Numerical Ability Test - 8

Calculation:

(x2 + 7x + 10) ÷ (x + 2)

⇒ \(\frac{x^2 + 7x + 10}{x + 2}\)

⇒ \(\frac{(x + 2)(x + 5)}{x + 2}\)

⇒ x + 5

∴ (x2 + 7x + 10) ÷ (x + 2) = (x + 5)

Given:

a - 12/a = 1

Calculation:

⇒ a - 12/a = 1

⇒ a² - a - 12 = 0

⇒ a² - 4a + 3a - 12 = 0

⇒ a(a - 4) + 3(a - 4) = 0

⇒ (a - 4)(a + 3) = 0

a = 4 or a = -3

but a > 0 so, a = 4

Then,

⇒ a² + 16/a² = 4² + 16/4² = 17

∴ a² + 16/a² = 17

Concept:

Equations has no solution

\(\frac{a1}{a2} = \frac{b1}{b2} ≠ \frac{c1}{c2}\)

Calculation:

2x + 4y = 6

2x + 4y - 6 = 0

a1 = 2

b1 = 4

c1 = -6

4x + 8y = 6

4x + 8y -6 = 0

a2 = 4

b2 = 8

c2 = -6

\(⇒ \frac{2}{4} = \frac{4}{8} ≠ \frac{-6}{-6}\)

\(⇒ \frac{1}{2} = \frac{1}{2} ≠ \frac{1}{1}\)

∴ 2x + 4y = 6 and 4x + 8y = 6 has no solution.

Given:

x2 + 7x - 10

Calculation:

Comparing the given equation with ax² + bx + c = 0, we get

a = 1, b = 7 and c = -10

We know, the discriminant is b² - 4ac

⇒ 7² - 4 × 1 × -10 = 89

As we can see, b² -  4ac  > 0

⇒ Roots are real

∴ The root of this equation is real

Important Points

If  b2 - 4ac > 0 

⇒ Root will be real

If  b2 - 4ac = 0 

⇒ Roots of equation will be equal

If  b2 - 4ac < 0 

⇒ Roots of equation will be imaginary.

Given:

α + β = 24     ----(1) and α - β = 8     ----(2)

Concept used:

The quadratic equation in terms of roots is x² - (α + β)x + αβ = 0

Calculation:

Adding both the equation,

α + β + α - β = 24 + 8

⇒ 2 α = 32

⇒ α = 16

From eq (1), β = 8 

Now, α β = 16 × 8 = 128

The quadratic equation in terms of roots is x2 - (α + β)x + αβ = 0

⇒ x2 - 24x + 128 = 0

Given:

Root = 2

Equation = 3x² - 2kx + 5 = 0

Concept:

Roots are the solution to the Quadratic equation i.e value of the equation is Zero.

Calculation:

By putting the Root (x = 2) in the equation 3x² - 2kx + 5, we get

= 3(2)² - 2k(2) + 5 = 0

= 3 × 4 - (2 × k × 2) + 5 = 0

= 12 - 4k + 5 = 0

Hence, k = 17/4

Additional Information

For a Quadratic equation: ax² + bx + c = 0

If Roots of the equation are: α and β

Sum of the roots = α + β = - (b/a)

Product of roots = α × β = (c/a)

Other representation: x2 - x(α + β) + α.β = (x - α) (x - β)

Given:

The sum of two numbers = 184

Calculation:

Let the numbers be x and (184 − x)

According to the question,

x × (1/3) - (184 − x)/7 = 8

⇒ (7x - 552 + 3x)/21 = 8

⇒ 7x - 552 + 3x = 8 × 21

⇒ 10x = 168 + 552

⇒ x = 720/10 = 72

One number = 72

Other number = 184 − x = 184 - 72 = 112

∴ The smaller number is 72.

Calculation:

\(\frac{a}{b}=\frac{3}{4}\)

⇒ 4a = 3b

⇒ 4a - 3b = 0      ----(1)

Also, 9a + 5b = 22      ----(2)

On multiplying equation (1) by 5, we get,

⇒ 20a - 15b = 0      ----(3)

On multiplying equation (2) by 3, we get,

⇒ 27a + 15b = 66      ----(4)

On adding equations (3) & (4), we get,

⇒ 47a = 66

⇒ a = 66/47

Given

The cost of 5 chairs and 8 tables is Rs. 6574

Concept used

Linear equation in two variables.

Calculation

Let the number of chairs be x and the table be y 

The cost of 5 chairs and 8 tables is Rs. 6574

⇒ 5x + 8y = Rs.6574

The cost of 10 chairs and 16 tables will be 

⇒ 10x + 16y = ?

It will be 6574 × 2 = Rs.13148

Because the asked quantity is double the quantity given.

Given:

7x + 3y = 4 and 2x + y = 2

Calculation:

7x + 3y = 4      ----(1)

2x + y = 2     ----(2)

By solving,

equation (1) - 3 × equation(2)

⇒ 7x + 3y - 3(2x + y) = 4 - (3 × 2)

⇒ x = -2

Putting the value of x in equation (1)

7x + 3y = 4

⇒ 7 × (-2) + 3y = 4

⇒ y = 6

∴ The value of x and y is -2 and 6.