Quantitative Aptitude PYQs

Shortcut Trick

According to Archimedes' principle, the volume of water displaced equals the volume of the submerged object.

Rise in water level (h) = Volume of sphere ÷ Base area of cylinder

⇒ h = [4/3 × π × (5)³] ÷ [π × (10)²]

⇒ h = (4/3 × 125) ÷ 100

⇒ h = 500 ÷ 300 = 5/3 cm

∴ The correct answer is 5/3 cm.

Alternate Method

Given:

Radius of the sphere (R) = 5 cm

Radius of the cylindrical vessel (r) = 10 cm

Let the level of water rise by 'h' cm.

Formula Used:

Volume of a sphere = 4/3 × π × R³

Volume of a cylinder = π × r² × h

Calculation:

According to the principle of volume displacement, the volume of water displaced is equal to the volume of the completely submerged sphere.

⇒ Volume of water raised = Volume of the sphere

⇒ π × r² × h = 4/3 × π × R³

⇒ π × (10)² × h = 4/3 × π × (5)³

⇒ 100 × h = 4/3 × 125

⇒ h = 500 ÷ (3 × 100)

⇒ h = 5/3 cm

∴ The correct answer is 5/3 cm.

Additional Information

Volume of a Cylinder

The volume of a right circular cylinder is given by V = π × r² × h, where r is the base radius and h is the height of the cylinder.

Volume of a Sphere

The volume of a sphere is given by V = 4/3 × π × r³, where r is the radius of the sphere.

Archimedes' Principle (Volume Displacement)

When a solid object is completely submerged in a liquid, it displaces a volume of liquid exactly equal to its own volume. Therefore, Volume of liquid displaced = Volume of submerged object.

Explanation:

Statement 1: 

We know that, any angle in a major sector is always acute.

Therefore statement 1 is correct.

Statement 2: 

We know that, if two sides of a pair of opposite sides of a cyclic quadrilateral are equal, then its diagonals are also equal.

∴ The correct option is 3.

Given:

(a + b + c)2 - 3(ab + bc + ca) + 3(x2 + y2 + z2) = 0

Calculation:

We have

(a + b + c)2 - 3(ab + bc + ca) + 3(x2 + y2 + z2) = 0

We solve this question by options 

If we put the value of option 3 then

a = b = c  = k (let) and x = y = z = 0

(k + k + k)² - 3(k² + k² + k²) + 3(0)

⇒ (3k)² - 3(3k²) 

⇒ 9k² - 9k² = 0

∴ The correct option is 3.

Given:

ABC is right angle triangle at angle B, BC = 10 cm

 AC = 12 cm, p is length of the perpendicular from B to AC

Formula used:

ArΔ = 1/2 × base × height

Calculation:

In an Δ ABC, by using the Pythagoras theorem

AC² = AB² + BC²

144 = AB² + 100

AB² = 44

AB = √44

Here, We can find the area in two ways,

1. By taking AC as the base & length p as the perpendicular.

2. By taking BC as base & AB as the perpendicular

As, Area (ΔABC) = Area (ΔABC)

⇒ 1/2 × 10 × √44 = 1/2 × 12 × p

⇒ 5 × 2√11 = 6p

⇒ p = (5√11)/3 cm

∴ The correct answer is (5√11)/3 cm

Shortcut Trick 

Mean = {(280 × 5) - (240 × 3)} / 2 = 340

∴ The mean of q and s = 340.

Alternate Method 

Formula Used:

Mean = The sum of numbers/The count of the numbers

Calculation:

According to the question

The mean of p, q, r, s, and t is 280

280 = (p + q + r + s + t)/5

⇒ p + q + r + s + t = 1400       ----(1)

Again, according to the question

The means of p, r, and t is 240     

240 = (p + r + t)/3

⇒ 720 = p + r + t

Put this value in the equation (1)

⇒ 720 + q + s = 1400

⇒ q + s = 1400 - 720 = 680

∴ The mean of q and s = 680/2 = 340.

Given:

Age relations:

D = 3 × A

B - C = E - G = D - E

(D + G)/2 = 16, (A + E)/2 = 11 and (B + C)/2 = 11

Age : Weight of D = 9 : 20 

Age : weight of A = 2 : 5 

Calculations:

Age : weight of D = 9 : 20

So, Weight will be a multiple of 20 and weight can not exceed 40

Let the weight of D = 40 kg

So, age of D = 9 × 2 = 18 years

 

Hence, the age of D is 18 years.

Given:

Age relations: D = 3 × A

B - C = E - G = D - E

(D + G)/2 = 16, (A + E)/2 = 11 and (B + C)/2 = 11

Age : Weight of D = 9 : 20 

Age : weight of A = 2 : 5 

Formula used:

Average = Sum of observations/Number of observations

Calculations:

Age : weight of D = 9 : 20

So, Weight will be a multiple of 20 and weight can not exceed 40

Let the weight of D = 40 kg

So, age of D = 9 × 2 = 18 years

Also age of A = D/3 = 18/3 = 6 years

So, the weight of A = 5 × 3 = 15 kg

Age of E - Age of G = 16 - 14 = 2 years

Also from the given conditions,

B - C = 2     -----(1)

B + C = 22     -----(2)

From equation(1) and equation(2)

⇒ 2C = 20

⇒ C = 20/2 = 10

So, Age of B = 22 - 10 = 12

 

Total age of B, C, D, G, and E = 12 + 10 + 18 + 14 + 16 = 70

Average age = 70/5 = 14

Hence, the required average age is 14 years.

Given:

Weight relations:

Weight of B = Weight of C

A = B - 10, D = E + 4, E = F + 4 and F = G + 4

Age : Weight of D = 9 : 20 

Age : weight of A = 2 : 5 

Formula used:

Average = Sum of observations/Number of observations

Calculations:

Age : weight of D = 9 : 20

So, Weight will be a multiple of 20 and weight can not exceed 40

Let the weight of D = 40 kg

So, the weight of A = 5 × 3 = 15 kg

Weight of B = A + 10 = 25 kg

Weight of B = Weight of C = 25 kg

Now, E = D - 4

⇒ 40 - 4 = 36

Similarly, F = 32 and D = 28 years

 

Now weight of G - weight of C = 28 - 25 = 3

Hence, the difference in weight of G and C is 3 kg.

Given:

Age relations:

D = 3 × A

B - C = E - G = D - E

(D + G)/2 = 16, (A + E)/2 = 11 and (B + C)/2 = 11

Weight relations:

Weight of B = Weight of C

A = B - 10, D = E + 4, E = F + 4 and F = G + 4

Age : Weight of D = 9 : 20 

Age : weight of A = 2 : 5 

Formula used:

Average = Sum of observations/Number of observations

Calculations:

Age : weight of D = 9 : 20

So, Weight will be a multiple of 20 and weight can not exceed 40

Let the weight of D = 40 kg

So, the age of D = 2 × 9 = 18 years

Also, age of A = D/3 = 18/3 = 6 years

So, the weight of A = 5 × 3 = 15 kg

Weight of B = A + 10 = 25 kg

Weight of B = Weight of C = 25 kg

Now, E = D - 4

⇒ 40 - 4 = 36

Similarly, F = 32 and D = 28 years

 

Total weight = 15 + 25 + 25 + 40 + 36 + 32 + 28 = 201

Average weight = 201/7

Hence, the average weight is 201/7 kg.

Given:

Age relations:

D = 3 × A

B - C = E - G = D - E

(D + G)/2 = 16, (A + E)/2 = 11 and (B + C)/2 = 11

Weight relations:

Weight of B = Weight of C

A = B - 10, D = E + 4, E = F + 4 and F = G + 4

Age : Weight of D = 9 : 20 

Age : weight of A = 2 : 5 

Formula used:

Average = Sum of observations/Number of observations

Calculations:

Age : weight of D = 9 : 20

So, Weight will be a multiple of 20 and weight can not exceed 40

Let the weight of D = 40 kg

So, age of D = 9 × 2 = 18 years

Also, age of A = D/3 = 18/3 = 6 years

So, the weight of A = 5 × 3 = 15 kg

Weight of B = A + 10 = 25 kg

Weight of B = Weight of C = 25 kg

Age of E - Age of G = 16 - 14 = 2 years

Also from the given conditions,

B - C = 2     -----(1)

B + C = 22     -----(2)

From equation(1) and equation(2)

⇒ 2C = 20

⇒ C = 20/2 = 10

So, Age of B = 22 - 10 = 12

 

Hence, the Age of F can not be determined.

Now, the Average weight of D and F = (40 + 32)/2

⇒ 72/2 = 36 = Age of E

Now, A has a minimum age, and D has a maximum age

So, the weight difference is the maximum for D and A.

Hence, 1, 2 and 3 are correct.

Given:

Statement 1

m = (1 - p) (p2 + p + 1)

n = (p + 1) (p2 - p + 1)

Statement 2

m = pn

Formula:

a³ - b³ = (a - b) (a² + b² + ab)

a³ + b³ = (a + b)(a² + b² - ab)

Calculation:

Statement: 1

m = (1 - p) (p2 + p + 1)

By using the above identities 

m = (1 - p3)

n = (p + 1) (p2 - p + 1) 

By using the above identities 

n = (p3 + 1)

Since, we don't know wether p is +ve, -ve, integer or fraction

We can not conclude the relation on m & n

Statement: 2  m = pn

Here also, relation b/w m & n depend on only P. therefore

∴ The correct answer is option D.

Given:

One of the roots is  \(\frac{-4-\sqrt{10}}{2}\) 

Statement-1 The product of the roots is \(-\frac{3}{2}(3+\sqrt{10})\)

Statement-2 The sum of roots of the quadratic equation is -1.

Calculation:

According to the first statement

Let, the other root = a

⇒ a ×  \(\frac{-4-\sqrt{10}}{2}\)  = \(-\frac{3}{2}(3+\sqrt{10})\)

⇒ a = \(-\frac{3}{2}(3+\sqrt{10})\) × \(\frac{2}{-4-\sqrt{10}}\) 

This means the first statement is sufficient to answer.

According to the second statement

Let, the other root = a

⇒  \(\frac{-4-\sqrt{10}}{2}\) + a = -1

⇒ a = -1 -(\(\frac{-4-\sqrt{10}}{2}\))

This means the second statement is sufficient to answer.

∴ The correct answer is option 2

Given:

A 3 digit number which is divisible by 10.

Calculation:

Let the three digit number be "xy0"

x = digit of hundred's place and y = digit of ten's place

According to Statement-1

When the digits of hundred's and ten's place interchanged, then

⇒ xy0 - yx0 = 180

⇒ (100x + 10y + 0) - (100y + 10x + 0) = 180

⇒ 90x - 90y = 180

⇒ x - y = 2     -----(1)

Statement-1 alone is not sufficient to get the values.

Now from Statement-2

Digit in hundred's place is halved and ten's and unit place interchanged,

⇒ xy0 - (x/2)0y = 336

⇒ (100x + 10y + 0) - (100x/2 + 10 × 0 + y) = 336

⇒ 100x + 10y - 50x - y = 336

⇒ 50x + 9y = 336     -----(2)

We got two linear equations using both the statement,

∴ Both statements are required to find the value of x and y.

∴ Question can be answered by using both the statements together.

Given:

x = y = z and x, y, and z are real numbers

Formula used:

x2 + y2 + z2 - xy - yz - zx = 1/2[(x - y)² + (y - z)² + (z - x)²]

x3 + y3 + z3 - 3xyz = (x + y + z) (x² + y² + z² - xy - yz - zx)

Calculation:

From statement (1)

We can write the identity as,

x2 + y2 + z2 - xy - yz - zx = 1/2[(x - y)2 + (y - z)2 + (z - x)2]

Now put x = y = z, we get

⇒x2 + y2 + z2 - xy - yz - zx  = 1/2 (0 + 0 + 0)

⇒ x2 + y2 + z2 - xy - yz - zx = 0

∴ The given question can be answered by this statement.

From statement (2)

x3 + y3 + z3 - 3xyz = (x + y + z) (x2 + y2 + z2 - xy - yz - zx) 

i.e.

(x + y + z) (x2 + y2 + z2 - xy - yz - zx) = 0

thus, We observe two cases:

Case (I): (x + y + z = 0)

This doesn't necessarily imply x = y = z.

e.g. x = 1, y = 1 and z = -2 satisfied x + y + z = 0 but not equal

case (II): (x2 + y2 + z2 - xy - yz - zx) = 0

This is the same as Statement-I, which we've shown leads to (x = y = z).

Thus, from the statement

(II) we can not say x = y = z

 

Hence, if the question can be answered by using one of the statements alone, but cannot be answered using the other statement alone.

 

∴ The correct answer is option (1).

Given:

Statements \(\rm \frac{x+z}{y}=\frac{z}{x}\) and \(\rm \frac{z-y}{x}=\frac{x}{z}\)

Calculation:

From statement-1 i.e.  \(\rm \frac{x+z}{y}=\frac{z}{x}\)

⇒ x(x + z) = yz     -----(1)

⇒ x² + xz = yz

⇒ x² = z(y - x)     -----(2)

Statement-1 alone is not sufficient to answer the question.

From statement-2 i.e. \(\rm \frac{z-y}{x}=\frac{x}{z}\)

⇒ x² = z(z - y)     -----(3)

Equating eq(2) and eq(3)

⇒ z(y - x) = z(z - y)

⇒ y - x = z - y

⇒ z + x = 2y      -----(4)

Divide eq(1) by eq(4)

⇒ \(\frac{x(x + z)}{z+x} = \frac{yz}{2y}\)

⇒ x = z/2, which means x : z = 1 : 2

Put x = z/2 in eq(4)

⇒ 2y = z + x = 1 + 2 = 3

⇒ y = 3/2

Hence, x : y : z = 1 : 3/2 : 2 = 2 : 3 : 4

∴ The question can be answered by using both statements together, but cannot be answered using either statement alone.

Concept:

For two number a & b

LCM × HCF = a × b

Calculation:

Given that

Statement 1: LCM of the two numbers is 144

 

Statement 2: One of the numbers is 72

 

We can see, to find the other natural number, we need HCF also so that we can apply in the relation

LCM × HCF = a × b

∴ The correct answer is option 4

Given:

Statement 1: The difference between the largest and the second largest numbers is less than the difference between the second smallest and the smallest of the numbers.

Statement 2: The difference between the largest and the smallest numbers is greater than the difference between the second largest and the second smallest of the numbers.

**Let's analyze each

Statement:

**

Statement-I: The difference between the largest and the second largest numbers is less than the difference between the second smallest and the smallest of the numbers.

This statement doesn't provide direct information about whether the average of the largest and smallest numbers is greater than the average of all four numbers.

Statement-II: The difference between the largest and the smallest numbers is greater than the difference between the second largest and the second smallest of the numbers.

Similarly, this statement also doesn't directly give us the comparison of the averages.

**Now, let's combine the

Statements:

**

By combining both statements, we understand that the differences between the largest and smallest numbers and between the second largest and second smallest numbers are related. However, this information alone doesn't allow us to determine whether the average of the largest and smallest numbers is greater than the average of all four numbers.

Therefore, the correct answer is option 4) if the question cannot be answered even by using both the statements together.

Calculation:

(a - b + c) > (a + b - c)

⇒ - b + c > b - c

⇒ -2b > -2c

⇒ b < c

Statement

1. b is negative.

Statement 2.  c is negative.

Both statements are not enough to give the correct answer.

∴ The correct answer is option 4

Given:

Statement 1 The cost of 7 pens, 6pencils, and 5 notebooks is 200

Statement 2 The cost of 3 pens, 8 pencils, and 7 notebooks is 210

Calculation:

Let, the cost of 1 pen, 1 pencil, and 1 notebook be Rs a, Rs b, and Rs c respectively

According to the question,

⇒ 7a + 6b + 5c = 200   ....(i)

⇒ 3a + 8b + 7c = 210   ....(ii)

If we add both equations

⇒ 10a + 14b + 12c = 410 ...(iii)

If we multiply 1.5 in the equation (iii)

⇒ 15a + 21b + 18c = 605

⇒ 15 pen + 21 pencil + 18 notebook = 615

∴ The correct answer is option 3

Given:

Statement 1 The diameter of semi- circle is 20 cm

Statement 2 Two shorter sides of the triangle are 12 cm and 16 cm

Formula used:

Area = 1/2 × (breath × height)

Calculation:

By using statement 2 alone 

In an ΔOAB, OA ⊥ OB

Area = (1/2) × (12 × 16)

Area = 6 × 16 = 96 cm

As we can see, we can answer by using the statement 2 alone but can not answer by using the statement  1 alone. Therefore, 

∴ The answer is option 1.