SSC CGL Tier 1 2016 Quantitative Aptitude Questions with Solutions

Explanation

To find the least number to be added to 13851 to make it exactly divisible by 87, we first divide 13851 by 87 to find the remainder.

13851 = (87 × 159) + 18

The division of 13851 by 87 yields a remainder of 18. For a number to be perfectly divisible by a divisor after adding a value, the required value is calculated using the formula:

Number to be added = Divisor - Remainder

Substituting the values into the formula: Required number = 87 - 18 = 69

Adding 69 to 13851 results in 13920, which is exactly 160 times 87.

Explanation

Let the numbers be 3x and 5x. According to the problem, (3x + 6) / (5x + 6) = 2/3, which gives 3(3x + 6) = 2(5x + 6), leading to 9x + 18 = 10x + 12, or x = 6. The numbers are 3(6) = 18 and 5(6) = 30.

Explanation

The weight of the new parcel can be determined by calculating the difference between the total weight of 11 parcels and the total weight of the initial 10 parcels. First, the total weight of the original 10 parcels is 10 × 1.7 = 17 kg. The problem states that the addition of a new parcel reduces the average weight by 60 grams, which is equivalent to 0.06 kg. Therefore, the new average weight for the 11 parcels is 1.7 - 0.06 = 1.64 kg.

The total weight of the 11 parcels is 11 × 1.64 = 18.04 kg. By subtracting the initial total weight from this new total, we find the weight of the added parcel: 18.04 - 17 = 1.04 kg.

A quicker approach uses the formula for an added value in an average: Weight = Old Average + (New Count × Change in Average). Substituting the given values where the change is negative due to a reduction: 1.7 + (11 × -0.06) = 1.7 - 0.66 = 1.04 kg.

Explanation

The numerator follows the identity x³ + y³ + z³ - 3xyz = (x + y + z)(x² + y² + z² - xy - yz - zx). When x, y, z are in an arithmetic progression with a common difference d, the term x² + y² + z² - xy - yz - zx simplifies to 3d².

Given x = 999, y = 1000, z = 1001, the values are in arithmetic progression with d = 1.

The numerator is calculated as: (x + y + z) · 3d² = (999 + 1000 + 1001) · 3(1)² = 3000 × 3 = 9000

The denominator is: x - y + z = 999 - 1000 + 1001 = 1000

The value of the expression is: ⁹⁰⁰⁰⁄₁₀₀₀ = 9

Explanation

To find the value of the expression ¹⁄₁₋₂ₐ + (1)/(1-2b) + (1)/(1-2c), we can rewrite each term using the components of the given equation. We observe the following algebraic identity for each term: ¹⁄₁₋₂ₐ = (1 - 2a + 2a)/(1-2a) = ¹⁻²ᵃ⁄₁₋₂ₐ + ²ᵃ⁄₁₋₂ₐ = 1 + 2 ( ᵃ⁄₁₋₂ₐ )

By applying the same logic to the terms involving b and c, we obtain: (1)/(1-2b) = 1 + 2 ( (b)/(1-2b) ) (1)/(1-2c) = 1 + 2 ( (c)/(1-2c) )

Summing these three equations gives: ¹⁄₁₋₂ₐ + (1)/(1-2b) + (1)/(1-2c) = (1 + 1 + 1) + 2 ( ᵃ⁄₁₋₂ₐ + (b)/(1-2b) + (c)/(1-2c) )

We are given that ᵃ⁄₁₋₂ₐ + (b)/(1-2b) + (c)/(1-2c) = ½. Substituting this value into our sum: Value = 3 + 2 ( ½ ) = 3 + 1 = 4

Explanation

In any triangle △ ABC, the point where the angle bisectors meet is the incenter. The angle formed at the incenter D by the bisectors of the base angles ∠ B and ∠ C is given by the formula ∠ BDC = 90° + (∠ A)/(2).

Given that ∠ A = 80°, substituting this value into the formula gives:

∠ BDC = 90° + (80°)/(2)

∠ BDC = 90° + 40°

∠ BDC = 130°

Alternatively, since AB = AC, the base angles are equal. Using the angle sum property of a triangle, ∠ B + ∠ C = 180° - 80° = 100°, so ∠ B = ∠ C = 50°. The bisectors create △ BDC where ∠ DBC = (50°)/(2) = 25° and ∠ DCB = (50°)/(2) = 25°. In △ BDC, the sum of angles is 180°, therefore ∠ BDC = 180° - (25° + 25°) = 130°.

Explanation

In a circle, a perpendicular line drawn from the centre to a chord bisects that chord. This construction creates a right-angled triangle where the hypotenuse is the radius r = 13 cm, the altitude is the distance from the centre d = 12 cm, and the base is half the length of the chord x. According to the Pythagorean theorem:

r² = d² + x²

Substituting the given values:

13² = 12² + x² 169 = 144 + x² x² = 169 - 144 = 25 x = √(25) = 5 cm

Since x is only half of the chord, the total length of the chord is 2x = 2 × 5 = 10 cm.

Explanation

The given equation is 7cos²θ + 3sin²θ = 4. Using the trigonometric identity sin²θ = 1 - cos²θ, we can express the entire equation in terms of cosθ:

7cos²θ + 3(1 - cos²θ) = 4

Expanding the term, we get:

7cos²θ + 3 - 3cos²θ = 4

4cos²θ = 4 - 3

4cos²θ = 1

cos²θ = ¼

Taking the square root for the positive acute angle θ:

cosθ = ½

The value of θ for which cosθ = ½ in the range of acute angles is 60°.

Explanation

The smallest six-digit number is 100,000. To find the smallest six-digit perfect square, we determine the square root of 100,000. Since √(100,000) ≈ 316.22, the smallest integer whose square yields a six-digit number is 317. Using the identity (a+b)² = a² + 2ab + b² with a = 310 and b = 7:

317² = (310 + 7)² 317² = 310² + 2(310)(7) + 7² 317² = 96,100 + 4,340 + 49 317² = 100,489

Alternatively, using a = 300 and b = 17: 317² = 300² + 2(300)(17) + 17² 317² = 90,000 + 10,200 + 289 = 100,489

This value represents the first perfect square that reaches the six-digit threshold.

Explanation

When a solid is melted and recast into another shape, the total volume remains constant. First, convert the units to centimeters: radius r = 3.2 dm = 32 cm, and the side of each cube a = 8 cm.

The volume of the cylindrical rod is given by V = π r² h, and the volume of 44 cubes is V = 44 × a³. Equating the two volumes:

²²⁄₇ × (32)² × h = 44 × 8³

²²⁄₇ × 1024 × h = 44 × 512

Dividing both sides by 512:

²²⁄₇ × 2 × h = 44

⁴⁴⁄₇ × h = 44

h = 7 cm

Explanation

Let the height of the tower be h. Let the two points be at distances a = 4 m and b = 9 m from the base of the tower along the same straight line. Let the angle of elevation from the point at 4 m be θ. Since the angles of elevation are complementary, the angle of elevation from the point at 9 m is 90° - θ.

From the first point, the relationship is tan θ = ʰ⁄₄. From the second point, the relationship is tan(90° - θ) = ʰ⁄₉. Using the trigonometric identity tan(90° - θ) = cot θ, the second equation becomes cot θ = ʰ⁄₉.

Multiplying these two equations together gives tan θ · cot θ = ʰ⁄₄ · ʰ⁄₉. Since tan θ · cot θ = 1, the equation simplifies to 1 = (h²)/(36), which leads to h² = 36. Taking the square root, h = 6 m. This follows the standard height and distance formula h = √(ab) for complementary angles of elevation from distances a and b from the base. Substituting a = 4 and b = 9 yields h = √(36) = 6 m.

Explanation

When a quantity undergoes two successive percentage changes of a% and b%, the net percentage change is calculated using the formula (a + b + ᵃᵇ⁄₁₀₀)%. In this scenario, the number is first decreased by 10%, making a = -10, and then increased by 10%, making b = +10.

Substituting these values into the formula: Net Change = (-10 + 10 + (-10 × 10)/(100))% Net Change = (0 - ¹⁰⁰⁄₁₀₀)% = -1%

The negative result signifies a decrease. This can also be visualized by assuming the initial value is 100. After a 10% decrease, the value becomes 90. Increasing this new value by 10% (which is 9) results in 99. The total change from 100 to 99 is a reduction of 1, which corresponds to a 1% decrease.

Explanation

In a right-angled triangle formed by the vertical pole and its shadow on the ground, the height of the pole acts as the perpendicular side and the shadow acts as the base. The angular elevation θ of the Sun is the angle at which the shadow meets the tip of the pole's projection.

Using the trigonometric ratio for tangent: tan θ = (Perpendicular)/(Base) = (Height of the pole)/(Length of the shadow)

Substituting the given values: tan θ = (15)/(15){√3} tan θ = 15 × √3{15} tan θ = √3

Since tan 60° = √3, the value of θ is 60°.

Explanation

The given equation 2y cosθ = x sinθ can be rearranged to express x in terms of y: x = (2y cosθ)/(sinθ). Substitute this expression for x into the second equation 2x secθ - y cscθ = 3:

2((2y cosθ)/(sinθ)) (1)/(cosθ) - (y)/(sinθ) = 3

The cosθ terms in the first part cancel out, simplifying the equation to:

(4y)/(sinθ) - (y)/(sinθ) = 3 (3y)/(sinθ) = 3 y = sinθ

Now, substitute y = sinθ back into the first equation to find x:

x sinθ = 2(sinθ) cosθ x = 2 cosθ

To find the value of x² + 4y², substitute the solved values of x and y:

x² + 4y² = (2 cosθ)² + 4(sinθ)² x² + 4y² = 4 cos²θ + 4 sin²θ x² + 4y² = 4(cos²θ + sin²θ)

Applying the fundamental trigonometric identity sin²θ + cos²θ = 1:

x² + 4y² = 4(1) = 4

Explanation

For a period of 2 years, the difference between compound interest and simple interest is given by the formula D = P × (ʳ⁄₁₀₀)², where D is the difference, P is the principal amount, and r is the annual rate of interest.

Substituting the given values into the formula: 96 = 15000 × (ʳ⁄₁₀₀)² 96 = 15000 × (r²)/(10000) 96 = 1.5 × r² r² = (96)/(1.5) r² = ⁹⁶⁰⁄₁₅ r² = 64 r = √(64) r = 8

The rate of interest per annum is 8%.

Explanation

The ratio of spirit to water is 3:2. The difference between the ratio parts of spirit and water is 3 - 2 = 1 unit. According to the problem, this 1 unit difference corresponds to an actual physical difference of 3 litres. Since 1 unit represents 3 litres, the quantity of any component in the mixture can be found by multiplying its ratio part by 3. The spirit component consists of 3 units in the ratio, so the total quantity of spirit is 3 × 3 = 9 litres.

Explanation

When the cost price of x articles is equal to the selling price of y articles, the profit or loss percentage is calculated using the formula (x - y)/(y) × 100. In this problem, the cost price of x = 25 pens is equal to the selling price of y = 20 pens. Substituting these values into the formula, the profit percentage is (25 - 20)/(20) × 100 = ⁵⁄₂₀ × 100. Simplifying this expression gives ¼ × 100, which results in 25 percent. Alternatively, if we assume the cost price of one pen is 1 unit, the total cost for 25 pens is 25 units. Since this is the revenue generated from selling only 20 pens, the selling price per pen is ²⁵⁄₂₀ = 1.25 units. The profit per pen is 1.25 - 1 = 0.25 units, leading to a profit percentage of (0.25)/(1) × 100 = 25%.

Explanation

The net percentage change for two successive percentage changes of x% and y% is calculated using the formula x + y + ˣʸ⁄₁₀₀.

Since the number is decreased in both instances, we take x = -10 and y = -20. Substituting these values into the formula gives:

Net percentage change = -10 + (-20) + ((-10 × -20))/(100)

Net percentage change = -30 + ²⁰⁰⁄₁₀₀

Net percentage change = -30 + 2 = -28%

The negative sign indicates a total decrease of 28%. Alternatively, if we assume the initial number is 100, the first decrease of 10% reduces it to 90. The second decrease of 20% is applied to 90, which is 0.20 × 90 = 18. The final value is 90 - 18 = 72. The total reduction from the original 100 is 100 - 72 = 28.

Explanation

When two objects move in the same direction, their relative speed is the difference between their individual speeds. Here, the speed of the car starting from A is V₁ = 35 km/h and the speed of the car starting from B is V₂ = 25 km/h.

Relative speed Vᵣₑₗ = V₁ - V₂ = 35 - 25 = 10 km/h.

The initial distance between the two cars is D = 60 km. The time taken for the cars to meet is calculated by dividing the initial distance by the relative speed:

Time = (Distance)/(Relative Speed)

Time = ⁶⁰⁄₁₀ = 6 hours.

Explanation

The simple interest SI is calculated using the formula SI = (P × R × T)/(100), where P is the principal sum, R is the rate of interest per annum, and T is the time in years.

Given that SI = 60, R = 6, and T = 5, we can substitute these values into the formula to find the principal:

60 = (P × 6 × 5)/(100)

Multiplying the values in the numerator: 60 = (30P)/(100)

Simplifying the fraction: 60 = (3P)/(10)

To solve for P, multiply both sides by 10 and divide by 3: 3P = 600 P = ⁶⁰⁰⁄₃ P = 200

Explanation

The square of any real number is always non-negative. Therefore, for the sum of several squares to be zero, each individual squared term must independently equal zero. Given the equation (x - 3)² + (y - 4)² + (z - 5)² = 0, where x, y, and z are real numbers, we can set each component to zero:

(x - 3)² = 0 ⟹ x = 3 (y - 4)² = 0 ⟹ y = 4 (z - 5)² = 0 ⟹ z = 5

To find the value of the expression (x + y + z), we substitute these determined values: x + y + z = 3 + 4 + 5 = 12

Explanation

From the given equation 2x + ¹⁄₃ₓ = 5, multiply both sides by 3x to eliminate the fraction: 3x(2x) + 3x(¹⁄₃ₓ) = 5(3x) 6x² + 1 = 15x

The expression to be evaluated is (5x)/(6x² + 20x + 1). Rearranging the terms in the denominator allows for direct substitution of the derived identity: (5x)/((6x² + 1) + 20x)

Substitute 6x² + 1 = 15x into the denominator: (5x)/(15x + 20x) = ⁵ˣ⁄₃₅ₓ

Dividing both the numerator and the denominator by 5x results in: ⅐

Explanation

The orthocentre of a triangle is the point of concurrency where the three altitudes of the triangle meet. An altitude is defined as a line segment drawn from a vertex of the triangle perpendicular to the opposite side. Depending on the type of triangle, the position of the orthocentre varies: it lies inside an acute triangle, at the vertex of the right angle in a right-angled triangle, and outside the triangle in an obtuse triangle. The existence of this single point of intersection for all three altitudes is a fundamental property of Euclidean geometry.

Explanation

In a cyclic quadrilateral ABCD, the sum of opposite angles is 180°. Therefore, ∠ ABC + ∠ ADC = 180°. Since AD ∥ BC, the line CD acts as a transversal, and the sum of the co-interior angles is 180°, which gives ∠ BCD + ∠ ADC = 180°.

By equating these two expressions: ∠ ABC + ∠ ADC = ∠ BCD + ∠ ADC

By subtracting ∠ ADC from both sides, we find that ∠ ABC = ∠ BCD. Given that ∠ ABC = 70°, the value of ∠ BCD must also be 70°. This relationship holds because any cyclic trapezium is necessarily an isosceles trapezium, meaning the angles adjacent to each parallel base are equal.

Explanation

The angles 63° and 27° are complementary because 63° + 27° = 90°. According to trigonometric complementary angle identities, tan(90° - θ) = cot θ, which means tan 63° = tan(90° - 27°) = cot 27°.

We are given cos 27° = x. Using the fundamental identity sin² θ + cos² θ = 1, we can determine the value of sin 27° as √(1 - cos² 27°) = √(1 - x²). Since cot θ = (cos θ)/(sin θ), substituting the values for 27° gives cot 27° = (cos 27°)/(sin 27°) = (x)/(√(1 - x²)). Therefore, tan 63° = (x)/(√(1 - x²)).

Explanation

The sum of n terms of an arithmetic progression is given by the formula Sₙ = ⁿ⁄₂[2a + (n-1)d] and the n-th term is defined as aₙ = a + (n-1)d, where a is the first term and d is the common difference.

Given the third term a₃ = 19: a + (3-1)d = 19 a + 2d = 19 (Equation 1)

Given the sum of 10 terms S₁₀ = 390: ¹⁰⁄₂[2a + (10-1)d] = 390 5[2a + 9d] = 390 2a + 9d = 78 (Equation 2)

To solve for a, we can express 2d from Equation 1 as 2d = 19 - a, which implies d = (19 - a)/(2). Substituting this into Equation 2: 2a + 9((19 - a)/(2)) = 78 Multiplying the entire equation by 2 to clear the fraction: 4a + 9(19 - a) = 156 4a + 171 - 9a = 156 -5a = 156 - 171 -5a = -15 a = 3

Explanation

Using the method of alligation, the ratio of passed candidates to failed candidates is determined by comparing their respective averages to the overall mean average. The difference between the overall average 30 and the failed average 10 is 20, while the difference between the passed average 35 and the overall average 30 is 5. Thus, the ratio of passed to failed candidates is 20 : 5, which simplifies to 4 : 1. Since the total number of candidates is 100, representing 4 + 1 = 5 total parts, the number of passed candidates is calculated as ⅘ × 100 = 80. Alternatively, using an algebraic approach, let x be the number of passed candidates. The total marks obtained by all candidates is 100 × 30 = 3000. This total is the sum of marks from passed and failed candidates: 35x + 10(100 - x) = 3000. Expanding the equation gives 35x + 1000 - 10x = 3000, which simplifies to 25x = 2000. Solving for x results in 80.

Explanation

The algebraic identity relating the square of the difference of two numbers to their sum and product is (a - b)² = (a + b)² - 4ab. Substituting the given values a + b = 10 and ab = 21 into this formula yields (a - b)² = (10)² - 4(21). Performing the arithmetic gives 100 - 84 = 16. Alternatively, the numbers a and b can be identified as 7 and 3, since 7 + 3 = 10 and 7 × 3 = 21. The value of (a - b)² is thus (7 - 3)² = 4² = 16.

Explanation

The given expression (x - 4)(x² + 4x + 16) matches the standard algebraic identity for the difference of two cubes, which is (a - b)(a² + ab + b²) = a³ - b³. In this specific case, by setting a = x and b = 4, the middle terms fit the required pattern as ab = 4x and b² = 4² = 16. Applying the identity, we get (x - 4)(x² + 4x + 16) = x³ - 4³. Calculating the cube of the constant, 4³ = 4 × 4 × 4 = 64. Comparing the resulting expression x³ - 64 with the provided form x³ - P, it is evident that P = 64.

Explanation

In any triangle, the centroid G divides the median AM (where M is the midpoint of BC) in the ratio 2:1. This means AG = ⅔ AM and GM = ⅓ AM. The problem states that AG = BC. By substituting this into the ratio, we get BC = ⅔ AM, which implies AM = ³⁄₂ BC.

Now, we calculate the length of GM in terms of BC: GM = ⅓ AM = ⅓ (³⁄₂ BC) = ½ BC

Since M is the midpoint of BC, the segments BM and MC are both equal to ½ BC. Therefore, in Δ BGC, we have the relation GM = BM = MC. This indicates that the point G lies on a circle with BC as the diameter and M as the center. In geometry, an angle subtended by the diameter at any point on the circumference is always a right angle. Consequently, ∠ BGC = 90°.

Explanation

In a circle, the radii connecting the center to the endpoints of a chord and the chord itself form an isosceles triangle. If the chord subtends a right angle at the center, this triangle is a right-angled isosceles triangle where the radii represent the two legs and the chord represents the hypotenuse. Applying the Pythagorean theorem, the relationship is r² + r² = c², where r is the radius and c is the chord length. Substituting the given value c = 5√2 cm into the equation, we get 2r² = (5√2)². This simplifies to 2r² = 25 × 2, which further simplifies to 2r² = 50. Dividing by 2, we find r² = 25, and taking the square root yields r = 5 cm.

Explanation

The given equation is cos² x + cos⁴ x = 1. By rearranging the terms, we get cos⁴ x = 1 - cos² x. Using the identity sin² x + cos² x = 1, this simplifies to cos⁴ x = sin² x. Dividing both sides of this equation by cos⁴ x, we obtain 1 = (sin² x)/(cos⁴ x). This expression can be factored as 1 = (sin² x)/(cos² x) · (1)/(cos² x). Substituting the trigonometric ratios tan² x = (sin² x)/(cos² x) and sec² x = (1)/(cos² x), the equation becomes 1 = tan² x · sec² x. Applying the identity sec² x = 1 + tan² x, we have 1 = tan² x (1 + tan² x), which expands to 1 = tan² x + tan⁴ x.

Explanation

The speed of an aeroplane flying at a constant height h when the angle of elevation changes from ₁ to ₂ in time t is given by the formula: v = (h(cot ₂ - cot ₁))/(t)

Given h = 2500 m, ₁ = 45°, ₂ = 30°, and t = 15 seconds, we substitute these values: v = (2500(cot 30° - cot 45°))/(15) v = 2500(√3 - 1){15} m/s

To convert the speed from m/s to km/hr, we multiply by the factor ¹⁸⁄₅: Speed in km/hr = 2500(√3 - 1){15} × ¹⁸⁄₅ Speed in km/hr = 500(√3 - 1){3} × ¹⁸⁄₅ Speed in km/hr = 100(√3 - 1) × 6 Speed in km/hr = 600(√3 - 1)

Explanation

The volume of a solid sphere is given by V = ⁴⁄₃ π r³, which can be expressed in terms of the diameter D as V = ⅙ π D³. When n small spheres are melted and recast into a single large sphere, the volume of the large sphere equals the sum of the volumes of the small spheres.

Let D be the diameter of the new sphere and d be the diameter of the small spheres. n × ⅙ π d³ = ⅙ π D³

Simplifying the equation by canceling common terms: n × d³ = D³ D = d × ³√n

Substituting the given values n = 1000 and d = 6 cm: D = 6 × ³√(1000) D = 6 × 10 D = 60 cm

Explanation

The relationship between workers, time, and work is expressed by the formula (M₁ D₁)/(W₁) = (M₂ D₂)/(W₂), where M represents the number of spiders, D represents the number of days, and W represents the number of webs.

Given that 7 spiders make 7 webs in 7 days, we have M₁ = 7, D₁ = 7, and W₁ = 7. To find the time D₂ it takes for 1 spider (M₂ = 1) to make 1 web (W₂ = 1), substitute these values into the formula:

(7 × 7)/(7) = (1 × D₂)/(1)

7 = D₂

Logically, if 7 spiders working simultaneously produce 7 webs in 7 days, it implies that each individual spider is working on its own web and takes the full 7 days to complete it. Therefore, one spider will take 7 days to finish one web.

Explanation

The list price of the fan is Rs. 150. After the first discount of 20% (Rs. 30), the price becomes 150 - 30 = Rs. 120. To reach a final net price of Rs. 108, an additional discount of Rs. 12 must be given (120 - 108 = 12). The percentage for this additional discount is calculated on the reduced price: (12 / 120) × 100 = 10%. Since 10% is not among options A, B, or C, the correct choice is "None of these".

Explanation

Let the three numbers be x, 2x, and 3x based on the given ratio 1 : 2 : 3. The sum of their cubes is expressed by the equation x³ + (2x)³ + (3x)³ = 4500.

Expanding the terms, we get: x³ + 8x³ + 27x³ = 4500 36x³ = 4500

To find x³, divide 4500 by 36: x³ = ⁴⁵⁰⁰⁄₃₆ x³ = 125

Taking the cube root of both sides: x = ³√(125) = 5

The smallest number corresponds to the first part of the ratio, which is x. Therefore, the smallest number is 5.

Explanation

The relationship between the Cost Price (CP), Selling Price (SP), and gain percentage is given by the formula SP = CP × (1 + (Gain%)/(100)). Given that the selling price is ₹ 390 and the gain is 20%, the equation can be written as 390 = CP × (1 + ²⁰⁄₁₀₀).

Simplifying the term in the bracket gives 390 = CP × ¹²⁰⁄₁₀₀, which further reduces to 390 = CP × 1.2. To find the cost price, divide the selling price by 1.2:

CP = (390)/(1.2)

CP = ³⁹⁰⁰⁄₁₂

CP = 325

Alternatively, using the fraction method, a 20% gain means the selling price is ⁶⁄₅ of the cost price. Therefore, CP = 390 × ⅚. Dividing 390 by 6 gives 65, and multiplying 65 by 5 results in 325.

Explanation

Let the initial volume of the liquid be 100 units. Since the volume increases by 4% when the liquid freezes, the volume of the resulting solid is: 100 + (4% of 100) = 100 + 4 = 104 units

When the solid melts back into a liquid, the volume returns from 104 units to the original 100 units. The absolute decrease in volume is: 104 - 100 = 4 units

The percentage decrease is calculated using the volume of the solid as the base value: Percentage Decrease = (Decrease)/(Initial Volume of Solid) × 100 Percentage Decrease = ⁴⁄₁₀₄ × 100

Simplifying the fraction: ⁴⁄₁₀₄ = ¹⁄₂₆ ¹⁄₂₆ × 100 = ⁵⁰⁄₁₃ = 3 ¹¹⁄₁₃%

Dividing 50 by 13 gives a quotient of 3 and a remainder of 11, resulting in the mixed fraction 3 ¹¹⁄₁₃.

Explanation

Let the time spent walking be t hours. Consequently, the time spent on the bicycle is 9 - t hours. The total distance covered is the sum of the distances traveled in both segments. Using the formula Distance = Speed × Time, we can establish the equation 4t + 9(9 - t) = 61.

Solving for t: 4t + 81 - 9t = 61 -5t = 61 - 81 -5t = -20 t = 4

The man spent 4 hours walking. The distance covered by walking is 4 km/hr × 4 hours = 16 km. Alternatively, using the rule of alligation, the ratio of time spent walking to cycling is (9 - 61/9) : (61/9 - 4), which simplifies to 20/9 : 25/9, or a ratio of 4:5. Since the total time is 9 hours, the time spent walking is 4 hours, resulting in a distance of 16 km.

Explanation

For a sum of money to double itself, the simple interest earned must be equal to the principal amount (P), so that the total amount A = P + SI = 2P. The rate of simple interest (R) can be calculated using the direct formula for the doubling of a sum:

R = (100(n - 1))/(T)

Where n is the number of times the sum increases (in this case, n = 2) and T is the time in years. Substituting the given values:

n = 2 T = 16½ = ³³⁄₂ years

R = (100(2 - 1))/(33){2} R = (100 × 2)/(33) R = ²⁰⁰⁄₃₃ R ≈ 6.06%

Rounding to the nearest value provided in the options, the rate of interest is 6%.

Explanation

The algebraic identity for the sum of three cubes states that a³ + b³ + c³ - 3abc = (a + b + c)(a² + b² + c² - ab - bc - ca). Given the condition a + b + c = 0, the right-hand side of the equation becomes zero because any expression multiplied by zero results in zero. Thus, the equation simplifies to a³ + b³ + c³ - 3abc = 0. By transposing the term -3abc to the other side of the equation, we obtain a³ + b³ + c³ = 3abc.

Explanation

The given expression (x² + x⁻²)(x² - 1 + x⁻²) appears to be a typographical error for the standard algebraic identity (x¹/³ + x⁻¹/³)(x²/³ - 1 + x⁻²/³). Applying the identity (a + b)(a² - ab + b²) = a³ + b³ where a = x¹/³ and b = x⁻¹/³, we get (x¹/³)³ + (x⁻¹/³)³ = x + x⁻¹. This matches the form of option D.

Explanation

In △ ABC, since DE ∥ AC, the Basic Proportionality Theorem (Thales' Theorem) states that the segments created on the other two sides are proportional. Specifically, the ratio of the segments on side AB must equal the ratio of the segments on side BC: (BD)/(DA) = (BE)/(EC)

The length of segment BD is determined by subtracting AD from the total length of AB: BD = AB - AD = 5 cm - 3 cm = 2 cm

By substituting the values of BD and AD into the proportionality equation: (BE)/(EC) = ⅔

This results in the ratio BE : EC = 2:3.

Explanation

In a circle, a tangent at any point is perpendicular to the radius through the point of contact. Therefore, the radius OT is perpendicular to the tangent PT at point T, making △ OTP a right-angled triangle with the right angle at T.

According to the Pythagoras theorem: OP² = OT² + PT²

Given that the radius OT = 6 cm and the tangent length PT = 8 cm, substituting these values into the formula gives: OP² = 6² + 8² OP² = 36 + 64 OP² = 100 OP = √(100) = 10 cm

Explanation

To find the value of the expression (3sinθ + 2cosθ)/(3sinθ - 2cosθ) , we can divide both the numerator and the denominator by cosθ . This transformation allows us to express the entire fraction in terms of tanθ because (sinθ)/(cosθ) = tanθ and (cosθ)/(cosθ) = 1 .

The expression becomes:

(3sinθ)/(cosθ) + 2(cosθ)/(cosθ){3(sinθ)/(cosθ) - 2(cosθ)/(cosθ)} = (3tanθ + 2)/(3tanθ - 2)

Given that tanθ = ⁴⁄₃ , we substitute this value into the simplified expression:

(3(4)/(3)) + 2{3(⁴⁄₃) - 2} = (4 + 2)/(4 - 2) = ⁶⁄₂ = 3

Alternatively, using the trigonometric ratio where tanθ = (perpendicular)/(base) = ⁴⁄₃ , we can assign the perpendicular p = 4k and the base b = 3k . The hypotenuse h is √((4k)² + (3k)²) = 5k . Therefore, sinθ = ⅘ and cosθ = ⅗ . Substituting these values into the original expression:

(3(4)/(5)) + 2(⅗){3(⅘) - 2(⅗)} = (12)/(5) + ⁶⁄₅{¹²⁄₅ - ⁶⁄₅} = (18)/(5){⁶⁄₅} = ¹⁸⁄₆ = 3

Explanation

To find the largest possible size for the classes such that each class of boys and girls contains the same number of students, we must calculate the Highest Common Factor (HCF) of the number of boys and the number of girls.

Given: Number of boys = 391 Number of girls = 323

We can find the HCF using the Euclidean division algorithm: 391 = 323 × 1 + 68 323 = 68 × 4 + 51 68 = 51 × 1 + 17 51 = 17 × 3 + 0

The HCF of 391 and 323 is 17. This is also evident from the prime factorization of the two numbers: 391 = 17 × 23 323 = 17 × 19

The largest common divisor is 17, which represents the maximum equal strength that each class can have while ensuring that both the boys and girls are divided exactly into classes of the same size.

Explanation

The difference between the total temperature of the second group (Tuesday to Friday) and the first group (Monday to Thursday) is calculated by multiplying the difference in their averages by the number of days: Difference = 4 × (63 - 60) = 4 × 3 = 12°

Since Tuesday, Wednesday, and Thursday are common to both groups, this difference represents the gap between Friday and Monday: Friday - Monday = 12°

Given the ratio of Monday to Friday is 21 : 25, let the temperatures be 21k and 25k. Substituting these into the difference equation: 25k - 21k = 12 4k = 12 k = 3

The temperature of Friday is calculated as: Friday = 25k = 25 × 3 = 75°

Explanation

Given the equation: ¹⁄ₚ + (1)/(q) = (1)/(p+q)

By taking the least common multiple on the left side, we get: (q+p)/(pq) = (1)/(p+q)

Cross-multiplying the terms results in: (p+q)² = pq p² + 2pq + q² = pq p² + pq + q² = 0

The algebraic expansion for the difference of two cubes is: p³ - q³ = (p-q)(p² + pq + q²)

By substituting the derived value p² + pq + q² = 0 into the expansion formula: p³ - q³ = (p-q)(0) p³ - q³ = 0

Explanation

The expression x² - y² + 10xz + 10yz can be simplified using algebraic factorization. By grouping the terms, we have (x² - y²) + (10xz + 10yz). Applying the difference of squares formula, x² - y² = (x + y)(x - y), and factoring out 10z from the second group, the expression becomes (x + y)(x - y) + 10z(x + y). Factoring out the common term (x + y) yields the simplified form (x + y)(x - y + 10z).

Given the values x = 93, y = 93, and z = 94, we substitute them into the factored expression. First, calculate the sums and differences: x + y = 93 + 93 = 186 and x - y = 93 - 93 = 0. Substituting these into the simplified formula results in 186 × (0 + 10 × 94), which further reduces to 186 × 940. To find the product, calculate 186 × 94 = 186 × (100 - 6) = 18600 - 1116 = 17484. Multiplying this by 10 gives the final value of 174840.