Quantitative Aptitude PYQs

Given:

Time taken to complete the journey = t hours

Time taken to complete the journey after increasing speed = (t - y) hours

Formula used:

Distance = Speed × Time

Calculation:

Let the distance be D and the initial speed of the car be a kmph.

According to the first condition,

D = at      ----(1)

According to the second condition,

D = (a + x)(t - y)      ----(2)

From equation (1) and equation (2),

at = (a + x)(t - y)

⇒ at = at + xt - ay - xy

⇒ ay = x(t - y)

⇒ a = x(t - y)/y

Put the value of a in equation 1,

D = x(t - y)ty^-1

∴ The distance is x(t - y)ty^-1 km.

Given:

Time α √Distance

Time required to cover 78.4-meter distance = 4 s

Concept used:

If time varies as the square root of the distance, then (t₁²/t₂²) = d₁/d₂.

Where, t₁ and t₂ is the time taken whereas d₁ and d₂ is the distance.

Calculation:

We know that,

(t12/t22) = d1/d₂

⇒ 4²/t₂² = 78.4/122.5

⇒ t₂² = (122.5/78.4) × 16

⇒ t₂ = √25

⇒ t₂ = 5

∴ The required time is 5 seconds.

Given:

63 - 4x 4x + 5 = 8 

Concept:

a^m⋅aⁿ = a^{m + n}

\(\rm {a^m\over a^n}\) = a^{m - n}

(a⋅b)ⁿ = aⁿ⋅bⁿ

log a^b = b⋅log a

Calculation:

Given: 63 - 4x 4x + 5 = 8 

⇒ 3^{3 - 4x}⋅ 2^{3 - 4x}⋅ 2^{2x + 10} = 2³                                         (∵(a⋅b)n = an⋅bⁿ)

⇒ 33 - 4x⋅ 23 - 4x + 2x + 10 = 2³                                          (∵am⋅an = a^{m + n})

⇒ 33 - 4x⋅ 2^{13 - 2x} = 2³

⇒ 3^{3 - 4x} = \(\rm {2^3\over {2^{13\ -\ 2x}}}\)

⇒ 3^{3 - 4x} = 2^{3 - (13 - 2x)}                                                     (∵\(\rm {a^m\over a^n}\) =  a^{m - n})

⇒ 3^{3 - 4x} = 2^{2x - 10}

Taking log both side

⇒ log (3^{3 - 4x}) = log (2^{2(x - 5)})

⇒ (3 - 4x)⋅log 3 = 2(x - 5)⋅log 2                                     (∵ log ab = b⋅log a)

⇒ (3 - 4x)(0.477) = 2(x - 5)(0.301)

⇒ 1.4313 - 1.9084x = 0.602x - 3.010

⇒ 1.4313 + 3.010 = 0.602x + 1.9084x

⇒ 4.4413 = 2.5104x

⇒ x = \(\rm {4.4413\over 2.5104}\) 

⇒ x = 1.7692

∴ The value of x lies in between 1 < x < 2.

Concept:

Euclid's algorithm: 

• Euclid's algorithm is basically used to find the GCD of two numbers
• If a and b are two positive integers, then there exist a unique positive integers q and r such that;

a = bq + r, where 0 ≤ r < b 

If b ∣ a then r = 0

Otherwise r must satisfy stronger inequality 0 < r < b.

Explanation:

Using the above concept, we can conclude that,

A euclidean algorithm is used to calculate HCF of two integers.

Given : 

Radius of the sphere is rational 

Concept used : 

If r is the radius of the sphere then, 

Surface area of sphere = 4π r²

Volume of the sphere = (4/3)π r³

π is an irrational number 

Product of a rational and irrational number is always irrational

Calculations : 

Let the radius of the sphere be x cm 

Surface area of the sphere = 4π x = 4x²π     

Volume of the sphere = (4/3)π x = (4/3)x³π 

We can conclude that both surface area and the volume are irrational 

∴ Option 4 is the correct answer.

Given : 

cosec θ - sin θ = m 

sec θ - cos θ = n 

Calculations :

Put θ = 45° 

cosec 45°  - sin 45° = m 

⇒ m = √2 - (1/√2) 

⇒ m = 1/√2 

sec 45° - cos 45° = n 

⇒ n = √2 - (1/√2) 

⇒ n = 1/√2   

Putting value of n and m in the equation 

m^4/3 . n^2/3 + m^2/3 . n^4/3 = (1/√2)^4/3 . (1/√2)^2/3 + (1/√2)^2/3 . (1/√2)^4/3

⇒ (1/√2)² + (1/√2)² 

⇒ (1/2) + (1/2) 

⇒ 1 

∴ Option 2 will be the correct answer.

Given:

cos θ + sec θ = k

Formula used:

sec θ = 1/cos θ 

a² + b² = (a + b)² - 2ab

Calculation:

cos θ + sec θ = k

⇒ cos θ + 1/cos θ = k

On squaring both sides, we get,

⇒ cos²θ + 1/cos²θ + 2 = k²

⇒ cos2θ + 1/cos2θ = k² - 2

According to the question,

sin2θ - tan2θ = (1 - cos²θ) - (sec²θ - 1)

⇒ 2 - (cos2θ + 1/cos2θ)

⇒ 2 - (k2 - 2)

⇒ sin2θ - tan2θ = 4 - k²

∴ The value of sin2θ - tan2θ is 4 - k².

Given:

ΔABC is in a semicircle where AB is a diameter.

Calculation:

ΔABC is in a semicircle where AB is a diameter.

An angle on semicircle = 90°

∠C = 90°

In a triangle, the sum of three angles is 180°

∠A + ∠B + ∠C = 180°

⇒ ∠A + ∠B + 90° = 180°

⇒ ∠A + ∠B = 180° - 90° = 90°

Now, 

cos(A+B) + sin(A + B) 

⇒ cos90° + sin90° = 1

∴ The value of cos(A+B) + sin(A + B)  is 1

Calculations :

Staement 1

sin θ = x + (1/x) 

⇒ sin θ = (x² + 1)/x

⇒ x.sin θ = x² + 1

⇒ x² - x.sin θ + 1 = 0 

For any quadratic equation(ax² + bx + c) real values exist when b² > 4ac

So, 

(- sin θ)² > 4 × 1 × 1 

⇒ sin² θ > 4 

⇒ sin θ > 2 

We know that -1 ≤ sin θ ≤ 1

So no value exist for some real value of x.

Statement 2 

cos θ = x + (1/x) 

⇒ cos θ = (x2 + 1)/x

⇒ x.cos θ = x2 + 1

⇒ x2 - x.cos θ + 1 = 0 

For any quadratic equation(ax2 + bx + c) real values exist when b2 > 4ac

So, 

(- cos θ)2 > 4 × 1 × 1 

⇒ cos2 θ > 4 

⇒ cos θ > 2 

We know that -1 ≤ cos θ ≤ 1

So no value exist for some real value of x. 

So no statement is following. 

∴ Option 4 is the correct choice.

Formula Used:

Interior angle of regular polygon = \(\rm {(n - 2)\cdot\pi\over n}\)

Calculation:

In pentagon n = 5

= \(\rm {(5\ -\ 2)\cdot\pi\over 5}\)

= \(\rm 3\pi \over 5 \) = 108°

∴ The magnitude (in radian) of the interior angle is \(\rm 3\pi \over 5 \)

Shortcut Trickα + β = 15°, α - β = 75°

α = 45° and β = 15° 

α/β = 45/15 = 3/2

α = (3/2)β, ∴ k = 3/2

Alternate Method 

Given :

The difference between the two angles is 15°

And the sum of the angles in radian is (5/12)π = 75° 

And the bigger angle is k times the smaller angle.

Calculations :

Let the bigger angle be α and smaller angle be β 

According to the questions

α - β = 15°     ....(1) 

α + β = 75°      ....(2)             (π = 180°)

Solving equation (1) and (2) we get 

α = 45° and β = 30° 

We get 

α = (3/2)β     (α = kβ, given in question) 

So, 

⇒ k = 3/2 

∴ Option 2 will be the correct answer.

Calculations :

Statement 1

2 sin2 θ - cos θ + 4 = 0

⇒ 2 (1 - cos² θ) - cos θ + 4 = 0 

⇒ 2 - 2 cos² θ - cos θ + 4 = 0 

⇒ 2 cos² θ + cos θ - 6 = 0 

⇒ 2 cos² θ + 4 cos θ - 3 cos θ - 6 = 0 

⇒ 2 cos θ (cos θ + 2) - 3 (cos θ + 2) = 0 

⇒ (2 cos θ - 3)(cos θ + 2) = 0 

⇒ cos θ = 3/2 or -2  

This statement is incorrect as cos θ lies between -1 and 1 

Now statement 2

tan θ + cot θ is more than or equal to 2 

(tan θ + cot θ)/2  ≥ √(tan θ .cot θ)       (Arithmetic mean ≥ Geometric mean)

(tan θ + cot θ)/2 ≥ 1

Tan θ + cot θ ≥ 2  

We can conclude that only statement 2 is correct. 

∴ Option 2 is the correct choice.

Shortcut Trick 

⇒ 44 = (42/360) × 2 × (22/7) × R

⇒ R = 60 m

∴ The radius to be used is 60 m.

Alternate Method 

Given:

Change in direction by angle = 42°

Distance of curve road = 44 m

Concept Used:

Arc Length: The arc length is the measure of the distance along the curved line making up the arc.

It is longer than the straight line distance between its endpoints (which would be a chord).

Formula Used:

Arc Length = 2πR(C/360°)

where:

R = Radius of the arc

C = Central angle of the arc in degrees

Calculation:

Observe the following diagram carefully,

According to the question,

Length of the curved road = Length of the arc of the circle

⇒ 44 = 2πR(C/360°)

⇒ 2πR(C/360°) = 44

⇒ 2 × (22/7) × R × (42°/360°) = 44

⇒ (44/7) × R × (42/360) = 44

⇒ R = 44 × (7/44) × (360/42)

⇒ R = 60 m

∴ The radius to be used is 60 m.

Concept used :

Maximum value of sinθ = 1 

Calculations :

Maximum value of 3 sin θ - 4 = 3 × 1 - 4                (For maximum value we must take sin θ greatest value which is 1)

⇒ 3 - 4 

⇒ -1   

∴ Option 2 will be the correct choice.

Given                       

sin θ + cos θ = √2

Formula used :

sin² θ + cos² θ = 1 

(x + y)³ = x³ + y³ + 3.x.y(x + y) 

Calculations :

(cos² θ + sin² θ)³ = cos⁶ θ + sin⁶ θ + 3.sin² θ.cos² θ(cos² θ + sin² θ) 

⇒ cos⁶ θ + sin⁶ θ + 3.sin² θ.cos² θ(cos² θ + sin² θ) = 1³ = 1 

⇒  sin6 θ + cos6 θ + 3 sin2 θ cos2 θ = 1

Now, 

sin θ + cos θ = √2 

Squarring both the sides 

Sin² θ + cos² θ + 2 sin θ cos θ = 2 

⇒ 2 sin θ cos θ = 1 

⇒ sin θ cos θ = 1/2 

⇒ sin² θ cos² θ = 1/4 

⇒ 3 sin2 θ cos2 θ  = 3/4 

Now,

sin6 θ + cos6 θ + 6 sin2 θ cos2 θ = sin6 θ + cos6 θ + 3 sin2 θ cos2 θ + 3 sin2 θ cos2 θ  

⇒ 1 + (3/4) 

⇒ 7/4 

∴ Option 4 is the correct answer. 

Shortcut Trick⇒ sin θ + cos θ = √2 

At θ = 45° satisfy the above equation.

⇒ sin6 θ + cos6 θ + 6 sin2 θ cos² θ

Put θ = 45° 

⇒  (1/√2)⁶ + (1/√2)⁶ + 6 × (1/√2)² × (1/√2)²

⇒ 1/8 + 1/8 + 6 × (1/2) × (1/2)

⇒ 7/4

∴ The correct answer is option (4).

Concept used :

sin2 θ = 1 - cos2 θ

Calculations :

9 sin2 θ + 16 cos2 θ = 9 sin2 θ + 16 ( 1 - sin2 θ) 

⇒ 9 sin2 θ + 16 - 16 sin2 θ

⇒ -7 sin² θ + 16 

⇒ -7 + 16         (for least value of the equation value of sin² θ must be equal to 1)

⇒ 9 

∴ Option 2 will be the correct choice.

Shortcut Trick Minimum value of a sin² θ + b cos² θ is b if b < a or a if a < b.

The least or minimum value will be either a or b, depending on which is lesser.

⇒ 9 sin2 θ + 16 cos2 θ 

Here, 9 < 16 so the least value is 9.

∴ The correct answer is option (2).

Given :

cos 47° + sin 47° = k

Concept used :

cos 2θ = cos² θ - sin² θ 

cos2 θ + sin2 θ = 1  

2 sin θ cos θ = sin 2θ 

Calculations :

We have

cos 47° + sin 47° = k    

Squaring on both side

(cos 47° + sin 47°)² = k²

We know that,

(a + b)² = a² + 2ab + b²

⇒ cos² 47° + sin² 47° + 2 cos 47° sin 47° = k²

1 + 2 cos 47° sin 47° = k²       (∵ cos2 θ + sin2 θ = 1)

2 cos 47° sin 47° = k² - 1

sin 94°  =  k2 - 1        (∵ 2 sin θ cos θ = sin 2θ )

We know that,

sin² θ + cos² θ = 1 

⇒ sin2 94° + cos2 94° = 1 

⇒ cos2 94° = 1 - (k² - 1)²       [∵ (a - b)2 = a2 - 2ab + b2]

⇒ cos2 94° = 1 - (k⁴ + 1 - 2k²)

⇒ cos2 94° = 1 - k⁴ - 1 + 2k² 

⇒ cos2 94°= 2k² - k⁴

⇒ cos 94° = ± k√(2 - k²),

We know that cos 94° lies on the second quadrant, and cos θ is 

negative in this quadrant. 

Therefore,  

cos 94° = - k√(2 - k2)

⇒ cos 2 × 47 = - k√(2 - k2)

Since cos 2θ = cos2 θ - sin2 θ 

⇒ cos2 47° - sin2 47° =  - k√(2 - k2)

∴ Option 2 will be the correct answer.

Given :

cosec θ - sin θ = p3 and sec θ - cos θ = q³

Concept used :

1 - sin2 θ = cos² θ 

1 - cos2 θ = sin² θ 

Calculations :

cosec θ - sin θ = 1/(sin θ) - sin θ 

⇒ (1 - sin² θ)/sin θ 

⇒  (cos² θ)/(sin θ)

So, 

(cos² θ)/(sin θ) = p³      ....(1) 

Now 

sec θ - cos θ = (1/cos θ) - cos θ 

⇒ (1 - cos² θ)/cos θ 

⇒  (sin² θ)/(cos θ)

So, 

(sin² θ)/(cos θ) = q³      ....(2) 

Divide equation (2) by (1) 

(sin² θ)/(cos θ) ÷ (cos² θ)/(sin θ) = q³/p³ 

⇒ sin³ θ/cos³ θ = q³/p³ 

⇒ tan θ = q/p 

∴ Option 2 is the correct answer.

Given :

0 ≤ α, β ≤ 90° and cos (α - β) = 1

Concept used :

cosθ = 1    at θ = 0°

Calculations :

cos (α - β) = 1

So, 

α - β = 0° 

⇒ α = β  

sin α - sin β + cos α - cos β

⇒ sin α - sin α  + cos α - cos α    (α = β)

⇒ 0 + 0 

⇒ 0 

∴ Option 2 will be the correct answer.

Concept used :

cos (90° - θ) = sin θ

tan (90° - θ) = cot θ  

Calculations :

Statement 1 

cos 61° + sin 29° = sin 29° + sin 29° 

⇒ 2 sin 29° 

⇒ 2 sin 29° < 1     (sin 29° is less than 0.5 as the value of sin θ increases from 0° to 90° and the value of sin 30° is 0.5)

Statement 2 

tan 23° - cot 67° = cot 67° - cot 67° 

⇒ 0  

Only statement 1 is correct 

∴ Option 1 is the correct choice.