Quantitative Aptitude PYQs

Calculation:

Given number is

4x² + x

This can be written as

⇒ 4x² + x + (\(1 \over 4\))² - (\(1 \over 4\))²

⇒ (2x + \(1 \over 4\))² - \(1 \over 16\)

Minimum value of (2x + \(1 \over 4\))²  = 0.  

The minimum value of the entire expression is (- \(1 \over 16\)).

To find the value of x that gives this minimum:

Set (2x + \(1 \over 4\))2 = 0

⇒ (2x + \(1 \over 4\)) = 0

⇒ x = - \(1 \over 8\) = - 0.125

∴ The real number is calculated as - 0.125

Alternate MethodCalculation:

According to the question,

⇒ f (x) = x + 4 × x2

For the critical point which is "x ", f ' (x) = 0

Differentiating we get,

⇒ f ' (x) = 1 + 4 × (2x) = 0

⇒ f ' (x) = 1 + 8x = 0 

⇒ 8x = - 1

⇒ x = - 1 / 8 = - 0.125.

∴ The real number is calculated as - 0.125.

Given:

m, n are positive natural numbers, also m > n

 m² - n² = 72 

Formula Used:

(a² -b²) = (a + b) (a - b)

Calculation:

According to the question,

⇒ m2 - n2 = 72

Where,

All the factor pairs of 72 are (1, 72) (2, 36) (3, 24) (4, 18) (6, 12) (8, 9) 

However, there are only three pairs of natural numbers (m, n) that satisfy the equation  m2 - n2 = 72, and m > n.

⇒ (19, 17), (11, 7), and (9, 3).

∴ The number of pairs that satisfy the above conditions is 3.

Given:

N is a 5-digit number.

When N is divided by 6, it leaves a remainder of 2.

When N is divided by 12, it leaves a remainder of 8.

When N is divided by 15, it leaves a remainder of 11.

When N is divided by 24, it leaves a remainder of 20.

Formula Used:

Chinese Remainder Theorem

Calculation:

We know:

\(N ≡ 2 ,(\text{mod}, 6)\)

Means N + 4 will be divisible by 6

\(N ≡ 8 ,(\text{mod}, 12)\)

Means N + 4 will be divisible by 8

\(N ≡ 11 ,(\text{mod}, 15)\)

Means N + 4 will be divisible by 15

\(N ≡ 20 ,(\text{mod}, 24)\)

Means N + 4 will be divisible by 24

Now, to find the common solution, we observe:

Since 6, 12, 15, and 24 have common factors, we need to find the least common multiple (LCM) of these numbers.

\(LCM(6, 12, 15, 24) = 120\)

99999/120 = 833.325

Largest 5 digit number divisible by 6, 12, 15, 24 is:

833 × 120 = 99960

since N + 4 is divisible by all these numbers,

N + 4 = 99960

N = 99960 - 4 = 99956

∴ The correct answer is 99956.

Given:

111²²² + 222³³³ + 333⁴⁴⁴ 

Concept Used:

Every unit digit's power repeats every fourth time except 4 and 6, which repeats every second time.

E.g., 23²⁵,

We first divide 25 by 4, we get the remainder = 1 

Now, unit place digit 3¹ ÷ 4, we get 3 as the remainder.

Calculation:

Similarly,

111²²² :

222 ÷ 4, we get remainder = 2, 

So unit place digit 1² = 1

222³³³ :

333 ÷ 4, we get remainder 1,

So unit place digit 2¹ = 2

333⁴⁴⁴ :

444 ÷ 4, we get remainder = 0 or say 4,

So unit place digit 3⁴ = 1

By summing all the unit place digits = 1 + 2 + 1 = 4

∴ When 4 is divided by 5 we finally get the remainder = 4

Shortcut Trick

Let's consider the last 3 digits of the numbers

⇒ 345 × 789 = 272205

∴ The last three digits of the multiplication is 2, 0, 5

Calculation:

When x = 1, y= 2, z = 3 then

(x2 + y2 + z2) = 1² + 2² + 3² = 14

⇒ p ∝ (x2 + y2 + z2)

⇒ 70 ∝ 14

⇒ 70 = k (14) 

⇒ k = 5 

When, x = -1, y = 1, z = 5, then 

(x2 + y2 + z2) = (1 + 1 + 25) = 27

p = k (x2 + y2 + z2)  

⇒ p = 5 × (27) = 135

∴ The right answer is 135.

Given:

N is a number which is divided by 11 and leaves a remainder of 5

Calculation:

LCM of  6, 12, 15, 18 is 180

According to the question,

180K + 5 is divided by 11

Put the value of K which is less than 11 

So, putting the value of K is 7

⇒ 180 × 7 + 5 = 1265, this is divisible by 11

∴ The answer is 1200 < N < 1300

Concept Used: 

\({1 \over √n + √(n-1)} = {1 \over √ n + √ (n-1)} \times {(√n - √(n-1)) \over(√n - √(n-1))}\)

\({1 \over √n + √(n-1)} = {√n - √(n-1) \over (n - (n-1))} = √ n - √ (n-1)\)

Calculation:

\({1 \over √10 + √9} = { (√10 - √9) \over (√ 10 + √9)(√10 - √9)}= {√ 10 - √9 \over (10 - 9 )}\) = √10 - √9

Similarly, 

\({1 \over √11 + √10} = \) √11 - √10, and \({1 \over √12 + √11} = √12 - √11\) 

putting the values in the equation, 

⇒ (√10 - √9) + (√11 - √10) + (√12 -√11) + ................ + (√196 - √195) 

⇒ (-√9 + √196)

⇒  (-3 + 14) = 11

∴ The value of this expression is 11

Shortcut Trick

Apply Alligation Method on the time taken by each train to cross a man.

Time for X = 24s, Time for Y = 18s, Combined relative time = 20s.

Ratio of speeds (V_X : V_Y) = |18 − 20| : |24 − 20|

Ratio = 2 : 4 = 1 : 2

∴ The correct answer is 1 : 2.

Alternate Method

Given: Time for Train X (t₁) = 24 s, Time for Train Y (t₂) = 18 s, Crossing time (t) = 20 s.

Let speed of Train X be v₁ and Train Y be v₂.

⇒ Length of Train X (L₁) = speed × time = v₁ × 24

⇒ Length of Train Y (L₂) = speed × time = v₂ × 18

When crossing in opposite directions, Relative Speed = v₁ + v₂.

⇒ Time = (L₁ + L₂) ÷ (v₁ + v₂)

⇒ 20 = (24v₁ + 18v₂) ÷ (v₁ + v₂)

⇒ 20v₁ + 20v₂ = 24v₁ + 18v₂

⇒ 20v₂ − 18v₂ = 24v₁ − 20v₁

⇒ 2v₂ = 4v₁

⇒ v₁ / v₂ = 2 / 4 = 1 / 2

∴ The correct answer is 1 : 2.

Additional Information

Relative Speed Principle

When two objects move in opposite directions, relative speed is the sum (S₁ + S₂). In the same direction, it is the difference |S₁ − S₂|.

Crossing Objects

When a train crosses a stationary point object (man, pole), distance = length of train. When it crosses another train/platform, distance = sum of lengths.

Alligation Application

The ratio of the speeds is inversely proportional to the deviation of individual crossing times from the combined crossing time.

Concept Used:

For the quadratic equation ax² + bx + c = 0

Sum of roots α + β = -b/a

Product of roots α.β = c/a

Calculation:

p, q be the roots of x2 + mx - n = 0

By using the above concept

⇒ p + q = -m

⇒ p = - m - q        ----(1)

⇒ pq = -n             -----(2)

Similarly,

m, n are the roots of x2 + px - q = 0 

⇒ m + n = -p        ------(3)

⇒ mn = -q            ------(4)

Now, Put the value of eq (1) in eq (3):

m + n = -p

⇒ m + n = -(-m - q)

⇒ m + n = m + q

⇒ n = q

From equation (4):

mn = - q

⇒ m (q) = - q

⇒ m = -1

From equation (2):

pq = -n

pq = - q

p = -1

Put the value of p = -1 and m = -1 in the equation (1):

p = - m - q 

⇒ -1 = - (-1) - q

⇒ -1 = 1 - q

⇒ q = 1 + 1 = 2

Since, n = q = 2

Now, check statements

Statement (I):  m (m + n) = -1

-1 (-1 + 2) = -1

-1 (1) = -1

-1 = -1 (true)

Statement (II): p + q = 1

• 1+ 2 = 1

1 = 1 (true)

Hence, both the options are correct.

Shortcut Trick The sinθ value lies between 0 ≤ sinθ ≤ 1

So, the max value of sinθ = 1 

Putting this to 8sinθ - 4sin²θ 

⇒  8 × 1 - 4 × 1² 

⇒  8 - 4 = 4

∴ The maximum value of  8sinθ - 4sin²θ is 4.

Alternate Method Let sinθ = x

Then, y =  8sinθ - 4sin²θ = 8x - 4x²

differentiate both sides, in order to find critical values

\( \frac {dy}{dx}\) = 8 - 4 × (2x) = 0

⇒ 8 - 8x = 0 

⇒ x = 1    (put this value in above polynomial)

Then, y = 8 - 4 = 4 

∴ The maximum value of  8sinθ - 4sin²θ is 4.

Given:

The expression is (1 + tanα tanβ)² + (tanα - tanβ)².

Concept:

We will expand the given expression using basic trigonometric identities and simplifications.

Calculation:

⇒ (1 + tanα tanβ)² + (tanα - tanβ)²

⇒ 1 + 2 tanα tanβ + tan²α tan²β + tan²α - 2 tanα tanβ + tan²β

⇒ 1 + tan2α tan2β + tan2α + tan2β

⇒  tan²α(1 + tan2β) + (1 + tan2β)

⇒  (1 + tan²α)(1 + tan2β)

⇒ sec²α sec²β

∴ The correct answer is sec²α sec²β.

Given:

tan 50° - cot 50° = ?

cot 25° - tan 25° = ?

Formula used:

In 1^st quadrant, all value are + ve

1. tan θ = sin θ / cos θ
2. cot θ = cos θ / sin θ

Calculations:

tan 50° - cot 50°

⇒ (sin 50° / cos 50°) - (cos 50° / sin 50°)

⇒ (sin²50° - cos²50°) / (sin 50° × cos 50°)

⇒ - (cos²50° - sin²50°) / (sin 50° × cos 50°)

⇒ - cos 100° / (sin 50° × cos 50°)

Since, - cos 100° = +ve, sin 50° × cos 50° = + ve

Thus, the overall value is Positive ⇒ True

cot 25° - tan 25°

⇒ (cos 25° / sin 25°) - (sin 25° / cos 25°)

⇒ (cos² 25° - sin² 25°) / (sin 25° × cos 25°)

⇒ cos 50° / (sin 25° × cos 25°)

Since, cos 50° = +ve, sin 25° × cos 25° = + ve

Thus, the overall value is Positive ⇒ False

Hence, the correct answer is option (1)

Given:

\(tan(α + β) = √3\)

\(tan(α - β) = 1/√3\)

Formula used:

tan(α + β) = tan(π/3) = √3

tan(α - β) = tan(π/6) = 1/√3

cot(2β) = 1 / tan(2β)

Calculation:

From the given information, we can find the values of α and β:

α + β = π/3

α - β = π/6

Solving these equations simultaneously:

Adding the two equations:

⇒ (α + β) + (α - β) = π/3 + π/6

⇒ 2α = π/2

⇒ α = π/4

Now, substitute α = π/4 in α + β = π/3:

⇒ π/4 + β = π/3

⇒ β = π/3 - π/4

⇒ β = π/12

Now, calculate tan(α) and cot(2β):

tan(α) = tan(π/4) = 1

cot(2β) = 1 / tan(2β)

tan(2β) = tan(2 × π/12) = tan(π/6) = 1/√3

⇒ cot(2β) = 1 / (1/√3) = √3

Finally, calculate tan(α) × cot(2β):

⇒ tan(α) × cot(2β) = 1 × √3 = √3

∴ The correct answer is Option 3.

Given:

sin²θ × cos²θ × (sec²θ + cosec²θ) = ?

Formula used:

1. sec²θ = 1 + tan²θ
2. cosec²θ = 1 + cot²θ
3. sin²θ + cos²θ = 1
4. sec²θ + cosec²θ = 1/sin²θ + 1/cos²θ

Calculations:

sec²θ + cosec²θ = (1/sin²θ) + (1/cos²θ)

⇒ sec²θ + cosec²θ = (sin²θ + cos²θ) / (sin²θ × cos²θ)

⇒ sec²θ + cosec²θ = 1 / (sin²θ × cos²θ)

Now, sin²θ × cos²θ × (sec²θ + cosec²θ)

⇒ sin²θ × cos²θ × [1 / (sin²θ × cos²θ)]

⇒ sin²θ × cos²θ cancels with the denominator.

⇒ The result = 1.

∴ The correct answer is option (2): 1.

Concept:

Calculation:

We have,

\(64^{\sin ^2 θ}+64^{\cos ^2 θ}=16\)

Given that, \(0 \leqslant θ \leqslant \frac{\pi}{2}\), 

As we know, 64^1/2 + 64^1/2 = 8 + 8 = 16,

⇒ Sin²θ = Cos²θ = 1/2

Taking square root on both sides, 

⇒ Sin θ = Cos θ = \(\frac {1}{\sqrt 2}\)

⇒ θ = 45° 

Then tan 45° = cot 45° = 1 

⇒ tan 45° + cot 45° = 1 + 1 = 2 

∴ The right answer is 2

Given:

If cosecθ - cotθ = m and secθ - tanθ = n

Concept used: 

If cosecθ - cotθ = a, then cosecθ + cotθ = 1/a

if secθ - tanθ = b, then secθ + tanθ = 1/b

Formula Used:

cosec²θ - cot²θ = 1

sec²θ - tan²θ = 1

Calculation:

cosecθ - cotθ = m          --(i)

So, cosecθ + cotθ = 1/m  --(ii)

Add equation (i) and (ii),

2 cosec θ = m + 1/m

⇒ cosec θ = 1/2 (m + 1/m)

Now,

secθ - tanθ = n    --(iii)

So, secθ + tanθ = 1/n   --(iv)

Add equations (iii) and (iv),

2 secθ = n + 1/n

⇒ sec θ = 1/2 ( n + 1/n)

So,

cosec θ + sec θ = 1/2 ( m + 1/m) + 1/2(n + 1/n)

⇒ 1/2 ( m + n + 1/m + 1/n)

∴ The correct answer is option (1).

Given:

Height of the point X above the surface of the river = h

Angles of depression of points P and Q = α and β respectively

α and β are complementary angles

Formula Used:

Width of the river = 2h(tanα + cotα)

Calculation:

Since α and β are complementary, β = 90° - α

Let the width of the river be W

From point X, tanα = h / d₁ and tanβ = h / d₂

Here, d₁ and d₂ are the distances from X to points P and Q respectively.

Since β = 90° - α, tanβ = cotα

So, tanα = h / d₁ and cotα = h / d₂

Width of the river W = d₁ + d₂

⇒ d₁ = h / tanα and d₂ = h / cotα

⇒ W = h / tanα + h / cotα

⇒ W = h(tanα + cotα) / tanα.cotα

⇒ tanα.cotα = 1

⇒ W = h(tanα + cotα)

⇒ W = h(\(\frac{\sin\alpha }{\cos \alpha} + \frac{cos \alpha}{\sin\alpha}\))

⇒ W = h(\(\frac{\sin^2\alpha + \cos^2\alpha}{\sin \alpha \cos \alpha}\))

⇒ W = h(\(\frac{1}{\sin \alpha \cos \alpha}\))

⇒ W = hsecα.cosecα

∴ The correct answer is option 4.

Given:

∠ABC = 60°, AB = 6cm and BC = 8 cm

Formula Used:

Area of triangle = 1/2 × AB × BC × sin∠ ABC

Calculations:

Since AB = 6 cm and BC = 8 cm 

∠ ABC = 60° 

⇒ Sin∠ ABC = sin 60° = √ 3/2

⇒ Area = 1/2 × AB × BC × sin 60°

⇒ Area = (1/2) × 6 × 8 × sin 60° 

⇒ Area = 1/2 × 6 × 8 × (√3)/2= 12√3

∴ The right answer is 12√3

Concept used:

ax2 + bx + c = 0, then roots p + q = -b/a, pq = c/a

Calculation:

x² - sin²θx - cos²θ = 0 , when compared to ax² + bx + c = 0,

we get a = 1, b = -sin²θ, c = - cos²θ 

⇒ p + q = -b/a = -(-sin²θ / 1) = sin²θ 

⇒ pq = (-cos²θ / 1) = - cos²θ 

⇒ p² + q² = (p + q)² - 2pq = (sin²θ)² - 2 × (-cos²θ )

⇒ p2 + q2 = (p + q)2 - 2pq = sin⁴θ + 2 cos²θ 

For minimum value, put θ = 0° , 90°, 45° 

At θ = 0°, sin4θ + 2 cos2θ = 0 + 2(1)² = 2

At θ = 90°, sin4θ + 2 cos2θ = (1)⁴ + 2(0) = 1

At θ = 45°, sin4θ + 2 cos2θ = (1/√2)⁴ + 2(1/√2)² = 5/4

we get minimum value at θ = 90° at which p² + q² = 1

∴ The correct answer is 1.