Mathematics and Science PYQs

Calculation:

By verifying the options we can see that,

⇒ xa × xb = xab = 2¹ × 2² ≠ 2²

⇒' xa × xb = xa + b = 21 × 22 = 2³

⇒ xa × yb = (xy)ab = 2² × 3³ ≠ (6)⁶

⇒ xa ÷ xb =   \(\rm x^{\frac{a}{b}}\) = 2¹ / 2² ≠ 2^1/2

∴ The correct answer is xa × xb = xa + b.

Calculation:

(2.05 + 68.5 - 6.321 + 8.18 - 1.5)

⇒ 70.55 + 8.18 - 7.821

⇒ 70.55 + 0.359

⇒ 70.909

∴ The correct answer is 70.909.

Calculation:

1. HCF and LCM of two numbers x and y can be respectively 28 and 84:         

Let's find the HCF and LCM of two numbers whose values are 28 and 84, respectively:

HCF(28, 84) = 28 LCM(28, 84) = 84 This statement is true. 2. HCF and LCM of two numbers x and y can be respectively 1 and 105: 

Let's find the HCF and LCM of two numbers whose values are 1 and 105, respectively:

HCF(1, 105) = 1 LCM(1, 105) = 105 This statement is true. 3. HCF and LCM of two numbers x and y can be respectively 150 and 900:

Let's find the HCF and LCM of two numbers whose values are 150 and 900, respectively:

HCF(150, 900) = 150 LCM(150, 900) = 900 This statement is true. 4. HCF and LCM of two numbers x and y can be respectively 16 and 196:

Let's find the HCF and LCM of two numbers whose values are 16 and 196, respectively: HCF(16, 196) = 4 LCM(16, 196) = 784

This statement is not true because the given HCF and LCM values are incorrect.

Therefore, the statement that is not true is HCF and LCM of two numbers x and y can be respectively 16 and 196.

Calculation:

Least number will be 44 which when added to 12500 then the result will be 12544 which will be the square root of 112.

And the sum of digits in number 44 will be 4 + 4 = 8

∴ The correct answer is 8.

Calculation:

Let's analyze each

Statement:

1. A number is irrational if it can be expressed in the form p/q where p and q are integers, q ≠ 0. This statement is incorrect. 

A number is irrational if it cannot be expressed in the form p/q where p and q are integers, q ≠ 0. 2. Sum of any two irrational numbers is sometimes a rational number and sometimes an irrational number. This statement is correct. 

The sum of two irrational numbers can be either rational or irrational, depending on the specific numbers involved.

For example, √2 + (-√2) = 0, which is a rational number. √2 + √3 which is an irrational
3. Product of any two irrational numbers is always an irrational number. This statement is incorrect.

The product of two irrational numbers is always an rational number.

For example, √2 × √2 = 2, which is rational. 4. The square of an irrational number is always a rational number. This statement is incorrect.

The square of an irrational number can be either rational or irrational, depending on the specific number involved.

For example, (√2)² = 2, which is rational, while (√3)² = 3, which is also rational. However, (1 + √3)²  = 4 + 2√3, which is irrational.

Calculation:

∴ The correct answer is option 4.

Calculation:

Let's denote the number as "x".

According to the given information, when the number is increased by 9, it becomes 3 times half of the number.

Mathematically, we can express this as: x + 9 = 3 ×  (x/2)

Simplifying the equation: x + 9 = (3/2) × x

Multiplying both sides of the equation by 2 to eliminate the fraction: 2x + 18 = 3x

Subtracting 2x from both sides of the equation: 18 = x

Therefore, the number is 18.

Shortcut Trick

For Simple Interest, use the formula: (n₁ − 1) / T₁ = (n₂ − 1) / T₂

Where n is the multiplier and T is the time.

⇒ (2 − 1) / 8 = (4 − 1) / T₂

⇒ 1 / 8 = 3 / T₂

⇒ T₂ = 8 × 3 = 24 years.

∴ The correct answer is 24 years.

Alternate Method

Given:

Time (T₁) = 8 years

Amount (A₁) = 2 × Principal (P)

Target Amount (A₂) = 4 × Principal (P)

Calculations:

⇒ Simple Interest (SI₁) = A₁ − P = 2P − P = P

⇒ This interest P is earned in 8 years.

⇒ For the sum to become 4 times, Interest (SI₂) = A₂ − P = 4P − P = 3P

⇒ In Simple Interest, the interest earned is directly proportional to time.

⇒ Time required (T₂) = (SI₂ / SI₁) × T₁

⇒ T₂ = (3P / P) × 8

⇒ T₂ = 3 × 8 = 24 years.

∴ The correct answer is 24 years.

Additional Information

Simple Interest Formula

The formula is SI = (P × R × T) / 100, where P is Principal, R is Rate per annum, and T is Time in years.

Rate of Interest Calculation

If a sum doubles in T years at SI, the rate R = 100 / T. In this case, R = 100 / 8 = 12.5%.

Amount in Simple Interest

The total Amount (A) is the sum of the Principal and the Interest earned: A = P + SI.

Proportionality Rule

In Simple Interest, the interest earned for any period is constant, meaning Interest is directly proportional to the Time duration.

Given:

Renu's mother age while her birth is 30 years.

Calculation:

Let the present age of renu is x.

According to question,

⇒ (30 + x - 5) = 4(x - 5)

⇒ 25 + x = 4x - 20

⇒ 3x = 45

⇒ x = 15

∴ The correct answer is 15.

Shortcut Trick

Initial Speed = 140 ÷ 3.5 = 40 km/h.

New Speed = 40 − 5 = 35 km/h.

Since distance is constant, Time ratio is the inverse of Speed ratio.

Time ratio = 35 : 40 = 7 : 8.

7 units = 3.5 hours ⇒ 1 unit (increase) = 0.5 hours = 30 minutes.

∴ The correct answer is 30 minutes.

Alternate Method

Given:

Distance = 140 km

Initial Time (T₁) = 3.5 hours

Decrease in Speed = 5 km/h

Distance = 140 km

Case 1: S = 40 km/h
T = 3.5 hours

Case 2: S = 35 km/h
T = 4 hours

Formula Used:

Speed = Distance ÷ Time

Time = Distance ÷ Speed

Calculations:

⇒ Initial Speed (S₁) = 140 ÷ 3.5 = 40 km/h

⇒ New Speed (S₂) = 40 − 5 = 35 km/h

⇒ New Time (T₂) = 140 ÷ 35 = 4 hours

⇒ Increase in time = T₂ − T₁ = 4 − 3.5 = 0.5 hours

⇒ Increase in time (in minutes) = 0.5 × 60 = 30 minutes

∴ The correct answer is 30 minutes.

Additional Information

Inverse Relation of Speed and Time

When the distance is constant, speed and time are inversely proportional. Ratio: S₁/S₂ = T₂/T₁.

Average Speed

Average Speed = Total Distance ÷ Total Time. If equal distances are covered at speeds a and b, Average Speed = 2ab ÷ (a + b).

Relative Speed

When two objects move in opposite directions, relative speed = a + b. When moving in the same direction, relative speed = a − b.

Concept use:

A constructible angle is one that can be drawn using only a compass and straightedge, and a limited set of operations (creating a line segment between two points, extending a line, drawing a circle with a given radius, etc.).

Explanation:

Some of the most elementary constructible angles are 90°, 60°, 45°, 30°, etc. These angles are easy to construct because they can be achieved through halving or trisecting certain angles.

Example:

 

A 45° angle can be obtained by bisecting a 90° angle

.

but 

For a 40-degree angle,

it is not a straightforward task with just a compass and straightedge because 40 is not a product of subtracting, adding, and halving these elementary values.

Hence, option 2 is correct.

Given:

Measure of an exterior angle of a regular polygon: 45°

Calculation:

Let's denote the number of sides of the regular polygon as "n".

According to the given information,

The measure of an exterior angle is 45°.

Using the formula mentioned above,

We can write the equation: 360° / n = 45°

To solve for "n", we can cross-multiply and simplify:

⇒ 360° = 45n

Dividing both sides by 45°: 360° / 45° = n

⇒ 8 = n

Therefore, the number of sides of the regular polygon is 8.

Calculkation:

According to question,

In ΔOAC = 180° - 130° 

∠AOC = 50° 

By vertically opposite angle

∠COA = ∠BOD = 50° 

Now In ΔOBD,

∠ ODB = 180° - (50  - 90° ) = 40° 

∴ The correct answer is 40°

Given:

Number of faces of a polyhedron E.

Number of edges of a polyhedron V.

Number of vertices of a polyhedron.

Concept used:

Euler's formula for a polyhedron states that the number of faces (F), edges (E), and vertices (V) of a polyhedron are related by the equation F + V - E = 2.

Calculation:

Using Euler's formula, we can see that the correct option is: F + V - E = 2

Therefore, the statement "F + V - E = 2" is true.

Hence, option 3 is correct.

Calculation:

OA = OB

∠OBA = ∠OAB = 54° 

In ΔOBA

⇒ ∠OBA + ∠OAB + ∠BOA = 180° 

⇒ 2 × 54° +  ∠BOA  = 180° 

⇒ ∠BOA = 72° 

Now ∠BOA + ∠COA = 180° 

⇒ ∠COA = 108° 

Now,

∠COA + ∠OCA + ∠OAC = 180° 

⇒ 108° + 2x = 180° 

⇒ 2x = 72° 

⇒ x = 36° 

∴ The correct answer is 36°.

Given:

Diagonals of rhombus are of 10 cm and 24 cm.

Calculation:

According to question,

⇒ (Side)² = 5² + 12²

⇒ (Side)2 = 25 + 144

⇒ (Side)2  = 169

⇒ Side = 13

Perimeter = Sum of all sides 

⇒ 4a = 4 × 13 = 52

∴ The correct answer is 52 cm.

Shortcut Trick

Volume remains constant: π × R² × H = π × r² × h

Convert radius to same unit (cm): 2 mm = 0.2 cm

1.2² × 22 = 0.2² × h

1.44 × 22 = 0.04 × h ⇒ h = 36 × 22 = 792 cm

∴ The correct answer is 792 cm.

Alternate Method

Given:

Radius of rod (R) = 1.2 cm

Length of rod (H) = 22 cm

Radius of wire (r) = 2 mm = 0.2 cm (since 10 mm = 1 cm)

Formula Used:

Volume of cylinder = πr²h

Calculations:

⇒ Volume of metallic rod = π × (1.2)² × 22

⇒ Volume of cylindrical wire = π × (0.2)² × L

⇒ π × 1.44 × 22 = π × 0.04 × L

⇒ 1.44 × 22 = 0.04 × L

⇒ L = (1.44 × 22) ÷ 0.04

⇒ L = 36 × 22

⇒ L = 792 cm

∴ The correct answer is 792 cm.

Additional Information

Recasting Concept

When a 3D solid is melted and recast into another shape, its Total Volume remains constant, provided there is no wastage.

Cylinder Properties

Curved Surface Area (CSA) = 2πrh | Total Surface Area (TSA) = 2πr(h + r).

Unit Conversion

Always ensure all dimensions are in the same unit. 1 cm = 10 mm | 1 m = 100 cm.

Given:

Five cubical wooden blocks having side 10 cm

Formula used:

Surface area of cuboid = 2(LB + BH + HL) where L is Length, B is Breadth and H is the Height

Calculation:

According to question,

L = 50, B = 10, H = 10

⇒ 2(500  +100 + 500)

⇒ 2 ×  (1100)

⇒ 2200

Cost of painting surface = 2200 × 10/100 = Rs.220

∴ The correct answer is Rs.220.

Given:

Mean of 10, 4, 5, 3, 2, 6, 7, 3 is m.

Mean of 2, 5, 3, 11, m, 10 is p.

Formula used : 

Mean = Sum of all observation/Number of observation

Calculation:

According to question,

⇒ 10 +  4 + 5 + 3 + 2 + 6 + 7 + 3 = 8m

⇒ m= 5

Now,

⇒ 2 + 5 + 3 + 11+ m + 10 = 6p

⇒ p = 6

Now, (m + p) = 11

∴ The correct answer is 11.

Given:

Numbers from 25 to 50 are written on different slips, one number on each slip.

Calculation:

The prime numbers between 25 and 50 are: 29, 31, 37, 41, 43, and 47. So, there are 6 prime numbers in this range.

To find the number of non-prime numbers, we subtract the number of prime numbers from the total number of outcomes: Total number of outcomes = 50 - 25 + 1 = 26 (since there are 26 numbers in the given range)

Number of non-prime numbers = Total number of outcomes - Number of prime numbers

Number of non-prime numbers = 26 - 6 = 20

Therefore, the probability of not getting a prime number is:

Probability = Number of non-prime numbers / Total number of outcomes

Probability = 20 / 26 = 10 / 13

∴ The correct answer is 10/13.