Quantitative Aptitude PYQs

Given:

2 sin⁴(α) + 2 cos⁴(α) - 1 = 0, where 0 < α < π/2

Formula used:

sin⁴(α) + cos⁴(α) = (sin²(α) + cos²(α))² - 2sin²(α) cos²(α)

We know that sin²(α) + cos²(α) = 1.

Calculation:

The given equation is:

2 sin⁴(α) + 2 cos⁴(α) - 1 = 0

⇒ 2 (sin⁴(α) + cos⁴(α)) = 1

⇒ sin⁴(α) + cos⁴(α) = 1/2

Now, using the identity:

sin⁴(α) + cos⁴(α) = (sin²(α) + cos²(α))² - 2sin²(α) cos²(α)

⇒ (1)² - 2sin²(α) cos²(α) = 1/2

⇒ 1 - 2sin²(α) cos²(α) = 1/2

⇒ 2sin²(α) cos²(α) = 1/2

⇒ sin²(α) cos²(α) = 1/4

Now, take the square root of both sides:

⇒ sin(α) cos(α) = 1/2

Using the identity sin(2α) = 2 sin(α) cos(α):

⇒ sin(2α) = 2 × 1/2 = 1

sin(2α) = 1 implies that: 2α = 90°

Therefore: α = 45°

Now, find sin(2α) + cos(2α):

Since α = 45°:

sin(2α) = sin(90°) = 1

cos(2α) = cos(90°) = 0

Thus: sin(2α) + cos(2α) = 1 + 0 = 1.

sin(2α) + cos(2α) = 1.

∴ The correct answer is option 2.

Given:

(I): 1 − sin⁶α = cos²α (cos⁴α − 3cos²α + 3)

(II): cos⁸α − sin⁸α = 2sin²α (1 − cos⁴α + sin²α cos²α)

Check (I):

LHS = 1 − sin⁶α = (1 − sin²α)(1 + sin²α + sin⁴α)

⇒ = cos²α (1 + (1 − cos²α) + (1 − cos²α)²)

⇒ = cos²α (3 − 3cos²α + cos⁴α)

⇒ RHS ✔

So, (I) is true.

Check (II):

LHS = cos⁸α − sin⁸α = (cos⁴α − sin⁴α)(cos⁴α + sin⁴α)

⇒ (cos²α − sin²α)(cos²α + sin²α)(cos⁴α + sin⁴α)

⇒ (cos 2α)(1)(cos⁴α + sin⁴α)

RHS = 2sin²α(1 − cos⁴α + sin²α cos²α)

This form does not simplify to LHS.

So, (II) is false.

Final Answer:

Only (I) is an identity.

Given:

p = 1/(cosecθ + cotθ)

q = cosecθ

Calculations:

We have,

p = 1/(cosecθ + cotθ) = 1/ (1/sinθ + cosθ/sinθ)

⇒ 1/{(1 + cosθ)/sinθ}

⇒ sinθ / (1 + cos θ)

q = cosec θ = 1/sinθ

So, p² - 2pq

⇒ {sinθ / (1 + cos θ)}² - 2 × {sinθ / (1 + cos θ)} × {1/sinθ}

⇒ {sin²θ / (1 + cos θ)2} - 2 / (1 + cos θ)

⇒ {sin²θ / (1 + cos θ)2 - 2 / (1 + cos θ)

⇒ (1 - cos²θ) /(1 + cos θ)2 - 2 / (1 + cos θ)

⇒ (1 - cosθ)(1 + cosθ) /(1 + cos θ)2 - 2 / (1 + cos θ)

⇒ (1 - cosθ) / (1 + cos θ) - 2 / (1 + cos θ)

⇒ (1 - cosθ - 2) / (1 + cos θ)

⇒ (-1 - cosθ) / (1 + cos θ)

⇒ - (1 + cosθ) / (1 + cos θ)

⇒ - 1

∴ The correct answer is option (1)

Given:

Statement I: (cosec α - sec α) is always positive in the first quadrant.

Statement II: (tan α - cot α) is always negative in the first quadrant.

Formula used:

1. In the first quadrant, all trigonometric functions are positive.
2. cosec α = 1/sin α, sec α = 1/cos α, tan α = sin α/cos α, cot α = cos α/sin α.

Calculation:

For Statement I:

(cosec α - sec α) = 1/sin α - 1/cos α

⇒ Taking LCM:

(cosec α - sec α) = (cos α - sin α)/(sin α × cos α)

Since cos α and sin α are both positive in the first quadrant, the numerator (cos α - sin α) can be positive, zero, or negative depending on the specific value of α. Thus, Statement I is not always true.

For Statement II:

(tan α - cot α) = (sin α/cos α) - (cos α/sin α)

⇒ Taking LCM:

(tan α - cot α) = (sin² α - cos² α)/(sin α × cos α)

In the first quadrant, sin α and cos α are positive. However, sin² α - cos² α is negative because cos α > sin α for α in the first quadrant. Hence, (tan α - cot α) is always negative in the first quadrant.

∴ The correct answer is option (4).

Given:

AB = h

Calculation:

Let OP be the height of the tower,

In Δ ABP,

tan 30° = p/b = AB / AP

⇒ 1/√3 = h / AP

⇒ AP = h √3

Now, In Δ AOP,

tan 60° = p/b = OP/AP

⇒ √3 = OP / h √3

⇒ OP = h √3 × √3

⇒ OP = 3 h

Hence, the height of the tower is 3h.

Given:

Expression = (sin θ) / (1 − cot θ) + (cos θ) / (1 − tan θ), where θ ≠ π/4.

Formula used:

cot θ = cos θ / sin θ, tan θ = sin θ / cos θ.

Simplify each term separately and then add.

Calculation:

First term:

(sin θ) / (1 − cot θ) = (sin θ) / (1 − cos θ/sin θ)

= (sin θ) / ((sin θ − cos θ)/sin θ)

= (sin²θ) / (sin θ − cos θ)

Second term:

(cos θ) / (1 − tan θ) = (cos θ) / (1 − sin θ/cos θ)

= (cos θ) / ((cos θ − sin θ)/cos θ)

= (cos²θ) / (cos θ − sin θ)

= − (cos²θ) / (sin θ − cos θ)

Now, add both terms:

(sin²θ)/(sin θ − cos θ) − (cos²θ)/(sin θ − cos θ)

= (sin²θ − cos²θ)/(sin θ − cos θ)

= −(cos²θ − sin²θ)/(sin θ − cos θ)

= −(cos θ − sin θ)(cos θ + sin θ)/(sin θ − cos θ)

= −(−1)(cos θ + sin θ)

= cos θ + sin θ

∴ The correct answer is (sin θ + cos θ).

Given:

Radius (r) = 4 cm

Length of arc (l) = π cm

Formula used:

Length of arc (l) = r × θ

Where θ = Angle subtended by the arc at the centre (in radians)

Calculation:

l = r × θ

π = 4 × θ

⇒ θ = π / 4

Hence, the magnitude of the angle subtended by the arc at the center is π/4.

The correct answer is option (4).

Given:

cot2θ - \(3\sqrt{3}\) cotθ + 6 = 0

Formula used:

cot2θ - \(3\sqrt{3}\) cotθ + 6 = 0

put cotθ = x

So, x² - \(3\sqrt{3}\)x + 6 = 0

⇒ x² - 2\(\sqrt3\)x - \(\sqrt3\)x + 6 = 0

⇒ x(x - 2\(\sqrt3\)) - \(\sqrt3\)(x - 2\(\sqrt3\)) = 0

⇒ (x - 2\(\sqrt3\)) (x - \(\sqrt3\)) = 0

so, x = \(\sqrt3\) & 2\(\sqrt3\)

i.e. cot θ = \(\sqrt3\) = cot 30°

So, θ = 30°

Now,

sin θ + cos 2θ

⇒ sin 30 + cos 60

⇒ 1/2 + 1/2 = 1

∴ The correct answer is option (2).

Given:

I. sin²θ = ((x + y)²) / (4xy), where x, y are positive unequal real numbers.

II. sin θ + cos θ = x + 1/x, where x is a positive real number.

Formula used:

For any real angle θ, sin²θ ≤ 1 and |sin θ + cos θ| ≤ √2.

Check whether the given equations satisfy these constraints.

Calculation:

For I: sin²θ = ((x + y)²) / (4xy)

Since x, y > 0 and x ≠ y, (x + y)² > 4xy ⇒ ((x + y)²)/(4xy) > 1

But sin²θ ≤ 1 always.

Hence equation I is not possible.

For II: sin θ + cos θ = x + 1/x

By AM–GM inequality, x + 1/x ≥ 2 for x > 0

Also, |sin θ + cos θ| ≤ √2 ≈ 1.414

So x + 1/x ≥ 2 > √2 ⇒ equation II is also not possible.

∴ Neither I nor II is possible.

Given:

Equation: m²(sin θ - 1) + n²(sin θ + 1) = 0

Range: 0 ≤ θ ≤ π/2

Expression to find: (m² + n²) cos θ - (m² - n²) cot θ

Formula used:

Basic trigonometric identities:

cot θ = cos θ / sin θ

Calculation:

From the given equation:

⇒ m²(sin θ - 1) + n²(sin θ + 1) = 0

⇒ m² sin θ - m² + n² sin θ + n² = 0

⇒ (m² + n²) sin θ + (-m² + n²) = 0

⇒ (m² + n²) sin θ = (m² - n²)

⇒ sin θ = (m² - n²) / (m² + n²)

Now, substitute sin θ into the given expression:

(m² + n²) cos θ - (m² - n²) cot θ

⇒ (m² + n²) cos θ - (m² - n²) × (cos θ / sin θ)

⇒ (m² + n²) cos θ - (m² - n²) × (cos θ / [(m² - n²) / (m² + n²)])

⇒ (m² + n²) cos θ - (m² - n²) × (cos θ × (m² + n²) / (m² - n²))

⇒ (m² + n²) cos θ - (m² + n²) cos θ

⇒ 0

∴ The correct answer is option (4).

Shortcut Trick

sin α + cos α = \(\sqrt{2}\)

It is valid when α = π/4

As, sin π/4 + cos π/4 = 1/\(\sqrt{2}\) + 1/\(\sqrt{2}\) = 2/\(\sqrt{2}\) = \(\sqrt{2}\)

So, sin α - cos α = sin π/4 - cos π/4 = 1/\(\sqrt{2}\) - 1/\(\sqrt{2}\) = 0

So, sin3α − cos3α

⇒ (sin α - cos α) (sin² α + cos²α - sinα cosα)

⇒ 0 (sin2 α + cos2α - sinα cosα)

⇒ 0

Hence, the correct answer is 0.

Alternate Method

Given:

sin α + cos α = √2, where 0 < α < π/2

Formula used:

sin³α - cos³α = (sin α - cos α)(sin²α + sin α cos α + cos²α)

Calculation:

sin α + cos α = √2

⇒ (sin α + cos α)² = (√2)²

⇒ sin²α + cos²α + 2sin α cos α = 2

⇒ 1 + 2sin α cos α = 2

⇒ 2sin α cos α = 1

⇒ sin α cos α = 1/2

Now, sin α - cos α = √(sin²α + cos²α - 2sin α cos α)

⇒ sin α - cos α = √(1 - 1)

⇒ sin α - cos α = 0

Plugging into the formula:

sin³α - cos³α = (sin α - cos α)(sin²α + sin α cos α + cos²α)

⇒ sin³α - cos³α = 0 × (1 + 1/2)

⇒ sin³α - cos³α = 0

∴ The correct answer is option (4).

Given:

Expression: (1 + cot α − cosec α)(1 + tan α + sec α)

Formula used:

cot α = cos α / sin α

cosec α = 1 / sin α

tan α = sin α / cos α

sec α = 1 / cos α

(x − 1)(x + 1) = x² − 1

sin²α + cos²α = 1

Calculations:

(1 + cot α − cosec α)(1 + tan α + sec α)

⇒ (1 + cos α / sin α − 1 / sin α)(1 + sin α / cos α + 1 / cos α)

⇒ ((sin α + cos α − 1) / sin α) × ((cos α + sin α + 1) / cos α)

Multiply numerators:

⇒ ((sin α + cos α − 1)(sin α + cos α + 1)) / (sin α × cos α)

Use difference of squares:

⇒ ((sin α + cos α)² − 1) / (sin α × cos α)

Expand and apply identity:

⇒ (sin²α + cos²α + 2 sin α cos α − 1) / (sin α × cos α)

⇒ (1 + 2 sin α cos α − 1) / (sin α × cos α)

⇒ 2 sin α cos α / (sin α × cos α)

⇒ 2

∴ (1 + cot α − cosec α)(1 + tan α + sec α) = 2

Given:

tan θ = (sin α - cos α) / (sin α + cos α)

θ and α are acute angles, α ≠ π/4.

Formula used:

√(2)sin θ = ?

Calculation:

tan θ = (sin α - cos α) / (sin α + cos α)

⇒ θ = tan^-1[(sin α - cos α) / (sin α + cos α)]

Using trigonometric identity: sin θ = tan θ / √(1 + tan²θ)

⇒ sin θ = [(sin α - cos α) / (sin α + cos α)] / √(1 + [(sin α - cos α) / (sin α + cos α)]²)

Let k = sin α - cos α and m = sin α + cos α.

⇒ sin θ = k / m / √[1 + (k / m)²]

⇒ sin θ = k / √(m² + k²)

⇒ √(2)sin θ = √(2) × (sin α - cos α) / √[(sin α + cos α)² + (sin α - cos α)²]

⇒ √(2)sin θ = √(2) × (sin α - cos α) / √[2(sin²α + cos²α)]

⇒ √(2)sin θ = √(2) × (sin α - cos α) / √(2)

⇒ √(2)sin θ = sin α - cos α

∴ The correct answer is option (1).

Given:

(sin α + 2)(sin α + 4)(sin α − 2)(sin α − 4) = 0

Formula used:

Expression = 0 ⇔ at least one factor = 0. But sin α ∈ [−1, 1]

Calculations:

⇒ sin α + 2 = 0 ⇒ sin α = −2 (not possible)

⇒ sin α + 4 = 0 ⇒ sin α = −4 (not possible)

⇒ sin α − 2 = 0 ⇒ sin α = 2 (not possible)

⇒ sin α − 4 = 0 ⇒ sin α = 4 (not possible)

Number of real values of α = 0.

∴ The correct answer is option 1.

Given:

tan3x × tan6x = 1, 0 ≤ x ≤ 30°

Formula used:

tanA × tanB = 1 ⇒ tanA = cotB = tan(90° − B + k × 180°)

Calculations:

⇒ tan3x = tan(90° − 6x + n × 180°)

⇒ 3x = 90° − 6x + n × 180°

⇒ 9x = 90° + n × 180°

⇒ x = 10° + n × 20°

Check n values for 0 ≤ x ≤ 30°:

n = 0 → x = 10° (valid)

n = 1 → x = 30° (but 3x = 90° → tan3x undefined → reject)

n = −1 → x = −10° (out of range)

∴ x = 10°.

Given:

Expression to evaluate:

\(\frac{(a - b)^2}{(b - c)(c - a)} + \frac{(b - c)^2}{(c - a)(a - b)} + \frac{(c - a)^2}{(a - b)(b - c)} - 3\)

Step 1: Let

x = \(\frac{(a - b)^2}{(b - c)(c - a)}\)

y = \(\frac{(b - c)^2}{(c - a)(a - b)}\)

z = \(\frac{(c - a)^2}{(a - b)(b - c)}\)

So expression becomes: x + y + z − 3

Step 2: Take LCM of all 3 terms:

LCM = (a − b)(b − c)(c − a)

Write all terms with common denominator:

x = \(\frac{(a - b)^2}{(b - c)(c - a)} = \frac{(a - b)^3}{(a - b)(b - c)(c - a)}\)

y = \(\frac{(b - c)^2}{(c - a)(a - b)} = \frac{(b - c)^3}{(b - c)(c - a)(a - b)}\)

z = \(\frac{(c - a)^2}{(a - b)(b - c)} = \frac{(c - a)^3}{(c - a)(a - b)(b - c)}\)

Numerators:

⇒ x + y + z = \(\frac{(a - b)^3 + (b - c)^3 + (c - a)^3}{(a - b)(b - c)(c - a)}\)

Use identity:

If x + y + z = 0 ⇒ x³ + y³ + z³ = 3xyz

Here, let: u = (a − b), v = (b − c), w = (c − a)

Then: u + v + w = 0

So: u³ + v³ + w³ = 3uvw

Apply:

Numerator = (a − b)³ + (b − c)³ + (c − a)³ = 3(a − b)(b − c)(c − a)

Denominator = (a − b)(b − c)(c − a)

So: x + y + z = \(\frac{3(a - b)(b - c)(c - a)}{(a - b)(b - c)(c - a)} = 3\)

Final Expression = x + y + z − 3 = 3 − 3 = 0

∴ The correct answer is: 0

Given:

Expression: \(\dfrac{100!}{100^n}\) is an integer

Formula used:

We need to find the largest n such that 100ⁿ divides 100!

100 = 2² × 5² ⇒ 100ⁿ = 2²ⁿ × 5²ⁿ

So, we must find maximum n such that both 2²ⁿ and 5²ⁿ divide 100!

Calculations:

Powers of 2 in 100!

⇒ [100/2] + [100/4] + [100/8] + [100/16] + [100/32] + [100/64]

⇒ 50 + 25 + 12 + 6 + 3 + 1 = 97

Powers of 5 in 100!

⇒ [100/5] + [100/25] + [100/125]

⇒ 20 + 4 + 0 = 24

Now: 2²ⁿ must ≤ 2⁹⁷ ⇒ 2n ≤ 97 ⇒ n ≤ 48.5

And: 5²ⁿ must ≤ 5²⁴ ⇒ 2n ≤ 24 ⇒ n ≤ 12

So, the limiting factor is powers of 5 ⇒ n ≤ 12

The largest value of n is 12.

∴ The correct answer is option 4.

Given:

Distance traveled East = x meters

Distance traveled South = 40 meters

Distance traveled West = 60 meters

Distance traveled North = 200 meters

PR = 200 meters

Formula used:

Pythagoras theorem: In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

Calculation:

PR = 200 meters

Horizontal distance between P and R = x - 60

Vertical distance between P and R = 200 - 40 = 160

Using Pythagoras theorem:

⇒ (PR)² = (Horizontal distance)² + (Vertical distance)²

⇒ 200² = (x - 60)² + 160²

⇒ 40000 = (x - 60)² + 25600

⇒ (x - 60)² = 40000 - 25600

⇒ (x - 60)² = 14400

⇒ x - 60 = √14400

⇒ x - 60 = 120

⇒ x = 120 + 60

⇒ x = 180

∴ The correct answer is option (2).

Given:

Equation: x² + y² + z² = 3

x, y, and z are integers.

Formula used:

The algebraic identity: (x + y + z)² = x² + y² + z² + 2(xy + yz + zx)

Calculations:

We are given x² + y² + z² = 3, where x, y, and z are integers.

Since x, y, and z are integers, their squares (x², y², z²) must be non-negative integers (0, 1, 4, 9, ...).

The only way to sum three non-negative integer squares to get 3 is if each square is 1

⇒ x² = 1, y² = 1, z² = 1

This implies that x, y, and z can only be either 1 or -1.

Now, let P = (xy + yz + zx) and S = (x + y + z).

Using the algebraic identity:

(x + y + z)² = x² + y² + z² + 2(xy + yz + zx)

⇒ S² = 3 + 2P

Rearrange the equation to solve for P:

⇒ 2P = S² − 3

⇒ P = \((\dfrac{S^2 - 3}{2})\)

Now, we will consider all possible combinations of x, y, and z from {1, -1} and calculate S and P for each case:

Case 1: All three integers are 1.

(x, y, z) = (1, 1, 1)

S = 1 + 1 + 1 = 3

P = \((\dfrac{3^2 - 3}{2})\) = \((\dfrac{9 - 3}{2})\) = \((\dfrac{6}{2})\) = 3

Case 2: Two integers are 1, and one is -1.

Possible combinations: (1, 1, -1), (1, -1, 1), (-1, 1, 1)

For any of these combinations, S = 1 + 1 − 1 = 1

P = \((\dfrac{1^2 - 3}{2})\) = \((\dfrac{1 - 3}{2})\) = \((\dfrac{-2}{2})\) = -1

Case 3: One integer is 1, and two are -1.

Possible combinations: (1, -1, -1), (-1, 1, -1), (-1, -1, 1)

For any of these combinations, S = 1 − 1 − 1 = -1

P = \((\dfrac{(-1)^2 - 3}{2})\) = \((\dfrac{1 - 3}{2})\) = \((\dfrac{-2}{2})\) = -1

Case 4: All three integers are -1.

(x, y, z) = (-1, -1, -1)

S = -1 − 1 − 1 = -3

P = \((\dfrac{(-3)^2 - 3}{2})\) = \((\dfrac{9 - 3}{2})\) = \((\dfrac{6}{2})\) = 3

The distinct values that (xy + yz + zx) can have are 3 and -1.

Therefore, there are 2 possible values for (xy + yz + zx).

Given:

x + y + z = 10

xy + yz + zx = 18

Formula used:

The expression for the sum of cubes of three numbers can be expressed as:

x³ + y³ + z³ - 3xyz = (x + y + z) [(x + y + z)² - 3(xy + yz + zx)]

Calculation:

Substituting the given values:

x³ + y³ + z³ - 3xyz = (10) [(10)² - 3(18)]

⇒ x³ + y³ + z³ - 3xyz = 10 [100 - 54]

⇒ x³ + y³ + z³ - 3xyz = 10 × 46

⇒ x³ + y³ + z³ - 3xyz = 460

∴ The correct answer is option (3).