Linear Equations

We are tasked with finding the point of intersection of the pair of linear equations:

Step 1: Solve the system of equations.

Rewrite the equations in standard form:

Substitute into the second equation:

Simplify:

Substitute into :

Final Answer: The point of intersection is , which corresponds to option .

Step 1: Define variables.

Let the present age of Ramu be and the present age of Somu be .

Step 2: Translate the conditions into equations.

• Five years ago, Ramu was thrice as old as Somu:
• Ten years later, Ramu will be twice as old as Somu:

Step 3: Simplify the equations.

From the first equation:

From the second equation:

Step 4: Solve the system of equations.

Set the two expressions for equal to each other:

Simplify:

Substitute into :

Final Answer: The present ages of Ramu and Somu are , which corresponds to option .

Step 1: Analyze the system of equations.

The first equation is:

The second equation is:

Step 2: Check if the equations are consistent or dependent.

Multiply the first equation by 2:

This shows that the second equation is simply a multiple of the first equation. Hence, the two equations represent the same line.

Step 3: Determine the number of solutions.

Since the two equations represent the same line, they have infinitely many solutions (every point on the line satisfies both equations).

Final Answer: The number of solutions is , which corresponds to option .

Step 1: Represent the problem using trigonometry.

Let the length of the string be . The height of the kite (60 m) is the opposite side, and the string forms the hypotenuse of a right triangle. Using the sine function:

Step 2: Substitute the value of .

The value of is . Substituting this into the equation:

Step 3: Solve for .

Rearrange the equation to isolate :

Rationalize the denominator:

Final Answer: The length of the string is , which corresponds to option .

The volume of soup for one patient is:

The volume for 100 patients is:

Converting to liters (1 liter = 1000 cm³):

Using :

So the answer is approximately 15.4 liters, which corresponds to option C.

We are given two equations:
1.
2.
These two equations represent two distinct, parallel lines on the coordinate plane. Since the lines are parallel and do not intersect, they have no solution.

The correct option is (D): no solution

The system of linear equations will have a unique solution if the determinant of the coefficient matrix is non-zero. The given system is: The coefficient matrix is: The determinant of this matrix is: For a unique solution, the determinant must not be zero: Solving for : Thus, the value of must not be 3 for the system to have a unique solution.

The correct option is (B):

Given equations:

...(1)

...(2)

Step 1: Convert the equations into the standard form

Comparing with and , we get:

Step 2: Check the consistency condition

We calculate the ratios:

, ,

Step 3: Analyze the results

Since , the given pair of equations has a unique solution.

Final Answer: Unique solution

Let:

• x be the number of 50 notes
• y be the number of 100 notes

We can set up a system of two equations:

1. Total Amount: 50x + 100y = 2000
2. Total Notes: x + y = 30

From equation (2), we can express x in terms of y:

x = 30 - y

Substitute this value of x into equation (1):

50(30 - y) + 100y = 2000

1500 - 50y + 100y = 2000

50y = 500

y = 10

Now substitute y = 10 back into x = 30 - y:

x = 30 - 10

x = 20

Nimra received 20 notes of 50 and 10 notes of 100.