Demo 1

Algebra Quiz

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1. If 5x + 3y = 15 and 2xy = 6 then the value of 5x - 3y will be:

3√5 3√4 3√3 3√2

Detailed Solution

Correct Answer: (a) 3√5

Given:
5x + 3y = 15
2xy = 6 => xy = 3

Formula: We use the algebraic identity (a - b)² = (a + b)² - 4ab.

Here, we apply a variation for our terms: (5x - 3y)² = (5x + 3y)² - 4(5x)(3y)

⇒ (5x - 3y)² = (5x + 3y)² - 60xy

Substitute the given values:

⇒ (5x - 3y)² = (15)² - 60(3)

⇒ (5x - 3y)² = 225 - 180

⇒ (5x - 3y)² = 45

Take the square root of both sides:

⇒ 5x - 3y = √45 = √(9 × 5)

⇒ 5x - 3y = 3√5

2. If 4x² + y² = 40 and xy = 6 then find the value of 2x+y.

4 8 6 5

Detailed Solution

Correct Answer: (b) 8

Formula: We use the identity (a + b)² = a² + b² + 2ab.

Applying this to our terms:

⇒ (2x + y)² = (2x)² + y² + 2(2x)(y)

⇒ (2x + y)² = 4x² + y² + 4xy

Given:
4x² + y² = 40
xy = 6

Substitute the given values into the expanded formula:

⇒ (2x + y)² = (40) + 4(6)

⇒ (2x + y)² = 40 + 24

⇒ (2x + y)² = 64

Take the square root of both sides:

⇒ 2x + y = √64 = 8

3. For what value of m will the system of equation 18x - 72y + 13 = 0 and 7x – my – 17 = 0 have no solutions?

24 12 9 28

Detailed Solution

Correct Answer: (d) 28

For a system of linear equations a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0 to have no solution, the condition is:

a₁/a₂ = b₁/b₂ ≠ c₁/c₂

Given equations:
1. 18x - 72y + 13 = 0 (Here a₁=18, b₁=-72, c₁=13)
2. 7x - my - 17 = 0 (Here a₂=7, b₂=-m, c₂=-17)

Apply the condition:

⇒ 18 / 7 = -72 / -m

⇒ 18 / 7 = 72 / m

Cross-multiply to solve for m:

⇒ 18 × m = 72 × 7

⇒ m = (72 × 7) / 18

⇒ m = 4 × 7

⇒ m = 28

4. Simplify the problem (3x + 2y)² – (3x – 2y)².

9x² - 4y² 12xy 18x² - 8y² 24xy

Detailed Solution

Correct Answer: (d) 24xy

Formula: We use the difference of squares identity: a² - b² = (a + b)(a - b).

Let a = (3x + 2y) and b = (3x - 2y).

Substitute these into the formula:

⇒ [(3x + 2y) + (3x - 2y)] × [(3x + 2y) - (3x - 2y)]

Simplify the terms inside each bracket:

⇒ (3x + 2y + 3x - 2y) × (3x + 2y - 3x + 2y)

⇒ (6x) × (4y)

⇒ 24xy

5. For what value of m will the system of equation 17x + my + 102 = 0 and 23x + 299y + 138 = 0 have an infinite number of solutions?

221 223 220 219

Detailed Solution

Correct Answer: (a) 221

For a system of linear equations a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0 to have an infinite number of solutions, the condition is:

a₁/a₂ = b₁/b₂ = c₁/c₂

Given equations:
1. 17x + my + 102 = 0 (Here a₁=17, b₁=m, c₁=102)
2. 23x + 299y + 138 = 0 (Here a₂=23, b₂=299, c₂=138)

Apply the condition:

⇒ 17 / 23 = m / 299

To solve for m, isolate it:

⇒ m = 17 × (299 / 23)

First, calculate 299 / 23 = 13.

⇒ m = 17 × 13

⇒ m = 221

6. If x + y + 3 = 0 then find the value of x³ + y³ – 9xy + 9.

18 -36 36 -18

Detailed Solution

Correct Answer: (d) -18

Given: x + y + 3 = 0

Formula: We use the identity: If a + b + c = 0, then a³ + b³ + c³ = 3abc.

In this case, let a = x, b = y, and c = 3. Since x + y + 3 = 0, the identity applies.

⇒ x³ + y³ + (3)³ = 3(x)(y)(3)

⇒ x³ + y³ + 27 = 9xy

Rearrange the equation to match the expression we need:

⇒ x³ + y³ - 9xy = -27

Now, we need to find the value of x³ + y³ - 9xy + 9.

Substitute the value we found:

⇒ (-27) + 9 = -18

7. If (22√2) / (4√2 - √(3+√5)) = a + √5b where a, b > 0, then what is the value of (ab) : (a+b)?

7:8 4:7 7:4 8:7

Detailed Solution

Correct Answer: (a) 7:8

First, let's simplify the term √(3+√5) in the denominator.

We multiply and divide by √2 inside the square root: `√( (6+2√5)/2 )`

Now, `6 + 2√5` can be written as `(√5)² + (1)² + 2(√5)(1)`, which is `(√5 + 1)²`.

So, `√( (√5 + 1)² / 2 ) = (√5 + 1) / √2`.

The original expression becomes: `(22√2) / [4√2 - (√5 + 1)/√2]`

Simplify the denominator: `(22√2) / [ (8 - (√5 + 1)) / √2 ]`

⇒ `(22√2 × √2) / (8 - √5 - 1)`

⇒ `44 / (7 - √5)`

Now, rationalize the denominator by multiplying the numerator and denominator by `(7 + √5)`:

⇒ `[44 × (7 + √5)] / [(7 - √5)(7 + √5)]`

⇒ `[44 × (7 + √5)] / (49 - 5)`

⇒ `[44 × (7 + √5)] / 44` = 7 + √5

We are given this equals `a + √5b`. By comparing terms, we get a = 7 and b = 1.

Finally, we need to find the ratio (ab) : (a+b).

⇒ (7 × 1) : (7 + 1) = 7 : 8

8. What will be the solution of the following system of linear equations?
3x - 9y + 4z = 5
2x + 7y + z = 12
3x - z = 0

x = 143/150, y = 31/30, z = 143/51 x = 143/50, y = 31/37, z = 143/50 x = 143/150, y = 31/30, z = 143/50 x = 137/150, y = 31/30, z = 143/50

Detailed Solution

Correct Answer: (c) x = 143/150, y = 31/30, z = 143/50

Given equations:
1. 3x - 9y + 4z = 5 .....(i)
2. 2x + 7y + z = 12 .....(ii)
3. 3x - z = 0 .....(iii)

From equation (iii), we can easily express z in terms of x: `z = 3x`.

Substitute `z = 3x` into equation (i):
3x - 9y + 4(3x) = 5
3x - 9y + 12x = 5
15x - 9y = 5 .....(iv)

Substitute `z = 3x` into equation (ii):
2x + 7y + (3x) = 12
5x + 7y = 12 .....(v)

Now we have a system of two equations with two variables. Multiply equation (v) by 3 to match the coefficient of x in equation (iv):
3 × (5x + 7y = 12) ⇒ 15x + 21y = 36 .....(vi)

Subtract equation (iv) from (vi):
(15x + 21y) - (15x - 9y) = 36 - 5
30y = 31 ⇒ y = 31/30

Substitute y back into equation (v):
5x + 7(31/30) = 12
5x = 12 - 217/30 = (360-217)/30 = 143/30
x = (143/30) / 5 ⇒ x = 143/150

Finally, find z using `z = 3x`:
z = 3 × (143/150) = 143/50 ⇒ z = 143/50

9. If 2a + b = 10 and 2ab = 9 then, the value of (2a - b) will be:

10 4 8 6

Detailed Solution

Correct Answer: (c) 8

Formula: We use the identity (x - y)² = (x + y)² - 4xy.

Let x = 2a and y = b. The formula becomes: (2a - b)² = (2a + b)² - 4(2a)(b) = (2a + b)² - 8ab.

Given:
2a + b = 10
2ab = 9

We need 8ab for our formula. We can get this by multiplying 2ab by 4: 8ab = 4 × (2ab) = 4 × 9 = 36.

Now substitute the values into the identity:

⇒ (2a - b)² = (10)² - 36

⇒ (2a - b)² = 100 - 36

⇒ (2a - b)² = 64

Take the square root of both sides:

⇒ (2a - b) = √64 = 8

10. If 2x + 3y - 5z = 18, 3x + 2y + z = 29 and x + y + 3z = 17, then what is the value of xy + yz + zx?

32 52 64 46

Detailed Solution

Correct Answer: (b) 52

Given equations:
1. 2x + 3y - 5z = 18 .....(i)
2. 3x + 2y + z = 29 .....(ii)
3. x + y + 3z = 17 .....(iii)

Step 1: Eliminate z.
Multiply equation (ii) by 5 and add it to equation (i):
(15x + 10y + 5z) + (2x + 3y - 5z) = (29 × 5) + 18
17x + 13y = 145 + 18 ⇒ 17x + 13y = 163 .....(iv)

Multiply equation (ii) by -3 and add it to equation (iii):
(-9x - 6y - 3z) + (x + y + 3z) = (29 × -3) + 17
-8x - 5y = -87 + 17 ⇒ 8x + 5y = 70 .....(v)

Step 2: Solve for x and y.
We have: 17x + 13y = 163 and 8x + 5y = 70.
Multiply (iv) by 5 and (v) by -13:
85x + 65y = 815
-104x - 65y = -910
Adding these two gives: -19x = -95 ⇒ x = 5.

Substitute x=5 into (v): 8(5) + 5y = 70 ⇒ 40 + 5y = 70 ⇒ 5y = 30 ⇒ y = 6.

Step 3: Solve for z.
Substitute x=5 and y=6 into (iii): 5 + 6 + 3z = 17 ⇒ 11 + 3z = 17 ⇒ 3z = 6 ⇒ z = 2.

Step 4: Calculate the final expression.
xy + yz + zx = (5)(6) + (6)(2) + (2)(5)
= 30 + 12 + 10 = 52