SSC CGL Tier 1 2015 Quantitative Aptitude Questions with Solutions

Explanation

The expression x³ + 27x² + 243x + 631 can be simplified by recognizing the expansion of the binomial (x + a)³ = x³ + 3ax² + 3a²x + a³. Comparing the given expression to this formula, we identify 3a = 27 and 3a² = 243, which both yield a = 9.

The expansion of (x + 9)³ is x³ + 27x² + 243x + 729. To represent the original expression, we adjust the constant term: x³ + 27x² + 243x + 631 = (x + 9)³ - 98

Substituting x = 2 into this simplified form: (2 + 9)³ - 98 = 11³ - 98

Since 11³ = 1331, the calculation becomes: 1331 - 98 = 1233

Explanation

The volume V of a sphere is given by the formula V = ⁴⁄₃ π r³ and its surface area S is given by S = 4 π r², where r is the radius. According to the problem, these two values are numerically equal:

⁴⁄₃ π r³ = 4 π r²

Dividing both sides by 4 π r² (since r cannot be zero for a sphere to exist):

ʳ⁄₃ = 1

r = 3

The diameter of the sphere is twice the radius:

d = 2r = 2 × 3 = 6 units

Explanation

The correct average can be calculated using the adjustment formula for the mean. The total marks were initially calculated using an incorrect value of 86 instead of the actual value of 68. This resulted in an error in the sum.

The net change in the total sum of marks is: Net Change = Correct Value - Incorrect Value Net Change = 68 - 86 = -18

The change in the average is the net change in the sum divided by the total number of subjects: Change in Average = ⁻¹⁸⁄₆ = -3

Therefore, the correct average is: Correct Average = Old Average + Change in Average Correct Average = 88 + (-3) = 85

Explanation

Let the number be x. The problem states that ¾ of the number exceeds ⅙ of the number by 7. This can be expressed as the linear equation:

¾x - ⅙x = 7

To solve for x, find the least common multiple of the denominators 4 and 6, which is 12. Multiply the terms to align with this common denominator:

⁹ˣ⁄₁₂ - ²ˣ⁄₁₂ = 7 ⁷ˣ⁄₁₂ = 7

Divide both sides by 7 and multiply by 12:

x = 12

The number is 12. The question requires finding ⁵⁄₃ of this number:

⁵⁄₃ × 12 = 5 × 4 = 20

Explanation

The efficiency ratio between A and B is derived from the statement that A's 2 days' work equals B's 3 days' work. This implies that A is more efficient than B. Let E_A and E_B be the efficiencies (work done per day) of A and B respectively. The given condition is: 2 × E_A = 3 × E_B E_A{E_B} = ³⁄₂

Total work can be calculated using A's completion time. Since A completes the work in 8 days, the total work is: Work = E_A × 8

The time taken by B to complete the same work (T_B) is the total work divided by B's efficiency: T_B = E_A × 8{E_B}

By substituting the efficiency ratio E_A{E_B} = ³⁄₂ into the equation: T_B = ³⁄₂ × 8 T_B = 3 × 4 = 12 days

Explanation

In any triangle Δ PQR, the angle formed at the incenter O by the internal bisectors of ∠ Q and ∠ R is given by the formula ∠ ROQ = 90° + ½∠ P. This relationship is derived from the fact that the sum of angles in Δ OQR is 180°, where ∠ OQR = ½∠ Q and ∠ ORQ = ½∠ R.

Substituting the given value ∠ ROQ = 96° into the formula: 96° = 90° + ½∠ P 96° - 90° = ½∠ P 6° = ½∠ P ∠ P = 12°

Therefore, the value of ∠ RPQ is 12°.

Explanation

To find the value of x + 1, first simplify the given expression for x by rationalizing the denominator: x = (1)/(√(2) + 1)

Multiply both the numerator and the denominator by the conjugate of the denominator, which is √2 - 1: x = (1)/(√(2) + 1) × √2 - 1{√2 - 1}

Using the algebraic identity (a + b)(a - b) = a² - b² for the denominator: x = √2 - 1{(√2)² - 1²} x = √2 - 1{2 - 1} x = √2 - 1

Now, substitute this simplified value of x into the required expression (x + 1): x + 1 = (√2 - 1) + 1 x + 1 = √2

By simplifying the rational expression and performing basic addition, the result is determined to be the square root of 2.

Explanation

Euler's formula for any convex polyhedron states that the relationship between the number of vertices v, edges e, and faces f is given by the equation v - e + f = 2. A rectangular parallelepiped, which is a three-dimensional figure formed by six rectangular faces, is a convex polyhedron. For this specific shape, the number of vertices v is 8, the number of edges e is 12, and the number of faces f is 6. Substituting these values into the formula yields 8 - 12 + 6 = 2. This characteristic topological property is invariant for all polyhedra that are homeomorphic to a sphere.

Explanation

The vertices of the triangle are determined by the intersection of the given lines: Intersection of x = 0 and 2x + 3y = 6: Substituting x = 0 into the equation gives 3y = 6, resulting in y = 2. This vertex is (0, 2). Intersection of x = 0 and x + y = 3: Substituting x = 0 into the equation gives y = 3. This vertex is (0, 3). Intersection of 2x + 3y = 6 and x + y = 3: Multiplying the second equation by 2 gives 2x + 2y = 6. Subtracting this from 2x + 3y = 6 yields y = 0. Substituting y = 0 into x + y = 3 gives x = 3. This vertex is (3, 0).

The base of the triangle lies along the y-axis (x = 0) between the points (0, 2) and (0, 3), providing a base length of 3 - 2 = 1 unit. The height of the triangle is the perpendicular distance from the third vertex (3, 0) to the y-axis, which is 3 units. Area = ½ × base × height = ½ × 1 × 3 = 1.5 = 1½ sq. units.

Explanation

Given the equations 5x + 9y = 5 and 125x³ + 729y³ = 120, we recognize that 125x³ = (5x)³ and 729y³ = (9y)³. Using the algebraic identity a³ + b³ = (a + b)³ - 3ab(a + b), let a = 5x and b = 9y.

Substituting the values into the identity: 125x³ + 729y³ = (5x + 9y)³ - 3(5x)(9y)(5x + 9y)

Using the given data 5x + 9y = 5 and 125x³ + 729y³ = 120: 120 = (5)³ - 3(45xy)(5) 120 = 125 - 15(45xy) 120 = 125 - 675xy

Rearranging the terms to solve for the product xy: 675xy = 125 - 120 675xy = 5 xy = ⁵⁄₆₇₅

Simplifying the fraction by dividing both the numerator and denominator by 5: xy = ¹⁄₁₃₅

Explanation

Let the number to be added to each term of the ratio be x. Based on the problem, adding x to both terms of the ratio 2:5 should result in a ratio of 5:6. This relationship can be expressed by the equation:

(2 + x)/(5 + x) = ⅚

By cross-multiplying the terms of the equation, we get:

6(2 + x) = 5(5 + x)

Expanding the expressions on both sides results in:

12 + 6x = 25 + 5x

To find the value of x, we rearrange the equation by moving all terms involving x to one side and constant terms to the other:

6x - 5x = 25 - 12

x = 13

When 13 is added to each term of the original ratio 2:5, the new terms become 2 + 13 = 15 and 5 + 13 = 18. The ratio 15:18 simplifies to 5:6 upon dividing both terms by 3.

Explanation

Using the trigonometric identity for complementary angles, sin(90° - θ) = cos θ, we can express sin 68° as sin(90° - 22°) = cos 22°. Substituting this into the first two terms of the expression gives sin² 22° + cos² 22°. Applying the Pythagorean identity sin² θ + cos² θ = 1, the sum of these two terms is 1. The value of cot 30° is √3, making cot² 30° = (√3)² = 3. Combining these values, the full expression evaluates to 1 + 3 = 4.

Explanation

The given expression is 2sin² 2Θ + 3cos² 2Θ. By utilizing the fundamental trigonometric identity sin² x + cos² x = 1, we can simplify the expression as follows:

2sin² 2Θ + 2cos² 2Θ + cos² 2Θ

2(sin² 2Θ + cos² 2Θ) + cos² 2Θ

2(1) + cos² 2Θ = 2 + cos² 2Θ

For any real angle x, the value of cos² x lies in the interval [0, 1]. To minimize the entire expression 2 + cos² 2Θ, we must substitute the minimum value of cos² 2Θ, which is 0.

Minimum value = 2 + 0 = 2

Alternatively, for any expression of the form asin² x + bcos² x, the minimum value is simply the smaller of the two coefficients a and b. In this case, since 2 < 3, the minimum value is 2.

Explanation

The given expression x²⁴ + 1{x¹²} = 7 can be simplified by dividing each term in the numerator by the denominator: x²⁴{x¹²} + (1)/(x¹²⁾ = 7 x¹² + (1)/(x¹²⁾ = 7

To find the value of x³⁶ + (1)/(x³⁶⁾, which is the simplified form of x⁷² + 1{x³⁶}, we use the algebraic identity for cubes: if k + ¹⁄ₖ = n, then k³ + (1)/(k³) = n³ - 3n. By substituting k = x¹² and n = 7, we get: (x¹²)³ + (1)/((x¹²⁾⁾^^³ = 7³ - 3(7) x³⁶ + (1)/(x³⁶⁾ = 343 - 21 x³⁶ + (1)/(x³⁶⁾ = 322

Explanation

The total floor area required for 5 persons is 5 × 16 = 80 m². The total volume of air required for 5 persons is 5 × 100 = 500 m³. For a conical tent, the floor area represents the base area of the cone, A = π r², and the volume is given by the formula V = ⅓ π r² h. Substituting the base area into the volume formula yields V = ⅓ A h. Plugging in the known values, we have 500 = ⅓ × 80 × h. Rearranging the equation to solve for height gives h = (500 × 3)/(80), which simplifies to h = ¹⁵⁰⁄₈ = 18.75 metres.

Explanation

Let the values of A and B be represented as 4x and 5x respectively, where x is a positive constant. The problem states that the difference of their squares is 81. This can be expressed by the equation (5x)² - (4x)² = 81. Simplifying the terms, we get 25x² - 16x² = 81, which reduces to 9x² = 81. Dividing both sides by 9 gives x² = 9, and taking the square root results in x = 3. To find the value of A, we substitute x = 3 into the original ratio A = 4x, yielding A = 4 × 3 = 12.

Explanation

The sum of the interior angles of any triangle is 180°. According to the given ratio 2:3:5, the measures of the three angles can be represented as 2x, 3x, and 5x. Using the angle sum property, we have 2x + 3x + 5x = 180°, which simplifies to 10x = 180°. Solving for x gives x = 18°. Consequently, the largest angle in the triangle is 5x = 5 × 18° = 90°. A triangle that contains an angle of exactly 90° is defined as a right-angled triangle.

Explanation

The amount A in compound interest is calculated using the formula A = P(1 + ʳ⁄₁₀₀)ⁿ, where P is the principal sum, r is the annual interest rate, and n is the number of years.

Given the amount A = 12,100, the rate r = 10%, and the time n = 2 years, the values can be substituted into the formula:

12100 = P(1 + ¹⁰⁄₁₀₀)² 12100 = P(1 + 0.1)² 12100 = P(1.1)² 12100 = P × 1.21

To find the principal sum P:

P = (12100)/(1.21) P = 10000

Explanation

To find the single equivalent discount for a series of successive discounts d₁, d₂, and d₃, we calculate the final multiplier by which the original price is reduced. The net multiplier is given by the product (1 - (d₁)/(100)) × (1 - (d₂)/(100)) × (1 - (d₃)/(100)). Substituting the given values of 10%, 20%, and 25%, the calculation becomes 0.9 × 0.8 × 0.75.

First, calculate the product of the first two factors: 0.9 × 0.8 = 0.72. Then, multiply this result by the third factor: 0.72 × 0.75 = 0.54. The single equivalent discount is found by subtracting this net multiplier from 1 and converting the result into a percentage: (1 - 0.54) × 100 = 46%.

Alternatively, applying the successive discount formula x + y - ˣʸ⁄₁₀₀ for the first two discounts of 10% and 20% gives 10 + 20 - (10 × 20)/(100) = 28%. Combining this 28% result with the final discount of 25% using the same formula yields 28 + 25 - (28 × 25)/(100) = 53 - 7 = 46%.

Explanation

In any triangle, the centroid G divides each median in the ratio 2:1 from the vertex to the midpoint of the opposite side. For the given medians AD and BE, the segments AG and BG are calculated as follows: AG = (2/3) × AD = (2/3) × 9 = 6 cm BG = (2/3) × BE = (2/3) × 12 = 8 cm

Since the medians intersect at right angles at G, the triangle AGB is a right-angled triangle with the right angle at vertex G. We can determine the length of the side AB by applying the Pythagorean theorem: AB = √(AG² + BG²) AB = √(6² + 8²) AB = √(36 + 64) AB = √(100) = 10 cm

Explanation

Using the trigonometric identity tan A = cot(90° - A), the relation tan A = cot B implies that A + B = 90°. In the given equation tan(4Θ - 50°) = cot(50° - Θ), we let A = 4Θ - 50° and B = 50° - Θ.

Summing the angles: (4Θ - 50°) + (50° - Θ) = 90°

Simplifying the equation: 4Θ - Θ - 50° + 50° = 90° 3Θ = 90° Θ = (90°)/(3) Θ = 30°

Explanation

The triangle with sides 3 cm, 4 cm, and 5 cm is a right-angled triangle because it satisfies the Pythagorean theorem 3² + 4² = 5². For any right-angled triangle with legs a and b and hypotenuse c, the inradius r of the inscribed circle C1 is calculated using the formula r = (a + b - c)/(2). Substituting the given values, we get r = (3 + 4 - 5)/(2) = ²⁄₂ = 1 cm.

The circumradius R of the circumscribed circle C2 for a right-angled triangle is exactly half of the hypotenuse, R = ᶜ⁄₂. Substituting the value of the hypotenuse, we get R = ⁵⁄₂ = 2.5 cm.

The area of a circle is given by π × radius². The ratio of the area of the inscribed circle C1 to the area of the circumscribed circle C2 is: (Area(C1))/(Area(C2)) = (π r²)/(π R²) = (r²)/(R²)

Plugging in the calculated radii: (1²)/((2.5)²) = (1)/(6.25) = ¹⁰⁰⁄₆₂₅ = ⁴⁄₂₅

Explanation

To calculate the time taken by a train to pass a stationary object like a post, the distance to be covered is equal to the length of the train. The length of the train is 180 metres. First, the speed of the train must be converted from km/h to m/s using the conversion factor ⁵⁄₁₈.

Speed in m/s = 90 × ⁵⁄₁₈ = 5 × 5 = 25 m/s.

Using the formula for time, Time = (Distance)/(Speed), we substitute the given values:

Time = ¹⁸⁰⁄₂₅ = 7.2 seconds.

Explanation

The single equivalent discount for two successive discounts of x% and y% is calculated using the formula x + y - ˣʸ⁄₁₀₀.

For the first case of 40% and 30%, the effective discount is 40 + 30 - (40 × 30)/(100) = 70 - 12 = 58%. For the second case of 45% and 20%, the effective discount is 45 + 20 - (45 × 20)/(100) = 65 - 9 = 56%.

The difference between these effective discounts is 58% - 56% = 2%. Given that this difference represents Rs. 12 on the marked price (MP), the relation is: ²⁄₁₀₀ × MP = 12 MP = (12 × 100)/(2) = 600

Explanation

Given the equation 5 sin Θ = 3, we can determine that sin Θ = ⅗. The expression to be evaluated is (sec Θ - tan Θ)/(sec Θ + tan Θ). We can simplify this expression using the basic trigonometric identities sec Θ = (1)/(cos Θ) and tan Θ = (sin Θ)/(cos Θ) as follows:

(sec Θ - tan Θ)/(sec Θ + tan Θ) = (1)/(cos Θ) - (sin Θ)/(cos Θ){(1)/(cos Θ) + (sin Θ)/(cos Θ)}

Multiplying both the numerator and the denominator by cos Θ simplifies the expression to:

(1 - sin Θ)/(1 + sin Θ)

By substituting the known value sin Θ = ⅗ into this simplified formula, we get:

(1 - 3)/(5){1 + ⅗} = (5 - 3)/(5){(5 + 3)/(5)} = (2)/(5){⁸⁄₅} = ²⁄₈ = ¼

The numerical value of the given expression is therefore 1/4.

Explanation

The time required to cover a distance at a constant speed is calculated using the formula t = d/v, where t is time, d is distance, and v is speed. Substituting the given values:

t = 1050 km / 75 km/hr

To solve this efficiently, divide both numbers by their common factors. Dividing both by 25: 1050 / 25 = 42 75 / 25 = 3

Then, 42 / 3 = 14.

The total time taken by the train to cover the distance of 1050 km at an average speed of 75 km/hr is 14 hours.

Explanation

The movement of the person from point L to point M creates a right-angled triangle where the eastward travel of 12 km and the northward travel of 5 km serve as the two legs. The shortest distance between the starting point L and the destination M is the hypotenuse of this triangle.

Using the Pythagoras theorem: LM² = Base² + Perpendicular² LM = √(12² + 5²) LM = √(144 + 25) LM = √(169) LM = 13 km

The result is derived directly from the Pythagorean triplet (5, 12, 13), which satisfies the condition a² + b² = c².

Explanation

The discount amount is the difference between the marked price and the selling price: Discount = 975 - 897 = 78

The discount percentage is calculated using the formula: Discount % = (Discount)/(Marked Price) × 100

Substituting the given values into the formula: Discount % = ⁷⁸⁄₉₇₅ × 100

By simplifying the fraction, we divide 78 and 975 by their greatest common divisor, which is 39: 78 ÷ 39 = 2 975 ÷ 39 = 25

The calculation becomes: Discount % = ²⁄₂₅ × 100 = 8%

Explanation

For an equilateral triangle with side length a, the altitude h is related to the side by the formula h = √3{2}a. Given the altitude is 12√3 cm, we can calculate the side:

12√3 = √3{2}a a = 12 × 2 = 24 cm

The area A of an equilateral triangle is determined by the formula A = √3{4}a². Substituting the calculated side length:

A = √3{4}(24)² = √3{4}(576) = 144√3 cm²

Alternatively, the area can be calculated directly from the altitude using the formula A = (h²)/(√(3)). Substituting the given altitude:

A = (12√3)²{√3} = (144 × 3)/(√(3)) = (432)/(√(3))

To rationalize the denominator, multiply the numerator and denominator by √3:

A = 432√3{3} = 144√3 cm²

Explanation

The first n odd natural numbers form an arithmetic progression where the first term a = 1 and the common difference d = 2. The sum of the first n terms of an arithmetic progression is given by the formula Sₙ = ⁿ⁄₂ [2a + (n - 1)d]. For the first n odd natural numbers, this simplifies to Sₙ = ⁿ⁄₂ [2(1) + (n - 1)2] = ⁿ⁄₂ [2 + 2n - 2] = n².

The arithmetic mean is calculated by dividing the sum of the numbers by the total count of numbers. Therefore, the mean of the first n odd natural numbers is Sₙ{n} = n²{n} = n. For the first 20 odd natural numbers, n = 20. Substituting this into the simplified formula, the arithmetic mean is 20. Alternatively, calculating the sum directly for n = 20 gives S₂₀ = 20² = 400, and dividing by 20 yields a mean of 20.

Explanation

To find the HCF of the given expressions, we factorize each polynomial. For the first expression, x⁸ - 1, we apply the difference of squares identity a² - b² = (a - b)(a + b) repeatedly: x⁸ - 1 = (x⁴ - 1)(x⁴ + 1) = (x² - 1)(x² + 1)(x⁴ + 1). This can be further broken down into linear factors as (x - 1)(x + 1)(x² + 1)(x⁴ + 1).

For the second expression, x⁴ + 2x³ - 2x - 1, we rearrange and group the terms: (x⁴ - 1) + (2x³ - 2x) = (x² - 1)(x² + 1) + 2x(x² - 1). Factoring out the common term (x² - 1), we get: (x² - 1)(x² + 1 + 2x) = (x² - 1)(x + 1)². Expanding the first part gives (x - 1)(x + 1)(x + 1)² = (x - 1)(x + 1)³.

The Highest Common Factor (HCF) is the product of the lowest powers of all common prime factors. Comparing the factored forms: First expression: (x - 1)¹ · (x + 1)¹ · (x² + 1)¹ · (x⁴ + 1)¹ Second expression: (x - 1)¹ · (x + 1)³ The common factors are (x - 1) and (x + 1). Taking the lowest power of each, the HCF is (x - 1)¹ · (x + 1)¹, which simplifies to x² - 1.

Explanation

The fundamental trigonometric identity sec² Θ - tan² Θ = 1 can be factored as (sec Θ + tan Θ)(sec Θ - tan Θ) = 1. Given the equation sec Θ + tan Θ = p, we substitute it into the identity to find p(sec Θ - tan Θ) = 1, which yields sec Θ - tan Θ = ¹⁄ₚ. To isolate sec Θ, we add the two equations together:

(sec Θ + tan Θ) + (sec Θ - tan Θ) = p + ¹⁄ₚ

2 sec Θ = p + ¹⁄ₚ

Dividing the entire equation by 2 results in:

sec Θ = ½ [p + 1 / p]

This solution holds true for all p ≠ 0 as given in the problem constraints.

Explanation

The distance traveled in each minute forms an arithmetic progression with the first term a = 500 m and common difference d = 125 m over n = 5 minutes. The total distance Sₙ is calculated using the formula Sₙ = ⁿ⁄₂ [2a + (n-1)d]. Substituting the values, S₅ = ⁵⁄₂ [2(500) + (5-1)125] = ⁵⁄₂ [1000 + 500] = 3750 meters. To find the average speed in km/hr, we convert the total distance to kilometers and the total time to hours. The distance is 3750 / 1000 = 3.75 km and the time is 5 / 60 = 1/12 hours. Average speed is the total distance divided by the total time: 3.75 / (1/12) = 3.75 × 12 = 45 km/hr.

Explanation

The relationship between principal, rate, time, and simple interest is defined by the formula SI = (P × R × T)/(100). According to the question, the ratio of principal P to simple interest SI for a time period T = 1 year is 25:1. This can be expressed as (P)/(SI) = ²⁵⁄₁. To find the rate of interest R, we rearrange the formula to R = (SI × 100)/(P × T). Substituting the given values and ratios into this equation, we get R = (1 × 100)/(25 × 1). Simplifying this calculation results in R = ¹⁰⁰⁄₂₅ = 4%.

Explanation

The length of the bridge corresponds to the total distance covered by the man while running across it. Using the formula Distance = Speed × Time, the units must be converted to metres for the final result.

First, convert the speed from km/h to metres per minute: Speed = (15 × 1000 metres)/(60 minutes) = 250 metres/minute

Next, multiply the speed by the time duration: Length of bridge = 250 metres/minute × 5 minutes = 1250 metres

Alternatively, converting time to hours: Time = ⁵⁄₆₀ hours = ¹⁄₁₂ hours Distance = 15 km/h × ¹⁄₁₂ hours = 1.25 km Converting km to metres: 1.25 × 1000 = 1250 metres

Explanation

The given equation is (m + 1) = √n + 3. By rearranging the terms, we get (m - 2) = √n.

The expression to be evaluated is ½ ((m³ - 6m² + 12m - 8)/(√(n)) - n).

Observe that the numerator m³ - 6m² + 12m - 8 is the expansion of (m - 2)³ following the algebraic identity (x - a)³ = x³ - 3ax² + 3a²x - a³. Substituting x = m and a = 2, we confirm (m - 2)³ = m³ - 6m² + 12m - 8.

Substituting (m - 2) = √n into the numerator, we get (√n)³, which simplifies to n√n. Substituting this back into the target expression:

½ (n√n{√n} - n)

Simplifying the fraction inside the parentheses results in:

½ (n - n) = ½ (0) = 0

Explanation

The given expression can be simplified using the principle of componendo and dividendo. For x = (a - b)/(a + b), we treat it as ˣ⁄₁ = (a - b)/(a + b) and calculate: (1 - x)/(1 + x) = ((a + b) - (a - b))/((a + b) + (a - b)) = ²ᵇ⁄₂ₐ = ᵇ⁄ₐ

Applying the same logic to the expressions for y and z: (1 - y)/(1 + y) = ((b + c) - (b - c))/((b + c) + (b - c)) = (2c)/(2b) = (c)/(b) (1 - z)/(1 + z) = ((c + a) - (c - a))/((c + a) + (c - a)) = (2a)/(2c) = (a)/(c)

The requested expression is the product of these three individual components: ((1 - x)(1 - y)(1 - z))/((1 + x)(1 + y)(1 + z)) = ((1 - x)/(1 + x)) ((1 - y)/(1 + y)) ((1 - z)/(1 + z)) Substituting the simplified fractions: ᵇ⁄ₐ · (c)/(b) · (a)/(c) = (abc)/(abc) = 1

Explanation

To simplify the expression √7 - 1{√7 + 1} - √7 + 1{√7 - 1}, we find a common denominator by multiplying the terms:

LHS = (√7 - 1)² - (√7 + 1)²{(√7 + 1)(√7 - 1)}

Using the algebraic identity (x - y)² - (x + y)² = -4xy for the numerator and (x + y)(x - y) = x² - y² for the denominator:

Numerator = -4(√7)(1) = -4√7 Denominator = (√7)² - (1)² = 7 - 1 = 6

The expression simplifies to: LHS = -4√7{6} = -⅔√7

Comparing this result with the given form a + √7b: 0 - ⅔√7 = a + b√7

By equating the rational and irrational components, we get a = 0 and b = -2/3.

Explanation

The expression consists of terms of the form (1)/(√(n) + n+1). By rationalizing the denominator for each term using the conjugate √(n+1) - √n, the general term Tₙ simplifies as follows: Tₙ = √(n+1) - √n{(√(n+1) + √n)(√(n+1) - √n)} = √(n+1) - √n{(n+1) - n} = √(n+1) - √n

Applying this transformation to all terms in the series results in a telescoping sum: (√2 - √1) + (√3 - √2) + (√4 - √3) + … + (√9 - √8)

In a telescoping series, the intermediate terms cancel out. Specifically, the positive part of one term cancels the negative part of the next term (e.g., √2 cancels -√2, √3 cancels -√3, and so on). This leaves only the first and the last components of the entire expansion: -√1 + √9 = -1 + 3 = 2

Explanation

Applying the componendo and dividendo rule to the expression √(a + 2b) + √(a - 2b){√(a + 2b) - √(a - 2b)} = √3{1} yields:

(√(a + 2b) + √(a - 2b)) + (√(a + 2b) - √(a - 2b)){(√(a + 2b) + √(a - 2b)) - (√(a + 2b) - √(a - 2b))} = √3 + 1{√3 - 1}

2√(a + 2b){2√(a - 2b)} = √3 + 1{√3 - 1}

Squaring both sides of the equation to eliminate the radicals gives:

(a + 2b)/(a - 2b) = (√3 + 1)²{(√3 - 1)²} = 3 + 1 + 2√3{3 + 1 - 2√3} = 4 + 2√3{4 - 2√3}

Simplifying the right side by dividing by 2 results in:

(a + 2b)/(a - 2b) = 2 + √3{2 - √3}

Applying componendo and dividendo once more to isolate a and b:

((a + 2b) + (a - 2b))/((a + 2b) - (a - 2b)) = (2 + √3) + (2 - √3){(2 + √3) - (2 - √3)}

(2a)/(4b) = (4)/(2√(3))

(a)/(2b) = (2)/(√(3))

(a)/(b) = (4)/(√(3))

Explanation

In the Cartesian coordinate system, the distance of a point (x, y) from the y-axis is determined by the absolute value of its x-coordinate, while its distance from the x-axis is determined by the absolute value of its y-coordinate. In the 4th quadrant, the x-coordinate is always positive (x > 0) and the y-coordinate is always negative (y < 0). Given that the point is 7 units away from the y-axis, the magnitude of the x-coordinate is 7, and since it is in the 4th quadrant, x = 7. Similarly, the distance of 6 units from the x-axis means the magnitude of the y-coordinate is 6, and the 4th quadrant location dictates that y = -6. Combining these values gives the coordinates (7, -6).

Explanation

In the cyclic pentagon PQRST, join the vertices P and S. This construction creates a cyclic quadrilateral PQRS and a triangle PST within the circle. For the cyclic quadrilateral PQRS, the sum of the opposite angles is 180°. Specifically, ∠ PQR + ∠ PSR = 180°.

Since PT is given as the diameter of the circle, the angle subtended by the diameter at any point on the circumference is a right angle. Therefore, in △ PST, the angle ∠ PST = 90°. The total angle ∠ RST of the pentagon is the sum of ∠ PSR and ∠ PST.

Substituting these relationships into the required sum, we get ∠ PQR + ∠ RST = ∠ PQR + (∠ PSR + ∠ PST). Rearranging the terms gives (∠ PQR + ∠ PSR) + ∠ PST = 180° + 90° = 270°.

Explanation

For an expression of the form x = √a + √b{√a - √b}, the sum of x and its reciprocal ¹⁄ₓ is calculated using the shortcut 2 ( (a + b)/(a - b) ). Substituting a = 3 and b = 2 from the given equation, we find x + ¹⁄ₓ = 2 ( (3 + 2)/(3 - 2) ) = 10. To evaluate the value of x³ + (1)/(x³⁾, we use the algebraic identity x³ + (1)/(x³⁾ = (x + ¹⁄ₓ)³ - 3(x + ¹⁄ₓ). By substituting the value 10 into this formula, we obtain 10³ - 3(10) = 1000 - 30 = 970.

Explanation

Two circles touch each other internally when one circle is positioned inside the other such that they share a single common point on their circumferences. In this configuration, the centers of both circles and the point of tangency are collinear. Let C₁ and C₂ be the centers of the circles with radii r₁ and r₂ respectively, and let d be the distance between C₁ and C₂. Assuming r₁ is the radius of the larger circle, the distance from C₁ to the point of contact on the circumference is r₁. This distance is equal to the sum of the distance between the two centers d and the radius of the smaller circle r₂. This geometric relationship is expressed by the equation r₁ = d + r₂. Rearranging the formula to solve for the distance between the centers gives d = r₁ - r₂. In general terms, the circles touch internally if the distance between their centers is exactly equal to the absolute difference of their radii, |r₁ - r₂|.

Explanation

In an isosceles triangle where AB = AC, the median AD drawn to the base BC is also the altitude to the base. This means ∠ ADB = 90° and D is the midpoint of BC. In the right-angled triangle ABD, we can apply the Pythagorean theorem to find the length of BD:

AB² = AD² + BD²

Substituting the given values: 10² = 8² + BD² 100 = 64 + BD² BD² = 100 - 64 = 36 BD = √(36) = 6 cm

Since D is the midpoint of BC, the total length of the base is: BC = 2 × BD = 2 × 6 = 12 cm

Explanation

Let O be the center of the circle and P be the point of tangency on the circle. In geometry, the radius OP is perpendicular to the tangent PQ at the point of contact. This forms a right-angled triangle △ OPQ, where OQ is the hypotenuse.

Using the Pythagorean theorem: OP² + PQ² = OQ²

Substituting the given values, where the tangent length PQ = 12 cm and the distance from the center OQ = 13 cm: r² + 12² = 13² r² + 144 = 169 r² = 25 r = 5 cm

The radius of the circle is 5 cm. The distance from Q to the nearest point on the circle is measured along the line segment OQ. This distance is calculated by subtracting the radius from the total distance of Q from the center: Distance = OQ - r Distance = 13 - 5 = 8 cm

Explanation

In a right-angled triangle with perpendicular sides AB and AC, the hypotenuse BC is calculated using the Pythagorean theorem: BC = √(AB² + AC²). Substituting the given values, BC = √(6² + 8²) = √(36 + 64) = √(100) = 10 cm.

For a right-angled triangle, the radius of the in-circle r is given by the formula r = (P + B - H)/(2), where P and B are the perpendicular sides and H is the hypotenuse. Substituting the lengths of the sides, r = (6 + 8 - 10)/(2) = ⁴⁄₂ = 2 cm.

Alternatively, the in-radius can be found using the relation r = (Area)/(s), where s is the semi-perimeter. The area of △ ABC is ½ × 6 × 8 = 24 cm² and the semi-perimeter s = (6 + 8 + 10)/(2) = 12 cm. Thus, r = ²⁴⁄₁₂ = 2 cm.

Explanation

Let the radius of the circle be r. Since AB is a diameter of the circle, its length is 2r, which implies r = (AB)/(2). The diameters AB and CD are perpendicular to each other and intersect at the centre O, meaning ∠ AOC = 90°. In the right-angled triangle AOC, the legs AO and CO are both radii of the circle, so AO = CO = r.

Applying the Pythagorean theorem to △ AOC: AC² = AO² + CO² AC² = r² + r² = 2r² AC = √2r

Substituting r = (AB)/(2) into the equation for AC: AC = √2 × (AB)/(2) AC = √2{2} AB AC = (1)/(√(2)) AB

Explanation

For any two similar triangles, the ratio of their perimeters is equal to the ratio of their corresponding sides, whereas the ratio of their areas is equal to the square of the ratio of their corresponding sides. Given that △ ABC ∼ △ DEF and the ratio of their perimeters is 4:1, the ratio of their corresponding sides is also 4:1. By the area-ratio theorem for similar triangles, the ratio of their areas is the square of the perimeter ratio, which is (4/1)² = 16/1.

Explanation

The angle θ between the hour hand and the minute hand of a clock at any given time H:M is calculated using the formula θ = |30H - ¹¹⁄₂M|. For the time 7:20, the value of H is 7 and the value of M is 20. Substituting these values into the formula yields θ = |30(7) - ¹¹⁄₂(20)|.

This simplifies to θ = |210 - 110|, which equals 100°. This result is derived from the fact that the hour hand moves 30° per hour and 0.5° per minute, while the minute hand moves 6° per minute. At 7:20, the minute hand is at the 120-degree position (calculated as 20 × 6°), and the hour hand is at the 220-degree position (calculated as 7 × 30° + 20 × 0.5°). The difference between these two positions is 220° - 120° = 100°.