Mathematics and Science PYQs

Shortcut Trick

Check the units digit: A × A must end in digit A.

Possible values for A are 0, 1, 5, and 6.

Testing A = 5: 25 × 5 = 125.

Matching with the form B 2 A, we get 1 2 5, which means B = 1.

Testing A = 6: 26 × 6 = 156 (Does not match B 2 A as the middle digit is 5, not 2).

∴ The correct answer is A = 5, B = 1.

Alternate Method

Given:

2A × A = B 2 A

Formula:

(10x + y) × z notation for digits.

⇒ In decimal form: (20 + A) × A = 100B + 20 + A

⇒ 20A + A² = 100B + 20 + A

⇒ A² + 19A − 20 = 100B

⇒ (A − 1)(A + 20) = 100B

⇒ Since B is a single digit, 100B must be a multiple of 100 (0, 100, 200, etc.).

⇒ If A = 5: (5 − 1) × (5 + 20) = 4 × 25 = 100

⇒ 100B = 100 ⇒ B = 1

⇒ If A = 1: (1 − 1) × (1 + 20) = 0 ⇒ 100B = 0 ⇒ B = 0 (But the product B2A is a 3-digit number, so B ≠ 0).

∴ The correct answer is A = 5, B = 1.

Additional Information

Cryptarithmetic Rules

Each letter represents a unique digit from 0 to 9, and the leading digit of a multi-digit number cannot be 0.

Units Digit Property

For any digit n, n² ends in the same digit n only if n ∈ {0, 1, 5, 6}. This is a critical starting point for solving multiplication puzzles.

Digit Range Constraints

The product of a 2-digit number (starting with 2) and a 1-digit number ranges from 20 × 0 = 0 to 29 × 9 = 261, which limits the possible value of B to 1 or 2.

Shortcut Trick

Direct relation: (a + b)² = (a − b)² + 4ab

Substitute values: (a + b)² = (11/5)² + 4 × (3/4)

(a + b)² = 121/25 + 3 = (121 + 75)/25 = 196/25

a + b = √(196/25) = 14/5 = 2 4/5

∴ The correct answer is \(2\frac{4}{5}\).

Alternate Method

Given:

a − b = 11/5

ab = 3/4

Formula used: (a + b)² = (a − b)² + 4ab

⇒ (a + b)² = (11/5)² + 4 × (3/4)

⇒ (a + b)² = 121/25 + 3

⇒ (a + b)² = (121 + 75) / 25

⇒ (a + b)² = 196/25

⇒ a + b = √(196/25)

⇒ a + b = 14/5

⇒ a + b = \(2\frac{4}{5}\)

∴ The correct answer is \(2\frac{4}{5}\).

Additional Information

Algebraic Identities

The square of sum and difference relate as: (a + b)² − (a − b)² = 4ab.

Sum of Squares

The sum of squares can be found using: a² + b² = (a − b)² + 2ab or a² + b² = (a + b)² − 2ab.

Difference of Squares

a² − b² = (a − b)(a + b). This identity is useful when both sum and difference are known.

Shortcut Trick

Inside bracket: 2x − (1−x)/2 = (4x − 1 + x)/2 = (5x − 1)/2.

Equation becomes: (7x − 1)/4 − (5x − 1)/6 = 10/3.

Multiply all terms by LCM 12: 3(7x − 1) − 2(5x − 1) = 40.

21x − 3 − 10x + 2 = 40 ⇒ 11x − 1 = 40 ⇒ 11x = 41.

Value required: (11x + 21)/2 = (41 + 21)/2 = 62/2 = 31.

∴ The correct answer is 31.

Alternate Method

Given: (7x − 1)/4 − 1/3(2x − (1 − x)/2) = 10/3

Formula used: Algebraic transposition and simplification.

⇒ (7x − 1)/4 − 1/3((4x − (1 − x))/2) = 10/3

⇒ (7x − 1)/4 − 1/3((4x − 1 + x)/2) = 10/3

⇒ (7x − 1)/4 − (5x − 1)/6 = 10/3

⇒ (3(7x − 1) − 2(5x − 1)) / 12 = 10/3

⇒ (21x − 3 − 10x + 2) / 12 = 10/3

⇒ (11x − 1) / 12 = 10/3

⇒ 11x − 1 = (10 × 12) / 3

⇒ 11x − 1 = 40

⇒ 11x = 41

⇒ Required value = (11x + 21) / 2

⇒ (41 + 21) / 2 = 62 / 2 = 31

∴ The correct answer is 31.

Additional Information

BODMAS Rule

Always solve Brackets, then Orders, Division/Multiplication, and finally Addition/Subtraction to simplify equations correctly.

Transposition Method

Moving a term from one side of an equation to the other changes its sign (− becomes + and × becomes ÷).

Linear Equations

An equation of the form ax + b = c, where x is the variable to be solved by isolating it on one side.

Shortcut Trick

Identify the middle term 4x² + 16y² − 16xy as the perfect square (2x − 4y)².

Substitute k = (2x − 4y) to rewrite the expression as 6k² + 5k − 6.

Factorize the quadratic 6k² + 5k − 6 into (3k − 2)(2k + 3).

Substituting k back, 3k − 2 = 3(2x − 4y) − 2 = 6x − 12y − 2 = 2(3x − 6y − 1).

One of the simplified factors is 3x − 6y − 1.

∴ The correct answer is 3x − 6y − 1.

Alternate Method

Given:

5(2x − 4y) + 6(4x² + 16y² − 16xy) − 6

Formula Used:

(a − b)² = a² + b² − 2ab

Calculations:

⇒ Expression = 5(2x − 4y) + 6[(2x)² + (4y)² − 2(2x)(4y)] − 6

⇒ Expression = 5(2x − 4y) + 6(2x − 4y)² − 6

⇒ Let k = 2x − 4y

⇒ Expression = 6k² + 5k − 6

⇒ Splitting the middle term:

⇒ 6k² + 9k − 4k − 6

⇒ 3k(2k + 3) − 2(2k + 3)

⇒ = (3k − 2)(2k + 3)

⇒ Substitute k = 2x − 4y into (3k − 2)

⇒ 3(2x − 4y) − 2

⇒ = 6x − 12y − 2

⇒ = 2(3x − 6y − 1)

⇒ Hence, 3x − 6y − 1 is a factor.

∴ The correct answer is 3x − 6y − 1.

Additional Information

Algebraic Identity

The identity (a − b)² = a² + b² − 2ab is used to simplify polynomials where squared terms and cross-products are present.

Quadratic Factorization

For a quadratic expression ax² + bx + c, find two numbers whose product is a × c and whose sum is b to split the middle term.

Factor Theorem

If a polynomial P(x) has a factor (ax − b), then substituting x = b ÷ a into the expression gives zero.

Shortcut Trick

Marked Price (MP) = ₹2,500 and Cost Price (CP) = ₹1,500.

Selling Price after 20% discount = 80% of 2,500 = ₹2,000.

Final Selling Price (SP) after additional ₹250 discount = 2,000 − 250 = ₹1,750.

Profit = Final SP − CP = 1,750 − 1,500 = ₹250.

∴ The correct answer is ₹250.

Alternate Method

Given:

Cost Price (CP) = ₹1,500

Marked Price (MP) = ₹2,500

Percentage Discount = 20%

Cash Discount = ₹250

Formula Used:

Selling Price (SP) = [MP × (100 − Discount%)/100] − Cash Discount

Profit = SP − CP

CP ₹1,500 MP ₹2,500 SP ₹1,750 -20% & -₹250 Profit = ₹250

Calculations:

⇒ Price after 20% discount = 2,500 × (100 − 20)/100

⇒ Price after 20% discount = 2,500 × 0.8 = ₹2,000

⇒ Final Selling Price (SP) = 2,000 − 250 = ₹1,750

⇒ Profit = 1,750 − 1,500 = ₹250

∴ The correct answer is ₹250.

Additional Information

Profit Percentage

Profit % = (Profit ÷ CP) × 100. In this case: (250 ÷ 1500) × 100 ≈ 16.67%.

Mark-up Percentage

Mark-up % = [(MP − CP) ÷ CP] × 100. Here: [(2500 − 1500) ÷ 1500] × 100 = 66.67%.

Successive Discounts

When two successive percentage discounts a% and b% are given, the single equivalent discount is (a + b − ab/100)%.

Shortcut Trick

For four numbers a, b, c, and d to be in proportion after adding x, use the formula:

x = (ad − bc) ÷ ((b + c) − (a + d))

Given a = 4, b = 10, c = 12, d = 24

x = (4 × 24 − 10 × 12) ÷ ((10 + 12) − (4 + 24)) = (96 − 120) ÷ (22 − 28) = −24 ÷ −6 = 4

Required value = 3x + 1 = 3(4) + 1 = 12 + 1 = 13

∴ The correct answer is 13.

Alternate Method

Given: 

Numbers: 4, 10, 12, 24

According to the problem, (4 + x), (10 + x), (12 + x), and (24 + x) are in proportion.

Formula:

(4 + x) ÷ (10 + x) = (12 + x) ÷ (24 + x)

4 + x 10 + x = 12 + x 24 + x Proportion: Product of Extremes = Product of Means

⇒ (4 + x)(24 + x) = (10 + x)(12 + x)

⇒ 96 + 4x + 24x + x² = 120 + 10x + 12x + x²

⇒ 96 + 28x = 120 + 22x

⇒ 28x − 22x = 120 − 96

⇒ 6x = 24 ⇒ x = 4

⇒ 3x + 1 = 3(4) + 1 = 12 + 1 = 13

∴ The correct answer is 13.

Additional Information

Mean Proportional

The mean proportional between two numbers a and b is √(a × b).

Third Proportional

If a, b, and c are in continued proportion (a:b = b:c), then c is the third proportional, c = b² ÷ a.

Fourth Proportional

If a, b, c, and d are in proportion (a:b = c:d), then d is the fourth proportional, d = (b × c) ÷ a.

Invertendo Property

If a : b = c : d, then b : a = d : c.

Shortcut Trick

Sum of supplementary angles = 180°

(3x − 30°) + (2x + 20°) = 180°

5x − 10° = 180°

5x = 190° → x = 38°

Required value = 2(38°) − 4° = 72°

∴ The correct answer is 72°.

Alternate Method

Given:

Angle 1 = (3x − 30°)

Angle 2 = (2x + 20°)

Formula Used:

Two angles are supplementary if their sum is exactly 180°.

 (3x − 30°) (2x + 20°) Total Sum = 180°

Calculations:

⇒ (3x − 30°) + (2x + 20°) = 180°

⇒ 3x + 2x − 30° + 20° = 180°

⇒ 5x − 10° = 180°

⇒ 5x = 180° + 10°

⇒ 5x = 190°

⇒ x = 190° ÷ 5

⇒ x = 38°

Now, find the value of (2x − 4°):

⇒ Value = 2(38°) − 4°

⇒ Value = 76° − 4°

⇒ Value = 72°

∴ The correct answer is 72°.

Additional Information

Supplementary Angles

Two angles are said to be supplementary if the sum of their measures is exactly 180°. They form a linear pair when placed adjacently.

Complementary Angles

Two angles are said to be complementary if the sum of their measures is exactly 90°.

Linear Pair of Angles

A pair of adjacent angles whose non-common sides are opposite rays, thus summing up to 180°.

Vertically Opposite Angles

Angles opposite each other when two lines intersect are always equal in measure.

Shortcut Trick

Angle between angle bisectors ∠POQ = 90° + (∠R ÷ 2)

In ΔPQR, ∠R = 180° − (30° + 50°) = 100°

∠POQ = 90° + (100° ÷ 2) = 90° + 50° = 140°

Required value = (2 × 140°) − 10° = 280° − 10° = 270°

∴ The correct answer is 270°.

Alternate Method

Given: ∠P = 30°, ∠Q = 50°. O is the intersection of angle bisectors of ∠P and ∠Q.

Formula Used: Sum of angles in a triangle = 180°.

 P (30°) Q (50°) R O 15° 25° ∠POQ

⇒ In ΔPOQ, the internal angles are formed by bisectors.

⇒ ∠OPQ = ∠P ÷ 2 = 30° ÷ 2 = 15°

⇒ ∠OQP = ∠Q ÷ 2 = 50° ÷ 2 = 25°

⇒ In ΔPOQ: ∠POQ + ∠OPQ + ∠OQP = 180°

⇒ ∠POQ + 15° + 25° = 180°

⇒ ∠POQ = 180° − 40° = 140°

⇒ Value of (2∠POQ − 10°) = (2 × 140°) − 10°

⇒ Value = 280° − 10° = 270°

∴ The correct answer is 270°.

Additional Information

Incenter Properties

The point where the internal angle bisectors of a triangle meet is the Incenter. The angle at the incenter is given by 90° + (opposite angle ÷ 2).

Excenter Properties

The angle formed by external bisectors of two angles (say Q and R) is given by 90° − (∠P ÷ 2).

Circumcenter Angle

The angle subtended by a side at the circumcenter is twice the angle at the opposite vertex (e.g., ∠QCR = 2 × ∠P).

Shortcut Trick

Check if the sides are in the ratio of standard Pythagorean triples like 3:4:5 or 5:12:13.

Option A: 12, 16, 20 → 4 × (3, 4, 5) → Right Triangle.

Option B: 4.5, 6, 7.5 → 1.5 × (3, 4, 5) → Right Triangle.

Option D: 2.5, 6, 6.5 → 0.5 × (5, 12, 13) → Right Triangle.

Option C: 8, 12, 16 → 4 × (2, 3, 4). Since 2² + 3² ≠ 4² (4 + 9 ≠ 16), it is not a right triangle.

∴ The correct answer is 8, 16, 12 cm.

Alternate Method

Given: Sides of triangles in cm: (A) 12, 16, 20; (B) 4.5, 6, 7.5; (C) 8, 16, 12; (D) 2.5, 6, 6.5.

Formula Used: Pythagoras Theorem: a² + b² = c², where c is the longest side.

Base (a) Perpendicular (b) Hypotenuse (c) a² + b² = c²

Check Option C: Sides are 8 cm, 12 cm, and 16 cm (Longest side c = 16).

⇒ a² + b² = 8² + 12²

⇒ 64 + 144 = 208

⇒ c² = 16² = 256

⇒ 208 ≠ 256

Check Option D: Sides are 2.5 cm, 6 cm, and 6.5 cm (Longest side c = 6.5).

⇒ 2.5² + 6² = 6.25 + 36 = 42.25

⇒ 6.5² = 42.25

⇒ 42.25 = 42.25 (Valid Right Triangle)

∴ The correct answer is 8, 16, 12 cm.

Additional Information

Pythagorean Triples

These are sets of three positive integers a, b, and c such that a² + b² = c². Common triples include (3, 4, 5), (5, 12, 13), (7, 24, 25), and (8, 15, 17).

Triangle Inequality Theorem

For any triangle with sides a, b, and c, the sum of any two sides must be greater than the third side: a + b > c.

Types of Triangles by Sides

If a² + b² > c², the triangle is acute; if a² + b² < c², the triangle is obtuse; if a² + b² = c², it is a right triangle.

Shortcut Trick

Rotational symmetry exists if a letter looks identical when turned 180° (upside down).

Letter O → Remains O after 180° rotation.

Letter N → Remains N after 180° rotation.

Letter H → Remains H after 180° rotation.

Letter E → Becomes an inverted shape after 180° rotation.

∴ The correct answer is E.

Alternate Method

Given: Letters O, E, N, H.

Definition: A figure has rotational symmetry if it coincides with its original position at an angle of rotation less than 360°.

 Original 'E'   180° Rotation     Rotated 'E' (No Match) E does not map onto itself at 180°

⇒ Letter O: Rotates 180° → Maps perfectly (Rotational Symmetry: Yes).

⇒ Letter N: Rotates 180° → Maps perfectly (Rotational Symmetry: Yes).

⇒ Letter H: Rotates 180° → Maps perfectly (Rotational Symmetry: Yes).

⇒ Letter E: Rotates 180° → Does not map onto itself (Rotational Symmetry: No).

∴ The correct answer is E.

Additional Information

Point Symmetry

A figure has point symmetry if it looks the same when rotated 180° about its center. Letters like S, Z, N, H, I, O, and X have point symmetry.

Line Symmetry

Also known as reflectional symmetry, where a figure can be divided into two identical halves by a line. Letter E has horizontal line symmetry but no rotational symmetry.

Order of Rotational Symmetry

The number of positions in which a figure looks identical to the original during a 360° rotation. For N, H, and I, the order is 2. For O (circle), the order is infinite.

Shortcut Trick

Property: Diagonals of a parallelogram bisect each other at their intersection point.

⇒ OQ = OS → 20 = y + 7 → y = 13

⇒ OP = OR → x + y = 16 → x + 13 = 16 → x = 3

⇒ Target value: (7x − y) = (7 × 3) − 13 = 21 − 13 = 8

∴ The correct answer is 8.

Alternate Method

Given:

In parallelogram PQRS, diagonals intersect at O.

OP = (x + y) cm, OR = 16 cm

OQ = 20 cm, OS = (y + 7) cm

Formula Used:

In a parallelogram, the diagonals bisect each other. Therefore, the intersection point O is the midpoint of both diagonals PR and QS.

⇒ OP = OR and OQ = OS

 P Q R S O x+y 16 20 y+7

Calculations:

⇒ OQ = OS

⇒ 20 = y + 7

⇒ y = 20 − 7 = 13

⇒ OP = OR

⇒ x + y = 16

⇒ x + 13 = 16

⇒ x = 16 − 13 = 3

Now, finding the value of (7x − y):

⇒ 7(3) − 13

⇒ 21 − 13 = 8

∴ The correct answer is 8.

Additional Information

Diagonal Properties of Quadrilaterals

In a parallelogram, diagonals bisect each other. In a rectangle, diagonals are also equal. In a rhombus, diagonals bisect each other at 90°.

Area of Parallelogram

The area can be calculated as Base × Height or using the product of adjacent sides and the sine of the included angle: Area = ab sinθ.

Sum of Squares of Diagonals

In any parallelogram with sides a and b and diagonals d₁ and d₂, the relation is: d₁² + d₂² = 2(a² + b²).

Shortcut Trick

Sum of parallel sides = Perimeter − Sum of non-parallel sides = 52 − 20 = 32 cm.

Area of trapezium = (1/2) × (Sum of parallel sides) × Height.

128 = (1/2) × 32 × Height.

128 = 16 × Height ⇒ Height = 128 ÷ 16 = 8 cm.

∴ The correct answer is 8 cm.

Alternate Method

Given:

Perimeter of trapezium = 52 cm

Area of trapezium = 128 cm²

Sum of non-parallel sides = 20 cm

Formula: Area = (1/2) × (Sum of parallel sides) × distance between them (h)

Parallel side (a) Parallel side (b) c d h Area = 128 cm

Calculations:

⇒ Perimeter = (Sum of parallel sides) + (Sum of non-parallel sides)

⇒ 52 = (a + b) + 20

⇒ Sum of parallel sides (a + b) = 52 − 20 = 32 cm

⇒ Area = (1/2) × (a + b) × h

⇒ 128 = (1/2) × 32 × h

⇒ 128 = 16 × h

⇒ h = 128 ÷ 16

⇒ h = 8 cm

∴ The correct answer is 8 cm.

Additional Information

Trapezium Properties

A quadrilateral with one pair of parallel sides. The sum of the angles on the same side of the non-parallel legs is 180°.

Median of a Trapezium

The segment connecting the midpoints of non-parallel sides is called the median. Length = (a + b) ÷ 2.

Isosceles Trapezium

If the non-parallel sides are equal in length, it is an isosceles trapezium where base angles are equal and diagonals are equal.

Shortcut Trick

Side 1 = 6 m, Side 2 = Area ÷ Side 1 = 48 ÷ 6 = 8 m.

Using Pythagorean triplet (6, 8, 10), the Diagonal = 10 m.

Time = Distance ÷ Speed = 10 m ÷ 20 m/min = 0.5 min.

0.5 minutes = 30 seconds.

∴ The correct answer is 30 seconds.

Alternate Method

Given:

Area of rectangular field = 48 m²

One side (Width) = 6 m

Speed = 20 m/min

Formula Used:

Area = Length × Width

Diagonal = √(Length² + Width²)

Time = Distance ÷ Speed

Length = 8 m 6 m Diagonal = 10 m A B C D Crossing Path: D → B

Calculations:

⇒ Area = Length × 6 = 48

⇒ Length = 48 ÷ 6 = 8 m

⇒ Diagonal = √(8² + 6²)

⇒ Diagonal = √(64 + 36) = √100 = 10 m

⇒ Time taken = Distance ÷ Speed = 10 ÷ 20

⇒ Time = 0.5 minutes

⇒ Time in seconds = 0.5 × 60 = 30 seconds

∴ The correct answer is 30 seconds.

Additional Information

Pythagoras Theorem

In a right-angled triangle, the square of the hypotenuse (diagonal) is equal to the sum of the squares of the other two sides: a² + b² = c².

Diagonal of a Rectangle

The diagonal of a rectangle with length 'l' and breadth 'b' is given by d = √(l² + b²).

Unit Conversion

To convert time from minutes to seconds, multiply by 60. To convert speed from m/min to m/sec, divide by 60.

Shortcut Trick

Total Volume = Sum of individual volumes.

Volume = 8³ + 6³ + 10³

Volume = 512 + 216 + 1000 = 1728 cm³.

Side of new cube = ³√1728 = 12 cm.

∴ The correct answer is 12 cm.

Alternate Method

Given:

Side of first cube (a₁) = 8 cm

Side of second cube (a₂) = 6 cm

Side of third cube (a₃) = 10 cm

Formula Used:

Volume of cube = (side)³

8 cm + 6 cm + 10 cm New Cube (12 cm) Melted

Calculations:

⇒ Volume of 1^st cube = 8³ = 512 cm³

⇒ Volume of 2^nd cube = 6³ = 216 cm³

⇒ Volume of 3^rd cube = 10³ = 1000 cm³

⇒ Total volume of the new cube = 512 + 216 + 1000

⇒ Total volume = 1728 cm³

⇒ Let the side of the new cube be 'A'.

⇒ A³ = 1728

⇒ A = ³√1728

⇒ A = 12 cm

∴ The correct answer is 12 cm.

Additional Information

Total Surface Area of Cube

The total surface area (TSA) of a cube with side 'a' is 6a².

Diagonal of a Cube

The length of the longest diagonal of a cube is given by a√3.

Volume Conservation Rule

When one or more solid objects are melted and recast into a new shape, the total volume remains constant (V_old = V_new).

Shortcut Trick

Arrange data: 20, 26, 28, 30, 45, 55, 56, 56, 60, 61

Median (center of 10 items) = average of 5^th and 6^th terms = (45 + 55) ÷ 2 = 50

Mode (most frequent value) = 56

Required Mean = (50 + 56) ÷ 2 = 53

∴ The correct answer is 53.

Alternate Method

Given observations: 30, 61, 55, 56, 60, 20, 26, 45, 28, 56

Number of observations (n) = 10

Formula Used: Median for even n = [(n/2)^th term + (n/2 + 1)^th term] ÷ 2; Mode = Most frequent observation; Mean = Sum ÷ Count

Median (50) Mode (56) 53 Required Mean 20 (Min) 61 (Max)

⇒ Sorting data in ascending order: 20, 26, 28, 30, 45, 55, 56, 56, 60, 61

⇒ Median = (5^th observation + 6^th observation) ÷ 2

⇒ Median = (45 + 55) ÷ 2 = 100 ÷ 2 = 50

⇒ Mode = 56 (since 56 occurs twice and others once)

⇒ Mean of Median and Mode = (50 + 56) ÷ 2 = 106 ÷ 2 = 53

∴ The correct answer is 53.

Additional Information

Empirical Relationship

For a moderately skewed distribution, the relationship is: Mode = 3 × Median − 2 × Mean.

Median for Odd N

If the number of observations n is odd, the median is the value at the ((n + 1)/2)^th position after sorting.

Range of Data

Range = Maximum Value − Minimum Value. For this data: 61 − 20 = 41.

Shortcut Trick

Total cards in a deck = 52

Number of 10s = 4 (one from each suit)

Number of red kings = 2 (Heart and Diamond)

Total favorable outcomes = 4 + 2 = 6

Probability = Favorable outcomes ÷ Total outcomes = 6 ÷ 52 = 3/26

∴ The correct answer is 3/26.

Alternate Method

Given: Total number of cards n(S) = 52

Let A be the event of drawing a 10 and B be the event of drawing a red king.

Formula: P(A or B) = P(A) + P(B) − P(A and B)

Sample Space (Total = 52) 10s (4) Red Kings (2) Mutually Exclusive Events

⇒ Total Outcomes n(S) = 52

⇒ Favorable 10s n(A) = 4

⇒ Favorable Red Kings n(B) = 2

⇒ Since no card is both a 10 and a King, n(A ∩ B) = 0

⇒ n(A ∪ B) = n(A) + n(B) = 4 + 2 = 6

⇒ P(A ∪ B) = 6/52

⇒ P = 3/26

∴ The correct answer is 3/26.

Additional Information

Addition Rule of Probability

For any two events A and B, P(A ∪ B) = P(A) + P(B) − P(A ∩ B). If they are mutually exclusive, P(A ∩ B) = 0.

Standard Deck Composition

A deck has 52 cards: 4 suits (Spades, Hearts, Diamonds, Clubs), each with 13 ranks (A, 2-10, J, Q, K). Hearts and Diamonds are red; Spades and Clubs are black.

Face Cards

There are 12 face cards in total (4 Jacks, 4 Queens, 4 Kings). Each suit contains 3 face cards.

Shortcut Trick

Subtract the remainders from the given numbers: 87 − 7 = 80 and 123 − 3 = 120.

Find the HCF of the resulting numbers: HCF(80, 120).

HCF(80, 120) = 40 (since 40 is the largest number dividing both 80 and 120).

Verification: 87 ÷ 40 gives remainder 7, and 123 ÷ 40 gives remainder 3.

∴ The correct answer is 40.

Alternate Method

Given:

First number = 87, Remainder = 7

Second number = 123, Remainder = 3

Formula Used:

The greatest number that divides N₁ and N₂ leaving remainders R₁ and R₂ is HCF(N₁ − R₁, N₂ − R₂).

HCF Visualization (80 & 120) 80 (87 − 7) 40 40 120 (123 − 3) 40 40 40 Greatest Common Divisor = 40

Calculations:

⇒ Number 1 = 87 − 7 = 80

⇒ Number 2 = 123 − 3 = 120

⇒ Prime factorization of 80 = 2 × 2 × 2 × 2 × 5 = 2⁴ × 5

⇒ Prime factorization of 120 = 2 × 2 × 2 × 3 × 5 = 2³ × 3 × 5

⇒ HCF(80, 120) = Common Factors = 2³ × 5 = 8 × 5 = 40

∴ The correct answer is 40.

Additional Information

HCF and LCM Relationship

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(a, b) × LCM(a, b).

Euclidean Algorithm

HCF can be found by dividing the larger number by the smaller number and then dividing the divisor by the remainder until the remainder is zero.

HCF of Fractions

To find the HCF of fractions, calculate the HCF of the numerators divided by the LCM of the denominators.

Shortcut Trick

Direct Rule: For single-digit recurring decimals, 0.x... = x÷9.

0.444... = 4/9

0.666... = 6/9

Total = 4/9 + 6/9 = 10/9 = \(1\frac{1}{9}\)

∴ The correct answer is \(1\frac{1}{9}\)

Alternate Method

Given: 0.444... and 0.666...

Formula: If x = 0.a... then x = a/9

4/9 6/9 1 Whole (9/9) Total = 10/9 = 1 1/9

⇒ Let x = 0.444... → (1)

⇒ 10x = 4.444... → (2)

⇒ Subtracting (1) from (2): 9x = 4 ⇒ x = 4/9

⇒ Let y = 0.666... → (3)

⇒ 10y = 6.666... → (4)

⇒ Subtracting (3) from (4): 9y = 6 ⇒ y = 6/9

⇒ Required value = x + y = 4/9 + 6/9

⇒ Result = 10/9 = \(1\frac{1}{9}\)

∴ The correct answer is \(1\frac{1}{9}\).

Additional Information

Pure Recurring Decimals

Decimals where the repetition starts immediately after the decimal point. Rule: 0.abc... = abc ÷ 999.

Mixed Recurring Decimals

Decimals where some digits after the decimal point do not repeat. Rule: 0.abcc... = (abc − ab) ÷ 900.

Fraction Types

Proper fractions have numerator < denominator. Improper fractions (like 10/9) can be converted to mixed fractions (1 1/9).

Shortcut Trick

Convert fractions to decimals: 4/7 ≈ 0.571, 9/16 = 0.5625, 1 2/7 = 9/7 ≈ 1.285, 3/5 = 0.60.

Identify extremes: Largest = 9/7 and Smallest = 9/16.

Difference = 9/7 − 9/16 = (9 × 16 − 9 × 7) / (7 × 16).

Difference = (144 − 63) / 112 = 81/112.

∴ The correct answer is 81/112.

Alternate Method

Given fractions: 4/7, 9/16, 1 2/7, 3/5

Formula Used: Difference = Largest Fraction − Smallest Fraction

9/16 (Min) 4/7 3/5 9/7 (Max) Difference

⇒ Mixed fraction conversion: 1 2/7 = (7 × 1 + 2) / 7 = 9/7

⇒ Comparing 4/7, 9/16, and 3/5 by taking LCM of denominators (7, 16, 5) = 560

⇒ 4/7 = 320/560; 9/16 = 315/560; 3/5 = 336/560

⇒ Smallest fraction = 9/16 (since 315 is the smallest numerator)

⇒ Since 9/7 is improper (9 > 7), it is > 1, while others are < 1. ∴ Largest = 9/7

⇒ Required Difference = 9/7 − 9/16

⇒ Difference = (9 × 16 − 9 × 7) / (7 × 16)

⇒ Difference = (144 − 63) / 112

⇒ Difference = 81/112

∴ The correct answer is 81/112.

Additional Information

Comparing Fractions

To compare two fractions a/b and c/d, cross-multiply: if ad > bc, then a/b > c/d. If ad < bc, then a/b < c/d.

Proper and Improper Fractions

A proper fraction has a numerator less than the denominator (value < 1). An improper fraction has a numerator greater than or equal to the denominator (value ≥ 1).

Subtraction of Fractions

To find the difference, use the formula: a/b − c/d = (ad − bc) / (bd). Always simplify the resulting fraction to its lowest terms.

Shortcut Trick

Let the units digit of a number be u and the units digit of its square be S.

Given: S = u + 2. Testing u → S: (2 → 4), (4 → 6), and (7 → 9).

If S = 5, then u = 5 − 2 = 3. However, 3² = 9, so S should be 9, not 5.

Therefore, 5 cannot be the units digit of the square under this condition.

∴ The correct answer is 5.

Alternate Method

Given:

Units digit of a number = u

Units digit of the square of the number = S

Condition: S = u + 2

Unit Digit (u) Square Unit (S) 2 4 2+2=4 4 6 4+2=6 7 9 7+2=9 3 5 3+2=5 ≠ 9

Calculations:

⇒ Let us test each digit u (0 to 9) and compare u² units digit (S) with u + 2.

⇒ If u = 2, S = 2² = 4. Also, u + 2 = 2 + 2 = 4. (Possible)

⇒ If u = 4, S = 4² = 16 → 6. Also, u + 2 = 4 + 2 = 6. (Possible)

⇒ If u = 7, S = 7² = 49 → 9. Also, u + 2 = 7 + 2 = 9. (Possible)

⇒ Checking Option A: S = 5. For S = u + 2, u must be 3. But 3² = 9, so S = 9 ≠ 5.

⇒ Checking remaining digits (0, 1, 3, 5, 6, 8, 9) shows no other matches for S = u + 2.

∴ The correct answer is 5.

Additional Information

Units Digit of Perfect Squares

A perfect square always ends in one of the following digits: 0, 1, 4, 5, 6, or 9. It can never end in 2, 3, 7, or 8.

Properties of Square Units

The units digit of the square of any number depends only on the units digit of the original number. For example, if a number ends in 3 or 7, its square will always end in 9.

Digit Sum and Squares

The digital root (sum of digits reduced to a single digit) of any perfect square can only be 1, 4, 7, or 9.