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Shortcut Trick

A quadratic polynomial ax² + bx + c is always positive if its parabola lies entirely above the x-axis.

This happens when the parabola opens upwards, which requires the leading coefficient to be positive (a > 0).

Additionally, it must have no real roots so that it never touches or crosses the x-axis, meaning the discriminant must be negative: b² − 4ac < 0.

∴ The correct answer is a is positive and b² − 4ac < 0.

Alternate Method

Given: Quadratic polynomial f(x) = ax² + bx + c

Formula Used: Minimum value of f(x) = ax² + bx + c is −D/4a at x = −b/2a (when a > 0).

Calculations:

⇒ For f(x) to be always positive for all real values of x, the curve must be concave upwards.

⇒ This condition is met when a > 0.

⇒ If a > 0, the function has a minimum value, and for f(x) to be always positive, this minimum value must be greater than zero.

⇒ Minimum value = (−b² + 4ac) / 4a > 0

⇒ Since a > 0, the denominator 4a is positive, so the numerator must also be positive: 4ac − b² > 0.

⇒ Multiplying the inequality by −1: b² − 4ac < 0.

⇒ Thus, both a > 0 and b² − 4ac < 0 must be satisfied.

∴ The correct answer is a is positive and b² − 4ac < 0.

Additional Information

Nature of Roots

The roots of a quadratic are determined by the discriminant (D = b² − 4ac). If D > 0, roots are real and distinct; if D = 0, roots are real and equal; if D < 0, roots are imaginary.

Always Negative Condition

A quadratic polynomial is always negative for all real x if the parabola opens downwards (a < 0) and has no real roots (D < 0).

Vertex of the Parabola

The vertex or the highest/lowest point of the graph y = ax² + bx + c is given by the coordinates (−b/2a, −D/4a).

Shortcut Trick

Identify the Sum (S) and Product (P) of the given roots.

Sum (S) = (√3 + 1) + (√3 − 1) = 2√3

Product (P) = (√3 + 1) × (√3 − 1) = 3 − 1 = 2

Equation pattern: x² − (Sum)x + (Product) = 0

∴ The correct answer is x² − 2√3x + 2 = 0.

Alternate Method

Given: The roots of the required equation are α = √3 + 1 and β = √3 − 1.

Formula Used: A quadratic equation can be expressed as x² − (α + β)x + αβ = 0, where (α + β) is the sum of roots and αβ is the product of roots.

⇒ Sum of roots (α + β) = (√3 + 1) + (√3 − 1)

⇒ Sum = √3 + 1 + √3 − 1 = 2√3

⇒ Product of roots (αβ) = (√3 + 1) × (√3 − 1)

⇒ Using the identity (a + b)(a − b) = a² − b²

⇒ Product = (√3)² − (1)² = 3 − 1 = 2

⇒ Substituting the Sum and Product in the formula: x² − (2√3)x + 2 = 0

∴ The correct answer is x² − 2√3x + 2 = 0.

Additional Information

General Form of Quadratic Equation

The standard form is ax² + bx + c = 0, where a ≠ 0.

Nature of Roots

Determined by the Discriminant D = b² − 4ac. If D > 0 roots are real and distinct; if D = 0 roots are real and equal; if D < 0 roots are imaginary.

Relation between Roots and Coefficients

For ax² + bx + c = 0, Sum of roots = −b/a and Product of roots = c/a.

Irrational Roots

If coefficients are rational, irrational roots always occur in conjugate pairs (e.g., if one root is a + √b, the other is a − √b).

Given:

Equation 1: y = 6x - 8

Equation 2: y = -2x + 4

Formula Used:

To find the point of intersection, equate the two equations:

6x - 8 = -2x + 4

Substitute the value of x into either equation to find y.

Calculation:

⇒ 6x - 8 = -2x + 4

⇒ 6x + 2x = 4 + 8

⇒ 8x = 12

⇒ x = 12 / 8

⇒ x = 1.5

Substituting x = 1.5 into y = 6x - 8:

⇒ y = 6(1.5) - 8

⇒ y = 9 - 8

⇒ y = 1

∴ The correct answer is (1.5, 1).

Shortcut Trick

Identify the most frequent age in the given data set.

Ages: 35, 40, 45, 40, 50, 55, 60, 40, 45, 55

The age 40 appears 3 times, while no other age appears more than twice.

Since 40 has the maximum frequency, it is the mode.

∴ The correct answer is 40.

Alternate Method

Given Data: Ages of 10 members: 35, 40, 45, 40, 50, 55, 60, 40, 45, 55.

Formula Used: Mode = The observation with the highest frequency in a data set.

⇒ Arrange observations in a frequency table:

⇒ Age 35: 1 time

⇒ Age 40: 3 times

⇒ Age 45: 2 times

⇒ Age 50: 1 time

⇒ Age 55: 2 times

⇒ Age 60: 1 time

⇒ Maximum frequency is 3, which corresponds to age 40.

∴ The correct answer is 40.

Additional Information

Definition of Mode

The mode is the value that appears most frequently in a data set. A set can be unimodal (one mode), bimodal (two modes), or multimodal.

Empirical Relationship

For a moderately skewed distribution, the relationship between mean, median, and mode is: Mode = 3 × Median − 2 × Mean.

Median Calculation

The median is the middle value when data is arranged in ascending order. If the count is even, it is the average of the two middle values.

Mean (Average)

Arithmetic Mean = (Sum of all observations) ÷ (Total number of observations).

Shortcut Trick

Identify a + b and a − b: (tan θ + sin θ) + (tan θ − sin θ) = 2 tan θ and (tan θ + sin θ) − (tan θ − sin θ) = 2 sin θ.

Apply Difference of Squares: a² − b² = (a + b)(a − b) = (2 tan θ) × (2 sin θ) = 4 tan θ sin θ.

Compare with √ab: √((tan θ + sin θ)(tan θ − sin θ)) = √(tan² θ − sin² θ) = √(tan² θ sin² θ) = tan θ sin θ.

Thus, a² − b² = 4 × (tan θ sin θ) = 4√ab.

∴ The correct answer is 4√ab.

Alternate Method

Given: a = tan θ + sin θ and b = tan θ − sin θ

Formula Used: a² − b² = (a + b)(a − b) and tan² θ − sin² θ = tan² θ sin² θ

⇒ a + b = (tan θ + sin θ) + (tan θ − sin θ) = 2 tan θ

⇒ a − b = (tan θ + sin θ) − (tan θ − sin θ) = 2 sin θ

⇒ a² − b² = (2 tan θ) × (2 sin θ) = 4 tan θ sin θ ... (i)

⇒ ab = (tan θ + sin θ)(tan θ − sin θ) = tan² θ − sin² θ

⇒ ab = (sin² θ / cos² θ) − sin² θ = sin² θ ((1 / cos² θ) − 1)

⇒ ab = sin² θ (sec² θ − 1) = sin² θ tan² θ

⇒ √ab = sin θ tan θ ... (ii)

⇒ Substituting (ii) in (i): a² − b² = 4√ab

∴ The correct answer is 4√ab.

Additional Information

Trigonometric Product Property

The expression tan² A − sin² A is always equal to the product tan² A × sin² A. Similarly, cot² A − cos² A = cot² A × cos² A.

Algebraic Identity

The difference of squares of two sums (x + y)² − (x − y)² simplifies directly to 4xy using the expansion method.

Square Root Relation

If M = tan θ + sin θ and N = tan θ − sin θ, then the area or product relation is often expressed as MN = tan² θ sin² θ.

Shortcut Trick

Check the divisibility rule of 3 (sum of digits) for each pair.

For the pair 57 and 33:

5 + 7 = 12 (divisible by 3) and 3 + 3 = 6 (divisible by 3).

Since both numbers share a common factor of 3, their HCF ≠ 1.

Therefore, they are not co-prime.

∴ The correct answer is 57, 33.

Alternate Method

Given: Four pairs of numbers to check for co-prime property.

Formula: Two numbers are co-prime if their Highest Common Factor (HCF) is exactly 1.

⇒ Prime factorization of 57 = 3 × 19

⇒ Prime factorization of 33 = 3 × 11

⇒ HCF(57, 33) = 3

⇒ Since HCF is not 1, the numbers are not co-prime.

⇒ For other pairs: HCF(17, 21) = 1, HCF(89, 77) = 1, and HCF(61, 83) = 1.

∴ The correct answer is 57, 33.

Additional Information

Co-prime Numbers

A set of numbers is co-prime if their only common positive factor is 1. They do not need to be prime numbers themselves.

Twin Primes

Two prime numbers are called twin primes if there is a difference of 2 between them, such as (3, 5) or (11, 13).

Consecutive Integers

Any two consecutive integers are always co-prime, as their HCF is always 1.

Prime vs Co-prime

All prime numbers are co-prime to each other, but composite numbers can also be co-prime if they share no common factors (e.g., 8 and 9).

Shortcut Trick

Parallel lines have identical slopes (m).

Slope of y = 4x + 1 is m = 4.

Equation of parallel line: y = 4x + k.

Substitute (2, −4): −4 = 4 × 2 + k ⇒ k = −12.

∴ The correct answer is y = 4x − 12.

Alternate Method

Given:

Point (x₁, y₁) = (2, −4)

Line L₁: y = 4x + 1

Formula Used:

Point-Slope Form: y − y₁ = m(x − x₁)

⇒ Comparing L₁ with y = mx + c, slope m₁ = 4

⇒ Since required line L₂ is parallel to L₁, slope m₂ = m₁ = 4

⇒ Applying point-slope form with m = 4 and (x₁, y₁) = (2, −4):

⇒ y − (−4) = 4 × (x − 2)

⇒ y + 4 = 4x − 8

⇒ y = 4x − 12

∴ The correct answer is y = 4x − 12.

Additional Information

Parallel Lines Condition

Two lines are parallel if their slopes are equal, m₁ = m₂.

Perpendicular Lines Condition

Two lines are perpendicular if the product of their slopes is −1, i.e., m₁ × m₂ = −1.

Slope-Intercept Form

The equation y = mx + c represents a line with slope m and y-intercept c.

Point-Slope Form

The equation of a line passing through (x₁, y₁) with slope m is y − y₁ = m(x − x₁).

Given:

Total men = 2

Total women = 4

Total members in the committee = 3

Formula Used:

Total ways to form a committee = Combination formula: nCk=n!k!(n−k)!nCk=n!k!(n−k)!

Probability = Favorable outcomes / Total outcomes

Calculation:

⇒ Total ways to form a committee = 6C3=6×5×43×2×1=206C3=6×5×43×2×1=20

⇒ Ways to form a committee with no men (only women) = 4C3=4×3×23×2×1=44C3=4×3×23×2×1=4

⇒ Ways to form a committee with at least one man = Total ways - Ways with no men

⇒ Ways to form a committee with at least one man = 20 - 4 = 16

⇒ Probability = Favorable outcomes / Total outcomes

⇒ Probability = 16 / 20

⇒ Probability = 4 / 5

∴ The correct answer is 4 / 5.

Shortcut Trick

Use formula: x² − (sum of roots)x + (product of roots) = 0

Sum = 3 + (−2) = 1

Product = 3 × (−2) = −6

Equation ⇒ x² − x − 6 = 0

∴ Answer = x² − x − 6 = 0

Alternate Method

Given:

Roots are 3 and −2

Formula used:

If roots are α and β, then equation is:

x² − (α + β)x + (αβ) = 0

Calculation:

⇒ α + β = 3 + (−2) = 1

⇒ αβ = 3 × (−2) = −6

⇒ Equation = x² − 1x − 6 = 0

∴ The required quadratic equation is x² − x − 6 = 0.

Additional Information

Standard Form

Quadratic equation is written as ax² + bx + c = 0.

Relation of Roots

Sum = −b/a and Product = c/a.

Factor Form

Equation can also be written as (x − α)(x − β) = 0.

Verification

Substitute roots to check correctness.

Shortcut Trick

Convert all units to meters: Breadth (B) = 1000 cm ÷ 100 = 10 m.

Surface Area = 2 × (Sum of products of adjacent sides taken two at a time).

Sum of products = (20 × 10) + (10 × 8) + (8 × 20) = 200 + 80 + 160 = 440.

Total Surface Area = 2 × 440 = 880 m².

∴ The correct answer is 880 m².

Alternate Method

Given:

Length (L) = 20 m

Breadth (B) = 1000 cm = 10 m (since 100 cm = 1 m)

Height (H) = 8 m

Formula Used:

Total Surface Area of Cuboid = 2(LB + BH + HL)

Calculations:

⇒ Total Surface Area = 2 × [(20 × 10) + (10 × 8) + (8 × 20)]

⇒ Total Surface Area = 2 × [200 + 80 + 160]

⇒ Total Surface Area = 2 × [440]

⇒ Total Surface Area = 880 m²

∴ The correct answer is 880 m².

Additional Information

Lateral Surface Area of Cuboid

The area of the four side walls is given by 2H(L + B).

Volume of Cuboid

The total space occupied by the cuboid is calculated as L × B × H.

Diagonal of Cuboid

The longest rod that can be placed inside the cuboid has a length of √(L² + B² + H²).

Surface Area of a Cube

If all sides are equal (a), the total surface area is 6a².