Grade what is the units digit of the simplest form of ? Answer: Solution

Answer: 20 cm²
Solution:
The area of ΔBIL is equal to the sum of the areas of ΔHAC and ΔCIL.
The area of ΔBIL is equal to the sum of the areas of ΔBGF, ΔDIL and quadrilaterals FEJG and JEDL.
The sum of the areas of ΔHAC and ΔCIL is equal to the sum of the areas of quadrilaterals FECH and FEJG and triangles AGJ, CID and DIL.
Thus, Area of ΔBGF + Area of FEJG + 480 + Area of ΔDIL = 60 + Area of FEJG + 420 + Area of ΔDIL + 20
Area of ΔBGF = 60 + 420 + 20 – 480 = 20 cm²

18. 
    Find the maximum value of 4 cos(x + ) + 5 sin(x + ).
    

Answer:
Solution:
4 cos(x + ) + 5 sin(x + ) = 4 cos(x + ) + 5 sin(x + )
= 4[cos (x + ) cos - sin(x + ) sin ] + 5 sin(x + )
= 4[cos (x + )( ) - sin(x + )( )] + 5 sin(x + )
= 2 cos (x + ) – 2 sin(x + ) + 5 sin(x + )
= 2 cos (x + ) + 3 sin(x + )

Hence, the maximum value is = = .

19. 
    In the figure shown below, lines PI, PL, and PS and straight lines. HO‖IL and MO‖SL. The area of triangles MOP and POH are 10 cm² and 20 cm², respectively. PH = 4cm and HI = 3cm. Find the sum of the areas of quadrilaterals SMOL and HILO.
    

Ans. 61.875 cm²

Solution:

= =
Area PLI = = 61.25 cm²
Area SLP = = 30.625 cm²
Area of SMOL + Area of HILO = (30.625 – 10) + (61.25 – 20)
Area of SMOL + Area of HILO = 61.875 cm²

20. 
    Three rhombus parquets are used to decorate a hall. These are glued edge to edge to form different shapes as in the given figure below. How many distinct shapes can be formed? (Shapes that look a reflection or rotation of certain shape are not considered distinct.)
    

Solution: 9 distinct shapes

21. 
    One day, Juan was tasked to bring equal number of candles to three magical chapels. He knows that to go to any chapel, he needs to pass through an enchanted bridge that doubles his candles. On his way to the first chapel, he passed through an enchanted bridge and observed his candles doubled. He has no more candle when he left the third chapel. What is the least possible number of candles does he have at first?