money for each

value . Apart from these three values suggest one more value which should be considered for

award .

25. The management committee of a residential colony decided to award some of its members (say x)

For honesty ,some (say y ) for helping others and some others (say z) for supervising the workers

to keep the colony neat and clean . The sum of all the awardees is 12 . Three times the sum of the

awardees for cooperation and supervision added to two times the number of awardees for honesty

is 33 . If the sum of the number of awardees for honesty and supervision is twice the number of

awardees for helping others, using matrix method, find the number of awardees of each category.

Apart from these values ,namely, honesty, cooperation and supervision , suggest one more value

which the management of the colony must include for awards .

26. 10 students were selected from a school on the basis of values for giving awards and were

divided into three groups. The first group comprises hard workers , the second has honest and

law abiding students and the third group contains vigilant and obedient students . Double the

number of students of the first group added to the number in the second group gives 13, while

the combined strength of the first group is four times that of the third group. Apart from the

values , hard work, honesty and respect for law, vigilance and obedience, suggest one more

value, which in your opinion , the school consider for awards.

Continuity and Differentiability and Applications of Derivatives :

1. Find dydx

for the following :

a) x− y=2π b) y = ax ,w here a is positive constant . c) sin y = x sin(x + y)

d) ¿ e) x = a(cost + t sin t) and y = a(sin t – t cos t )

2. Prove that the function f(x) = 5x -3 is continuous at x = -3 .

3. Discuss the continuity of the function f(x) at x = 12 , when f(x) is defined as follows :

f ( x )=12+ x ,0≤x< 1

2

= 1 , x = 12

= 32 + x ,

12<x ≤1

4. If x √1+ y+ y √1+x=0 , (x ≠ y ) , t h en prove t hat dydx

= −1(1+x )2 .

5. If x = a( cos t+log tan t2¿ and y=asint find i)

dydx ii) d

2 yd2 x

.

6. Show that f ( x )=|x−3|∀ xϵR is continuous but not differentiable at x = 3.

7. If x y+ yx=ab find dydx

8. If √1−x2+√1− y2=a ( x− y ) ,prove that dydx

=√ 1− y2

1−x2

9. Find all the points of discontinuity of the function f ( x )= [x2 ] on [1,2) , where [ . ] denotes the

greatest integer function.

10. Differentiate sin−1(2x √1−x2¿)w .r .¿cos−1( 1−x2

1+x2 ¿)¿ .

11. For what value of k is the function defined by f ( x )=k (x2+2 ) if x ≤0

= 3x+1 if x¿0

is continuous at x = 0 ? Also write whether the function is continuous at x = 1 .

12. If y=¿¿ , then find dydx

.

13. Show that the semi vertical angle of the cone of the maximum volume and of given slant height is

cos−1 1√3

.

14. Of all closed right circular cylindrical cans of volume 128π cm3. Find the dimensions of the can

which has minimum surface area.

15. Prove that the semi –vertical angle at the right circular cone of given volume and least curved

surface area is cot−1√2 .

16. Prove that the height of the cylinder of maximum volume that can be inscribed in a sphere of

radius R is 2R√3

. Also , find the maximum Volume .

17. If the sum of the lengths of the hypotenuse and a side of a right triangle is given , then show that

the area of the triangle is maximum , when the angle between them is 60 ° .

18. A tank capacity 250 m3has to be dug out . The cost of the land is .50 per m2 . The cost of digging

increases with depth and the cost of whole tank is 400 (dept h)2 . Find the dimensions of the tank for

least cost .

19. Find the maximum area of an isosceles triangle inscribed in the ellipse x2

a2 + y2

b2 =1with its vertex at

one end of the major axis .

20. Show that the volume of the greatest cylinder which can be inscribed in a cone of height h and

semi-vertical angle 3 0 °is 481

π h3 .

21. Prove that y=4 sin θ

2+cosθ−θ is an increasing function of θ in [0 , π2 ]

22. Find the point on the curve x2=4 y , which is nearest to the point (-1,2).

23. Find the points on the curve y=x3−3x2+2 x at which the tangent to the curve is parallel to the line

y−2x+3=0.

24. Find the equation of the tangent and normal to the curve x=1−cosθ∧ y=θ−sin θ at ¿π4 .

25 . Find the equations of tangents to the curve 3 x2− y2=8 , which passes through the point ( 43,0)

26. Find the equation of tangent to the curve x=sin 3 t , y=cos2t at t π4

27. Show that the rectangle of maximum area that can be inscribed in a circle is square.

28. Show that the height of the cylinder of maximum volume that can be inscribed in a cone of height

h is 13h .

29. If y=√x2+1− log ( 1x+√1+ 1

x2 ) find dydx

30. If y=cot−1( √1+sin x+√1−sin x√1+sin x−√1−sin x ) find

dydx

31. Discuss the continuity of the following function at x=0

f ( x )= x4+2 x3+x2

tan−1 x¿

,¿ x≠0 ,

¿0 x=0 32. Verify Lagrange’s mean value theorem for the following function : f ( x )=x2+2 x+3 , for [ 4 ,6 ]

33. If f ( x )=√ sec x−1sec x+1

, find f ' ( x ) . Also find f ‘( π2 )

34. If x √1+ y+ y √1+x=0 , find dydx

35. Prove that the curves x= y2 and xy=k intersect at right angles if 8k2=1.

36 . Find dydx for the following functions :

a) ¿¿ b) xsin x+¿ c) y = ¿¿

d) xsin x+¿

37. Find the intervals in which the function f given by f ( x )=x3+ 1x3 , x ≠0is

a) increasing b) decreasing .

38. Find the equation of the tangent to the curve y = √3 x−2 which is parallel to the line

4 x−2 y+5=0

Grade : 12 Science Worksheet – Mathematics - Integral Calculus

1. Integrate the following :

a) ∫ 1−sin xcos2 x

dx b) ∫ 2−3sin xcos2 x

dx c) ∫(x2+1)3dx d) ∫ √ tan xsin x .cos x

dx

2. Integrate the following :

a) ∫sin 2 x dx b) ∫ xlogx dx c) ∫ 1−x2

1+x4 dx d) ∫ x2−1x4+x2+1

dx

3. Evaluate the following :

a) ∫ x+2(x2+3 x+3)√ x+1

dx b) ∫ dx5+4 cos x c) ∫sin−1( 2x

1+x2 ¿¿)dx ¿¿

d) ∫sin x .sin ¿¿¿¿ d) ∫ tan−1(√ 1−sin x1+sin x

¿)dx¿ e) ∫ 3 x−1( x−1 ) ( x−2 )(x−3)

dx

f) ∫ x cos−1 x dx g) ∫ xlog 2x dx h) ∫ dx√ x2+4 x+6

dx

4. Find a¿∫√1−4 x−x2dx b) ∫√1+ x2

9dx c) ∫ dx

(x¿¿2+1)(x2+4)¿

5. Evaluate : a¿∫ dx√x+3−√x+2

b) ∫ dx√ x(1+√x ) c) ∫ sin (x−a)

sin (x+a)dx

6. Evaluate : a) ∫0

2π 11+esin x dx b) ∫

2

8

|x−5|dx c) ∫0

1

cot−1 ( 1−x+ x2 )dx

d) ∫0

π xsin x1+cos2 x

dx e) ∫0

π2

logsin x dx f) ∫0

π2

cos2xcos2x+4 sin2 x

dx

7. Evaluate : ∫0

2

(x¿¿2+2x+1)dx ¿ as limit of sum . 8. Evaluate : ∫ x √x4−1dx

9. Evaluate : ∫0

π

log ¿¿ 10. Evaluate ∫ x2+1( x−1 )2(x+3)

dx

Grade : 12 Science Worksheet – Mathematics - Inverse Trigonometric Functions

1. 1. If f ( x )=x+7∧g ( x )=x−7 , x∈R , find ( fog) (7 )

2. Evaluate : sin [ π3 −sin−1 (−12 )]

4. Prove the following :

a) tan−1 13+tan−1 1

5+ tan−1 1

7+ tan−1 1

8=π

4

b) cot−1( √1+sin x+√1−sin x√1+sin x−√1−sin x )= x

2 , xϵ (0 ,π4) c) sin−1( 4

5 )+¿ sin−1( 513 )+¿ sin−1( 16

65 ¿)=π2

¿¿¿

5. Solve for x 2 tan−1¿¿

6. Solve for x : tan−1 (2x )+ tan−1 (3x )= π4

7. Solve for x : tan−1( x−1x−2 )+ tan−1( tan−1 x+1

x+2 )=¿ π4

¿

8. Solve : cos¿¿

9. Solve for x : tan−1 1−x1+x

=12

tan−1 x ; x>0

10. Prove that tan( π4 + 12

cos−1 ab )+ tan( π4 + 1

2cos−1 a

b )=2ba

11. Solve tan−1 (x+1 )+ tan−1 (x−1 )=tan−1 831

12. Using the principal value evaluate the following :

a) cos−1( 12¿)−2 sin−1(−1

2 )¿ b) sin−1¿¿ c) cos−1¿¿

13. Prove the following :

a) tan−1 13+tan−1 1

5+ tan−1 1

7+ tan−1 1

8+ π

4

b) cot−1( √1+sin x+√1−sin x√1+sin x−√1−sin x )= x

2 , xϵ (0 ,π4)

c) sin−1( 45 )+¿sin−1( 5

13 )+¿ sin−1( 1665 ¿)=

π2

¿¿¿

14. Solve for x : tan−1 (2x )+ tan−1 (3x )= π4

15. Solve for x : tan−1( x−1x−2 )+ tan−1( tan−1 x+1

x+2 )=¿ π4

¿

16. Solve for x 2 tan−1¿¿

17. Solve for x : tan−1 1−x1+x

=12

tan−1 x ; x>0

18. Prove that tan( π4 + 12

cos−1 ab )+ tan( π4 + 1

2cos−1 a

b )=2ba

19. Solve tan−1 (x+1 )+ tan−1 (x−1 )=tan−1 831

20. Evaluate : sin [ π3 −sin−1 (−12 )]

HOTS : Applications of Integrals

1. Find the area of the region bounded by the curve y2=x and the lines x = 1 , x = 4 and the x-axis .

2. Find the area of the region bounded by y2 = 9x , x = 2 , x=4 and the x – axis in the first quadrant .

3. Find the area of the region bounded by the ellipse x2

16 + y2

9=1

4. Find the area of the region bounded by the ellipse x2

4 + y2

9=1

5. Find the area of the region in the first quadrant enclosed by x-axis , line x=√3 y and the circle

x2+ y2=4

6. Find the area of the region bounded by the curve x2=4 y∧the line x=4 y−2.

7. Find the area of the region bounded by the curve y2=4 x and the lines y=4 x−2.

8. Find the area of the region bounded by y2 = 9x , x = 2 , x=4 and the x – axis in the first quadrant .

9. Find the area of the region bounded by y = x2 and the line x = 2y-1

10. Find the area of the circle 4 x2+4 y2=9which is interior to the parabola x2=4 y .

11. Find the area bounded by the curves (x−1)2+ y2=1∧x2+ y2=1

12. Find the area of the region bounded by the curves y=x2+2 , y=x , x=0∧x=3

13.Using integration find the area of region bounded by the triangle whose vertices are (-1,0) ,

(1,3) and (3,2)

14.Using integration find the area of the triangular region whose sides have the equations

y=2x+1 , y=3x+1∧x=4.

HOTS : DIFFERENTIAL EQUATIONS

1. If y=log [ x+√x2+1¿] , prove t h at(x¿¿2+1) d2 ydx2 +x dy

dx=0 .¿¿

2. Find the particular solution of the differential equation :

(1+e2 x )dy+(1+ y2 )ex dx=0 , given t hat y=1 ,w hen x=0 .

3. Solve the differential equation : dydx

+ ycot x=4 xcosec x , given that y = 0 when x=π2 .

4. Verify whether y=aex+be−x+x2is a solutionof x d2 ydx2 +2 dy

dx−xy+x2−2=0.

5 . Form the DE representing the family of curves y=Acos (x+B) A∧Bare parameters .

6. Form the DE corresponding to y2=a (b−x ) (b+x ) be eliminating parameters a and b.

7. Find the DE of all the circles in the first quadrant which touches the coordinate axes .

8. Form the DE of family of parabolas having vertex at the origin and axis along positive y –axis .

9. Solve the following DE :

a) (1+x2 ) dydx

−x=2 tan−1x b) (x3+x2+x+1 ) dydx

=2x2+x

c) x ( 1+ y2)dx− y (1+x2 )dy=0 d) y−x dydx

=a( y2+ dydx

)

e) dydx

=ex− y+x2e− y f) dydx

= ysin2 x given that y(0) = 1 .

g) xdydx

+ y= y2 h) (1+x)(1+ y¿¿2)dx+ (1+ y ) (1+x2 )dy=0¿

10 . Solve : (x¿¿3−3 x y2)dx=( y¿¿3−3 x2 y )dy¿¿

11. Find the particular solution of the DE dydx=1+ y2

1+x2 when y(0) = 1

12. Find the equation of a curve passing through the point (-2,3) given the slope of the tangent to the

curve at any point (x,y) is 2xy2 . =

13 Find the general solution of the DE dydx

= x2− y2

xy dydx

+ y=sin x .

14. Solve : xdydx

+ y−x+xycot x=0 , x ≠0

15. Solve : ydx+(x− y3 )dy=0.

16. Show that the DE [ xsin2( yx )− y ]dx+ xdy=0is homogeneous. Find the particular solutions of

this DE given that x = 1 and y = π4 .

17. Solve : ( x+1 ) dydx

=2e− y−1 , y=0when x=0 .

18. Show that the DE xdydx

sin ( yx )+x− y sin( yx )=0is homogeneous. Find the particular solutions

of this DE given that x = 1 and y = π2 .

19. Solve : (1+e2x )dy+(1+ y2 )ex dx=0when x=0∧ y=1

20. Find the particular solution of the DE dydx

=1+x+ y+xy , giventhat y=0when x=1 .

21. Solve : (1+x2 ) dydx

+ y=e tan−1x

22. Solve : dydx

+ ycot x=2 cos x , given y=0∧x= π2 .

23. Solve : (x2− y x2)dy+( y2+x2 y2 )dx=0given y=1∧x=1

1. Using integration, Find the area of the region bounded by the line x− y+2=0 , the

curve x=√ y and y – axis .

2. Using integration , prove that the curves y2=4 x∧x2=4 y divide the area of the

square bounded by x=0 , x=4 , y=4∧ y=0into three equal parts .

3. Using the method of integration , find the area of the region bounded by the lines :

2 x+ y=4 ,3 x−2 y=6 , x−3 y+5=0

4. Find the area bounded by the curves y=sin x between the ordinates x = 0 and x = π and

the x – axis .

5. Find the area of the region { (x , y ) : x2≤ y ≤|x|}

6. Using integration , find the area of the region bounded by the parabola y2=4 xand the

circle 4 x2+4 y2=9.

7. Find the area of the region in the first quadrant enclosed by the x-axis , the line y = x

and the circle x2+ y2=32

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