The given figure shows sector OAB with centre O and radius 54cm. Another circle XYZ with centre P, is enclosed by the sector OAB.the angle AOB=60 degree . Find the area of OXPY.

Solⁿ: Given: sector OAB with centre O and radius 54cm.

                      <AOB=60⁰

                        OZ=OB=OA=54cm

Construction: join O and P.

Let XP=r, then OP=54-r

In ΔOXP  right angled at X,

        <XOP=30⁰

          Sin30⁰=XP/OP=r/54-r=1/2     then r=18cm-----------(1)

          OP=54-18=36cm

          So,XP²+OX²=OP²

          18²+OX²=36² => OX=18√3cm---------------(2)

Area of OXPY quadrilateral=2xarea of ΔOXP

                                              =2x ½ x 18√3 x 18 = 561.2cm(ans.)

Comment: The radius of a circle drawn inside a sector is 1/3 of the mother circle

       

If the roots of the quadratic equation x^2+Px+q=0 are tan30 and tan15 respectively ,then the value of 2+q-p is --------------------------------

Solution :tan(A+B)=tanA+tanB/1-tanA.tanB
then ,tan45 degree=tan30+tan15/1-tan30.tan15
then tan30+tan15+tan30.tan15=1
Then  2+q-p=2+tan30+tan15+tan30.tan15=3(Answer)

Three containers have their volume in ratio 3:4:5. They are full of mixtures of milk and water. The mixtures contain milk and water in the ratio of (4:1), (3:1) and (5:2) respectively. The contents of all these three containers are poured into a fourth container. The ratio of milk and water in the fourth container is

                                                                                                                                                                                                                                                                                                                                        Three containers have their volume in ratio 3:4:5. They are full of mixtures of milk and water. The     mixtures contain milk and water in the ratio of (4:1), (3:1) and (5:2) respectively. The contents of all these three containers are poured into a fourth container. The ratio of milk and water in the fourth container is?

A. 4:1

B. 151:48

C. 157:53

D. 5:2

                                                                                                                                                                                                                                                                                                                                        Ans: Let the ratio of milk and water in the first container be,

                                   M1/W1=4/1

        The ratio of milk and water in the second container be,              

                                   M2/W2=3/1

        The ratio of milk and water in the third container be,

                                   M3/W3=5/2

        Since, the three containers are in the ratio 3:4:5,

        

         In the first container,

                                   M1+W1=3x---------(1)

                                   M1/W1=4/1

                               => M1=4W1-------------------(2)

              Substituting (2) in (1),

                                   4W1+W1=3x

                               => 5W1=3x

                               => W1=3x/5-----------(3)

               Substituting (3) in (1) we get,

                                   M1=12x/5

         In the second container,

                                 M2+W2=4x---------(4)

                                   M2/W2=3/1

                               => M2=3W2------------------(5)

               Substituting (5) in (4) we get,

                                   3W2+W2=4x

                                   4W2=4x

                                   W2=4x/4=x---------(6)

                Substituting (6) in (4) we get,

                                   M2=3x

       In the third container,

                                   M3+W3=5x---------(7)

                                   M3/W3=5/2

                               => 2M3=5W3

                               => M3=5W3/2---------(8)

                Substituting (8) in (7) we get,

                                   5W3/2+W3=5x

                               => 5W3 +2W3/7=5x

                               => 7W3=10x

                               => W3=10x/7----------(9)

                Substituting (9) in (7),

                                   M3=25x/7

       Since all three mixtures are poured into the fourth container,

                            M4=M1+M2+M3 and

                            W4=W1+W2+W3

.: M4=12x/5 + 3x + 25x/7

       =84x+105x+125x/35

       =314x/35

.: W4=3x/5 + x +10x/5

        =21x + 35x + 50x/35

        =106x/35

M4/W4=314x/35÷106x/35

            =157/53

           =157:53(answer)

(3x-1)^7, then addition of all cofactors is =?

Question :(3x-1)⁷, then addition of all cofactors is =? if cofactors are A7,A6,A5,A4,A3,A2,A1,A0

Solution

1.       3x-1=3x-1

2.       (3x-1)²=9x²−6x+1

3.       (3x-1)³=27x³−27x²+9x−1

4.       (3x-1)⁴=81x⁴−108x³+54x²−12x+1

5.       (3x-1)⁵ =243x⁵−405x⁴+270x³−90x²+15x−1

6.       (3x-1)⁶ =729x⁶−1458x⁵+1215x⁴−540x³+135x²−18x+1

7.       (3x-1)⁷ =2187x⁷−5103x⁶+5103x⁵−2835x⁴+945x³−189x²+21x−1
  Key to answer :Addition of An,A(n-1),A(n-2).............A0=2^n

Algebra	Cofactors for the algebraic expression (3x-1)7
	A7		A6	A5	A4	A3		A2	A1	A0	Total
3x-1									3	-1	2
(3x-1)2								9	-6	1	4
(3x-1)3							27	-27	9	-1	8
(3x-1)4					81		-108	54	-12	1	16
(3x-1)5				243	-405		270	-90	15	-1	32
(3x-1)6		729		-1458	1215		-540	135	-18	1	64
(3x-1)7	2187	-5103		5103	-2835		945	-189	21	-1	128

If the polynomial 16X4—24X3+41X2-mx+16 is a perfect square, then find the value of m (A)12 (B) -12 (C) 24 (D) -24

If the polynomial 16X⁴—24X³+41X²-mx+16 is a perfect square, then find the value of m

(A)12 (B) -12 (C) 24 (D) -24

Solution :Let find it using division method to find square root of an algebraic expression

A does half as much work as B in three-fourth of the time. If together they take 18 days to complete the work, how much time shall B take to do it?

A does half as much work as B in three-fourth of the time. If together they take 18 days to complete the work, how much time shall B take to do it?

(A) 30 days

(B) 35 days

(C) 40 days

(D) None of these

Solution:

1.  Let B does the work in B days, then B in 1 day will do 1/B part of work

2.  A takes three-fourth of working days of B to do half of work done by B, therefore A in 3/4XB days will do 1/2 work, then in 1 day A would do 1/2÷3/4XB=2/3B part of work

3.  A and B together in 1 day will do =1/B+2/3B=5/3B part of work

4.  As given, A and B do 1 work in 18days, then A and B together in 1 day will do 1/18 part of the work

5.  Therefore, looking at step 3 and 4 ,5/3B=1/18, THEN B=30DAYS (Answer)

The ratio of the incomes of P and Q is 5 : 4 and the ratio of their expenditures is 3 : 2. If at the end of the year, each saves 1600, then the income of P is

The ratio of the incomes of P and Q is 5 : 4 and the ratio of their expenditures is 3 : 2. If at the end of the year, each saves 1600, then the income of P is=?

Solution :

1.       Let income of P and Q are PI and QI,then PI/QI=5/4,THEN QI=4/5.PI

2.       Let expenditure of P and Q are PE and QE,then PE/QE=3/2 ,THEN PE=3/2.QE

3.       Saving =Income -expenditure

4.       Saving of P is PI-PE=1600 AND QI-QE=1600, then QE=QI-1600

5.       We need income of P=PI

   From above (4) we know PI-PE=1600, Then replacing PE value from above (2), PI-3/2XQE=1600, then replacing QE from above (4), PI-3/2(QI-1600) =1600, then replacing QI from above (1), PI-3/2(4PI/5-1600) =1600, then PI-12PI/10+2400=1600, then -2PI/10=-800, then 2PI=8000, then PI=4000(ANSWER).