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1. If  $21y5$ is a multiple of 9, where y is a digit, what is the value of $y$?

If each edge of a cube is doubled,

(ii) how many times will its volume increase?

Volume of cube =   Since l becomes 2l, so new volume is :     . Hence volume becomes 8 times.

2. Express the following numbers in usual form.

(v) $5.8\times 10^{12}$

this is the usual from

1. Express the following numbers in standard form.

(v) 31860000000

The standard form is

1. Write the following numbers in standard form.

(iv) 15240000

The standard form 15240000

Q3. Find the value of.

(v) $\left \{ \left (\frac{-2}{3} \right )^{-2} \right \}^{2}$

The detailed explanation for the above-written question is as follows .............. BY using these form of exponential  ......... use this

Q2. Simplify and express the result in power notation with a positive exponent.

(iv)  $(3^{-7}\div 3^{-10})\times 3^{-5}$

The detailed explanation for the above-written question is as follows As we know the exponential form  By using these two form we get,

Q2. Simplify and express the result in power notation with a positive exponent.

(iii) $(-3)^4\times \left(\frac{5}{3} \right )^{4}$

The detailed solution for the above-written question is as follows, We know the exponential formula So,

(iii) Can you draw a rhombus ZEAL where ZE = 3.5 cm, diagonal EL = 5 cm? Why?

Yes, we can draw a rhombus ZEAL where ZE = 3.5 cm, diagonal EL = 5 cm. Because all sides of a rhombus are equal.  Therefore we have four sides and a diagonal to construct the rhombus uniquely.

8  Mohan wants to buy a trapezium shaped field. Its side along the river is parallel to and twice the side along the road. If the area of this field is 10500 $m^{2}$ and the perpendicular distance between the two parallel sides is 100 m, find the length of the side along the river.

Let the length of the side along road be x m. Then according to question, lenght of side along river will be 2x m.             So equation becomes :                               or                               or                                So the length of the side along the river is 2x = 140m.

The diagonals of a rhombus are 7.5 cm and 12 cm. Find its area.

Area of rhombus =

4    The diagonal of a quadrilateral shaped field is 24 m and the perpendiculars dropped on it from the remaining opposite vertices are 8 m and 13 m. Find the area of the field.

The diagonal of a quadrilateral shaped field is 24 m  the perpendiculars are 8 m and 13 m.  the area of the field =

By repeated subtraction of odd numbers starting from 1, find whether the following numbers are perfect squares or not? If the number is a perfect square then find its square root.

(v) 90

We have ,  then  90 - 1 = 89 89 - 3 = 86 86 - 5 =  81 81 - 7 = 74 74 - 9 = 65 65 - 11 = 54 54 - 13 = 41 41 - 15 = 26 ;        26 - 17 = 9. So from the all above calculation, we can say that the given number is not a perfect square.

3. Can a quadrilateral ABCD be a parallelogram if

(ii) $AB = DC = 8 cm, AD = 4 cm$ and $BC = 4.4 cm$?

Opposite sides of a parallelogram are equal in length. Since,   and  are opposite sides and have different length. No, it is not a parallelogram.

Q.2    Mrs. Kaushik has a square plot with the measurement as shown in the figure. She wants to construct a house in the middle of the plot. A garden is developed around the house. Find the total cost of developing a garden around the house at the rate of Rs. 55 per $m^{2}$.

Area of plot =   Area of house =  Area of garden around house =  the rate = 55 per . the total cost of developing a garden around the house=

Q1.  A square and a rectangular field with measurements as given in the figure have the same perimeter. Which field has a larger area?

Given:      Perimeter of square = perimeter of rectangle                                                  Area of square  Area of rectangle =  Hence, area of square is greater than area of rectangle.

Q6. Find the angle measure $x$ in the following figures.

The solution for the above-written question is mentioned below, Sum of angles of a quadrilateral

Q5. What is a regular polygon?
State the name of a regular polygon of
(i) 3 sides

A regular polygon is a polygon which has equal sides and equal angles. The name of a regular polygon of 3 sides is an equilateral triangle. All sides of the equilateral triangle are equal and angles are also equal. Each angle =

Q.The perimeter of a rectangular swimming pool is 154 m. Its length is 2 m more than twice its breadth. What are the length and the breadth of the pool?

Let the breadth of the pool be x m. According to the question, the length of the pool = 2x + 2 m. Perimeter of recatangle = 2(l + b) = 154 m . i.e., 2(2x + 2 + x) = 154     2(3x + 2) = 154 Dividing both sides by 2, we get 3x + 2 = 77 Now transposing 2 to the RHS, we get  3x = 77 - 2 = 75  Dividing both sides by 3, we get Thus breadth of pool = 25 m     and lenght of the pool = 225 + 2 = 52 m

Write the rational number for each point labelled with a letter:

(i) In this, we can see that 1 is divided into 5 parts each, so when we are moving from zero to the right-hand side, it is easy to observe that     All the numbers should contain 5 in their denominator. Thus,   A is equal to   ,  B is equal to ,  C is equal to    ,   D is equal to  ,  E is equal to    (ii) Here we see that 1 is divided in 6 parts each. So when we move from zero towards left we...
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