Question number 37 olympiad 2013 section B

Solution :
Income on the nth day is =n^2+2..........(1)
Expenditure on nth day=2n+1..............(2)
Therefore ,the saving on nth day=n^-2n+1.....(3)
The savings on day1 to nth day when summed  up =1240......(Given)
The saving series from day-1  to nth day as follows like 0,1,4,9,16,----------n^2
Sum of this type of series= n(n+1)(2n+1)/6....................................Not A.P or G.P
Here the value of n=n-1 as first day saving is 0
Threfore ,(n-1)(n)(2n-2+1)/6=1240,=>n=16(Answer)

The given figure is created by using the arcs of quadrants with radii 1cm,2cm and 3cm.Find the total area of the shaded region.Take pi is equal to 3.14

Solution:To find
1.Find the area of all the smallest petals =8xarea of the smallest arc
2.Find the area of all the smallest petals =8xarea of the intermediate  arc
3.Find the area of all the smallest petals =8xarea of the biggest arc
4.The area of shaded region =8xarea of the smallest arc+8xarea of the biggest arc-8xarea of the intermediate  arc
5.The area of an arc=area of sector-area of the triangle inside =theta/360xx(radiu)^2-1/2x(radius)^2
6.Therefore ,
         1.The area of all the smallest petals=2.28cm^2
         2.The area of all the medium size  petals=9.12cm^2
         3.The area of all the largest petals=20.25cm^2
7.The entire shaded area in the figure=2.28cm^2+(20.25cm^2-9.12cm^2)=13.41cm^2(Answer)

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)