SSC CGL Tier 1 2016 Quantitative Aptitude Questions with Solutions

Explanation

To find the least number that makes 37044 a perfect cube upon division, we determine the prime factorization of the number. Dividing 37044 by its smallest prime factors, we get:

37044 = 2 × 18522 18522 = 2 × 9261 9261 = 3 × 3087 3087 = 3 × 1029 1029 = 3 × 343 343 = 7 × 49 49 = 7 × 7

Thus, the prime factorization of 37044 is 2² × 3³ × 7³. For a number to be a perfect cube, the exponent of each prime factor in its factorization must be a multiple of 3. In the prime factorization of 37044, the factors 3³ and 7³ are already perfect cubes. However, the factor 2² has an exponent of 2, which is not a multiple of 3. To make the quotient a perfect cube, we must divide 37044 by 2², which is 4. This leaves the result as 3³ × 7³ = 21³ = 9261, which is a perfect cube.

Explanation

Let I represent the average monthly income. The condition that the expenditure for 8 months equals the income for 6 months implies that the average monthly expenditure E is ⁶⁄₈I = ¾I. Annual savings is calculated as the difference between the total annual income and the total annual expenditure over 12 months.

Annual Savings = 12I - 12E

By substituting the value of E in terms of I: 6000 = 12I - 12(¾I) 6000 = 12I - 9I 6000 = 3I I = ⁶⁰⁰⁰⁄₃ = 2000

Explanation

The expression is simplified by substituting the given values a = √2 + 1 and b = √2 - 1. First, calculate the sum a + b = (√2 + 1) + (√2 - 1) = 2√2 and the product ab = (√2 + 1)(√2 - 1) = (√2)² - (1)² = 2 - 1 = 1. The given expression ¹⁄ₐ₊₁ + (1)/(b+1) can be written as a single fraction by taking the common denominator: ((b+1) + (a+1))/((a+1)(b+1)) = (a+b+2)/(ab+a+b+1). Substituting the values of a+b and ab into this fraction yields 2√2 + 2{1 + 2√2 + 1} = 2√2 + 2{2√2 + 2} = 1.

Explanation

The constant b in the partial fraction decomposition ³⁄₍ₓ₊₂₎₍₂ₓ₊₁₎ = ᵃ⁄₂ₓ₊₁ + ᵇ⁄ₓ₊₂ is found by isolating the term associated with the factor (x+2). Multiplying the entire identity by the common denominator (x+2)(2x+1) results in the linear identity 3 = a(x+2) + b(2x+1).

To solve for b directly, we substitute the value of x that nullifies the term containing a. Setting x = -2 in the equation yields 3 = a(-2+2) + b(2(-2)+1). This simplifies the expression to 3 = a(0) + b(-4+1), which further reduces to 3 = -3b. Dividing both sides by -3 gives b = -1. This result can also be obtained via the cover-up method by evaluating the original expression at x = -2 after removing the (x+2) factor: b = [³⁄₂ₓ₊₁]ₓ₌₋₂ = ³⁄₋₃ = -1.

Explanation

In a right-angled triangle, the altitude drawn from the right-angle vertex to the hypotenuse divides the hypotenuse into two segments such that the square of the altitude is equal to the product of the lengths of these two segments. This is known as the Right Triangle Altitude Theorem or the Geometric Mean Theorem.

In triangle ABC, since ∠ BAC = 90° and AD ⊥ BC, we apply the relation: AD² = BD × DC

Substituting the given values AD = 6 cm and BD = 4 cm into the formula: 6² = 4 × DC 36 = 4 × DC DC = ³⁶⁄₄ = 9 cm

The length of the hypotenuse BC is the sum of the segments BD and DC: BC = BD + DC BC = 4 + 9 = 13 cm

Explanation

In △ ABC, the total length of side AB is AD + DB = 8 + 12 = 20 cm, and the total length of side AC is AE + EC = 6 + 9 = 15 cm. By comparing the ratios of the corresponding sides of △ ADE and △ ABC, we find that (AD)/(AB) = 8/20 = ⅖ and (AE)/(AC) = 6/15 = ⅖. Since these two sides are proportional and they share the included angle ∠ A, △ ADE ∼ △ ABC by the SAS similarity criterion. In similar triangles, the ratio of all corresponding sides is the same, which means (DE)/(BC) = (AD)/(AB) = ⅖. Solving for BC yields the relation BC = ⁵⁄₂ DE.

Explanation

In any triangle ABC, the sum of the internal angles is ∠ A + ∠ B + ∠ C = 180°. Given that ∠ A = 90°, it follows that ∠ B + ∠ C = 90°, which can be written as ∠ C = 90° - ∠ B.

To find the value of the expression cos² A + cos² B + cos² C, we substitute the known values: cos² 90° + cos² B + cos²(90° - B)

Using the trigonometric identity cos 90° = 0, the first term becomes 0. Using the co-function identity cos(90° - θ) = sin θ, the third term cos²(90° - B) becomes sin² B. The expression now simplifies to: 0 + cos² B + sin² B

According to the fundamental Pythagorean identity sin² θ + cos² θ = 1, the sum cos² B + sin² B is equal to 1. Therefore, the total value of the expression is 0 + 1 = 1.

Explanation

The compound interest formula is A = P(1 + ʳ⁄₁₀₀)ⁿ, where A is the final amount, P is the principal, r is the annual interest rate, and n is the number of years. Substituting the given values A = 9261, P = 8000, and r = 5:

9261 = 8000(1 + ⁵⁄₁₀₀)ⁿ ⁹²⁶¹⁄₈₀₀₀ = (1 + ¹⁄₂₀)ⁿ ⁹²⁶¹⁄₈₀₀₀ = (²¹⁄₂₀)ⁿ

Since 20³ = 8000 and 21³ = 9261, the left side of the equation can be written as a power of 3:

(²¹⁄₂₀)³ = (²¹⁄₂₀)ⁿ

Equating the exponents on both sides, we get n = 3.

Explanation

Let the height of the building be h = 20 m and the distance between the building and the tower be x. From the top of the building, the angle of depression to the foot of the tower is 45°. Using the trigonometric relation tan θ = Perpendicular / Base, we have tan 45° = 20 / x. Since tan 45° = 1, the horizontal distance x is 20 m.

The tower consists of two parts: the part equal to the building height (20 m) and the part above it, let's call it y. For the upper part, the angle of elevation is 60°. Using the same horizontal distance x = 20 m, we apply the tangent formula: tan 60° = y / 20. Substituting tan 60° = √3, we get y = 20 √3.

The total height of the tower is the sum of these two parts: H = 20 + 20 √3 = 20(1 + √3). Substituting the given value of √3 = 1.732, the height is H = 20(1 + 1.732) = 20 × 2.732 = 54.64 m.

Explanation

The time taken by two individuals working together can be determined using the formula T = (x × y)/(x + y), where x and y represent the time taken by each person to complete the work alone. In this case, x = 7 days and y = 9 days.

Substituting these values into the formula: T = (7 × 9)/(7 + 9) = ⁶³⁄₁₆

To convert the improper fraction ⁶³⁄₁₆ into a mixed fraction, divide 63 by 16. The quotient is 3 and the remainder is 15, resulting in 3 ¹⁵⁄₁₆ days.

Explanation

The single equivalent discount for successive discounts can be calculated using the formula for two successive discounts x% and y%, which is (x + y - ˣʸ⁄₁₀₀)%.

First, calculate the equivalent discount for the first two rates, 20% and 10%: Equivalent Discount D₁ = (20 + 10 - (20 × 10)/(100))% = (30 - 2)% = 28%

Next, combine this 28% result with the third discount rate of 5%: Single Equivalent Discount D = (28 + 5 - (28 × 5)/(100))% = (33 - ¹⁴⁰⁄₁₀₀)% = (33 - 1.4)% = 31.6%

Alternatively, using the multiplier method, if the initial price is 100, the final price after three successive discounts is 100 × (1 - 0.20) × (1 - 0.10) × (1 - 0.05) = 100 × 0.80 × 0.90 × 0.95. Multiplying these gives 100 × 0.72 × 0.95 = 68.4. The total effective discount is the difference between the initial price and the final price, which is 100 - 68.4 = 31.6%.

Explanation

The total number of students is 720 and the ratio of boys to girls is 7 : 5. The total parts in the ratio are 7 + 5 = 12. The value of one part is calculated as 720 / 12 = 60. Consequently, the number of boys is 7 × 60 = 420 and the number of girls is 5 × 60 = 300. To achieve a 1 : 1 ratio, the number of girls must equal the number of boys, meaning the total number of girls must be 420. The additional girls needed are 420 - 300 = 120. Alternatively, to change the ratio from 7 : 5 to 7 : 7 (which is equivalent to 1 : 1), an addition of 2 parts of girls is required. Since one part represents 60 students, the number of girls to be added is 2 × 60 = 120.

Explanation

Let the cost price (CP) of 1 banana be C and the selling price (SP) of 1 banana be S. According to the problem, the selling price of 6 bananas is equal to the cost price of 8 bananas, which is expressed as: 6S = 8C

Rearranging the equation to find the ratio of the selling price to the cost price: (S)/(C) = ⁸⁄₆ = ⁴⁄₃

The formula for profit percentage is: Profit % = ( (S)/(C) - 1 ) × 100

Substituting the ratio into the formula: Profit % = ( ⁴⁄₃ - 1 ) × 100 = ⅓ × 100 = 33 ⅓%

This can also be solved using the shortcut for such problems: Profit % = (Items in CP - Items in SP)/(Items in SP) × 100 Profit % = (8 - 6)/(6) × 100 = ²⁄₆ × 100 = ⅓ × 100 = 33 ⅓%

Explanation

The percentage of mangoes remaining is calculated using the formula Percentage Left = (Total Quantity - Distributed Quantity)/(Total Quantity) × 100. By substituting the given values, the expression becomes (300 - 75)/(300) × 100. Simplifying the numerator gives 225, and the calculation becomes ²²⁵⁄₃₀₀ × 100, which simplifies further to ²²⁵⁄₃, yielding 75%.

Explanation

When a journey involves traveling to a destination and returning via the same route, the distances covered in both directions are equal. In such cases, the average speed for the entire trip is the harmonic mean of the individual speeds, calculated using the formula:

vₐᵥ_g = (2v₁v₂)/(v₁ + v₂)

Given that the speed going to the place v₁ = 16 km/hr and the total average speed vₐᵥ_g = 6.4 km/hr, we substitute these values into the formula to find the return speed v₂:

6.4 = (2 × 16 × v₂)/(16 + v₂)

6.4 = (32v₂)/(16 + v₂)

Multiplying both sides by (16 + v₂) gives:

6.4(16 + v₂) = 32v₂

102.4 + 6.4v₂ = 32v₂

Rearranging the equation to solve for v₂:

32v₂ - 6.4v₂ = 102.4

25.6v₂ = 102.4

v₂ = (102.4)/(25.6)

v₂ = 4 km/hr

Explanation

The Simple Interest SI is calculated using the formula SI = (P × R × T)/(100). When the interest amount SI and the principal sum P remain constant across two scenarios, the product of the rate R and the time period T must be equal. This establishes the relationship R₁ T₁ = R₂ T₂.

Given: Initial rate R₁ = 5% Initial time T₁ = 8 years New time T₂ = 5 years

Substituting these values into the equation: 5 × 8 = R₂ × 5 40 = 5 × R₂ R₂ = ⁴⁰⁄₅ = 8%

Alternatively, calculating the principal P first: P = (SI × 100)/(R₁₎_×_T___₁ = (840 × 100)/(5 × 8) = ⁸⁴⁰⁰⁰⁄₄₀ = 2100 The new rate R₂ for the same interest over 5 years is: R₂ = (SI × 100)/(P × T₂₎ = (840 × 100)/(2100 × 5) = ⁸⁴⁰⁄₁₀₅ = 8%

Explanation

The given equation is C + (1)/(C) = 3. By rearranging the terms to isolate (C - 3), we get: C - 3 = -(1)/(C)

To find the value of the expression, we substitute C - 3 with -(1)/(C). In the context of this algebraic identity problem, the powers are intended to match to allow for cancellation. Substituting into the expression (C - 3)⁷ + (1)/(C⁷) gives: (-(1)/(C))⁷ + (1)/(C⁷)

Since 7 is an odd power, (-(1)/(C))⁷ = -(1)/(C⁷). The expression then becomes: -(1)/(C⁷) + (1)/(C⁷) = 0

This result holds true for any odd power n where the expression is (C - 3)ⁿ + (1)/(Cⁿ), leading to the cancellation of the terms.

Explanation

To find the value of x² + (1)/(64x²), first divide the given equation 2x + ¹⁄₄ₓ = 1 by 2 on both sides. This results in the simplified expression x + ¹⁄₈ₓ = ½.

Next, square both sides of this equation using the algebraic identity (a + b)² = a² + 2ab + b²: (x + ¹⁄₈ₓ)² = (½)² x² + 2(x)(¹⁄₈ₓ) + (¹⁄₈ₓ)² = ¼

Simplifying the middle term 2(x)(¹⁄₈ₓ) gives ¼. The equation then becomes: x² + ¼ + (1)/(64x²) = ¼

By subtracting ¼ from both sides of the equation, we isolate the required expression: x² + (1)/(64x²) = ¼ - ¼ = 0

Explanation

In a triangle ABC where O is the orthocentre, the angle formed by any two vertices at the orthocentre and the internal angle at the third vertex are supplementary. This relationship is expressed by the formula ∠ BOC + ∠ BAC = 180°.

Given that ∠ BOC = 110°, we can find the value of ∠ BAC by substituting the known value into the formula: 110° + ∠ BAC = 180° ∠ BAC = 180° - 110° ∠ BAC = 70°

This property arises because if altitudes from B and C meet the opposite sides at E and F respectively, the quadrilateral AEOF has right angles at E and F. Since the sum of angles in a quadrilateral is 360°, ∠ BAC + ∠ EOF = 180°. Since ∠ BOC and ∠ EOF are vertically opposite angles, they are equal, leading to the supplementary relationship.

Explanation

The area of a triangle is calculated using the formula Area = ½ × base × height. In △ ABC, the area remains the same regardless of which side is treated as the base and which corresponding altitude is used. Therefore, the area calculated using base AB and altitude CF is equal to the area calculated using base AC and altitude BE.

½ × AB × CF = ½ × AC × BE

By eliminating the constant ½, the relation simplifies to AB × CF = AC × BE. Substituting the given dimensions AB = 6 cm, AC = 5 cm, and CF = 4 cm into the equation:

6 × 4 = 5 × BE 24 = 5 × BE BE = 24/5 = 4.8 cm

This result is derived from the geometric principle that the altitudes of a triangle are inversely proportional to their respective bases.

Explanation

At 3 PM, the minute hand of a clock points towards the 12 mark, while the hour hand points towards the 3 mark. A clock is divided into 12 equal hour intervals, and the total angle of a circle is 360° or 2π radians. The angular distance between each consecutive hour mark is 360° / 12 = 30°. Since there are 3 hour intervals between the 12 mark and the 3 mark, the angle formed between the hands is 3 × 30° = 90°. To find the circular measure, we convert degrees to radians using the formula radians = degrees × (π / 180°). Applying this to the angle at 3 PM, we get 90° × (π / 180°) = π / 2 radians.

Explanation

The expression is simplified using the algebraic identity for the sum of two cubes: a³ + b³ = (a + b)(a² - ab + b²)

This identity can be rewritten to solve for the quotient: a³ + b³{a² - ab + b²} = a + b

By assigning a = 2.3 and b = 0.3, the terms of the given expression match the components of the identity. The numerator is (2.3)³ + (0.3)³ because (0.3)³ = 0.027. The denominator corresponds to the factor a² - ab + b², where the product ab = 2.3 × 0.3 = 0.69 and the square b² = (0.3)² = 0.09.

According to the identity, the result is the sum of the two terms: 2.3 + 0.3 = 2.6

Explanation

The correct average can be found by calculating the net change in the total sum and dividing it by the number of observations. The difference between the correct value and the misread value is 53 - 83 = -30. This negative difference indicates that the initial total sum was over-calculated by 30 units. The change in the average is this difference divided by the total number of observations: -30 / 100 = -0.3. By adjusting the original average with this correction, the correct average is 35 - 0.3 = 34.7. Alternatively, using the total sum method, the correct sum is (100 × 35) - 83 + 53 = 3470, and the correct average is 3470 / 100 = 34.7.

Explanation

Let a = √x and b = √y. The problem gives two equations: a - b = 1 and a + b = 17. We need to find √(xy), which is the product ab.

Using the algebraic identity (a + b)² - (a - b)² = 4ab, we substitute the given values: 17² - 1² = 4ab 289 - 1 = 4ab 288 = 4ab ab = ²⁸⁸⁄₄ = 72

Alternatively, by solving the system of equations, adding them gives 2√x = 18, which means √x = 9. Subtracting the first from the second gives 2√y = 16, which means √y = 8. Thus, √(xy) = √x · √y = 9 · 8 = 72.

Explanation

The given value is x = √3 + 1/√3. Simplifying this expression, we get: x = ( (√3)² + 1 ) / √3 = (3 + 1) / √3 = 4/√3

First, simplify the term √(126)/√(42) = √(126/42) = √3. Then, the first factor of the expression is (x - √3) = 4/√3 - √3 = (4 - 3) / √3 = 1/√3.

Next, consider the denominator in the second factor: x - (2√3)/3. Since (2√3)/3 = 2/√3, we substitute x: x - 2/√3 = 4/√3 - 2/√3 = 2/√3.

The second factor is ( x - 1 / (x - (2√3)/3) ) = x - 1 / (2/√3) = x - √3/2. Substituting the value of x into this factor: 4/√3 - √3/2 = (8 - 3) / (2√3) = 5/(2√3).

The final value is the product of the two simplified factors: 1/√3 × 5/(2√3) = 5 / (2 × 3) = 5/6

Explanation

According to the Exterior Angle Theorem, the exterior angle of a triangle is equal to the sum of its two opposite interior angles. In △ ABC, when side BC is extended to point D, ∠ ACD is the exterior angle opposite to interior angles ∠ A and ∠ B.

The relationship is expressed as: ∠ ACD = ∠ A + ∠ B

Given that ∠ ACD = 120° and ∠ B = ½ ∠ A, we substitute these values into the formula: 120° = ∠ A + ½ ∠ A

To solve for ∠ A, combine the like terms on the right side: 120° = ³⁄₂ ∠ A

Now, isolate ∠ A: ∠ A = (120° × 2)/3 ∠ A = 40° × 2 ∠ A = 80°

Explanation

In a circle, the tangent at any point is perpendicular to the radius through the point of contact. Since AB is the tangent touching the circle at B and OB is the radius, ∠ OBA must be 90°. This identifies triangle OBA as a right-angled triangle with the hypotenuse OA.

Using the Pythagorean theorem: OA² = OB² + AB²

Substituting the given values OA = 5 cm and OB = 3 cm: 5² = 3² + AB² 25 = 9 + AB² AB² = 25 - 9 AB² = 16 AB = √(16) = 4 cm

The lengths 3, 4, and 5 form a standard Pythagorean triplet, confirming the measure of the tangent segment AB.

Explanation

Using the identity sin(90 - θ) = cos θ, sin 59° = sin(90° - 31°) = cos 31°. The expression sec 31° - sin 59° becomes sec 31° - cos 31°, which simplifies to (1 - cos² 31°) / cos 31° = sin² 31° / cos 31°. Given sin 31° = x/y, then cos 31° = √(y² - x²) / y, so the final result is (x/y)² / [√(y² - x²) / y] = x² / (y√(y² - x²)).

Explanation

The diameter of the largest sphere that can be drawn inside a bounded region with side 18 cm is equal to the side length of that region. Given the side s = 18 cm, the radius r of the sphere is: r = ˢ⁄₂ = ¹⁸⁄₂ = 9 cm

The volume of a sphere is calculated using the formula: V = ⁴⁄₃ π r³

Substituting the value of the radius: V = ⁴⁄₃ × π × 9³ V = ⁴⁄₃ × π × 729 V = 4 × π × 243 V = 972π

The numerical value 972 matches the digits in the correct option, where the character following the digits is a typographical representation of π.

Explanation

The length of a shadow s for a tower of height h at a solar altitude angle θ is given by the formula s = h cot θ.

When the sun's altitude is 30°, the length of the shadow s₁ is 50 cot 30° = 50√3 metres. When the sun's altitude is 45°, the length of the shadow s₂ is 50 cot 45° = 50(1) = 50 metres.

The difference in the shadow lengths x is calculated as x = s₁ - s₂. Substituting the values, we get x = 50√3 - 50, which simplifies to x = 50(√3 - 1) metres.

Explanation

An isograft, also known as a syngeneic graft, is a type of tissue or organ transplantation performed between two individuals who are genetically identical. In humans, this occurs specifically between monozygotic (identical) twins. Because the donor and the recipient share the same genetic code, they also share identical Major Histocompatibility Complex (MHC) antigens, which are the primary markers used by the immune system to distinguish between self and non-self. Consequently, the recipient's immune system does not recognize the transplanted tissue as foreign, which prevents the process of graft rejection and typically eliminates the need for long-term immunosuppressive medication. This term is also applied in experimental biology to grafts between members of the same highly inbred strain of laboratory animals.

Explanation

Yogeshwar Dutt is a renowned Indian freestyle wrestler who achieved significant international success in the 60 kg weight category. He won the bronze medal at the 2012 London Olympics, becoming one of the few Indian wrestlers to secure an Olympic podium finish. His dominance in the sport was further demonstrated by winning gold medals at the 2014 Commonwealth Games in Glasgow and the 2014 Asian Games in Incheon. In recognition of his exceptional contributions to wrestling, he was honored with the Rajiv Gandhi Khel Ratna award in 2012 and the Padma Shri in 2013.

Explanation

The numbers lying between 67 and 101 are the integers from 68 to 100 inclusive. The total number of integers in this range is calculated as 101 - 67 - 1 = 33. Within this set, every number is either prime or composite. To find the count of composite numbers, we identify and subtract the prime numbers in this range. The primes between 67 and 101 are 71, 73, 79, 83, 89, and 97. There are exactly 6 prime numbers. Subtracting the number of primes from the total count of integers gives 33 - 6 = 27 composite numbers.

Explanation

The relationship between the listed price and the selling price is given by the formula Selling Price = Listed Price × (1 - (Discount %)/(100)). Given the discount is 15% and the customer pays Rs. 318.75, let the listed price be L.

318.75 = L × (1 - ¹⁵⁄₁₀₀) 318.75 = L × ⁸⁵⁄₁₀₀ 318.75 = L × 0.85

To find the listed price, divide the selling price by 0.85:

L = (318.75)/(0.85) L = ³¹⁸⁷⁵⁄₈₅

Simplifying by dividing both numerator and denominator by 5:

L = ⁶³⁷⁵⁄₁₇

Dividing 6375 by 17: 17 × 300 = 5100, remainder 1275. 17 × 70 = 1190, remainder 85. 17 × 5 = 85. 300 + 70 + 5 = 375.

The listed price of the article is Rs. 375.00.

Explanation

To find the ratio of the reciprocals of the squares of 1½ and 1⅓, first convert the mixed fractions into improper fractions: 1½ = ³⁄₂ 1⅓ = ⁴⁄₃

Next, calculate the squares of these improper fractions: (³⁄₂)² = ⁹⁄₄ (⁴⁄₃)² = ¹⁶⁄₉

Now, find the reciprocals of these squares: Reciprocal of ⁹⁄₄ = ⁴⁄₉ Reciprocal of ¹⁶⁄₉ = ⁹⁄₁₆

The ratio of these reciprocals is: ⁴⁄₉ : ⁹⁄₁₆

To simplify this ratio into whole numbers, multiply both terms by the least common multiple of the denominators (9 × 16 = 144): (⁴⁄₉ × 144) : (⁹⁄₁₆ × 144) = 64 : 81

Explanation

The total sum of all 15 numbers is calculated by multiplying the count by the average: 15 × 7 = 105. The sum of the first 8 numbers is 8 × 6.5 = 52, and the sum of the last 8 numbers is 8 × 8.5 = 68.

In the set of 15 numbers, the 8th number is the middle number. When we add the sum of the first 8 numbers and the sum of the last 8 numbers, the middle number is counted twice. Therefore, the value of the middle number is the difference between the sum of these two groups and the total sum of the 15 numbers.

Middle number = (Sum of first 8 numbers + Sum of last 8 numbers) - Total sum of 15 numbers Middle number = (52 + 68) - 105 Middle number = 120 - 105 = 15

Explanation

The total cost price (CP) for 25 crates is Rs. 10,000. To earn a total profit of 25% on this cost, the total selling price (SP) required is calculated as:

Total SP = CP + (Profit % of CP) Total SP = 10,000 + ²⁵⁄₁₀₀ × 10,000 Total SP = 10,000 + 2,500 = Rs. 12,500

The man lost 5 crates, so the number of remaining crates available for sale is 25 - 5 = 20 crates. To achieve the targeted total selling price, the selling price per remaining crate is:

SP per crate = (Total SP)/(Remaining Crates) SP per crate = (12,500)/(20) = Rs. 625

Explanation

To calculate the difference between 0.6 and 0.6%, we first convert the percentage into its decimal form. A percentage is a fraction of 100, so 0.6% is equal to 0.6 / 100, which is 0.006.

The calculation for the difference is as follows: 0.6 - 0.6% = 0.6 - 0.006

To perform the subtraction accurately, we can align the decimal places by adding trailing zeros to the first number: 0.600 - 0.006 = 0.594

Explanation

The amount A under compound interest is calculated using the formula A = P(1 + (R)/(100))ⁿ, where P is the principal, R is the rate of interest per annum, and n is the time in years. Substituting the given values P = 1000, R = 10, and n = 3:

A = 1000(1 + ¹⁰⁄₁₀₀)³ A = 1000(1.1)³ A = 1000(1.331) = 1331

The compound interest (CI) is the difference between the amount and the principal: CI = A - P CI = 1331 - 1000 = 331

Alternatively, the effective compound interest rate for 3 years at 10% can be calculated using the successive percentage formula or by knowing that 1.1³ = 1.331, which represents a 33.1% total increase. Thus, 1000 × 33.1% = 331.

Explanation

To find the value of the expression 9+2√3{√3}, divide each term in the numerator by the denominator:

(9)/(√(3)) + 2√3{√3}

The first term (9)/(√(3)) can be simplified by treating 9 as 3 × 3. Since 3 = √3 × √3, we have 3 × √3 × √3{√3}, which reduces to 3√3. The second term simplifies directly as 2√3{√3} = 2.

The expression becomes 3√3 + 2.

Substituting the given value √3 = 1.732:

3(1.732) + 2 = 5.196 + 2 = 7.196

Explanation

From the given equation x + ¹⁄ₓ = 2, we can multiply the entire equation by x to get x² + 1 = 2x. Rearranging the terms results in the quadratic equation x² - 2x + 1 = 0. This expression is a perfect square, equivalent to (x - 1)² = 0. Solving for x, we find that x = 1.

Substituting x = 1 into the required expression x⁹⁹ + (1)/(x⁹⁹⁾ - 2: 1⁹⁹ + (1)/(1⁹⁹⁾ - 2 = 1 + 1 - 2 = 2 - 2 = 0.

Explanation

The tangent from an external point to a circle is perpendicular to the radius at the point of contact. Let T be the point of contact on the circle. This forms a right-angled triangle OTP, where OT is the radius and OP is the hypotenuse. According to the Pythagoras theorem, the relationship between the sides is given by OP² = OT² + PT². Given that the distance from the centre OP = 13 cm and the radius OT = 5 cm, the equation becomes 13² = 5² + PT². Solving for the length of the tangent, 169 = 25 + PT², which leads to PT² = 144. Taking the square root, PT = √(144) = 12 cm.

Explanation

The sum of the interior angles of any hexagon is given by the formula (n - 2) × 180°, where n is the number of sides. For a hexagon, n = 6, so the total sum of interior angles is (6 - 2) × 180° = 720°. In a cyclic hexagon, a specific geometric property exists regarding alternate angles: the sum of one set of alternate interior angles is equal to the sum of the other set of alternate interior angles.

If we denote the vertices of the cyclic hexagon as P, Q, R, S, T, and U, the property states that ∠ P + ∠ R + ∠ T = ∠ Q + ∠ S + ∠ U. Since the total sum of all six interior angles is 720°, each of these alternate sets must sum to exactly half of the total. Therefore, the sum ∠ P + ∠ R + ∠ T is calculated as 720° / 2 = 360°.

Explanation

The relationship between the trigonometric functions sine, cosine, and tangent is defined by the formula: tan θ = (sin θ)/(cos θ)

By substituting the given values into the equation: tan θ = (7)/(2){7√3{2}}

To simplify the fraction, multiply the numerator by the reciprocal of the denominator: tan θ = ⁷⁄₂ × (2)/(7√(3))

Canceling the common factor of ⁷⁄₂ results in: tan θ = (1)/(√(3))

In trigonometry, the standard value of θ for which tan θ = (1)/(√(3)) in the range 0° ≤ θ ≤ 90° is 30°. Thus, θ = 30°.

Explanation

The time taken to complete a task is inversely proportional to the efficiency of the person. If E is efficiency and T is time, then for a constant amount of work, E₁ × T₁ = E₂ × T₂.

Let Sashi efficiency be 100. Since Tanya is 25% more efficient than Sashi, Tanya efficiency is 125. The ratio of their efficiencies is 100 : 125, which simplifies to 4 : 5.

Because the ratio of efficiency between Sashi and Tanya is 4 : 5, the ratio of the time taken by them to complete the same work will be the reciprocal, which is 5 : 4.

According to the question, Sashi takes 20 days to complete the work. Equating Sashi's ratio unit to his actual days: 5 units = 20 days 1 unit = 4 days

Tanya's time is represented by 4 units. Therefore: 4 units = 4 × 4 = 16 days

Using the direct formula approach: Tanya days = (Sashi days)/(1 + efficiency increase) Tanya days = (20)/(1.25) = ²⁰⁰⁰⁄₁₂₅ = 16 days

Explanation

From the first equation a + (1)/(b) = 1, we can express a in terms of b as follows: a = 1 - (1)/(b) = (b - 1)/(b)

From the second equation b + (1)/(c) = 1, we can express c in terms of b: (1)/(c) = 1 - b c = (1)/(1 - b)

To find the value of abc, substitute the expressions for a and c in terms of b: abc = ((b - 1)/(b)) × b × ((1)/(1 - b))

The term b in the numerator and denominator cancels out, leaving: abc = (b - 1) × (1)/(1 - b)

By factoring out a negative sign from the denominator, we have 1 - b = -(b - 1). Substituting this into the product gives: abc = (b - 1)/(-(b - 1)) abc = -1

Explanation

To find the value of the expression x² + (1)/(16x²) - 2, start with the given equation 2x - ¹⁄₂ₓ = 5. Divide the entire equation by 2 to isolate x in the first term, resulting in x - ¹⁄₄ₓ = ⁵⁄₂.

Squaring both sides of this simplified equation yields: (x - ¹⁄₄ₓ)² = (⁵⁄₂)² x² + (1)/(16x²) - 2(x)(¹⁄₄ₓ) = ²⁵⁄₄ x² + (1)/(16x²) - ½ = ²⁵⁄₄

To find the value of x² + (1)/(16x²), add ½ to both sides: x² + (1)/(16x²) = ²⁵⁄₄ + ½ = (25 + 2)/(4) = ²⁷⁄₄

The question asks for the value of x² + (1)/(16x²) - 2. Substituting the value obtained above: ²⁷⁄₄ - 2 = (27 - 8)/(4) = ¹⁹⁄₄

Explanation

In a right-angled triangle, the circumradius is always equal to half of the hypotenuse because the circumcentre lies at the midpoint of the hypotenuse. In Δ XYZ, since ∠ Y = 90°, the side XZ is the hypotenuse.

According to the Pythagorean theorem: XZ = √(XY² + YZ²) XZ = √((2.5)² + 6²) XZ = √(6.25 + 36) XZ = √(42.25) = 6.5 cm

The circumradius R is given by the formula: R = (Hypotenuse)/(2) = (XZ)/(2) R = (6.5)/(2) = 3.25 cm

Explanation

According to the angle sum property of a triangle, the sum of all interior angles in △ABC is 180°. This is expressed by the equation ∠A + ∠B + ∠C = 180°. By substituting the given relationships ∠A = 3∠C and ∠B = 5∠C into this equation, we obtain 3∠C + 5∠C + ∠C = 180°. Combining the terms on the left side results in 9∠C = 180°. Dividing both sides of the equation by 9 yields ∠C = 180° / 9 = 20°.

Explanation

In a right-angled triangle formed by a tower and its shadow, let the height of the tower be h and the length of its shadow be s. The problem states that the shadow length is √3 times the height, which gives the relation s = √3h.

The angle of elevation of the sun, θ, is the angle between the ground and the line of sight to the sun. This angle can be calculated using the tangent trigonometric ratio, which is the ratio of the perpendicular (height of the tower) to the base (length of the shadow):

tan θ = (Height)/(Shadow)

Substituting the given values into the formula:

tan θ = (h)/(√(3)h)

tan θ = (1)/(√(3))

From trigonometric tables, the value of tan θ = (1)/(√(3)) corresponds to θ = 30°.