The area of a quadrilateral can be calculated by dividing it into two triangles using one of its diagonals. The area of a triangle is given by half the product of its base and corresponding height. In this problem, the diagonal AC serves as the common base for two triangles, \(\triangle ABC\) and \(\triangle ADC\), and the given perpendiculars are their respective heights.
Understand the given information:
Express the area of the quadrilateral:
The area of quadrilateral ABCD is the sum of the areas of \(\triangle ABC\) and \(\triangle ADC\).
\[ \text{Area}(\text{ABCD}) = \text{Area}(\triangle ABC) + \text{Area}(\triangle ADC) \]Calculate the area of each triangle:
The formula for the area of a triangle is \( \frac{1}{2} \times \text{base} \times \text{height} \).
Substitute the areas into the quadrilateral area equation:
\[ 299 = \left( \frac{1}{2} \times d \times 11 \right) + \left( \frac{1}{2} \times d \times 12 \right) \]Simplify and solve for \(d\):
Factor out \( \frac{1}{2} \times d \):
\[ 299 = \frac{1}{2} \times d \times (11 + 12) \] \[ 299 = \frac{1}{2} \times d \times 23 \]Multiply both sides by 2:
\[ 299 \times 2 = d \times 23 \] \[ 598 = 23d \]Divide by 23:
\[ d = \frac{598}{23} \] \[ d = 26 \]So, the length of the diagonal AC is 26 cm.
A) 26 cm. This matches our calculated value for the length of diagonal AC.
The problem presents a series of three-letter codes. We need to identify the pattern in the alphabetical positions of the letters within each code and how these positions change from one code to the next. By finding this pattern, we can determine the missing code.
A) IBD is the correct option, derived by applying the consistent pattern of subtracting 2 from the alphabetical position of each letter in the preceding term.
To solve this problem, we need to track Mr. Chotu's movements in the North-South and East-West directions separately. We will calculate the net displacement in each direction from the starting point (Point A) to the final stopping point (Point P). The shortest distance to return to Point A will be the magnitude of this net displacement, and the direction will be opposite to the net displacement from A to P.
B) 6 km to the West
After tracking all movements, Mr. Chotu ends up 6 km to the East of his starting point A. Therefore, to return to Point A, he must drive 6 km towards the West.
Rajat ranked 16th from the top and 55th from the bottom in his class. How many students are there in his class?
This problem involves calculating the total number of students in a class given a student's rank from both the top and the bottom. The key is to avoid double-counting the student whose rank is provided.
Therefore, there are 70 students in the class.
Correct Option: A) 70
The problem asks us to identify the relationship between the numbers in the given sets and then find an option that follows the same relationship. We are given two example sets: (2, 20, 4) and (5, 61, 6). We need to find a pattern involving the first, second, and third numbers in each set.
Let the numbers be \(A\), \(B\), and \(C\). So, \(A=2\), \(B=20\), \(C=4\).
We need to find a relationship that connects \(A\), \(B\), and \(C\). Let's try operations involving squares or multiplication.
Consider the squares of the first and third numbers:
\[ A^2 = 2^2 = 4 \]\[ C^2 = 4^2 = 16 \]Adding them: \(A^2 + C^2 = 4 + 16 = 20\). This matches the middle number \(B\).
So, the pattern seems to be \(A^2 + C^2 = B\).
Here, \(A=5\), \(B=61\), \(C=6\).
Using the proposed pattern:
\[ A^2 + C^2 = 5^2 + 6^2 = 25 + 36 = 61 \]This also matches the middle number \(B\).
The established pattern is: The middle number is the sum of the squares of the first and third numbers.
D) (3, 25, 4)
Let \(A=3\), \(B=25\), \(C=4\).
Check the pattern: \(A^2 + C^2 = 3^2 + 4^2 = 9 + 16 = 25\).
This matches the middle number \(B\). Therefore, option D follows the same relationship.
Here, \(A=7\), \(B=45\), \(C=2\).
Check the pattern: \(A^2 + C^2 = 7^2 + 2^2 = 49 + 4 = 53\).
Since \(53 \neq 45\), this option does not follow the pattern.
Here, \(A=4\), \(B=40\), \(C=6\).
Check the pattern: \(A^2 + C^2 = 4^2 + 6^2 = 16 + 36 = 52\).
Since \(52 \neq 40\), this option does not follow the pattern.
Here, \(A=1\), \(B=8\), \(C=3\).
Check the pattern: \(A^2 + C^2 = 1^2 + 3^2 = 1 + 9 = 10\).
Since \(10 \neq 8\), this option does not follow the pattern.
We need to find the principal amount invested given the difference between compound interest (CI) and simple interest (SI) for a specific period and rate. We can use the formula for the difference between CI and SI for 2 years or calculate CI and SI separately and then find the principal.
Let the principal amount be \(P\).
The rate of interest \(R = 4\%\) per annum.
The time period \(T = 2\) years.
The difference between CI and SI for 2 years is given as ₹44.
Calculate Simple Interest (SI):
The formula for Simple Interest is:
\[ \text{SI} = \frac{P \times R \times T}{100} \]Substituting the given values:
\[ \text{SI} = \frac{P \times 4 \times 2}{100} = \frac{8P}{100} = \frac{2P}{25} \]Calculate Compound Interest (CI):
The formula for Amount under Compound Interest is:
\[ \text{Amount} = P \left(1 + \frac{R}{100}\right)^T \]Substituting the given values:
\[ \text{Amount} = P \left(1 + \frac{4}{100}\right)^2 = P \left(1 + \frac{1}{25}\right)^2 = P \left(\frac{26}{25}\right)^2 = P \frac{676}{625} \]Compound Interest (CI) is Amount - Principal:
\[ \text{CI} = P \frac{676}{625} - P = P \left(\frac{676 - 625}{625}\right) = P \frac{51}{625} \]Find the difference between CI and SI:
Given that the difference is ₹44:
\[ \text{CI} - \text{SI} = 44 \]\[ P \frac{51}{625} - P \frac{2}{25} = 44 \]To subtract, find a common denominator (625):
\[ P \frac{51}{625} - P \frac{2 \times 25}{25 \times 25} = 44 \]\[ P \frac{51}{625} - P \frac{50}{625} = 44 \]\[ P \frac{51 - 50}{625} = 44 \]\[ P \frac{1}{625} = 44 \]Solve for P:
\[ P = 44 \times 625 \]\[ P = 27500 \]Alternatively, for 2 years, the difference between CI and SI is given by the formula:
\[ \text{Difference} = P \left(\frac{R}{100}\right)^2 \]Substituting the given values:
\[ 44 = P \left(\frac{4}{100}\right)^2 \]\[ 44 = P \left(\frac{1}{25}\right)^2 \]\[ 44 = P \frac{1}{625} \]\[ P = 44 \times 625 \]\[ P = 27500 \]Correct Option: C) ₹27,500
The problem requires us to evaluate a mathematical expression after interchanging certain operators. We need to replace '+' with '−', '−' with '+', '×' with '÷', and '÷' with '×'. After performing these substitutions, we will evaluate the new expression following the order of operations (BODMAS/PEMDAS).
\(187 \div 3 + 168 \times 2 - 174 = ?\)
Substitute these into the original expression:
\(187 \times 3 - 168 \div 2 + 174 = ?\)
First, perform multiplication and division from left to right.
Now, substitute these values back into the expression:
\(561 - 84 + 174 = ?\)
C) 651 is the correct answer after interchanging the operators and performing the calculations according to the order of operations.
This is a linear arrangement puzzle where seven boxes are stacked one above the other. We need to deduce the exact order of the boxes based on the given clues and then identify the box second below T.
Clue 1: No box is kept above F.
This means F is the topmost box.
Clue 2: Only three boxes are kept between F and S.
Since F is at the top, S must be at position 5 from the top.
Clue 3: E is kept immediately above S.
This places E at position 4 from the top.
Clue 4: Only one box is kept between D and E.
Since E is at position 4, D can be at position 2 (above E) or position 6 (below E). Let's consider both possibilities.
Case 1: D is above E
\[ \begin{array}{|c|} \hline \text{F} \ \hline \text{D} \ \hline \ \hline \text{E} \ \hline \text{S} \ \hline \ \hline \ \hline \end{array} \]Clue 5: Only four boxes are kept between D and G.
In Case 1, D is at position 2. Counting four boxes down from D, G would be at position 7. This is a valid placement.
Clue 6: T is kept at some place above U.
The remaining boxes are T and U. The empty slots are position 3 and position 6. For T to be above U, T must be at position 3 and U at position 6.
Let's check if Case 2 (D is below E) is possible. If D is at position 6, then there are 4 boxes between D and G. This would place G at position 1 (F) or position 11 (not possible as there are only 7 boxes). So, Case 2 is not possible.
The final arrangement is:
The question asks: Which box is kept second below T?
T is at position 3. The box immediately below T is E (position 4). The box second below T is S (position 5).
Correct Option: D) S
S is the box kept second below T in the final arrangement.
The question asks for a key feature of Business Process Outsourcing (BPO), which emerged due to globalisation. We need to identify the characteristic that best defines BPO in the context of global economic trends.
Correct Option: C) Providing services to foreign companies from India