SSC CGL Tier 1 2016 Quantitative Aptitude Questions with Solutions

Explanation

In any triangle ABC, the sum of the interior angles is A + B + C = 180°. The given condition is sin A = cos B. Using the trigonometric identity cos θ = sin(90° - θ), the equation can be written as sin A = sin(90° - B). Equating the arguments, we get A = 90° - B, which simplifies to A + B = 90°. Substituting this relation into the angle sum property of the triangle, we have 90° + C = 180°, which results in C = 90°. Consequently, the value of cos C is cos 90° = 0.

Explanation

When a rectangle is inscribed in a circle, its diagonal serves as the diameter of the circle. This is because each vertex of the rectangle forms a 90° angle, and according to circle geometry, an angle of 90° subtended on the circumference must be subtended by the diameter. For a rectangle with adjacent sides a = 5 cm and b = 12 cm, the length of the diagonal d is found using the Pythagorean theorem:

d = √(5² + 12²) = √(25 + 144) = √(169) = 13 cm

The diameter of the circle is therefore 13 cm. The radius r is half of the diameter:

r = ¹³⁄₂ = 6.5 cm

Explanation

To find the value of sinθ - cosθ, we use the algebraic identity (a - b)² = a² + b² - 2ab. Let a = sinθ and b = cosθ. Thus, (sinθ - cosθ)² = sin²θ + cos²θ - 2sinθ. Applying the trigonometric identity sin²θ + cos²θ = 1 and substituting the given value sinθ = ½, we get (sinθ - cosθ)² = 1 - 2(½) = 1 - 1 = 0. Taking the square root of both sides results in sinθ - cosθ = 0. Alternatively, multiplying the given equation by 2 gives 2sinθ = 1, which is the formula for sin(2θ). Since sin(2θ) = 1 and 0° < θ < 90°, it follows that 2θ = 90°, meaning θ = 45°. At 45°, the values of sinθ and cosθ are both (1)/(√(2)), making their difference zero.

Explanation

The correct mean is determined by adjusting the initial mean for the error in recording the data. The change in the total sum is 53 - 83 = -30. Since there are 100 observations, the impact of this error on the mean is ⁻³⁰⁄₁₀₀ = -0.3. Applying this correction to the original mean, the correct mean is 40 - 0.3 = 39.7. This can be calculated directly using the formula for the adjusted mean:

xₙₑ_w = xₒₗ_d + x_cₒᵣᵣₑ_cₜ - xᵢₙ_cₒᵣᵣₑ_cₜ{n}

Substituting the values:

xₙₑ_w = 40 + (53 - 83)/(100) = 40 - ³⁰⁄₁₀₀ = 40 - 0.3 = 39.7

This calculation shows that because the value was recorded as higher than it actually was, the initial mean was overestimated, and the correction results in a lower average.

Explanation

For any positive real number x, the equation x + ¹⁄ₓ = 2 is satisfied if and only if x = 1. This is a standard result derived from the fact that (x - 1)² ≥ 0, which implies x² - 2x + 1 ≥ 0, or x + ¹⁄ₓ ≥ 2 for x > 0, where equality holds only at x = 1. In the given expression a³ + (1)/(a³) = 2, let x = a³. Since a is a positive number, a³ = 1, which gives a = 1.

The value to be calculated is (a² + 1)/(a). We can simplify this expression by dividing each term of the numerator by the denominator: (a²)/(a) + ¹⁄ₐ = a + ¹⁄ₐ

Substituting the value a = 1 into the simplified expression: 1 + ¹⁄₁ = 1 + 1 = 2

Alternatively, using the algebraic identity (a + ¹⁄ₐ)³ = a³ + (1)/(a³) + 3(a + ¹⁄ₐ), let k = a + ¹⁄ₐ. The equation transforms into k³ = 2 + 3k, or k³ - 3k - 2 = 0. Factoring this yields (k - 2)(k + 1)² = 0. Given a > 0, k must be positive, hence k = 2.

Explanation

The given equation is x(3 - ²⁄ₓ) = ³⁄ₓ. Expanding the left side results in 3x - 2 = ³⁄ₓ. By rearranging the terms, we get 3x - ³⁄ₓ = 2. Dividing the entire equation by 3 yields the simplified form x - ¹⁄ₓ = ⅔.

To find x² + (1)/(x²), we apply the algebraic identity (x - ¹⁄ₓ)² = x² + (1)/(x²) - 2. Rearranging this identity gives x² + (1)/(x²) = (x - ¹⁄ₓ)² + 2. Substituting the value of x - ¹⁄ₓ into the formula, we get x² + (1)/(x²) = (⅔)² + 2. This simplifies to ⁴⁄₉ + 2, which is 2 ⁴⁄₉.

Explanation

When a solid object is melted and recast into another shape, its volume remains constant. The volume of a sphere is given by the formula Vₛ = ⁴⁄₃ π r³ and the volume of a cone is given by V_c = ⅓ π r² h.

Setting the volume of the sphere equal to the volume of the cone: ⁴⁄₃ π r³ = ⅓ π r² h

Since the radius of the sphere and the base radius of the cone are given as the same (r = 5 cm), we can simplify the equation by dividing both sides by ⅓ π r²: 4r = h

Substituting the value of r = 5 cm into the simplified formula: h = 4 × 5 = 20 cm

Explanation

The difference (D) between compound interest and simple interest for a period of 2 years is determined by the formula D = P × (r/100)², where P is the principal sum and r is the annual rate of interest. Given that the difference is Rs. 41 and the rate is 5% per annum, the values can be substituted into the formula as follows:

41 = P × (5/100)²

41 = P × (1/20)²

41 = P × (1/400)

P = 41 × 400

P = 16400

Explanation

Let h be the height of the tree and w be the breadth of the river. According to the first observation, the angle of elevation is 60°, giving the relation tan 60° = (h)/(w), which simplifies to h = w√3. After the man moves 36 m further away from the bank, the total distance from the tree becomes w + 36 and the angle of elevation becomes 30°. This provides the second relation tan 30° = (h)/(w + 36), or h = (w + 36)/(√(3)).

By equating the two expressions for h: w√3 = (w + 36)/(√(3))

Multiplying both sides by √3 yields: 3w = w + 36 2w = 36 w = 18 m

Alternatively, using the distance formula for change in angles of elevation d = w((tan α)/(tan β) - 1), where d = 36, α = 60°, and β = 30°: 36 = w(√3{1/√3} - 1) 36 = w(3 - 1) 36 = 2w w = 18 m

Explanation

To form the greatest five-digit number using the digits 2, 5, 0, 6, and 8 without repetition, we arrange the digits in descending order. This gives us 86520.

To form the least five-digit number, we arrange the digits in ascending order. However, a five-digit number cannot begin with the digit 0, as it would then become a four-digit number. Therefore, we place the smallest non-zero digit, 2, in the ten-thousands place, followed by 0 in the thousands place, and then the remaining digits in ascending order. This results in 20568.

The difference between the greatest and the least number is calculated as: 86520 - 20568 = 65952

Explanation

The relationship between the Marked Price (MP), the discount percentage (d), and the Selling Price (SP) is given by the formula SP = MP × (1 - ᵈ⁄₁₀₀). In this problem, the Selling Price after the discount is Rs. 119 and the discount rate is 15%. Substituting these values into the formula, we get 119 = MP × (1 - ¹⁵⁄₁₀₀), which simplifies to 119 = MP × ⁸⁵⁄₁₀₀.

To solve for the Marked Price, we rearrange the equation as MP = (119 × 100)/(85). Dividing both 119 and 85 by their greatest common divisor, 17, the expression becomes MP = (7 × 100)/(5). Further simplification gives MP = 7 × 20 = 140. Thus, the marked price of the shirt before the discount was Rs. 140.

Explanation

Based on the symmetry of the given variables, we define the equal ratios such that (a)/(q-r) = ᵇ⁄ᵣ₋ₚ = (c)/(p-q) = k. This allows us to express a as k(q-r), b as k(r-p), and c as k(p-q). Substituting these expressions into the target sum pa + qb + rc, we get p(k(q-r)) + q(k(r-p)) + r(k(p-q)). Factoring out the constant k yields k(pq - pr + qr - qp + rp - rq). Within the parentheses, the terms cancel out in a cyclic manner: pq cancels with -qp, qr cancels with -rq, and -pr cancels with rp. This results in k(0), which simplifies to 0.

Explanation

The sum of a set of values is equal to the average multiplied by the number of observations. From the given averages, we can establish two equations: a + b + c = 20 × 3 = 60 and b + c + d = 25 × 3 = 75. Subtracting the first equation from the second eliminates the common terms b and c, resulting in (b + c + d) - (a + b + c) = 75 - 60, which simplifies to d - a = 15. Substituting the given value d = 30 into this relation gives 30 - a = 15. Rearranging the terms to solve for a, we get a = 30 - 15 = 15.

Explanation

Let the cost price (CP) of the watch be 100. Given that the profit is 25% of the cost price, the profit amount is 0.25 × 100 = 25. The selling price (SP) is calculated by adding the profit to the cost price: SP = CP + Profit = 100 + 25 = 125.

To determine the profit percentage relative to the selling price, we use the formula: Profit % on SP = ( (Profit)/(SP) ) × 100

Substituting the known values into the formula: Profit % on SP = ( ²⁵⁄₁₂₅ ) × 100 Profit % on SP = ( ⅕ ) × 100 = 20%

Explanation

The relationship between the three variables can be determined by sequential substitution. Given A = 20% of B and B = 25% of C, we can write these as mathematical equations: A = 0.20 × B and B = 0.25 × C.

Substituting the value of B into the first equation: A = 0.20 × (0.25 × C) A = 0.05 × C

To express this as a percentage, we multiply by 100: A = (0.05 × 100)% of C A = 5% of C

Alternatively, using the assume-value method, let C = 100. Then B = 25% of 100 = 25. Consequently, A = 20% of 25, which is ⅕ × 25 = 5. Since A is 5 when C is 100, A is 5% of C.

Explanation

The speed of sound can be calculated using the basic formula for speed, which is the ratio of distance traveled to the time taken:

v = ᵈ⁄ₜ

In this problem, the distance (d) is given as 1.7 km. To obtain the speed in meters per second (m/s), this value must first be converted into meters:

d = 1.7 × 1000 = 1700 meters

The time (t) taken for the sound to reach Ram is 25 seconds. Substituting these values into the formula:

v = ¹⁷⁰⁰⁄₂₅

To simplify the calculation, 1700 can be viewed as 17 × 100. Since 100 divided by 25 is 4, the calculation becomes:

v = 17 × 4 = 68 m/s

Explanation

The time duration can be calculated using the standard formula for simple interest: T = (SI × 100)/(P × R), where SI is the simple interest, P is the principal amount, R is the rate of interest per annum, and T is the time in years. Substituting the given values SI = 1080, P = 3000, and R = 12:

T = (1080 × 100)/(3000 × 12)

T = (1080)/(30 × 12)

T = ¹⁰⁸⁰⁄₃₆₀ = 3 years

Alternatively, calculating the interest for one year at 12% on ₹ 3000 gives 3000 × ¹²⁄₁₀₀ = 360. Dividing the total interest of ₹ 1080 by the annual interest of ₹ 360 yields ¹⁰⁸⁰⁄₃₆₀ = 3 years.

Explanation

From the given equation (a + ¹⁄ₐ)² = 3, taking the square root on both sides provides the relation a + ¹⁄ₐ = √3.

To find the value of a³ + (1)/(a³), we utilize the standard algebraic identity for the sum of cubes: x³ + y³ = (x + y)³ - 3xy(x + y)

By substituting x = a and y = ¹⁄ₐ, the formula simplifies to: a³ + (1)/(a³) = (a + ¹⁄ₐ)³ - 3(a · ¹⁄ₐ)(a + ¹⁄ₐ) a³ + (1)/(a³) = (a + ¹⁄ₐ)³ - 3(a + ¹⁄ₐ)

Now, substitute the value a + ¹⁄ₐ = √3 into the expression: a³ + (1)/(a³) = (√3)³ - 3(√3) Since (√3)³ = √3 × √3 × √3 = 3√3, the calculation becomes: a³ + (1)/(a³) = 3√3 - 3√3 a³ + (1)/(a³) = 0

This result is consistent with the algebraic property that whenever a + ¹⁄ₐ = √3, the expression a⁶ + 1 = 0 holds true, which means a³ + (1)/(a³) = (a⁶ + 1)/(a³) = 0.

Explanation

The property of equal ratios states that if (x₁)/(y₁) = (x₂)/(y₂) = … = (xₙ)/(yₙ) = λ, then λ is also equal to the ratio of the sum of the numerators to the sum of the denominators, such that λ = (Σ xᵢ)/(Σ yᵢ). Applying this property to the given equation:

¹⁄ₖ = ((a² + b²) + (b² + c²) + (c² + a²))/(c² + a² + b²)

The numerator consists of two instances of each squared term, which can be factored:

¹⁄ₖ = (2a² + 2b² + 2c²)/(a² + b² + c²)

¹⁄ₖ = (2(a² + b² + c²))/(a² + b² + c²)

Provided that a² + b² + c² ≠ 0, the common term in the numerator and denominator can be cancelled:

¹⁄ₖ = 2

By taking the reciprocal of both sides to solve for k:

k = ½

Explanation

To maximize the area of a triangle inscribed in a semicircle, the base of the triangle must coincide with the diameter of the semicircle. In this configuration, the base b is equal to 2r, where r is the radius. The area of a triangle is given by the formula A = ½ × base × height. For any point on the arc of the semicircle, the height h is the perpendicular distance to the diameter. This height is maximized when the vertex is at the midpoint of the arc, making the height equal to the radius r. Substituting these values into the area formula gives A = ½ × (2r) × r = r². Given the radius r = 6 m, the area is 6² = 36 m².

Explanation

To simplify the expression (sinθ)/(1+cosθ) + (sinθ)/(1-cosθ), we first find a common denominator by multiplying the two denominators:

(1+cosθ)(1-cosθ) = 1 - cos²θ

Using the fundamental trigonometric identity sin²θ + cos²θ = 1, we can substitute 1 - cos²θ with sin²θ. Now, we combine the fractions over this common denominator:

(sinθ(1-cosθ) + sinθ(1+cosθ))/(sin²θ)

Expanding the terms in the numerator gives:

(sinθ - sinθ + sinθ + sinθ)/(sin²θ)

The terms -sinθ and +sinθ cancel each other out, leaving:

(2sinθ)/(sin²θ)

By dividing both the numerator and denominator by sinθ, the expression reduces to:

(2)/(sinθ)

Since the reciprocal of sinθ is defined as cscθ, the expression is equivalent to 2cscθ.

Explanation

The total work performed is the product of the number of individuals, the time duration, and the individual capacity or efficiency. Let M be the capacity of a man and W be the capacity of a woman. Since the same work is being completed in both instances, the total work units are equated as follows:

20 × W × 16 = 16 × M × 15

Dividing both sides of the equation by 16 simplifies the expression to:

20 × W = 15 × M

To determine the ratio of the capacity of a man to that of a woman, the equation is rearranged to solve for M/W:

(M)/(W) = ²⁰⁄₁₅

By dividing both the numerator and denominator by their greatest common divisor, 5, the fraction simplifies to:

(M)/(W) = ⁴⁄₃

Explanation

Given the equation 2x + ²⁄₉ₓ = 4, we first simplify it by dividing both sides by 2 to get x + ¹⁄₉ₓ = 2. To transform this into the required form for the target expression 27x³ + (1)/(27x³), which is equivalent to (3x)³ + (1)/((3x)³), we multiply the entire simplified equation by 3:

3 · (x + ¹⁄₉ₓ) = 3 · 2 3x + ¹⁄₃ₓ = 6

Using the algebraic identity a³ + b³ = (a+b)³ - 3ab(a+b), where a = 3x and b = ¹⁄₃ₓ, we can substitute the values. For any expression in the form a + ¹⁄ₐ = k, the sum of their cubes is given by k³ - 3k. Substituting k = 6:

27x³ + (1)/(27x³) = (3x + ¹⁄₃ₓ)³ - 3(3x)(¹⁄₃ₓ)(3x + ¹⁄₃ₓ) 27x³ + (1)/(27x³) = 6³ - 3(1)(6) 27x³ + (1)/(27x³) = 216 - 18 27x³ + (1)/(27x³) = 198

Explanation

In a cyclic quadrilateral, the sum of opposite angles is 180°. Given ∠ BCD = 120°, the opposite angle ∠ BAD is calculated as: ∠ BAD = 180° - ∠ BCD = 180° - 120° = 60°

The problem states that a side passes through the centre of the circle, which implies it is a diameter. Taking AB as the diameter, the angle subtended by the diameter at any point on the circle's circumference is a right angle (90°). Thus: ∠ ADB = 90°

In △ ABD, the sum of all internal angles is 180°. Using the values for ∠ BAD and ∠ ADB: ∠ ABD = 180° - (∠ BAD + ∠ ADB) ∠ ABD = 180° - (60° + 90°) ∠ ABD = 180° - 150° = 30°

Explanation

According to the Midpoint Theorem, the line segment joining the midpoints of two sides of a triangle is parallel to the third side and is half the length of that third side. In triangle ABC, since X and Y are the midpoints of AB and AC, we have the relationship XY = ½ BC, or BC = 2XY.

Given that BC + XY = 12 units, we can substitute BC with 2XY: 2XY + XY = 12 3XY = 12 XY = 4 units.

Substituting the value of XY back into the relationship BC = 2XY, we find: BC = 2 × 4 = 8 units.

The value of BC - XY is calculated as: 8 - 4 = 4 units.

Explanation

In an isosceles Δ ABC where AB = AC, the median AD to the unequal side BC is also the altitude. Therefore, AD ⊥ BC, which implies ∠ ADB = ∠ ADC = 90°. Since DP is the angle bisector of ∠ ADB, we have ∠ ADP = ½ ∠ ADB = ½(90°) = 45°.

By the Angle Bisector Theorem in Δ ADB, the ratio of the sides is AP/PB = AD/DB. Since PQ ∥ BC, by the Basic Proportionality Theorem, we have AP/PB = AQ/QC. Given that D is the midpoint of BC, we know DB = DC. Substituting these into the ratios, we get AQ/QC = AD/DC.

According to the converse of the Angle Bisector Theorem in Δ ADC, since AQ/QC = AD/DC, the line segment DQ must be the angle bisector of ∠ ADC. Thus, ∠ ADQ = ½ ∠ ADC = ½(90°) = 45°. The measure of ∠ PDQ is the sum of ∠ ADP and ∠ ADQ, which results in 45° + 45° = 90°.

Explanation

The height of the cliff, h, can be determined using the trigonometric ratio for tangent in a right-angled triangle. In this scenario, the height of the cliff represents the perpendicular (opposite) side and the distance from the foot of the cliff represents the base (adjacent) side. Given the angle of elevation θ = 30° and the distance d = 129 metres:

tan θ = (h)/(d)

tan 30° = ʰ⁄₁₂₉

Substituting the value tan 30° = (1)/(√(3)):

(1)/(√(3)) = ʰ⁄₁₂₉

h = (129)/(√(3))

To simplify, rationalise the denominator by multiplying both the numerator and the denominator by √3:

h = 129 × √3{√3 × √3}

h = 129√3{3}

h = 43√3 metres

Explanation

The volume V of a hollow cylindrical pipe is given by the formula V = π(R² - r²)h, where R is the external radius, r is the internal radius, and h is the length of the pipe. Substituting the given values V = 748 cm³, R = 9 cm, and h = 14 cm into the formula, we get:

748 = ²²⁄₇ × (9² - r²) × 14

748 = 22 × (81 - r²) × 2

748 = 44 × (81 - r²)

Dividing 748 by 44:

17 = 81 - r²

r² = 81 - 17 = 64

r = √(64) = 8 cm

The thickness of the pipe is the difference between the external radius and the internal radius:

Thickness = R - r = 9 - 8 = 1 cm

Explanation

The work done is constant and can be calculated using the chain rule formula M₁ × D₁ × H₁ = M₂ × D₂ × H₂, where M is the number of men, D is the number of days, and H is the number of hours worked per day. Substituting the given values:

20 × 21 × 8 = 48 × 7 × H₂

To find H₂, rearrange the equation:

H₂ = (20 × 21 × 8)/(48 × 7)

Simplifying the calculation, 21 divided by 7 is 3, and 48 divided by 8 is 6. The equation becomes:

H₂ = (20 × 3)/(6)

H₂ = ⁶⁰⁄₆

H₂ = 10

Explanation

In triangle △ ABD, the sum of interior angles is 180°. Therefore, ∠ DAB = 180° - (∠ DBA + ∠ ADB). Substituting the given values, we get ∠ DAB = 180° - (50° + 33°) = 97°. In a cyclic quadrilateral, the sum of opposite angles is 180°. Since ∠ DAB and ∠ BCD are opposite angles, ∠ BCD = 180° - ∠ DAB. Substituting the calculated value, ∠ BCD = 180° - 97° = 83°. This can also be simplified by observing that ∠ BCD = 180° - [180° - (∠ DBA + ∠ ADB)], which reduces to ∠ BCD = ∠ DBA + ∠ ADB = 50° + 33° = 83°.

Explanation

The relationship between Cost Price (CP) and Printed Price (MP) when a discount (D%) and profit (P%) are given is expressed by the ratio: (CP)/(MP) = (100 - D)/(100 + P)

Given that the discount is 10% and the profit earned is 12%, substituting these values into the formula gives: (CP)/(MP) = (100 - 10)/(100 + 12) (CP)/(MP) = ⁹⁰⁄₁₁₂

By simplifying the fraction by dividing both terms by their greatest common divisor, 2, we obtain: (90 ÷ 2)/(112 ÷ 2) = ⁴⁵⁄₅₆

This results in the ratio 45:56.

Explanation

The total number of pupils is 55, and they are divided into the ratio of male to female as 5:6. To find the number of female pupils, calculate the fraction of females relative to the total ratio parts. The sum of the ratio parts is 5 + 6 = 11. Using the proportion formula, the number of female pupils is given by ⁶⁄₁₁ × 55. Dividing 55 by 11 gives 5, and multiplying 6 by 5 results in 30.

Explanation

The difference between the two selling prices is Rs. 350 - Rs. 340 = Rs. 10. According to the problem, this difference of Rs. 10 represents a 5% increase in gain. Since profit or gain percentage is always calculated on the cost price (CP), we can establish the following relationship:

5% of CP = Rs. 10

To find the cost price, we use the formula:

⁵⁄₁₀₀ × CP = 10

CP = (10 × 100)/(5)

CP = 2 × 100 = 200

By selling the watch for Rs. 350 instead of Rs. 340, the seller earns an extra Rs. 10, which corresponds exactly to 5% of the Rs. 200 cost price.

Explanation

Let the number be x. The given condition can be written as the algebraic equation x + ¹⁄ₓ = 2. Multiplying the entire equation by x to eliminate the fraction results in x² + 1 = 2x. By shifting all terms to one side, we obtain the quadratic equation x² - 2x + 1 = 0.

This quadratic expression is a perfect square following the algebraic identity (a - b)² = a² - 2ab + b², where a = x and b = 1. Therefore, the equation simplifies to (x - 1)² = 0. Solving for x by taking the square root of both sides, we find that x - 1 = 0, which yields x = 1.

Alternatively, applying the Arithmetic Mean-Geometric Mean (AM-GM) inequality for a positive number x, the sum x + ¹⁄ₓ is always greater than or equal to 2√(x · ¹⁄ₓ) = 2. The equality x + ¹⁄ₓ = 2 holds if and only if the two terms are equal, meaning x = ¹⁄ₓ, which implies x² = 1. Given the sum is positive 2, x must be 1.

Explanation

Let the length and width of the rectangle be l and w respectively. The perimeter is given as 2(l + w) = 34 cm, which simplifies to l + w = 17 cm. The area is given as lw = 60 cm². The length of the diagonal d is determined by the Pythagorean theorem as d = √(l² + w²).

Using the algebraic identity (l + w)² = l² + w² + 2lw, we can express the square of the diagonal as d² = l² + w² = (l + w)² - 2lw. Substituting the known values:

d² = 17² - 2(60) d² = 289 - 120 d² = 169

Taking the square root of both sides, d = √(169) = 13 cm. Alternatively, this corresponds to the Pythagorean triplet (5, 12, 13) where 5 + 12 = 17 and 5 × 12 = 60.

Explanation

The rebate offered per metre of cloth is 25% of its cost price. Given the cost is ₹ 32 per metre, the rebate per metre is calculated as: Rebate per metre = 32 × ²⁵⁄₁₀₀ = 32 × ¼ = ₹ 8

To achieve a total rebate of ₹ 40, the length of cloth required is determined by dividing the total desired rebate by the rebate available per metre: Length of cloth = ⁴⁰⁄₈ = 5 metres

Explanation

The new average salary can be calculated using the deviation method. The new member brings Rs. 20000, which is Rs. 4000 more than the existing average of Rs. 16000. This surplus amount is distributed equally among all members of the new group, which now consists of 20 people (19 + 1 = 20).

Increase in average = (Surplus Salary)/(Total Members) = (20000 - 16000)/(20) = ⁴⁰⁰⁰⁄₂₀ = 200.

New average = Old Average + Increase = 16000 + 200 = 16200.

Alternatively, using the sum method: Total salary of 19 members = 19 × 16000 = 304000. Total salary of 20 members = 304000 + 20000 = 324000. New Average = ³²⁴⁰⁰⁰⁄₂₀ = 16200.

Explanation

The cost price is determined by the relationship between the incremental profit and the difference in profit percentages. The formula for the cost price in such cases is: Cost Price = (Change in Profit Amount)/(Change in Profit Percentage) × 100

Given that the profit increases by Rs. 1000 when the profit percentage increases from 5% to 10%, the change in profit percentage is 10% - 5% = 5%. Substituting these values into the formula: Cost Price = ¹⁰⁰⁰⁄₅ × 100 Cost Price = 200 × 100 Cost Price = 20000

Explanation

The speed of the train can be directly calculated using the formula v = Lₚ{t₁ - t₂}, where Lₚ is the length of the platform, t₁ is the time taken to cross the platform, and t₂ is the time taken to cross the lamp post. This is because the additional distance covered in the extra time is exactly the length of the platform.

Substituting the given values: v = (50)/(14 - 10) = ⁵⁰⁄₄ = 12.5 m/s

To convert the speed from m/s to km/h, multiply by the conversion factor ¹⁸⁄₅: Speed in km/h = 12.5 × ¹⁸⁄₅ = 2.5 × 18 = 45 km/h

Explanation

Using the algebraic identity (x - 1/x)² = x² + 1/x² - 2(x)(1/x), we simplify the expression to (x - 1/x)² = x² + 1/x² - 2. Substituting the given value x² + 1/x² = 2 into this identity, we get (x - 1/x)² = 2 - 2 = 0. Taking the square root of both sides leads to x - 1/x = 0. Alternatively, the condition x² + 1/x² = 2 is satisfied only when x² = 1, which implies x = 1 or x = -1. Substituting x = 1 gives 1 - 1/1 = 0, and substituting x = -1 gives -1 - 1/(-1) = -1 + 1 = 0.

Explanation

The given equation is pq(p + q) = 1. Dividing both sides by pq, we obtain the relation p + q = (1)/(pq).

To find the value of the target expression, we cube both sides of this relation: (p + q)³ = ((1)/(pq))³

Using the identity (a + b)³ = a³ + b³ + 3ab(a + b), the left side expands as follows: p³ + q³ + 3pq(p + q) = (1)/(p³q³)

We now substitute the original condition pq(p + q) = 1 into the equation: p³ + q³ + 3(1) = (1)/(p³q³) p³ + q³ + 3 = (1)/(p³q³)

Rearranging the terms to isolate the expression (1)/(p³q³) - p³ - q³: (1)/(p³q³) - p³ - q³ = 3

Explanation

The total percentage of students in a school is 100%. Given that 60% of the students are boys, the percentage of girls must be 100% - 60% = 40%.

The problem states that the actual number of girls is 812. Since 40% of the total students is equal to 812, we can find the value of 1% of the total student population by dividing 812 by 40: 1% of students = ⁸¹²⁄₄₀ = 20.3

To find the number of boys, which constitutes 60% of the total students, we multiply this 1% value by 60: Number of boys = 60 × 20.3 = 1218

Alternatively, using the direct ratio between boys and girls: Number of boys = (60%)/(40%) × 812 = ³⁄₂ × 812 = 3 × 406 = 1218

Explanation

A perpendicular line drawn from the center of a circle to a chord always bisects that chord. This construction creates a right-angled triangle where the hypotenuse is the radius of the circle (r = 10 cm), one leg is half the length of the chord (16/2 = 8 cm), and the other leg is the distance of the chord from the center (d).

Using the Pythagorean theorem a² + b² = c²:

d² + 8² = 10² d² + 64 = 100 d² = 100 - 64 d² = 36 d = √(36) = 6 cm

This result can also be determined using the standard 3:4:5 Pythagorean triplet. Since the hypotenuse is 10 (5 × 2) and one leg is 8 (4 × 2), the required distance must be the third side of the triplet, 3 × 2 = 6 cm.

Explanation

The Inscribed Angle Theorem states that the angle subtended by an arc at the center of a circle is twice the angle subtended by the same arc at any point on the circle's circumference. In a semicircle, the arc is a diameter, and the angle subtended by a diameter at the center of the circle is a straight angle, which measures 180°. Therefore, the angle subtended by the diameter at any point on the circumference of the semicircle is half of the central angle, calculated as ½ × 180° = 90°. This specific case of the inscribed angle theorem is also widely known as Thales's Theorem.

Explanation

By squaring both equations, we get x² = a² cos² θ + b² sin² θ + 2ab sin θ cos θ and y² = b² cos² θ + a² sin² θ - 2ab sin θ cos θ. Adding them yields x² + y² = (a² + b²)(cos² θ + sin² θ), which simplifies to a² + b² because cos² θ + sin² θ equals 1.

Explanation

The share of money earned is proportional to the efficiency of the workers when they work for the same amount of time. Efficiency is the reciprocal of the time taken to complete the work. The efficiency of A is 1/12 and the efficiency of B is 1/15.

The ratio of the efficiency of A to B is: Efficiency_A : Efficiency_B = ¹⁄₁₂ : ¹⁄₁₅

To simplify this ratio, multiply both terms by the least common multiple of 12 and 15, which is 60: Ratio = ⁶⁰⁄₁₂ : ⁶⁰⁄₁₅ = 5 : 4

The total number of ratio parts is 5 + 4 = 9. A's share corresponds to 5 parts out of the total 9 parts of the Rs. 450 payment: A's share = ⁵⁄₉ × 450 A's share = 5 × 50 = 250

A receives Rs. 250 as his share of the total amount.

Explanation

Given x = √2 + 1, the reciprocal is 1/x = (1)/(√(2) + 1). Rationalizing the denominator gives 1/x = √2 - 1{(√2 + 1)(√2 - 1)} = √2 - 1{2 - 1} = √2 - 1.

We calculate the sum and difference of x and 1/x: x + 1/x = (√2 + 1) + (√2 - 1) = 2√2 x - 1/x = (√2 + 1) - (√2 - 1) = 2

To find x⁴ - 1/x⁴, we use the algebraic identity for the difference of squares: x⁴ - 1/x⁴ = (x² + 1/x²)(x² - 1/x²)

First, find x² + 1/x² using the relation (x + 1/x)² - 2: x² + 1/x² = (2√2)² - 2 = 8 - 2 = 6

Next, find x² - 1/x² using (x + 1/x)(x - 1/x): x² - 1/x² = (2√2)(2) = 4√2

Substituting these values back into the factored expression: x⁴ - 1/x⁴ = 6 × 4√2 = 24√2

Explanation

The algebraic identity (a + b)² = a² + b² + 2ab can be applied here by substituting a = 3x and b = 4y. Squaring the given equation 3x + 4y = 6 results in:

(3x + 4y)² = (3x)² + (4y)² + 2(3x)(4y)

Substituting the known values into the expanded equation:

6² = 9x² + 16y² + 24xy

Since 9x² + 16y² = 60, the equation becomes:

36 = 60 + 24xy

Rearranging the terms to solve for xy:

24xy = 36 - 60 24xy = -24 xy = -24 / 24 xy = -1

Explanation

In Δ ABC, since DE ∥ BC, Δ ADE ∼ Δ ABC based on the AA (Angle-Angle) similarity criterion. Consequently, the ratios of their corresponding sides are equal:

(AD)/(AB) = (DE)/(BC)

Given AB = 7.5 cm and BD = 6 cm, the length of AD is: AD = AB - BD = 7.5 - 6 = 1.5 cm

Substituting the known values into the similarity ratio: (1.5)/(7.5) = (2)/(BC)

Simplifying the ratio: ⅕ = (2)/(BC)

Solving for BC: BC = 2 × 5 = 10 cm

Explanation

In any triangle, the medians intersect at a single point called the centroid (G). A fundamental property of the centroid is that it divides each median in a 2:1 ratio, where the segment originating from the vertex is twice as long as the segment reaching the midpoint of the opposite side.

This can be proven using the section formula in coordinate geometry. Let the vertices be A(x₁, y₁), B(x₂, y₂), and C(x₃, y₃). The midpoint F of side BC is ((x₂+x₃)/(2), (y₂+y₃)/(2)). The coordinates of the centroid G are ((x₁+x₂+x₃)/(3), (y₁+y₂+y₃)/(3)). If G divides the median AF in the ratio k:1, then by the section formula:

G = (kF + 1A)/(k+1)

Substituting the x-coordinates:

(x₁+x₂+x₃)/(3) = (k(x₂+x₃)/(2)) + x₁{k+1}

For this identity to hold for all x₁, x₂, x₃, the coefficients must match. Comparing the x₁ terms gives k+1 = 3, which results in k = 2. Thus, the ratio AG:GF is 2:1.