SSC CGL Tier 1 2016 Quantitative Aptitude Questions with Solutions

Explanation

Given the equation (x + ¹⁄ₓ)² = 3, taking the square root on both sides yields x + ¹⁄ₓ = ±√3. To find the value of x³ + (1)/(x³), we use the algebraic identity a³ + b³ = (a + b)³ - 3ab(a + b). Substituting a = x and b = ¹⁄ₓ, the expression becomes x³ + (1)/(x³) = (x + ¹⁄ₓ)³ - 3(x)(¹⁄ₓ)(x + ¹⁄ₓ), which simplifies to (x + ¹⁄ₓ)³ - 3(x + ¹⁄ₓ).

Substituting x + ¹⁄ₓ = √3 into the identity, we get (√3)³ - 3(√3) = 3√3 - 3√3 = 0. Similarly, substituting x + ¹⁄ₓ = -√3 results in (-√3)³ - 3(-√3) = -3√3 + 3√3 = 0. In both cases, the result is consistently 0.

Explanation

The product ab is calculated using the algebraic identity (a - b)² = a² + b² - 2ab. This identity allows for the direct derivation of the product when the difference and the sum of squares are known.

Rearranging the identity to solve for the term ab: ab = ((a² + b²) - (a - b)²)/(2)

Substituting the provided values a² + b² = 25 and a - b = 3 into the rearranged formula: ab = (25 - 3²)/(2) ab = (25 - 9)/(2) ab = ¹⁶⁄₂ ab = 8

Explanation

In Δ ABC, the sum of all internal angles is 180°. Given ∠ B = 70° and ∠ C = 60°, the third angle ∠ A is calculated as: ∠ A = 180° - (∠ B + ∠ C) = 180° - (70° + 60°) = 50°

Comparing the three angles, the two smallest angles are ∠ A = 50° and ∠ C = 60°. The internal bisectors of these two angles meet at point O, which is the incenter of the triangle. According to the property of the incenter, the angle formed by the internal bisectors of two angles (say ∠ A and ∠ C) is given by the formula: ∠ AOC = 90° + (∠ B)/(2)

Substituting the value of the third angle ∠ B: ∠ AOC = 90° + (70°)/(2) = 90° + 35° = 125°

Alternatively, in Δ AOC, the base angles are half of the original angles: (∠ A)/(2) = (50°)/(2) = 25° and (∠ C)/(2) = (60°)/(2) = 30°. Using the angle sum property for Δ AOC: ∠ AOC = 180° - (25° + 30°) = 180° - 55° = 125°

Explanation

The given equation 5cosθ + 12sinθ = 13 can be analyzed using the property of the expression acosθ + bsinθ = c. When c = √(a² + b²), the equation represents a specific case where the coefficients a, b, and c form a Pythagorean triple. Here, 5² + 12² = 25 + 144 = 169, which is exactly 13².

Dividing the original equation by 13 yields ⁵⁄₁₃cosθ + ¹²⁄₁₃sinθ = 1. This can be compared to the fundamental trigonometric identity cos²θ + sin²θ = 1, which is cosθ(cosθ) + sinθ(sinθ) = 1. By direct comparison of the coefficients, we identify cosθ = ⁵⁄₁₃ and sinθ = ¹²⁄₁₃. This result is consistent for a positive acute angle θ as both values are positive and their squares sum to unity.

Explanation

When sand is poured from a cylindrical container to form a conical heap, the volume of the sand remains constant. The volume of a cylinder is calculated using the formula V₁ = π r₁² h₁, and the volume of a cone is calculated using V₂ = ⅓ π r₂² h₂.

Given the dimensions for the cylinder are r₁ = 18 cm and h₁ = 32 cm, and for the cone h₂ = 24 cm, we equate the volumes: π × 18² × 32 = ⅓ × π × r₂² × 24

Dividing both sides by π: 18 × 18 × 32 = 8 × r₂²

Solving for r₂²: r₂² = (18 × 18 × 32)/(8) r₂² = 18² × 4

Taking the square root of both sides to find the radius of the conical heap: r₂ = √(18² × 2²) r₂ = 18 × 2 r₂ = 36 cm.

Explanation

Let H be the height of the pillar and d be the horizontal distance between the building and the pillar. From the foot of the 20 m high building, the angle of elevation to the top of the pillar is 60°. Using the trigonometric ratio tan θ = perpendicular / base, we have tan 60° = H/d, which gives d = H/√3.

From the top of the building, the elevation angle to the top of the pillar is 30°. The vertical height from the level of the building's top to the pillar's top is H - 20. Thus, tan 30° = (H - 20)/d. Substituting the expression for d into this equation, we get 1/√3 = (H - 20) / (H/√3). Cross-multiplying leads to H/3 = H - 20. Rearranging the terms gives H - H/3 = 20, which simplifies to 2H/3 = 20. Solving for H results in H = 30 m.

Explanation

Efficiency and the time taken to complete a job are inversely proportional to each other. Let the efficiency of A be 100%. Since B is 60% more efficient than A, the efficiency of B is 100% + 60% = 160%.

Using the relation E₁ × T₁ = E₂ × T₂: 100 × 12 = 160 × T_B T_B = (100 × 12)/(160) T_B = (10 × 12)/(16) = ¹²⁰⁄₁₆ Dividing both the numerator and the denominator by 8: T_B = ¹⁵⁄₂ = 7 ½ days

Alternatively, the ratio of the efficiency of A to B is 100 : 160, which simplifies to 5 : 8. Because time taken is inversely proportional to efficiency, the ratio of time taken by A to B is 8 : 5. Given that A takes 12 days, we can equate 8 units to 12 days. This means 1 unit equals 1.5 days. Therefore, the time taken by B is 5 × 1.5 = 7.5 days.

Explanation

When a rectangle is inscribed in a circle, the diagonal of the rectangle coincides with the diameter of the circle. Given the radius r = 2.5 cm, the diameter d is calculated as: d = 2r = 2 × 2.5 = 5 cm.

Let the sides of the rectangle be a and b. According to the Pythagorean theorem, the relationship between the sides and the diagonal is: a² + b² = d²

Substituting the known side a = 4 cm and the diameter d = 5 cm into the equation: 4² + b² = 5² 16 + b² = 25 b² = 25 - 16 = 9 b = 3 cm.

The area of the rectangle is the product of its adjacent sides: Area = a × b = 4 cm × 3 cm = 12 cm².

Explanation

The net price is calculated by applying the successive discounts to the marked price. Using the formula for successive discounts, the final price is MP × (1 - d₁/100) × (1 - d₂/100).

Substituting the given values: Net Price = 50 × (1 - 15/100) × (1 - 20/100) Net Price = 50 × (85/100) × (80/100) Net Price = 50 × 0.85 × 0.8

Calculating step-by-step: 50 × 0.8 = 40 40 × 0.85 = 34

Alternatively, the single equivalent discount for two successive discounts of x% and y% is given by the formula x + y - (xy/100). Equivalent discount = 15 + 20 - (15 × 20)/100 = 35 - 3 = 32%. Net price = 50 - (32% of 50) = 50 - 16 = 34.

Explanation

Let the share of the second brother be S₂ and the combined share of the first and third brothers be S₁₊₃. The problem states that S₂ = ⁵⁄₁₃ S₁₊₃, which establishes the ratio S₂ : S₁₊₃ = 5 : 13. Since the total amount of Rs. 1620 represents the sum of all three shares, it corresponds to 5 + 13 = 18 equal parts in this ratio. The second brother's share is 5 parts out of these 18 total parts. Using the formula for proportional division, the share is calculated as ⁵⁄₁₈ × 1620. Dividing 1620 by 18 gives 90, and multiplying 90 by 5 yields Rs. 450.

Explanation

Let the Cost Price (CP) of the goods be 100. The dealer marks the goods 20% above the CP, so the Marked Price (MP) becomes 100 + 20 = 120. The dealer wants to achieve a profit of 8%, which means the Selling Price (SP) must be 100 + 8 = 108.

The discount is the difference between the Marked Price and the Selling Price: Discount = MP - SP = 120 - 108 = 12

The rate of discount is calculated on the Marked Price using the formula: Discount Rate = (Discount)/(MP) × 100 Discount Rate = ¹²⁄₁₂₀ × 100 = 10%

Explanation

Let the original salary of the person be 100. After a reduction of 20%, the present salary becomes 100 - 20 = 80. To restore the salary to its original value of 100, an increase of 100 - 80 = 20 is required on the present salary of 80.

The required percentage increase is calculated using the formula: Percentage Increase = (Increase)/(Present Salary) × 100 Substituting the values: Percentage Increase = ²⁰⁄₈₀ × 100 = ¼ × 100 = 25%

Alternatively, if a value is decreased by r%, the percentage increase needed to restore the original value is given by the formula (r)/(100 - r) × 100. Substituting r = 20: Required Increase = (20)/(100 - 20) × 100 = ²⁰⁄₈₀ × 100 = 25%

Explanation

The distance d for a journey where two different speeds v₁ and v₂ result in a time difference Δ t is calculated using the formula d = v₁ × v₂{v₂ - v₁} × Δ t. Substituting the given values v₁ = 45 km/hr, v₂ = 60 km/hr, and Δ t = 5.5 hours:

d = (45 × 60)/(60 - 45) × 5.5

d = ²⁷⁰⁰⁄₁₅ × 5.5

d = 180 × 5.5

d = 990

Alternatively, using the ratio method, the ratio of speeds is 45:60, which simplifies to 3:4. Since distance is constant, the ratio of time taken is the inverse of the speed ratio, which is 4:3. The difference in time units is 4 - 3 = 1 unit. According to the problem, this 1 unit equals 5.5 hours. Therefore, the time taken by the first bus is 4 × 5.5 = 22 hours. The total distance is calculated by multiplying the speed of the first bus by its time: 45 km/hr × 22 hours = 990 km.

Explanation

The given ratio can be expressed as the equation (x + 1)/(x){x - ¹⁄ₓ} = ⁵⁄₃. To simplify the expression, multiply the numerator and the denominator by x: (x² + 1)/(x² - 1) = ⁵⁄₃

Applying cross-multiplication: 3(x² + 1) = 5(x² - 1) 3x² + 3 = 5x² - 5

Rearrange the terms to solve for x²: 5x² - 3x² = 3 + 5 2x² = 8 x² = 4

Taking the square root of both sides: x = ± 2

Explanation

According to the Exterior Angle Theorem, the measure of an exterior angle of a triangle is equal to the sum of the measures of its two interior opposite angles. Given that the exterior angle is 115° and one interior opposite angle is 45°, the second interior opposite angle is calculated as 115° - 45° = 70°.

Additionally, the interior angle adjacent to the exterior angle forms a linear pair with it, meaning their sum is 180°. Therefore, the adjacent interior angle is 180° - 115° = 65°. The other two angles of the triangle are thus 65° and 70°.

Explanation

The sum of the interior angles of a triangle is always 180°, which gives the relation ∠ A + ∠ B + ∠ C = 180°. The problem provides two equations: ∠ A + ∠ B = 75° and ∠ B + ∠ C = 140°. By adding these two given equations together, we obtain ∠ A + 2∠ B + ∠ C = 75° + 140° = 215°. This expression can be partitioned into (∠ A + ∠ B + ∠ C) + ∠ B = 215°. Substituting the standard triangle angle sum of 180° into this equation yields 180° + ∠ B = 215°. Solving for the unknown angle, we find ∠ B = 215° - 180° = 35°.

Explanation

The given expression is (1+tan²θ)(1-sin²θ). According to fundamental trigonometric identities, 1 + tan²θ = sec²θ and 1 - sin²θ = cos²θ. By substituting these identities into the expression, we obtain sec²θ · cos²θ. Since the secant function is the reciprocal of the cosine function, where secθ = (1)/(cosθ), the product simplifies to (1)/(cos²θ) · cos²θ, which yields a final value of 1.

Explanation

A quadratic expression ax² + bx + c is a perfect square if its discriminant D = b² - 4ac is equal to zero. For the given expression 4x² - 12x + k, the coefficients are a = 4, b = -12, and c = k.

Setting the discriminant to zero: (-12)² - 4(4)(k) = 0 144 - 16k = 0 16k = 144 k = ¹⁴⁴⁄₁₆ = 9

Alternatively, the expression can be compared to the algebraic identity (A - B)² = A² - 2AB + B². Here, A² = 4x², which means A = 2x. The middle term 2AB corresponds to 12x, so 2(2x)(B) = 12x. This simplifies to 4xB = 12x, resulting in B = 3. The constant term k must equal B², which gives k = 3² = 9.

Explanation

The given expression is in the form of the algebraic identity a² + b² + c² - ab - bc - ca. This expression can be simplified using the standard identity: a² + b² + c² - ab - bc - ca = ½ [(a - b)² + (b - c)² + (c - a)²]

By assigning the values a = 113, b = 115, and c = 117, we can calculate the differences between the terms: a - b = 113 - 115 = -2 b - c = 115 - 117 = -2 c - a = 117 - 113 = 4

Now, substitute these differences into the simplified identity: ½ [(-2)² + (-2)² + (4)²] ½ [4 + 4 + 16] ½ [24] 12

Using this identity avoids the tedious calculation of large squares and products, providing the result directly through the differences of the numbers.

Explanation

The runs required in the 100th innings can be calculated by finding the difference between the total runs after 100 innings and the total runs after 99 innings. The total runs scored in 99 innings is the product of the number of innings and the average, which is 99 × 99 = 9801. To achieve a target average of 100 runs over 100 innings, the total runs must reach 100 × 100 = 10000. The runs to be scored in the 100th innings is 10000 - 9801 = 199.

A faster approach uses the concept of average deviation. To raise the average of the previous 99 innings from 99 to 100, the cricketer must score the new average of 100 plus 1 additional run for each of the 99 previous innings. This is calculated as 100 + (99 × 1) = 199.

Explanation

To find the value of r², we use the trigonometric identity sin²θ + cos²θ = 1. Squaring and adding the two given equations, rsinθ = 1 and rcosθ = √3, yields (rsinθ)² + (rcosθ)² = 1² + (√3)². This simplifies to r²(sin²θ + cos²θ) = 1 + 3, which gives r² = 4.

To find tanθ, we divide the first equation by the second: (rsinθ)/(rcosθ) = (1)/(√(3)), which simplifies to tanθ = (1)/(√(3)).

The required value is the product of these two results: r²tanθ = 4 × (1)/(√(3)) = (4)/(√(3)).

Explanation

The standard algebraic identity for the difference of cubes is p³ - q³ = (p - q)(p² + pq + q²). The expression given in the question is p³ - q³ = (p - q)((p - q)² + x pq). Expanding the term inside the second bracket, we have (p - q)² + x pq = p² - 2pq + q² + x pq = p² + (x - 2)pq + q².

By comparing the resulting expression (p - q)(p² + (x - 2)pq + q²) with the identity (p - q)(p² + pq + q²), the coefficients of the pq term must be equal. Therefore, x - 2 = 1, which simplifies to x = 3. Alternatively, using the identity p³ - q³ = (p - q)³ + 3pq(p - q), we can factor out (p - q) to get (p - q)((p - q)² + 3pq). Comparing this directly with the given form (p - q)((p - q)² + x pq) confirms that x = 3.

Explanation

The given equation (a + ¹⁄ₐ)² = 3 implies that a + ¹⁄ₐ = √3. Using the algebraic identity (x + y)³ = x³ + y³ + 3xy(x + y), we cube both sides of the relation a + ¹⁄ₐ = √3 to find a relationship for a⁶.

(a + ¹⁄ₐ)³ = (√3)³ a³ + (1)/(a³) + 3(a)(¹⁄ₐ)(a + ¹⁄ₐ) = 3√3 a³ + (1)/(a³) + 3√3 = 3√3 a³ + (1)/(a³) = 0

Multiplying the entire equation by a³ gives a⁶ + 1 = 0, which means a⁶ = -1. The expression to be evaluated is a¹⁸ + a¹² + a⁶ + 1. By substituting a⁶ = -1 into this expression, we get:

(a⁶)³ + (a⁶)² + (a⁶) + 1 = (-1)³ + (-1)² + (-1) + 1 -1 + 1 - 1 + 1 = 0

Explanation

The standing portion of the tree and the distance from the base to the fallen top form the two legs of a right-angled triangle, while the broken part of the tree acts as the hypotenuse. Using the Pythagoras theorem, where h = 8 m is the height of the break and d = 15 m is the distance from the base to the top:

Length of broken part L = √(h² + d²) L = √(8² + 15²) = √(64 + 225) = √(289) = 17 m

The actual height of the tree is the sum of the standing part and the broken part that fell over: Total height = h + L = 8 + 17 = 25 m

Explanation

In Δ PQR, since XY is parallel to the base QR, Δ PXY is similar to Δ PQR by the AA (Angle-Angle) similarity criterion, as ∠ P is common and ∠ PXY = ∠ PQR are corresponding angles. For similar triangles, the ratio of corresponding sides is equal:

(XY)/(QR) = (PX)/(PQ)

The problem provides the ratio PX:XQ = 5:6. Let PX = 5k and XQ = 6k. The length of the side PQ is the sum of its segments PX and XQ:

PQ = PX + XQ = 5k + 6k = 11k

By substituting the expressions for PX and PQ into the similarity ratio, we get:

(XY)/(QR) = ⁵ᵏ⁄₁₁ₖ = ⁵⁄₁₁

This confirms that the ratio XY:QR is 5:11.

Explanation

Let the principal amount borrowed be P. The simple interest (SI) paid by the man over 4 years at a rate of 5% per annum is calculated as SI = (P × 5 × 4)/(100) = 0.20P.

The compound interest (CI) earned by the man by lending the same amount at 10% per annum for 4 years is determined by the effective rate. For a rate of 10% compounded annually for 4 years, the total interest is CI = P[(1 + ¹⁰⁄₁₀₀)⁴ - 1]. This simplifies to CI = P[(1.1)⁴ - 1] = P[1.4641 - 1] = 0.4641P.

The profit made by the man is the difference between the interest he received and the interest he paid. Therefore, Profit = 0.4641P - 0.20P = 0.2641P. Given that the total profit is ₹ 26,410, we equate the two: 0.2641P = 26410. Solving for P yields P = (26410)/(0.2641), which results in P = 100,000.

Explanation

In a right-angled triangle formed by a pillar of height h and a point on the ground at a horizontal distance x, the relation between the angle of elevation α and the sides is given by cot α = (base)/(perpendicular) = ˣ⁄ₕ. Therefore, the distance of any point from the foot of the pillar is x = h cot α.

Let the distances of the two points from the foot of the pillar be x₁ and x₂. For the point with the angle of elevation Φ, the distance is x₁ = h cot Φ. For the point with the angle of elevation θ, the distance is x₂ = h cot θ. Since both points lie on the same side of the pillar and θ < Φ, the point with the smaller angle θ is further away from the pillar (as the cotangent function is decreasing in the first quadrant).

The distance between the two points is the difference between their respective distances from the pillar: Distance = x₂ - x₁ Distance = h cot θ - h cot Φ Distance = h(cot θ - cot Φ) m

Explanation

White phosphorus exists as discrete P₄ molecules where four phosphorus atoms are arranged at the corners of a regular tetrahedron. In this structure, the P-P-P bond angle is only 60°, which deviates significantly from the preferred bonding angles. This creates intense angular strain within the molecule, making the P-P bonds exceptionally weak and unstable. Due to this high thermodynamic instability and ring strain, white phosphorus is chemically the most reactive allotrope and spontaneously catches fire in air at room temperature to form phosphorus pentoxide.

Explanation

Zubin Mehta is a world-renowned Indian conductor of Western classical music. Born in Mumbai, he has held prestigious positions such as the Music Director of the New York Philharmonic, the Los Angeles Philharmonic, and the Montreal Symphony Orchestra. He is particularly celebrated for his long-standing association with the Israel Philharmonic Orchestra, where he was appointed Music Director for Life in 1981. His contribution to the international music scene has been recognized with numerous awards, including the Padma Vibhushan, India's second-highest civilian honor, and the Praemium Imperiale from the Japan Art Association.

Explanation

To solve the equation 3¹⁰ × 27² = 9² × 3ⁿ, we convert all terms to a common base of 3 using the laws of exponents.

Substituting 27 = 3³ and 9 = 3² into the equation: 3¹⁰ × (3³)² = (3²)² × 3ⁿ

Using the power of a power rule, (aᵐ)ᵖ = aᵐᵖ, the terms are simplified as: 3¹⁰ × 3⁶ = 3⁴ × 3ⁿ

Applying the product rule for exponents, aᵐ × aᵖ = aᵐ⁺ᵖ, we add the exponents on both sides of the equation: 3¹⁰⁺⁶ = 3⁴⁺ⁿ 3¹⁶ = 3⁴⁺ⁿ

Since the bases are equal on both sides, we can equate the exponents: 16 = 4 + n n = 16 - 4 n = 12

Explanation

The volume of a cylinder is calculated by the formula V = π r² h. The ratio of the volumes of two cylinders is expressed as V₁/V₂ = (r₁/r₂)² × (h₁/h₂). Substituting the provided ratios r₁:r₂ = 3:2 and h₁:h₂ = 3:7 into this expression yields (3/2)² × (3/7). Squaring the first term results in 9/4, and multiplying by the second term gives (9 × 3) / (4 × 7) = 27/28.

Explanation

The selling price SP of an article subject to successive discounts d₁ and d₂ on its marked price MP is calculated using the formula SP = MP × (1 - d₁{100}) × (1 - d₂{100}). Substituting the given values MP = 900, d₁ = 20, and d₂ = 10 into the expression yields SP = 900 × (1 - 0.20) × (1 - 0.10). Simplifying the terms within the parentheses gives SP = 900 × 0.8 × 0.9. Multiplying these together results in 900 × 0.72, which equals 648. This can also be viewed as applying a single equivalent discount of d₁ + d₂ - d₁d₂{100} = 20 + 10 - ²⁰⁰⁄₁₀₀ = 28%. Subtracting 28% of 900 (which is 252) from the marked price 900 confirms the final price as 648.

Explanation

The total investment of the partnership is Rs. 6000, of which A contributes Rs. 1000. This leaves the combined investment of B and C as 6000 - 1000 = 5000. Given that B and C invest in the ratio of 2:3, the investment of C is calculated as ³⁄₂₊₃ × 5000 = ⅗ × 5000 = 3000.

The ratio of the investments made by A, B, and C is 1000:2000:3000, which simplifies to 1:2:3. In a partnership where the time period for each partner is the same, the profit is shared in the direct ratio of their capital investments. Therefore, the profit share of C is determined by the formula:

Profit of C = (Ratio of C)/(Total Ratio) × Total Profit Profit of C = ³⁄₁₊₂₊₃ × 2400 = ³⁄₆ × 2400 = ½ × 2400 = 1200

Explanation

The sum of the first two numbers is calculated by multiplying their count by their average: 2 × 8 = 16. Similarly, the sum of the other three numbers is 3 × 3 = 9. The total sum of all five numbers is the sum of these two groups, which is 16 + 9 = 25. To find the average of the five numbers, divide the total sum by the total count: 25 / 5 = 5.

Explanation

The cost price (CP) is calculated using the formula CP = (SP × 100)/(100 + Profit %), where SP is the selling price. Given that SP = 240 and the profit percentage is 20, substituting these values into the formula gives CP = (240 × 100)/(100 + 20). This simplifies to CP = ²⁴⁰⁰⁰⁄₁₂₀, which results in CP = 200. Alternatively, a profit of 20 percent means the selling price is 120 percent of the cost price. Setting up the equation 1.2 × CP = 240 and dividing 240 by 1.2 yields the cost price of Rs. 200.

Explanation

Using the principle of inclusion-exclusion for sets, the percentage of students who failed in at least one subject is calculated by the formula P(A ∪ B) = P(A) + P(B) - P(A ∩ B). Here, P(A) represents the percentage of students who failed in Hindi (35%), P(B) represents the percentage of students who failed in English (45%), and P(A ∩ B) represents the percentage of students who failed in both subjects (20%).

Substituting the values into the formula gives 35 + 45 - 20 = 60%. This 60% is the total percentage of students who failed in at least one subject. The percentage of students who passed in both subjects is found by subtracting the percentage of those who failed in at least one subject from the total percentage of students (100%). Thus, the percentage of students who passed in both subjects is 100% - 60% = 40%.

Explanation

Average speed is defined as the total distance traveled divided by the total time taken. The formula is:

Average Speed = (Total Distance)/(Total Time)

First, calculate the time taken for each segment of the journey using the relation Time = (Distance)/(Speed):

Time for the first segment t₁ = (50 km)/(25 km/h) = 2 hours. Time for the second segment t₂ = (40 km)/(20 km/h) = 2 hours. Time for the third segment t₃ = (90 km)/(15 km/h) = 6 hours.

Now, find the total distance and total time:

Total Distance = 50 + 40 + 90 = 180 km. Total Time = 2 + 2 + 6 = 10 hours.

Substitute these values into the average speed formula:

Average Speed = (180 km)/(10 hours) = 18 km/h.

Explanation

Using the algebraic identity a³ + b³ = (a + b)³ - 3ab(a + b), let k = x + ¹⁄ₓ. The given equation x³ + (1)/(x³) = 0 can be rewritten as k³ - 3(x)(¹⁄ₓ)(k) = 0, which simplifies to k³ - 3k = 0. Factoring this expression gives k(k² - 3) = 0. Because the problem states that x + ¹⁄ₓ ≠ 0, we know k ≠ 0, therefore k² - 3 = 0, which leads to k² = 3. The value of (x + ¹⁄ₓ)⁴ is k⁴, which is calculated by squaring k². Thus, k⁴ = (k²)² = 3² = 9.

Explanation

The algebraic identity for the square of a sum states that (x + ¹⁄ₓ)² = x² + 2(x)(¹⁄ₓ) + (1)/(x²), which simplifies to the relation x² + (1)/(x²) = (x + ¹⁄ₓ)² - 2. By substituting the given value x + ¹⁄ₓ = 2 into this formula, we calculate x² + (1)/(x²) = 2² - 2 = 4 - 2 = 2. Alternatively, using pattern recognition for the equation x + ¹⁄ₓ = 2, it is evident that x = 1 is the unique solution because 1 + ¹⁄₁ = 2. Substituting x = 1 into the required expression gives 1² + (1)/(1²) = 1 + 1 = 2.

Explanation

The orthocentre of a triangle is the point where the three altitudes intersect. Its location depends on the measures of the triangle's angles. In an acute-angled triangle, the orthocentre is located in the interior, while in an obtuse-angled triangle, it lies in the exterior. For a right-angled triangle, the two sides that form the right angle are themselves altitudes to each other. If a triangle has a right angle at a specific vertex, the altitude from one acute vertex to the opposite side is the leg of the triangle meeting at that right-angled vertex. Similarly, the altitude from the other acute vertex is the other leg. These two altitudes intersect precisely at the vertex where the right angle is formed. Therefore, if the orthocentre lies on one of the sides of a triangle, the triangle must be right-angled, and the orthocentre must coincide with the vertex of the right angle.

Explanation

According to the Triangle Inequality Theorem, the length of any side of a triangle must be strictly greater than the absolute difference between the lengths of the other two sides and strictly less than the sum of the lengths of the other two sides. This is represented by the inequality |a - b| < c < a + b. Substituting the given side lengths 5 cm, 9 cm, and x cm, we get |9 - 5| < x < 9 + 5, which simplifies to 4 < x < 14. Since x must be an integer, the set of possible values for x is {5, 6, 7, …, 13}. The minimum value in this set is 5.

Explanation

Using the complementary angle relationship cos θ = sin(90° - θ), the term cos² 70° can be expressed as cos²(90° - 20°), which simplifies to sin² 20°. Substituting this value back into the original expression gives cos² 20° + sin² 20°. According to the fundamental trigonometric identity cos² θ + sin² θ = 1, the sum of the squares of the cosine and sine of the same angle is always equal to 1.

Explanation

The relationship between the number of workers, time taken, and the work completed is given by the formula (M₁ × T₁)/(W₁) = (M₂ × T₂)/(W₂), where M is the number of persons, T is the time in hours, and W is the work done (number of mats).

Substituting the initial values: M₁ = 5 persons T₁ = 5 hours W₁ = 5 mats

And the second set of values: M₂ = 10 persons T₂ = 10 hours W₂ = ?

The equation becomes: (5 × 5)/(5) = (10 × 10)/(W₂)

Simplifying the left side: 5 = (100)/(W₂)

Solving for W₂: W₂ = ¹⁰⁰⁄₅ W₂ = 20

By doubling both the number of workers and the time provided, the total man-hours increase fourfold, leading to the production of four times as many mats as the original group.

Explanation

Using the algebraic identity (a + b)² + (a - b)² = 2(a² + b²), we substitute the given values a + b = 17 and a - b = 9:

17² + 9² = 2(a² + b²)

289 + 81 = 2(a² + b²)

370 = 2(a² + b²)

To find the value of 4a² + 4b², we multiply the entire equation by 2:

2 × 370 = 2 × 2(a² + b²)

740 = 4a² + 4b²

Explanation

Using the standard algebraic identities (x + y)² = x² + y² + 2xy and (x - y)² = x² + y² - 2xy, we can derive expressions for 4xy and 2(x² + y²).

First, calculate 4xy by subtracting the squares: 4xy = (x + y)² - (x - y)² Substituting the given values: 4xy = (√3)² - (√2)² = 3 - 2 = 1

Next, calculate 2(x² + y²) by adding the squares: 2(x² + y²) = (x + y)² + (x - y)² Substituting the given values: 2(x² + y²) = (√3)² + (√2)² = 3 + 2 = 5

The expression 8xy(x² + y²) can be rewritten as the product of these two derived components: 8xy(x² + y²) = (4xy) × [2(x² + y²)] Substituting the calculated values: 8xy(x² + y²) = 1 × 5 = 5

Explanation

In a circle, the perpendicular drawn from the center to a chord bisects the chord. This creates a right-angled triangle where the hypotenuse is the radius of the circle (r = 13 cm), one leg is the distance from the center to the chord (d = 5 cm), and the other leg is half the length of the chord (x).

According to the Pythagorean theorem: r² = d² + x²

Substituting the given values: 13² = 5² + x² 169 = 25 + x² x² = 169 - 25 x² = 144 x = √(144) = 12 cm

Since x represents half the length of the chord, the total length is calculated as: Length of chord = 2x = 2 × 12 = 24 cm

Explanation

The sum of the angles in a triangle is 180°. Given the ratio 1:2:3, let the angles be x, 2x, and 3x. Solving x + 2x + 3x = 180° yields 6x = 180°, so x = 30°. The angles of the triangle are 30°, 60°, and 90°. For any triangle, the ratio of the sides is equal to the ratio of the sines of their opposite angles according to the Sine Rule. Thus, the side lengths follow the ratio sin 30° : sin 60° : sin 90°.

Substituting the standard trigonometric values, the ratio becomes 1/2 : √3/2 : 1, which is equivalent to 1 : √3 : 2. The smallest side is always opposite the smallest angle (30°), and the longest side is opposite the largest angle (90°). Since the length of the smallest side is given as 10 cm and the ratio between the smallest and longest sides is 1:2, the length of the longest side is 10 × 2 = 20 cm.

Explanation

Given the trigonometric equation cos A + cos² A = 1, we can rearrange the terms to isolate cos A:

cos A = 1 - cos² A

Using the fundamental identity sin² A + cos² A = 1, we substitute sin² A for 1 - cos² A:

cos A = sin² A

Squaring both sides of this equality, we obtain:

cos² A = (sin² A)² = sin⁴ A

The expression to be evaluated is sin² A + sin⁴ A. By substituting the equivalent cosine terms derived above into this expression:

sin² A + sin⁴ A = cos A + cos² A

From the initial equation, we know that cos A + cos² A = 1. Thus, the value of the required expression is 1.

Explanation

The amount A after n years under compound interest is calculated using the formula A = P(1 + ʳ⁄₁₀₀)ⁿ, where P is the principal and r is the interest rate.

Given that Rs. 3000 amounts to Rs. 6000 in two years: 6000 = 3000(1 + ʳ⁄₁₀₀)² (1 + ʳ⁄₁₀₀)² = ⁶⁰⁰⁰⁄₃₀₀₀ = 2

For compound interest, the growth factor remains constant over equal intervals of time. Since the amount doubles in the first two years (growth factor of 2), it will double again in the next two years. The amount after four years A₄ can be expressed as: A₄ = P[(1 + ʳ⁄₁₀₀)²]² Substituting the known values: A₄ = 3000 × (2)² = 3000 × 4 = 12000

The total interest for four years is the difference between the final amount and the original principal: Interest = A₄ - P Interest = 12000 - 3000 = 9000

Explanation

Let h be the height of the lighthouse. The angles of depression to the ships are 45° and 30°, which are equal to the angles of elevation from the ships to the top of the lighthouse.

In the right-angled triangle formed with the first ship at 45°, the distance from the base is d₁ = h cot(45°) = h(1) = h. In the right-angled triangle formed with the second ship at 30°, the distance from the base is d₂ = h cot(30°) = h√3.

The distance between the two ships is d₂ - d₁ = 200m. Substituting the expressions for d₁ and d₂ gives:

h√3 - h = 200 h(√3 - 1) = 200 h = (200)/(√(3) - 1)

To simplify, rationalize the denominator by multiplying the numerator and denominator by (√3 + 1):

h = 200(√3 + 1){(√3 - 1)(√3 + 1)} = 200(√3 + 1){3 - 1} = 200(√3 + 1){2} = 100(√3 + 1)

Using the value √3 = 1.73:

h = 100(1.73 + 1) = 100(2.73) = 273m