Saturday, 30 July 2022

H.C.F & L.C.M - A to Z

Greatest Common Divisor(G.C.D) Or H.C.F

To find H.C.F follow , following Steps:
A) Obtain the polynomials. Lat the polynomials be p(x) and q(x).

B) Factorise the polynomials p(x) and q(x).

C) Express p(x) and q(x) as a product of powers of irreducible factors. Also, express the numerical factor as a product of powers of primes.

D) Identify common irreducible divisors and find the smallest (least) exponents of these common divisors in the given polynomials.

E) Raise the common irreducible divisors to the smallest exponents found in step D and multiply them to get the GCD(HCF). If there is no common divisors, the GCD (HCF) is 1.

The following formulae will be helpful in factorising polynomials

1) (x+ y)²= x² + 2xy + y²
2)  (x- y)²= x² - 2xy + y²
3)  (x+ y+ z)²= x² + y²+ z²+ 2xy+ 2yz+ 2zx.
4) (x+ y)³ = x³ + 3xy(x+ y) + y³
5) (x- y)³ = x³ - 3xy(x- y) - y³
6) x²- y² = (x+ y)(x-y)
7) x³+ y³ = (x+ y)(x² - xy +y²)
8) x³- y³ = (x- y)(x² + xy +y²)
9) x³+ y³ + z³ - 3xyz= (x+ y+ z)(x² +y²+ z²- xy - yz - zx)
10) x⁴- y⁴ = (x²+ y²)(x² - y²)
                = (x+ y)(x - y)(x² +y²)
11) x⁸- y⁸ = (x⁴- y⁴)(x⁴  +y⁴)
                = (x+ y)(x- y)(x² +y²)(x⁴ + y⁴)
12) x⁴+ y⁴+ x²y² = (x⁴+ y⁴+ 2x²y²) - x²y²
                          = (x² +y²)² -(xy)²
                         = (x²++ xy y²)(x² - xy +y²)



Find HCF of the following:

1) (x -2)²(x-3)²(x+1)² and (x +1)³(x+2)³(x- 3).                                   (x+2)²(x- 3)(x+1)²

2) 30(x²- 3x+2)(x -2) & 50(x²- 2x+1)(x²- 5x+6).                                 10(x- 1)(x -2)

3) (x -1)²(x+ 2)(x+3)³ and (x -1)(x-2)³(x+ 3)².                                 (x- 1)(x +3)²

4) (2x² -18) and (x² - 2x -3).        x - 3

5) 2x² - x -1 and 4x² + 8x +3.         2x+1

6) ⁸(x³ -x²+ x) and 28(x³ +1).        4(x- 2).

7) 24(6x⁴- x³ -2x²) and 20(2x⁶+ 3x⁵+ x⁴).           4x²(2x+1)

8) x² + x-2 and x³ + 4x²+ x - 6.      (x-1)(x +2)

9) 4x³ -16x² + 20x-8 and 8x³ - 48x²+ 72x - 32.                       4(x -1)² 

10) 33(2x+3)(3x -4)³(5- 4x)⁴ and 22(x +1) (2x +3)(5- 4x)³(4x- 9).             11(2x+3)²(4x -5)³

11) 10(x+1)²(x -3)³(x-2) and 15(x -2)³(x -3)²(x+5) and 75(x+5)(x-3)³(x-2)².      5(x -2)(x -3)²

12) 8x⁴ - 16x³ - 40x² + 48x and 16x⁵ + 64x⁴ + 80x³ + 32x².                 8x(x +2)

13) 2x⁴ - 2y⁴ and 3x³ + 6x²y - 3xy² - 6y³.     (x - y)(x + y).

14) 4(x - 3)²(x -1)(x+1)³ and 6(x- 1)²(x +1)²(x+7).                            2(x-1)(x+1)²

15) 16 - 4x² and x² + x- 6.              x-2

16) xy - y and x⁴y - xy.                  (x-1)y

17) x³ + 64 and x² - 16.                   x+4

18) 3 + 13x - 30x² and 25x² - 30x +9.    5x -3

19) 56(x⁶y² - x²y⁶) and 72(x⁵y³+ 3y⁵x³ + 2y⁷x).                                     8xy²(x²+ y²)

20) (x - 2)²(x +3)(x-4) and (x- 2)(x +2)(x -5).                                                    (x-2)

21) (2x - 7)(3x +4) and (2x- 7)²(x +3).     (2x -7)                     

22) (3x - 2)²(2x+3)³(x -1) and (3x- 2)³(2x +3)(x-1)³.                       (2x+3)(x-1)(3x-2)²

23) (x +1)³(x -1) and (x- 1)³(x +1).         (x-1)(x+1)

24) (x +4)²(x -3)³ and (x- 1)(x -3)²(x+4).                  (x+4)(x-3)²

25) 24(x - 3)(x -2)² and 15(x- 2)(x -3)³.                      3(x-3)(x-2)

26) 2x² - 7x +3 and 3x²-7x - 6.         (x-3)

27) x³+ 2x²- 3x and 2x³ + 5x² -3x.    x(x +3)

28) 22x(x+1)² and 36x²(2x²+ 3x+1).   2x(x +1).                       2x(x+1)

29) 3+ 13x - 50x² and 25x²-30x +9.    (5x-3)

30) x³ - y³ and x⁴+ x²y² + y⁴.      (x² + xy+ y²)

31) 2x² + 7xy + 3y² and 2x²+ 6xy + x + 3y.      x + 3y

32) 56(x⁶y² - x²y⁶) and 72(x⁵y³+ 3y⁵x³+ 2y⁷x).                                      8xy²(x²+ y²)

33) 54(x³+ 8y³) and 90(x³ + 7x²y+ 16xy²+ 12y³).                                         18(x+2y)

34) (4x⁴+ y⁴) and (2x³ - xy²+ y³).   2x² + 2xy + y²

35) x³ - y³ , x³y - y⁴, y²(x - y)²(x² + xy + y²).    x³ - y³

36) (2x² - 3xy)², (4x - 6y)³ and (8x³ - 27y³).                   2x - 3y

37) (4x⁴ + y⁴), (2x³ - xy² - y³) and (2x² + 2xy + y²).                           2x² + 2xy + y²

38) x⁴ - x³ + x -1 and x⁴+ x² +1.       x² - x +1

39) (8x⁶ - 32x⁵+ 128x³ - 128x²) and (12x⁶ - 36x⁵ + 48x³).        4x²(x -2)

40) (8x³ - x²- x +1) and (x⁴ - 2x³ + 2x -1).                       (x- 1)²(x+1)

41) 2x²y(x²- y²) and 35xy²(x - y).      xy(x- y)

42) (x² + 4x -21) and (x³+ 7x² - 9x -63).     (x+7)(x-3)

43) (6x⁴- 13x³ +6x²) and (8x⁴- 3x³+ 54x² - 27).                            x(2x-3)

44) (4x⁵+ 16x⁴ - 44x² - 24x) and (2x⁵ - x³ - 2x).                  2x(x² - x -1)

45) (3x³ - 14x² + 9x+ 10) and 15(x³-34x²+ 21x-10).                           3x -5

46) 4x²(x² - a²) and 9x²(x³ - a³).     x²(x - a)

47) 2(a² - b²) and 3(a³ - b³).           a-- b

48) 45(2x⁴ - x³ - x²) and 75(8x⁵ + x²).

49) (x²+3x-4) and (x³- 2x²- 2x+3).      x-1

50)12(x⁴ - 25) and 8(x⁴+4x²- 5).   4(x²+5) 











) If x- a is the HCF of x² - x-6 and x²+ 3x - 18 find the value of a.                          3

) For what value of a of x² - 2x -24 and x²- ax - 6 is x-6.                                       5

) x² + x- 2 is the HCF of (x -1)(2x²+ ax +2) and (x+2)(3x²+ bx +1). Find the value of a and b.                                            5, -4

) If x- k is the HCF of x² + x- 12 and 2x²- kx - 9. find the value of k.                    3

) Find the value of a and b so that the polynomials p(x) and q(x) have (x-1)(x +4) as their HCF.                             12,4

) If x- a is the HCF of ax² + bx + c and cx²+ ax + b such that c ≠ 0, then show that a³+ b³ + c³ = 3abc.

) If x- a is the HCF of x² + ax + b= 0 and x²+ bx + a= 0, then find the value of a+ b+ 1.                                                       

) If x- a is the HCF of x² + mx +1 and x²- 3x +2 find the value of m.          -5/2, -2





EXERCISE 

Find the LCM of following:

1) 12(x- 1)³(x³+ 6x + 8) and 150(x- 1)(x +2)(x² + 7x +10).                300(x-1)³ (x+2)²(x+4)(x+5)

2) 18x⁴ - 36x³+ 18x² and 45x⁶ - 45x³.     90x³(x -1)²(x²+ x+1)

3) 15(4x³-4x²+x) and 352x²- 7x+3).       105x(2x -1)²(x-3)

4) x³+ x²+x+1 and x³ +2x²+ x+2    (x+1)(x +2)(x²+1)

5) 7x³+ 2x²- 16x- 31 and x³+6x²+11x+ 6.     (x+1)(x +2)(x-2)(x+3)(7x²+ 16x+16)

6) 3(x⁴- x²y²), 6xy(x²- y²) and 96(x³y + y⁴).    90x²y(x - y)(x+ y)(x²- xy+ y²)

7)  a² - b² - c²+ 2bc, (a+ b+ c)², a² - b² + c² + 2ac.                 (a+ b+ c)(A-- b+ c)(a+ b- c)²

8) - x² - x+6 and -x²+ x +2.      (x-2)(x+1)(x+3)

9) x³+ x²+ x+1, x³+2x² +x+2.       -(x+3)²(x-5)²

10) 13(x -1)(x-2)², 7(x-2)²(x+3)² and (x-1)²(x+3).                    91(x -1)²(x-2)²(x+3)²

11) 8(1- x)³+(x+2)(x-3) and 12(x+2)² (x+1) (x+3).         

12) 12(x⁴+324x), 36x³+ 90x²- 54x.    36x(2x-1)(x+3)(x²- 3x+9)

13) 18x³+45x²-27x, 15x⁴ - 135x².      45x²(x-3)(x+3)(2x-1)

14) 20(2x³+3x² - 2x), 45(x⁴+ 8x).      180(x+2)(2x-1)(x²- 2x+4)

15) 35(x⁴- 27x), 40(2x³ - 5x² -3x).      280x(x-3)(2x-1)(x² + 3x+9)

16) 22x(x+1)², 36x²(2x²+ 3x+1).    396x² (x+1)²(2x+1)    

17) x³ - 1, x⁴+x²+ 1, x⁶ - 1.     (x³ - 1)(x³+ 1)

18) x³ - 8, x²-5x+ 6, x³ - 4x² + 4x.       x(x - 2)²(x-3)(x² + 2x+4)

19) x² - 3x+2, (x-1)², (x⁴ - 1).      (x+1)(x-1)²(x-2)(x²+1)

20) 15x³ - 75x²- 90x, 6x⁴ - 18x³ - 108x².      30x²(x-6)(x+1)(x+3)

21) a³+ 2a²-5a-6, a³- 3a²+4.       (a+1)(a+3)(A-- 2)²

22) a² - 8a + 7, a² - 12a + 35, a³ - 5a² - a +5.                          (a-1)(a+1)(a-5)(a-7)

23)  x² + 2xy + y², xy²+ x²y.     xy(x - y)²

24) (x³- y³), x² - y².          (x² - y²)(x² +xy+ y²)

25) x² + 2xy + y², x³+ y³, xy²+ x²y.     xy(x + y)²(x²- xy+ y²)

26) 6x² - 13xa + 6a², 6x²+11xa-10a², 6x²+ 2ax - 4a².        2(x+a)(3x - 2a)(2x+5a)(2x - 3a).

27) x² + 2xy + y²- z², y²+z ²- x²+ 2yz, z²+ x² - y²+ 2zx.               (x+ y+z)(x+y-z)(y+ z- x)(z+ x - y)

28) 6a³b²- 6ab⁴, 8a³b + 16a²b²+ 8ab³, 4a³- 2a²b - 6ab².          24ab²(a+b)(a-b)( 2a- 3b) 

29) 4(a⁴ - b⁴), 2(a⁴ + b⁴ - 2a²b²), 6(a⁴ - 3a²b² - 4b²).                 12(a + b)²(a- b)²(a² +b²)(a+ 2b)(a- 2b).

30) 11x³(x+ 1)³, 121x(2x² + 3x+1).      121x³(x+ 1)³(2x+1).

31) 25(x²+7x+ 12), 15x(x² - 16).    75x (x+ 3)(x+4)(x-4).

32) x(8x³- 27), 2x²(2x² +9x+9).    2x²(8x³+27)(x+3).

33) 6(x +2)²(x²-x+1) and 15(x+1)(x+2)².         30(x+1)(x+2)⁴(x²-x+1)

34) (x -3)²(x+2), (x-3)⁴, (x+2)³, (x+1)(x-3).        (x - 3)⁴(x+2)³(x+1)

35) (1- x²), (x³- 1) and (x⁴-1).        (x⁴-2)(x²+x+1)






Misc-

1) The LCM of two polynomial (x+ 3)(- 6x² + 5x+ 4) and (2x²+ 7x +3)(x+3) is (x+3)(2x+1)(4- 3x). Find their HCF.

2) The HCF and LCM of two polynomial are (x+ a) and 12x²(x+ a)(x² - a²) respectively. The first polynomial is 4x(x+a)². Find the other polynomials.

3) The HCF of two polynomials is (x -3)(x² +x -2) and (x²- 5x +6) is (x -3). Find their HCF.

4) The LCM of two polynomial 10 (x² - 9)(x - 1) and 10x(x - 1)(x+3) (x - 2) is 20x(x² - 9)(x² - 3x+2). Find their HCF.

5) Find the LCM and HCF of the polynomial (2x² -8) and (x² - 5x+ 6). Verify that the product of the two polynomials is equal to the product of their HCF and LCM.

6) The HCF of two polynomials is (x+ 1) and their LCM is (x⁶ -1). If one of them is (x³+1), find the other.

7) The LCM of two polynomials is (x +3)(x -2)²(x -6) and their HCF is (x -2). If one of them is (x+3)(x -2)², find the other.

8) Find GCD (x⁶y² - x²y⁶) and 72(x⁵ + 3y⁵x³+ 2y⁷x) 

9) Find the LCM of (4a⁴ - b⁴), 2(a⁴ + b⁴ - 2a²b²) and 6(a⁴ - 3a²b² - 4b²).

10) The HCF and LCM of two polynomials are 5(x+3)(x -1) and 20(x² -9)(x² - 3x+2) respectively. If one polynomial is 10(x² -9)(x-1), Find thei other polynomial.



Wednesday, 27 July 2022

SET THEORY (2)


1) Express each of the following statements in set theoretical notation:
A) x belongs to the set A.
B) Set A is a subset of set B.
C) P is a proper subset of Q.
D) a is not an element of set X
E) Y is not a proper subset of Z.
F) A Set whose elements are contained in U but not contained in B.

2) Suppose P and Q are two given sets and the corresponding Universal set is U; Express each of the following statement in symbols :
A) A set whose elements are contained either in P or in Q.
B) A set which contains the common elements of P and Q.
C) A set whose elements belongs to U but do not belong to Q.
D) A set whose elements belongs to Q but do not belong to P.
E) A set whose elements do not belong to any of the set P and Q.
F) A set which does not contain the common elements of P and Q.

3) Represent the following sets in Roster form :
A) set of vowels in English alphabet.
B) set a factors of the number 36.
C) A={x: x is a prime integer and 6< x ≤ 29}.
D) P={x| x ∈ N and x≤ 12}.
E) set of letters in the word 'statistics'.

4) Rewrite the following set in set builder notation form:
A) A={......, -3,-2,-1,0,1,2,3}
B) P={1,3,5,7,9,11}
C) set of roots of the equation x⁴ - 13x²+ 36= 0.
D) set of even positive integers greater than 4 and less than or equals to 19

5) State weather each of the following sets is infinite or finite:
A) D={x: x is the number of people living on the earth}
B) W={x:x is the time a person waits for a bus}
C) V={x:x is an odd integer exceeding 889}.

6) State with reasons whether the sets defined in each of the following are equal sets:
A) X= {∅} ; Y= ∅.
B) A={-2, -5}.
     B={x:x is a root of the equation x²+ 7x + 10= 0}.
C) P={b, i, e, n, s, u}
     Q={x:x is a letter in the word ' Business'.
D) A={x:x is a digit in the number 30255}
    B={x:x is an integer and 0≤x≤5}
    C={1,0,2,3,4,5}.

7) Some well-defined sets are given below; identify the null sets:
A) A={0}
B) B={∅}
C) C= ∅
D) Set of boy students in a girls' college.
E) P={x:x is an integer and 1<x<2}
F) Q={x:x is an integer and 1<x≤2}
G) R={x:x is positive and x² + 7x +12= 0}
H) A={x:x is an integer and 6x² - 5 +1= 0}
I) {x: 3x² -4= 0, x is an integer}
M) {x: (x+3(x+3)= 9, x is a real number}
N) A∩ B - A.

8) State with reasons which of the following statements are correct/ incorrect:
A) If P U Q={a, b, c, d}, then a ∈ P And a ∈ Q
B) If P∩ Q={a,b,c,d}, then a ∈ P and a ∈ Q.
C) If A∩ B=∅, then A and B are disjoint sets.
D) If A is super subset of B, then B is super subset of A.
E) If A is super subset of B, and B is super subset of C, then A is the super subset of C. 
F) x ∈ A U B => x ∈ A.
G) If A is super subset of B, and B is super subset of A, then A = B.
G) If A= {2,4, 6, 8}, then {2,4}∈A.
H) If A= {2,4, 6, 8}, then {2,6,8} is subset of A.
I) If A= {2,4, 6, 8}, then ∅ is the subset of A.
M) If A= {2,4, 6, 8}, then {2,4}is the subset of P(A).
N) (A - B), A ∩ B and (B - A) are mutually disjoint.
O) If A= {2,4,6,8}, then {2,6,8} ∈ P(A).

9) If a ∈ A and a ∈ B, does it follow A is super subset of B ? Give reasons.

10) State with the reasons, which of the following statements are true or false :
A) {a} ∈ {a, b, c}
B) a ∈ {a, b, c}.
C) a⊂ {a, b, c}
D) a not belong to {a, b, c}.
E) 3 ⊂ {1,3,5}
F) 3 ∈ {1,3,5}
G) {3} ⊂ {1,3,5} 
H) {3} ∈ {1,3,5}

11) If A = {a,b,c} Name
A) the subsets of A
B) the proper subsets of A.

12) Define power set of a set A. Find the power state of a A{a, b c}, If B be the power set of A, state with the reason, which of the following statement is correct:
A> B, A ∈ B, A⊂B, A= B, A is not subset of B.

13) Fill in the gaps:
A) The number of subsets in a set consisting of four distinct elements are -----.
B) The number of proper subsets in set consisting of n district elements are----.
C) If x ∈ A => x ∈ B then ----.
D) if A is super subset of B and B is super subset of A then-----.
E) the set of P whose elements are all subsets of the set {1,2} is given by
  P={__, ___, ___, ___}
F) if A and B are disjoint set then n(A U B)= ____.
G) If A U B= A ∩ B then____.
H) The dual of A U (B ∩C)= (A U B) ∩ (A U C) is _____.
I) The dual of A U U= U is ___.
J) If A is a given set and ∅ is the null set then ___.

14) Let A= {a, b, c}, B={a,b}, C={a,b,d}, D{c, d} and E={d}. State which of the following statements are correct and give reasons: 
A) B ⊂ A
B) E is no subset of E
C) D⊂B
D) {a} ⊂A

15) Let A={a,b, c,d,e,f,g,h,i},
B={b,d,f,h}.
C={a,c,e,g,i}, D{c,d,e}, E={c,e}. Which set can equal X if we are given the following informations?
A) X and B are disjoint
B) X ⊂A but X is not subset of C.
C) X ⊂ D but X is not subset of B.
D) X⊂ C but X is not subset of A.

16) List the sets A U B, A ∩ B A∩ (B U C). given that,
A={p,q,r,s}, B={q,r,s,t}, C={q,r,t}

17) If P{a,b,c,d,e} and Q={a,e,i,o,u}, prove P⊂ P U Q and P ∩Q ⊂ P.

18) If A= {2,3,4,5} and B={1,2,3,4} show that B - A ≠ A- B.

19) Let S={1,2,3,4,5} be the Universal set and let A={3,4,5} and B={1,4,5} be two of its subsets. Verify (AU B)'= A' ∩B'.

20) If A{1,2,3,4}, B={2,3,4,5}, C={1,3,4,5,6,7}, find
A) A - B
B) A - C
and hence verify A - (B∩C)= (A - B) U (A - C).

21) If P={a,b, c, d, e, f} and Q={a, c,e,f}, show (P - Q) U (P ∩Q)= P.

22) If A={x : x is an integer and 1 ≤ x ≤ 10} and B={x: x is a multiple of 3 and 5 ≤ x ≤ 30}; find A U B, A∩B, A - B and B - A.

23) Let U={1,2,3,4,5,6,7,8,9,10} be the universal set. suppose, A={1,2,3,4,5,6} and B {5,6,7} are its two subsets. Write down the elements of A - B and A ∩ B'.

24) If X={x : x is an even integer and 6< x ≤ 20} and Y={x : x is a multiple of 3 and 0≤ x≤25}, find
A) X U Y 
B) X ∩ Y
C) Y - X
D) X - Y

25) Let S={1,2,3,4,5,6} be the Universal set, let A U B= {2,3,4}; find A' ∩ B' where A' , B' are complements of A and B respectively. also show that A UB, and A' ∩ B' are disjoint sets.

26) if P={p,q,r,s,t,u} and Q={q,r, v,w}; find
A) (P U Q) ∩(P U R)
B) (P - Q)U(P - R).

27) if A,B, C be three subsets of the universal set S where S={1,2,3,4,5,6,7}, A={1,3,5,6} and B ∩C ={1,2,6}, find
A) (A U B)∩(A U C)
B) B' U C'

28) Given, A={x: 0< x≤2} and B={x: 1< x < 3}; find
A) A∩B
B) A U B
C) A - B

29) Let A= {x:2 ≤ x < 5} and B={x : 3 < x < 7} be two subsets of the universal set S={x: 0 < x≤ 10}; verify that (A UB)'= A'∩B'.

30) Given X U Y={1,2,3,4}, X U Z = {2,3,4,5}, X ∩Y={2,3} and X∩Y= {2,4}; find X, Y Z.

31) If aN={ax: x ∈N}, Describe 3N∩ 7N where N is the set of natural numbers.

32) Using set operation show that the numbers 231 and 260 are prime to each other.

33) Applying set operation prove that 3 + 4 = 7.

34) Using Venn diagram or otherwise, solve the following problems:
 In a class of 70 students. Each students has taken either English or Hindi or both. 45 students have taken English and 30 students have taken Hindi. How many students have taken both English and Hindi ?

35) A market research group conducted a survey of 1000 consumers reported that 700 consumers liked product A and 480 consumers liked product B. What is the least number that must have liked both products ?

36) In a town of 60% read magazine A, 25% do not read magazine A but read magazine B. Calculate the percentage of those who do not read any magazine. Also find the highest in the lowest possible figures of those who read magazine B.

37) In a statistical investigation of 1003 families of Calcutta, it was found that 63 families had neither a radio nor a T.V., 794 families had a radio and 187 had a television. How many families in that group had both radio and TV ?

38) Three daily newspaper, E, B, H are published in a certain city. 62% of the citizens read E, 59% read B, 41% read H, 40% read both E and B, 28% read both B and H, 24% read both E and H. Find the percentage of citizens who read all the three papers.

39) In a survey of college students, it was found that 40% use their own books, 50% use library books, 30% use borrowed books, 20% use both their own books and library books, 15% use their own books and borrowed books, 10% use library books and borrowed books, and 4% use their own books, library books and borrowed books. Calculate the percentage of students who do not use a book at all.

40) In a city 3 daily news papers X, Y, Z are published; 65% of the citizens read X, 54% read Y, 45% read Z; 38% read X and Y; 32% read Y and Z; 28% read X and Z; 12% do not read any of these three papers If the total number of people in the City be 10000000, find the number of citizens who read all the three newspapers.

41) A company studies the product preferences of 300 consumers. It was found that 226 liked product A, 51 liked product B, 54 laked product C; 21 liked products A and B, 54 liked products A and C, 39 liked products B and C and 9 likwd all the three products. Prove that, the study results are not correct. [Asume that each consumers liked at least one of the three products]

42) The production manager of Sen, Sarkar and Lahiri company examined 100 items produced by the workers and furnished the following report to his boss.
Defect in measurement 50, defect in colouring 30, defect in quality 25, defect in quality and colouring 10, defect in measurement and colouring 8, defect in measurement and quality 20 and 5 are defective in all respect. The manager was penalised for the report. Using appropriate results of set theory, explain the reason for the penal measure.

43) In a survey of 150 students, it was found that 40 students studied Economics, 50 students studies Mathematics, 60 students studied Accountancy and 15 student studied all the three subjects. It was also found that 27 students studied Economics and Accountancy, 35 students studied Accountancy and mathematics. Find the number who studied only economics and the number who studied none of these subjects.

44) Out of thousand students in a college 540 played football, 465 plyed cricket and 370 played volleyball; of the total 325 played football and cricket, 260 played football and volleyball, 235 played cricket and volleyball, 125 played all the three games. How many students 
A) did not play any game
B) played only one game.
C) played just two games.

45) A group of consist of a number of students and each Student of the group can speak at least one of the languages Bengali, Hindi and English. 65 can speak Bengali, 54 Hindi and 37 English; 31 can speak both Bengali and Hindi, 17 both Hindi and English, and 18 both Bengali and English. Determine the greatest and least number of students in the group.

46) An investigator interviewed 100 students to determine their preference for the three drinks; Milk(M), coffee (C) and tea(T). He reported the following:
 10 students had all the three drinks M, C and R; 20 had M and C; 30 had C and T, 25 had M and T; 12 had M only, 5 had C only and 8 had T only.Find how many did not take any of the three drinks.

Wednesday, 6 July 2022

APPLICATION OF DERIVATIVES IN COMMERCE AND ECONOMICS

EXERCISE -1

1) A manufacturer finds that the production cost of each article produced by his firm is ₹25 and the other fixed costs are ₹25000. If each article is sold for ₹35, Find
A) cost function.                     C(x)= 25000+ 25x
B) Revenue function.        R(x)= 35x
C) break even point.                 2500

2) The fixed cost of a new product is ₹35000 and the variable cost per unit ₹500. If the demand function p(x)= 5000 - 10x, find the break-even values.                          10, 35

3) The fixed cost in the variable cost of x units of a product of a company are ₹ 30000 and ₹75x respectively. If each unit is sold for ₹125, find break-even point.       600

4) A company decides to set up a small production plant for manufacturing clocks. The total cost of initial set up(fixed cost) is ₹9 lakhs. The additional cost (variable cost) for producing each clock is ₹300. Each clock is sold at ₹750. During the first month 1500 clocks are produced and sold.
A) determine the cost function for C(x) for producing x clocks.       300x + 900000
B) determine the revenue function R(x) for the sale of x clocks.     750x
C) determine the profit function p(x) for the sale of x clocks.          450x - 900000
D) what profit or loss the company incurs during the first month when all 1500 clocks are sold ?        225000
E) determine the break even point.           2000

5) The printing cost of 1900 and 1300 copies of a book are 5100 and 3700 respectively. Find the equation of the cost curve of printing assuming it to be linear. If the selling price is ₹3 per copy, find the number of copies that must be printed so that
A) there is no profit or loss.      y= 7x/3 + 2000/3, 1000 copies 
B) a profit of ₹ 40.         1600 copies

6) A publishing house finds that the production cost directly attributed to each book is ₹30 and that the fixed costs are ₹15000. If each book can be sold for ₹45 then find
A) cost function.           30x+15000 
B) the revenue function.            45x
C) break-even point.                 1000

7) A firm knows that the demand function for one of its products is linear. It also knows that it can sell 1000 units when the price is ₹4 per u6, and it can sell 1500 units when the price is ₹2 per unit. Find
A) demand function.       8 - x/250
B) total revenue function.   8x -x²/250
C) marginal revenue function.      8 - x/125

8) ABC Co. Ltd. is planning to market a new model of shaving razor. Rather than set the selling price of the razor based only on production cost estimates, management pulls the retailer of the razors to see how many razors they would buy for various prizes. From this survey it is determined that the unit demand function (the relationship between the amount x each retailer would buy and the price p he is willing to pay) is 
x= 1500 p + 30000
The fixed costs to the company for production of the razors are found to be ₹28000 and the cost for material and labour to produce each razor is estimated to be ₹8.00 per unit. What price should the company charge retailer in order to obtain a maximum profit?       at 14, ₹26000, 9000

9) 


EXERCISE -2

1) The unit demand function is x= (25- 2p)/3, where x is the number of units and p is the price. Let the average cost per unit be ₹40. Find
A) the revenue function R in terms of price p.                        (25 - 2p)/3
B) the cost function C.    40(25- 2p)/3
C) the profit function P.      (-2p² + 105p - 1000)/3
D) the price per unit that maximizes the profit function.          When p= 105/4
E) the maximum profit.    

2) The demand function faced by a firm is p= 500 - 0.2x and its cost function is C= 25x + 10000 (p= price, x= output and C= cost). Find
A) the output at which the profits of the firm are maximum.      1187.50
B) the price it will charge.    262.50

3) 

7) For a manufacturer of dry cells, the daily cost of production C for x cells is given by C(x)= ₹(2.05x + 550). If the price of a cell is ₹ 3. Determine the minimum number of cells those must be produced and Sold daily to ensure no loss.       579

9) The daily cost of production C for x units of an assembly is given by C(x)= ₹(12.5 x + 6400).
A) if each units is sold for ₹25, determine the minimum number of units that should be produced and Sold to ensure no loss.            513
B) if the selling price is reduced by ₹2.50 per unit, what would be the break-even point.                    640
C) if it is known that 500 units can be sold daily, what price per unit should be charged to guarantee no loss ?                    25.30

10) A firm produces x tonnes of output per week at a total cost of ₹(x³/10 - 5x² + 60x +100). Find
a) average cost.     x²/10- 5x + 60 + 100/x
b) average variable cost.    x²/10 - 5x + 60
c) marginal cost.      3x²/10- 10x+60

11) A calculator manufacturing company introduces production bonus to the workers that increases the cost of a calculator. The daily cost of production C for y calculators is given by C(y)= ₹ 2.05y + ₹ 550.
A) If each calculator is sold for ₹3, determine the minimum number that must be produced and sold daily to ensure no loss.          579
B) if the selling price is increased by 30 paise per piece, what would be the break even point ?             440
C) if it is known that at least 500 calculator can be sold daily, what price the company should charge per piece of calculator to guarantee no loss?                3.15

12) The total cost and the total revenue of a company that produces and sales x units of a product are respectively C(x)= 10x + 400 and R(x)= 60x - x³.  Find
A) the break-even values.     10 & 40
B) the values of x that gives a profit.       10< x < 40
C) the values of x that results in a loss.                        x< 10 and x> 40

13)  The total cost and the total revenue functions of a company that produces and sales x units of a particular product are given by C(x)= 5x + 350 and R(x)= 50x - x² respectively. Find
A) the break even values of x.      10 & 35
B) the values of x that produces a profit.                          10< x < 35
C) the values of x that result in a loss.           x< 10 and x > 35

14) The total cost and the total revenue of a company that produces and sells x units of a product are respectively C(x)= 5x +350 and R(x)= 50x - x². Find
A) the break-even values.       10 or 35
B) the value of x that produces a profit.                          10< x < 35
C) the value of x that results in a loss.                x< 10 or x> 35

15) The cost function C(x) of a firm is given by C(x)= 2x² - 4x + 5. Find
A) The average cost.       
B) the marginal cost when x= 10.            16.5 units, 36 units

16) A firm produces x units of a article at a total cost of ₹(5+ 48/x + 3x²). Find the minimum value of the total cost.                At x= 2, ₹ 41

17) A firm produces x units of output at a total cost of ₹(2x/3 + 35/2).  Find the cost when the output is 4 units, the average cost of output of 10 units, and the marginal cost when output is 3 units.           ₹20.16, ₹2.42, ₹0.67

18) A firm produces x units of output per week at a total cost of ₹(x³/3 - x² + 5x + 3). Find the output levels at which the marginal cost and the average variable cost attains their respective minima.                1, 1.5

19) A firm produces x tons of a valuable metal per month at a total cost C given by C= ₹(x³/3 - 5x² + 75x +10). Find at what level of output the marginal cost attains it's minimum.                        5 tons.

20) Let the cost function of a firm be given by the equation: C(x)= 300x - 10x² + x³/3, where C(x) stands for cost function and x for output. Calculate the output at which
A) the marginal cost is minimum.   10
B)  the average cost is minimum.    15
C) average cost is equals to Marginal cost.              15

21) The efficiency E of a small manufacturing concern depends on the number of workers w and is given by 10E = - w³/40 + 39w - 392. Find the strength of the workers which gives maximum efficiency.   20

22) A company after examining its cost structure and revenue structure has determined that the following functions approximately describe its cost and revenues:
C= 100 + 0.015x² and R= 2x where C= total cost, R= total revenue and x= number of units produced and Sold. Find the output rate which will maximum profits for the firm.    66.67

23) A firm can sell x items per week at a price p= (300 - 2x) rupees. Producing items cost the firm y rupees where y= 2x + 1000. How much production will yield maximum profits ?                74

24) The total revenue function and the total cost function of a company are given by R= 21q - q² and C= q³/3 - 3q² - 7q + 16 respectively, where q is the output of the company. Find the output at which the total revenue is maximum and the output at which the total cost is minimum.      10.5, 7

25) The demand function of a firm is p= 500 - 0.2x and its cost function is c= 25x - 10000, where p is the price and x is the output. Find the output at which the profit of the firm will maximum. Also find the price it will charge.          1187.5, 262.5

26) The demand function of a producer is 3q= 98 - 4p and its average cost is 3q +2, where q is the output and p is the price. Find the maximum profit of the producer.                33.75 units

27) A radio manufacturer finds that he can sell x radio per week at ₹p each, where p= 2(100 - x/4). His cost of production of x radios per week is ₹ (120x + x²/2). Show that his profit is maximum when the production is 40 radios per week. Find also his maximum profit per week.       1600

28) A manufacturer produces x units per month at a total cost of ₹(x²/25 + 3x + 100). There is no competition in the market and the demand follows the rule x= 75 - 3p, where p is the selling price per article. Find x such that the net revenue is maximum,. also find the monopoly price.                 30, 15

29) A firm produces x units of output at a total cost of ₹(300x - 10x² + x³/3). Find
A) output at which marginal cost is minimum.                      10
B) output at which average cost is minimum.                         15
C) output at which average cost is equal to marginal cost.             0, 15

30) The demand function of a monopolist is given by p= 1500 - 2x - x². Find the marginal revenue for any level of output x. Also, find marginal revenue when x= 0.     1160

31) A firm produces x tons of output per week of a total cost of ₹(x³/8 - 4x² + 12x +3). Find the level of output at which average variable cost attains minimum value.     16

32) The manufacturing cost of an item consists of ₹900 as overheads, the material cost is ₹3.00 per item and the labour cost is ₹ x²/100 for x items produced. How many items must be produced to have average cost minimum.                  300

33) The total cost function of a firm is C= x³/3 - 5x² + 28x +10. Where C is total cost and x is output. A tax at the rate of ₹2 per unit of output is imposed and the producer adds it to his cost. If the market demand function is given by
p= 2530 - 5x.
Where ₹ p is the price per unit of output, find the profit maximizing output and price.               50, 2280

34) Given the demand and cost functions:
p= 10 - 4x
C= 4x
Find
A) the maximum quantity, price and the profit on this level.              12
B) what will be the new equilibrium after a tax of ₹0.50 is imposed ?    12.25
*C) the tax rate that will maximizing tax revenue and determine that tax revenue.                 8

35)