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14.5E: Exercises for Section 14.5 - Mathematics


In exercises 1 - 8, evaluate the triple integrals (displaystyle iiint_E f(x,y,z) , dV) over the solid (E).

1. (f(x,y,z) = z, quad B = ig{(x,y,z), | ,x^2 + y^2 leq 9, quad x leq 0, quad y leq 0, quad 0 leq z leq 1ig})

Answer:
(frac{9pi}{8})

2. (f(x,y,z) = xz^2, space B = ig{(x,y,z), | ,x^2 + y^2 leq 16, space x geq 0, space y leq 0, space -1 leq z leq 1ig})

3. (f(x,y,z) = xy, space B = ig{(x,y,z), | ,x^2 + y^2 leq 1, space x geq 0, space x geq y, space -1 leq z leq 1ig})

Answer:
(frac{1}{8})

4. (f(x,y,z) = x^2 + y^2, space B = ig{(x,y,z), | ,x^2 + y^2 leq 4, space x geq 0, space x leq y, space 0 leq z leq 3ig})

5. (f(x,y,z) = e^{sqrt{x^2+y^2}}, space B = ig{(x,y,z), | ,1 leq x^2 + y^2 leq 4, space y leq 0, space x leq ysqrt{3}, space 2 leq z leq 3 ig})

Answer:
(frac{pi e^2}{6})

6. (f(x,y,z) = sqrt{x^2 + y^2}, space B = ig{(x,y,z), | ,1 leq x^2 + y^2 leq 9, space y leq 0, space 0 leq z leq 1ig})

7. a. Let (B) be a cylindrical shell with inner radius (a) outer radius (b), and height (c) where (0 < a < b) and (c>0). Assume that a function (F) defined on (B) can be expressed in cylindrical coordinates as (F(x,y,z) = f(r) + h(z)), where (f) and (h) are differentiable functions. If (displaystyle int_a^b ar{f} (r) ,dr = 0) and (ar{h}(0) = 0), where (ar{f}) and (ar{h}) are antiderivatives of (f) and (h), respectively, show that (displaystyle iiint_B F(x,y,z) ,dV = 2pi c (bar{f} (b) - a ar{f}(a)) + pi(b^2 - a^2) ar{h} (c).)

b. Use the previous result to show that ( displaystyle iiint_B left(z + sin sqrt{x^2 + y^2} ight) ,dx space dy space dz = 6 pi^2 ( pi - 2),) where (B) is a cylindrical shell with inner radius (pi) outer radius (2pi), and height (2).

8. Let (B) be a cylindrical shell with inner radius (a) outer radius (b) and height (c) where (0 < a < b) and (c > 0). Assume that a function (F) defined on (B) can be expressed in cylindrical coordinates as (F(x,y,z) = f(r) g( heta) f(z)), where (f, space g,) and (h) are differentiable functions. If (displaystyleint_a^b ilde{f} (r) , dr = 0,) where ( ilde{f}) is an antiderivative of (f), show that (displaystyleiiint_B F (x,y,z),dV = [b ilde{f}(b) - a ilde{f}(a)] [ ilde{g}(2pi) - ilde{g}(0)] [ ilde{h}(c) - ilde{h}(0)],) where ( ilde{g}) and ( ilde{h}) are antiderivatives of (g) and (h), respectively.

b. Use the previous result to show that (displaystyleiiint_B z sin sqrt{x^2 + y^2} ,dx space dy space dz = - 12 pi^2,) where (B) is a cylindrical shell with inner radius (pi) outer radius (2pi), and height (2).

In exercises 9 - 12, the boundaries of the solid (E) are given in cylindrical coordinates.

a. Express the region (E) in cylindrical coordinates.

b. Convert the integral (displaystyle iiint_E f(x,y,z) ,dV) to cylindrical coordinates.

9. (E) is bounded by the right circular cylinder (r = 4 sin heta), the (r heta)-plane, and the sphere (r^2 + z^2 = 16).

Answer:

a. (E = ig{(r, heta,z), | ,0 leq heta leq pi, space 0 leq r leq 4 sin heta, space 0 leq z leq sqrt{16 - r^2}ig})

b. (displaystyleint_0^{pi} int_0^{4 sin heta} int_0^{sqrt{16-r^2}} f(r, heta, z) r , dz space dr space d heta)

10. (E) is bounded by the right circular cylinder (r = cos heta), the (r heta)-plane, and the sphere (r^2 + z^2 = 9).

11. (E) is located in the first octant and is bounded by the circular paraboloid (z = 9 - 3r^2), the cylinder (r = sqrt{3}), and the plane (r(cos heta + sin heta) = 20 - z).

Answer:

a. (E = ig{(r, heta,z) , | , 0 leq heta leq frac{pi}{2}, space 0 leq r leq sqrt{3}, space 9 - r^2 leq z leq 10 - r(cos heta + sin heta)ig})

b. (displaystyleint_0^{pi/2} int_0^{sqrt{3}} int_{9-r^2}^{10-r(cos heta + sin heta)} f(r, heta,z) r space dz space dr space d heta)

12. (E) is located in the first octant outside the circular paraboloid (z = 10 - 2r^2) and inside the cylinder (r = sqrt{5}) and is bounded also by the planes (z = 20) and ( heta = frac{pi}{4}).

In exercises 13 - 16, the function (f) and region (E) are given.

a. Express the region (E) and the function (f) in cylindrical coordinates.

b. Convert the integral (displaystyle iiint_B f(x,y,z) ,dV) into cylindrical coordinates and evaluate it.

13. (f(x,y,z) = x^2 + y^2), (E = ig{(x,y,z), | ,0 leq x^2 + y^2 leq 9, space x geq 0, space y geq 0, space 0 leq z leq x + 3ig})

Answer:

a. (E = ig{(r, heta,z), | ,0 leq r leq 3, space 0 leq heta leq frac{pi}{2}, space 0 leq z leq r space cos heta + 3ig},)
(f(r, heta,z) = frac{1}{r space cos heta + 3})

b. (displaystyle int_0^3 int_0^{pi/2} int_0^{r space cos heta+3} frac{r}{r space cos heta + 3} , dz space d heta space dr = frac{9pi}{4})

14. (f(x,y,z) = x^2 + y^2, space E = ig{(x,y,z) |0 leq x^2 + y^2 leq 4, space y geq 0, space 0 leq z leq 3 - x ig})

15. (f(x,y,z) = x, space E = ig{(x,y,z), | ,1 leq y^2 + z^2 leq 9, space 0 leq x leq 1 - y^2 - z^2ig})

Answer:

a. (y = r space cos heta, space z = r space sin heta, space x = z,space E = ig{(r, heta,z), | ,1 leq r leq 3, space 0 leq heta leq 2pi, space 0 leq z leq 1 - r^2ig}, space f(r, heta,z) = z);

b. (displaystyle int_1^3 int_0^{2pi} int_0^{1-r^2} z r space dz space d heta space dr = frac{356 pi}{3})

16. (f(x,y,z) = y, space E = ig{(x,y,z), | ,1 leq x^2 + z^2 leq 9, space 0 leq y leq 1 - x^2 - z^2 ig})

In exercises 17 - 24, find the volume of the solid (E) whose boundaries are given in rectangular coordinates.

17. (E) is above the (xy)-plane, inside the cylinder (x^2 + y^2 = 1), and below the plane (z = 1).

Answer:
(pi)

18. (E) is below the plane (z = 1) and inside the paraboloid (z = x^2 + y^2).

19. (E) is bounded by the circular cone (z = sqrt{x^2 + y^2}) and (z = 1).

Answer:
(frac{pi}{3})

20. (E) is located above the (xy)-plane, below (z = 1), outside the one-sheeted hyperboloid (x^2 + y^2 - z^2 = 1), and inside the cylinder (x^2 + y^2 = 2).

21. (E) is located inside the cylinder (x^2 + y^2 = 1) and between the circular paraboloids (z = 1 - x^2 - y^2) and (z = x^2 + y^2).

Answer:
(pi)

22. (E) is located inside the sphere (x^2 + y^2 + z^2 = 1), above the (xy)-plane, and inside the circular cone (z = sqrt{x^2 + y^2}).

23. (E) is located outside the circular cone (x^2 + y^2 = (z - 1)^2) and between the planes (z = 0) and (z = 2).

Answer:
(frac{4pi}{3})

24. (E) is located outside the circular cone (z = 1 - sqrt{x^2 + y^2}), above the (xy)-plane, below the circular paraboloid, and between the planes (z = 0) and (z = 2).

25. [T] Use a computer algebra system (CAS) to graph the solid whose volume is given by the iterated integral in cylindrical coordinates (displaystyle int_{-pi/2}^{pi/2} int_0^1 int_{r^2}^r r , dz , dr , d heta.) Find the volume (V) of the solid. Round your answer to four decimal places.

Answer:

(V = frac{pi}{12} approx 0.2618)

26. [T] Use a CAS to graph the solid whose volume is given by the iterated integral in cylindrical coordinates (displaystyle int_0^{pi/2} int_0^1 int_{r^4}^r r , dz , dr , d heta.) Find the volume (E) of the solid. Round your answer to four decimal places.

27. Convert the integral (displaystyleint_0^1 int_{-sqrt{1-y^2}}^{sqrt{1-y^2}} int_{x^2+y^2}^{sqrt{x^2+y^2}} xz space dz space dx space dy) into an integral in cylindrical coordinates.

Answer:
(displaystyleint_0^1 int_0^{pi} int_{r^2}^r zr^2 space cos heta , dz space d heta space dr)

28. Convert the integral (displaystyle int_0^2 int_0^y int_0^1 (xy + z) , dz space dx space dy) into an integral in cylindrical coordinates.

In exercises 29 - 32, evaluate the triple integral (displaystyle iiint_B f(x,y,z) ,dV) over the solid (B).

29. (f(x,y,z) = 1, space B = ig{(x,y,z), | ,x^2 + y^2 + z^2 leq 90, space z geq 0ig})

[Hide Solution]

Answer:
(180 pi sqrt{10})

30. (f(x,y,z) = 1 - sqrt{x^2 + y^2 + z^2}, space B = ig{(x,y,z), | ,x^2 + y^2 + z^2 leq 9, space y geq 0, space z geq 0ig})

31. (f(x,y,z) = sqrt{x^2 + y^2}, space B ) is bounded above by the half-sphere (x^2 + y^2 + z^2 = 9) with (z geq 0) and below by the cone (2z^2 = x^2 + y^2).

Answer:
(frac{81pi(pi - 2)}{16})

32. (f(x,y,z) = sqrt{x^2 + y^2}, space B ) is bounded above by the half-sphere (x^2 + y^2 + z^2 = 16) with (z geq 0) and below by the cone (2z^2 = x^2 + y^2).

33. Show that if (F ( ho, heta,varphi) = f( ho)g( heta)h(varphi)) is a continuous function on the spherical box (B = ig{( ho, heta,varphi), | ,a leq ho leq b, space alpha leq heta leq eta, space gamma leq varphi leq psiig}), then (displaystyleiiint_B F space dV = left(int_a^b ho^2 f( ho) space dr ight) left( int_{alpha}^{eta} g ( heta) space d heta ight)left( int_{gamma}^{psi} h (varphi) space sin varphi space dvarphi ight).)

34. A function (F) is said to have spherical symmetry if it depends on the distance to the origin only, that is, it can be expressed in spherical coordinates as (F(x,y,z) = f( ho)), where ( ho = sqrt{x^2 + y^2 + z^2}). Show that (displaystyleiiint_B F(x,y,z) ,dV = 2pi int_a^b ho^2 f( ho) ,d ho,) where (B) is the region between the upper concentric hemispheres of radii (a) and (b) centered at the origin, with (0 < a < b) and (F) a spherical function defined on (B).

Use the previous result to show that (displaystyleiiint_B (x^2 + y^2 + z^2) sqrt{x^2 + y^2 + z^2} dV = 21 pi,) where (B = ig{(x,y,z), | ,1 leq x^2 + y^2 + z^2 leq 2, space z geq 0ig}).

35. Let (B) be the region between the upper concentric hemispheres of radii a and b centered at the origin and situated in the first octant, where (0 < a < b). Consider F a function defined on B whose form in spherical coordinates (( ho, heta,varphi)) is (F(x,y,z) = f( ho)cos varphi). Show that if (g(a) = g(b) = 0) and (displaystyleint_a^b h ( ho) , d ho = 0,) then (displaystyleiiint_B F(x,y,z),dV = frac{pi^2}{4} [ah(a) - bh(b)],) where (g) is an antiderivative of (f) and (h) is an antiderivative of (g).

Use the previous result to show that (displaystyle iiint_B = frac{z cos sqrt{x^2 + y^2 + z^2}}{sqrt{x^2 + y^2 + z^2}} , dV = frac{3pi^2}{2},) where (B) is the region between the upper concentric hemispheres of radii (pi) and (2pi) centered at the origin and situated in the first octant.

In exercises 36 - 39, the function (f) and region (E) are given.

a. Express the region (E) and function (f) in cylindrical coordinates.

b. Convert the integral (displaystyle iiint_B f(x,y,z), dV) into cylindrical coordinates and evaluate it.

36. (f(x,y,z) = z; space E = ig{(x,y,z), | ,0 leq x^2 + y^2 + z^2 leq 1, space z geq 0ig})

37. (f(x,y,z) = x + y; space E = ig{(x,y,z), | ,1 leq x^2 + y^2 + z^2 leq 2, space z geq 0, space y geq 0ig})

Answer:

a. (f( ho, heta, varphi) = ho space sin varphi space (cos heta + sin heta), space E = ig{( ho, heta,varphi), | ,1 leq ho leq 2, space 0 leq heta leq pi, space 0 leq varphi leq frac{pi}{2}ig});

b. (displaystyle int_0^{pi} int_0^{pi/2} int_1^2 ho^3 cos varphi space sin varphi space d ho space dvarphi space d heta = frac{15pi}{8})

38. (f(x,y,z) = 2xy; space E = ig{(x,y,z), | ,sqrt{x^2 + y^2} leq z leq sqrt{1 - x^2 - y^2}, space x geq 0, space y geq 0ig})

39. (f(x,y,z) = z; space E = ig{(x,y,z), | ,x^2 + y^2 + z^2 - 2x leq 0, space sqrt{x^2 + y^2} leq zig})

Answer:

a. (f( ho, heta,varphi) = ho space cos varphi; space E = ig{( ho, heta,varphi), | ,0 leq ho leq 2 space cos varphi, space 0 leq heta leq frac{pi}{2}, space 0 leq varphi leq frac{pi}{4}ig});

b. (displaystyleint_0^{pi/2} int_0^{pi/4} int_0^{2 space cos varphi} ho^3 sin varphi space cos varphi space d ho space dvarphi space d heta = frac{7pi}{24})

In exercises 40 - 41, find the volume of the solid (E) whose boundaries are given in rectangular coordinates.

40. (E = ig{ (x,y,z), | ,sqrt{x^2 + y^2} leq z leq sqrt{16 - x^2 - y^2}, space x geq 0, space y geq 0ig})

41. (E = ig{ (x,y,z), | ,x^2 + y^2 + z^2 - 2z leq 0, space sqrt{x^2 + y^2} leq zig})

Answer:
(frac{pi}{4})

42. Use spherical coordinates to find the volume of the solid situated outside the sphere ( ho = 1) and inside the sphere ( ho = cos varphi), with (varphi in [0,frac{pi}{2}]).

43. Use spherical coordinates to find the volume of the ball ( ho leq 3) that is situated between the cones (varphi = frac{pi}{4}) and (varphi = frac{pi}{3}).

Answer:
(9pi (sqrt{2} - 1))

44. Convert the integral (displaystyle int_{-4}^4 int_{-sqrt{16-y^2}}^{sqrt{16-y^2}} int_{-sqrt{16-x^2-y^2}}^{sqrt{16-x^2-y^2}} (x^2 + y^2 + z^2) , dz , dx , dy) into an integral in spherical coordinates.

45. Convert the integral (displaystyle int_0^4 int_0^{sqrt{16-x^2}} int_{-sqrt{16-x^2-y^2}}^{sqrt{16-x^2-y^2}} (x^2 + y^2 + z^2)^2 , dz space dy space dx) into an integral in spherical coordinates.

Answer:
(displaystyleint_0^{pi/2} int_0^{pi/2} int_0^4 ho^6 sin varphi , d ho , dphi , d heta)

47. [T] Use a CAS to graph the solid whose volume is given by the iterated integral in spherical coordinates (displaystyle int_{pi/2}^{pi} int_{5pi}^{pi/6} int_0^2 ho^2 sin varphi space d ho space dvarphi space d heta.) Find the volume (V) of the solid. Round your answer to three decimal places.

Answer:

(V = frac{4pisqrt{3}}{3} approx 7.255)

48. [T] Use a CAS to graph the solid whose volume is given by the iterated integral in spherical coordinates as (displaystyle int_0^{2pi} int_{3pi/4}^{pi/4} int_0^1 ho^2 sin varphi space d ho space dvarphi space d heta.) Find the volume (V) of the solid. Round your answer to three decimal places.

49. [T] Use a CAS to evaluate the integral (displaystyle iiint_E (x^2 + y^2) , dV) where (E) lies above the paraboloid (z = x^2 + y^2) and below the plane (z = 3y).

Answer:
(frac{343pi}{32})

50. [T]

a. Evaluate the integral (displaystyle iiint_E e^{sqrt{x^2+y^2+z^2}}, dV,) where (E) is bounded by spheres (4x^2 + 4y^2 + 4z^2 = 1) and (x^2 + y^2 + z^2 = 1).

b. Use a CAS to find an approximation of the previous integral. Round your answer to two decimal places.

51. Express the volume of the solid inside the sphere (x^2 + y^2 + z^2 = 16) and outside the cylinder (x^2 + y^2 = 4) as triple integrals in cylindrical coordinates and spherical coordinates, respectively.

Answer:
(displaystyle int_0^{2pi}int_2^4int_{−sqrt{16−r^2}}^{sqrt{16−r^2}}r,dz,dr,dθ) and (displaystyle int_{pi/6}^{5pi/6}int_0^{2pi}int_{2csc phi}^{4} ho^2sin ho , d ho , d heta , dphi)

52. Express the volume of the solid inside the sphere (x^2 + y^2 + z^2 = 16) and outside the cylinder (x^2 + y^2 = 4) that is located in the first octant as triple integrals in cylindrical coordinates and spherical coordinates, respectively.

53. The power emitted by an antenna has a power density per unit volume given in spherical coordinates by (p( ho, heta,varphi) = frac{P_0}{ ho^2} cos^2 heta space sin^4 varphi), where (P_0) is a constant with units in watts. The total power within a sphere (B) of radius (r) meters is defined as (displaystyle P = iiint_B p( ho, heta,varphi) , dV.) Find the total power (P).

Answer:
(P = frac{32P_0 pi}{3}) watts

54. Use the preceding exercise to find the total power within a sphere (B) of radius 5 meters when the power density per unit volume is given by (p( ho, heta,varphi) = frac{30}{ ho^2} cos^2 heta sin^4 varphi).

55. A charge cloud contained in a sphere (B) of radius (r) centimeters centered at the origin has its charge density given by (q(x,y,z) = ksqrt{x^2 + y^2 + z^2}frac{mu C}{cm^3}), where (k > 0). The total charge contained in (B) is given by (displaystyle Q = iiint_B q(x,y,z) , dV.) Find the total charge (Q).

Answer:
(Q = kr^4 pi mu C)

56. Use the preceding exercise to find the total charge cloud contained in the unit sphere if the charge density is (q(x,y,z) = 20 sqrt{x^2 + y^2 + z^2} frac{mu C}{cm^3}).

Contributors

Gilbert Strang (MIT) and Edwin “Jed” Herman (Harvey Mudd) with many contributing authors. This content by OpenStax is licensed with a CC-BY-SA-NC 4.0 license. Download for free at http://cnx.org.


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5E-14.1421. Identification Card - Training Verification

1 (1) The licensee shall maintain written training records 9 for both the initial five (5) day (40 hour) training required in Section 22 482.091(3), F.S., 24 and the continuing training required in Section 31 482.091(10), F.S., 33 on all identification cardholders within their employ and make those records available during routine inspection or upon request of the department. Licensees must maintain the training record for at least a two year period. 67 The training required for Section 72 482.091(3), F.S., 74 must be conducted by a certified operator or a person under the supervision of the certified operator in charge who has been designated in writing as responsible for training. The 40 hour initial training shall be verified by:

112 (a) Completion of the Verification Record of Initial Employee Training, (FDACS-13665, Rev. 10/15), which is 127 hereby adopted and incorporated by reference and 134 available online at 137 https://www.flrules.org/Gateway/reference.asp?No=Ref-07322 , 139 or

140 (b) A written record of 40 hours of attendance in a training course with a written course syllabus and copies of all training materials used in the course available for department inspection.

172 (2) The department will accept either of the following as documentation of verifiable training as required under Section 190 482.091(10), F.S. 192 :

193 (a) Written record of attendance on Identification Cardholder Training Verification, (FDACS-13662, Rev. 10/15), which is incorporated by reference and available online at 215 https://www.flrules.org/Gateway/reference.asp?No=Ref-07323 217 and provided by the licensee or trainer, with a complete copy of all training materials used during the training session that covers the training topics required by Section 245 482.091(10), F.S., 247 or

248 (b) Written record of attendance at a department approved certified operator continuing education course on the Record of Attendance for Continuing Education Units (CEUs), (FDACS-13325, Rev. 05/04), as adopted in Rule 279 5E-9.029, 280 F.A.C., and provided by the trainer, only if the course content covers the training topics as required by Section 299 482.091(10), F.S.

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464 (4) Licensees or certified operators applying for Wood-Destroying Organism Inspector Identification cards for employees in compliance with Section 482 482.091(9), F.S., 484 may complete the application process online at http://www.FreshFromFlorida.com, or shall submit the Special Training to Perform Wood-Destroying Organisms Inspections and Control Training Verification Record, (FDACS-13642, Rev. 10/15), which is hereby adopted and incorporated by reference and available online at 523 https://www.flrules.org/Gateway/reference.asp?No=Ref-07324 , 525 to 526 the address as instructed on the form 533 .

534 (5) Fumigation employees that participate in fumigations using a residential fumigant must complete Initial and Annual Stewardship Training as required by the label and Stewardship Policy for the residential fumigant(s) used as defined in Rule Chapter 5E-2, F.A.C. Information verifying Continuing Education Units for Stewardship Training (Initial or Annual) for all residential fumigant(s) used by the fumigation employee must be submitted to the department annually through 600 http://ceu.freshfromflorida.com , 601 or by submtting the Record of Attendance for Continuing Education Units (CEUs), Form FDACS-13325, Rev. 10/13, which is incorporated by reference in Rule 624 5E-9.029, 625 F.A.C., by 627 electronic mail to [email protected], 631 or facsimile to (850)617-7968 by the renewal date of the fumigation employee’s identification card.

645 (6) The licensee or certified operator in charge of fumigation must apply for an identification card that identifies that employee as having received the training specified in paragraph 673 5E-14.108(3)(b), 674 F.A.C., to assist as the second trained person during the use of a residential fumigant as described in subsection 693 5E-14.108(2), 694 F.A.C. The application for such identification card with a fumigation endorsement must be accompanied by an affidavit, signed by the prospective identification cardholder and by the licensee or certified operator in charge for fumigation, which states that the prospective identification cardholder has received the training required by paragraph 742 5E-14.108(3)(b), 743 F.A.C. Application shall be made online at http://www.FreshFromFlorida.com, or by submitting the Special Training to Perform Fumigation Affidavit (FDACS-13002, 01/17), which is hereby adopted and incorporated by reference and available online at 775 https://www.flrules.org/Gateway/reference.asp?No=Ref-07905 , 777 to the address as instructed on the form in order to receive a Fumigation Identification Card endorsement on the employee’s identification card as required by Section 803 482.091, F.S. 805 The identification cardholder must complete stewardship training as required by the label and Stewardship Policy for the residential fumigant(s) used within 60 days of receiving an identification card with a fumigation endorsement.

837 Rulemaking Authority 839 482.051, 840 482.091, 841 570.07(23) FS. 843 Law Implemented 845 482.091, 846 482.151, 847 570.07(22) FS. 849 History–New 6-12-02, Amended 2-24-09, 1-9-17, 5-7-17.


14.5E: Exercises for Section 14.5 - Mathematics

NCERT Solutions for Class 10 Maths

Solutions for all the questions from Maths NCERT, Class 10th

EUCLIDS DIVISION ALGORITHM

EUCLIDS DIVISION ALGORITHM

Doubtnut.com presents complete NCERT solutions for class 10 maths in video tutorial format. To make it easier for you to learn and understand maths the video tutorials are prepared by our esteemed mathematicians from the renowned IITs of India. These are some of the best online lectures on maths, where our experts have discussed a wide array of class 10 maths topics. Class 10th Maths as a subject is vast therefore, we’ve listed every important topic by segregating its chapters into subsequent exercises. In our video tutorials, we've discussed Real Numbers, Polynomials, Pair of Linear Equations in Two Variables, Quadratic Equations, Arithmetic Progressions, Triangles, Coordinate Geometry, Trigonometry, Some Applications of Trigonometry, Circles, Constructions, Area Related to Circles, Surface Areas and Volumes, Statistics and Probability and more…

Cengage Chapterwise Maths Solutions

PAIR OF LINEAR EQUATIONS IN TWO VARIABLES

INTRODUCTION TO TRIGONOMETRY

SOME APPLICATIONS OF TRIGONOMETRY

SURFACE AREAS AND VOLUMES

Key Highlights of NCERT Solutions for Class 10 Maths:

1. Strictly as per the CBSE examination guidelines.

2. Fully solved exercises of NCERT Math textbook for Class 10.

3. The questions have been answered by the IITians and best mathematicians of the country.

4. Solutions are as per the CBSE marking scheme.

5. Videos are full of tips and tricks to help you gain the competitive edge to ace the board examination.

Diagrams are given to help the students in visualizing the solutions.

All the questions from each chapter are covered.

Videos are enriched with expert advice, math's tips and step-by-step maths solutions to give a proper justification to every math's problem. It would provide students detailed insights on every important topic. The NCERT solutions for class 10 maths in Hindi are prepared to keep the student's need in mind. You can easily learn either using your mobile or personal computer. The given video tutorials provide students detailed insights on every important topic.

Do you know natural numbers, whole numbers, integers, fractions, rational numbers and irrational numbers are all real numbers? Any real number that can be found on the number line is known as the real number. The which we use and apply in the real world calculations are the real numbers. In this chapter, you'll be learning some interesting concepts related to a real number and their useful applications. The video tutorial covers every important topic related to real numbers and gives some great examples. The exercise wise NCERT solution class 10 maths book are listed below: Exercise 1.1 Introduction Exercise 1.2 Euclid’s Division Lemma Exercise 1.3 The Fundamental Theorem of Arithmetic Exercise 1.4 Revisiting Irrational Numbers Exercise 1.5 Revisiting Rational Numbers and Their Decimal Expansions Exercise 1.6 Summary

In this chapter, you'll be learning about the polynomials in-depth. Starting from the exercise 2.1 Introduction to polynomials, you’ll be learning how to simplify and evaluate polynomials. Then you’ll learn about the zeros or roots of polynomials, roots of quadratic equations and cubic equation and it’s coefficient. In the final exercise 2.5, you'll learn about how to do long division of the polynomials. Exercise-wise solutions we’ve covered include: Exercise 2.1 Introduction Exercise 2.2 Geometrical Meaning Of The Zeroes Of A Polynomial Exercise 2.3 Relationship Between Zeroes And Coefficients Of A Polynomial Exercise 2.4 Division Algorithm For Polynomials Exercise 2.5 Polynomials Exercise 2.6 Summary

Chapter 3: Pair of Linear Equations in Two Variables

In NCERT class 10 Chapter 3, you'll be learning about the interpretation of linear equations in two variables. With given NCERT solutions for class 10, maths students can easily model the linear equations into real-world problems. You’ll enjoy learning to solve the linear equation problems both algebraically and graphically. In the algebraic method, you will learn some important concepts like the elimination method, substitution method, and cross-multiplication method. Then finally we will move on to equations reducible to a pair of linear equations in two variables. The exercise-wise NCERT solutions we’ve covered include: Exercise 3.1 Introduction Exercise 3.2 Pair of Linear Equations in Two Variables Exercise 3.3 Graphical Method of Solution of a Pair of Linear Equations Exercise 3.4 Algebraic Methods of Solving a Pair of Linear Equations Exercise 3.4.1 Substitution Method Exercise 3.4.2 Elimination Method Exercise 3.4.3 Cross-Multiplication Method Exercise 3.5 Equations Reducible To a Pair of Linear Equations in Two Variables Exercise 3.6 Summary

Chapter 4: Quadratic Equations

So far you have learned about the linear equations and linear equations in two variables. In chapter 4, you'll be learning some new concepts, which includes variables raise to the second power and how to take square roots on both sides. You'll also learn to solve the factored equation like (x-1) (x+3) = 0 and using the factorization method, solution of quadratic equation by completing the square and finally the nature of roots. Listed below are the exercise wise NCERT solutions for chapter 4 quadratic equations: Exercise 4.1 Introduction Exercise 4.2 Quadratic Equations Exercise 4.3 Solution of a Quadratic Equation by Factorization Exercise 4.4 Solution of a Quadratic Equation by Completing the Square Exercise 4.5 Nature of Roots Exercise 4.6 Summary

Chapter 5: Arithmetic Progressions

Find here detailed and in-depth answers to Class 10 Maths Chapter 5 Arithmetic Progressions (AP). If the difference between two consecutive terms is constant in a progression, it is known as the arithmetic progression. In this lesson five, you’ll be introduced important concepts of arithmetic progression and you’ll be studying about constructing an arithmetic progression (A.P). After learning the basics of AP, you can learn to calculate the Nth term of an AP. Then you will see how to find the sum of n terms of an AP. It is interesting and fun to learn the concept of mathematics. The exercise wise NCERT solutions for chapter 5 Arithmetic Progressions we’ve covered are: Exercise 5.1 Introduction Exercise 5.2 Arithmetic Progressions Exercise 5.3 Nth Term of An AP Exercise 5.4 Sum Of First N Terms Of An AP Exercise 5.5 Summary

Find here complete study material on class 10 chapter 6 triangles. Triangle is an interesting 3 cornered shape that has some unique properties to explore. In this chapter 6 triangles, you'll be studying all about triangles, similarity criterion of triangles and their properties. Itemized below are the exercises that we’ve covered here: Exercise 6.1 Introduction Exercise 6.2 Similar Figures Exercise 6.3 Similarity Of Triangles Exercise 6.4 Criteria for Similarity of Triangles Exercise 6.5 Areas of Similar Triangles Exercise 6.6 Pythagoras Theorem Exercise 6.7 Summary

Chapter 7: Coordinate Geometry

Get Free NCERT Solutions for Class 10 Maths Chapter 7 Coordinate Geometry. In lesson 7 you’ll learn about finding distances between the two points whose coordinates are given. You’ll also learn how to find the coordinates of the point using the distance formula, section formula, area of triangle etc. Given NCERT Solutions were prepared by experienced mathematicians to helps students build a stronger foundation in mathematics. Detailed exercise wise answers to Chapter 7 Maths Class 10 Coordinate Geometry are provided below: Exercise 7.1 Introduction Exercise 7.2 Distance Formula Exercise 7.3 Section Formula Exercise 7.4 Area of a Triangle Exercise 7.5 Summary

Chapter 8: Introduction to Trigonometry

In chapter eight introductions to trigonometry, you'll learn about some important concepts like trigonometric ratios, trigonometric ratios of some specific angles, Trigonometric Ratios of Complementary Angles and finally Trigonometric Identities. This lesson is restricted to the acute angles only, however, ratios can be extended to other angles as well. We will also define and calculate trigonometric ratios and some simple identities involving these ratios, called trigonometric identities. The exercise wise NCERT solution for Class 10 Maths Chapter 8 Introduction to Trigonometry we’ve covered includes: Exercise 8.1 Introduction Exercise 8.2 Trigonometric Ratios Exercise 8.3 Trigonometric Ratios Of Some Specific Angles Exercise 8.4 Trigonometric Ratios Of Complementary Angles Exercise 8.5 Trigonometric Identities Exercise 8.6 Summary

Chapter 9: Some Applications of Trigonometry

In this lesson, you’ll be learning about the practical application of trigonometry. The NECRT solutions will teach about calculating the heights and distances of various objects, without actually quantifying them. The exercise covered includes: Exercise 9.1 Introduction Exercise 9.2 Heights and Distances Exercise 9.3 Summary

With NCERT Class 10 chapter 10 Circles you’ll be learning various types of situations that can arise when a circle and a line are given in a plane. You'll also learn the concept of a tangent and number of tangents from a point on a circle. The exercise wise NCERT solution for Class 10 Maths Chapter 10 Circles that we’ve covered includes: Exercise 10.1 Introduction Exercise 10.2 Tangent to a circle Exercise 10.3 Number of tangents from a point on a circle Exercise 10.4 Summary

Class 10 Chapter 11 constructions are an important part of the geometry that has some useful real-world applications. In this chapter, you be learning about various types of constructions in geometry. The exercise wise NCERT solutions we’ve covered are listed below: Exercise 11.1 Introduction Exercise 11.2 Division of A Line Segment Exercise 11.3 Construction Of Tangents To A Circle Exercise 11.4 Summary

Chapter 12: Area Related to Circles

Chapter 12 will begin with the concepts of circumference and area of a circle. Then you'll be shifted to the area of sector and segment of a circle and apply this you’ll be extending your knowledge to sector and segment of circles. In the last exercise, you are to learn about the areas of combinations of plane figures. Here are the exercise wise solutions that we’ve included: Exercise 12.1 Introduction Exercise 12.2 Perimeter and area of a circle – a review Exercise 12.3 Areas of sector and segment of a circle Exercise 12.4 Areas of combinations of plane figures Exercise 12.5 Summary

Chapter 13: Surface Areas and Volumes

NCERT chapter 13 surface areas and volumes are an important part of geometry in which you'll be learning to calculate the surface area, volume, or perimeter of a variety of geometrical solid shapes such as circles, rectangle, pyramid, cube, triangle etc. Each has its own specific formulas and method to meet the solution. Listed below are the exercise wise NCERT solutions for chapter 13 surface areas and volumes: Exercise 13.1 Introduction Exercise 13.2 Surface Area of a Combination Of Solids Exercise 13.3 Volume of a Combination Of Solids Exercise 13.4 Conversion of Solid from One Shape to Another Exercise 13.5 Frustum of a Cone Exercise 13.6 Summary

In this lesson, you’ll be learning about the grouped data such as mean, median and mode and the concept of cumulative frequency. The NCERT solutions will provide a comprehensive and in-depth study of the statistics. Find here detailed answers to Chapter 14 Maths Class 10 Statistics: Exercise 14.1 Introduction Exercise 14.2 Mean of Grouped Data Exercise 14.3 Mode of Grouped Data Exercise 14.4 Median of Grouped Data Exercise 14.5 Graphical Representation Of Cumulative Frequency Distribution Exercise 14.6 Summary

Probability is the final chapter of NCERT class 10th maths books. In this chapter, you'll be introduced to theoretical, also known as the classical probability of an event. The video tutorials talk about some of the very simple problems based on the concept of probability. Listed below are the exercise wise solutions that we’ve covered: Exercise 15.1 Introduction Exercise 15.2 A Theoretical Approach Exercise 15.3 Summary

Keep visiting DoubtNut for latest NCERT Solutions for Class 10 Maths as well as other interactive study material!


Advanced Calculus I. Math 451, Fall 2011

Class meets: MWF 2:10 - 3:00 pm in 4088 EH.

Office hours: M: 12:45 - 2:00 pm, W: 10:45 am - 12:00 noon, in 4844 EH. Please do come to my office hours, I will be there to help you. It is an effective way to improve your command of the material. You can also grab me after a class if you have a quick question. I won't be able to hold office hours at any other times than those posted. Instead you can come to the office hours of Professor Mark Rudelson who is teaching a different section of Math 451 this Fall. Professor Rudelson's office hours are T 2:00 - 3:00 pm, W: 5:30 - 7:30 pm, in 3834 EH. We have an agreement with Prof. Rudelson so you don't need to ask him in advance, just come.

Prerequisites: A thorough understanding of Calculus and one of 217, 312, 412.

Course Description: This course has two complementary goals: (1) a rigorous development of the fundamental ideas of Calculus, and (2) a further development of the student's ability to deal with abstract mathematics and mathematical proofs. The key words here are rigor and proof almost all of the material of the course consists in understanding and constructing definitions, theorems (propositions, lemmas, etc.), and proofs. This is considered one of the more difficult among the undergraduate mathematics courses, and students should be prepared to make a strong commitment to the course. In particular, it is strongly recommended that some course which requires proofs (such as Math 412) be taken before Math 451.

Textbook: Kenneth A. Ross, Elementary analysis: the theory of calculus. Springer, Corr. 10, 1998. ISBN: 9780387904597. The course covers most of the material in the book except the starred (optional) sections, in the following order (tentative): 1 - 20, 28, 29, 32 - 34, 23 - 26, 31, concluding with a full week of review.

  • Homework (25%). Will be assigned every class. Due every Friday before class. One homework with the lowest score will be dropped. Scroll down to see the homework. (20%). October 26, in class. Covers the material in Sections 1 - 12 that we covered in class. Books and calculators are allowed. Solutions. (20%), take-home. Due November 21 at the beginning of class. Solutions. (35%). Friday, December 16, 1:30 - 3:30 pm, in class. Covers the whole course. Books and calculators are allowed. Solutions.Warning: Statement of Problem 3 is false.

Missing/late work: Missed exams - there will be no make-up for the exams for any reason. A missed midterm exam counts as zero points, with the following exception. If you miss a midterm exam due to a documented medical or family emergency, the exam's weight will be added to the weight of the final exam. Late homework (and late take-home exam) can not be accepted. In extenuating circumstances, you may e-mail me your scanned or typed homework (as a single pdf file) on the day when the homework is collected by 8:00 p.m.

Lecture Schedule and Homework: It is a useful practice to read ahead the sections to be covered. Solutions are posted for some (but not all) homework problems. Please come to the office hours to discuss other problems.


14.5E: Exercises for Section 14.5 - Mathematics

Latest version of the final adopted rule presented in Florida Administrative Code (FAC):

Effective Date: 5/7/2017
History Notes: Rulemaking Authority 482.051, 570.07(23) FS. Law Implemented 482.051(4) FS. History–New 1-1-77, Amended 6-27-79, 6-22-83, 10-25-90, Formerly 10D-55.110, Amended 7-5-95, 9-17-08, 9-6-10, 5-7-17.
References in this version: Ref-08120 Notification of Fumigation (FDACS-13667, Rev. 03/17
History of this Rule since Jan. 6, 2006
Notice /
Adopted
Section Description ID Publish
Date
Proposed
5E-14.1025
.
Proposed changes consist primarily of updates to program forms and website URL references compelled by substantive form modifications and the Department’s transition to the FDACS.gov domain. Further proposed changes include . 24434511 4/20/2021
Vol. 47/76
Development
5E-14.1025
.
Proposed changes consist primarily of updates to program forms and website URL references compelled by substantive form modifications and the Department’s transition to the FDACS.gov domain. Further proposed changes include . 24408030 4/13/2021
Vol. 47/71
Final
5E-14.110
Fumigation Requirements - Notices 18884462 Effective:
05/07/2017
Correction
5E-14.110
Fumigation Requirements - Notices 18797841 3/30/2017
Vol. 43/62
Change
5E-14.108
.
Fumigation Requirements - General Fumigation, Fumigation Requirements - Notices, Responsibilities and Duties - Records, Reports, Advertising, Applications, Identification Card - Training Verification 18714712 3/16/2017
Vol. 43/52
Correction
5E-14.102
.
Definitions, Prohibited Acts, Fumigation Requirements - General Fumigation, Fumigation Requirements - Notices, Responsibilities and Duties - Records, Reports, Advertising, Applications, Identification . 18714130 3/16/2017
Vol. 43/52
Proposed
5E-14.102
.
The purpose of this rulemaking is to implement new statutory authority to adopt safety procedures for the clearance of residential structures after a fumigation. The effect will be improved structural fumigation safety and . 18487829 1/13/2017
Vol. 43/09
Workshop
5E-14.102
.
Registrant stewardship training requirements, continuing education training requirements, quality assurance reviews, and stop sale of product. .
September 23, 2016, 10:00 a.m.
Mid Florida Research and Education Center, 2725 S. Binion Road, .
17957530 9/1/2016
Vol. 42/171
Development
5E-14.102
.
Proposed changes to Chapter 5E-14, F.A.C., are in response to statutory changes that will improve structural fumigation safety and regulatory oversight. The effect of this rulemaking will: prohibit performing fumigation . 17754606 7/14/2016
Vol. 42/136
Final
5E-14.110
Fumigation Requirements - Notices 9065055 Effective:
09/06/2010
Proposed
5E-14.110
.
To provide electronic submission of required 24 hour prior notice of structural fumigations to the Department and to clarify and delineate precautionary language directing fumigator’s to visually inspect and secure the space . 8829830 7/2/2010
Vol. 36/26
Development
5E-14.110
.
To provide an electronic web-based means of complying with the requirement to notify the Department of a structural fumigation 24 hours in advance and to clarify precautions requiring inspections and sealing between fumigated . 8183228 1/22/2010
Vol. 36/03
Final
5E-14.110
Fumigation Requirements - Notices 6116352 Effective:
09/17/2008
Change
5E-14.102
.
Definitions, Contractual Agreements in Public's Interest - Control and Preventive Treatment for Wood-Destroying Organisms, Fumigation Requirements - Notices, Fumigation Requirements - Application Restrictions . 5852609 7/18/2008
Vol. 34/29
Proposed
5E-14.102
.
To clarify the definition of a “connected structure” and delineate requirements for structural connections which have not previously been provided in rule for fumigation pest control, alert consumers to the possibility that . 5448507 4/4/2008
Vol. 34/14
Development
5E-14.102
.
The purpose of the rule amendment is to clarify the definition of a “connected structure” and delineate requirements for structural connections which have not previously been provided in rule for fumigation pest control, . 4621873 9/21/2007
Vol. 33/38
Final
5E-14.110
Fumigation Requirements - Notices 1013667 Effective:
07/05/1995

Under Florida law, E-mail addresses are public records. If you do not want your E-mail address released in response to a public records request, do not send electronic mail to this entity. Instead, contact this office by phone or in writing.


Mathematics - Class 5 / Grade 5

Till now we have learned various types of measurement such length, weight, capacity and time. But none of these measurements can be used to find how hot or cold any object is.

When we touch any hot or cold object, we can make out which is hot and which is cold. But the question here is how much cold or how much hot. At the same time, it is difficult to know the exact level of hotness or coldness by touch because the sense of touch varies from one person to another. To find exact answer, we need some measure of hotness or coldness.

Temperature is the measure of the hotness or coldness of an object with reference to some standard value. The instrument which measures the temperature of body is known as thermometer.

Each thermometer has a scale. Two different temperature scales that are commonly used are:

The Fahrenheit scale is marked from 32° F to 212° F. 32° F indicates the freezing point of water and 212° F indicates the boiling point of water.

Celsius scale is also called centigrade scale and is marked from 0° C to 100° C. 0° C indicates the freezing point of water and 100° C indicates the boiling point of water.


Comparison of two scales

Conversion of Temperature

When the temperature is given in degree Celsius:

Step 1: Multiply the given temperature in degree by 9

Step 2: Divide the product obtained by 5.

Step 3: Add 32 to the quotient obtained in step 2 to get the temperature in degree Fahrenheit.

Here given temperature is 55° C.

To convert into degree Fahrenheit first multiply 55 by 9 and we get product as 495.

Then, divide 495 by 5 and answer we get is 99.

Now, add 32 to 99 and final answer we get is 131.

When the temperature is given in degree Fahrenheit:

Step 1: Subtract 32 from the given temperature in degree

Step 2: Multiply the difference obtained from step 1 by 5.

Step 3: Divide the product obtained from step 2 by 9 to get the temperature in degree Celsius.

Here given temperature is 90° F.

To convert into degree Celsius first we need to subtract 32 from 90 and we get 58.

Then, multiply 58 by 5 we get answer as 290.

Now divide 290 by 9 and we get answer as 32.2.


A medical thermometer, also known as a clinical thermometer is a thermometer used to measure body temperature. Most of these thermometers have both the two scales, Celsius scale and Fahrenheit scale. The scale runs from 35 degree Celsius to 42 degree Celsius.

The temperature of a healthy human body is approximately 37 °C or 98.6 °F. The tip of the thermometer is inserted into the mouth under the tongue or under the armpit for a minute. Then, the number against the point at which the mercury column stops rising further indicates the body temperature of the person.


Mathematics– Class 2 / Grade 2

Many times we use picture or symbol to represent quantities of objects. Such representation is called pictograph. Pictograph makes information easy and clear to understand. Information that we collect is called data.

Example 1: Aditya decided to conduct a study on most popular car color. So he stood outside his house and recorded the observation about 50 cars passed from there.

This was what he observed.


Now, read the above table and answer the following questions:

Q1. How many cars did Aditya count?

Q2. How many blue cars did Aditya count?

Q3. How many green cars did Aditya count?

Q4. What is the most popular color car according to his observation?

Q5. What is the least popular color car according to his observation?

Reading of Pictograph Table

Example 2: Ria interviews 20 people and asks them which ice-cream flavor is their favorite. She made a pictogram with the details she gathered:


Now, read the above table and answer the following questions:

Q1. Which ice cream flavor do you think is most popular choice?

Q2. Which ice cream flavor do you think is least popular choice?

Q3. How many people chose vanilla as their favorite ice cream flavor?

Q4. How many people chose peach as their favorite ice cream flavor?

Q5. Which ice cream flavor does you like the most?

Now, read the above table and answer the following questions:


ACT Math Problems

Instructions: This practice test contains 20 questions. You will see your score and the answers to all of the questions when you have completed the test.

This page provides ACT math problems for the examination.

You will find problems and formulas for all of the parts of the math test.

The problems include algebra, coordinate geometry, plane geometry, and trigonometric functions.

ACT Math Problems for Algebra

The algebra questions on the exam cover basic algebraic concepts such as:

  • mean (arithmetic average)
  • median
  • mode
  • range
  • algebraic fractions
  • absolute values
  • square roots
  • factoring and simplifying polynomials
  • algebraic inequalities, expressions, and equations
  • rational and radical expressions

You will encounter combinations of these concepts in certain math problems on the intermediate algebra part on the test.

So you will often need more than one type of math formula to solve these questions.

For instance, you may see a question that has a fraction that contains another fraction in its numerator or denominator.

Another common type of problem is where an erroneous result is given.

Then you have to use the data provided in order to correct the result.

For example, an average grade may have been calculated for a class, but the score for one student was omitted.

In these problems, you will be given the erroneous average and the omitted score.

Then you will have to calculate the correct average.

ACT – Geometry

The geometry questions page gives you the math formulas that you need for the test.

On our geometry page, you will learn how to solve coordinate geometry problems, like midpoints, slope, and distance.

Our geometry page also gives you common formulas for plane geometry.

You will see all of the formulas for figures like circles, triangles, squares, rectangles, and cones.

On the geometry page, you will also see formulas for the circumference of circles.

Formulas are also given for the area, angles, perimeter, and volume of various geometric figures.

ACT – Trigonometry

You will also see the equations that you need in order to calculate trigonometric functions on the math test.

The trigonometry section illustrates the concepts of sine, cosine, and tangent.

It also gives you the math formulas that you need for these trigonometric relationships.

More ACT Math Problems

Need more math problems? If so, click on these links:

Copyright © 2018-2020. Exam SAM Study Aids & Media.
All rights reserved.

ACT is a trademark of ACT, Inc, which is not affiliated with nor endorses this website.


14.5E: Exercises for Section 14.5 - Mathematics

Notes for Class 11 Mathematics Concepts, have been designed in the most basic and detailed format possible, covering nearly all domains such as differential calculus, arithmetic, trigonometry, and coordinate geometry. Preparing from these notes will help students achieve high marks in their 11th grade as well as competitive exams such as JEE Mains and JEE Advanced. These notes provided by GeeksforGeeks would assist students in easily grasping every idea and properly revising before the exams. These notes were written by subject experts which have a significant benefit in that students would be well qualified to answer any kind of question that could be posed in the exams.

Chapter 1: Sets

The chapter explains the concept of sets along with their representation. The topics discussed are empty set, equal sets, subsets, finite and infinite sets, power set, and universal set. There are six exercises in the chapter, in which Exercise 1.1 is based on the introduction of sets and their representations Exercise 1.2 is based on the concept of the empty set, finite and infinite sets and equal sets Exercise 1.3 is based on the concept of subsets, power set, and universal set Exercise 1.4 is based on the concept of Venn diagrams and operations on sets Exercise 1.5 is based on the complement of a set and their properties and Exercise 1.6 is based on the concept of union and intersection of two sets.

Chapter 2: Relations & Functions

Chapter 3: Trigonometric Functions

This chapter explains the concept of the Principle of Mathematical Induction. The topics discussed are the process to prove the induction and motivating the application taking natural numbers as the least inductive subset of real numbers. There is only one exercise in the chapter which is based on the Principle of Mathematical Induction with its simple applications.

Chapter 4: Principle of Mathematical Induction

As the name suggests, the chapter explains the concept of the Principle of Mathematical Induction. The topics discussed are the process to prove the induction and motivating the application taking natural numbers as the least inductive subset of real numbers. There is only one exercise in the chapter which is based on the Principle of Mathematical Induction with its simple applications.

Chapter 5: Complex Numbers and Quadratic Equations

As the name of the chapter suggests, therefore, this chapter explains the concept of complex numbers and quadratic equations and their properties. The topics discussed are the square root, algebraic properties, argand plane and polar representation of complex numbers, solutions of quadratic equations in the complex number system. There are four exercises in the chapter, in which Exercise 5.1 is based on the introduction, algebraic functions, the modulus and the conjugate of a complex number Exercise 5.2 is based on the argand plane and polar representation of a complex number Exercise 5.3 is based on the quadratic equations with real coefficients and miscellaneous exercise based on all the topics discussed in the chapter.

Chapter 6: Linear Inequalities

The chapter explains the concept of Linear Inequalities. The topics discussed are algebraic solutions and graphical representation of Linear Inequalities in one variable and two variables respectively. There are four exercises in the chapter, in which Exercise 6.1 is based on the introduction to linear inequalities, algebraic solution, and graphical representation of linear inequalities in one variable, Exercise 6.2 is based on the graphical representation of Linear Inequalities in two variables Exercise 6.3 is based on the graphical method to find a solution of the system of Linear Inequalities in two variables and a miscellaneous exercise based on the problems of inequalities in one variable only.

Chapter 7: Permutations and Combinations

The present chapter explains the concepts of permutation (an arrangement of a number of objects in a definite order) and combination (a collection of the objects irrespective of the order). The topics discussed are the fundamental principle of counting, factorial, permutations, combinations and their applications. There are five exercises in the chapter, in which Exercise 7.1 is based on the introduction to permutations and combinations along with the fundamental principle of counting Exercise 7.2 is based on the application of the permutation for all distinct objects and factorial notation Exercise 7.3 is based on the application of the permutation when all the objects are not distinct and derivation of the formula of permutation Exercise 7.4 is based on the introduction to combinations where the order doesn’t matter and its applications and a miscellaneous exercise based on the introduction to permutations and combinations and fundamental principle of counting.

Chapter 8: Binomial Theorem

This chapter discusses the binomial theorem for positive integers used to solve complex calculations. The topics discussed are the history, statement, and proof of the binomial theorem and its expansion along with Pascal’s triangle. There are three exercises in the chapter, in which Exercise 8.1 is based on the introduction to the binomial theorem, the theorem for positive integral indices, and Pascal’s triangle Exercise 8.2 is based on the general and middle term in the binomial expansion and their simple applications and a miscellaneous exercise based on all the topics discussed in the chapter.

Chapter 9: Sequences and Series

The chapter Sequences and Series discuss the concepts of a sequence (an ordered list of numbers) and series (the sum of all the terms of a sequence). The topics discussed are the sequence and series, arithmetic and geometric progression, arithmetic and geometric mean. There are five exercises in the chapter, in which Exercise 9.1 is based on the introduction to sequence and series Exercise 9.2 is based on the arithmetic progression, arithmetic mean and general term of the progression Exercise 9.3 is based on the finite and infinite geometric progression, geometric mean, general term of the progression, the sum of n terms of geometric progression and relation between arithmetic and geometric mean Exercise 9.4 is based on the sum of the special series sums to n terms and a miscellaneous exercise based on all the topics discussed in the chapter.

Chapter 10: Straight Lines

Straight lines defined the concept of the line, its angle, slope, and general equation. The topics discussed are the slope of a line, the angle between two lines, various forms of line equations, general equation of a line, and family of lines respectively. There are four exercises in the chapter, in which Exercise 10.1 is based on the introduction to straight lines, the slope of a line for given coordinates of two points, parallel and perpendicular lines with the axis, the angle between two lines, and collinearity of three points Exercise 10.2 is based on the various form of equations of a line in terms of point-slope form, slope-intercept form, two-point form, intercept form and normal form Exercise 10.3 is based on the general line equation, equation of the family of lines that passes through the two lines intersection point and the distance of a point from a line and a miscellaneous exercise based on all the topics discussed in the chapter.

Chapter 11: Conic Sections

The topics discussed in the present chapter are the sections of a cone, the degenerate case of a conic section along the equations and properties of conic sections. There are five exercises in the chapter, in which Exercise 11.1 is based on the introduction of a cone, sections of a cone (generated and degenerated) and circle Exercise 11.2 is based on the introduction to a parabola, its standard equations and latus rectum Exercise 11.3 is based on the introduction to ellipse, its standard equations, eccentricity, latus rectum, focus, semi-major and semi-minor axis Exercise 11.4 is based on the introduction to hyperbola, its standard equations, eccentricity and latus rectum and a miscellaneous exercise based on all the topics discussed in the chapter.

Chapter 12: Introduction to Three-dimensional Geometry

As the name suggests, the chapter explains the concepts of geometry in three-dimensional space. The topics discussed are the coordinate axes and planes respectively, points coordinate, distance, and section for points. There are four exercises in the chapter, in which Exercise 12.1 is based on the introduction to three-dimensional geometry, coordinate axes and coordinate planes in three dimensions and coordinates of a point in space Exercise 12.2 is based on the distance between two points Exercise 12.3 is based on the section formula to find a coordinate of a point that divides the line in a ratio and a miscellaneous exercise based on all the topics discussed in the chapter.

Chapter 13: Limits and Derivatives

The chapter explains the concept of calculus that deals with the study of change in the value of a function when the change occurs in the domain points. The topics discussed are the definition and algebraic operations of limits and derivatives respectively. There are three exercises in the chapter, in which Exercise 13.1 is based on the introduction to limits and derivatives, algebra of limits, limits of trigonometric functions, polynomial and rational functions Exercise 13.2 is based on the algebra of derivative of functions, derivative of polynomial and trigonometric functions and a miscellaneous exercise based on the intuitive idea of derivatives, limits and derivatives and limits of trigonometric functions.

Chapter 14: Mathematical Reasoning

As the name suggests, the chapter explains the concepts of mathematical reasoning (a critical skill to analyze any given hypothesis in the context of mathematics). The topics discussed are the statements, inductive reasoning and deductive reasoning. There are six exercises in the chapter, in which Exercise 14.1 is based on the simple statements, application of “implies” condition Exercise 14.2 is based on the negation and true-false statement and application of “and/or” condition Exercise 14.3 is based on the compound statement, application of “and” and “or” condition Exercise 14.4 is based on the If-then statement, application of “if and only if” condition Exercise 14.5 is based on the If-then statement, application of “implied by” condition and a miscellaneous exercise based on all the topics discussed in the chapter.

Chapter 15: Statistics

This chapter explains the concepts of statistics (data collected for specific purposes), dispersion, and methods of calculation for ungrouped and grouped data. The topics discussed are range, mean deviation, variance and standard deviation, and analysis of frequency distributions. There are four exercises in the chapter, in which Exercise 15.1 is based on the range and mean deviation about mean and median for the data Exercise 15.2 is based on the mean, variance, and standard deviation for the data and analysis of frequency distribution Exercise 15.3 is based on the mean, variance and coefficient of variance for the data and a miscellaneous exercise based on all the topics discussed in the chapter.

Chapter 16: Probability

The chapter discusses the concept of probability (a measure of uncertainty of various phenomena or a chance of occurrence of an event). The topics discussed are the random experiments., outcomes, sample spaces, event, and their type. There are four exercises in the chapter, in which Exercise 16.1 is based on the introduction of probability, possible outcomes and sample spaces Exercise 16.2 is based on the introduction of events, the occurrence of events, ‘not’, ‘and’ and ‘or’ events, exhaustive events, mutually exclusive events Exercise 16.3 is based on the probability of an event and random experiments and a miscellaneous exercise based on the advanced probability problems and axiomatic probability.