Write all the subsets of \(A\) having a single element.[1]
Which elements of set \(B\) are to be removed to make the sets \(A\) and \(B\) disjoint sets? Write it.[1]
Let A and B are two sets, the A is subset of B if every element of A is also an element of B, it is denoted as \(A \subseteq B\).
From the Venn diagram, \(A = \{3, 4, 5\}\).
The subsets of \(A\) having a single element are \(\{3\}, \{4\}, \{5\}\)
Sets \(A\) and \(B\) overlap because they share the elements \(4\) and \(5\).
To make them disjoint, remove the common elements \(4\) and \(5\) from set \(B\)
The marked price of a laptop is Rs 80000. The shopkeeper allows a \(15\%\) discount on it.
If marked price and discount are represented by MP and D respectively, write the formula to find the discount percentage.[1]
Find the discount amount given on the laptop.[1]
Find the profit or loss of the shopkeeper if he bought the laptop at Rs 65000.[2]
Formula to find discount percentage is Discount Percent = \( \dfrac{\text{Discount}}{\text{MP}} \times 100\% \)
orDiscount Percent = \( \dfrac{\text{MP} - \text{SP}}{\text{MP}} \times 100\% \)
Finding the discount amount
Marked Price (MP) = Rs. \(80,000\)
Discount = \(15\%\) of MP
Therefore Discount = \( \dfrac{15}{100} \times 80,000 = 12,000 \)
So, the discount amount is Rs. \(12,000\).
Finding profit or loss
Cost Price (CP) = Rs. \(65,000\)
Selling Price (SP) = MP – Discount = \(80,000 - 12,000 = 68,000\)
Since SP > CP, there is a profit. Profit = SP – CP = \(68,000 - 65,000 = 3,000\)
So, the shopkeeper makes a profit of Rs. \(3,000\).
A man deposited Rs 60,000 in a bank. If he received Rs 16,200 as interest from the bank after 3 years,
Find the total amount he received from the bank.[1]
Find the rate of interest.[2]
If he divided the received interest between two daughters Rusmita and Susmita in the ratio 2:3, then how much money did Rusmita and Susmita get?[2]
Total amount received:
Principal (P) = Rs. 60,000
Interest (I) = Rs. 16,200
Thus, Amount (A) = P + I = 60,000 + 16,200 = 76,200
So, the total amount received is Rs. 76,200.
Rate of interest:
Principal (P) = Rs. 60,000
Time (T) = 3 years
Interest (I) = Rs. 16,200
We know, Rate (R) = \( \dfrac{I \times 100}{P \times T} = \dfrac{16,200 \times 100}{60,000 \times 3} = 9\% \)
So, the rate of interest is 9% per annum.
Money received by daughters:
Total Interest = Rs. 16,200
Ratio = 2:3 (Rusmita : Susmita)
Sum of ratio = 2 + 3 = 5
Thus, Rusmita's share = \( \dfrac{2}{5} \times 16,200 = 6,480 \) Susmita's share = \( \dfrac{3}{5} \times 16,200 = 9,720 \)
So, Rusmita got Rs. 6,480 and Susmita got Rs. 9,720.
There are \(530\) students in a school.
Express the total number of students in scientific notation.[1]
Find the total number of copies required to distribute to all students at the rate of \(3\) copies per student.[1]
Convert \(0.94\) into a fraction.[1]
A teacher writes his salary in quinary number in the expanded form as
\[
2 \times 5^{6} + 1 \times 5^{5} + 4 \times 5^{4} + 2 \times 5^{3} + 3 \times 5^{2} + 0 \times 5^{1} + 0 \times 5^{0}.
\]
Write the short form of his salary in the quinary system.[1]
In the figure, a rectangular field is shown and a square cottage is constructed in its one corner.
Write the formula to find the area of a rectangle. [1]
Find the area of the cottage. [1]
Find the area of the field excluding the cottage. [2]
How much does it cost to fence the field at the rate of Rs 50 per meter? [1]
Area of rectangle formula
We know that, Area (\( A \)) = \( l \times b \)
Area of the cottage
Given that, The cottage is square with side (\( l \)) = 8 m
Therefore, Area (\( A_1 \)) = \( l^2 = 8^2 = \) 64 m²
Area of the field excluding the cottage
From the figure, Length of field (\( l \)) = 40 m Breadth of field (\( b \)) = 30 m
Calculating total area of field, Total Area (\( A_2 \)) = \( 40 \times 30 = \) 1200 m²
Now to find the area excluding cottage, Required Area = \( A_2 - A_1 = 1200 - 64 = \) 1136 m²
Cost of fencing the field
We know that, Perimeter (\( P \)) = \( 2(l + b) = 2(40 + 30) = 2 \times 70 = \) 140 m
According to the question, the rate is Rs 50 per meter, so Total Cost = \( 140 \times 50 = \) Rs 7,000
Express \(\dfrac{x^{m}}{x^{n}}\) as a power of \(x\).[1]
Let \(x\) and \(y\) be two numbers whose sum is \(4\) and difference is \(2\).
Form the linear equations to represent the given statements.[1]
Solve the equations using the graphical method.[2]
If two algebraic expressions are \(x^{2} + x - 20\) and \(x^{2} - 25\),
Find the Highest Common Factor (H.C.F.) of the given algebraic expressions.[2]
At what values of \(x\) is the expression \(x^{2} + x - 20\) equal to zero?[2]
In the adjoining figure, lines \(DE \parallel BC\). Also, \(\angle BAC = 70^{\circ}\), \(\angle ABC = 3x\), \(\angle ACB = 2x\), and \(\angle EAC = y\) are given.
Write the alternate angle of \(\angle DAB\).[1]
Find the values of \(x\) and \(y\) from the figure.[2]
Compare the values of \(x\) and \(y\).[1]
Construct a parallelogram \(ABCD\) with \(AB = 8cm\), \(BC = 6cm\), and \(\angle ABC = 45^{\circ}\).[3]
In the given figure, if \(AB \parallel CD\), prove that \(\triangle ADB \sim \triangle COD\).[2]
Write down the bearing of point \(P\) from point \(O\).[1]
Find the value of \(m\) if the distance between \(A(2, -1)\) and \(B(m, -5)\) is \(4\sqrt{2}\) units.[2]
Draw a triangle \(ABC\) with vertices \(A(4, 2)\), \(B(3, 5)\), and \(C(6, 5)\) on graph paper. Rotate it through \(+90^{\circ}\) about the origin and show the image on the same graph paper.[3]
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