Search Previous Year Questions

Shortcut Trick

Apply BODMAS: First solve the bracket (36 + 29) = 65.

Perform multiplication: 65 × 2 = 130.

Solve addition and subtraction from left to right: 88 + 47 − 59 + 130 − 94.

Result: 135 − 59 + 130 − 94 = 76 + 130 − 94 = 206 − 94 = 112.

∴ The correct answer is 112.

Alternate Method

Given: 88 + 47 − 59 + (36 + 29) × 2 − 94

Formula Used: BODMAS Rule (Brackets, Orders, Division, Multiplication, Addition, Subtraction)

⇒ 88 + 47 − 59 + (65) × 2 − 94

⇒ 88 + 47 − 59 + 130 − 94

⇒ 135 − 59 + 130 − 94

⇒ 76 + 130 − 94

⇒ 206 − 94

⇒ 112

∴ The correct answer is 112.

Additional Information

BODMAS Rule

It is an acronym to help remember the order of operations in calculations: Brackets, Orders, Division, Multiplication, Addition, and Subtraction.

Sign Priority

When operations like addition and subtraction are at the same priority level, they are typically solved from left to right to avoid errors.

Brackets Priority

If multiple brackets are present, solve in the order: Round Brackets (), then Curly Brackets {}, and finally Square Brackets [].

Shortcut Trick

Solve using BODMAS: 11 of 2 = 22, 18 ÷ 3 = 6, and 12 × 2 = 24.

Expression simplifies to: 88 − 27 + 24 − 6 + 22.

Combine positive and negative terms: (88 + 24 + 22) − (27 + 6) = 134 − 33.

Final calculation: 134 − 33 = 101.

∴ The correct answer is 101.

Alternate Method

Given: 88 − 27 + 12 × 2 − 18 ÷ 3 + (11 of 2)

Method Used: BODMAS Rule (Brackets, Of, Division, Multiplication, Addition, Subtraction).

⇒ 88 − 27 + 12 × 2 − 18 ÷ 3 + 22 (Solving the 'Of' part: 11 × 2)

⇒ 88 − 27 + 12 × 2 − 6 + 22 (Solving Division: 18 ÷ 3 = 6)

⇒ 88 − 27 + 24 − 6 + 22 (Solving Multiplication: 12 × 2 = 24)

⇒ 61 + 24 − 6 + 22 (Solving Subtraction: 88 − 27 = 61)

⇒ 85 − 6 + 22 (Solving Addition: 61 + 24 = 85)

⇒ 79 + 22 = 101 (Final Subtraction and Addition)

∴ The correct answer is 101.

Additional Information

BODMAS Priority

The rule dictates the order: Brackets first, then Of (powers/roots/multiplication), followed by Division and Multiplication (left-to-right), and finally Addition and Subtraction (left-to-right).

The 'Of' Operator

In mathematical expressions, 'Of' acts as a multiplication but must be performed before division and multiplication according to the BODMAS hierarchy.

Simplification Tip

Always group all positive numbers and all negative numbers separately at the end to minimize errors during the final Addition and Subtraction steps.

Shortcut Trick

Apply the BODMAS rule: Multiplication → Addition → Subtraction.

0.5 × 0.3 = 0.15

0.15 + 0.9 = 1.05

1.05 − 0.55 = 0.5

∴ The correct answer is 0.5.

Alternate Method

Given:

0.5 × 0.3 + 0.9 − 0.55

Formula/Rule Used:

BODMAS (Brackets, Orders, Division, Multiplication, Addition, Subtraction)

⇒ Step 1: Perform Multiplication

⇒ 0.5 × 0.3 = 0.15

⇒ Step 2: Perform Addition

⇒ 0.15 + 0.9 = 1.05

⇒ Step 3: Perform Subtraction

⇒ 1.05 − 0.55 = 0.5

∴ The correct answer is 0.5.

Additional Information

BODMAS Rule

Standard order of operations: Brackets, Orders (powers/roots), Division and Multiplication (left to right), Addition and Subtraction (left to right).

Decimal Multiplication

When multiplying decimals, count the total number of decimal places in the factors and place the decimal point in the product accordingly.

Decimal Addition/Subtraction

Always align the decimal points vertically before performing addition or subtraction to ensure units are matched correctly.

Shortcut Trick

For numbers ending in 0.25, if the integer part is n × (n + 1), the square root is n.5.

√90.25 (9 × 10 = 90) → 9.5

√6.25 (2 × 3 = 6) → 2.5

√2.25 (1 × 2 = 2) → 1.5

√0.25 (0 × 1 = 0) → 0.5

Expression: 9.5 + 2.5 − 1.5 + 0.5 = 11

∴ The correct answer is 11.

Alternate Method

Given:

Expression = √90.25 + √6.25 − √2.25 + √0.25

Calculations:

⇒ √90.25 = √(9025 ÷ 100) = 95 ÷ 10 = 9.5

⇒ √6.25 = √(625 ÷ 100) = 25 ÷ 10 = 2.5

⇒ √2.25 = √(225 ÷ 100) = 15 ÷ 10 = 1.5

⇒ √0.25 = √(25 ÷ 100) = 5 ÷ 10 = 0.5

⇒ Result = 9.5 + 2.5 − 1.5 + 0.5

⇒ Result = 12 − 1.5 + 0.5

⇒ Result = 10.5 + 0.5 = 11

∴ The correct answer is 11.

Additional Information

Square of Numbers Ending in 5

To square a number ending in 5, multiply the digits before 5 by the next consecutive integer and append 25. Example: (45)² = (4 × 5) and 25 = 2025.

Square Root of Decimal Places

When finding the square root of a decimal, the number of decimal places in the result is exactly half of the decimal places in the original number. (e.g., √0.0001 = 0.01).

BODMAS Rule

Always follow the BODMAS order (Brackets, Orders, Division, Multiplication, Addition, Subtraction) for simplification to ensure accuracy.

Shortcut Trick

The perimeter of a square is the total length of its boundary.

Since all four sides of a square are equal, the formula is 4 × side.

Perimeter = 4 × 12 = 48 cm.

∴ The correct answer is 48 cm.

Alternate Method

Given:

Side (a) of the square = 12 cm

Formula Used:

Perimeter (P) = 4 × a

Calculations:

⇒ Perimeter = 12 + 12 + 12 + 12

⇒ Perimeter = 4 × 12

⇒ Perimeter = 48 cm

∴ The correct answer is 48 cm.

Additional Information

Area of a Square

The total space occupied inside the boundary of the square, calculated as side × side or a².

Diagonal of a Square

The straight line connecting opposite vertices. Length = Side × √2.

Properties of a Square

All four sides are equal, and each interior angle is exactly 90°.

Shortcut Trick

The expression follows the Algebraic Identity: √((a + b)² − 4ab) = (a − b).

Identify a = 36 and b = 29.

Result = 36 − 29 = 7.

∴ The correct answer is 7.

Alternate Method

Given: √((36 + 29)² − 4 × 36 × 29)

Solve the terms inside the square root step by step:

⇒ √((65)² − (4 × 1044))

⇒ √(4225 − 4176)

⇒ √49

⇒ 7

∴ The correct answer is 7.

Additional Information

Square of a Sum

(a + b)² = a² + b² + 2ab

Square of a Difference

(a − b)² = a² + b² − 2ab

Key Mathematical Relation

(a + b)² − (a − b)² = 4ab, which simplifies the given square root expression directly to (a − b).

Shortcut Trick

Side (a) of rhombus = Perimeter ÷ 4 = 68 ÷ 4 = 17 cm.

One half-diagonal (d₁/2) = 16 ÷ 2 = 8 cm.

Using Pythagorean triplet (8, 15, 17): the other half-diagonal (d₂/2) = 15 cm.

Second diagonal (d₂) = 15 × 2 = 30 cm.

Sum of diagonals = 16 + 30 = 46 cm.

∴ The correct answer is 46 cm.

Alternate Method

Given: Perimeter = 68 cm, Diagonal d₁ = 16 cm.

Formula Used: Side of rhombus a = Perimeter/4; Relationship: a² = (d₁/2)² + (d₂/2)²

⇒ Side a = 68 ÷ 4 = 17 cm

⇒ Let half of the diagonals be h₁ and h₂.

⇒ h₁ = 16 ÷ 2 = 8 cm

⇒ a² = h₁² + h₂²

⇒ 17² = 8² + h₂²

⇒ 289 = 64 + h₂²

⇒ h₂² = 289 − 64 = 225

⇒ h₂ = √225 = 15 cm

⇒ Second diagonal d₂ = 2 × h₂ = 2 × 15 = 30 cm

⇒ Sum of both the diagonals = d₁ + d₂ = 16 + 30 = 46 cm

∴ The correct answer is 46 cm.

Additional Information

Properties of a Rhombus

All sides are equal, and diagonals bisect each other at right angles (90°).

Area of a Rhombus

Area = 1/2 × d₁ × d₂, where d₁ and d₂ are the lengths of the diagonals.

Side-Diagonal Relationship

4a² = d₁² + d₂², which is derived from the Pythagorean theorem applied to the four internal right-angled triangles.

Shortcut Trick

Side of Rhombus (s) = Perimeter ÷ 4 = 68 ÷ 4 = 17 cm.

Half of the given diagonal (d₁/2) = 30 ÷ 2 = 15 cm.

Using Pythagoras triplet (8, 15, 17), the half of other diagonal (d₂/2) = 8 cm.

Ratio of diagonals is the same as the ratio of their halves.

Required Ratio = 15 : 8.

∴ The correct answer is 15 : 8.

Alternate Method

Given: Perimeter = 68 cm, Diagonal 1 (d₁) = 30 cm.

Formula Used: Side² = (d₁/2)² + (d₂/2)²

⇒ Side of Rhombus = 68 ÷ 4 = 17 cm

⇒ Let half of the diagonals be 15 cm and x cm.

⇒ In right-angled triangle AOB, AB² = AO² + OB²

⇒ 17² = 15² + x²

⇒ 289 = 225 + x²

⇒ x² = 289 − 225 = 64

⇒ x = √64 = 8 cm

⇒ Second diagonal (d₂) = 2 × 8 = 16 cm

⇒ Ratio of diagonals = d₁ : d₂ = 30 : 16

⇒ Ratio = 15 : 8

∴ The correct answer is 15 : 8.

Additional Information

Area of Rhombus

The area of a rhombus can be calculated using diagonals: Area = 1/2 × d₁ × d₂.

Properties of Diagonals

The diagonals of a rhombus bisect each other at right angles (90°).

Relation between Side and Diagonals

For a rhombus with side 's', the relation is 4s² = d₁² + d₂².

Shortcut Trick

To find the Highest Common Factor (HCF) of numbers in prime power form, identify the common bases and pick the lowest exponent for each.

Common base 5: The powers are 7 and 5. The minimum power is 5.

Common base 11: The powers are 3 and 6. The minimum power is 3.

HCF = 5⁵ × 11³.

∴ The correct answer is 5⁵ × 11³.

Alternate Method

Given:

Number 1 (N₁) = 5⁷ × 11³

Number 2 (N₂) = 5⁵ × 11⁶

Formula Used:

For any two numbers expressed as product of primes p₁^a × p₂^b and p₁^c × p₂^d:

HCF = p₁^{min(a, c)} × p₂^{min(b, d)}

Calculations:

⇒ Compare exponents of prime factor 5: 7 and 5. The smaller one is 5.

⇒ Compare exponents of prime factor 11: 3 and 6. The smaller one is 3.

⇒ HCF = 5^{min(7, 5)} × 11^{min(3, 6)}

⇒ HCF = 5⁵ × 11³

∴ The correct answer is 5⁵ × 11³.

Additional Information

Least Common Multiple (LCM)

For numbers in prime power form, LCM is found by taking the highest power of every prime factor present across the numbers.

HCF and LCM Relation

For any two positive integers a and b, the product of the numbers is equal to the product of their HCF and LCM: a × b = HCF(a, b) × LCM(a, b).

Co-prime Numbers

If the Highest Common Factor of two numbers is 1, they are called co-prime or relatively prime numbers.

Shortcut Trick

The Highest Common Factor (HCF) of any two numbers is always a factor of their Least Common Multiple (LCM).

Therefore, the LCM must be exactly divisible by the HCF without any remainder.

Divide 84 by the given options:

84 ÷ 21 = 4 (Exactly divisible)

Other options (18, 30, 16) do not divide 84 completely.

∴ The correct answer is 21.

Alternate Method

Given: LCM of two numbers = 84

Property used: LCM is always a multiple of HCF. If L is the LCM and H is the HCF, then L/H must be an integer.

Testing each option:

⇒ Case 1: 84 ÷ 18 = 4.66 (Not an integer)

⇒ Case 2: 84 ÷ 21 = 4 (Integer)

⇒ Case 3: 84 ÷ 30 = 2.8 (Not an integer)

⇒ Case 4: 84 ÷ 16 = 5.25 (Not an integer)

⇒ Since only 21 divides 84 completely, it can be the HCF.

∴ The correct answer is 21.

Additional Information

Product Rule of LCM and HCF

For any two numbers a and b, the product of the numbers is equal to the product of their HCF and LCM: a × b = HCF × LCM.

Co-prime Numbers

Two numbers are called co-prime if their HCF is 1. In this case, their LCM is simply the product of the two numbers.

HCF of Fractions

To find the HCF of fractions, calculate the HCF of numerators and divide it by the LCM of denominators.