Quantitative Aptitude PYQs

Formula used:

When interest is compounded half-yearly

Time = 2 × original time

Rate = (1/2) × original rate.

Calculation:

Rate = 20/2 = 10%

Time = 2 × 2 = 4 half-year

 

S.I = 1000 + 1000 + 1000 + 1000 = 4000

C.I = 4000 + 600 + 40 + 1 = 4641

∴ C.I - S.I = 4641 - 4000 = 641

Calculation:

We have

a(b - c) (x - b) (x - c) + b(c - a) (x - c) (x - a) + c(a - b) (x - a) (x - b)

Let's solve the only first term

a(b - c) (x - b) (x - c)

(ab - ac)[x² - x(b + c) + bc]

(ab - ac)x² - x(ab - ac)(b + c) + (ab -ac)bc

As we need the coefficient of x² & x only. Therefore, ignore the constant term.

(ab - ac)x2 - x(ab - ac)(b + c)

(ab - ac)x2 - x(ab² + abc - abc - ac²)

(ab - ac)x2 - x(ab2 - ac2)

Therefore, a simplified form of the whole expression will be as below

[(ab - ac)x2 - x(ab2 - ac2)] +

[(bc - ba)x2 - x(bc2 - ba2)] +

[(ca - cb)x2 - x(ca2 - cb2)]

Statement:1 Coefficient of x²

⇒ x² (ab - ac + bc - ba + ca - cb) = 0

Statement:1 Coefficient of x

x(ab² - ac² + bc² - ba² + ca² - cb²) 

∴ Option (1) is correct.

Concept:

The factor of (aⁿ - bⁿ) is always divisible by (a - b) & (a + b).

Calculation:

1 - x - xn + xn+1

(1 - x) - xn(1 - x)

(1 - x)(1 - xⁿ)

Since (1 - xⁿ) is the factor. Therefore

Statement 2 is correct.

By using the above concept

(1 - xⁿ) = (1 - x)(1 + x). k

Where k is the remaining factor.

1 - x - xn + xn+1 = (1 - x)(1 - x)(1 + x).k 

1 - x - xn + xn+1 = (1 - 2x + x²)(1 + x).k 

∴ Statement 1 is also correct.

Concept:

A quadratic to be a perfect square, its discriminant must be zero, as this

implies it has exactly one root (counting multiplicity).

In other words, when b² - 4ac = 0.

Calculation:

We have

mx² + (m + 8)x + 9

Here, a = m, b = m + 8, and c = 9

So now we can set up the equation b² - 4ac = 0

(m + 8)² - 4m9 = 0

But, (a + b)² = a² + 2ab + b²

⇒ m² + 16m + 64 = 36m

⇒ m² - 20m + 64 = 0

⇒ m² - 16m - 4m + 64 = 0

⇒ m (m - 16) - 4(m - 16) =

⇒ (m - 4)(m - 16) = 0

∴ m = 4, 16

Concept used:

In question there are 4 variables (a, b, x and y) and only 2 equation. Hence we put any value to a and b to get value of x and y

Calculation:

Let a = 0 and b = 1

\(x=a+b+\frac{(a-b)^2}{4 a+4 b}\)

⇒ \(x=0+1+\frac{(0-1)^2}{0+4 \times1}\)

⇒ \(x=\frac{5}{4}\)

Again,

\(y=\frac{a+b}{4}+\frac{a b}{a+b}\)

⇒ \(y=\frac{0+1}{4}+0\)

⇒ \(y=\frac{1}{4}\)

Now,

(x - a)2 - (y - b)2 = \((\frac{5}{4}-0)^2 - (\frac{1}{4}-1)^2\)

⇒ \((\frac{25}{16})-(\frac{9}{16}) =\frac{25-9}{16} = 1\)

Also, b² = (1)² = 1

∴ The required value is b2.

Formula used:

cos 2θ = cos² θ - sin²θ = \(\frac{1-\tan ^2 θ}{1+\tan ^2 }\)

cosec²θ - cot²θ = 1

(a² - b²) = (a - b)(a + b)

Calculation:

Statement: 1

\(\cos ^4 θ-\sin ^4 θ\)

(cos²θ + sin²θ) (cos2θ - sin2θ)

(cos2θ - sin2θ) = \(\frac{2\tan θ}{1-\tan ^2θ }\)

cos 2θ = tan 2θ

which is incorrect.

Statement: 2

cosec²θ - cot²θ = 1

(cosecθ - cotθ)(cosecθ + cotθ) = 1

\( \operatorname{cosec} θ+\cot θ=\frac{1}{\operatorname{cosec} θ-\cot θ} \)

Statement: 3

As discussed in the formula used section

\(\cos ^2 θ-\sin ^2 θ=\frac{1-\tan ^2 θ}{1+\tan ^2 θ} = cos2θ\)

∴ Option (2) is correct.

Shortcut Trick

In a right-angled triangle with sin θ = 12/13, the Pythagorean triplet is (5, 12, 13).

Here, Perpendicular (P) = 12, Hypotenuse (H) = 13, and Base (B) = 5.

(tan θ + sec θ) = (P + H) / B = (12 + 13) / 5 = 5.

(cosec θ − cot θ) = (H − B) / P = (13 − 5) / 12 = 8/12 = 2/3.

Expression = 5² × (2/3)⁻² = 25 × (3/2)² = 25 × 9/4 = 225/4.

∴ The correct answer is 225/4.

Alternate Method

Given: sin θ = 12/13 and 0 < θ < π/2 (First Quadrant).

Formula Used: sin θ = P/H, cos θ = B/H, tan θ = P/B, sec θ = H/B, cosec θ = H/P, cot θ = B/P.

⇒ Base (B) = √(H² − P²) = √(13² − 12²)

⇒ B = √(169 − 144) = √25 = 5

⇒ tan θ = 12/5 and sec θ = 13/5

⇒ (tan θ + sec θ) = 12/5 + 13/5 = 25/5 = 5

⇒ cosec θ = 13/12 and cot θ = 5/12

⇒ (cosec θ − cot θ) = 13/12 − 5/12 = 8/12 = 2/3

⇒ Result = (5)² × (2/3)⁻²

⇒ Result = 25 × (3/2)² = 25 × 9/4 = 225/4

∴ The correct answer is 225/4.

Additional Information

Pythagorean Triplets

Common integer sets satisfying a² + b² = c² include (3, 4, 5), (5, 12, 13), (8, 15, 17), and (7, 24, 25).

Reciprocal Trigonometric Identities

sec θ = 1/cos θ, cosec θ = 1/sin θ, and cot θ = 1/tan θ = cos θ/sin θ.

Negative Exponents

Any expression a⁻ⁿ is equivalent to 1/aⁿ. Thus (x/y)⁻² = (y/x)².

Fundamental Trigonometric Identity

sin² θ + cos² θ = 1, which helps find any ratio if one is given.

Formula used:

(x + \(\frac{1}{x}\))² = (x² + \(\frac{1}{x^2}\)) + 2

(x + \(\frac{1}{x}\)) = \(\sqrt{[(x^2 + \frac{1}{x^2}) + 2]}\)

Calculation:

Given that 

tan8 θ + cot8 θ = m

Since, cot θ = 1/tan θ. Therefore,

tan⁸θ + (1/tan²θ) = m

Let tan θ = n then 

n⁸ + 1/n⁸ = m      ---(1)

We know that , (x + \(\frac{1}{x}\)) = \(\sqrt{[(x^2 + \frac{1}{x^2}) + 2]}\)

⇒ (n⁴ + 1/n⁴) = \(\sqrt{[(n^8 + \frac{1}{n^8}) + 2]}\)

⇒ (n4 + 1/n4) = \(\sqrt{[m + 2]}\)

Again applying the same identity

⇒ (n² + 1/n²) = \(\sqrt{(\sqrt{(m + 2)}+2)}\)

Again applying the same identity

⇒ (n + 1/n) = \(\sqrt {\sqrt{(\sqrt{(m + 2)}+2)}+2}\)

But, n = tan θ 

⇒ tan θ + 1/tan θ = \(\sqrt {\sqrt{(\sqrt{(m + 2)}+2)}+2}\)

∴ tan θ + cot θ = \(\sqrt{\sqrt{\sqrt{m+2}+2}+2}\)

Concept:

-1 ≤ sin θ ≤ 1 When θ ∈ [0, π/2]

Calculation:

f(θ) = 6- 4 sin θ 

To make it minimum, (4 sinθ) should be maximum.

Since, θ ∈ [0 ,π/2]

Maximum value of sin θ = 1 

f(θ)_min = 6 - 4 × 1 = 2

∴ Minimum value of 6- 4 sin θ is 2.

Formula used:

cos(30°) = √3/2

sin(30°) = 1/2

cot(30°) = √3

tan (90° - θ) = cot θ 

tan θ × cot θ = 1

Calculation:

4 cos2 30° + 2x sin 30° - cot2 30° - 6 tan 15° tan 75° = 0

Since, tan (90° - θ) = cot θ 

⇒ 4 cos2 30° + 2x sin 30° - cot2 30° - 6 tan (90° - 75°) tan 75° = 0

⇒ 4 cos2 30° + 2x sin 30° - cot2 30° - 6 cot 75° tan 75° = 0

But, tan θ × cot θ = 1

⇒ 4 cos2 30° + 2x sin 30° - cot2 30° - 6 = 0

Using the trigonometric values

⇒ 4(√3/2)² + 2x(1/2) - (√3)² - 6= 0

⇒ 3 + x - 3 - 6 = 0

∴ x = 6

Given:

\(\frac{\cos ^2 32^{\circ}+\cos ^2 58^{\circ}}{\sec ^2 50^{\circ}-\cot ^2 40^{\circ}}\) +  4 tan 13° tan 37° tan 53° tan 77°

Concept used:

1. cos(90° - x) =  sinx
2. tan(90° - x) = cotx
3. tanx . cotx = 1
4. sin²x + cos²x = 1
5. sec²x - tan²x = 1

 

Calculation:

\(\frac{\cos ^2 32^{\circ}+\cos ^2 58^{\circ}}{\sec ^2 50^{\circ}-\cot ^2 40^{\circ}}\) + 4 tan 13° tan 37° tan 53° tan 77°

​\(\frac{\cos ^2 32^{\circ}+\cos ^2 (90^{\circ}-32^{\circ})}{\sec ^2 50^{\circ}-\tan ^2 (90^{\circ}-50^{\circ})}\) + 4 tan(90° - 77°) tan (90° - 53°) tan 53° tan 77°

​\(\frac{\cos ^2 32^{\circ}+\sin ^2 32^{\circ}}{\sec ^2 50^{\circ}-\tan ^2 50^{\circ}}\) + 4 cot 77° cot 53° tan 53° tan 77°

By using the above formula,

⇒ 1/1 + 4 × 1 × 1

⇒ 1 + 4 = 5

∴ The required value is 5.

Formula used:

1 + cot2 θ =  cosec² θ 

1 + tan² θ = sec² θ 

cosec θ = 1/sin θ and sec θ = 1/cos θ 

Calculation:

(1 + cot2 θ) (1 + cos θ) (1 - cos θ) - (1 + tan2 θ) (1 + sin θ) (1 - sin θ)

⇒ cosec2 θ (1 - cos² θ) - sec² θ (1 - sin² θ) (∵ (a+b)(a-b) = a² - b²)

⇒ cosec2 θ . sin² θ - sec2θ . cos² θ   

⇒ 1 - 1 = 0

Shortcut Trick 

Put θ = π/6 from option

⇒ 2 × (√3/2)² + (1/2) - 2 = 0

⇒ 3/2 + (1/2) - 2 = 0

⇒ 0 = 0

⇒ LHS = RHS

∴ The correct answer is θ = \(\frac{π}{6}\).

Alternate Method 

Formula used:

sin²θ + cos²θ = 1

Calculation:

⇒ 2cos²θ + sinθ - 2 = 0

⇒ 2(1 - sin²θ) + sinθ - 2 = 0

⇒ 2 - 2sin²θ + sinθ - 2 = 0

⇒ -2sin²θ + sinθ = 0

⇒ sinθ(-2sinθ + 1) = 0

⇒ sinθ = 0 or -2sinθ + 1 = 0

⇒ sinθ = 0, sinθ = 1/2

But  0 < θ ≤ \(\frac{π}{2}\), Therefore,

⇒ sin θ = 1/2

∴ The correct answer is θ = \(\frac{π}{6}\).

Shortcut Trick

Distance from tower base for 45° = h and for 30° = h√3.

Distance covered in 6 min = h√3 − h = h(√3 − 1).

At uniform speed, Time is proportional to Distance.

Time for distance h(√3 − 1) is 6 min; so time for distance h is t.

t = 6 ÷ (√3 − 1) = 3(√3 + 1) ≈ 3 × 2.732 = 8.196 minutes.

∴ The correct answer is 8.196 minutes.

Alternate Method

Given:

Initial angle of depression = 30°

Final angle of depression = 45°

Time taken for this change = 6 minutes

Speed of the car is uniform.

Formula Used:

tan θ = Perpendicular ÷ Base

Distance = Speed × Time

Calculations:

⇒ Let height of tower AB = h.

⇒ In right ΔABD, tan 45° = AB ÷ BD ⇒ 1 = h ÷ BD ⇒ BD = h.

⇒ In right ΔABC, tan 30° = AB ÷ BC ⇒ 1/√3 = h ÷ BC ⇒ BC = h√3.

⇒ Distance covered CD = BC − BD = h√3 − h = h(√3 − 1).

⇒ Speed of car (v) = Distance ÷ Time = h(√3 − 1) ÷ 6.

⇒ Time t to cover distance BD = BD ÷ v = h ÷ [h(√3 − 1) ÷ 6].

⇒ t = 6 ÷ (√3 − 1).

⇒ Rationalizing: t = [6 × (√3 + 1)] ÷ [(√3 − 1)(√3 + 1)] = 6(√3 + 1) ÷ (3 − 1).

⇒ t = 3(√3 + 1) = 3(1.732 + 1) = 3 × 2.732 = 8.196 minutes.

⇒ Since 8.196 lies between 8 and 8.3, option 2 is correct.

∴ The correct answer is 8.196 minutes.

Additional Information

Angle of Depression and Elevation

The angle of depression from the top of the tower is numerically equal to the angle of elevation from the car to the top due to alternate interior angles.

Trigonometric Ratio Shortcut

For a tower of height h, distance at 30° is h√3, at 45° is h, and at 60° is h/√3.

Uniform Speed and Time

When speed is constant, the ratio of times taken to cover two distances is equal to the ratio of the distances themselves (t₁/t₂ = d₁/d₂).

Formula used:

tan θ = Perpendicular / Base

Calculation:

From the figure,

AB = DE  & BD = AE

∠EAD = ∠ADB = 30° & ∠CAE = 60°

In an Δ ABD

tan 30° = h/BD = 1/√3

⇒ BD = h√3 = AE

In an ΔAEC

tan 60° =  CE/AE = CE/h√3

⇒ √3 = CE/h√3

⇒ CE =  3h

Therefore, the height of the tower

CD = CE + ED = 3h + h = 4h

∴ The height of the tower is 4h.

Formula used:

Area of triangle = (1/2) × base × height

Caculation:

Area of ΔABC = Area of ΔABC 

⇒ (1/2) × AC × AB = (1/2) × p × BC

⇒ AC × AB = p × BC

⇒ 5 × 12 = p × 13

∴ 13p = 60

Formula used:

Area of equilateral triangle = (√3/4) (side)²

Calculation:

OA = OB = OC = radius (r)

Since OABC is a rhombus

BC = AB = r

Therefore, ΔOAB & ΔOBC are equilateral trainge 

According to the question

Area of rhombus = 32√3

Area of ΔOAB + Area of ΔOBC = 32√3

(√3/4)r² + (√3/4)r² = 32√3

2 [(√3/4) × r²] = 32√3

r² = 64

r = 8 cm

∴ The radius of the circle is 8 cm²

Formula used:

The area of each surface of the cube = (side)²  [each surface of a cube is a square]

Calculation:

Let the initial surface area of the cube = 600

then the initial area of each surface = 100 

Initial length of side = √100 = 10 unit

Final Surface area of the cube = 600 × 125/100 = 750

Then the final area of each surface = 125

(Final length)² = 125

(Final length) = √125 

Therefore, according to the question

P = [(√125 - 10)/10] × 100 

P = 10(5√5 - 10) 

P = 10(5 × 2.236 - 10) = 11.8

∴ 10 < P < 12

Formula used:

Volume of cube of side a = a³

Total Surface area of cube = 6a²

Volume of cuobid of dimensione L, B & H = LBH 

Total surface area = 2 (LB + BH + HL)

Calculation:

Side of the cube = a.

Volume of the cube = a³

Volume of each cuboid = a³/2

Dimensions of each cuboid = a × a × a/2

Total surface area of the cube = 6 × a²

Total surface area of one cuboid

⇒ 2[a × a + a × (a/2) + a × (a/2)] = 4a² 

∴ Required ratio = 6a² : 4a² = 3 : 2

Shortcut Trick

For any cuboid, the sum of its dimensions can be found directly using the formula: Sum = √(Diagonal² + Surface Area)

⇒ Sum = √(11² + 240)

⇒ Sum = √(121 + 240)

⇒ Sum = √361 = 19 cm

∴ The correct answer is 19 cm.

Alternate Method

Given:

Length of diagonal (d) = 11 cm

Total Surface Area (S) = 240 cm²

Let the length, breadth, and height of the cuboid be l, b, and h respectively.

        l h b d = 11 cm Area = 240 cm²

Formula Used:

Diagonal of a cuboid = √(l² + b² + h²)

Total Surface Area = 2(lb + bh + hl)

Algebraic Identity: (l + b + h)² = l² + b² + h² + 2(lb + bh + hl)

Calculations:

⇒ √(l² + b² + h²) = 11

Squaring both sides:

⇒ l² + b² + h² = 121

We are given the Surface Area:

⇒ 2(lb + bh + hl) = 240

Substituting these values into the standard algebraic identity:

⇒ (l + b + h)² = 121 + 240

⇒ (l + b + h)² = 361

⇒ l + b + h = √361 = 19

∴ The correct answer is 19 cm.

Additional Information

Volume of a Cuboid

The volume of a cuboid is the product of its length, breadth, and height: V = l × b × h.

Sum of Length of All Edges

Since a cuboid has exactly 12 edges (4 each of length, breadth, and height), the sum of their lengths is calculated as 4(l + b + h).

Cube Properties

For a cube with an edge side equal to 'a', the body diagonal is given by a√3, and the total surface area is 6a².