Vardaan Learning Institute

1. The pair of equations $x+2y-5=0$ and $-3x-6y+15=0$ have:

1. A unique solution
2. Exactly two solutions
3. Infinitely many solutions
4. No solution

2. If the lines given by $3x + 2ky = 2$ and $2x + 5y = 1$ are parallel, then the value of k is:

1. -5/4
2. 2/5
3. 15/4
4. 3/2

3. The value of k for which the system of equations $x + 2y = 3$ and $5x + ky + 7 = 0$ has no solution is:

1. 10
2. 6
3. 3
4. 1

4. The solution of the equations $x-y=2$ and $x+y=4$ is:

1. 3, 1
2. 4, 0
3. 1, 3
4. 2, 2

5. If a pair of linear equations is consistent, then the lines will be:

1. Parallel
2. Always coincident
3. Intersecting or coincident
4. Always intersecting

6. The sum of the digits of a two-digit number is 9. If 27 is added to it, the digits get reversed. The number is:

1. 27
2. 72
3. 45
4. 36

7. The pair of equations $y = 0$ and $y = -7$ has:

1. One solution
2. Two solutions
3. Infinitely many solutions
4. No solution

8. If the lines $3x+2ky-2=0$ and $2x+5y+1=0$ are parallel, then what is the value of $k$?

1. $4/15$
2. $15/4$
3. $4/5$
4. $5/4$

9. The value of $c$ for which the pair of equations $cx-y=2$ and $6x-2y=3$ will have infinitely many solutions is:

1. 3
2. -3
3. -12
4. No value

10. Assertion (A): The pair of linear equations $x+2y-4=0$ and $2x+4y-12=0$ are inconsistent.
Reason (R): If $a_1/a_2 = b_1/b_2 \neq c_1/c_2$, then the pair of linear equations has no solution.

1. Both A and R are true and R is the correct explanation of A.
2. Both A and R are true but R is not the correct explanation of A.
3. A is true but R is false.
4. A is false but R is true.

11. Solve for x and y: $2x + 3y = 11$ and $2x - 4y = -24$.

12. For what value of k will the following system of linear equations have infinite solutions?
$kx + 3y = (k-3)$, $12x + ky = k$

13. Find the values of $\alpha$ and $\beta$ for which the following pair of linear equations has infinitely many solutions:
$2x + 3y = 7$, $2\alpha x + (\alpha + \beta)y = 28$

14. Solve for x and y: $\frac{2}{x} + \frac{3}{y} = 13$, $\frac{5}{x} - \frac{4}{y} = -2$.

15. A fraction becomes 9/11 if 2 is added to both the numerator and the denominator. If 3 is added to both the numerator and the denominator it becomes 5/6. Find the fraction.

16. Five years hence, the age of Jacob will be three times that of his son. Five years ago, Jacob's age was seven times that of his son. What are their present ages?

17. Yarn A costs 4 rupees per meter and Yarn B costs 5 rupees per meter. A total of 100 meters of yarn was purchased for 440 rupees. How much of each yarn was purchased?

18. Solve graphically the system of linear equations: $x - y + 1 = 0$ and $3x + 2y - 12 = 0$. Determine the coordinates of the vertices of the triangle formed by these lines and the x-axis.

19. The area of a rectangle gets reduced by 9 square units, if its length is reduced by 5 units and breadth is increased by 3 units. If we increase the length by 3 units and the breadth by 2 units, the area increases by 67 square units. Find the dimensions of the rectangle.

20. Places A and B are 100 km apart on a highway. One car starts from A and another from B at the same time. If the cars travel in the same direction at different speeds, they meet in 5 hours. If they travel towards each other, they meet in 1 hour. What are the speeds of the two cars?

21. 2 men and 7 boys can do a piece of work in 4 days. The same work is done in 3 days by 4 men and 4 boys. How long would it take one man and one boy to do it?

22. Roohi travels 300 km to her home partly by train and partly by bus. She takes 4 hours if she travels 60 km by train and the remaining by bus. If she travels 100 km by train and the remaining by bus, she takes 10 minutes longer. Find the speed of the train and the bus separately.

23. Case Study: Taxi Charges
A taxi charge in a city consists of a fixed charge together with the charge for the distance covered. For a distance of 10 km, the charge paid is ₹105 and for a journey of 15 km, the charge paid is ₹155.

(i) Form the linear equations representing the situation. (1 Mark)
(ii) Find the fixed charge. (1 Mark)
(iii) Find the charge per kilometer. (1 Mark)
(iv) How much does a person have to pay for travelling a distance of 25 km? (1 Mark)