SSC CGL Tier 1 2014 Quantitative Aptitude Questions with Solutions

Explanation

In any right-angled triangle, the orthocentre is the vertex where the right angle is formed. This occurs because the two legs of the triangle serve as the altitudes for each other. For △ PQR, the right angle is specified at vertex P. By examining the coordinates, we see that PQ is a horizontal line segment since P(3, 4) and Q(7, 4) share the same y-coordinate, and PR is a vertical line segment since P(3, 4) and R(3, 8) share the same x-coordinate. The intersection of these two perpendicular segments is the vertex P(3, 4), which necessarily acts as the orthocentre of the triangle.

Explanation

For the given angle Θ = 45°, the values of the trigonometric functions are sec 45° = √2 and csc 45° = √2.

Substituting these values into the expression sec Θ + csc Θ:

sec 45° + csc 45° = √2 + √2 = 2√2

Explanation

The value of x⁶ + 1/x⁶ can be calculated by first finding x³ + 1/x³ and then squaring the result. Given the standard identity that if x + 1/x = k, then x³ + 1/x³ = k³ - 3k, we substitute k = 5 to get x³ + 1/x³ = 5³ - 3(5) = 125 - 15 = 110.

To find x⁶ + 1/x⁶, we use the identity (a + 1/a)² = a² + 1/a² + 2 by letting a = x³. Substituting the previously calculated value, we have (x³ + 1/x³)² = x⁶ + 1/x⁶ + 2. Inserting 110 into the equation gives 110² = x⁶ + 1/x⁶ + 2, which simplifies to 12100 = x⁶ + 1/x⁶ + 2. Subtracting 2 from both sides results in x⁶ + 1/x⁶ = 12098.

Another method involves first squaring the original term to get x² + 1/x² = 5² - 2 = 23, and then cubing that result. Using the relation (x² + 1/x²)³ = x⁶ + 1/x⁶ + 3(x² + 1/x²), we substitute 23 to get 23³ = x⁶ + 1/x⁶ + 3(23). This leads to 12167 = x⁶ + 1/x⁶ + 69, which also yields 12098.

Explanation

The trigonometric identity tan² Θ = sec² Θ - 1 is used to evaluate the expression (tan² Θ)/(sec Θ + 1) - sec Θ. By substituting this identity into the numerator, the expression becomes:

(sec² Θ - 1)/(sec Θ + 1) - sec Θ

The numerator sec² Θ - 1 can be factored using the algebraic difference of squares formula, a² - b² = (a - b)(a + b), which gives (sec Θ - 1)(sec Θ + 1). Substituting this back into the fraction provides:

((sec Θ - 1)(sec Θ + 1))/(sec Θ + 1) - sec Θ

Canceling the common factor (sec Θ + 1) from the numerator and denominator simplifies the expression to:

(sec Θ - 1) - sec Θ

Subtracting sec Θ results in -1.

Explanation

A line graph is a graphical representation of data that connects a series of points to illustrate how variables change. Its fundamental analytical purpose is to allow for the comparison of numerical values across a continuous scale or between different sets of data. By plotting data series on a single coordinate system, the line graph facilitates the identification of relative differences, similarities, and intersections. This visual format helps in understanding the relationship between independent and dependent variables, making it a primary tool for comparing the behavior and performance of different data groups within a single visual framework.

Explanation

For two similar triangles, the ratio of any two corresponding linear dimensions, such as sides, heights, or medians, is constant. Therefore, the ratio of the corresponding sides LM and OP is equal to the ratio of their respective heights.

Using the property of similarity for Δ LMN ∼ Δ OPQ:

(LM)/(OP) = (Height of Δ LMN)/(Height of Δ OPQ)

Substituting the given values:

(LM)/(12) = ⁴⁸⁄₃₆

To solve for LM, simplify the ratio of the heights:

⁴⁸⁄₃₆ = ⁴⁄₃

Now, multiply by the length of OP:

LM = 12 × ⁴⁄₃ = 4 × 4 = 16~cm

Explanation

An ogive is a specialized graph used in statistics to represent cumulative frequency distributions. It is constructed by plotting the cumulative frequency of each class interval against its corresponding class limit. Specifically, a less than ogive is created by plotting cumulative frequencies against the upper class limits, while a more than ogive uses the lower class limits. The resulting curve shows the total number of observations that fall below or above specific values within a dataset. Because it tracks the running total of frequencies, the ogive is the primary graphical method for determining positional averages such as the median, quartiles, and percentiles directly from the plotted data.

Explanation

To find the value of a, express each term of the equation as a power of 5 using the laws of indices: xᵐ × xⁿ = xᵐ⁺ⁿ and xᵐ ÷ xⁿ = xᵐ⁻ⁿ.

The first term 5√5 is interpreted as the fifth root of 5 (based on the provided answer), which is 5¹^/⁵ or 5⁰^.². The second term is 5³. The third term 5³^/² is 5¹^.⁵.

Substituting these into the left-hand side (LHS) of the equation: LHS = 5⁰^.² × 5³ ÷ 5¹^.⁵ LHS = 5⁽⁰^.²⁺³⁻¹^.⁵⁾ LHS = 5⁽³^.²⁻¹^.⁵⁾ LHS = 5¹^.⁷

Equating the LHS with the right-hand side (RHS): 5¹^.⁷ = 5ᵃ⁺²

Since the bases are equal, we equate the exponents: 1.7 = a + 2 a = 1.7 - 2 a = -0.3

Converting the decimal to a fraction: a = -3/10

Explanation

The given equation is x² - 3x + 1 = 0. Dividing the entire equation by x, we get x - 3 + ¹⁄ₓ = 0, which implies x + ¹⁄ₓ = 3. The expression to be evaluated is (x⁶ + x⁴ + x² + 1)/(x³). By dividing each term in the numerator by x³, the expression simplifies to x³ + x + ¹⁄ₓ + (1)/(x³).

This can be rearranged as (x³ + (1)/(x³)) + (x + ¹⁄ₓ). Using the algebraic identity a³ + b³ = (a + b)³ - 3ab(a + b), we can find the value of the first part: x³ + (1)/(x³) = (x + ¹⁄ₓ)³ - 3(x)(¹⁄ₓ)(x + ¹⁄ₓ). Substituting x + ¹⁄ₓ = 3, we calculate 3³ - 3(3) = 27 - 9 = 18. Finally, adding the remaining part (x + ¹⁄ₓ = 3) to this result, the total value is 18 + 3 = 21.

Explanation

Let the speed of the boat in still water be u km/hr and the speed of the stream be v km/hr. The upstream speed is x = u - v and the downstream speed is y = u + v. Based on the given conditions, we can establish the following equations using the formula Time = Distance / Speed:

Case 1: ²⁴⁄ₓ + (28)/(y) = 6 Case 2: ³⁰⁄ₓ + (21)/(y) = 6.5

To simplify, let a = ¹⁄ₓ and b = (1)/(y). The equations become:

1. 24a + 28b = 6 ⟹ 12a + 14b = 3
2. 30a + 21b = 6.5

Multiplying the first simplified equation by 3 and the second by 2 to eliminate b: 36a + 42b = 9 60a + 42b = 13

Subtracting the first result from the second gives: 24a = 4 ⟹ a = ⅙ Thus, x = 6 km/hr.

Substituting a = ⅙ into 12a + 14b = 3: 12(⅙) + 14b = 3 ⟹ 2 + 14b = 3 ⟹ 14b = 1 ⟹ b = ¹⁄₁₄ Thus, y = 14 km/hr.

The speed of the boat in still water is the average of the downstream and upstream speeds: u = (x + y)/(2) = (6 + 14)/(2) = ²⁰⁄₂ = 10 km/hr.

Explanation

For a two-year period, the relationship between compound interest (CI) and simple interest (SI) on the same principal at a rate r is given by the formula CI = SI(1 + ʳ⁄₂₀₀). This relation accounts for the extra interest earned on the first year interest in a compounding scenario. Substituting the given values CI = 410 and r = 5 into the equation, we get 410 = SI(1 + ⁵⁄₂₀₀), which simplifies to 410 = SI(⁴¹⁄₄₀). Solving for SI, we find SI = (410 × 40)/(41) = 400.

Alternatively, using effective percentage rates, the simple interest rate over 2 years at 5% per annum is 2 × 5% = 10%. The compound interest rate is calculated using the successive percentage formula x + y + ˣʸ⁄₁₀₀, giving 5 + 5 + (5 × 5)/(100) = 10.25%. If 10.25% of the principal corresponds to 410, then the 10% simple interest is calculated as ((410)/(10.25)) × 10 = 400.

Explanation

The equation x = a represents a vertical line parallel to the y-axis where every point on the line has an x-coordinate of a. Similarly, the equation y = b represents a horizontal line parallel to the x-axis where every point on the line has a y-coordinate of b. The point of intersection of two such lines must satisfy both equations simultaneously. Therefore, the x-coordinate of the intersection point must be a and the y-coordinate must be b, resulting in the coordinates (a, b).

Explanation

In a circle, a perpendicular drawn from the center to a chord bisects that chord. Since OM ⊥ AB and AB = 20 cm, the length of the segment AM is half of the chord length, which gives AM = ²⁰⁄₂ = 10 cm.

In the right-angled triangle OMA, the radius OA (denoted as r) acts as the hypotenuse, while OM = 2√(11) cm and AM = 10 cm are the two legs. Applying the Pythagoras theorem:

r² = OM² + AM² r² = (2√(11))² + 10² r² = (4 × 11) + 100 r² = 44 + 100 r² = 144 r = √(144) = 12 cm

Explanation

The sum of all internal angles in a triangle is always 180°. Given that the angles ∠ A, ∠ B, and ∠ C are in the ratio 2:3:1, we can represent them as 2x, 3x, and x respectively, where x is a common constant.

According to the angle sum property: 2x + 3x + x = 180° 6x = 180° x = (180°)/(6) = 30°

By substituting the value of x back into the expressions for the angles: ∠ A = 2x = 2 × 30° = 60° ∠ B = 3x = 3 × 30° = 90° ∠ C = 1x = 1 × 30° = 30°

Explanation

The point of intersection of the perpendicular bisectors of the sides of a triangle is known as the circumcenter. A fundamental property of the circumcenter O is that the angle it subtends with any side at the center is twice the angle at the opposite vertex of the triangle. Specifically, ∠ BOC = 2 ∠ BAC.

In Δ ABC, the sum of all internal angles is 180°. Given ∠ ABC = 70° and ∠ BCA = 40°, the third angle is calculated as: ∠ BAC = 180° - (∠ ABC + ∠ BCA) ∠ BAC = 180° - (70° + 40°) = 70°

Applying the circumcenter property: ∠ BOC = 2 × ∠ BAC = 2 × 70° = 140°

Explanation

For a triangle with sides a, b, and c to be right-angled, the sum of the squares of the two shorter sides must equal the square of the longest side according to the Pythagorean theorem. Given the sides a = x² - 1, b = 2x, and c = x² + 1, we can verify this relationship by calculation.

Calculating the sum of the squares of the first two sides: a² + b² = (x² - 1)² + (2x)² Expanding the binomial and the square: a² + b² = (x⁴ - 2x² + 1) + 4x² Simplifying the expression: a² + b² = x⁴ + 2x² + 1

Now, calculating the square of the third side: c² = (x² + 1)² Expanding the binomial: c² = x⁴ + 2x² + 1

Since a² + b² = c² holds true, the measures of the sides satisfy the Pythagorean identity. This specific algebraic form is a standard generator for Pythagorean triples, where the longest side x² + 1 acts as the hypotenuse.

Explanation

The given equation 2ˣ = 4ʸ = 8ᶻ can be rewritten by expressing all bases as powers of 2: 2ˣ = (2²)ʸ = (2³)ᶻ, which leads to 2ˣ = 2²ʸ = 2³ᶻ. Comparing the exponents, we get the relationship x = 2y = 3z. This implies y = x/2 and z = x/3.

Substituting these into the second given equation xyz = 288: x · ˣ⁄₂ · ˣ⁄₃ = 288 (x³)/(6) = 288 x³ = 1728 x = ³√(1728) = 12

To find the value of the expression ¹⁄₂ₓ + (1)/(4y) + (1)/(8z), we use the relationships 4y = 2(2y) = 2x and 8z = 8(x/3) = ⁸ˣ⁄₃: ¹⁄₂ₓ + ¹⁄₂ₓ + ³⁄₈ₓ = ¹⁄ₓ + ³⁄₈ₓ = (8 + 3)/(8x) = ¹¹⁄₈ₓ

Substituting x = 12: (11)/(8 · 12) = ¹¹⁄₉₆

Explanation

Given the equation x⁴ + (1)/(x⁴) = 119, we first transform it to find x² + (1)/(x²) by adding 2 to both sides of the equation: x⁴ + (1)/(x⁴) + 2 = 119 + 2. This simplifies to (x² + (1)/(x²))² = 121. Taking the positive square root, we obtain x² + (1)/(x²) = 11.

To find the value of x³ - (1)/(x³), we need to determine the value of x - ¹⁄ₓ. We do this by subtracting 2 from both sides of the equation x² + (1)/(x²) = 11, which gives x² + (1)/(x²) - 2 = 11 - 2, or (x - ¹⁄ₓ)² = 9. Taking the square root yields x - ¹⁄ₓ = ± 3.

The algebraic identity for the difference of cubes is x³ - (1)/(x³) = (x - ¹⁄ₓ)(x² + (1)/(x²) + 1). Substituting the values we have found, we get x³ - (1)/(x³) = (± 3)(11 + 1) = ± 3(12) = ± 36. The question specifically asks for the positive value of the expression, so we select 36. The condition x < 1 is satisfied by values such as x = 3 - √(13){2} ≈ -0.302, which results in the positive value for the required expression.

Explanation

Let x = 3 + 2√2 and y = 3 - 2√2. First, find the product of these conjugate terms: xy = (3 + 2√2)(3 - 2√2) = 3² - (2√2)² = 9 - 8 = 1

Since xy = 1, it follows that x⁻¹ = y and y⁻¹ = x. Therefore, the given expression (x)⁻³ + (y)⁻³ can be rewritten as: y³ + x³

Using the algebraic identity x³ + y³ = (x + y)³ - 3xy(x + y), we calculate the sum: x + y = (3 + 2√2) + (3 - 2√2) = 6

Substituting the values of (x + y) and xy into the identity: x³ + y³ = 6³ - 3(1)(6) x³ + y³ = 216 - 18 = 198

Explanation

The expression cos⁴Θ - sin⁴Θ can be factored as a difference of squares: cos⁴Θ - sin⁴Θ = (cos²Θ + sin²Θ)(cos²Θ - sin²Θ)

According to the fundamental trigonometric identity, cos²Θ + sin²Θ = 1 for any angle Θ. Substituting this identity and the given value cos²Θ - sin²Θ = 1/3 into the factored expression: cos⁴Θ - sin⁴Θ = (1) × (1/3) = 1/3

Explanation

Given tan θ = 1/√(11) for 0 < θ < π/2, the problem requires evaluating the expression csc² θ - sec² θ{csc² θ + sec² θ}.

Using trigonometric identities, the expression can be simplified as follows: csc² θ - sec² θ{csc² θ + sec² θ} = (1)/(sin²⁾^θ - (1)/(cos²⁾^θ{(1)/(sin²⁾^θ + (1)/(cos²⁾^θ} = cos² θ - sin² θ{cos² θ + sin² θ}

Since cos² θ + sin² θ = 1, the expression reduces to cos² θ - sin² θ, which is the identity for cos 2θ. The formula for cos 2θ in terms of tan θ is: cos 2θ = 1 - tan² θ{1 + tan² θ}

Substituting the given value tan θ = 1/√(11): tan² θ = (1/√(11))² = 1/11 cos 2θ = (1 - 1/11)/(1 + 1/11) = (10/11)/(12/11) = ¹⁰⁄₁₂ = ⅚

Explanation

According to the Angle Bisector Theorem, the internal bisector of an angle of a triangle divides the opposite side into segments whose lengths are in the same ratio as the other two sides of the triangle. In a triangle ABC, let the bisector of ∠ A intersect the opposite side BC at point D. The theorem establishes the relationship AB/AC = BD/DC. The problem states that the angle bisector also bisects the opposite side, meaning BD = DC. Substituting this equality into the formula yields AB/AC = 1, which simplifies to AB = AC. By definition, a triangle with at least two equal sides is an isosceles triangle.

Explanation

The area of a square is proportional to the square of its side, expressed as A = s². When the side of the square is reduced by 50%, the new side length becomes 0.5s. Consequently, the new area Aₙₑ_w is (0.5s)² = 0.25s². The reduction in area is calculated by subtracting the new area from the original area: s² - 0.25s² = 0.75s². To express this as a percentage, we calculate (0.75s²{s²}) × 100 = 75%. Alternatively, using the net percentage change formula for the product of two variables x + y + ˣʸ⁄₁₀₀, where x = -50 and y = -50, the result is -50 - 50 + ⁽⁻⁵⁰⁾⁽⁻⁵⁰⁾⁄₁₀₀ = -100 + 25 = -75%. The negative sign indicates a 75% reduction in the total area.

Explanation

By definition, two triangles are congruent if they are identical in both shape and size. When Δ ABC ≅ Δ PQR, all corresponding sides and angles are equal, meaning the triangles can be superimposed perfectly on one another. Since the physical dimensions and proportions of congruent triangles are exactly the same, they must occupy the same amount of space. Therefore, the area of Δ ABC is equal to the area of Δ PQR. Given that the area of Δ ABC is 60 sq. cm, the area of Δ PQR is also 60 sq. cm.

Explanation

To find the least number that must be added to 1728 to make it a perfect square, we identify the perfect square immediately greater than 1728. Calculating squares of integers near the value, we find that 41² = 1681 and 42² = 1764. Since 1728 is greater than 1681 but less than 1764, the next perfect square is 1764. The required number to be added is the difference between this square and the given number, calculated as 1764 - 1728 = 36.

Explanation

In triangle ABC, points D and E are located on sides AB and AC such that AD = ¼ AB and AE = ¼ AC. This establishes that the ratios of the corresponding sides are equal: (AD)/(AB) = (AE)/(AC) = ¼. Since the two triangles ADE and ABC share a common angle at vertex A, they are similar by the SAS (Side-Angle-Side) similarity criterion.

According to the properties of similar triangles, the ratio of all corresponding sides is constant. Therefore, the ratio of the side DE to the base BC must be equal to the ratio of the other sides: (DE)/(BC) = (AD)/(AB) = ¼

Given that BC = 12 cm, we substitute this value into the proportionality equation: DE = ¼ × BC DE = ¼ × 12 DE = 3 cm

Explanation

Given the trigonometric equation tan Θ + cot Θ = 2, we can substitute cot Θ = (1)/(tan Θ) to get tan Θ + (1)/(tan Θ) = 2. Let x = tan Θ, which transforms the equation into x + ¹⁄ₓ = 2. Multiplying by x gives x² + 1 = 2x, or x² - 2x + 1 = 0. This is a perfect square equivalent to (x - 1)² = 0, leading to x = 1. Therefore, tan Θ = 1, which corresponds to Θ = 45° in the first quadrant. Alternatively, substituting 45° into the original expression gives tan 45° + cot 45° = 1 + 1 = 2, which satisfies the condition.

Explanation

The expression 121a² + 64b² can be identified as the sum of two squares, where 121a² = (11a)² and 64b² = (8b)². To transform this into a perfect square trinomial of the form (x + y)² = x² + 2xy + y², we must add the middle term 2xy. Substituting x = 11a and y = 8b into this formula, the required term is 2 × (11a) × (8b) = 176ab. This results in the perfect square (11a + 8b)².

Explanation

The combined rate of work for A, B, and C per day is the sum of their individual rates: ⅙ + ⅒ + ¹⁄₁₅. Finding a common denominator of 30, this becomes (5 + 3 + 2)/(30) = ¹⁰⁄₃₀ = ⅓ of the work per day. Since the entire work is completed in 3 days for a total payment of 300, the daily wage for the three individuals combined is ³⁰⁰⁄₃ = 100. Consequently, the sum of their wages for 2 days is 100 × 2 = 200.

Explanation

Let C's share be x. According to the problem, B gets 6 more than C, so B's share is x + 6. A gets 7 more than B, so A's share is (x + 6) + 7 = x + 13. The total sum is Rs 76, which leads to the following equation:

A + B + C = 76 (x + 13) + (x + 6) + x = 76 3x + 19 = 76 3x = 57 x = 19

Using the value of x, the shares of each individual are calculated as: C = 19 B = 19 + 6 = 25 A = 25 + 7 = 32

The ratio of the shares A : B : C is 32 : 25 : 19.

Explanation

A quadrilateral with opposite sides equal is a parallelogram. Let the quadrilateral be ABCD where AB = CD and AD = BC. If the diagonals are also equal, such that AC = BD, the parallelogram must be a rectangle. This is proven by observing triangles ABC and DCB. In these triangles, AB = DC, BC is a common side, and AC = DB. By the Side-Side-Side (SSS) congruence criterion, △ ABC ≅ △ DCB. Consequently, the corresponding angles ∠ ABC and ∠ DCB are equal. Since ABCD is a parallelogram, the sum of adjacent angles is 180°, meaning ∠ ABC + ∠ DCB = 180°. Setting them equal gives 2∠ ABC = 180°, resulting in ∠ ABC = 90°. In a parallelogram, opposite angles are equal and consecutive angles are supplementary, so all four angles of the quadrilateral must be 90°.

Explanation

In any triangle, the sum of the interior angles is 180°. If we denote the three angles as A, B, and C, then A + B + C = 180°. According to the given condition, each angle is less than the sum of the other two. Taking the first angle A, the condition is expressed as A < B + C. From the angle sum property, we know that B + C = 180° - A. Substituting this into the inequality gives A < 180° - A. Solving for A, we add A to both sides to get 2A < 180°, which means A < 90°. By symmetry, the same calculation applies to the other angles, leading to B < 90° and C < 90°. Since every individual angle in the triangle is less than 90°, the triangle must be an acute-angled triangle.

Explanation

The given equation is cos π x = x² - x + ⁵⁄₄. By completing the square for the quadratic expression on the right-hand side, we get x² - x + ⁵⁄₄ = (x² - x + ¼) + 1 = (x - ½)² + 1. Since the term (x - ½)² is always greater than or equal to zero for any real value of x, the minimum possible value of the right-hand side is 1, which occurs when x = ½. On the left-hand side, the range of the cosine function cos π x is [-1, 1], meaning its maximum possible value is 1. For the equation to have a solution, both sides must be equal to 1 simultaneously. This requires x = ½ for the right-hand side to be 1. However, substituting x = ½ into the left-hand side gives cos (π)/(2) = 0. Since 0 ≠ 1, there is no real value of x that satisfies the equation.

Explanation

The volume of the earth dug out from the tank is calculated using the formula for the volume of a rectangular prism, V = l × b × d. Substituting the given dimensions of the tank, the volume is 40 m × 30 m × 12 m = 14400 m³. This earth is spread over the remaining area of the field. The total area of the field is 1000 m × 30 m = 30000 m². Since the tank is dug inside the field, the area occupied by the tank must be subtracted from the total area to find the surface where the earth is actually spread. The area of the tank top is 40 m × 30 m = 1200 m².

The effective area for spreading the earth is 30000 m² - 1200 m² = 28800 m². The rise in the level of the field (h) is found by dividing the volume of the dug-out earth by this effective area: h = (14400 m³)/(28800 m²). This simplifies to h = ½ metre, which is 0.5 metre.

Explanation

In an equilateral triangle, the symmetry of the figure ensures that all internal lines drawn from the vertices to the opposite sides perform multiple roles simultaneously. Specifically, each median is also an altitude, an angle bisector, and a perpendicular bisector of the opposite side. The centroid is defined as the intersection of medians, the orthocentre as the intersection of altitudes, the incentre as the intersection of angle bisectors, and the circumcentre as the intersection of perpendicular bisectors. Because these four sets of lines are identical in an equilateral triangle, their respective points of concurrency must coincide at the exact same geometric center. This condition of coincidence among all four centers serves as a defining property that distinguishes equilateral triangles from all other types of triangles.

Explanation

Given a = 2 + √3, we first find the reciprocal ¹⁄ₐ by rationalizing the denominator. Multiplying the numerator and denominator by the conjugate 2 - √3 gives: ¹⁄ₐ = (1)/(2 + √(3)) × 2 - √3{2 - √3} = 2 - √3{2² - (√3)²} = 2 - √3{4 - 3} = 2 - √3

Adding a and its reciprocal, we get: a + ¹⁄ₐ = (2 + √3) + (2 - √3) = 4

To find the value of a² + (1)/(a²), we use the algebraic identity (x + ¹⁄ₓ)² = x² + (1)/(x²) + 2. Rearranging this identity for the required term: a² + (1)/(a²) = (a + ¹⁄ₐ)² - 2

Substituting the calculated sum a + ¹⁄ₐ = 4: a² + (1)/(a²) = 4² - 2 = 16 - 2 = 14

Explanation

An expression in the form Ax² + Bx + C is a perfect square if its discriminant D = B² - 4AC is equal to zero. Let x = √p, so the expression becomes x² + ¼x + k². Here, A = 1, B = 1/4, and C = k². Setting the discriminant to zero:

(1/4)² - 4(1)(k²) = 0 1/16 - 4k² = 0 4k² = 1/16 k² = 1/64 k = ± √(1/64) k = ± 1/8

Alternatively, using the identity (a + b)² = a² + 2ab + b², we let a = √p and a² = p. The middle term 2ab must equal ¼√p. Solving 2(√p)b = ¼√p gives 2b = 1/4, so b = 1/8. The constant term k² must correspond to b², which means k² = (1/8)², leading to k = ± 1/8.

Explanation

The given condition is 15% of x = 20% of y. In mathematical terms, this is expressed as: ¹⁵⁄₁₀₀ · x = ²⁰⁄₁₀₀ · y Canceling out the common denominator of 100 from both sides gives: 15x = 20y To find the ratio x:y, we rearrange the equation to solve for (x)/(y): (x)/(y) = ²⁰⁄₁₅ Simplifying the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 5, results in: (x)/(y) = ⁴⁄₃ Therefore, the ratio x:y is 4:3.

Explanation

Given the equation: ᵇ⁻ᶜ⁄ₐ + (a+c)/(b) + (a-b)/(c) = 1

Rearrange the equation by grouping the terms involving the denominators a and c on one side and the term with denominator b on the other: ᵇ⁻ᶜ⁄ₐ + (a-b)/(c) = 1 - (a+c)/(b)

Simplify the expression on the right-hand side: ᵇ⁻ᶜ⁄ₐ + (a-b)/(c) = (b - (a+c))/(b) = (b-a-c)/(b) = -(a-b+c)/(b)

Now, find a common denominator for the left-hand side: (c(b-c) + a(a-b))/(ac) = -(a-b+c)/(b) (bc - c² + a² - ab)/(ac) = -(a-b+c)/(b)

Factorize the numerator on the left-hand side by grouping terms: ((a² - c²) - (ab - bc))/(ac) = -(a-b+c)/(b) ((a-c)(a+c) - b(a-c))/(ac) = -(a-b+c)/(b) ((a-c)(a+c-b))/(ac) = -(a-b+c)/(b)

Since a - b + c ≠ 0, we can divide both sides by (a-b+c): (a-c)/(ac) = -(1)/(b)

Split the fraction on the left: (a)/(ac) - (c)/(ac) = -(1)/(b) (1)/(c) - ¹⁄ₐ = -(1)/(b)

Rearrange the terms to isolate 1/b: (1)/(b) = ¹⁄ₐ - (1)/(c)

Explanation

The total distance covered by a train while crossing a platform is the sum of the length of the train and the length of the platform. Using the formula Distance = Speed × Time, we first convert the speed from km/h to m/s by multiplying by the factor 5/18: Speed = 60 × ⁵⁄₁₈ = ⁵⁰⁄₃ m/s

Substituting the given values into the distance equation where Lₚ represents the length of the platform: Length of Train + Lₚ = Speed × Time 200 + Lₚ = ⁵⁰⁄₃ × 30 200 + Lₚ = 50 × 10 200 + Lₚ = 500 Lₚ = 500 - 200 = 300 metres

Explanation

In any triangle, the Triangle Inequality Theorem states that the sum of the lengths of any two sides must be greater than the length of the third side. Let the sides of Δ FGH be FG, GH, and FH. We are given that FG < 3 cm and GH = 8 cm. Since the triangle is isosceles, at least two sides must be equal.

If we assume FG = GH, then FG = 8 cm, which contradicts the given condition FG < 3 cm. If we assume FG = FH, then FH < 3 cm. In this case, the sum of the two shorter sides FG + FH would be less than 3 + 3 = 6 cm. However, for a triangle to exist, FG + FH must be greater than the third side GH. Since 6 < 8, the inequality FG + FH > 8 is not satisfied, meaning this configuration is impossible.

The only remaining possibility for the triangle to be isosceles is GH = FH. Setting FH = 8 cm satisfies the triangle inequality because 8 + 8 > FG (where FG < 3) and 8 + FG > 8 are both true. Therefore, the relation GH = FH must hold.

Explanation

Using the reciprocal trigonometric identities, we know that 1 / cot² 63° = tan² 63° and 1 / sin² 63° = csc² 63°. Substituting these into the given expression, we get: 1 + tan² 63° - sec² 27° + csc² 63° - csc² 27°

We use the fundamental identity sec² θ = 1 + tan² θ. Applying this to the first two terms: sec² 63° - sec² 27° + csc² 63° - csc² 27°

Since 63° and 27° are complementary angles (63° + 27° = 90°), we can apply the co-function identities: sec 63° = csc (90° - 63°) = csc 27° csc 63° = sec (90° - 63°) = sec 27°

Substituting these values into the expression: (csc² 27° - csc² 27°) + (sec² 27° - sec² 27°) 0 + 0 = 0

Explanation

In any circle, the angle subtended by a diameter at any point on the circumference is 90°. Given that ∠ BAC = 90° and points A, B, C lie on the circumference, the side BC must be the diameter of the circle. According to the Pythagorean theorem for the right-angled triangle ABC, the length of the diameter BC is calculated as BC = √(AB²⁾⁺^A^C^^^². Substituting the given values AB = AC = 5√2, we get BC = √((52))² + (5√2)² = √(50 + 50) = √(100) = 10 cm. The radius of the circle is half of the diameter, therefore R = (BC)/(2) = ¹⁰⁄₂ = 5 cm.

Explanation

The efficiency relationship is given as 10 men = 20 women = 40 children, which simplifies to 1 man = 4 children and 1 woman = 2 children. The total work capacity required to complete the task in 7 months is equivalent to 40 children working for that duration.

The combined group of 5 men, 5 women, and 5 children is converted into an equivalent number of children to standardize the calculation: 5 men + 5 women + 5 children = 5(4) + 5(2) + 5 = 20 + 10 + 5 = 35 children.

Using the work equivalence formula M₁ D₁{W₁} = M₂ D₂{W₂}, where M is the number of children, D is the time in months, and W is the work units: (40 × 7)/(1) = (35 × T)/(1/2) 280 = 70T T = ²⁸⁰⁄₇₀ = 4 months.

Explanation

Given x = √3 + √2, we first find the value of 1/x by rationalizing the denominator: 1/x = 1 / (√3 + √2) = (√3 - √2) / ((√3 + √2)(√3 - √2)) = (√3 - √2) / (3 - 2) = √3 - √2.

Now, calculate the difference: x - 1/x = (√3 + √2) - (√3 - √2) = 2√2.

The algebraic identity for x³ - 1/x³ is: x³ - 1/x³ = (x - 1/x)³ + 3(x - 1/x).

Substituting the value of x - 1/x = 2√2 into the identity: x³ - 1/x³ = (2√2)³ + 3(2√2) x³ - 1/x³ = (8 × 2√2) + 6√2 x³ - 1/x³ = 16√2 + 6√2 = 22√2.

Explanation

For a quadratic equation in the standard form ax² + bx + c = 0, Vieta's formulas state that the product of the roots is given by the ratio ᶜ⁄ₐ. In the given equation x² - √3 = 0, the coefficient of the x² term, a, is 1. The constant term, c, is -√3. Applying the formula, the product of the roots is:

Product of roots = ᶜ⁄ₐ = -√3{1} = -√3

Alternatively, solving for x directly yields the roots by taking the square root of both sides. Starting with x² = √3, the two roots are x₁ = √(3) and x₂ = -√(3). Multiplying these two roots together results in (√(3)) × (-√(3)) = -(√(3))² = -√3. Both the algebraic property of coefficients and direct calculation confirm the product.

Explanation

The relationship between the given terms is derived from the fundamental trigonometric identity cos² θ + sin² θ = 1, which can be rewritten as cos² θ = 1 - sin² θ. Factoring the right side using the difference of squares formula, we get cos² θ = (1 - sin θ)(1 + sin θ). By rearranging this equation to separate the factors, we have:

(cos θ)/(1 - sin θ) = (1 + sin θ)/(cos θ)

Given that x = (cos θ)/(1 - sin θ), it follows by substitution that x = (1 + sin θ)/(cos θ). To find the value of the required expression (cos θ)/(1 + sin θ), we take the reciprocal of both sides of this equation:

¹⁄ₓ = (cos θ)/(1 + sin θ)

Alternatively, multiplying the given expression x by the target expression results in (cos² θ)/(1 - sin² θ). Since the numerator and denominator are identical based on the Pythagorean identity, the product is 1, proving the two expressions are reciprocals of each other.

Explanation

The difference between Compound Interest (C.I.) and Simple Interest (S.I.) for a period of 2 years can be directly calculated using the formula D = P × ((R)/(100))², where D is the difference, P is the principal sum, and R is the annual rate of interest.

Substituting the given values into the formula: 25 = P × (⁵⁄₁₀₀)²

Simplifying the term inside the parentheses: 25 = P × (¹⁄₂₀)² 25 = P × ¹⁄₄₀₀

To find the principal P, multiply both sides by 400: P = 25 × 400 P = 10,000

Explanation

In the right-angled triangle ABC, where ∠ B = 90°, let the lengths of the sides be AB = 2k and BC = k based on the ratio AB:BC = 2:1. Using the Pythagoras theorem, the hypotenuse AC is calculated as: AC = √(AB²⁾⁺^B^C^^^² = √((2k)²⁾⁺ᵏ^^^² = √(5k²⁾ = k√5

To find the value of sin A + cot C, we determine the individual trigonometric ratios: sin A = (Opposite)/(Hypotenuse) = (BC)/(AC) = (k)/(k√(5)) = (1)/(√(5)) cot C = (Adjacent)/(Opposite) = (BC)/(AB) = ᵏ⁄₂ₖ = ½

Adding these values together: sin A + cot C = (1)/(√(5)) + ½ = 2 + √5{2√5}

Explanation

Let the original cost price (CP) of the article for Anand be 100x. Anand marks up the price by 50%, so the marked price is 150x. He then allows a discount of 20% on this marked price. Therefore, Anand's selling price (SP), which is Balaji's purchase price, is calculated as: CP_Bₐₗₐⱼᵢ = 150x × (1 - 0.20) = 150x × 0.80 = 120x

It is given that Balaji's selling price is 30% more than the original cost price. Thus: SP_Bₐₗₐⱼᵢ = 100x × (1 + 0.30) = 130x

Balaji's profit in absolute terms is SP_Bₐₗₐⱼᵢ - CP_Bₐₗₐⱼᵢ = 130x - 120x = 10x. To find Balaji's profit percentage, we use the formula: Profit % = (Profit)/(CP) × 100 Profit % = ¹⁰ˣ⁄₁₂₀ₓ × 100 = ¹⁄₁₂ × 100 = 8.33%