SSC CGL Tier 1 2020 Quantitative Aptitude Questions with Solutions

Explanation

In an arithmetic progression, the average of n consecutive terms is the middle value. For five consecutive even numbers, the average M corresponds to the third term. Let the numbers be x, x+2, x+4, x+6, x+8. Thus, M = x+4. When the next five even numbers (x+10, x+12, x+14, x+16, x+18) are included, the set contains ten terms. The average of ten consecutive terms in an arithmetic progression is the mean of the two middle terms, which are the fifth and sixth terms. The fifth term is x+8 and the sixth term is x+10. The new average is ((x+8) + (x+10))/(2) = x+9. Since M = x+4, the new average x+9 can be written as (x+4) + 5, which equals M+5. More generally, for any arithmetic progression with a common difference d, if k subsequent terms are added, the average increases by (k × d)/(2). In this case, d=2 and k=5, leading to an increase of (5 × 2)/(2) = 5.

Explanation

According to the BODMAS rule, the operations must be performed in the order of Division, Multiplication, then Addition/Subtraction.

First, perform the division: ³⁄₂ ÷ 6 = ³⁄₂ × ⅙ = ¼

Next, perform the multiplication with the resulting quotient: ¼ × ½ = ⅛

Now, substitute this back into the original expression: -⁵⁄₂ + ⅛

To add these fractions, find a common denominator, which is 8: -(5 × 4)/(2 × 4) + ⅛ = -²⁰⁄₈ + ⅛

Combining the numerators over the common denominator gives: (-20 + 1)/(8) = -¹⁹⁄₈

Explanation

The given expression follows the algebraic identity X² - Y² = (X + Y)(X - Y).

Let X = (a + b - c + d) and Y = (a - b + c - d).

First, calculate the sum (X + Y): X + Y = (a + b - c + d) + (a - b + c - d) X + Y = a + a + b - b - c + c + d - d X + Y = 2a

Next, calculate the difference (X - Y): X - Y = (a + b - c + d) - (a - b + c - d) X - Y = a - a + b - (-b) - c - c + d - (-d) X - Y = 2b - 2c + 2d X - Y = 2(b - c + d)

Now, multiply the sum and the difference: (X + Y)(X - Y) = (2a) · [2(b - c + d)] Result = 4a(b - c + d)

By rearranging the terms inside the parentheses to match the format of the options, we obtain 4a(b + d - c).

Explanation

The given equation is 2sin θ - 8cos² θ + 5 = 0. Using the identity cos² θ = 1 - sin² θ, we can rewrite the equation in terms of sin θ:

2sin θ - 8(1 - sin² θ) + 5 = 0 2sin θ - 8 + 8sin² θ + 5 = 0 8sin² θ + 2sin θ - 3 = 0

Factoring the quadratic equation by splitting the middle term: 8sin² θ + 6sin θ - 4sin θ - 3 = 0 2sin θ (4sin θ + 3) - 1(4sin θ + 3) = 0 (2sin θ - 1)(4sin θ + 3) = 0

This gives sin θ = 1/2 or sin θ = -3/4. Since 0° < θ < 90°, we have sin θ = 1/2, which implies θ = 30°.

The expression to be evaluated is tan 2θ + cosec 2θ. Substituting θ = 30°: tan(2 × 30°) + cosec(2 × 30°) = tan 60° + cosec 60°

tan 60° = √3 cosec 60° = (2)/(√(3))

Adding these values: √3 + (2)/(√(3)) = (3 + 2)/(√(3)) = (5)/(√(3))

Multiplying the numerator and denominator by √3 to rationalize the result: (5)/(√(3)) × √3{√3} = 5√3{3}

Explanation

Let the distance of the race be D meters. Using the formula Time = Distance / Speed, the difference in time is D/100 - D/120 = 10, which simplifies to (6D - 5D)/600 = 10, resulting in D = 6000 meters or 6 km.

Explanation

Substitute the standard values for the trigonometric ratios into the expression: tan 30° = (1)/(√(3)), tan 60° = √3, and cos 30° = √3{2}.

First, simplify the numerator: tan 30° + tan 60° = (1)/(√(3)) + √3 = 1 + (√3 · √3){√3} = (1 + 3)/(√(3)) = (4)/(√(3))

Now, divide the simplified numerator by the denominator: (4)/(√(3)){√3{2}} = (4)/(√(3)) × (2)/(√(3))

Multiplying the numerators and denominators results in: (4 × 2)/(√(3) × 3) = ⁸⁄₃

Explanation

A's work rate is 1/20 per day and B's is 1/25 per day, so their combined rate is 9/100 per day. In 5 days they complete 45/100 of the work, leaving 55/100 (or 11/20) for A to finish alone. A takes (11/20) / (1/20) = 11 more days to complete the remaining task.

Explanation

The formula for the curved surface area of a hemisphere is 2πr². Substituting the given values r = 7 cm and π = 22/7 results in 2 * (22/7) * 7 * 7 = 308 cm².

Explanation

To find the value of the expression, follow the BODMAS (Brackets, Of, Division, Multiplication, Addition, Subtraction) rules.

Numerator calculation:

1. Term 1: 36 ÷ 42 of 6 × 7 = 36 ÷ (42 × 6) × 7 = 36 ÷ 252 × 7 = (36 / 252) × 7 = (1/7) × 7 = 1.
2. Term 2: 24 × 6 ÷ 18 = 24 × (6 / 18) = 24 × (1/3) = 8.
3. Term 3: 3 ÷ (2 - 6) = 3 ÷ (-4) = -3/4 or -0.75.
4. Term 4: (4 + 3 × 2) ÷ 8 = (4 + 6) ÷ 8 = 10 ÷ 8 = 5/4 or 1.25. Numerator = 1 + 8 - 0.75 - 1.25 = 9 - 2 = 7.

Denominator calculation:

1. 21 ÷ 3 of 7 = 21 ÷ (3 × 7) = 21 ÷ 21 = 1.

Final value: Result = Numerator / Denominator = 7 / 1 = 7.

Explanation

In an isosceles triangle where AB = AC, the angles opposite those sides are equal, meaning angle B = angle C. Since the sum of angles in a triangle is 180 degrees and angle BAC is 40 degrees, the sum of the remaining two equal angles is 140 degrees, making angle B equal to 70 degrees.

Explanation

The divisibility rule for 11 states that the difference between the sum of digits at odd positions and the sum of digits at even positions must be 0 or a multiple of 11. For the number 94*2357, the sum of digits at odd positions is 9 + * + 3 + 7 = 19 + *, and the sum of digits at even positions is 4 + 2 + 5 = 11. The difference (19 + *) - 11 = 8 + * must be 11, which gives * = 3.

Explanation

The two numbers are in the ratio 5 : 7. Given that the first number is 20, we can establish the relationship by finding the multiplier between the ratio value and the actual value. Since 5 × 4 = 20, the common multiplier for both parts of the ratio is 4. To find the second number, we multiply its corresponding ratio value by the same multiplier: 7 × 4 = 28. Alternatively, using the property of proportions, ⁵⁄₇ = ²⁰⁄ₓ leads to 5x = 140, which yields x = 28.

Explanation

The difference between 80% and 62% is 18%, which is given as 198. Since 18% equals 198, then 1% equals 11. The required difference between 92% and 56% is 36%, which equals 36 multiplied by 11, resulting in 396.

Explanation

The cost price (CP) of the 6 sewing machines is calculated by dividing the selling price (SP) by the profit factor: 63,000 / 1.05 = 60,000. Therefore, the CP per machine is 60,000 / 6 = 10,000. For 8 machines at a 15% profit, the calculation is (10,000 * 8) * 1.15 = 80,000 * 1.15 = 92,000.

Explanation

The expression 27a³ - 2√2b³ follows the algebraic identity for the difference of two cubes: x³ - y³ = (x - y)(x² + xy + y²).

First, identify the cube roots of the given terms: x³ = 27a³ ⟹ x = ³√(27a³⁾ = 3a y³ = 2√2b³ ⟹ y = ³√((√(2))³⁾ᵇ^^^³ = √2b

Now, calculate the terms for the second factor: x² = (3a)² = 9a² y² = (√2b)² = 2b² xy = (3a)(√2b) = 3√2ab

Substituting these into the identity (x - y)(x² + y² + xy) , we get: (3a - √2b)(9a² + 2b² + 3√2ab)

Explanation

In a circle, a perpendicular line drawn from the center to a chord bisects the chord. This construction forms a right-angled triangle where the radius of the circle is the hypotenuse, the distance from the center is one leg, and half the length of the chord is the other leg. Applying the Pythagorean theorem:

r² = d² + ((L)/(2))²

Given the radius r = 5 cm and the distance from the center d = 3 cm, we substitute these values into the formula:

5² = 3² + ((L)/(2))² 25 = 9 + ((L)/(2))² ((L)/(2))² = 25 - 9 = 16 (L)/(2) = √(16) = 4 cm

The calculation identifies the side as 4 cm based on the standard (3, 4, 5) Pythagorean triplet. Since this value represents half the chord, the total length of the chord L is 2 × 4 = 8 cm.

Explanation

The net profit percentage when a markup and a discount are applied successively can be calculated using the formula for successive percentage changes: Net Change = x + y + ˣʸ⁄₁₀₀.

In this transaction, the markup percentage x = +40% and the discount percentage y = -20%. Substituting these values into the formula:

Net Profit % = 40 - 20 + (40 × (-20))/(100) Net Profit % = 20 - ⁸⁰⁰⁄₁₀₀ Net Profit % = 20 - 8 = 12%

Alternatively, if we assume the Cost Price (CP) is 100, the Marked Price (MP) becomes 140 to allow for a 40% profit. A 20% discount on the MP is 0.20 × 140 = 28. The Selling Price (SP) is then 140 - 28 = 112. Since the SP is 112 and the CP is 100, the profit is 12, which corresponds to a 12% profit percentage.

Explanation

The simple interest SI for 3 years is 1,200, meaning the interest for each year is SI/3 = 1,200/3 = 400. To find the compound interest CI for the same period, we calculate the interest earned on previous interest payments at the given rate of 5%.

Using the splitting method for 3 years, CI is given by the formula 3x + 3y + z, where x is the simple interest for one year, y is the interest on x, and z is the interest on y.

First, x = 400. Second, y = 5% of 400 = (5/100) × 400 = 20. Third, z = 5% of 20 = (5/100) × 20 = 1.

Substituting these values into the compound interest structure: CI = 3(400) + 3(20) + 1(1) CI = 1,200 + 60 + 1 = 1,261

Alternatively, the principal P can be found using SI = (P × R × T) / 100. 1,200 = (P × 5 × 3) / 100 ⟹ P = (1,200 × 100) / 15 = 8,000. The amount A after 3 years is P(1 + R/100)³ = 8,000(1 + 5/100)³ = 8,000(21/20)³. A = 8,000 × (9,261 / 8,000) = 9,261. CI = A - P = 9,261 - 8,000 = 1,261.

Explanation

In the right-angled △ ABC where ∠ A = 90°, the hypotenuse BC is calculated using the Pythagorean theorem: BC = √(AB² + AC²) = √(7² + 24²) = √(49 + 576) = √(625) = 25 cm.

Since M is the midpoint of the hypotenuse BC, the segment AM is the median to the hypotenuse. In any right-angled triangle, the length of the median to the hypotenuse is equal to half the length of the hypotenuse, meaning AM = (BC)/(2) = ²⁵⁄₂ = 12.5 cm.

The altitude AD dropped from the right angle to the hypotenuse BC is found by equating the area of the triangle calculated in two ways: Area = ½ × AB × AC = ½ × BC × AD. Substituting the known values gives 7 × 24 = 25 × AD, which results in AD = ¹⁶⁸⁄₂₅ cm.

The ratio AD:AM is therefore ¹⁶⁸⁄₂₅ : ²⁵⁄₂. Multiplying the terms to clear the denominators yields 168 × 2 : 25 × 25, which simplifies to 336:625.

Explanation

Given the equation 5x + ¹⁄₃ₓ = 4, we first transform the expression to match the coefficients of the required value 9x² + (1)/(25x²⁾. To obtain a 3x term from 5x, multiply both sides of the equation by ⅗:

⅗ ( 5x + ¹⁄₃ₓ ) = 4 × ⅗ 3x + ¹⁄₅ₓ = ¹²⁄₅

Now, square both sides of the resulting equation:

( 3x + ¹⁄₅ₓ )² = ( ¹²⁄₅ )²

Using the algebraic identity (a + b)² = a² + b² + 2ab:

9x² + (1)/(25x²⁾ + 2(3x)( ¹⁄₅ₓ ) = ¹⁴⁴⁄₂₅ 9x² + (1)/(25x²⁾ + ⁶⁄₅ = ¹⁴⁴⁄₂₅

To find the value of 9x² + (1)/(25x²⁾, subtract ⁶⁄₅ from the right side:

9x² + (1)/(25x²⁾ = ¹⁴⁴⁄₂₅ - ⁶⁄₅ 9x² + (1)/(25x²⁾ = (144 - 30)/(25) 9x² + (1)/(25x²⁾ = ¹¹⁴⁄₂₅

Explanation

Total computers sold by dealer A from February to June is the sum of sales for those five months: 94 + 85 + 108 + 98 + 95 = 480

Total computers sold by all dealers (A, B, C, and D) in June is the sum of the values in the June row: 95 + 108 + 102 + 91 = 396

To find how much more the sales of dealer A are compared to the June total, calculate the difference: Difference = 480 - 396 = 84

The percentage more is calculated by dividing this difference by the total sales in June: Percentage more = ⁸⁴⁄₃₉₆ × 100

Simplifying the fraction by dividing both numerator and denominator by 12: ⁷⁄₃₃ × 100 = ⁷⁰⁰⁄₃₃ = 21.2121...

Rounding to one decimal place, the percentage is 21.2.

Explanation

To find the value of the expression, substitute the standard trigonometric ratios: tan 30° = (1)/(√(3)), csc 60° = (2)/(√(3)), tan 60° = √3, sec 30° = (2)/(√(3)), sin 30° = ½, cot 45° = 1, and sec 60° = 2.

The numerator of the expression is: tan 30°csc 60° + tan 60°sec 30° = ((1)/(√(3)) × (2)/(√(3))) + (√3 × (2)/(√(3))) = ⅔ + 2 = ⁸⁄₃

The denominator of the expression is: sin²30° + 4cot²45° - sec²60° = (½)² + 4(1)² - (2)² = ¼ + 4 - 4 = ¼

Combining these results gives the final value: (8/3)/(1/4) = ⁸⁄₃ × 4 = ³²⁄₃

Explanation

Let the initial income of Sonu be 100. Since she saves 15%, her initial savings are 15 and her initial expenditure is 100 - 15 = 85.

After the income increases by 20%, the new income becomes 120. It is given that her savings remain the same as before, so her new savings are still 15. Therefore, the new expenditure is the difference between the new income and the savings: 120 - 15 = 105.

The increase in expenditure is 105 - 85 = 20. To find the percentage increase in expenditure, we use the formula:

Percentage Increase = (Increase)/(Original Expenditure) × 100

Percentage Increase = ²⁰⁄₈₅ × 100 = ⁴⁄₁₇ × 100 = ⁴⁰⁰⁄₁₇

Calculating the division: 400 ÷ 17 ≈ 23.529. Rounding to one decimal place, we get 23.5%.

Explanation

The division algorithm states that Dividend = (Divisor × Quotient) + Remainder, where the divisor must be greater than the remainder. Given that 732 divided by x leaves a remainder of 12, we have 732 = qx + 12. This simplifies to qx = 720. Thus, x must be a factor of 720 such that x > 12.

To find the total number of factors of 720, we use its prime factorization: 720 = 72 × 10 = (2³ × 3²) × (2¹ × 5¹) = 2⁴ × 3² × 5¹

The total number of factors is calculated using the formula (a+1)(b+1)(c+1) for prime powers: Total factors = (4+1)(2+1)(1+1) = 5 × 3 × 2 = 30

Next, we identify the factors of 720 that are less than or equal to 12 to exclude them, as x > 12. These factors are: 1, 2, 3, 4, 5, 6, 8, 9, 10, 12. Counting these, we find there are 10 such values.

The number of possible values for x is the total number of factors minus the number of factors less than or equal to the remainder: 30 - 10 = 20

Explanation

The circumference of the base of the conical tent is given by 2 π r = 66 m. Substituting π = ²²⁄₇, we have 2 × ²²⁄₇ × r = 66, which simplifies to r = (66 × 7)/(44) = 10.5 m.

The slant height l of the cone is calculated using the Pythagorean relation l = √(r² + h²). Given the height h = 36 m and radius r = 10.5 m, we find l = √(10.5² + 36²) = √(110.25 + 1296) = √(1406.25) = 37.5 m.

The area of the canvas used to make the tent corresponds to the curved surface area of the cone, which is calculated as CSA = π r l. Substituting the values, we get CSA = ²²⁄₇ × 10.5 × 37.5. Simplifying the calculation, 22 × 1.5 × 37.5 = 33 × 37.5 = 1237.5 m².

Explanation

The value of the expression a³ + b³ + c³ can be determined using the standard algebraic identity a³ + b³ + c³ - 3abc = (a + b + c)((a + b + c)² - 3(ab + bc + ca)).

Substitute the given values into the formula: a + b + c = 11 ab + bc + ca = 3 abc = -135

Plugging these into the identity: a³ + b³ + c³ - 3(-135) = 11 × (11² - 3 × 3) a³ + b³ + c³ + 405 = 11 × (121 - 9) a³ + b³ + c³ + 405 = 11 × 112 a³ + b³ + c³ + 405 = 1232

To find a³ + b³ + c³, subtract 405 from 1232: a³ + b³ + c³ = 1232 - 405 = 827

Explanation

Let the length of side AB be x cm. Since △ ABC is isosceles with AB = AC, the length of AC is also x cm. Given that D is the midpoint of AC, the length of the segment AD is ˣ⁄₂ cm. According to the tangent-secant theorem, when a tangent segment and a secant segment are drawn to a circle from an external point, the square of the length of the tangent segment is equal to the product of the lengths of the whole secant segment and its external part. In this case, from external point A, AD is the tangent segment and ABP is the secant segment. The theorem provides the relation AD² = AP · AB. Substituting the known values into the formula, we have (ˣ⁄₂)² = 2.5 · x. This simplifies to (x²)/(4) = 2.5x. Dividing both sides by x (as the length x must be positive) gives ˣ⁄₄ = 2.5, resulting in x = 10 cm. Therefore, the length of AB is 10 cm.

Explanation

The expression is evaluated using the BODMAS rule, which dictates the order of operations: Brackets, Of, Division, Multiplication, Addition, and Subtraction.

Numerator: 7 - [4 + 3(2 - 2 × 2 + 5) - 8] ÷ 5 First, solve the expression inside the small parentheses: 2 - 4 + 5 = 3 Next, solve the expression inside the square brackets: 4 + 3(3) - 8 = 4 + 9 - 8 = 5 Finally, evaluate the numerator: 7 - 5 ÷ 5 = 7 - 1 = 6

Denominator: 2 ÷ 2 of(4 + 4 ÷ 4 of4) First, solve the expression inside the parentheses: The of operator is evaluated before division: 4 of 4 = 16. Then perform the division: 4 ÷ 16 = 0.25. Then perform the addition: 4 + 0.25 = 4.25. Now, evaluate the full denominator: The of operator is evaluated before division: 2 of 4.25 = 8.5. Perform the division: 2 ÷ 8.5 = (2)/(8.5) = ⁴⁄₁₇.

Combining the numerator and the denominator: 6 ÷ ⁴⁄₁₇ = 6 × ¹⁷⁄₄ = ¹⁰²⁄₄ = 25.5

Considering the standard integer options provided, the value 25.5 is closest to 25.

Explanation

When four numbers a, b, c, and d are in proportion, the relation is defined by a/b = c/d. Given the numbers 2x + 1, x + 2, 2, and 5 are in proportion, we set up the equation:

(2x + 1) / (x + 2) = 2 / 5

Cross-multiplying gives 5(2x + 1) = 2(x + 2), which simplifies to 10x + 5 = 2x + 4. Solving for x results in 8x = -1, or x = -1/8.

The mean proportional between two numbers p and q is calculated as √(pq). Substituting x = -1/8 into the given expressions:

The first term is 3.5(1 - x) = 3.5(1 + 1/8) = (7/2) × (9/8) = 63/16. The second term is 8(1 + x) = 8(1 - 1/8) = 8 × (7/8) = 7.

The mean proportional is √((63/16) × 7) = √((9 × 7 × 7) / 16). Taking the square root gives (3 × 7) / 4 = 21/4 = 5.25.

Explanation

In a cyclic quadrilateral ABCD, if one diagonal AC bisects the other diagonal BD at point O (such that BO = OD), the product of the pairs of sides meeting at the vertices of the bisected diagonal are equal. This follows from the fact that the triangles ABC and ADC share the same base AC and have equal altitudes from B and D because O is the midpoint of BD.

The area of △ ABC is ½ · AB · BC · sin(∠ ABC) and the area of △ ADC is ½ · AD · CD · sin(∠ ADC). Since ∠ ABC + ∠ ADC = 180° in a cyclic quadrilateral, their sines are equal. Given the areas are equal due to the diagonal bisection, we have:

AB · BC = AD · CD

Substituting the given values AB = 16.5, BC = x, CD = 11, and AD = 19.8:

16.5 · x = 19.8 · 11

x = (19.8 · 11)/(16.5)

x = (217.8)/(16.5)

x = 13.2 cm

Explanation

In △ ABC, the sum of all internal angles is 180°. Given ∠ B = 68° and ∠ C = 32°, the measure of ∠ A is calculated as: ∠ A = 180° - (68° + 32°) = 180° - 100° = 80°

When the sides AB and AC are produced to D and E, and the external bisectors of ∠ DBC and ∠ BCE meet at F, the point F is the excenter of the triangle relative to vertex A. The angle formed by these external bisectors at the excenter is given by the standard geometric formula: ∠ BFC = 90° - (∠ A)/(2)

Substituting the value of ∠ A into the formula: ∠ BFC = 90° - (80°)/(2) ∠ BFC = 90° - 40° = 50°

Explanation

To find the markup percentage, we use the standard relationship between Cost Price (CP) and Marked Price (MP) involving profit and discount percentages:

(MP)/(CP) = (100 + Profit%)/(100 - Discount%)

Given that the discount is 18% and the profit is 6.6%, we substitute these values into the formula:

(MP)/(CP) = (100 + 6.6)/(100 - 18) (MP)/(CP) = (106.6)/(82)

To find the markup percentage above the cost price, we calculate:

Markup% = ( (MP)/(CP) - 1 ) × 100 Markup% = ( (106.6)/(82) - 1 ) × 100 Markup% = (106.6 - 82)/(82) × 100 Markup% = (24.6)/(82) × 100 Markup% = ²⁴⁶⁰⁄₈₂ = 30%

Explanation

The total number of computers sold by dealer B in April, May, and June is calculated as 97 + 102 + 108 = 307.

To find the total sales by all dealers in February and April, we sum the values for all four dealers in those months: February total = 94 + 96 + 104 + 106 = 400 April total = 108 + 97 + 99 + 96 = 400 Combined total for February and April = 400 + 400 = 800

The required percentage is determined by the formula: Percentage = (Sales by B (Apr+May+Jun))/(Total Sales (Feb+Apr)) × 100 Percentage = ³⁰⁷⁄₈₀₀ × 100 = ³⁰⁷⁄₈ = 38.375%

Rounding the result to the nearest whole number provides 38.

Explanation

Let the speed of the boat upstream be u km/min and downstream be v km/min. Based on the given conditions, we can form two linear equations for the time taken:

3/u + 5/v = 55 4/u + 9/v = 85

To eliminate 1/u, multiply the first equation by 4 and the second equation by 3:

12/u + 20/v = 220 12/u + 27/v = 255

Subtracting the first modified equation from the second:

(27 - 20)/v = 255 - 220 7/v = 35 1/v = 5 minutes/km

The downstream speed v is 1/5 km per minute. To convert this to km per hour:

v = (1/5) × 60 = 12 km/hr

The time required to travel 43.2 km downstream is calculated by dividing the distance by the downstream speed:

Time = 43.2 / 12 = 3.6 hours

Explanation

Using the fundamental trigonometric identity sec² θ - tan² θ = 1, we can write (sec θ + tan θ)(sec θ - tan θ) = 1. Given the relationship expressed in the options and the constraints, we define sec θ - tan θ = (x)/(y). Consequently, its reciprocal is sec θ + tan θ = ʸ⁄ₓ.

To find sin θ, we first determine the values of sec θ and tan θ by solving these two equations: Adding the equations: 2 sec θ = ʸ⁄ₓ + (x)/(y) = (y² + x²)/(xy) ⟹ sec θ = (y² + x²)/(2xy) Subtracting the first from the second: 2 tan θ = ʸ⁄ₓ - (x)/(y) = (y² - x²)/(xy) ⟹ tan θ = (y² - x²)/(2xy)

We then use the relation sin θ = (tan θ)/(sec θ): sin θ = (y² - x²)/(2xy) ÷ (y² + x²)/(2xy) = (y² - x²)/(y² + x²)

Given the condition 0 < x < y, the value of sin θ is positive and less than 1, which is consistent with the quadrant constraint 0° < θ < 90°.

Explanation

Let the total cost price of 5 tables be T and the total cost price of 9 chairs be C. Based on the given information, the sum of these costs is T + C = 15,400. The profit earned from selling the tables is 10% and from the chairs is 20%, resulting in a total profit expressed as 0.10T + 0.20C = 2,080. To solve this system of equations, multiply the profit equation by 10 to get T + 2C = 20,800.

Subtracting the first equation from this new equation gives (T + 2C) - (T + C) = 20,800 - 15,400, which results in C = 5,400. This value represents the total cost price of all 9 chairs. To find the cost price of 3 chairs, calculate the price for one chair by dividing the total cost of chairs by 9, which is 5,400 / 9 = 600, and then multiply by 3. Thus, the cost price for 3 chairs is 600 × 3 = 1,800.

Explanation

To find the average number of computers sold by dealer C over the six months, calculate the sum of sales from January to June and divide by 6. Average of C = (95 + 104 + 100 + 99 + 100 + 102)/(6) = ⁶⁰⁰⁄₆ = 100

Now, compare the monthly sales of dealer B with this average value of 100: In January, dealer B sold 92 computers, which is less than 100. In February, dealer B sold 96 computers, which is less than 100. In March, dealer B sold 94 computers, which is less than 100. In April, dealer B sold 97 computers, which is less than 100. In May, dealer B sold 102 computers, which is more than 100. In June, dealer B sold 108 computers, which is more than 100.

The months in which dealer B's sales were less than the average of dealer C are January, February, March, and April, making a total of 4 months.

Explanation

The total number of computers sold by dealer A in February, April, and May is calculated by summing the respective values from the table: 94 + 108 + 98 = 300.

The total number of computers sold by dealer D in March, May, and June is found by summing the corresponding values: 90 + 89 + 91 = 270.

The required ratio of the sales of dealer A to dealer D for these specific months is 300 : 270. By dividing both components of the ratio by 30, the expression simplifies to 10 : 9.

Explanation

Let the efficiency of A be 100. Since B is 25% more efficient than A, the efficiency of B is 125. C is 20% more efficient than B, so the efficiency of C is 125 × 1.20 = 150. The ratio of the efficiencies of A, B, and C is 100 : 125 : 150, which simplifies to 4 : 5 : 6.

The total work is calculated using A's efficiency and the time A takes: Total Work = Efficiency_A × Time_A = 4 × 30 = 120 units.

When A, B, and C work together, their combined efficiency is 4 + 5 + 6 = 15 units per day. In 3 days, the amount of work they complete is 15 × 3 = 45 units. The remaining work is 120 - 45 = 75 units.

B completes the remaining work alone. The time taken by B is Remaining Work / Efficiency_B = 75 / 5 = 15 days.

Explanation

Let v be the speed of the boat in still water and w be the speed of the stream. We are given w = 3 km/h. Using the relation Time = Distance / Speed, and knowing that the time for both journeys is equal, we can set up the following equation:

13 / (v + 3) = 7 / (v - 3)

Cross-multiplying gives: 13(v - 3) = 7(v + 3) 13v - 39 = 7v + 21 6v = 60 v = 10 km/h

The time required to travel a distance of 44.8 km in still water is: T = 44.8 / 10 = 4.48 hours

To express 4.48 as a mixed fraction: 4.48 = 4 + 48/100 = 4 + 12/25

This results in a total time of 4 ¹²⁄₂₅ hours.

Explanation

To find the number of years in which the revenue of product D exceeded the average revenue of product A, first calculate the average revenue for product A over the six-year period from 2012 to 2017.

Average Revenue of A = (98 + 94 + 80 + 95 + 110 + 115)/(6) = ⁵⁹²⁄₆ ≈ 98.67

Now, compare the revenue of product D for each year against this average value of 98.67: In 2012, revenue of D is 74, which is less than 98.67. In 2013, revenue of D is 102, which is more than 98.67. In 2014, revenue of D is 92, which is less than 98.67. In 2015, revenue of D is 93, which is less than 98.67. In 2016, revenue of D is 97, which is less than 98.67. In 2017, revenue of D is 102, which is more than 98.67.

The revenue of product D is greater than the average revenue of product A in exactly two years: 2013 and 2017.

Explanation

In △ PQR, the sum of angles is 180°, so ∠ P = 180° - (∠ Q + ∠ R) = 180° - (85° + 65°) = 30°. Since points S and T lie on PQ and PR respectively, ∠ PTS and ∠ STR are supplementary angles on the straight line PR. Thus, ∠ PTS = 180° - ∠ STR = 180° - 95° = 85°.

Now, considering △ PST and △ PQR, we have ∠ P as a common angle and ∠ PTS = ∠ PQR = 85°. By the AA (Angle-Angle) similarity criterion, △ PTS ∼ △ PQR. In similar triangles, the ratios of corresponding sides are equal, which gives the relation PT/PQ = ST/QR.

The problem provides the ratio QR:ST = 9:5, which implies ST/QR = 5/9. Substituting the known values into the side-ratio equation, we get PT/21.6 = 5/9. Solving for PT, we calculate PT = 21.6 × (5/9) = 2.4 × 5 = 12 cm.

Explanation

Given the quadratic equation 30x² - 15x + 1 = 0, divide every term by 6x to align the equation with the components of the required expression 25x² + (1)/(36x²⁾.

30x²{6x} - ¹⁵ˣ⁄₆ₓ + ¹⁄₆ₓ = 0 5x - ⁵⁄₂ + ¹⁄₆ₓ = 0 5x + ¹⁄₆ₓ = ⁵⁄₂

To find the value of 25x² + (36x²)⁻¹, apply the algebraic identity a² + b² = (a + b)² - 2ab, where a = 5x and b = ¹⁄₆ₓ.

25x² + (1)/(36x²⁾ = (5x + ¹⁄₆ₓ)² - 2(5x)(¹⁄₆ₓ)

Substitute the value obtained from the first step:

25x² + (1)/(36x²⁾ = (⁵⁄₂)² - ¹⁰⁄₆ 25x² + (1)/(36x²⁾ = ²⁵⁄₄ - ⁵⁄₃

To subtract these fractions, find a common denominator, which is 12:

(25 × 3)/(4 × 3) - (5 × 4)/(3 × 4) = ⁷⁵⁄₁₂ - ²⁰⁄₁₂ = ⁵⁵⁄₁₂

Explanation

Let E_B be the expenditure of B. According to the problem, A's income is given by I_A = 0.7 E_B and A's expenditure is given by E_A = 0.6 E_B. Since the income of A is 60% less than that of B, we have I_A = 0.4 I_B. By substituting the value of I_A, we find B's income as 0.7 E_B = 0.4 I_B, which simplifies to I_B = 1.75 E_B.

The savings of A are S_A = I_A - E_A = 0.7 E_B - 0.6 E_B = 0.1 E_B. The savings of B are S_B = I_B - E_B = 1.75 E_B - E_B = 0.75 E_B. To find the ratio of their savings, we calculate S_A : S_B = 0.1 E_B : 0.75 E_B. Multiplying both sides by 100 to clear the decimals, the ratio becomes 10 : 75, which simplifies to 2 : 15 after dividing both terms by 5.

Explanation

The expression is simplified using the BODMAS rule, which prioritizes Brackets, Of, Division, Multiplication, Addition, and Subtraction.

For the numerator, first convert the mixed fractions to improper fractions: 5½ = ¹¹⁄₂, 3⅔ = ¹¹⁄₃, 5⅑ = ⁴⁶⁄₉, 7⅞ = ⁶³⁄₈, and 9⁹⁄₂₀ = ¹⁸⁹⁄₂₀. The first part of the numerator is ¹¹⁄₂ ÷ (¹¹⁄₃ of ¼). Evaluating the of operation first, we get ¹¹⁄₂ ÷ ¹¹⁄₁₂ = ¹¹⁄₂ × ¹²⁄₁₁ = 6. The bracketed term is (⁴⁶⁄₉ - ⁶³⁄₈ ÷ ¹⁸⁹⁄₂₀). Performing the division first: ⁶³⁄₈ × ²⁰⁄₁₈₉ = ½ × ⁵⁄₃ = ⅚. Then, ⁴⁶⁄₉ - ⅚ = (92 - 15)/(18) = ⁷⁷⁄₁₈. Multiplying this by ⁹⁄₁₁ gives ⁷⁷⁄₁₈ × ⁹⁄₁₁ = ⁷⁄₂. The total numerator is 6 + ⁷⁄₂ = ¹⁹⁄₂.

For the denominator, evaluate the of operation followed by division: 5 ÷ (5 of ⅒) - (10 × 10 ÷ 20) = 5 ÷ ½ - (100 ÷ 20) = 10 - 5 = 5.

Dividing the numerator by the denominator, we get (19/2)/(5) = ¹⁹⁄₁₀ = 1⁹⁄₁₀.

Explanation

The volume of water remains constant when transferred from the cylindrical vessel to the rectangular tub. The volume of the cylinder is given by V = π r² H and the volume of the rectangular tub is V = L × B × h. By equating these two volumes, the height of the water in the tub can be determined: π r² H = L × B × h

Substitute the given values r = 30 cm, H = 42 cm, L = 75 cm, B = 44 cm, and π = ²²⁄₇ into the equation: ²²⁄₇ × 30 × 30 × 42 = 75 × 44 × h

Simplify the left side of the equation by dividing 42 by 7: 22 × 900 × 6 = 3300 × h 118800 = 3300 × h

Now, solve for h by dividing the total volume by the base area of the tub: h = ¹¹⁸⁸⁰⁰⁄₃₃₀₀ h = ¹¹⁸⁸⁄₃₃ h = 36 cm

Alternatively, using simplification through cancellation: h = (22 × 30 × 30 × 6)/(75 × 44) h = (1 × 900 × 6)/(75 × 2) h = (900 × 6)/(150) h = 6 × 6 = 36 cm

Explanation

To find the required percentage, first calculate the total revenue from the sale of products B, C, and D in the year 2014. From the table, these values are 92, 96, and 92 respectively. Total revenue (B + C + D) in 2014 = 92 + 96 + 92 = 280

Next, calculate the total revenue from the sale of products C and D over the entire six-year period from 2012 to 2017. Total revenue of C = 82 + 98 + 96 + 88 + 93 + 103 = 560 Total revenue of D = 74 + 102 + 92 + 93 + 97 + 102 = 560 Total revenue (C + D) in 6 years = 560 + 560 = 1120

The required percentage is calculated using the formula: Percentage = (Revenue (B+C+D) in 2014)/(Total Revenue (C+D) in 6 years) × 100 Percentage = ²⁸⁰⁄₁₁₂₀ × 100 Percentage = ¼ × 100 = 25

Explanation

Substitute cos² θ = 1 - sin² θ into the given equation 7 sin² θ - cos² θ + 2 sin θ = 2 to express it in terms of sin θ: 7 sin² θ - (1 - sin² θ) + 2 sin θ = 2 8 sin² θ + 2 sin θ - 3 = 0

Solving this quadratic equation by factoring: 8 sin² θ + 6 sin θ - 4 sin θ - 3 = 0 2 sin θ (4 sin θ + 3) - 1 (4 sin θ + 3) = 0 (2 sin θ - 1)(4 sin θ + 3) = 0

Since 0° < θ < 90°, the value of sin θ must be positive. Thus, sin θ = 1/2, which means θ = 30°.

Now, substitute 2θ = 60° into the required expression: (sec 60° + cot 60°)/(csc 60° + tan 60°) = (2 + 1)/(√(3)){(2)/(√(3)) + √3}

To simplify, multiply both the numerator and the denominator by √3: 2√3 + 1{2 + (√3)(√3)} = 2√3 + 1{2 + 3} = 2√3 + 1{5} = ⅕(1 + 2√3)

Explanation

When interest is compounded yearly for one year, the effective rate is 10%. When compounded half-yearly, the interest is calculated for two periods of six months each at a rate of 5% per period. The effective annual rate is calculated using the successive percentage formula a + b + ᵃᵇ⁄₁₀₀, giving 5 + 5 + (5 × 5)/(100) = 10.25%.

The difference in effective rates is 10.25% - 10% = 0.25%. We are given that this difference equals ₹88.80. To find the principal P: 0.25% of P = 88.80 (P)/(400) = 88.80 P = 88.80 × 400 = 35,520

The simple interest on the same sum for 1⅔ years (or ⁵⁄₃ years) at the same rate of 10% is: SI = (P × R × T)/(100) SI = (35520 × 10 × 5)/(3){100} SI = (3552 × 5)/(3) = 1184 × 5 = 5,920

Explanation

Using the fundamental trigonometric identity sec²θ - tan²θ = 1, we can derive tan²θ - sec²θ = -1. By applying the cubic algebraic identity (a - b)³ = a³ - b³ - 3ab(a - b) and substituting a = tan²θ and b = sec²θ, we get:

(tan²θ - sec²θ)³ = (tan²θ)³ - (sec²θ)³ - 3tan²²θ(tan²θ - sec²θ)

Substituting the value tan²θ - sec²θ = -1 into this equation:

(-1)³ = tan⁶θ - sec⁶θ - 3tan²²θ(-1)

-1 = tan⁶θ - sec⁶θ + 3tan²²θ

For a faster result in competitive exams, substituting θ = 0° (where tan 0° = 0 and sec 0° = 1) into the original expression yields 3(1)²(0)² + 0⁶ - 1⁶ = 0 + 0 - 1 = -1.