SSC CGL Tier 1 2016 Quantitative Aptitude Questions with Solutions

Explanation

Using the simple interest formula SI = (PRT)/100, for a sum to double, the interest earned must equal the principal (SI = P). Substituting values gives P = (P * R * 15) / 100, which simplifies to R = 100/15 = 20/3 = 6 2/3%.

Explanation

The given equation is (1)/(√(a)) - (1)/(√(b)) = 0. By transposing the term, we get: (1)/(√(a)) = (1)/(√(b)) Taking the reciprocal of both sides leads to √a = √b. Squaring both sides yields a = b.

The expression to be evaluated is ¹⁄ₐ + (1)/(b). Substituting the condition b = a into this expression, we obtain: ¹⁄ₐ + ¹⁄ₐ = ²⁄ₐ

Since a = b, the product ab is equal to a², and consequently, √(ab) = √(a²) = a. Substituting a = √(ab) back into the result of the expression: ²⁄ₐ = (2)/(√(ab))

Explanation

The algebraic identity for the square of a trinomial is (x+y+z)² = x²+y²+z² + 2(xy+yz+zx). By substituting the given values x²+y²+z² = 14 and xy+yz+zx = 11 into this formula, we get (x+y+z)² = 14 + 2(11). Calculating the sum results in 14 + 22, which equals 36.

Explanation

The given equation is √2 tan 2θ = √6. Dividing both sides by √2, we obtain tan 2θ = √6{√2} = √3. Since 0° < θ < 45°, the angle 2θ lies in the first quadrant (0° < 2θ < 90°). We know that tan 60° = √3, so 2θ = 60°, which gives θ = 30°.

Substitute θ = 30° into the expression sinθ + √3cosθ - 2tan²θ: E = sin 30° + √3cos 30° - 2tan² 30°

Using the standard trigonometric values sin 30° = ½, cos 30° = √3{2}, and tan 30° = (1)/(√(3)): E = ½ + √3 · √3{2} - 2 · ((1)/(√(3)))² E = ½ + ³⁄₂ - 2 · ⅓ E = ⁴⁄₂ - ⅔ E = 2 - ⅔ = ⁴⁄₃

Explanation

Let the positive number be x. According to the given condition, when the number is decreased by 4, it equals 21 times its reciprocal. This can be expressed as the equation x - 4 = 21/x. To solve for x, multiply the entire equation by x to get x² - 4x = 21, which rearranges into the quadratic form x² - 4x - 21 = 0.

Factorizing the quadratic equation, we look for two numbers that multiply to -21 and add to -4, which are -7 and 3. Thus, (x - 7)(x + 3) = 0. This gives two possible solutions for x, which are 7 and -3. Since the problem specifies that the number is positive, we take x = 7. Verifying this, 7 - 4 = 3 and 21/7 = 3, satisfying the condition.

Explanation

To evaluate the expression (sin α)/(sin³ α + cos³ α) using the given tan α = 2, divide both the numerator and the denominator by cos³ α to express the terms in terms of tan α.

The numerator is (sin α)/(cos³ α) = (sin α)/(cos α) · (1)/(cos² α) = tan α sec² α. By applying the trigonometric identity sec² α = 1 + tan² α, the numerator becomes tan α (1 + tan² α).

The denominator is (sin³ α + cos³ α)/(cos³ α) = (sin³ α)/(cos³ α) + (cos³ α)/(cos³ α) = tan³ α + 1.

The expression is now rewritten as: (tan α (1 + tan² α))/(tan³ α + 1)

Substituting the given value tan α = 2 into the simplified expression: (2 (1 + 2²))/(2³ + 1) = (2 (1 + 4))/(8 + 1) = (2 × 5)/(9) = ¹⁰⁄₉

Explanation

The algebraic identity for the difference of two cubes is x³ - y³ = (x - y)(x² + xy + y²). From this, we can express the fractional term in the given expression as:

(1)/(x² + xy + y²) = (x - y)/(x³ - y³)

Given x = ³√(28) and y = ³√(27), we find their cubes: x³ = (³√(28))³ = 28 y³ = (³√(27))³ = 27

Calculating the denominator x³ - y³: x³ - y³ = 28 - 27 = 1

Substituting this back into the simplified fractional term: (1)/(x² + xy + y²) = (x - y)/(1) = x - y

Now, substitute this result into the original expression: x + y - (x - y) = x + y - x + y = 2y

Since y = ³√(27) = 3, the final value is: 2 × 3 = 6

Explanation

Substitute the given values x = 12 and y = 4 into the expression (x + y)⁽ˣ⁾^/⁽ʸ⁾ to find its value.

First, calculate the base of the exponent: x + y = 12 + 4 = 16

Next, calculate the exponent: (x)/(y) = ¹²⁄₄ = 3

Now, substitute these results back into the original expression: 16³

To solve 16³, multiply 16 by itself three times: 16 × 16 = 256 256 × 16 = 4096

Alternatively, expressing the base as a power of 2 simplifies the calculation: 16³ = (2⁴)³ = 2¹² Since 2¹⁰ = 1024, then 2¹² = 1024 × 4 = 4096.

Explanation

In triangle PQR, given PQ = PR, the angles opposite to these sides must be equal according to the properties of an isosceles triangle. Therefore, ∠ R = ∠ Q. The problem also states that ∠ Q = 2∠ P.

Let ∠ P = x. It follows that ∠ Q = 2x and, because ∠ R = ∠ Q, then ∠ R = 2x.

The sum of the interior angles of any triangle is 180°. Applying this to triangle PQR: ∠ P + ∠ Q + ∠ R = 180° x + 2x + 2x = 180° 5x = 180° x = 36°

Since ∠ Q = 2x, we calculate its value as: ∠ Q = 2 × 36° = 72°

Explanation

The perpendicular drawn from the center of a circle to a chord always bisects the chord. Let the center of the circle be O and the chord be AB. If M is the point where the perpendicular from O meets the chord, then AM = MB = 16 / 2 = 8 cm. The distance OM is given as 15 cm. These segments form a right-angled triangle OMA, where the radius OA is the hypotenuse.

Using the Pythagoras theorem: OA² = OM² + AM² r² = 15² + 8² r² = 225 + 64 r² = 289 r = √(289) = 17 cm

Explanation

The average maximum temperature for the four cities is calculated using the formula Average = (Sum)/(Number of observations). Given the average is 35 and there are 4 cities, the total sum of their maximum temperatures must be 35 × 4 = 140.

Let the maximum temperature of Chennai be x. The sum of the temperatures of Delhi, Kolkata, and Mumbai is 35 + 33 + 34 = 102. Therefore, we set up the equation 102 + x = 140. Solving for x gives x = 140 - 102 = 38.

Using the deviation method, the temperatures of Delhi, Kolkata, and Mumbai relative to the average of 35 are 0, -2, and -1 respectively. The sum of these deviations is -3. For the average to remain 35, the sum of all deviations must be zero. Therefore, Chennai must have a deviation of +3 from the average, which results in a temperature of 35 + 3 = 38° C.

Explanation

Let the radii of the first and second spheres be R₁ and R₂ respectively. Since the diameter of the first sphere is twice the diameter of the second, their radii are in the same ratio, giving R₁ = 2R₂. The problem states that the curved surface area of the first sphere is numerically equal to the volume of the second sphere.

Using the formulas for the surface area of a sphere (4π R²) and the volume of a sphere (⁴⁄₃π R³): 4π R₁² = ⁴⁄₃π R₂³

Substituting R₂ = (R₁)/(2) into the equation: 4π R₁² = ⁴⁄₃π ((R₁)/(2))³ 4π R₁² = (4π)/(3) · (R₁³)/(8)

Dividing both sides by 4π R₁²: 1 = (R₁)/(3 × 8) 1 = (R₁)/(24) R₁ = 24

Explanation

In the right-angled triangle formed by the kite's position, the point on the ground directly below it, and the point where the thread is held, the thread acts as the hypotenuse. The vertical height h is the side opposite to the given angle of elevation. Using the trigonometric sine ratio:

sin(θ) = (Opposite)/(Hypotenuse)

Given that the angle θ = 60° and the length of the thread (hypotenuse) is 80 m, the relationship is:

sin(60°) = ʰ⁄₈₀

Using the standard trigonometric value sin(60°) = √3{2}:

√3{2} = ʰ⁄₈₀

h = 80 × √3{2}

h = 40√3 m

Explanation

The total work performed is governed by the formula M₁ × D₁ × E₁ = M₂ × D₂ × E₂, where M represents the number of people, D represents the number of days, and E represents the efficiency level. In the first scenario, 10 people working for 20 days with a standard efficiency of 1 unit results in a total work requirement of 10 × 20 × 1 = 200 units.

In the second scenario, the number of people increases to 20 and their efficiency is doubled to 2 units per person. To find the number of days D₂ required to complete the same 200 units of work, we set up the equation 200 = 20 × D₂ × 2. This simplifies to 200 = 40 × D₂. Solving for D₂ gives 200 / 40, which results in 5 days.

Explanation

In any triangle, the sum of the lengths of any two sides is greater than the length of the third side. This fundamental property is known as the triangle inequality theorem. In quadrilateral ABCD, the diagonals AC and BD create several triangles. By applying the theorem to the triangles defined by these diagonals, we obtain the following relationships:

AB + BC > AC from △ABC CD + DA > AC from △ADC BC + CD > BD from △BCD DA + AB > BD from △ABD

By adding these four inequalities together, the terms on the left side combine to form: 2AB + 2BC + 2CD + 2DA > 2AC + 2BD

Factoring out the constant 2 from both sides of the inequality gives: 2(AB + BC + CD + DA) > 2(AC + BD)

Dividing both sides of the expression by 2 simplifies the relation to: AB + BC + CD + DA > AC + BD

This mathematical derivation proves that the perimeter of a quadrilateral is always strictly greater than the sum of its diagonals.

Explanation

To find the equivalent discount of two successive discounts x% and y%, we use the net percentage change formula. If a value is changed by a% and then by b%, the effective percentage change is given by a + b + ᵃᵇ⁄₁₀₀. Since a discount represents a negative percentage change in price, we substitute a = -x and b = -y into the formula.

Net change = -x + (-y) + ⁽⁻ˣ⁾⁽⁻ʸ⁾⁄₁₀₀ Net change = -x - y + ˣʸ⁄₁₀₀ Net change = -(x + y - ˣʸ⁄₁₀₀)

The negative sign indicates a total reduction or discount. Therefore, the single equivalent discount percentage is (x + y - ˣʸ⁄₁₀₀)%. This can also be visualized by assuming an initial price of 100. The first discount reduces the price by x, leaving 100 - x. The second discount y% is applied to this remainder, reducing it by y - ˣʸ⁄₁₀₀. Adding the two reductions gives the total discount of x + y - ˣʸ⁄₁₀₀.

Explanation

The sweets are distributed between A and B in the ratio 3:4. Let the shares of A and B be 3x and 4x respectively. It is given that A received 36 sweets, so we can set up the equation 3x = 36. Solving for x gives x = 36 / 3 = 12. The total number of sweets in the box is the sum of the shares of A and B, which is 3x + 4x = 7x. Substituting the value of x, the total number of sweets is 7 × 12 = 84.

Explanation

Let the cost price be 4x and the selling price be 5x. The profit earned is the difference between the selling price and the cost price, which is 5x - 4x = x. The profit percentage is calculated using the standard formula:

Profit Percentage = (Profit)/(Cost Price) × 100

Substituting the respective values into the formula: Profit Percentage = ˣ⁄₄ₓ × 100 = ¼ × 100 = 25

Thus, when the ratio of cost price to selling price is 4 : 5, the profit percentage is 25.

Explanation

To find the percentage decrease in the value of the stock, we use the formula for percentage change:

Percentage Decrease = (Initial Value - Final Value)/(Initial Value) × 100

Given the initial value of the stock is Rs. 25 and the final value is Rs. 21, the absolute decrease in the price per share is:

25 - 21 = 4

Applying these values to the percentage formula:

Percentage Decrease = ⁴⁄₂₅ × 100

Since 100 divided by 25 equals 4, the calculation simplifies to:

4 × 4 = 16%

Explanation

Let the time spent traveling on foot be t hours. Consequently, the time spent traveling on the bicycle is 9 - t hours. Using the relationship Distance = Speed × Time, the total distance of 61 km can be expressed as: 4t + 9(9 - t) = 61 4t + 81 - 9t = 61 -5t = 61 - 81 -5t = -20 t = 4 The time spent traveling on foot is 4 hours. Therefore, the distance traveled on foot is 4 km/hr × 4 hours = 16 km.

Alternatively, applying the method of alligation, the average speed for the entire journey is 61/9 km/hr. The ratio of the time spent on foot to the time spent on the bicycle is (9 - 61/9) : (61/9 - 4), which simplifies to 20/9 : 25/9 or 4:5. Given the total travel time is 9 hours, the time spent on foot is exactly 4 hours. Multiplying this by the walking speed of 4 km/hr yields a distance of 16 km.

Explanation

For the sum of squares of real numbers to be equal to zero, each individual squared term must independently equal zero. This provides two equations: (2x - y)² = 0 ⟹ 2x - y = 0 ⟹ 2x = y ⟹ (x)/(y) = ½ (3y - 2z)² = 0 ⟹ 3y - 2z = 0 ⟹ 3y = 2z ⟹ (y)/(z) = ⅔

From these results, we have the ratios x:y = 1:2 and y:z = 2:3. Since the term for y is represented by the same value (2) in both individual ratios, they can be combined directly to form the continuous ratio x:y:z = 1:2:3.

Explanation

In a cyclic quadrilateral, the sum of opposite angles is always 180°. Let the four consecutive angles be A, B, C, and D. Given the ratio of the first three consecutive angles A:B:C = 1:4:5, we can represent these angles as x, 4x, and 5x respectively.

Since A and C are opposite angles, their sum must satisfy A + C = 180°. Substituting the ratio values, we get x + 5x = 180°, which simplifies to 6x = 180°, resulting in x = 30°.

The measure of the second angle B is 4x, which is 4 × 30° = 120°. In a cyclic quadrilateral, the second and fourth angles B and D are also opposite and must sum to 180°. Therefore, B + D = 180°. Substituting the value of B, we have 120° + D = 180°, which gives D = 180° - 120° = 60°.

Explanation

Let the average contribution of the 9 students be A. The total contribution of the 9 students is 9A. According to the problem, seven students gave Rs. 50 each, the eighth student gave A + 50, and the ninth student gave A + 90. Equating the sum of individual contributions to the total average-based sum:

7 × 50 + (A + 50) + (A + 90) = 9A 350 + 2A + 140 = 9A 490 + 2A = 9A 7A = 490 A = 70

Alternatively, using the concept of deviation, the combined surplus of the last two students over the average is 50 + 90 = 140. This surplus must compensate for the total deficit of the first seven students compared to the average. The deficit for each of the seven students is 140 / 7 = 20. Since they each paid Rs. 50, the average is 50 + 20 = 70.

Explanation

Given x = 2 + √3, we find its reciprocal 1/x by rationalizing the denominator: 1/x = (1)/(2 + √(3)) × 2 - √3{2 - √3} = 2 - √3{4 - 3} = 2 - √3

The sum of x and its reciprocal is: x + 1/x = (2 + √3) + (2 - √3) = 4

To find the value of the given expression (x² - x + 1)/(x² + x + 1), we divide both the numerator and the denominator by x: (x²)/(x) - ˣ⁄ₓ + ¹⁄ₓ{(x²)/(x) + ˣ⁄ₓ + ¹⁄ₓ} = (x - 1 + 1/x)/(x + 1 + 1/x) = ((x + 1/x) - 1)/((x + 1/x) + 1)

Substituting x + 1/x = 4 into the simplified expression: (4 - 1)/(4 + 1) = ⅗

Explanation

In △ ABC, the sum of all interior angles is 180°. Thus, ∠BAC = 180° - (∠B + ∠C) = 180° - (35° + 65°) = 180° - 100° = 80°. Since AD is the bisector of ∠BAC, it divides the angle into two equal parts, meaning ∠BAD = ∠CAD = 80° / 2 = 40°.

In △ ABD, the sum of angles is also 180°. Therefore, ∠ADB = 180° - (∠B + ∠BAD) = 180° - (35° + 40°) = 180° - 75° = 105°. Alternatively, using the exterior angle theorem for △ ADC, the exterior angle ∠ADB is equal to the sum of the two opposite interior angles, ∠CAD + ∠ACD = 40° + 65° = 105°.

Explanation

In a circle, the tangent at any point is perpendicular to the radius or diameter at the point of contact. Since AB is the diameter and the tangent is drawn at point A, the angle ∠BAC = 90°. Therefore, △ ABC is a right-angled triangle with AB and AC as the legs and BC as the hypotenuse.

Using the Pythagoras theorem: BC² = AB² + AC² BC² = 8² + 6² BC² = 64 + 36 BC² = 100 BC = √(100) = 10 cm

The lengths 6, 8, and 10 form a Pythagorean triplet, as they are a multiple of the basic 3-4-5 triplet. Thus, the distance from point B to point C is exactly 10 cm.

Explanation

In the right-angled triangle formed by the tower and the level ground, the height of the tower is the perpendicular and the distance from point P to the foot of the tower is the base. The relationship between these sides and the angle of elevation is given by the tangent trigonometric ratio: tan(θ) = (Perpendicular)/(Base)

Substituting the given height of 100 m and the angle of elevation of 30°: tan(30°) = (100)/(d)

The value of tan(30°) is (1)/(√(3)). Therefore: (1)/(√(3)) = (100)/(d) d = 100 × √3

Using the value √3 = 1.73 as specified in the question: d = 100 × 1.73 = 173 m

Explanation

Any integer between 100 and 1000 is a three-digit number, which can be expressed in decimal notation as 100x + 10y + z, where x, y, and z are the digits. The sum of these digits is x + y + z. Subtracting the sum of the digits from the original integer gives:

(100x + 10y + z) - (x + y + z)

By grouping like terms, the expression simplifies to: 99x + 9y

This can be rewritten by factoring out 9: 9(11x + y)

Since x and y are integers, the term (11x + y) is also an integer. This demonstrates that the resulting number is always a multiple of 9. Consequently, the result is always divisible by 9 regardless of the specific digits of the original integer. This mathematical property stems from the fact that any power of 10 is congruent to 1 modulo 9, meaning 10ⁿ ≡ 1 9, which ensures that N - S(N) is always divisible by 9 for any positive integer N.

Explanation

In simple interest, the interest earned is directly proportional to the time duration. If a sum of money P doubles in 6 years, the interest earned in that period is equal to the principal, I₆ = 2P - P = P.

Using the formula for simple interest I = (P · R · T)/(100), the interest earned in 12 years (I₁₂) is: I₁₂ = (P · R · 12)/(100) = 2 × ((P · R · 6)/(100)) = 2 × I₆

Substituting the value of I₆: I₁₂ = 2 × P = 2P

The total amount after 12 years is the sum of the principal and the interest earned over that period: A = P + I₁₂ = P + 2P = 3P

Explanation

To determine the value of the first part, the fractional ratio ½ : ⅔ : ¾ must be converted into a whole number ratio. This is done by multiplying each term by the Least Common Multiple (LCM) of the denominators 2, 3, and 4, which is 12.

The first term becomes ½ × 12 = 6, the second term becomes ⅔ × 12 = 8, and the third term becomes ¾ × 12 = 9. Thus, the ratio is 6 : 8 : 9.

The total number of units in the ratio is 6 + 8 + 9 = 23. Given the total sum is Rs 782, the value of a single unit is ⁷⁸²⁄₂₃ = 34. Since the first part corresponds to 6 units, its value is 6 × 34 = 204.

Explanation

According to the data table, the production of E type cycles in the year 2001 is 80 units (in thousands). The production of B type cycles in the year 2000 is 160 units (in thousands). To find what percent the production of E in 2001 was of the production of B in 2000, the calculation is as follows:

Required Percentage = (Production of E in 2001)/(Production of B in 2000) × 100

Required Percentage = ⁸⁰⁄₁₆₀ × 100

Required Percentage = ½ × 100 = 50%

The production level of 80 is exactly half of 160, which corresponds to 50 percent.

Explanation

From the given equation (4a - 3)² = 0, we find that 4a - 3 = 0, which implies 4a = 3.

The expression 64a³ - 48a² + 12a + 13 can be rewritten in terms of 4a as follows: (4a)³ - 3(4a)² + 3(4a) + 13

Substituting 4a = 3 into the expression: 3³ - 3(3²) + 3(3) + 13 27 - 3(9) + 9 + 13 27 - 27 + 9 + 13 22

Alternatively, using the identity (x - 1)³ = x³ - 3x² + 3x - 1 with x = 4a, the expression can be viewed as (4a - 1)³ + 14. Substituting 4a = 3 gives (3 - 1)³ + 14 = 2³ + 14 = 8 + 14 = 22.

Explanation

The expression can be solved using the trigonometric identity cos(90° - θ) = sin θ. Here, 70° can be written as (90° - 20°), so cos 70° = cos(90° - 20°) = sin 20°. Substituting this into the given expression, we get:

cos² 20° + cos² 70° = cos² 20° + sin² 20°

According to the fundamental Pythagorean identity sin² θ + cos² θ = 1, for any angle θ, the sum of the squares of the sine and cosine of the same angle is always 1. Thus, cos² 20° + sin² 20° = 1.

Explanation

The expression a(a² - 3a + 3) can be expanded as a³ - 3a² + 3a. This closely resembles the binomial expansion of (a - 1)³, which is given by the formula (x - y)³ = x³ - 3x²y + 3xy² - y³. Substituting x = a and y = 1, we get (a - 1)³ = a³ - 3a² + 3a - 1.

By adding 1 to both sides, the expression simplifies to a³ - 3a² + 3a = (a - 1)³ + 1. Substituting the given value a = 101 into this simplified form:

(101 - 1)³ + 1 = 100³ + 1

Since 100³ = 1000000, the final value is 1000000 + 1 = 1000001.

Explanation

Pradhan Mantri Ujjwala Yojana (PMUY) was announced in the Union Budget 2016-17 with the objective of providing clean cooking fuel to underprivileged households. The scheme aims to provide deposit-free LPG (Liquefied Petroleum Gas) connections to women belonging to Below Poverty Line (BPL) households. By replacing traditional and hazardous cooking fuels such as firewood, coal, and dried cow-dung cakes with clean energy, the initiative seeks to protect women and children from the adverse health effects of indoor air pollution. This scheme is implemented by the Ministry of Petroleum and Natural Gas and serves as a major step toward women's empowerment and environmental conservation.

Explanation

Agronomy is derived from the Greek words agros, meaning field, and nomos, meaning to manage. It is a comprehensive branch of agricultural science that encompasses the study of the principles and practices of soil, water, and crop management. In a broad agricultural context, it involves the application of scientific techniques to manage the environment of the field for the production of food and fiber. This practice effectively integrates the cultivation of field crops with the management of livestock that interact with the land, ensuring a sustainable and productive ecosystem that involves raising both plants and animals.

Explanation

The algebraic identity for the difference of squares of two numbers x and y is given by the formula x² - y² = (x + y)(x - y).

In this problem, the sum of the two integers is provided as x + y = 80 and the difference between them is provided as x - y = 20. By substituting these values directly into the identity, we get:

x² - y² = 80 × 20 = 1600

Explanation

The difference between the selling prices of an article sold at two different profit percentages is equivalent to the difference in those profit percentages calculated on the cost price.

Difference in profit percentage = 4% - 3% = 1%

Given that the absolute difference in selling prices is ₹ 3, this 1% of the cost price (CP) must be equal to ₹ 3. Using the percentage formula:

1% of CP = 3 ¹⁄₁₀₀ × CP = 3 CP = 3 × 100 = 300

Explanation

Total students are 1500. The percentage of girls is 44%, which means the percentage of boys is 56%. The monthly fee for each boy is ₹ 540. The monthly fee for each girl is 25% less than the boy's fee: Girl fee = 540 × (1 - 0.25) = 540 × 0.75 = 405 The total fee collection T is the sum of fees from boys and girls: T = (1500 × 0.56 × 540) + (1500 × 0.44 × 405) To simplify the arithmetic, factor out 1500 × 540: T = 1500 × 540 × (0.56 + 0.44 × 0.75) T = 810000 × (0.56 + 0.33) T = 810000 × 0.89 T = 8100 × 89 Calculating 81 × 89 using the identity (a-b)(a+b) = a² - b²: 81 × 89 = (85 - 4)(85 + 4) = 85² - 4² = 7225 - 16 = 7209 Multiplying by the factor of 100 gives 720900.

Explanation

To calculate the time required for a train to cross a bridge, the total distance covered is the sum of the length of the train and the length of the bridge. Therefore, the total distance D is 110 + 132 = 242 metres.

The speed of the train is given in km/hr and must be converted to m/s to match the units of distance. Using the conversion factor ⁵⁄₁₈, the speed v is 72 × ⁵⁄₁₈ = 20 m/s.

Using the formula for time t = (Distance)/(Speed), we substitute the values to get t = ²⁴²⁄₂₀. Dividing 242 by 20 results in 12.1 seconds.

Explanation

The expression is in the form of the algebraic identity (a³ + b³)/(a² - ab + b²). According to the sum of cubes formula, a³ + b³ = (a + b)(a² - ab + b²). By rearranging this identity, we get:

(a³ + b³)/(a² - ab + b²) = a + b

In the given problem, let a = 0.73 and b = 0.27. Substituting these values into the simplified identity:

0.73 + 0.27 = 1.00

The terms in the denominator exactly match the quadratic factor of the numerator's expansion, allowing them to cancel out and leaving only the sum of the two base decimal values.

Explanation

Given the equation x + ¹⁄ₓ = -2, we can multiply both sides by x to obtain x² + 1 = -2x. Rearranging the terms gives x² + 2x + 1 = 0, which is a perfect square trinomial equivalent to (x + 1)² = 0. Solving for x yields x = -1.

Substituting x = -1 into the required expression x⁷ + (1)/(x⁷), we note that any negative number raised to an odd power remains negative. Since 7 is an odd integer, (-1)⁷ = -1. The expression evaluates to -1 + ¹⁄₋₁ = -1 - 1 = -2.

Explanation

The angles subtended by the arcs are proportional to the lengths of the arcs, and their ratio is given as 5:3. Let the two angles be 5x and 3x. The problem states that the smaller angle is 45 degrees, which corresponds to the smaller part of the ratio. Therefore, we can establish the equation 3x = 45°. Solving for x gives x = (45°)/(3) = 15°. To find the measure of the other angle, we multiply the larger part of the ratio by the value of x. This results in 5 × 15° = 75°.

Explanation

The orthocentre is the point of intersection of the three altitudes of a triangle. In an obtuse-angled triangle, one interior angle is greater than 90°. To construct the altitudes from the two acute-angled vertices, the sides opposite to them must be extended. Because these altitudes are perpendicular to the extended base lines, they diverge from the interior of the triangle and eventually intersect at a point in the exterior region. This point of intersection lies outside the triangle, specifically behind the vertex containing the obtuse angle.

Explanation

The value of a can be found by substituting the specific trigonometric ratios into the equation a sin 45° cos 45° tan 60° = tan² 45° - cos 60°. The standard values for these ratios are sin 45° = (1)/(√(2)), cos 45° = (1)/(√(2)), tan 60° = √3, tan 45° = 1, and cos 60° = ½. Substituting these into the equation yields:

a ( (1)/(√(2)) ) ( (1)/(√(2)) ) √3 = 1² - ½

Simplifying the left-hand side, the product (1)/(√(2)) · (1)/(√(2)) equals ½, so the expression becomes a √3{2}. On the right-hand side, 1 - ½ equals ½. Setting the two sides equal gives the linear equation a √3{2} = ½. Multiplying both sides by 2 eliminates the denominator, leaving a √3 = 1. Solving for a results in a = (1)/(√(3)).

Explanation

The time taken by Sohan can be determined using the work efficiency method. Let the total work be the Least Common Multiple (LCM) of 16 and 24, which is 48 units. The combined efficiency of Ganesh, Ram, and Sohan is 48 / 16 = 3 units per day. The combined efficiency of Ganesh and Ram is 48 / 24 = 2 units per day.

Sohan efficiency is calculated by subtracting the efficiency of Ganesh and Ram from the efficiency of all three together: 3 - 2 = 1 unit per day. The number of days Sohan alone takes to finish the work is the total work divided by his efficiency: 48 / 1 = 48 days. This can also be expressed using the formula 1/T_S = 1/T_G_R_S - 1/T_G_R, where 1/16 - 1/24 = (3 - 2) / 48 = 1/48.

Explanation

The height h of the right circular cone is found using the Pythagorean theorem relationship between the slant height l and the radius r: h = √(l² - r²). Substituting the given values r = 6 cm and l = 10 cm, the height is h = √(10² - 6²) = √(100 - 36) = √(64) = 8 cm.

The volume V of the cone is calculated using the formula V = ⅓ π r² h. Using π = 22/7:

V = ⅓ × ²²⁄₇ × 6² × 8 V = ⅓ × ²²⁄₇ × 36 × 8 V = ²²⁄₇ × 12 × 8 V = ²¹¹²⁄₇

Performing the division, V ≈ 301.71428... cm³. When rounded to two decimal places, the volume is 301.71 cm³.

Explanation

The selling price (SP) is the price charged by the shopkeeper after the discount. In this case, SP = ₹ 740 and the discount rate is 20% of the marked price (MP).

The formula relating the selling price and the marked price is: SP = MP × (1 - (Discount %)/(100))

Substituting the given values into the formula: 740 = MP × (1 - ²⁰⁄₁₀₀) 740 = MP × ⁸⁰⁄₁₀₀ 740 = MP × 0.8

To find the marked price, rearrange the equation: MP = (740)/(0.8) MP = ⁷⁴⁰⁰⁄₈ MP = 925

Alternatively, if a 20% discount is given, the customer pays 80% of the marked price. Since 80% of the MP equals ₹ 740, then 100% of the MP is calculated as (740 / 80) × 100 = 9.25 × 100 = 925.

Explanation

The speed of an object is determined by the formula Speed = (Distance)/(Time). To compare the two speeds, they must be converted into the same units. The truck covers 550 metres in 1 minute, so its speed is 550 m/min. The bus covers 33 km in 3/4 hours. Converting these values to metres and minutes, 33 km equals 33,000 metres and 3/4 hours equals 45 minutes. Thus, the speed of the bus is ³³⁰⁰⁰⁄₄₅ = ²²⁰⁰⁄₃ m/min. The ratio of the truck's speed to the bus's speed is 550 : ²²⁰⁰⁄₃. Multiplying both parts of the ratio by 3 to eliminate the fraction gives 1650 : 2200. Dividing both terms by 550 results in the simplified ratio of 3 : 4.

Explanation

Let the present ages of A, B, and C be A, B, and C years respectively. The average age of A, B, and C seven years ago was 51 years, so the sum of their ages at that time was 51 × 3 = 153 years. To find the sum of their present ages, we add 7 years for each person: 153 + (3 × 7) = 174 years.

The problem states that A = B + 3 and B = C + 3. Substituting the value of B into the first equation gives A = (C + 3) + 3 = C + 6. Substituting these expressions into the sum of their present ages: (C + 6) + (C + 3) + C = 174 3C + 9 = 174 3C = 165 C = 55

Using C = 55, we find B = 55 + 3 = 58 and A = 58 + 3 = 61. Thus, the present ages of A, B, and C are 61, 58, and 55 years respectively.