Pair of Linear Equations in Two Variables Test

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Pair of Linear Equations in Two Variables Test - 55

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Weekly Quiz Competition

Question 1
1 / -0

$$\displaystyle \begin{cases} \dfrac{3}{4}x-\dfrac{1}{2}y=12\\ \\ kx-2y=22 \end{cases}$$
If the above system of equations has no solution, find the value of $$k$$.

A
$$\displaystyle -\frac{4}{3}$$

B
$$\displaystyle -\frac{3}{4}$$

C
$$3$$

D
$$4$$

Solution

Given: $$\dfrac 34 x - \dfrac 12 y -12=0$$ and $$kx-2y-22 = 0$$

$$\therefore a_1 = \dfrac 34, b_1 = -\dfrac 12, c_1 = -12$$

And, $$a_2 = k, b_2 = -2, c_2 = -22$$

Since the given system of equations has no solution,
$$\dfrac {a_1}{a_2} = \dfrac {b_1}{b_2} \neq \dfrac {c_1}{c_2}$$

$$\dfrac {3}{4k} = \dfrac {1}{4} \neq \dfrac {12}{22}$$

Hence, $$k = 3$$
Question 2
1 / -0

$$3x+y=19$$, and $$x+3y=1$$. Find the value of $$2x+2y$$.

A
$$20$$

B
$$18$$

C
$$11$$

D
$$10$$

E
None of the above

Solution
Question 3
1 / -0

Two bus tickets from city A to B and three tickets from city A to C cost Rs. 77 but three tickets from city A to B and two tickets from city A to C cost Rs. 73. What are the fares for cities B and C from A ?

A
Rs. $$4$$, Rs. $$23$$

B
Rs. $$13$$, Rs. $$17$$

C
Rs. $$15$$, Rs. $$14$$

D
Rs. $$17$$, Rs. $$13$$

Solution

Let Rs. $$x$$ be the fare of city B from city A and Rs. $$y$$ be the fare of city C from city A.
Then,
$$2x + 3y = 77 $$...(i)
and
$$3x + 2y = 73$$ ...(ii)
Multiplying (i) by $$3$$ and (ii) by $$2$$ and subtracting,
we get: $$5y = 85$$ or $$y = 17.$$
Putting $$y = 17$$ in (i), we get:$$ x = 13.$$
Question 4
1 / -0

A sum of $$Rs. 6.25$$ is made up of $$80$$ coins which are either $$10$$ paise or $$5$$ paise coins. The number of $$10$$ paise and $$5$$ paise coins are _____ and ____ respectively.

A
$$45 , 35$$

B
$$42 , 38$$

C
$$39, 41$$

D
$$37 , 43$$

Solution
Question 5
1 / -0

Srikant bought two cows for Rs. $$30000$$. By selling one at a loss of $$15\%$$ and the other at a gain of $$19\%$$, he found that selling price of both cows is the same. Find the cost price of each ( in Rs).

A
$$10000, 20000$$

B
$$15000, 15000$$

C
$$17500, 12500$$

D
$$17000, 12500$$

Solution

Let $$x,y$$ be the $$C.P$$ of the $$2$$ cows
$$x+y=3000-1$$ (given)
One is sold at $$15$$% loss and other at gain of $$19$$% S.P of these $$2$$ are equal
$$x-\cfrac { 15 }{ 100 } x=y+\cfrac { 19 }{ 100 } y$$
$$85x=119y\Rightarrow x=\cfrac { 119 }{ 85 } y$$
Using $$1$$ and above result
$$\cfrac { 119 }{ 85 } y+y=30000$$
$$204y=85\times 30000$$
$$y=12,500$$
$$x=30000-12500=17500$$
Cost price of each cow is $$17500,12500$$
Question 6
1 / -0

Solve the following pairs of equations.
$$4x-y-5=0$$; $$x+y-5=0$$

A
$$\left( 2,3 \right) $$

B
$$\left( 3,3 \right) $$

C
$$\left( 4,3 \right) $$

D
None of these

Solution

$$x+y=5$$ .......eq1; $$4x-y=5$$ ......eq2
Add both the equations, we will get $$x=2$$
Put $$x= 2$$ in eq1, we get
$$y=3$$
Question 7
1 / -0

Solve the following pairs of equations.
$$3x-y=0$$; $$x-2=0$$

A
$$x=2,y=6$$

B
$$x=3,y=6$$

C
$$x=4,y=6$$

D
None of these

Solution

As we are give $$3x-y=0$$ & $$x-2= 0 $$
$$\therefore$$ $$x=2$$ &, on substituting the value of $$x$$ in $$3x-y=0$$
$$\therefore y=6$$
Question 8
1 / -0

Solve the following pairs of equations.
$$x-y=0$$; $$y+3=0$$

A
$$\left( -3,-3 \right) $$

B
$$\left( 3,-3 \right) $$

C
$$\left( 2,-3 \right) $$

D
None of these

Solution

Consider $$x-y=0$$
$$\Rightarrow x=y$$
$$y+3=0 \implies y =-3$$
$$\Rightarrow x= y =-3$$
Hence solution for given equations is $$(-3,-3)$$
Question 9
1 / -0

Solve the following pairs of equations.
$$2x=y+1$$; $$x+2y-8=0$$

A
$$x= 2,y=3 $$

B
$$x= 3,y=3 $$

C
$$x= 2,y=2 $$

D
None of these

Solution

$$2x=y+1$$.....(ii) ;

and
$$x+2y=8$$
$$x=8-2y$$.....(i)
Put (i) in (ii)
$$16-4y=y+1$$
$$y=3$$

$$y=3$$ & $$x=2$$
Question 10
1 / -0

Solve the following pairs of equations.
$$x+y=5$$; $$x-y=1$$

A
$$\left( 3,2 \right) $$

B
$$\left( 2,2 \right) $$

C
$$\left( 4,2 \right) $$

D
None of these

Solution

$$x-y = 1$$ ....eq1
$$x+y=5$$.....eq2
Add & subtract both the equations, we will get
$$2X=6$$
$$\implies x=3$$ &
Put the value of x in equ 1
$$y=2$$

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Pair of Linear Equations in Two Variables Test - 55

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Pair of Linear Equations in Two Variables Test - 55

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Question 1

$$\displaystyle -\frac{4}{3}$$

$$\displaystyle -\frac{3}{4}$$

$$3$$

$$4$$

Solution

Question 2

$$20$$

$$18$$

$$11$$

$$10$$

None of the above

Solution

Question 3

Rs. $$4$$, Rs. $$23$$

Rs. $$13$$, Rs. $$17$$

Rs. $$15$$, Rs. $$14$$

Rs. $$17$$, Rs. $$13$$

Solution

Question 4

$$45 , 35$$

$$42 , 38$$

$$39, 41$$

$$37 , 43$$

Solution

Question 5

$$10000, 20000$$

$$15000, 15000$$

$$17500, 12500$$

$$17000, 12500$$

Solution

Question 6

$$\left( 2,3 \right) $$

$$\left( 3,3 \right) $$

$$\left( 4,3 \right) $$

None of these

Solution

Question 7

$$x=2,y=6$$

$$x=3,y=6$$

$$x=4,y=6$$

None of these

Solution

Question 8

$$\left( -3,-3 \right) $$

$$\left( 3,-3 \right) $$

$$\left( 2,-3 \right) $$

None of these

Solution

Question 9

$$x= 2,y=3 $$

$$x= 3,y=3 $$

$$x= 2,y=2 $$

None of these

Solution

Question 10

$$\left( 3,2 \right) $$

$$\left( 2,2 \right) $$

$$\left( 4,2 \right) $$

None of these

Solution

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