Quantitative Ability Test - 4

Slope of line passing through points (x₁, y₁) and (x₂, y₂) is given by

The orthocenter is the point where all the three altitudes of the triangle cut or intersect each other. Here, the altitude is the line drawn from the vertex of the triangle and is perpendicular to the opposite side.

Slope of line passing through (3, 0) and (0, 3) = (3 − 0) / (0 − 3) = −1

Slope of line passing through (−3, 0) and (0, 3) = (3 − 0 / (0 + 3) = 1

∴ The above two lines are perpendicular to each other since the product of their slopes is equal to −1.

Now, if we draw perpendiculars from the vertex (−3, 0) and vertex (3, 0), they will intersect at the vertex (0,3).

∴ (0,3) is the orthocentre of the triangle.

Hence, the correct option is (C).

Let us check for each option

(A) Given:

f(x) = |x|, ∀ x ∈ R

As we know that,

f(x) = |x|

⇒ f(x) = {−x, x< 0x ,x ≥ 0

So, f(−1) = −(−1) = 1 and f(1) = 1

⇒f(−1) = f(1), but −1 ≠ 1

∴ The property, f(x₁) = f(x₂) ⇒x1 = x₂, does not hold true ∀ x₁, x₂ ∈ R.

Therefore, the function f(x) = |x|, ∀ x ∈ R is not an injective function.

(B) Given:

f(x) = x₂, ∀ x ∈ R

Let x₁ = 1 and x₂ = −1

⇒ f(x¹) = x₁² = 1

and, f(x²) = x₂² = 1

⇒ f(x₁) = f(x₂), but −1 ≠ 1

∴ The property, f(x₁) = f(x₂) ⇒ x₁ = x₂, does not hold true ∀ x₁, x₂ ∈ R.

Therefore, the function f(x) = x₂, ∀ x ∈ R is not an injective function.

(C) Given:

f(x) = −x,∀ x ∈ R

Let x₁ and x₂ be any two real numbers.

⇒ f(x₁) = −x1 and f(x₂) = −x₂

If f(x1) = f(x₂)

⇒ −x₁ = x₂

⇒ x₁ = x₂

∴ The property, f(x₁) = f(x₂)

⇒ x₁ = x₂, holds true ∀x₁, x₂ ∈ R

Therefore, the function f(x)=−x, ∀x ∈ R is an injective function.

Hence, the correct option is (C).

Given

Percentage of girls in the college = 45%

Percentage of girls who voted for Anushka = 60%

Percentage of boys who voted for Anushka = 80%

Let the number of students be 100.

Then according to the question,

Number of girls = 45% of 100 = 45

Number of boys = 100 – 45 = 55

Now, Votes cast by girls = 60 / 100 × 45 = 27

Votes cast by boys = 80 / 100 × 55 = 44

Votes cast by both = (27 + 44) = 71

% of votes cast by both = 71 / 100 × 100 = 71%

∴ Votes obtained by Anushka is 71%.

Hence, the correct option is (C).

Given,

The odds against an event A are 5 : 3

Probability of not occurring the individual event A = 5/8

Probability of occurring the individual event A = 3/8

Again,

Odds in favor of another independent event B and 6 : 5

Probability of occurring the individual event B = 6 / (6 + 5) = 6 / 11

Probability of not occurring the individual event B = 5 / (6 + 5) = 5 / 11

The chances that neither A nor B occurs is = Probability of not occurring event A× probability of not occurring event B [As the two events are independent therefore multiplication will occur]

= 5 / 8 × 5 / 11

= 25 / 88

Hence, the correct option is (B).

Given equation

2ⁿ⁻¹ + 2ⁿ⁺¹ = 320

⇒ 2ⁿ⁻¹(1 + 2²) = 320

⇒ 5 × 2ⁿ⁻¹ = 320

⇒ 2ⁿ⁻¹ = 320 / 5

⇒ 2ⁿ⁻¹ = 64

⇒ 2ⁿ⁻¹ = 26

⇒ n−1 = 6

n = 7

Hence, the correct option is (D).

Given

vertex = (3,0)

Therefore, h = 3 and k = 0.

For focus = (5, 0)

h + a = 5

3 + a = 5

a = 2

So, equation of parabola:

(y − k)² = 4a(x − h)

(y − 0)² = 4 × 2(x − 3)

y² = 8x − 24

Hence, the correct option is (C).

Given:

Measurement of the floor (l × b) = 25 m × 16 m

Measurement of the brick surface (l × b) = 20 cm × 10 cm

Concept:

Area of rectangle = l × b

Let n number of bricks be used to fill the floor.

Area of floor = 25 × 16 = 400 sq. m

Area of one brick surface = 20 × 10 = 200 sq. cm

200 sq. cm = 200 / (100 × 100) = 0.02 sq. m

0.02 × n = 400

n = 400 / 0.02 = 20,000 bricks

Hence, the correct option is (D).

Given that

R = {(a, b) : a² ≥ b}

As we know:

A relation R on a set A is said to be an equivalence relation if R is reflexive, symmetric and transitive.

We know that:

a² ≥ a

Therefore, (a, a) ∈ R, for all a ∈ Z.

Thus, relation R is reflexive.

Let (a, b) ∈ R

⇒ a² ≥ b but b² ≱ a for all a, b ∈ Z.

So, if (a, b) ∈ R, then it does not implies that (b,a) also belongs to R.

Thus, relation R is not symmetric.

Now, let (a, b) ∈ R and (b, c) ∈ R.

⇒ a² ≥ b and ^b2 ≥ c

This does not implies that a² ≥ c, therefore (a, c) does not belong to R for all a, b, c ∈ Z.

Thus, relation R is not transitive.

Hence, the correct option is (D).

Number of males visiting the restaurant on Thursday and Saturday

= 135 + 75 = 210

Number of females visiting the restaurant on Monday and Tuesday = 90 + 125 = 215

Required percentage = (215 − 210)/ 215 × 100

= 5 / 215 × 100

= 100 / 43 ≈ 2.33%

∴ Total number of males visiting the restaurant on Thursday and Saturday is 2.33% less than the number of females visiting the restaurant on Monday and Tuesday.

Hence, the correct option is (C).

Let the speeds of the two trains be x m/sec

and y m/sec respectively.

Then, length of the first train = 27x metres,

and length of the second train = 17y metres.

∴ 27x + 17y / x + y = 23

⇒ 27x + 17y = 23x + 23y

⇒ 4x = 6y

⇒ x/y = 32

Hence, the correct option is (B).