Quantitative Aptitude PYQs

Formula used:

The Inradius of a right-angled triangle can be given as

r = (P + B - H)/2

Area of circle = πr2

Calculation:

By using the above formula 

r = (12 + 16 - 20)/2 = 8/2 = 4 cm

Area of circle = πr²

Area of circle = π (4)² ≈ 50.27 cm² or 50 cm²

Formula used:

The total surface area (A_Total) = 2πr(h + r)

The curved (or lateral) surface area (A_curved) = 2πrh

Calculation:

Given that

2 × Total Surface Area = 3 × Curved Surface Area

By using the above formula

⇒ 2 × 2πr(h + r) = 3 × 2πrh

⇒ 4πr(h + r) = 6πrh

⇒ 2(h + r) = 3h

⇒ 2h + 2r = 3h

∴  h = 2r.

Formula used:

HCF = The product of these common factors is the HCF of the given numbers.

Calculation:

Length of the floor = 30 m 60 cm = 3060 cm

Width of the floor = 23 m 40 cm = 2340 cm

Factor of 3060 = 2² × 3² × 5 × 17

Factor of 2430 = 2² × 3² × 5 × 13

HCF of 3060 & 2340 = 2² × 3² × 5 = 180

Total number of tiles required = (3060 × 2340) / (180× 180)

∴ Total number of tiles required = 221

Formula used:

Area = (side)2

Perimeter of the square field = 4 × sides

Time = Distance/Speed

Calculation:

The area of the field is 25 hectares (250,000 m²).

For a square,

Araa = (side)²

Side length = √area = √250,000 m = 500 m.

Perimeter of the square field = 4 × sides

Perimeter of the square field = 4 × 500 m = 2000 m.

Walking speed is 5 km/hr = 5000 m/hr.

Time to walk the perimeter = Distance/Speed

⇒ 2000 m / 5000 m/hr = 0.4 hr

But 1 hr = 60 min

Therefore,

Time to walk the perimeter = 60 × 0.4 = 24 minutes.

Shortcut Trick 

⇒ x = (√2R)² = 2R²

⇒ Height of triangle = R + R/2 = 3R/2

⇒ Y = (3R/2)² / √3 = 3√3R² / 4

⇒ x²/y² = 64 / 27

∴ The correct answer is 27x2 = 64y2**.**

Alternate Method 

Formula used:

Pythagoras theorem:

Hypotenuse² = Base² + Perpendicular²

Area of square = (side)²

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

Calculation:

Case:1 

By using the Pythagoras' theorem 

⇒ a² + a² = (2R)²

⇒ 2a² = 4R²

⇒ a = R√2

Area of square = a²

⇒ x = 2R²

⇒ R = √(x/2)            -----(1)

Case:2

In an Δ BOD

Since a center angle will always be double of the angle on the edge of a circumference. Therefore, ∠BOC = 120° 

ΔBOC is isosceles triangle. Therefore, ∠BOD = ∠COD = 60° 

BD = R sin 60° = R (√3/2)

Therefore, side of triangle = 2 × R(√3/2) = R√3

⇒ y = (√3/4) (side)²

⇒ y = (√3/4) (R√3)2 

⇒ y = (3√3/4)R²

From equation (1)

⇒ y = (3√3/4) × (x/2) 

⇒ (3√3)x = 8y

∴ The correct answer is 27x² = 64y².

Shortcut Trick 

⇒ % decrease in width = (2/5) × 100 = 40%

∴ The required value is 40%.

Alternate Method 

Given:

Increased length = Original length + (2/3) of original length

Concept used:

New area = Original area

% Change = [(Final value - Initial value)/Initial value ] × 100

Calculation:

Let the length and width of the rectangle be L and W respectively.

Original area = Length × Width = L × W

New length = Original length + (2/3) of original length

New length = L + (2/3)L = (5/3)L

New area = New length × New width

New area = (5/3)L × New width

According to the question, New area = Original area

(5/3)L × New width = L × W

New width = (3/5)W

Now, the percent decrease in width

⇒ [(W - 3W/5) / W] × 100 

Percent decrease in width = (2/5) × 100 = 40%

∴ The required value is 40%.

Formula used:

Area of circle = πr²

Calculation:

To achieve the maximum coverage, we will arrange the circles such that the

larger circle (with a radius of 4 cm) is inscribed within the rectangle, and the

two smaller circles (each with a radius of 2 cm) are positioned at opposite

corners of the rectangle.

As shown in the figure, radius of three circle will be 4 cm, 2 cm, 2 cm.

Therefore, total area

⇒ 16π cm² + 8π cm²

⇒  24π cm²

∴ Required area is 24π cm².

Given:

AB = x

∠APO = θ

Concept used:

Diagonal of square = √2a

cotθ = base/perpendicular

Calculation:

According to the above concept,

Diagonal of square = √2a

⇒ AC = x√2

⇒ OA = \(\frac{x√2}{{2}}\) = \(\frac{x}{{√2}}\)

Now, according to the above figure,

In ΔOAP,

⇒ cotθ = \(\frac{OP}{{OA}}\) = \(\frac{2x}{{x/√ 2}}\)

⇒ cotθ = 2**√2**

∴ The required value of cotθ is 2√2

Formula used:

Volume of sphere = (4/3) π (radius)³

Surface  Area of sphere = 4π(radius)² 

Calculation:

Let's assume the radius of the larger solid iron ball is R, and the radius of

each of the smaller solid balls is r.

By using the above formula

V_larger = (4/3)πR³

V_smaller = (4/3)πr³

Since the entire volume of iron is used to make the smaller balls, we have:

V_larger = 64V_smaller

(4/3)πR³ = 64 × (4/3)πr³

R³ = 64r³

R = 4r

Now, let's calculate the surface area of the larger ball and the sum of the surface areas of the smaller balls.

Ratio = (4πR2) / (64 (4πr2)

Ratio = (4π(4r)2) / (256πr2)

Ratio = (64πr²) / (256πr²)

Ratio = 64/256 = 1/4

∴ The required ratio is 1/4.

Formula used:

The perimeter of the triangle = Sum of three sides

Calculation:

Required perimeter = Perimeter of (P + R + S) - Perimeter of Q

Required perimeter = (16 + 4 + 12) - 12 = 20 cm.

Alternate MethodGiven:

Perimeter of triangle P = 16 cm

Perimeter of triangle R = 4 cm

Perimeter of triangle S = 12 cm

Perimeter of triangle Q = 12 cm

Corrected calculation based on your instructions:

Required perimeter = (Perimeter of P + Perimeter of R + Perimeter of S) - Perimeter of Q

So, Required perimeter = (16 cm + 4 cm + 12 cm) - 12 cm = 32 cm - 12 cm = 20 cm.

Therefore, the perimeter of the triangle ABC is indeed 20 cm. 

Additional Information

To find the perimeter of triangle ABC, we simply add the perimeters of triangles P, R, and S, and then we subtract the perimeter of triangle Q. This is because the sides of triangles P, R, and S form the outside of triangle ABC, but Q is inside and doesn't add to the outside length.

So, we calculate it like this: Add the perimeters of P (16 cm), R (4 cm), and S (12 cm) to get 32 cm. Then, we subtract the perimeter of Q (12 cm) because it's on the inside.

The final perimeter of triangle ABC is 20 cm (32 cm - 12 cm = 20 cm). This gives us the total length around triangle ABC.

Concept:

Divisibility rule of 25: Last two digits of the number should be divisible by 25

Divisibility rule of 9: A number is divisible by 9 if and only if the sum of its digits is divisible by 9.

Calculation:

Given that,

A number 277XY5 (where X, Y are digits) is divisible by 25.

By using the above concept, the possible values of Y are 2 & 7.

Statement: I  The given number is divisible by 9.

The sum of digits of 277XY5 = 21 + X + Y

If Y = 2  then sum = 21 + X + 2 = 23 + X

If it is divisible by 9, then X = 4

If Y = 7  then sum = 21 + X + 7 = 28 + X

If it is divisible by 9, then X = 8

So, we have two possible values of X 

Statement: II  X > 5

If we consider only statement II only, we will have information

In 277XY5, Y = 2 or 7

277X25 & 277X75 are divisible by 25. 

Clearly, both numbers are divisible by 25 independent of X.

Let's combine both statements 

From statement I, we have X = 4 & 8

From statement II, we have X > 5

Therefore, the value of X = 8.

Choose this option if the Question can be answered by using both Statements together, but cannot be answered by using either statement alone.

Concept:

For the quadratic equation ax² + bx + c = 0

Sum of root (α + β) = -b/a

Produt of root (αβ)   = c/a

The quadratic equation can also be written as 

k[x² - x (α + β) + α β] = 0

Where k is any integer.   

Calculation:

We have 

ax2 + bx + c = 0      ----(1)

One of the roots is 2 (let α)

Statement: 1 Ratio of c to a is 1.

Product of root 

αβ = c/a = 1

2 × β = 1 

⇒ β = 1/2

The quadratic equation will be 

k[x² - x (2 + 1/2) + 2 × (1/2)] = 0

k[x2 - 5x/2 + 1] = 0

k[2x² - 5x + 2] = 0

On comparing it to equation (1)

a =  2k, b = -5k, c = 2k

So there is no unique value of a, b & c.

Statement:2  Ratio of b to a is (-5/2)

2 + β = 5/2

β = 1/2 

So again there is no unique value of a, b & c.

∴ Option 4 will be correct.

Calculation:

Statement:1  \(\frac{m}{n} > 1\)

⇒ m > n

Statement:2  m > 2n

Let n = 1

If m > 2 × 1 then m > 1

∴ Option 2 is correct.

Concept:

The sum of the length of the two sides of a triangle is greater than the length of the third side.

Explanation:

Let's first check whether these points can form a triangle or not (i.e. collinear point).

Statement:1  AB = 5 cm, BC = 5 cm, CA = 6 cm

5 + 5 > 6

5 + 6 > 5

So, A, B & C can form the tirangle.

Therefore, the circle can be drawn from these points.

Statement:2  AB = 3 cm, BC = 4 cm, CA = 7 cm. 

3 + 4 = 7

So A, B & C are collinear points & a circle can not be drawn.

Clearly, we can see, we are able to answer the given question with Statement I but not with Statement

II. ∴ The correct answer will be option A.

Alternate MethodConcepts or formulas:

A circle can be drawn through three points if they form a triangle, and the sum of any two sides must be greater than the third side (Triangle Inequality Theorem).

Calculation steps:

Statement I:

⇒ 5 cm + 5 cm > 6 cm (True)

⇒ 5 cm + 6 cm > 5 cm (True)

⇒ 6 cm + 5 cm > 5 cm (True)

All inequalities hold; hence, a circle can be drawn.

Statement II:

⇒ 3 cm + 4 cm > 7 cm (False)

The inequality does not hold; hence, a circle cannot be drawn.

Conclusion:

From Statement I, a circle can be drawn. Statement II fails to satisfy the triangle inequality.

Therefore, the correct answer is: 1) Choose this option if the Question can be answered by one of the Statements alone but not by the other.

Calculation:

Statement I: (x + y)4 = 256

(x + y)⁴ = 16⁴

(x + y) = ± 4

So, no unique values of x, and y are possible from this statement alone.

Statement II: (x + y)3 < 16

No unique values of x, and y are possible from this statement alone.

Let's combine both statements

Let x = 4

(4)³ < 16 

64 < 16 which is wrong

If x = -4

-64 < 16 which is correct.

So, (x + y) = - 4 

∴ Optipon 3 is correct.

Calculation:

We have p2 + q2 + q 

Statement I: 2p + q is odd

For any value of p, 2p will be even 

Since 2p + q is odd,

q must be odd.

Statement II: q - 2p is odd

Here also, q must be odd to make q  - 2p even.

∴ Option 4 is correct.

Concept used:

Area of the circle = πr2

Here r = radius of the circle

Statement I:

As we know,

Length of an arc = 2πr × θ/360°

Where r is the radius and θ = angle subtended by the arc

7 = 2πr × 30°/360°

⇒ 7 = 2πr × 1/12

⇒ r = 42/π 

Area of the circle C = π(42/π)² = π × 1764/π²

Area of the circle C = 1764/π

So, the statement I alone is sufficient to answer the question,

Statement II:

Let O be the center of the circle

So, AO = OB = radius of the circle

Now, AO² + OB² = AB²

⇒ r2 + r2 = 100

⇒ 2r2 = 100 ⇒ r = 5√2

Area of the circle C = πr² = (5√2)²π = 50π

So, statement II alone is sufficient to answer the question,

∴ The required answer is Option 2.

Calculation:

Statement I:

The length of the line segment joining the mid-points of two sides of Δ is half of the third side of Δ

From this, we cannot say it is a right-angle triangle

So, the statement I is not sufficient to answer the question,

Statement II:

The angles of Δ are in the ratio 1 : 2 : 3

As we know in a triangle sum of all the angles is 180°

So, The angles of Δ are 30°, 60°, 90°

So, from statement II we can say the triangle is a right-angle triangle.

∴ The required answer is Option 1.

Concept:

1) If in a right-angle triangle

Ratio of three angle = 30° : 60° : 90°  (or 1 : 2 : 3), then

The ratio of it's side opposite to this angle = 1: √3 : 2

2) AC × BD = AB × AD 

Calculation:

Statement I:

The angles of Δ are in the ratio 1 : 2 : 3

As we know in a triangle sum of all the angles is 180°

So, The angles of Δ are 30°, 60°, 90°

Therefore,

So, the sides should be in the ratio 1: √3 : 2

By using the Pythagoras' theorem

x2 + 242 = 252

⇒ x2 + 576 = 625

⇒ x2 = 625 - 576

⇒ x2 = 49

⇒ x = 7

So, the shortest side = 7 cm

But, 7, 24 & 25 are not in the ratio 1 : √3 : 2, but as per the property,

If the ratio of angle = 1 : 2 : 3 then the ratio of side = 1 : √3 : 2. 

Therefore, the information given in Statement I is not sufficient. 

Statement II:

From the Pythagoras theorem, we got x = 7 

By using the above property

BC × AD = AB × AC

7 × 24 = 25 × AD

AD = 168/25 = 6.72

Therefore, if we take the length of the shortest side equal to 7 cm, we got the length of the perpendicular drawn on the longest side of Δ from its opposite vertex is 6.72 cm.  

Statement 2 is sufficient to answer this question.

∴ Option (1) is correct.

Formula Used:

Area of Circle = πr²

Calculation:

According to the question,

4 × (Area of minor segment) = 3 × (Area of major segment)

⇒ (Area of minor segment)/(Area of major segment) = 3/4

Thus, if the area of the minor segment is given, we can find the area of the major segment and vice versa.

Area of Circle = Area of minor segment + Area of major segment

As we know, Area of Circle = πr2

Thus, we can find the radius of the circle if the area of the minor segment or the area of the major segment is given.

∴ Option 02 is the correct answer.