Linear Equations- 1 - Free MCQ Practice Test with solutions, SSC CGL

Given the equations:

• x – 2y + 6 = 0 (Equation a)
• 4y - 2x - 14 = 0 (Equation b)

To find the solution, follow these steps:

• Multiply Equation a by 2:
  2x - 4y + 12 = 0
• Add this result to Equation b:
  4y - 2x - 14 = 0
• This leads to:
  -2 = 0

This result indicates that the system of equations does not have a unique solution.

Given the equations:

• -2x - ky - 9 = 0 (Equation a)
• 4x - 10y + 18 = 0 (Equation b)

To find the value of k for which this system has infinite solutions, follow these steps:

• Multiply Equation (a) by 2:
  -4x - 2ky - 18 = 0
• Add this result to Equation (b):
  -4x - 2ky - 18 + 4x - 10y + 18 = 0
• This simplifies to:
  -2ky - 10y = 0
• Factor out y:
  -2ky = 10y
• Divide both sides by y (assuming y ≠ 0):
  -2k = 10
• Thus, solving for k gives:
  k = -5

⇒ 2(z-1)³ + 6(1-z)³ = 32

⇒ 2 [ z³ -1 - 3z(z-1) ] + 6 [1 - z³ - 3z(1-z)] = 32

⇒ 2 [ z³ - 1 -3z² + 3z ] + 6 [ 1 - z³ -3z + 3z² ] = 32

⇒  2z³ - 2 - 6z² + 6z + 6 - 6z³ -18z² + 18z = 32

⇒  -4z³ + 12z² - 12z + 4 = 32

Substract 32 from both sides we get,

⇒  -4z³ + 12z² - 12z + 4 - 32 = 32 - 32

⇒  -4z³ + 12z² - 12z - 28 = 0

⇒ -4( z + 1 )( z² - 4z -7) = 0

⇒ ( z + 1 )( z² - 4z -7) = 0

Then, 

( z + 1 ) = 0

z = -1

or

( z² - 4z -7) = 0

z = 2 + √3i,  2 - √3i

In a period of 1 year, Mike visits Alan 3 times (12 months divided by 4). However this excludes the first time visit and takes into consideration the subsequent visits only. So starting on his 25th birthday, Mike will visit Alan 3(n-25) times up till his nth birthday. However we have to add the first visit as well. So the final answer would be 3n-74 .

We are given

2yz = 1

Observe that there is   in the denominator of (II).

From (I) we know that

Using this in (II) we get,

Correct Answer: A

Let the price per unit of Coffee, Burger, and Ice Cream be C, B and R dollars respectively.

We need to find the value of B.

We are given that Charles spent $28 to buy 4 cups of coffee, 2 burgers and 4 cones of ice cream. Therefore:

4C + 2B + 4R = 28  …………… (I)

We are also given Coffee costs $2 less than Ice Cream. Therefore:

C = R - 2  …………….. (II)

The final piece of information given to us is that price of burger is half the price of ice cream.

B=R/2  ……………. (III)

Using (II) and (III) in (I), we get:

4(R−2)+2∗R/2+4R=28

• 4R−8+R+4R=28

• 9R=36
• R=4

Therefore from (III), we get

B=4/2

• B = 2

Correct Answer: B

Let’s say the number of strawberry flavored cakes and number of chocolate flavored cakes are S and C respectively.

⇒ S + C = 12 …………… (I)
We need to find the value of C
Step 1: Analyze Statement 1
Statement 1 says that for every strawberry-flavored cake, two chocolate-flavored cakes were baked.
This means:
if S straw-berry flavored cakes were baked, then 2S chocolate-flavored cakes would be baked.
Therefore, number of chocolate-flavored cakes, C = 2S
⇒ C = 2S ……………… (II)
Using (II) in (I), we get
S + 2S = 12
⇒ 3S = 12
⇒ S = 4
Therefore from (II)
C = 2 * 4
⇒ C = 8
Therefore statement 1 alone is sufficient to arrive at a unique answer.

Step 2: Analyze Statement 2
Statement 2 says that both varieties of cakes are sold at $2 apiece and that the total revenue was $24.
Therefore
2 * S + 2 * C = 24
⇒  2 * (S + C) = 24
⇒  S + C =12
But we already know this from (I).
Statement 2 doesn’t provide us with any new information.
So statement 2 is not sufficient to arrive at a unique answer.
   Correct Answer: A

Let the current ages of Jay, Mary and Hans be J, M and H years respectively. We need to find the age of Mary after four years. In other words, we need to find the value of M+4

It is given that presently, Jay’s age is twice the age of Mary.

• J = 2M ……………….. (I)

It is also given that four years later, Jay’s age will be twice the age of Hans at that time.

• J + 4 = 2(H + 4)
• J + 4 = 2H + 8
• J - 2H - 4 = 0  ………….. (II)

The final piece of information given to us is that Jay is 20 years older than Hans today.

• J = H + 20
• H = J - 20  ……………….. (III)

Using (III) in (II), we get

J -2(J - 20) - 4 = 0

• J - 2J + 40 - 4 = 0
• J = 36

So we have the value of J. Now using this in (I), we get:

2M = J = 36

• M = 18

Therefore, M + 4 = 22

• Correct Answer: C

Step 1: Question statement and Inferences

Note that   which means we can divide both the sides of the equation by y. That would leave us with an equation in x alone, which we can solve.

Step 2: Finding required values

Given:

Dividing both sides of the equation with y, we have

Step 3: Finding the final answer

Answer: Option (E)