SSC CGL Tier 1 2016 Quantitative Aptitude Questions with Solutions

Explanation

According to the Midpoint Theorem, the line segment connecting the midpoints of two sides of a triangle is parallel to the third side and equal to half its length. For △ ABC with midpoints D, E, F of sides BC, CA, AB, it follows that EF = ½ BC, FD = ½ CA, and DE = ½ AB. Since the corresponding sides are proportional, △ DEF is similar to △ ABC with a similarity ratio of k = ½. The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides. Therefore, the ratio of Area(△ DEF) to Area(△ ABC) is (½)² = ¼. Geometrically, the segments DE, EF, FD divide the triangle ABC into four congruent triangles: △ AFE, △ FBD, △ EDC, and △ DEF. Each of these four triangles has an area exactly equal to one-fourth of the area of the original triangle △ ABC.

Explanation

The radius OA is perpendicular to the tangent PA at the point of contact A. Therefore, triangle OAP is a right-angled triangle with the distance from the center PO acting as the hypotenuse. Using the Pythagoras theorem:

OA² + PA² = PO²

Substituting the given values PA = 4 cm and PO = 5 cm:

r² + 4² = 5² r² + 16 = 25 r² = 9 r = 3 cm

This follows the standard Pythagorean triplet (3, 4, 5), where the side lengths of a right triangle are in that specific ratio. Since the hypotenuse is 5 cm and one side is 4 cm, the remaining side representing the radius must be 3 cm.

Explanation

The trigonometric identity for complementary angles states that sec θ = csc(90° - θ). This implies that if sec A = csc B, then A + B = 90° (for acute angles). Given the equation sec(4x - 50°) = csc(50° - x), we equate the sum of the angles to 90 degrees:

(4x - 50°) + (50° - x) = 90°

Simplifying the expression by combining like terms:

4x - x - 50° + 50° = 90° 3x = 90°

Solving for x:

x = (90°)/(3) x = 30°

Substituting x = 30° back into the original expressions confirms that the angles are 70° and 20°, which are indeed complementary since 70° + 20° = 90°.

Explanation

The volume of water that overflows from the vessel is equal to the total volume of the spherical balls dropped into it. The volume of the conical vessel is given by V = ⅓π r² h. Since 25% of the water overflows, the volume of the overflowed water is: Vₒᵥₑᵣ_fₗₒ_w = ¼ × ⅓ × π × 5² × 8 = (50π)/(3)

Let n be the number of spherical balls. The volume of a single spherical ball with radius rₛ = ½ cm is: V_bₐₗₗ = ⁴⁄₃ π rₛ³ = ⁴⁄₃ π (½)³ = ⁴⁄₃ π × ⅛ = (π)/(6)

Equating the total volume of n balls to the volume of the overflowed water: n × (π)/(6) = (50π)/(3) n = (50π)/(3) × (6)/(π) = 50 × 2 = 100

Explanation

The total work is considered as one whole unit. Since A and B together complete ⁹⁄₁₁ of the work, the remaining fraction of the work performed by C is: Work done by C = 1 - ⁹⁄₁₁ = ²⁄₁₁

Payment is distributed in direct proportion to the share of work completed. Given the total contract value is ₹ 440, the share of C is calculated by multiplying the total amount by the fraction of work C completed: Share of C = 440 × ²⁄₁₁ = 40 × 2 = 80

Explanation

The selling price (SP) is calculated by subtracting the discount amount from the marked price (MP). Using the formula SP = MP × (1 - (D)/(100)), where D is the discount percentage:

SP = 7500 × (1 - ⁶⁄₁₀₀) SP = 7500 × ⁹⁴⁄₁₀₀ SP = 75 × 94 SP = 7050

Alternatively, the discount amount is 6% of 7500, which is 0.06 × 7500 = 450. Subtracting this from the marked price gives 7500 - 450 = 7050.

Explanation

Let the two numbers be 3x and 4x based on the given ratio 3:4. According to the problem, the sum of their cubes is 5824. This can be expressed as the equation (3x)³ + (4x)³ = 5824. Expanding the terms gives 27x³ + 64x³ = 5824. Combining the like terms results in 91x³ = 5824. Dividing both sides by 91 gives x³ = 64. Taking the cube root of both sides, we find x = 4. The sum of the two numbers is 3x + 4x = 7x. Substituting the value of x, the sum is 7 × 4 = 28.

Explanation

The conversion between the loss percentage on the selling price (Lₛ) and the loss percentage on the cost price (x) depends on the fundamental relation where the cost price equals the selling price plus the loss amount. The specific formula used to calculate x directly from Lₛ is: x = Lₛ{100 + Lₛ} × 100

Given the loss on the selling price is 20%, we substitute Lₛ = 20 into the equation: x = (20)/(100 + 20) × 100 x = ²⁰⁄₁₂₀ × 100

Simplifying the fraction: x = ⅙ × 100 = 16 ⅔

Explanation

To find the combined percentage of defective parts, first determine the total count of defective items from both motors. In the first motor of 120 parts, 5% are defective, which equals 120 × 0.05 = 6 parts. In the second motor of 80 parts, 10% are defective, which equals 80 × 0.10 = 8 parts.

The total number of defective parts is 6 + 8 = 14, and the total number of parts in both motors is 120 + 80 = 200. The combined percentage is calculated by dividing the total defects by the total parts and multiplying by 100:

Combined Percentage = ¹⁴⁄₂₀₀ × 100 = 7%

This can also be solved using the weighted average formula:

Average = ((n₁ × p₁) + (n₂ × p₂))/(n₁ + n₂) = ((120 × 5) + (80 × 10))/(120 + 80) = (600 + 800)/(200) = ¹⁴⁰⁰⁄₂₀₀ = 7

Explanation

Using the power of a power rule (aᵐ)ⁿ = aᵐ^×ⁿ, the left side of the equation simplifies as follows: (2³)² = 2³^ײ = 2⁶ = 64

The given equation is (2³)² = 4ˣ. Substituting the calculated value: 64 = 4ˣ

Expressing 64 as a power of 4, we have 64 = 4³. Therefore: 4³ = 4ˣ

By comparing the exponents of the same base, we find x = 3. The question asks for the value of 3x: 3x = 3 × 3 = 9

Explanation

To calculate the time taken to cross a bridge, the train must cover a total distance equal to the sum of its own length and the length of the bridge. The total distance D is 200 + 175 = 375 meters. The speed of the train is provided in km/hr and needs to be converted to m/s using the conversion factor ⁵⁄₁₈. The speed v is 54 × ⁵⁄₁₈ = 15 m/s. Using the formula Time = (Distance)/(Speed), the time t is ³⁷⁵⁄₁₅ = 25 seconds.

Explanation

A right-angled triangle must satisfy the Pythagorean theorem, which states that the sum of the squares of the two shorter sides must equal the square of the longest side, represented by the formula a² + b² = c². For the ratio 13:12:5, the longest side is 13. Squaring the two shorter sides gives 5² = 25 and 12² = 144. Their sum is 25 + 144 = 169. The square of the longest side is 13² = 169. Since the relation 5² + 12² = 13² holds true, these sides form a Pythagorean triplet and thus define a right-angled triangle.

Explanation

According to the fundamental theorem of geometry, there is one and only one circle passing through three given non-collinear points. For a circle to pass through three points A, B, and C, its center must be equidistant from all three points. The locus of points equidistant from A and B is the perpendicular bisector of the line segment AB. Similarly, the locus of points equidistant from B and C is the perpendicular bisector of the line segment BC. Since the points are non-collinear, the line segments AB and BC are not parallel, which ensures that their perpendicular bisectors will intersect at exactly one unique point. This point of intersection is the circumcenter of the triangle formed by the three points. Because the center and the radius (the distance from the center to any of the points) are uniquely determined, only one unique circle can be drawn.

Explanation

The trigonometric identity for complementary angles states that cos θ = sin(90° - θ). In this expression, we have θ = 53°. Applying the identity, cos 53° = sin(90° - 53°) = sin 37°. Substituting this value into the given expression, cos 53° - sin 37° = sin 37° - sin 37° = 0.

Explanation

The given equation is 4x + ¹⁄ₓ = 5. By multiplying both sides of the equation by x, we derive the quadratic relation 4x² + 1 = 5x. The expression to be evaluated is (5x)/(4x² + 10x + 1). This expression can be rewritten by grouping the terms in the denominator as (5x)/((4x² + 1) + 10x). By substituting the previously derived value 4x² + 1 = 5x into the denominator, the expression becomes (5x)/(5x + 10x). Simplifying the denominator gives ⁵ˣ⁄₁₅ₓ. Since the problem states x ≠ 0, we can cancel x from the numerator and the denominator, leaving ⁵⁄₁₅, which simplifies to ⅓.

Explanation

The value of the expression C³ + (1)/(C³) can be determined using the algebraic identity a³ + b³ = (a + b)³ - 3ab(a + b). By substituting a = C and b = (1)/(C), the product ab becomes C × (1)/(C) = 1. The formula then simplifies to:

C³ + (1)/(C³) = (C + (1)/(C))³ - 3(C · (1)/(C))(C + (1)/(C))

C³ + (1)/(C³) = (C + (1)/(C))³ - 3(C + (1)/(C))

Given that C + (1)/(C) = √3, we substitute this value into the equation:

C³ + (1)/(C³) = (√3)³ - 3(√3)

Since (√3)³ = √3 × √3 × √3 = 3√3, the calculation becomes:

C³ + (1)/(C³) = 3√3 - 3√3 = 0

Explanation

A number is divisible by 9 if the sum of its digits is divisible by 9. For the number 451*603, the sum of the digits is 4 + 5 + 1 + * + 6 + 0 + 3 = 19 + *. The smallest digit that makes this sum a multiple of 9 is 8, since 19 + 8 = 27.

Explanation

The correct average can be determined by calculating the net change in the sum and dividing it by the total number of observations.

Correct Average = Old Average + (Correct Value - Incorrect Value)/(n)

Given that the old average is 35, the number of observations n is 9, the correct value is 81, and the misread value is 18, we substitute these into the formula:

Correct Average = 35 + (81 - 18)/(9) Correct Average = 35 + ⁶³⁄₉ Correct Average = 35 + 7 Correct Average = 42

Since the value was under-recorded (18 instead of 81), the total sum was lower than it should have been by 63. Distributing this deficit across the 9 observations results in an increase of 7 units to the initial average.

Explanation

The algebraic identity for the given expression is x³ + y³ + z³ - 3xyz = ½(x + y + z)[(x - y)² + (y - z)² + (z - x)²].

First, calculate the sum of the variables: x + y + z = 222 + 223 + 225 = 670

Next, find the squares of the differences between the terms: (x - y)² = (222 - 223)² = (-1)² = 1 (y - z)² = (223 - 225)² = (-2)² = 4 (z - x)² = (225 - 222)² = (3)² = 9

Substitute these values into the formula: x³ + y³ + z³ - 3xyz = ½ × 670 × (1 + 4 + 9) x³ + y³ + z³ - 3xyz = ½ × 670 × 14 x³ + y³ + z³ - 3xyz = 670 × 7 = 4690

Explanation

Given x = 3¹^/³ - 3⁻¹^/³, we cube both sides using the identity (a - b)³ = a³ - b³ - 3ab(a - b). By setting a = 3¹^/³ and b = 3⁻¹^/³, the expansion is:

x³ = (3¹^/³)³ - (3⁻¹^/³)³ - 3(3¹^/³ · 3⁻¹^/³)(3¹^/³ - 3⁻¹^/³)

Substituting the simplified terms where (3¹^/³)³ = 3, (3⁻¹^/³)³ = ⅓, and the product 3¹^/³ · 3⁻¹^/³ = 1:

x³ = 3 - ⅓ - 3(1)(x)

x³ = ⁸⁄₃ - 3x

Multiplying every term by 3 to eliminate the fraction:

3x³ = 8 - 9x 3x³ + 9x = 8

Explanation

When two circles touch each other internally, the distance between their centers is equal to the difference between their radii. Let R be the radius of the larger circle and r be the radius of the smaller circle. The distance d between the centers is given by the formula d = R - r. Substituting the given values where r = 6 cm and d = 3 cm, we get 3 = R - 6. Solving for R yields R = 3 + 6, which results in a radius of 9 cm for the larger circle.

Explanation

In an equilateral triangle PQR, all interior angles are 60°. When MN is drawn parallel to QR with M on PQ and N on PR, △ PMN is formed. Since MN is parallel to QR, the corresponding angles are equal, so ∠ PMN = ∠ PQR = 60° and ∠ PNM = ∠ PRQ = 60°. Since △ PMN has three angles of 60°, it is an equilateral triangle. In any equilateral triangle, the lengths of all sides are equal, which means PM = MN = PN. Given that the length of PN is 6 cm, the length of MN must also be 6 cm.

Explanation

The given expression is csc θ + sin θ = ⁵⁄₂. We use the algebraic identity (a - b)² = (a + b)² - 4ab to find the required value. Let a = csc θ and b = sin θ. Since csc θ and sin θ are reciprocals, their product ab = csc θ · sin θ = 1.

Substituting the known values into the identity: (csc θ - sin θ)² = (csc θ + sin θ)² - 4(csc θ · sin θ) (csc θ - sin θ)² = (⁵⁄₂)² - 4(1) (csc θ - sin θ)² = ²⁵⁄₄ - 4 (csc θ - sin θ)² = (25 - 16)/(4) = ⁹⁄₄

Taking the square root of both sides, we get csc θ - sin θ = ± ³⁄₂. Because csc θ + sin θ is positive and their product is positive, both terms must be positive. For any real θ where sin θ is positive, csc θ ≥ 1 and sin θ ≤ 1, which implies that csc θ - sin θ ≥ 0. Thus, the value is ³⁄₂.

Explanation

In this problem, the tower and the positions of the two men form two right-angled triangles sharing a common side, which is the height of the tower h = 50 m. Let x₁ be the distance of the first man from the tower and x₂ be the distance of the second man. Since they are on opposite sides, the total distance between them is x₁ + x₂.

For the first man with an angle of elevation of 30°, we use the relation tan(30°) = (Height)/(Base) = (50)/(x₁). This gives x₁ = (50)/(tan(30°)) = 50√3 m.

For the second man with an angle of elevation of 45°, the relation is tan(45°) = (50)/(x₂). Since tan(45°) = 1, we have x₂ = 50 m.

The total distance between the two men is 50√3 + 50 = 50(√3 + 1) m. Substituting the given value √3 = 1.73, the distance becomes 50(1.73 + 1) = 50 × 2.73 = 136.5 m.

Explanation

The volume of a cone is given by the formula V = ⅓ π r² h, where r is the radius and h is the height. Since ⅓ π is a constant, the volume is directly proportional to r² h. When both the radius and the height are increased by 20%, the new radius becomes 1.2r and the new height becomes 1.2h.

The new volume V' can be calculated as follows: V' = ⅓ π (1.2r)² (1.2h) V' = ⅓ π (1.44r²) (1.2h) V' = 1.728 × (⅓ π r² h) V' = 1.728V

The fractional increase in volume is 1.728 - 1 = 0.728. To find the percentage increase, multiply by 100, resulting in 72.8%. Alternatively, using the method of successive percentage increases for the three dimensions involved (r, r, h), the net change is calculated by first finding the effect on r², which is 20 + 20 + (20 × 20)/(100) = 44%. Combining this 44% increase with the 20% increase in height yields 44 + 20 + (44 × 20)/(100) = 64 + 8.8 = 72.8%.

Explanation

The clock gains 15 minutes in a 24-hour period. To find the gain per hour, we divide the total daily gain by 24: Gain per hour = ¹⁵⁄₂₄ = ⅝ minutes

The time interval from 12 noon to 4 AM the next day consists of 12 hours (noon to midnight) plus 4 hours (midnight to 4 AM), totaling 16 hours.

The total gain over these 16 hours is calculated by multiplying the hourly gain by the elapsed time: Total Gain = 16 × ⅝ = 2 × 5 = 10 minutes

Since the clock gains 10 minutes over the 16-hour period, it will show 10 minutes past the actual time. Therefore, at 4 AM, the clock will indicate 4.10 AM.

Explanation

The selling price (SP) after successive discounts can be calculated using the formula SP = MP × (1 - (d₁)/(100)) × (1 - (d₂)/(100)), where MP is the marked price and d₁, d₂ are the discount percentages. Given MP = 4950, d₁ = 20%, and d₂ = 15%, the calculation is as follows:

SP = 4950 × (1 - ²⁰⁄₁₀₀) × (1 - ¹⁵⁄₁₀₀) SP = 4950 × 0.80 × 0.85 SP = 3960 × 0.85 SP = 3366

Alternatively, the effective discount rate for two successive discounts a% and b% is a + b - ᵃᵇ⁄₁₀₀. Substituting the values, the effective discount is 20 + 15 - (20 × 15)/(100) = 35 - 3 = 32%. The selling price is therefore 100% - 32% = 68% of the marked price. Calculating 68% of 4950 results in 0.68 × 4950 = 3366.

Explanation

The problem states that 15% of x is three times 10% of y. This relationship can be written as an algebraic equation: ¹⁵⁄₁₀₀x = 3 × ( ¹⁰⁄₁₀₀y )

Multiplying both sides of the equation by 100 to eliminate the denominators gives: 15x = 3 × 10y 15x = 30y

To find the ratio x:y, we divide x by y and simplify the numerical fraction: (x)/(y) = ³⁰⁄₁₅ (x)/(y) = ²⁄₁

By converting the fraction into a ratio format, we obtain x:y = 2:1.

Explanation

The bookseller wants to earn a profit equivalent to the cost of 50 books. The total cost price corresponds to the 500 books purchased. Therefore, the profit percentage is simply the ratio of the number of books representing the profit to the total number of books purchased, multiplied by 100.

Profit Percentage = ⁵⁰⁄₅₀₀ × 100

By simplifying the ratio, we get: ⅒ × 100 = 10%

Calculating using the absolute monetary value yields the same result. The cost price of one book is ²⁰⁰⁰⁰⁄₅₀₀ = ₹ 40. The desired profit is the cost of 50 books, which is 50 × 40 = ₹ 2000. Using the profit percentage formula: (Total Profit)/(Total Cost Price) × 100 = ²⁰⁰⁰⁄₂₀₀₀₀ × 100 = 10%

Explanation

The total percentage of the man's salary allocated to rent, living expenses, and savings is 20% + 60% + 10% = 90%. The remaining portion of the salary, which is spent on education, represents 100% - 90% = 10% of the total income. Given that this remaining amount is ₹ 30, we can find the total salary S using the relation: 10% of S = 30 ¹⁰⁄₁₀₀ × S = 30 S = (30 × 100)/(10) = 300

Explanation

The speed of sound is calculated using the basic formula for speed v = ᵈ⁄ₜ, where d is the distance travelled and t is the time taken.

The given distance is 6.64 km. Converting this into meters gives 6.64 × 1000 = 6640 m. The time taken to hear the sound is 20 seconds. Substituting these values into the formula:

v = (6640 m)/(20 s)

v = 332 m/s

Explanation

The interest rate is specified as Rs. 75 per Rs. 1000 per annum. To find the interest for one year on a principal of Rs. 2000, we multiply the base rate by two, as Rs. 2000 contains two units of Rs. 1000. This gives an annual interest of 75 × 2 = 150. Since the simple interest is required for a period of 2 years, the total interest is calculated as 150 × 2 = 300. Alternatively, expressing the rate as a percentage gives R = ⁷⁵⁄₁₀₀₀ × 100 = 7.5% per annum. Applying the formula for simple interest SI = (P × R × T)/(100), we get SI = (2000 × 7.5 × 2)/(100) = 20 × 15 = 300.

Explanation

To find the value of x² - 2x - 4 given x = 1 + √2 + √3, we first rearrange the equation by subtracting 1 from both sides to simplify the squaring process: x - 1 = √2 + √3

Next, we square both sides of the equation: (x - 1)² = (√2 + √3)²

Expanding both sides using the algebraic identities (a - b)² = a² - 2ab + b² and (a + b)² = a² + 2ab + b²: x² - 2x + 1 = (√2)² + 2(√2)(√3) + (√3)² x² - 2x + 1 = 2 + 2√6 + 3 x² - 2x + 1 = 5 + 2√6

To isolate the required expression x² - 2x - 4, we subtract 5 from both sides of the equation: x² - 2x + 1 - 5 = 2√6 x² - 2x - 4 = 2√6

Explanation

Given the equation (a+b)/(√(ab)) = ²⁄₁, we cross-multiply to obtain a + b = 2√(ab). By moving all terms to one side, the equation becomes a + b - 2√(ab) = 0. This is the expansion of the algebraic identity (√a - √b)² = 0. Taking the square root of both sides, we find √a - √b = 0, which simplifies to √a = √b. Squaring both sides yields a = b. Consequently, the value of the expression a - b is a - a = 0.

Explanation

The length of the altitude h of an equilateral triangle with side a is given by the formula h = √3{2} a. This relationship is derived from the Pythagorean theorem, where the altitude bisects the base into two equal segments of length a/2, forming a right-angled triangle with the side as the hypotenuse.

Substituting the given side length a = 16 cm into the formula: h = √3{2} × 16 h = 8√3 cm

Alternatively, using the properties of a 30°-60°-90° triangle, the altitude opposite the 60° angle is √3{2} times the hypotenuse. With the side of 16 cm acting as the hypotenuse, the altitude is 16 × √3{2} = 8√3 cm.

Explanation

The circumcentre of a triangle is the center of the circumcircle, which is the unique circle that passes through all three vertices of the triangle. By the geometric definition of a circle, the distance from the center to any point on its circumference is constant and is called the radius. In the case of Δ ABC with circumcentre O, the line segments AO, BO, and CO all represent the circumradius R of the triangle. Consequently, the relationship AO = BO = CO = R must hold true. Given that the length of AO is 8 cm, the length of BO is also 8 cm.

Explanation

The given equation is tan(θ + 15°) = √3. In trigonometry, the tangent of an angle equals √3 when the angle is 60°, since tan(60°) = √3. By equating the arguments of the tangent function, we have θ + 15° = 60°. Subtracting 15° from both sides gives θ = 60° - 15°, which simplifies to θ = 45°.

Explanation

To find the number of tiles required, we first convert the dimensions of the ceiling into centimeters. The length is 15 m 17 cm = (15 × 100) + 17 = 1517 cm, and the breadth is 9 m 2 cm = (9 × 100) + 2 = 902 cm.

The number of tiles needed to cover an area is calculated by dividing the total area of the ceiling by the area of a single square tile. The formula is:

Number of tiles = (Length × Breadth)/(Side × Side)

Substituting the given values, where the side of the square tile is 41 cm:

Number of tiles = (1517 × 902)/(41 × 41)

We can simplify this by dividing each dimension by the tile side: 1517 / 41 = 37 902 / 41 = 22

The total number of tiles is the product of these two quotients: 37 × 22 = 814

Explanation

The total number of fruits offered during the entire week is calculated by multiplying the weekly average by the number of days: 75 × 7 = 525. Similarly, the total number of fruits offered during the six days excluding Tuesday is calculated by multiplying the six-day average by those six days: 72 × 6 = 432. By subtracting the total for the six days from the total for the full week, we find the number of fruits offered on Tuesday: 525 - 432 = 93.

Explanation

The problem is based on the algebraic identity (u + v)³ = u³ + v³ + 3uv(u + v).

To arrive at the result (y²)/(x), we define the terms u and v such that their cubes and product align with the target expression. Let: u = x¹^/³ v = ( (y²)/(x) )¹^/³

Now, we calculate the cubes: u³ = (x¹^/³)³ = x v³ = [ ( (y²)/(x) )¹^/³ ]³ = (y²)/(x)

Next, we calculate the product uv: uv = x¹^/³ · ( (y²)/(x) )¹^/³ = ( x · (y²)/(x) )¹^/³ = (y²)¹^/³ = y²^/³

Substituting a = u + v into the identity (u + v)³ = u³ + v³ + 3uv(u + v), we get: a³ = x + (y²)/(x) + 3(y²^/³)(a)

Rearranging the terms to match the target expression a³ - 3ay - x: a³ - 3ay²^/³ - x = (y²)/(x)

Given the typographical errors in the original question (where x³ likely represents x¹^/³ and the coefficient y in 3ay corresponds to the product term y²^/³), this identity confirms that the resulting value is (y²)/(x).

Explanation

To find the value of the expression (6x)/(x² + x + 1), we divide both the numerator and the denominator by x. This simplifies the fraction to a form that utilizes the given value:

(6x / x)/((x² + x + 1) / x) = (6)/(x + 1 + 1){x}

Rearranging the terms in the denominator, we have:

(6)/((x + 1){x) + 1}

Substituting the given value x + ¹⁄ₓ = 5 into the expression:

(6)/(5 + 1) = ⁶⁄₆ = 1

Explanation

In a triangle ABC, the relationship between the sides AB and BC can be analyzed using the Pythagorean identity. Consider the sum of the squares of the given side lengths: AB² + BC² = (a - b)² + (2√(ab))²

Expanding the algebraic terms: AB² + BC² = a² - 2ab + b² + 4ab AB² + BC² = a² + 2ab + b² AB² + BC² = (a + b)²

The expression (a + b)² represents the square of the side AC in a standard Pythagorean triplet where AB = a - b, BC = 2√(ab), and AC = a + b. According to the converse of the Pythagoras theorem, if AB² + BC² = AC², the triangle is a right-angled triangle with the right angle located at the vertex common to sides AB and BC. Thus, ∠ ABC = 90°.

Explanation

In the triangle POR, it is given that ∠ POR = 90°, which makes it a right-angled triangle with PR as the hypotenuse. Since PR is a tangent to the circle at point Q and O is the center, the radius OQ is perpendicular to PR. Thus, OQ is the altitude from the right angle O to the hypotenuse PR.

According to the Pythagorean theorem in △ POR: PR² = OP² + OR²

Substituting the given values OP = 20/3 cm and OR = 5 cm: PR² = (20/3)² + 5² PR² = 400/9 + 25 PR² = (400 + 225) / 9 PR² = 625 / 9

Taking the square root of both sides: PR = √(625/9) = 25/3 cm

This result is consistent with the given radius OQ = 4 cm. In a right triangle, the altitude h to the hypotenuse is related to the sides by 1/h² = 1/a² + 1/b². Here, 1/4² = 1/5² + 1/(20/3)², which simplifies to 1/16 = 1/25 + 9/400. Since 1/25 + 9/400 = (16 + 9)/400 = 25/400 = 1/16, the geometry is perfectly consistent.

Explanation

In a right-angled triangle formed by the tower, the ground, and the line of sight to the top, the height h and the distance from the base d are related to the angle of elevation θ by the formula tan θ = (Height)/(Base). Given the distance d = 40 m and the angle θ = 60°, we substitute these values into the equation:

tan 60° = ʰ⁄₄₀

Using the trigonometric value tan 60° = √3, the equation becomes:

√3 = ʰ⁄₄₀

Solving for h, we get h = 40√3. Since the distance from the foot of the tower was provided in meters, the height must also be expressed in meters, resulting in 40√3 m.

Explanation

The total surface area of a sphere with radius r is given by the formula 4π r². For a hemisphere of the same radius r, the total surface area consists of the curved surface area, 2π r², plus the area of the flat circular base, π r², resulting in a total of 3π r².

Since the radii are identical, the ratio of the total surface area of the sphere to the total surface area of the hemisphere is calculated as:

Ratio = 4π r²{3π r²} = ⁴⁄₃

This simplifies to the ratio 4:3.

Explanation

The length of an arc s in a circle of radius r subtending an angle θ at the centre is given by the formula s = rθ, where θ is measured in radians. Given that the arc lengths of the two circles are equal (s₁ = s₂), we have the relation r₁₁ = r₂₂.

The ratio of the radii can be expressed as r₁/r₂ = ₂/₁. Although the formula requires radians, the conversion factor π/180 cancels out when taking the ratio of two angles in degrees. Substituting the given values ₁ = 60° and ₂ = 75°, the ratio becomes r₁/r₂ = 75/60. Simplifying this fraction by dividing both the numerator and the denominator by their greatest common divisor, 15, yields 5/4.

Explanation

Let the total work be the Least Common Multiple (LCM) of the time taken by A and B together and A alone. The LCM of 6 and 9 is 18 units. The combined efficiency of A and B is calculated as 18 / 6 = 3 units per day. The efficiency of A alone is 18 / 9 = 2 units per day.

To find the efficiency of B, subtract A's efficiency from the combined efficiency: 3 - 2 = 1 unit per day. The number of days B will take to complete the work alone is the total work divided by B's efficiency, which is 18 / 1 = 18 days. This can also be solved using the work rate formula 1/B = 1/(A+B) - 1/A = 1/6 - 1/9. Finding a common denominator gives (3 - 2) / 18 = 1/18, meaning B completes the work in 18 days.

Explanation

The area of a trapezium is given by the formula A = ½ × (a + b) × h, where a and b are the lengths of the parallel sides and h is the height. Substituting the given values a = 16 m, b = 20 m, and h = 10 m into the formula, we get A = ½ × (16 + 20) × 10. This simplifies to A = ½ × 36 × 10. Dividing 36 by 2 gives 18, and multiplying 18 by 10 results in 180 square metres.

Explanation

To find the single equivalent discount of a series, use the successive discount formula D = x + y - ˣʸ⁄₁₀₀. First, calculate the equivalent discount for the first two rates, 15% and 20%.

D₁ = 15 + 20 - (15 × 20)/(100) = 35 - 3 = 32%

Next, combine this result with the third discount of 25% to find the final equivalent discount.

D_fᵢₙₐₗ = 32 + 25 - (32 × 25)/(100) = 57 - ⁸⁰⁰⁄₁₀₀ = 57 - 8 = 49%

Alternatively, using the multiplier method, if the initial price is 100, the final price after the series of discounts is 100 × (1 - 0.15) × (1 - 0.20) × (1 - 0.25). This simplifies to 100 × 0.85 × 0.80 × 0.75. Calculating this gives 100 × 0.85 × 0.6 = 100 × 0.51 = 51. The total discount is the difference between the initial price and the final price, which is 100 - 51 = 49%.

Explanation

The relationship between the shares of A, B, and C can be expressed as a ratio. Given that A's share is half of B's, the ratio A:B is 1:2. Since A's share is also thrice that of C's, the ratio A:C is 3:1. To find the combined ratio A:B:C, we equalize the value of A in both ratios. Multiplying the A:B ratio by 3 gives A:B = 3:6. Now, the combined ratio is A:B:C = 3:6:1.

The total amount of Rs. 490 is divided into 3 + 6 + 1 = 10 equal parts. The value of one part is calculated by dividing the total amount by the sum of the ratio terms: Value per part = ⁴⁹⁰⁄₁₀ = 49

As C's share corresponds to exactly 1 part in the ratio 3:6:1, C's share is 1 × 49 = 49.