SSC CGL Tier 1 2016 Quantitative Aptitude Questions with Solutions

Explanation

Given the quadratic equation x² - 3x + 1 = 0, where x ≠ 0. To find the value of x + ¹⁄ₓ, divide every term in the equation by x:

(x²)/(x) - ³ˣ⁄ₓ + ¹⁄ₓ = ⁰⁄ₓ

This simplifies to:

x - 3 + ¹⁄ₓ = 0

By transposing the constant term -3 to the right-hand side of the equation, we obtain:

x + ¹⁄ₓ = 3

Explanation

According to the Midpoint Theorem in a triangle, the line segment joining the midpoints of any two sides is parallel to the third side and is half the length of that third side. Since X and Y are the midpoints of sides AB and AC in △ ABC, the length of XY is half of BC, which can be expressed as: XY = ½ BC or BC = 2XY

We are given the equation: BC + XY = 12

Substituting BC = 2XY into the equation: 2XY + XY = 12 3XY = 12 XY = 4 units

Using this value to find BC: BC = 2 × 4 = 8 units

The question asks for the value of BC - XY: BC - XY = 8 - 4 = 4 units

Explanation

In an isosceles triangle, the perimeter P is defined as the sum of its three sides. Since two of these sides are equal in length, let each equal side be represented by s and the base by b. The formula for the perimeter is P = 2s + b.

The given values are P = 4x - 2y + 6z and b = 2x - 2y + 4z. Substituting these into the formula, we have: 4x - 2y + 6z = 2s + (2x - 2y + 4z)

To isolate 2s, we subtract the base from the perimeter: 2s = (4x - 2y + 6z) - (2x - 2y + 4z) 2s = 4x - 2x - 2y + 2y + 6z - 4z 2s = 2x + 2z

By dividing both sides by 2, we determine the length of each equal side: s = (2x + 2z)/(2) s = x + z

Explanation

The equation (cos²θ - 3cosθ + 2)/(sin²θ) = 1 is solved by utilizing the trigonometric identity sin²θ = 1 - cos²θ. Substituting this identity into the equation gives: cos²θ - 3cosθ + 2 = 1 - cos²θ By consolidating all terms on one side of the equation, we obtain a quadratic equation in terms of cosθ: 2cos²θ - 3cosθ + 1 = 0 Factoring this quadratic expression by splitting the linear term yields: (2cosθ - 1)(cosθ - 1) = 0 This results in two possible solutions: cosθ = 1/2 or cosθ = 1. The problem specifies that θ is an acute angle and θ > 0. The condition θ > 0 excludes cosθ = 1 because it corresponds to θ = 0°. For the remaining value, cosθ = 1/2, the corresponding acute angle is 60°.

Explanation

The average charge per person is calculated using the weighted average formula, where the attendance ratios act as weights for the respective daily charges. Let the number of visitors on the first, second, and third days be 2x, 5x, and 13x.

Total attendance = 2x + 5x + 13x = 20x

Total collection = (15 × 2x) + (7.50 × 5x) + (2.50 × 13x) Total collection = 30x + 37.50x + 32.50x = 100x

Average charge per person = (Total Collection)/(Total Attendance) = ¹⁰⁰ˣ⁄₂₀ₓ = 5

By applying the weighted average directly: Average = ((2 × 15) + (5 × 7.50) + (13 × 2.50))/(2 + 5 + 13) = (30 + 37.5 + 32.5)/(20) = ¹⁰⁰⁄₂₀ = 5

Explanation

The given equation is ²⁺ᵃ⁄ₐ + (2+b)/(b) + (2+c)/(c) = 4. This can be simplified by dividing each term in the numerator by the denominator:

(²⁄ₐ + ᵃ⁄ₐ) + ((2)/(b) + (b)/(b)) + ((2)/(c) + (c)/(c)) = 4 (²⁄ₐ + 1) + ((2)/(b) + 1) + ((2)/(c) + 1) = 4 ²⁄ₐ + (2)/(b) + (2)/(c) + 3 = 4

Subtracting 3 from both sides gives: ²⁄ₐ + (2)/(b) + (2)/(c) = 1 2(¹⁄ₐ + (1)/(b) + (1)/(c)) = 1 ¹⁄ₐ + (1)/(b) + (1)/(c) = ½

The expression to be evaluated is (ab+bc+ca)/(abc). Dividing each term in the numerator by the denominator abc: (ab)/(abc) + (bc)/(abc) + (ca)/(abc) = (1)/(c) + ¹⁄ₐ + (1)/(b)

Since we have already determined that ¹⁄ₐ + (1)/(b) + (1)/(c) = ½, the value of the required expression is ½.

Explanation

The given equation ¹⁄ₓ + (1)/(y) + (1)/(z) = 1 can be simplified by taking the common denominator xyz, which gives (xy + yz + zx)/(xyz) = 1. Substituting the value xyz = -1 into this expression, we find that xy + yz + zx = -1.

To find x³ + y³ + z³, we use the algebraic identity x³ + y³ + z³ - 3xyz = (x + y + z)(x² + y² + z² - (xy + yz + zx)). By substituting x² + y² + z² = (x + y + z)² - 2(xy + yz + zx), the identity becomes x³ + y³ + z³ - 3xyz = (x + y + z)((x + y + z)² - 3(xy + yz + zx)).

Now, substitute the known values x + y + z = 1, xy + yz + zx = -1, and xyz = -1 into the identity: x³ + y³ + z³ - 3(-1) = 1 · (1² - 3(-1)) x³ + y³ + z³ + 3 = 1 · (1 + 3) x³ + y³ + z³ + 3 = 4 x³ + y³ + z³ = 4 - 3 = 1

Explanation

According to the Basic Proportionality Theorem (also known as Thales's Theorem), if a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio. In Δ PQR, since the line LM is parallel to QR, the segments on side PQ and PR are proportional.

The relationship is expressed as: (PL)/(LQ) = (PM)/(MR)

Substituting the given values into the equation: ²⁄₆ = (1.5)/(MR)

Simplifying the fraction on the left side: ⅓ = (1.5)/(MR)

Solving for MR: MR = 1.5 × 3 = 4.5 cm

Explanation

The perpendicular drawn from the center of a circle to a chord bisects that chord. Let OM be the perpendicular distance from the center O to the chord AB. In this geometry, M is the midpoint of AB, so AM = (AB)/(2) = ⁸⁄₂ = 4 cm.

By connecting the center O to point A, we form a right-angled triangle OMA where the radius OA = 5 cm is the hypotenuse. According to the Pythagoras theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides: OA² = OM² + AM². Substituting the given values, we get 5² = OM² + 4², which leads to 25 = OM² + 16. Subtracting 16 from both sides gives OM² = 9. Taking the square root of both sides, we find OM = 3 cm. This value represents the perpendicular distance of the chord AB from the center O.

Explanation

Let h represent the height of the tree from the ground to the point where it broke, and let L represent the length of the broken part that is now leaning and touching the ground. These segments, along with the distance from the foot of the tree to where the top touches the ground, form a right-angled triangle.

In this triangle, the distance from the foot of the tree is the base (b = 10 m), the vertical part of the tree is the perpendicular (h), and the broken part is the hypotenuse (L). The angle between the hypotenuse and the base is given as 60°.

Using the tangent ratio to find the vertical height: tan(60°) = ʰ⁄₁₀ √3 = ʰ⁄₁₀ h = 10√3 m

Using the cosine ratio to find the length of the broken part: cos(60°) = (10)/(L) ½ = (10)/(L) L = 20 m

The original height of the tree is the sum of the vertical part and the broken part: Original Height = h + L Original Height = 10√3 + 20 Original Height = 10(2 + √3) m

Explanation

The compound interest (CI) for two years can be efficiently calculated using the effective rate of interest formula x + y + ˣʸ⁄₁₀₀, where x and y are the rates for the first and second years respectively. For an annual rate of 5%, the effective interest rate over two years is 5 + 5 + (5 × 5)/(100) = 10.25%.

Applying this effective rate to the principal amount of Rs. 25000: CI = 25000 × (10.25)/(100) CI = 250 × 10.25 CI = 2562.5

Alternatively, using the standard formula A = P(1 + ʳ⁄₁₀₀)ⁿ, the amount A = 25000(1 + ⁵⁄₁₀₀)² = 25000 × (1.05)² = 25000 × 1.1025 = 27562.5. The compound interest is the difference between the amount and the principal: 27562.5 - 25000 = 2562.5.

Explanation

The expression is cot 17° (cot 73° cos² 22° + (1)/(cot 17° sec² 68°)). Distributing cot 17° into the parentheses results in cot 17° cot 73° cos² 22° + (cot 17°)/(cot 17° sec² 68°).

In the first term, since 17° and 73° are complementary angles (17° + 73° = 90°), we use the identity cot(90° - θ) = tan θ to get cot 73° = tan 17°. Because cot θ tan θ = 1, the first term simplifies to 1 · cos² 22° = cos² 22°.

In the second term, the cot 17° in the numerator and denominator cancels out, leaving (1)/(sec² 68°), which is equivalent to cos² 68°.

The expression now stands as cos² 22° + cos² 68°. Using the complementary angle identity cos(90° - θ) = sin θ, we have cos 68° = sin 22°. Substituting this gives cos² 22° + sin² 22°. Applying the fundamental trigonometric identity sin² θ + cos² θ = 1, the final value is 1.

Explanation

The Central Government of India derives its taxing authority from the Union List (List I) of the Seventh Schedule of the Constitution. Excise Duty is levied on the manufacture or production of goods within India under Entry 84 of the Union List. Customs Duty is an indirect tax imposed on goods imported into or exported out of the country as per Entry 83. Income Tax is a direct tax levied on the income of individuals and corporations, excluding agricultural income, according to Entry 82. In the fiscal framework existing in 2016, these taxes were collected and managed by the Union Government through the Ministry of Finance to fund national expenditures and development projects.

Explanation

Manganese is a naturally occurring mineral commonly found in groundwater sources. When water containing dissolved manganese is exposed to oxygen or chemical oxidants like chlorine, it undergoes a chemical reaction to form manganese dioxide. This insoluble precipitate appears as a dark brown or black sediment that adheres to surfaces. This process is responsible for the characteristic brown or blackish staining found on laundry, porcelain fixtures, and glassware. Even in concentrations as low as 0.05 mg/L, manganese can cause noticeable discoloration and staining in household water systems.

Explanation

The volume of a sphere with radius r is given by the formula Vₛₚₕₑᵣₑ = ⁴⁄₃ π r³. The volume of a right circular cylinder with radius r and height h is given by the formula V_c_yₗᵢₙ_dₑᵣ = π r² h.

Given that the volumes of the sphere and the cylinder are equal, we can set the two formulas equal to each other:

⁴⁄₃ π r³ = π r² h

To find the ratio of the height and radius of the cylinder, we solve for ʰ⁄ᵣ. First, divide both sides of the equation by π r²:

⁴⁄₃ r = h

Rearranging the terms to isolate the ratio ʰ⁄ᵣ, we get:

ʰ⁄ᵣ = ⁴⁄₃

Thus, the ratio of the height to the radius of the cylinder is 4:3.

Explanation

Let the marked price be M. The problem states that after a discount of Rs. 42, the price becomes 86% of the original marked price. This implies that the discount of Rs. 42 represents the difference between the full price and the reduced price, which is 100% - 86% = 14% of the marked price.

Using the relationship between the discount value and its percentage: 14% of M = 42 ¹⁴⁄₁₀₀ × M = 42 M = (42 × 100)/(14) M = 3 × 100 M = 300

Alternatively, if 14% corresponds to 42, then 1% corresponds to 42 / 14 = 3. Therefore, the 100% marked price is 3 × 100 = 300. The marked price of the article is Rs. 300.

Explanation

The total Cost Price (CP) of the milk is calculated by multiplying the quantity of milk by the price per litre: CP = 25 × 12 = ₹ 300. When 5 litres of water are added to the 25 litres of milk, the total quantity of the mixture becomes 25 + 5 = 30 litres. The merchant sells this entire mixture at a rate of ₹ 10.40 per litre, so the total Selling Price (SP) is SP = 30 × 10.40 = ₹ 312. The profit earned is the difference between the selling price and the cost price: Profit = 312 - 300 = ₹ 12. To find the gain percentage, we use the formula (Profit)/(CP) × 100. Substituting the values, we get ¹²⁄₃₀₀ × 100 = 4%.

Explanation

Let the man's total income be represented by I. The problem states that the expenditure is 15% of this income. Given that the expenditure is Rs. 75, we can set up the following equation:

15% of I = 75 ¹⁵⁄₁₀₀ × I = 75

To find the value of I, rearrange the equation to isolate the variable:

I = (75 × 100)/(15) I = 5 × 100 I = 500

Alternatively, using the unitary method, if 15% of the income corresponds to Rs. 75, then 1% of the income is equivalent to 75 / 15 = 5. Since the total income represents 100%, the total income is calculated as 100 × 5 = 500.

Explanation

When the distance covered is constant, speed is inversely proportional to time, as expressed by the relation S₁ T₁ = S₂ T₂. In this problem, the initial speed S₁ is 60 km/hr and the initial time T₁ is 15 hours. To find the required speed S₂ for a time T₂ of 12 hours, we set up the equation 60 × 15 = S₂ × 12. Rearranging to solve for S₂ gives S₂ = (60 × 15) / 12. Since 60 divided by 12 equals 5, the calculation simplifies to 5 × 15, resulting in 75 km/hr. Alternatively, calculating the total distance first gives 60 × 15 = 900 km, and dividing this distance by the new time of 12 hours yields 900 / 12 = 75 km/hr.

Explanation

For any odd number of consecutive integers, the middle integer is equivalent to the arithmetic mean of the series. Let the three consecutive integers be x-1, x, and x+1, where x represents the middle integer. The sum of these integers is (x-1) + x + (x+1) = 3x. According to the problem, this sum is 51. By setting up the equation 3x = 51 and dividing both sides by 3, we find x = 17. Thus, the middle integer is 17.

Explanation

Given the equation x + ¹⁄ₓ = 5, we can find the value of the required expression by dividing both the numerator and the denominator by x.

The expression is (5x)/(x² + 5x + 1). Dividing every term by x, we get:

(5x)/(x){(x²)/(x) + ⁵ˣ⁄ₓ + ¹⁄ₓ} = (5)/(x + 5 + 1){x}

By rearranging the terms in the denominator, the expression becomes:

(5)/((x + 1){x) + 5}

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

(5)/(5 + 5) = ⁵⁄₁₀ = ½

Explanation

The centroid is the point of concurrency where the three medians of a triangle intersect. A median is defined as a line segment that joins a vertex of the triangle to the midpoint of the opposite side. Every triangle has exactly three medians, and they always meet at a single internal point. The centroid serves as the geometric center or center of mass of the triangle. A fundamental property of the centroid is that it divides each median into two parts in the ratio of 2:1, with the portion from the vertex to the centroid being twice as long as the portion from the centroid to the midpoint of the side. For a triangle with vertices at (x₁, y₁), (x₂, y₂), and (x₃, y₃), the coordinates of the centroid are calculated as (x₁ + x₂ + x₃{3}, y₁ + y₂ + y₃{3}).

Explanation

The vertices O(0, 0), A(0, 8), and B(5, 0) form a right-angled triangle with the right angle at the origin O. Since vertex A lies on the y-axis and vertex B lies on the x-axis, the legs of the triangle are perpendicular. The length of the base OB is the distance from (0, 0) to (5, 0), which is 5 units. The height OA is the distance from (0, 0) to (0, 8), which is 8 units. The area of a triangle is calculated using the formula Area = ½ × base × height. By substituting the values, we get Area = ½ × 5 × 8 = 20 sq. units. Alternatively, applying the coordinate geometry formula Area = ½ |x₁(y₂ - y₃) + x₂(y₃ - y₁) + x₃(y₁ - y₂)| for the given points gives ½ |0(0 - 8) + 5(8 - 0) + 0(0 - 0)| = ½ |40| = 20 sq. units.

Explanation

To find a number that leaves a remainder of 6 when divided by 9, 11, and 13, we first calculate the least common multiple (LCM) of these divisors. Since 9, 11, and 13 are co-prime to each other, their LCM is simply their product: LCM(9, 11, 13) = 9 × 11 × 13 = 1287.

Any number that leaves a remainder of 6 when divided by these three values must be of the form 1287k + 6, where k is an integer. For k = 1, the smallest such number greater than 1287 is 1287 + 6 = 1293. To find the least number that must be subtracted from 1294 to obtain this result, we perform the calculation 1294 - 1293 = 1.

Explanation

Let the total work be the LCM of 72, 120, and 90, which is 360 units.

The daily efficiencies (work done per day) are: Efficiency of A + B = ³⁶⁰⁄₇₂ = 5 units/day Efficiency of B + C = ³⁶⁰⁄₁₂₀ = 3 units/day Efficiency of A + C = ³⁶⁰⁄₉₀ = 4 units/day

Adding these efficiencies gives the combined efficiency of the pairs: 2(A + B + C) = 5 + 3 + 4 = 12 units/day Combined efficiency A + B + C = ¹²⁄₂ = 6 units/day

To find the efficiency of A alone, subtract the efficiency of (B + C) from the combined efficiency: Efficiency of A = (A + B + C) - (B + C) = 6 - 3 = 3 units/day

Time taken by A alone = (Total Work)/(Efficiency of A) = ³⁶⁰⁄₃ = 120 days.

Explanation

The relationship between the shares of A and B is given as A:B = 3:4. The relationship between the shares of B and C is given as B:C = 3.5:3. To simplify the second ratio, we multiply both terms by 2, resulting in B:C = 7:6.

To find the combined ratio A:B:C, we equate the value of B in both ratios by finding the least common multiple of 4 and 7, which is 28. Multiplying the first ratio by 7 and the second ratio by 4, we get: A:B = 21:28 B:C = 28:24 Therefore, A:B:C = 21:28:24.

The sum of the ratio parts is 21 + 28 + 24 = 73. Since the total amount to be divided is Rs. 730, each ratio unit is worth 730 / 73 = 10. The share of B is 28 × 10 = 280 and the share of C is 24 × 10 = 240. The difference between the share of B and the share of C is 280 - 240 = 40. Alternatively, the difference in ratio parts is 28 - 24 = 4 units, which corresponds to 4 × 10 = 40.

Explanation

The total sum of marks for the class is calculated by multiplying the number of students by the class average: 50 × 70 = 3500. The sum of marks for the first 25 students is 25 × 60 = 1500. The sum of marks for the next 24 students is 24 × 80 = 1920. To find the marks of the last student, we subtract the combined sum of the first 49 students from the class total: 3500 - (1500 + 1920) = 3500 - 3420 = 80.

Explanation

The given equation is ¹⁄ₐ (a² + 1) = 3, which simplifies to a + ¹⁄ₐ = 3 by dividing each term in the parentheses by a.

The expression to be evaluated is (a⁶ + 1)/(a³), which can be rewritten as a³ + (1)/(a³).

Using the algebraic identity for the sum of cubes, x³ + y³ = (x + y)³ - 3xy(x + y), we let x = a and y = ¹⁄ₐ. Substituting these into the identity gives a³ + (1)/(a³) = (a + ¹⁄ₐ)³ - 3(a)(¹⁄ₐ)(a + ¹⁄ₐ), which simplifies to (a + ¹⁄ₐ)³ - 3(a + ¹⁄ₐ).

Substituting the value a + ¹⁄ₐ = 3 into the formula, we get 3³ - 3(3) = 27 - 9 = 18.

Explanation

To find the value of the expression ¹⁄ₐ₊₁ + (1)/(b+1), first substitute the given values of a = √2 + 1 and b = √2 - 1 into the denominators: a + 1 = (√2 + 1) + 1 = √2 + 2 b + 1 = (√2 - 1) + 1 = √2

Substitute these back into the original expression: (1)/(√(2) + 2) + (1)/(√(2))

To add these fractions, find a common denominator by multiplying the terms: √2 + (√2 + 2){(√2 + 2)(√2)}

Simplify the numerator and the denominator: Numerator: √2 + √2 + 2 = 2√2 + 2 Denominator: (√2 × √2) + (2 × √2) = 2 + 2√2

The expression becomes: 2√2 + 2{2 + 2√2} = 1

Explanation

The centroid of a triangle divides each median in the ratio 2:1, where the distance from the vertex to the centroid is twice the distance from the centroid to the midpoint of the opposite side. If d is the distance from a vertex to the centroid, the total length of the corresponding median m is calculated using the formula m = ³⁄₂ d. For the given vertex distances of 4 cm, 6 cm, and 8 cm, the lengths of the three medians are ³⁄₂ × 4 = 6 cm, ³⁄₂ × 6 = 9 cm, and ³⁄₂ × 8 = 12 cm. By comparing these results, the length of the smallest median is 6 cm.

Explanation

The given equation is sinθ - cosθ = 0, which implies sinθ = cosθ. Dividing both sides by cosθ, we get tanθ = 1. For a positive acute angle, this condition is satisfied when θ = 45°. To find the value of secθ + cosecθ, we substitute θ = 45° into the expression to get sec 45° + cosec 45°. Since sec 45° = √2 and cosec 45° = √2, the sum is √2 + √2 = 2√2.

Explanation

The expression utilizes trigonometric identities of complementary angles, specifically tan(90° - θ) = cot θ and cot(90° - θ) = tan θ.

In the first term, since 37° = 90° - 53°, we have cot 37° = cot(90° - 53°) = tan 53°. Substituting this into the fraction, we get: (2 tan 53°)/(tan 53°) = 2

In the second term, since 10° = 90° - 80°, we have tan 10° = tan(90° - 80°) = cot 80°. Substituting this into the fraction, we get: (cot 80°)/(cot 80°) = 1

Subtracting the results of the two terms yields 2 - 1 = 1.

Explanation

When interest is compounded semi-annually, the annual rate is halved and the number of time periods is doubled. Given the principal P = Rs. 24000, the annual rate R = 10%, and the time T = 1.5 years, the adjusted values for calculation are:

Rate per half-year r = (10%)/(2) = 5% Number of half-yearly periods n = 1.5 × 2 = 3

The formula for the Amount A is: A = P(1 + ʳ⁄₁₀₀)ⁿ A = 24000(1 + ⁵⁄₁₀₀)³ A = 24000(²¹⁄₂₀)³ A = 24000 × ⁹²⁶¹⁄₈₀₀₀ A = 3 × 9261 = Rs. 27783

The Compound Interest CI is calculated as: CI = A - P CI = 27783 - 24000 = Rs. 3783

Explanation

Let h be the height of the observation tower. From the initial position of the boat at a distance of 50m, the angle of depression is 60°. Using the trigonometric ratio tan θ = height/base, we have: tan 60° = h/50 √3 = h/50 ⟹ h = 50√3 meters.

After 8 seconds, the boat moves to a new distance D from the tower where the angle of depression becomes 30°. Using the same ratio: tan 30° = (h)/(D) (1)/(√3) = 50√3/D ⟹ D = 50 × 3 = 150 meters.

The distance traveled by the boat in 8 seconds is the difference between the final and initial distances: Distance = 150 - 50 = 100 meters. Speed in m/s = (Distance)/(Time) = ¹⁰⁰⁄₈ = 12.5 m/s.

To convert the speed into km/hr, multiply by 18/5: Speed in km/hr = 12.5 × ¹⁸⁄₅ = 2.5 × 18 = 45 km/hr.

Explanation

The efficiency of a worker defines the amount of work done per unit of time. Given that A is twice as good as B, the ratio of the efficiency of A to B is 2:1. If we assume B completes 1 unit of work per day, then A completes 2 units of work per day. Together, their combined efficiency is 2 + 1 = 3 units of work per day.

Since they complete the work together in 16 days, the total work is calculated as: Total Work = Combined Efficiency × Time = 3 × 16 = 48 units.

The time required for A to finish the work alone is determined by dividing the total work by A's specific efficiency: Time_A = (Total Work)/(Efficiency_A₎ = ⁴⁸⁄₂ = 24 days.

Explanation

The single equivalent discount for multiple successive discounts is calculated by determining the final multiplier of the price. If the initial marked price is 100, the price after three successive discounts of 15%, 20%, and 25% is calculated as follows:

Final Price = 100 × (1 - ¹⁵⁄₁₀₀) × (1 - ²⁰⁄₁₀₀) × (1 - ²⁵⁄₁₀₀) Final Price = 100 × ⁸⁵⁄₁₀₀ × ⁸⁰⁄₁₀₀ × ⁷⁵⁄₁₀₀ Final Price = 100 × 0.85 × 0.8 × 0.75 Final Price = 85 × 0.8 × 0.75 = 68 × 0.75 = 51

The single equivalent discount is the total reduction from the original price: Equivalent Discount = 100 - 51 = 49%

Alternatively, using the formula for two successive discounts x% and y%, which is x + y - ˣʸ⁄₁₀₀: Combining 15% and 20%: 15 + 20 - (15 × 20)/(100) = 35 - 3 = 32% Combining the result 32% with 25%: 32 + 25 - (32 × 25)/(100) = 57 - 8 = 49%

Explanation

Given the ratio x:y:z = 2:3:5 and their sum is 80, we calculate the parts as x = 16, y = 24, and z = 40. Substituting z = 40 and x = 16 into the equation z = ax - 8 results in 40 = 16a - 8, which simplifies to 48 = 16a and gives a = 3.

Explanation

The profit percentage is determined by the ratio of the profit to the cost price (CP). Given the ratio of cost price to selling price (SP) is 10:11, we can assume CP = 10 and SP = 11. The profit is the difference between the selling price and the cost price, which is 11 - 10 = 1. Applying the formula for profit percentage:

Profit Percentage = ( (Profit)/(CP) ) × 100

Substituting the values, we get:

( ⅒ ) × 100 = 10%

Explanation

Let the original price of rice be 100 units. After an increase of 60%, the new price is calculated as 100 + (60% of 100) = 160 units. To restore the original price of 100 units, the new price must be reduced by 160 - 100 = 60 units.

The percentage reduction required is calculated relative to the new price of 160. Using the formula for percentage decrease: Percentage Reduction = ((Amount of Reduction)/(New Price)) × 100 Percentage Reduction = (⁶⁰⁄₁₆₀) × 100

Simplifying the fraction ⁶⁰⁄₁₆₀ by dividing both numerator and denominator by 20 yields ⅜. Converting this fraction into a percentage: ⅜ × 100 = 3 × 12.5% = 37.5% The value 37.5% is equivalent to the mixed fraction 37 ½%.

Alternatively, this can be solved directly using the percentage change formula for restoring initial values: ((r)/(100 + r)) × 100, where r is the percentage increase. Substituting r = 60 into the formula gives ((60)/(100 + 60)) × 100 = (⁶⁰⁄₁₆₀) × 100 = 37 ½%.

Explanation

The average speed is calculated as the ratio of the total distance covered to the total time taken for the journey. For the first part of the journey, the distance covered is d₁ = speed × time = 20 km/hr × 2 hours = 40 km. Since the total distance is 100 km, the remaining distance for the second part is d₂ = 100 km - 40 km = 60 km.

The time taken for the second part of the journey at a speed of 10 km/hr is t₂ = (distance₂)/(speed₂) = (60 km)/(10 km/hr) = 6 hours. Thus, the total time taken for the entire journey is T = 2 hours + 6 hours = 8 hours.

Using the formula for average speed: Average Speed = (Total Distance)/(Total Time) = (100 km)/(8 hours) = 12.5 km/hr

In mixed fraction form, 12.5 km/hr is equal to 12 ½ km/hr.

Explanation

Expanding the left side of the equation (x - 2)(x - p) gives x² - (p + 2)x + 2p. By comparing the coefficients of this expansion with the given expression x² - ax + 6, we equate the coefficients of x and the constant terms. Comparing the coefficients of x gives a = p + 2. By rearranging this equation to isolate the terms requested in the problem, we directly obtain a - p = 2. Additionally, comparing the constant terms gives 2p = 6, which implies p = 3. Substituting this back into the first relation gives a = 3 + 2 = 5, confirming that a - p = 5 - 3 = 2.

Explanation

Dividing the given equation 2x + ²⁄ₓ = 3 by 2 results in x + ¹⁄ₓ = ³⁄₂.

Using the algebraic identity for the sum of cubes, if x + ¹⁄ₓ = k, then x³ + (1)/(x³) = k³ - 3k. Substituting k = ³⁄₂:

x³ + (1)/(x³) = (³⁄₂)³ - 3(³⁄₂)

x³ + (1)/(x³) = ²⁷⁄₈ - ⁹⁄₂

To subtract these fractions, find a common denominator:

x³ + (1)/(x³) = ²⁷⁄₈ - ³⁶⁄₈ = -⁹⁄₈

The value of the expression x³ + (1)/(x³) + 2 is then found by adding 2 to this result:

-⁹⁄₈ + 2 = (-9 + 16)/(8) = ⅞

Explanation

In any triangle ABC, the sum of the interior angles is 180°, which gives the relation ∠ A + ∠ B + ∠ C = 180°. Given the first equation ∠ A + ∠ C = 140°, we can find the value of ∠ B by substituting this into the triangle angle sum property: ∠ B = 180° - (∠ A + ∠ C) = 180° - 140° = 40°.

The problem also provides the equation ∠ A + 3∠ B = 180°. Substituting the calculated value of ∠ B = 40° into this equation yields: ∠ A + 3(40°) = 180° ∠ A + 120° = 180° ∠ A = 180° - 120° = 60°.

Explanation

The line segment OP bisects the angle between the two tangents PA and PB, resulting in ∠ APO = ½ ∠ APB = (80°)/(2) = 40°. Based on the geometric theorem that a tangent to a circle is perpendicular to the radius at the point of contact, ∠ OAP = 90°. In △ OAP, the sum of the interior angles is 180°. Therefore:

∠ AOP + ∠ OAP + ∠ APO = 180° ∠ AOP + 90° + 40° = 180° ∠ AOP = 180° - 130° = 50°

Alternatively, in quadrilateral OAPB, since ∠ OAP = ∠ OBP = 90°, the angles ∠ AOB and ∠ APB are supplementary. Thus, ∠ AOB = 180° - 80° = 100°. Because △ OAP and △ OBP are congruent, OP bisects ∠ AOB, leading to ∠ AOP = (100°)/(2) = 50°.

Explanation

The trigonometric identity for complementary angles states that tan A = cot B if A + B = 90°. Given the equation tan (5x - 10°) = cot (5y + 20°), we can equate the sum of the angles to 90°:

(5x - 10°) + (5y + 20°) = 90°

Rearranging the terms to group the variables and constants:

5x + 5y + 10° = 90°

Subtracting 10° from both sides:

5(x + y) = 80°

Dividing by 5:

x + y = (80°)/(5) = 16°

Explanation

The pairs of positive integers whose product is 24 are (1, 24), (2, 12), (3, 8), and (4, 6). Their respective sums are 25, 14, 11, and 10, making 10 the minimum possible sum among the given options. While negative integers would yield a lower sum, none are present in the options, implying the question refers to positive integers.

Explanation

The total marks of the class are calculated by multiplying the total number of students by their average marks: 60 × 65 = 3900. The sum of marks for the first half of the students (30 students) is 30 × 85 = 2550. The total marks for the remaining 30 students is the difference between the total marks of the class and the marks of the first half: 3900 - 2550 = 1350. Therefore, the average marks of the remaining students is 1350 / 30 = 45.

Alternatively, since the students are divided into two equal groups, the overall average is the simple arithmetic mean of the averages of the two halves. Let x be the average of the remaining students. The relationship is expressed as 65 = (85 + x) / 2. Solving for x gives 130 = 85 + x, which leads to x = 45.

Explanation

The expression x⁴ + y⁴ - 2x²y² is a perfect square that can be simplified using the algebraic identity (m - n)² = m² + n² - 2mn. By substituting m = x² and n = y², the expression becomes: x⁴ + y⁴ - 2x²y² = (x² - y²)²

Using the difference of squares identity x² - y² = (x + y)(x - y), the expression further simplifies to: (x² - y²)² = [(x + y)(x - y)]²

Now, calculate the sum and difference of the given values for x and y: x + y = (√a + (1)/(√(a))) + (√a - (1)/(√(a))) = 2√a x - y = (√a + (1)/(√(a))) - (√a - (1)/(√(a))) = (2)/(√(a))

Substitute these results back into the simplified identity: (x + y)(x - y) = (2√a) × ((2)/(√(a))) = 4

Squaring this value gives the final result: 4² = 16

Explanation

The given equation is x = ³√(x² + 11) - 2. To find the value of the required expression, first isolate the cube root by adding 2 to both sides of the equation: x + 2 = ³√(x² + 11)

Now, cube both sides of the equation to eliminate the radical: (x + 2)³ = (³√(x² + 11))³

Applying the algebraic identity (a + b)³ = a³ + 3a²b + 3ab² + b³, the expansion becomes: x³ + 3(x²)(2) + 3(x)(2²) + 2³ = x² + 11 x³ + 6x² + 12x + 8 = x² + 11

Rearranging the terms to match the required expression x³ + 5x² + 12x, subtract x² and 8 from both sides: x³ + (6x² - x²) + 12x = 11 - 8 x³ + 5x² + 12x = 3

Explanation

According to the Triangle Inequality Theorem, for any three side lengths to form a triangle, the sum of the lengths of any two sides must be strictly greater than the length of the third side (a + b > c). In this case, the two shorter sides are 5 cm and 8 cm. Calculating their sum gives 5 + 8 = 13 cm. Since 13 cm is less than the length of the longest side, 15 cm (13 < 15), the triangle inequality is violated, and a triangle cannot be formed.