Pair of Linear Equations in Two Variables Test

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Pair of Linear Equations in Two Variables Test - 20

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Question 1
1 / -0

The solution of $$\sqrt{2}x+\sqrt{3}y=0$$ and $$\sqrt{3}x+\sqrt{8}y=0$$ is

A
$$(1, 2)$$

B
$$(\sqrt{2}, \sqrt{8})$$

C
$$(0, 0)$$

D
$$(\sqrt{3}, \sqrt{2})$$

Solution

$$\sqrt 2 x + \sqrt 3 y = 0 - - - \,\,\,\,\,\left( 1 \right) \times \sqrt 3 $$
$$\sqrt 3 x + \sqrt 8 y = 0 - - - \,\,\,\,\,\left( 2 \right) \times \sqrt 2 $$
subtracting both
$$\left( {\sqrt 6 x + 3y} \right) - \left( {\sqrt 6 x + 4y} \right) = 0 - 0$$
$$ - y = 0 \Rightarrow y = 0$$
From (1) $$\sqrt 2 x + \sqrt 3 y = 0$$
$$ \Rightarrow \sqrt 2 x + 0 = 0$$
$$ \Rightarrow x = 0$$
So, point of intersection is $$(0,0)$$
Question 2
1 / -0

If $$7$$ pens and $$3$$ books cost $$Rs 1395$$, while $$5$$ pens and $$4$$ books cost $$Rs 1665$$. Find the cost of a book

A
$$360$$

B
$$45360$$

C
$$35350$$

D
$$none\ of\ these$$

Solution

Let the cost of $$1$$ pen be $$x$$
and the cost of $$1$$ book be $$y$$
Then $$7x+3y=1395$$-----$$(1) \times 4$$
$$5x+4y=1665--(2) \times 3$$
$$4\times (1)$$ -------- $$3 \times (2)$$
$$(28x+12y)-(15x+12y)=5580-4995$$
$$13x=585$$
$$x=45$$
Putting $$x=45$$ in (1)
$$7 \times 45+3y=1395$$
$$\Rightarrow 315+3y=1395$$
$$\Rightarrow 3y=1395-315$$
$$\Rightarrow 3y=1080$$
$$\Rightarrow y=360$$
Therefore Cost of one pen is Rs $$45$$ and cost of one book is Rs $$360$$
Question 3
1 / -0

The sum of two numbers is $$2$$ and their difference is $$1$$. Find the numbers.

A
$$2,0$$

B
$$1.5,0.5$$

C
$$1,1$$

D
$$-0.5,2.5$$

Solution

Let two numbers be $$x$$ and $$y$$
According to question sum of $$x$$ and $$y$$ is $$2$$ and difference is $$1$$
$$\therefore x+y=2......(1)$$ and $$x-y=1......(2)$$

Now adding equation $$(1)$$ and $$(2)$$
$$x+y=2$$
$$\underline{x-y=1}$$
$$2x\quad=3$$
$$ \Rightarrow x=\dfrac{3}{2}$$
$$\Rightarrow x=1.5$$

Now substitute the value of $$x$$ in equation $$(1),$$
$$1.5+y=2$$
$$\Rightarrow y=2-1.5$$
$$\Rightarrow y=0.5$$
Question 4
1 / -0

To eliminate x, from $$3x+y=7$$ and $$-x+2y=2$$ second equation is multiplied by ______

A
$$1$$

B
$$3$$

C
$$2$$

D
$$-1$$

Solution

$$by\>multiplying\>second\>equation\>with\>3\>,\>we\>get\\-3x+6y=6\\now\>by\>adding\>it\>with\>equation\\3x+y=7,\>values\>of\>unknowns\>can\>be\>calculated$$
Question 5
1 / -0

Solve:
$$4x + \frac{6}{y} = 15$$
$$6x - \frac{8}{y} = 14$$

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

B
$$x=2,y=3$$

C
$$x=3,y=-2$$

D
$$x=3,y=2$$

Solution

Given: $$4x+\frac{6}{y}=15......(1)$$
$$6x-\frac{8}{y}=14......(2)$$
Multiplying equation (1) by 4 and equation (2) by 3 and then adding them,
$$16x+\frac{24}{y}=60$$
$$\underline{18x-\frac{24}{y}=42}$$
$$34x\quad=102\Rightarrow x=3$$
Now putting the value of x in equation (1)
$$4\times3+\frac{6}{y}=15\Rightarrow\frac{6}{y}=15-12\Rightarrow\frac{6}{y}=3\Rightarrow y=\dfrac{6}{3}=2$$
Question 6
1 / -0

Solve for $$x$$ and $$y$$ by using method of substitution :
$$0.2x+0.3y=1.3$$; $$0.4x+0.5y=2.3$$

A
$$-2,-3$$

B
$$2,-3$$

C
$$2,3$$

D
$$-2,3$$

Solution

$$0.2x+0.3y=1.3 \Rightarrow 2x+3y=13$$
$$\Rightarrow x=\dfrac{13-3y}{2}$$
and
$$0.4x+0.5y=2.3\Rightarrow 4x+5y=23$$
$$\Rightarrow 4\left ( \dfrac{13-3y}{2} \right )+5y=23$$
$$\Rightarrow 26-6y+5y=23$$
$$\Rightarrow 3=y$$
$$\therefore x=\dfrac{13-3\times 3}{2}=2$$
Question 7
1 / -0

Solve
$$5x-4y+8=0$$
$$7x+6y-9=0$$.

A

$$x = \cfrac{6}{29}$$ and $$y = -\cfrac{101}{58}$$

B

$$x = -\cfrac{6}{29}$$ and $$y = \cfrac{101}{58}$$

C

$$x = -\cfrac{8}{29}$$ and $$y = \cfrac{101}{58}$$

D
None of these

Solution

$$5x-4y+8 = 0$$...$$\times 7$$
$$7x+6y-9 = 0$$....$$\times 5$$
$$35x-28y+56 = 0$$
$$35x+30y-45 = 0$$
$$58y = 101$$
$$y = \cfrac{101}{58}$$
$$x = -\cfrac{6}{29}$$
Question 8
1 / -0

Solve : $$ ax + by = a - b$$
$$bx - ay = a + b$$

A
$$x=a,y=-b$$

B
$$x=1,y=-1$$

C
$$x=-1,y=1$$

D
$$x=\dfrac {1}{a},y=-\dfrac {1}{b}$$

Solution

$$ax+by=a-b$$ ............ (1)
$$bx-ay=a+b$$ .............. (2)

Multiply equation (1) with $$a$$ and equation (2) with $$b$$ ,

$$a^{2}x+aby=a^{2}-ab$$ ................. (3)

$$b^{2}x-aby=ab+b^{2}$$ ................. (4)

Adding equation (3) and (4), we get

$$a^{2}x+b^{2}x=a^{2}+b^{2}$$

$$\left ( a^{2}+b^{2} \right )x=a^{2}+b^{2}$$

$$\Rightarrow x=1$$

Substitute $$x=1$$ in equation (1),

$$a(1)+by=a-b$$

$$by=-b$$

$$y=-\cfrac{b}{b}=-1$$
Question 9
1 / -0

Solve $$0.2x+0.3y=1.3$$
$$0.4x+0.5y=2.3$$

A
$$x=-2$$ and $$y=3$$

B
$$x=3$$ and $$y=3$$

C
$$x=2$$ and $$y=3$$

D
None of these

Solution

Let $$0.2x+0.3y=1.3$$ ....$$\left(1\right)$$
$$0.4x+0.5y=2.3$$ ....$$\left(2\right)$$
Multiply eqn$$\left(1\right)$$ by $$10$$ we get
$$2x+3y=13$$ ....$$\left(3\right)$$
Multiply eqn$$\left(2\right)$$ by $$10$$ we get
$$4x+5y=23$$ ....$$\left(4\right)$$
Solving equations $$\left(3\right)$$ and $$\left(4\right)$$
$$5\times \left(3\right)\Rightarrow 10x+15y=65$$
$$3\times \left(4\right)\Rightarrow 12x+15y=69$$
Subtracting we get $$12x+15y-10x-15y=69-65=4$$
or $$2x=4$$ or $$x=2$$
Substituting $$x=2$$ in eqn$$\left(3\right)$$ we get
$$2x+3y=13\Rightarrow 2\times 2+3y=13$$ or $$3y=13-4=9$$ or $$y=3$$
Hence $$x=2,y=3$$
Question 10
1 / -0

Given two straight lines $$x + y - 7 = 0$$ and $$x - y + 3 = 0$$ . Find point of intersection of them.

A
$$(2,5)$$

B
$$(-2,-5)$$

C
$$(2,-5)$$

D
$$(-2,5)$$

Solution

$$x+y-7=0.......eq1\\x-y+3=0......eq2$$
Add both the equations
Thus we get, $$2x-4=0\\2x=4\\x=\dfrac{4}{2}\\x=2\\2+y-7=0\\y=7-2=5\\(x,y)=(2,5)$$

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Re-Attempt Weekly Quiz Competition

Pair of Linear Equations in Two Variables Test - 20

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Pair of Linear Equations in Two Variables Test - 20

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SHARING IS CARING

Question 1

$$(1, 2)$$

$$(\sqrt{2}, \sqrt{8})$$

$$(0, 0)$$

$$(\sqrt{3}, \sqrt{2})$$

Solution

Question 2

$$360$$

$$45360$$

$$35350$$

$$none\ of\ these$$

Solution

Question 3

$$2,0$$

$$1.5,0.5$$

$$1,1$$

$$-0.5,2.5$$

Solution

Question 4

$$1$$

$$3$$

$$2$$

$$-1$$

Solution

Question 5

$$x=-3,y=2$$

$$x=2,y=3$$

$$x=3,y=-2$$

$$x=3,y=2$$

Solution

Question 6

$$-2,-3$$

$$2,-3$$

$$2,3$$

$$-2,3$$

Solution

Question 7

$$x = \cfrac{6}{29}$$ and $$y = -\cfrac{101}{58}$$

$$x = -\cfrac{6}{29}$$ and $$y = \cfrac{101}{58}$$

$$x = -\cfrac{8}{29}$$ and $$y = \cfrac{101}{58}$$

None of these

Solution

Question 8

$$x=a,y=-b$$

$$x=1,y=-1$$

$$x=-1,y=1$$

$$x=\dfrac {1}{a},y=-\dfrac {1}{b}$$

Solution

Question 9

$$x=-2$$ and $$y=3$$

$$x=3$$ and $$y=3$$

$$x=2$$ and $$y=3$$

None of these

Solution

Question 10

$$(2,5)$$

$$(-2,-5)$$

$$(2,-5)$$

$$(-2,5)$$

Solution

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