Question : A square of side $p$ is taken. A rectangle is cut out from this square such that the length of one side of the rectangle is equal to half of the length of one side of the square and the length of another side of the rectangle is equal to $\frac{1}{3}$rd of the length of the first side of the rectangle. What is the area of the portion of the square that remained after the rectangle was cut out?
Option 1: $\frac{7}{8} p^2$
Option 2: $\frac{3}{4} p^2$
Option 3: $\frac{11}{12} p^2$
Option 4: $\frac{15}{16} p^2$
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Correct Answer: $\frac{11}{12} p^2$
Solution : Use the formula, The area of the rectangle = Length × Breadth The area of the square = Side2 According to the question, The length of the rectangle $=\frac{1}{2}\times p=\frac{p}{2}$ The breadth of the rectangle $=\frac{1}{3}\times \frac{p}{2}=\frac{p}{6}$ The area of the rectangle $=\frac{p}{2} \times \frac{p}{6}=\frac{p^2}{12}$ The area of the square $=p^2$ The area of the portion of the square that remained after the rectangle was cut out = The area of the square – The area of the rectangle ⇒ $p^2-\frac{p^2}{12}=\frac{11}{12} p^2$ Hence, the correct answer is $\frac{11}{12} p^2$.
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Question : The breadth of a rectangle is $\frac{4}{5}$th of the radius of a circle. The radius of the circle is $\frac{1}{5}$ of the side of a square, whose area is 625 cm2. What is the area of the rectangle if the length of the rectangle is 20 cm?
Option 1: 150 cm2
Option 2: 600 cm2
Option 3: 100 cm2
Option 4: 80 cm2
Question : If the length of one side of a rectangle is twice that of the other side, and its perimeter is 54 cm, then what is the length of the longer side of this rectangle?
Option 1: 14 cm
Option 2: 9 cm
Option 3: 27 cm
Option 4: 18 cm
Question : Each side of a square is 12 cm long. The perimeter of this square is equal to the perimeter of a rectangle whose length is 16 cm. What will be the area of this rectangle?
Option 1: 128 cm2
Option 2: 112 cm2
Option 3: 184 cm2
Option 4: 156 cm2
Question : The total surface area of a closed cube is 128 cm2 more than the area of a square. If the length of each side of the cube is 8 cm, then what is the length of each side of the square?
Option 1: 16 cm
Option 2: 12 cm
Option 3: 18 cm
Option 4: 9 cm
Question : The side of a square is equal to 40% of the radius of a sphere. If the volume of the sphere is $\frac{500 \pi}{3} \mathrm{~m}^3$, then what is the area of the square?
Option 1: 2 m2
Option 2: 1 m2
Option 3: 9 m2
Option 4: 4 m2
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