Solutions: Pair of Linear Equations - Vardaan Learning Institute

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

Ans: (c) Infinitely many solutions
$a_1/a_2 = 1/-3$, $b_1/b_2 = 2/-6$ $= -1/3$. $c_1/c_2 = -5/15 = -1/3$.
Since all ratios are equal, infinite solutions.

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

Ans: (c) 15/4
For parallel lines, $a_1/a_2 = b_1/b_2 \neq c_1/c_2$.
$3/2 = 2k/5$.
$2k \times 2 = 15$.
$4k = 15$ $\Rightarrow k = 15/4$.

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

Ans: (a) 10
No solution: $a_1/a_2 = b_1/b_2 \neq c_1/c_2$.
$1/5 = 2/k$. $k = 10$.

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

Ans: (a) 3, 1
Adding both: $2x = 6 \Rightarrow x = 3$.
$3 + y = 4 \Rightarrow y = 1$.

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

Ans: (c) Intersecting or coincident

6. The sum of the digits of a two-digit number is 9...

Ans: (d) 36
Number = $10x + y$. $x+y=9$.
$10x+y+27 = 10y+x$
$\Rightarrow 9y-9x=27$
$\Rightarrow y-x=3$.
Adding: $2y=12 \Rightarrow y=6, x=3$. Number 36.

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

Ans: (d) No solution
$y=0$ is x-axis. $y=-7$ is parallel to x-axis. Parallel lines meet nowhere.

8. If the lines $3x+2ky-2=0$ and $2x+5y+1=0$ are parallel...

Ans: (b) 15/4
$a_1/a_2 = b_1/b_2 \Rightarrow 3/2 = 2k/5 \Rightarrow 15 = 4k \Rightarrow k = 15/4$.

9. Value of c for conditions of infinite solutions...

Ans: (d) No value
$c/6 = -1/-2 = 2/3$.
$1/2 \neq 2/3$. Condition $a_1/a_2 = b_1/b_2 = c_1/c_2$ is impossible.

10. Assertion: Inconsistent. Reason: Condition...

Ans: (a)
$1/2 = 2/4 = 1/2$. $c_1/c_2 = -4/-12 = 1/3$.
$1/2 = 1/2 \neq 1/3$. Inconsistent (Parallel lines).

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

Ans: x = -2, y = 5
Subtracting eq 2 from eq 1: $7y = 35 \Rightarrow y = 5$.
$2x + 15 = 11 \Rightarrow 2x = -4 \Rightarrow x = -2$.

12. For what value of k will the following system of linear equations have infinite solutions?

Ans: k = 6
$k/12 = 3/k$ $= (k-3)/k$.
$k^2 = 36 \Rightarrow k = \pm 6$.
Check k=6: $6/12 = 3/6 = 3/6$ (True).
Check k=-6: $-6/12 = 3/-6 \neq -9/-6$ (False).

13. Find the values of $\alpha$ and $\beta$...

Ans: $\alpha = 4, \beta = 8$
$2/2\alpha = 3/(\alpha+\beta)$ $= 7/28$ $= 1/4$.
$1/\alpha = 1/4 \Rightarrow \alpha = 4$.
$3/(4+\beta) = 1/4 \Rightarrow 12 = 4 + \beta \Rightarrow \beta = 8$.

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

Ans: x = 1/2, y = 1/3
Let $1/x = u, 1/y = v$. $2u+3v=13$, $5u-4v=-2$.
Solving, $u=2, v=3$. Thus $x=1/2, y=1/3$.

15. A fraction becomes 9/11 if 2 is added...

Ans: 7/9
$(x+2)/(y+2) = 9/11$ $\Rightarrow$ $11x - 9y = -4$.
$(x+3)/(y+3) = 5/6$ $\Rightarrow$ $6x - 5y = -3$.
Solving: $x=7, y=9$.

16. Five years hence, the age of Jacob...

Ans: Jacob: 40, Son: 10
Eq 1: $J+5 = 3(S+5)$ $\Rightarrow$ $J-3S = 10$.
Eq 2: $J-5 = 7(S-5)$ $\Rightarrow$ $J-7S = -30$.
Solving: $J=40, S=10$.

17. Yarn A costs 4 rupees per meter...

Ans: A: 60m, B: 40m
$x+y=100$.
$4x+5y=440$.
Solving: $x=60, y=40$.

18. Solve graphically... Vertices?

Ans: Vertices: (2, 3), (-1, 0), (4, 0)
Intersection (2,3).
Line 1 on x-axis (y=0): $x+1=0 \Rightarrow x=-1$. (-1,0).
Line 2 on x-axis (y=0): $3x-12=0 \Rightarrow x=4$. (4,0).

19. Dimensions of rectangle...

Ans: Length 17, Breadth 9
$(L-5)(B+3) = LB-9$ $\Rightarrow$ $3L-5B-6=0$.
$(L+3)(B+2) = LB+67$ $\Rightarrow$ $2L+3B-61=0$.
Solving: $L=17, B=9$.

20. Cars 100 km apart...

Ans: 60 km/h, 40 km/h
Let speeds be x and y (x > y).
Same direction: $5(x-y)=100 \Rightarrow x-y=20$.
Opposite direction: $1(x+y)=100 \Rightarrow x+y=100$.
Adding: $2x=120 \Rightarrow x=60$. $y=40$.

21. 2 men 7 boys in 4 days...

Ans: Man 15 days, Boy 60 days
$2/m + 7/b = 1/4$.
$4/m + 4/b = 1/3$.
Let $1/m=u, 1/b=v$. $2u+7v=1/4$, $4u+4v=1/3$.
Solving: $u=1/15, v=1/60$.
Man takes 15 days, Boy takes 60 days.

22. Roohi travels 300 km...

Ans: Train 60 km/h, Bus 80 km/h
$60/x + 240/y = 4$.
$100/x + 200/y = 4 + 10/60 = 25/6$.
Solving: $1/x = 1/60 \Rightarrow x=60$. $1/y = 1/80 \Rightarrow y=80$.

23. Case Study: Taxi Charges

Detailed Solution:
(i) $x + 10y = 105$,
$x + 15y = 155$.
(ii) Subs Eq1 from Eq2: $5y = 50$ $\Rightarrow$ $y = 10$. Subs in Eq1: $x = 5$. Fixed charge: ₹5.
(iii) Charge per km: ₹10.
(iv) For 25 km: $x + 25y = 5 + 25(10) = 255$. ₹255.