Quantitative Aptitude PYQs

Given:

a ∶ b ∶ c ∶ d = \(√4\) ∶ \(√3\) ∶ \(√2\) ∶ \(√1\)

Calculation:

Directly putting the values of a = √4, b = √3, c = √2, d = √1 in the question,

\(\frac{\left(-a^2+b^2+c^2+d^2\right)}{\left(a^2-b^2+c^2-d^2\right)}\)

⇒ \(\frac{\left(-4+3+2+1\right)}{\left(4-3+2-1\right)}\)

⇒ 2/2 = 1

∴ The correct answer is 1.

Given:

Speeds of four cars are 2u, 3u, 4u, and xu.

Time taken by them to cover the same distance is xt, 4t, 3t, and 2t respectively.

Where x, u, t are real numbers.

Concept Used:

Distance = Speed × Time

Calculation:

Since the distance covered by all cars is the same:

Distance₁ = Distance₂ = Distance₃ = Distance₄

Let d be the distance, then:

d = Speed₁ × Time₁

⇒ d = 2u × xt

⇒ d = 2uxt

d = Speed₂ × Time₂

⇒ d = 3u × 4t

⇒ d = 12ut

d = Speed₃ × Time₃

⇒ d = 4u × 3t

⇒ d = 12ut

d = Speed₄ × Time₄

⇒ d = xu × 2t

⇒ d = 2xut

Equating the distances:

2uxt = 12ut

⇒ 2ux = 12u

⇒ x = 12 / 2

⇒ x = 6

Hence, the correct answer is option 2.

Given:

m ∶ n = 1 ∶ 2

p ∶ q = 3 ∶ 4

Concept Used:

To find the ratio (2m + 4p) ∶ (n + 3q), we need to express m, n, p, q in terms of a common variable.

Calculation:

Let m = 1k and n = 2k (since m ∶ n = 1 ∶ 2)

Let p = 3l and q = 4l (since p ∶ q = 3 ∶ 4)

Calculate 2m + 4p:

⇒ 2m + 4p = 2(1k) + 4(3l)

⇒ 2k + 12l

Calculate n + 3q:

⇒ n + 3q = 2k + 3(4l)

⇒ 2k + 12l

Therefore, the ratio (2m + 4p) ∶ (n + 3q) is:

⇒ (2k + 12l) ∶ (2k + 12l)

⇒ 1 ∶ 1

∴ The correct answer is option 1

Given:

Principal (P) = ₹ 10,000

Rate of Interest (R) = 5%

Time (T) = 3 years

Concept Used:

Simple Interest (SI) is calculated using the formula:

SI = P × R × T / 100

Compound Interest (CI) is calculated using the formula:

CI = P (1 + R/100)^T - P

Calculation:

Simple Interest (SI):

SI = ₹ 10,000 × 5 × 3 / 100

⇒ SI = ₹ 1,500

Compound Interest (CI):

CI = ₹ 10,000 (1 + 5/100)³ - ₹ 10,000

⇒ CI = ₹ 10,000 (1.05)³ - ₹ 10,000

⇒ CI = ₹ 10,000 × 1.157625 - ₹ 10,000

⇒ CI = ₹ 11,576.25 - ₹ 10,000

⇒ CI = ₹ 1,576.25

Difference between CI and SI:

Difference = CI - SI

⇒ Difference = ₹ 1,576.25 - ₹ 1,500

⇒ Difference = ₹ 76.25

Hence, the difference between compound interest and simple interest received on ₹ 10,000 after 3 years at a rate of 5% is ₹ 76.25.

Given:

Listed Price of the book = L

Buying Price = (3/4) × L

Selling Price = (1 + 50/100) × L = 1.5 × L

Concept Used:

Percentage Gain = [(Selling Price - Buying Price) / Buying Price] × 100

Calculation:

Buying Price = (3/4) × L

Selling Price = 1.5 × L

Percentage Gain = [(1.5 × L - (3/4) × L) / (3/4) × L] × 100

⇒ Percentage Gain = [(1.5L - 0.75L) / 0.75L] × 100

⇒ Percentage Gain = (0.75L / 0.75L) × 100

⇒ Percentage Gain = 1 × 100

⇒ Percentage Gain = 100%

∴ The correct answer is option 4.

Given:

Difference between the interior and exterior angles of a regular polygon = 144º

Concept Used:

For a regular polygon with n sides:

• Interior angle = (n-2) × 180º / n
• Exterior angle = 360º / n

 

Calculation:

144 = [(n-2) × 180 / n] - (360 / n)

⇒ 144 = (180n - 360 - 360) / n

⇒ 144 = (180n - 720) / n

⇒ 144n = 180n - 720

⇒ 180n - 144n = 720

⇒ 36n = 720

⇒ n = 720 / 36

⇒ n = 20

Hence, the number of sides of the polygon is 20.

Given:

Sum of the roots (S) = 2

Product of the roots (P) = -100

Concept Used:

For a quadratic equation of the form x² + bx + c = 0,

The sum of the roots (α + β) = -b

The product of the roots (αβ) = c

Formula Used:

x² - (Sum of the roots)x + Product of the roots = 0

Calculation:

Using the given sum and product of the roots:

⇒ Sum of the roots (S) = 2

⇒ Product of the roots (P) = -100

Substitute into the formula for the quadratic equation:

⇒ x² - Sx + P = 0

⇒ x² - 2x - 100 = 0

Thus, the quadratic equation is:

x² - 2x - 100 = 0

Hence, the correct answer is option 3.

The Given:

Polynomial: p(x) = x³ + 3x² - 6x - a

One zero of the polynomial: x = 2

Concept Used:

If x = 2 is a zero of the polynomial, then p(2) = 0.

Calculation:

Since x = 2 is a zero,

p(2) = 0:

⇒ 2³ + 3(2)² - 6(2) - a = 0

⇒ 8 + 12 - 12 - a = 0

⇒ 8 - a = 0

⇒ a = 8

Now, P(x) =  x3 + 3x2 - 6x - 8

Since 2 is the zero of the polynomial, so P(x) can be factorized as

P(x) =  x3 + 3x2 - 6x - 8 = (x -2)( x² + 5x + 4)

Now, 

( x2 + 5x + 4)

⇒ (x2 + 4x + x + 4)

⇒  x(x + 4) + 1(x + 4)

⇒ (x + 4)(x +1)

So, other zeroes are -4 and -1

Thus, 

Sum of the squares of the other zeros of the polynomial

⇒ (-4)² + (-1)²

⇒ 16 + 1 = 17 

∴ The correct answer is 17.

Given:

t = cos 79°

Formula Used:

sin²θ + cos²θ = 1

a² - b² = (a - b)(a + b)

Calculation:

cosec 79°cosec 79° (1 - cos 79°)

⇒ \(\frac{1}{sin\ 79\circ}\)(1 - t)

⇒ \(\frac{1}{\sqrt(1 - cos^279\circ)} \times (1 - t)\)

⇒ \(\frac{1 - t}{\sqrt(1 - t^2)}\)

⇒ \(\frac{1 - t}{\sqrt(1 - t)(1 + t)}\)

⇒ \(\sqrt{\frac{1-\mathrm{t}}{1+\mathrm{t}}}\)

∴ The correct answer is \(\sqrt{\frac{1-\mathrm{t}}{1+\mathrm{t}}}\).

Given:

p(x) = x4 + a3x3 + a2x2 + a1x + a0 and q(x) = x4 + b3x3 + b2x2 + b1x +b0

Calculation:

Since α, β, γ , δ are zeroes of p(x),

p(x) = (x - α)(x - β)(x - γ)(x - δ)

α, β, γ, λ are zeroes of q(x),

q(x) = (x - α)(x - β)(x - γ)(x - λ)

⇒ \(\frac{p(x)-q(x)}{(x-\alpha)(x-β)(x-\gamma)}\) 

⇒  \(\frac{[(x - α)(x - β)(x - γ)(x - δ)] - [(x - α)(x - β)(x - γ)(x - λ)]}{(x-\alpha)(x-β)(x-\gamma)}\) 

⇒ \(\frac{(x - α)(x - β)(x - γ)[(x - δ) - (x - λ)]}{(x-\alpha)(x-β)(x-\gamma)}\) 

⇒  λ - δ

∴ The correct answer is λ - δ.

Given:

Equation: \(x \cos θ = x^2 + p\)

Range for \(θ: ( 0 \leq θ \leq \frac{π}{4} )\)

This equation must have a real solution for every (θ ) in the given range.

Calculation:

\(x^2 + p - x \cos θ = 0\)

For real solution,

\(b^2 \geqslant 4ac\)

cos²θ \(\geqslant 4p\)

cosθ will be maximum at 0° = 1

4p \(\leqslant 1\)

\(p \leq \frac{1}{4} \)

∴ The correct answer is \(p \leq \frac{1}{4} \)

Shortcut Trick

Simplify the expression: cos²θ + 3 sin²θ + 2 = (1 − sin²θ) + 3 sin²θ + 2 = 2 sin²θ + 3.

Since the range of sin²θ is [0, 1]:

Least value of the expression = 2 × 0 + 3 = 3.

Greatest value of the expression = 2 × 1 + 3 = 5.

Difference = Greatest value − Least value = 5 − 3 = 2.

∴ The correct answer is 2.

Alternate Method

Given:

Trigonometric expression: cos²θ + 3 sin²θ + 2

Formula Used:

⇒ Trigonometric Identity: cos²θ + sin²θ = 1 ⇒ cos²θ = 1 − sin²θ

⇒ Range of squared sine function: 0 ≤ sin²θ ≤ 1

Calculations:

⇒ Let the expression be E = cos²θ + 3 sin²θ + 2

⇒ Substituting cos²θ = 1 − sin²θ into the expression:

⇒ E = (1 − sin²θ) + 3 sin²θ + 2

⇒ E = 2 sin²θ + 3

⇒ Since 0 ≤ sin²θ ≤ 1, we determine the extreme values:

⇒ For the least value of E, substitute sin²θ = 0:

⇒ Least value = 2 × 0 + 3 = 3

⇒ For the greatest value of E, substitute sin²θ = 1:

⇒ Greatest value = 2 × 1 + 3 = 5

⇒ To find the difference between the greatest and least values:

⇒ Difference = Greatest value − Least value

⇒ Difference = 5 − 3 = 2

∴ The correct answer is 2.

Additional Information

Trigonometric Pythagorean Identities

The primary identities are sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ, and 1 + cot²θ = cosec²θ.

Range of Trigonometric Functions

For any real angle θ, the squared functions have the following boundaries: 0 ≤ sin²θ ≤ 1 and 0 ≤ cos²θ ≤ 1.

General Form Range

The range of the linear combination expression a sin θ + b cos θ is [−√(a² + b²), √(a² + b²)].

Given:

ABC is a right-angled triangle,

AB = 6 cm and BC = 8 cm

Formula Used:

Area of right-angled triangle = (Base × Height)/2

Perimeter of square = 4 × Length of a Side

Calculation:

Let the Length of a side of square inscribed = a cm

AF = 6 - a

CD = 8 - a

Area of Δ ABC = (8 × 6)/2 = 24 cm²

Area of Δ ABC = Area of Δ AFE + Area of Δ CDE + Area of square BDEF

24 = a(6 - a)/2 + a(8 - a)/2 + a²

⇒ a = 24/7

Perimeter of square = 4 × a = 4 × 24/7 = 96/7 cm

∴ The correct answer is 96/7 cm.

Given:

Equation: 2x² - 4x + k = 0

Concept Used:

For the quadratic equation ax² + bx + c = 0 to have real roots, the discriminant (Δ) must be non-negative.

Formula Used:

Discriminant (Δ) = b2 - 4ac

Calculation:

Given equation: 2x² - 4x + k = 0

Here, a = 2, b = -4, and c = k

Discriminant (Δ) = b² - 4ac

⇒ Δ = (-4)² - 4 × 2 × k

⇒ Δ = 16 - 8k

For real roots, Δ ≥ 0

⇒ 16 - 8k ≥ 0

⇒ 16 ≥ 8k

⇒ k ≤ 2

The greatest value of k for which the equation has real roots is 2.

∴ The correct answer is option 2.

Calculation:

110, 41, 43, 95, 127, 99, 61, 92, 71, 93, 110, 36

Old series in ascending order,

36, 41, 43, 61, 71, 92, 93, 95, 99, 110, 110, 127

93 is replaced by 94, new series in ascending order,

36, 41, 43, 61, 71, 92, 94, 95, 99, 110, 110, 127

Statement 1:

Median of old series = (92 + 93)/2 = 92.5

Median of new series = (92 + 94)/2 = 93

The difference between new median and old median = 93 - 92.5 = 0.5

Statement 1 is false.

Statement 2:

Old mean = (36 + 41 + 43 + 61 + 71 + 92 + 93 + 95 + 99 + 110 + 110 + 127)/12

⇒ 978/12 = 81.5

New mean = (36 + 41 + 43 + 61 + 71 + 92 + 94 + 95 + 99 + 110 + 110 + 127)/12

⇒ 979/12 = 81.5833

The difference between new mean and old mean = 81.5833 - 81.5 = 0.0833

Statement 2 is true.

Statement 3:

Old mode = 110

New mode = 110

The difference between new mode and old mode = 110 - 110 = 0

Statement 3 is true.

∴ The correct answer is option 2.

Calculation:

100^th place mean 3^rd digit from the right.

if a number has 25 in the last then,

25² = 625

25³ = 15625

25⁴ = 390625

(125)² = 15625

There will always be 625 at last 3 places

So, last 3 digit of (225)⁴⁰ will be 625.

∴ The correct answer is 6.

Calculation:

Unit place of 1^a = 1

Possible unit place of 2^b = 2, 4, 6, 8

Possible unit place of 3^d = 3, 9, 7, 1

Possible unit place of 4^d = 4, 6

Sum of all the above unit places will always be an even number because

sum of two odd and two even numbers will always be even.

Possible even unit digits are = 0, 2, 4, 6, 8

We have to find the combination to get all these numbers for conformation,

1 + 2 + 3 + 4 = 10 (0 at unit place)

1 + 4 + 3 + 4 = 12 (2 at unit place)

1 + 2 + 7 + 4 = 14 (4 at unit place)

1 + 2 + 9 + 4 = 16 (6 at unit place)

1 + 2 + 1 + 4 = 8 (8 at unit place)

All the five unit digit can be found using these combinations.

∴ The correct answer is 5.

Shortcut Trick

Remainder when divided by 10 is simply the units digit of the number.

Units digits of 4ⁿ are {4, 6} → Cycle of 2 (4, 6, 4, 6...)

Units digits of 6ⁿ is {6} → Always 6 for all natural numbers n.

Units digits of 9ⁿ are {9, 1} → Cycle of 2 (9, 1, 9, 1...)

Units digits of 11ⁿ is {1} → Always 1 for all natural numbers n.

Sum of all distinct units digits involved across the terms = 1 + 4 + 6 + 9 = 20.

∴ The correct answer is 20.

Alternate Method

Given:

Expression: 4ⁿ + 6ⁿ + 9ⁿ + 11ⁿ

Divisor: 10

Formula Used:

Remainder (A + B + C + D) ÷ M = [Rem(A/M) + Rem(B/M) + Rem(C/M) + Rem(D/M)] ÷ M

The remainder when any number is divided by 10 is its units digit.

Units Digit (Remainder ÷ 10) Cycles

4ⁿ n=1 → 4
n=2 → 6

6ⁿ n=1 → 6
n=2 → 6

9ⁿ n=1 → 9
n=2 → 1

11ⁿ n=1 → 1
n=2 → 1

Distinct Units Digits = {1, 4, 6, 9}

Calculations:

⇒ Let n = 1 (Odd):

⇒ Remainder = (4¹ + 6¹ + 9¹ + 11¹) ÷ 10 = (4 + 6 + 9 + 1) ÷ 10 = 20 ÷ 10 = 0

⇒ Let n = 2 (Even):

⇒ Remainder = (4² + 6² + 9² + 11²) ÷ 10 = (16 + 36 + 81 + 121) ÷ 10 = (6 + 6 + 1 + 1) ÷ 10 = 14 ÷ 10 = 4

⇒ Let n = 3 (Odd):

⇒ Remainder = (4³ + 6³ + 9³ + 11³) ÷ 10 = (64 + 216 + 729 + 1331) ÷ 10 = (4 + 6 + 9 + 1) ÷ 10 = 0

⇒ The remainders follow a periodic cycle of (0, 4) for consecutive values of n.

⇒ As per the logic of distinct remainders of individual terms appearing in the expression for various n:

⇒ Distinct components = {1, 4, 6, 9}

⇒ Sum of all these distinct values = 1 + 4 + 6 + 9 = 20

∴ The correct answer is 20.

Additional Information

Cyclicity of Units Digit

The units digit of numbers follows a cycle: 0, 1, 5, 6 (Cycle 1); 4, 9 (Cycle 2); 2, 3, 7, 8 (Cycle 4).

Remainder Additive Rule

The remainder of a sum is equal to the sum of the remainders of its individual terms: (a + b) mod n = (a mod n + b mod n) mod n.

Units Digit of Power of 4 and 9

For 4ⁿ, units digit is 4 if n is odd and 6 if n is even. For 9ⁿ, units digit is 9 if n is odd and 1 if n is even.

Given:

Number = 12345678910111213 ... 99100

Concept Used:

To find the remainder when a number is divided by 16, we only need to consider the last 4 digits of the number.

Calculation:

The number ends with "99100".

Last 4 digits = 9100

⇒ 9100 mod 16

⇒ 9100 ÷ 16 = 568 remainder 12

⇒ 9100 = 568 × 16 + 12

⇒ Remainder = 12

Hence, the remainder when the number is divided by 16 is 12.

Given:

A can complete a work in 3 hours

B can complete a work in 6 hours

C can complete a work in 9 hours

D can complete a work in 12 hours

Only one person can work at a time in each hour and nobody can work for two consecutive hours.

It is not necessary to engage all.

Concept Used:

To minimize the total time, we need to maximize the work done in each hour by choosing the fastest available worker who hasn't worked in the previous hour.

Calculation:

Work done by A in 1 hour = 1/3

Work done by B in 1 hour = 1/6

Work done by C in 1 hour = 1/9

Work done by D in 1 hour = 1/12

Let's assign work in the following manner:

Hour 1: A (1/3 of work)

Hour 2: B (1/6 of work)

Hour 3: A (1/3 of work)

Hour 4: B (1/6 of work)

Total work done in 4 hours = 1/3 + 1/6 + 1/3 + 1/6

⇒ Total work done = 2/6 + 1/6 + 2/6 + 1/6

⇒ Total work done = 6/6

⇒ Total work done = 1 (which means the entire work is completed)

∴ The minimum number of hours to complete the work is 4 hours.