SSC CGL Tier 1 2015 Quantitative Aptitude Questions with Solutions

Explanation

To find the value of the given expression, substitute the standard trigonometric values: cos 30° = √3{2}, sin 60° = √3{2}, tan 45° = 1, sec 60° = 2, and cos 0° = 1.

Substituting these into the expression: cos² 30° + sin² 60° + tan² 45° + sec² 60° + cos 0° = (√3{2})² + (√3{2})² + (1)² + (2)² + 1 = ¾ + ¾ + 1 + 4 + 1 = ⁶⁄₄ + 6 = ³⁄₂ + 6 = 1½ + 6 = 7½

The calculation yields 7.5, which is equivalent to the mixed fraction 7½.

Explanation

The relationship between the height of an object, its shadow length, and the angle of elevation of the sun is given by the trigonometric ratio tan θ = (Height)/(Shadow), where θ represents the angle of elevation.

Given: Height of the building h = 48 m Length of the shadow s = 48√3 m

Substituting the values into the formula: tan θ = (48)/(48√(3)) tan θ = (1)/(√(3))

In trigonometry, the value of tan θ is equal to (1)/(√(3)) when θ = 30°. Therefore, the angle of elevation of the sun is 30°.

Explanation

From the given equation cos x + cos²x = 1, we can write cos x = 1 - cos²x. Using the trigonometric identity sin²x + cos²x = 1, it follows that cos x = sin²x. By squaring both sides of this relation, we obtain cos²x = sin⁴x.

The expression to be evaluated is sin⁸x + 2sin⁶x + sin⁴x. This expression follows the algebraic identity (a + b)² = a² + 2ab + b², where a = sin⁴x and b = sin²x. Thus, the expression can be rewritten as:

sin⁸x + 2sin⁶x + sin⁴x = (sin⁴x + sin²x)²

Substituting the previously derived relations sin⁴x = cos²x and sin²x = cos x into this perfect square, we get:

(cos²x + cos x)²

According to the initial equation provided in the question, cos x + cos²x = 1. Substituting this value into the expression yields:

(1)² = 1

Explanation

In △ ABC, the sum of all internal angles is 180°. Given ∠ BAC = 85° and ∠ BCA = 75°, we find ∠ ABC = 180° - (85° + 75°) = 20°.

The circumcenter O is the center of the circle passing through the vertices of the triangle. According to the property of circles, the angle subtended by an arc at the center is twice the angle subtended by it at any point on the remaining part of the circle. Therefore, ∠ AOC = 2 × ∠ ABC = 2 × 20° = 40°.

In △ OAC, the sides OA and OC are radii of the circumcircle, so OA = OC. This makes △ OAC an isosceles triangle where ∠ OAC = ∠ OCA. Using the angle sum property for △ OAC: ∠ OAC + ∠ OCA + ∠ AOC = 180° 2 × ∠ OAC + 40° = 180° 2 × ∠ OAC = 140° ∠ OAC = 70°

Explanation

Given the equations x = p csc θ and y = q cot θ, we can rearrange them to isolate the trigonometric functions as ˣ⁄ₚ = csc θ and (y)/(q) = cot θ. Squaring both equations gives (x²)/(p²) = csc² θ and (y²)/(q²) = cot² θ.

Substituting these values into the required expression: (x²)/(p²) - (y²)/(q²) = csc² θ - cot² θ

Using the fundamental trigonometric identity 1 + cot² θ = csc² θ, we know that csc² θ - cot² θ = 1. Therefore, the value of the expression is 1.

Explanation

Let the height of the aeroplane be h. The distances of the two stones from the point directly below the plane are h·cot(60°) and h·cot(30°). Since the stones are on opposite sides, the total distance between them is h/√3 + h·√3 = 1 km, which simplifies to h(4/√3) = 1, giving h = √3/4 = 1.732/4 = 0.433 km.

Explanation

In the right-angled triangle Δ ABC with ∠ C = 90°, the side AB = c is the hypotenuse, BC = a is the side opposite to ∠ A, and CA = b is the side opposite to ∠ B.

The trigonometric ratios for the angles A and B are defined as: csc B = (Hypotenuse)/(Opposite side to B) = (c)/(b) cos A = (Adjacent side to A)/(Hypotenuse) = (b)/(c)

Substituting these values into the required expression: csc B - cos A = (c)/(b) - (b)/(c) = c² - b²{bc}

By the Pythagorean theorem, a² + b² = c², which can be rearranged as c² - b² = a². Replacing the numerator with a² gives: c² - b²{bc} = a²{bc}

Explanation

For problems involving or and and conditions, the time taken can be calculated using the formula: Time = (D)/(m₂){m₁ + (w₂)/(w₁) + (b₂)/(b₁)}. In this formula, D is the number of days given, m₁, w₁, b₁ are the initial number of men, women, and boys from the or statement, and m₂, w₂, b₂ are the required number of workers from the and statement.

Given the values D = 44, m₁ = 1, w₁ = 2, b₁ = 3 and the target group m₂ = 1, w₂ = 1, b₂ = 1, the calculation is:

Time = (44)/(1){1 + ½ + ⅓}

Simplifying the denominator by finding the common multiple of 1, 2, and 3:

¹⁄₁ + ½ + ⅓ = (6 + 3 + 2)/(6) = ¹¹⁄₆

Placing this back into the formula:

Time = (44)/(11/6) = (44 × 6)/(11) = 4 × 6 = 24 days

Explanation

The unit digit of the given number 66049 is 9. A perfect square ends in 9 only if the unit digit of its square root is either 3 or 7, because 3² = 9 and 7² = 49. To determine which digit is correct, we estimate the square root by grouping the number as 660 and 49. The largest square less than or equal to 660 is 25² = 625. This indicates the square root starts with 25, making the possible roots 253 or 257.

We compare the prefix 660 with the product of the first part (25) and its successor (26). Since 25 × 26 = 650 and the given prefix 660 > 650, we select the larger unit digit. Alternatively, comparing the number to the square of the midpoint 255² = 65025 shows that 66049 > 65025. Thus, the square root is 257, and the digit in the unit place is 7.

Explanation

The work done in building a wall is proportional to its volume, which is calculated as length × breadth × height. To find the number of workers, we use the work-time relationship formula:

M₁ × D₁ × H₁{W₁} = M₂ × D₂ × H₂{W₂}

where M is the number of workers, D is the number of days, H is the hours per day, and W is the work (volume of the wall).

From the problem, we have: M₁ = 8 D₁ = 10 H₁ = 9 W₁ = 18 × 2 × 12 = 432

For the second scenario: M₂ = ? D₂ = 8 H₂ = 6 W₂ = 32 × 3 × 9 = 864

Substituting these values into the formula:

(8 × 10 × 9)/(432) = M₂ × 8 × 6{864}

⁷²⁰⁄₄₃₂ = M₂ × 48{864}

Simplifying the fractions:

⁵⁄₃ = M₂{18}

M₂ = (5 × 18)/(3) = 5 × 6 = 30

Explanation

Let the capacity of the tank be C litres. The leak empties the tank in 6 hours, so its rate of emptying is ⅙ of the tank per hour. When the inlet pipe is open, the tank empties in 8 hours, so the net rate of emptying is ⅛ of the tank per hour.

The rate of the inlet pipe in terms of the tank's capacity per hour is the difference between these two rates: Inlet rate = ⅙ - ⅛ = (4 - 3)/(24) = ¹⁄₂₄ tank per hour

The inlet pipe fills water at 4 litres per minute. Converting this to an hourly rate: Inlet rate = 4 × 60 = 240 litres per hour

Since ¹⁄₂₄ of the tank capacity equals 240 litres, the total capacity C is calculated as: C = 240 × 24 = 5760 litres

Explanation

The area of a rectangle is determined by the product of its length and breadth. When both dimensions are increased by a specific percentage, the resulting net percentage increase in the area can be calculated using the successive percentage change formula: x + y + ˣʸ⁄₁₀₀, where x and y represent the percentage changes in length and breadth respectively. Since each side of the rectangle is increased by 20%, we set x = 20 and y = 20. Substituting these values into the formula gives the total percentage increase as 20 + 20 + (20 × 20)/(100) = 40 + ⁴⁰⁰⁄₁₀₀ = 40 + 4 = 44%. This mathematical relation efficiently combines the linear increases of the sides into the composite increase of the two-dimensional area.

Explanation

Total Surface Area (TSA) of a cuboidal brick is calculated using the formula 2(lb + bh + hl). Substituting the given dimensions l = 22.5 cm, b = 10 cm, and h = 7.5 cm: TSA = 2(22.5 × 10 + 10 × 7.5 + 7.5 × 22.5) TSA = 2(225 + 75 + 168.75) = 2(468.75) = 937.5 cm²

The total area that can be painted is given as 9.375 m². To perform the calculation, convert this area into square centimeters using the relation 1 m² = 10000 cm²: Total Paint Area = 9.375 × 10000 = 93750 cm²

The number of bricks that can be painted is the quotient of the total paint area and the surface area of one brick: n = (93750 cm²)/(937.5 cm²) = 100

Explanation

The distance covered by the man in one round represents the perimeter of the rectangular park. First, convert the speed from kmph to m/min to match the units of the final answer: Speed v = 12 kmph = (12 × 1000)/(60) m/min = 200 m/min.

The distance (perimeter P) is calculated as: P = Speed × Time = 200 m/min × 8 min = 1600 m.

The ratio of length L to breadth B is given as 3:2. Let L = 3x and B = 2x. The formula for the perimeter of a rectangle is P = 2(L + B). Substituting the values: 1600 = 2(3x + 2x) 1600 = 2(5x) 1600 = 10x x = 160.

Now, find the dimensions of the park: L = 3 × 160 = 480 m B = 2 × 160 = 320 m.

The area of the park is: Area = L × B = 480 × 320 = 153600 m².

Explanation

The total percentage discount resulting from two successive discounts d₁ and d₂ is calculated using the formula d₁ + d₂ - (d₁ × d₂)/(100). Given the initial discount on second-grade shoes is d₁ = 20% and the additional shareholder's discount is d₂ = 15%, the total discount is 20 + 15 - (20 × 15)/(100). This simplifies to 35 - ³⁰⁰⁄₁₀₀, which equals 35 - 3 = 32%.

Alternatively, if the original price of the shoes is assumed to be 100, a 20% discount reduces the price to 80. Applying a further 15% discount on this reduced price of 80 amounts to a reduction of 0.15 × 80 = 12. The final price after both discounts is 80 - 12 = 68. The overall discount relative to the original price is 100 - 68 = 32%.

Explanation

Let the Cost Price (CP) be 100. Since a 20% profit is made, the Selling Price (SP) is 120. This price represents the Marked Price (MP) after a 20% discount. Using the relationship MP × (1 - Discount%) = SP: MP × 0.8 = 120 MP = 150

When the discount is changed to 30%, it is applied to the same Marked Price of 150. The new Selling Price (SPₙₑ_w) is: SPₙₑ_w = 150 × (1 - 0.30) = 150 × 0.7 = 105

The profit is determined by comparing the new Selling Price to the original Cost Price: Profit = SPₙₑ_w - CP = 105 - 100 = 5 Profit percentage = ⁵⁄₁₀₀ × 100 = 5%

Explanation

The purchase price after successive discounts is calculated using the formula P = LP × (1 - d₁) × (1 - d₂), where LP is the listed price and d₁, d₂ are the discount rates. Substituting the given values, P = 1500 × (1 - 0.20) × (1 - 0.10) = 1500 × 0.8 × 0.9 = 1080.

The total cost price includes the purchase price plus the transportation charges. Thus, Total Cost Price = 1080 + 20 = 1100. To find the selling price at a 20% profit, we use the formula Selling Price = Total Cost Price × (1 + (Profit%)/(100)).

Calculating the final value, Selling Price = 1100 × (1 + 0.20) = 1100 × 1.2 = 1320.

Explanation

To find the final ratio when mixing equal quantities of the three mixtures, we first identify the milk and water components in each. The ratios are given as 6:1, 5:2, and 3:1. The total parts for each mixture are 7 (6 + 1), 7 (5 + 2), and 4 (3 + 1) respectively. To equate the total quantity in each utensil, we find the least common multiple (LCM) of the total parts 7 and 4, which is 28.

In the first utensil, the milk is 28 × ⁶⁄₇ = 24 units and the water is 28 × ⅐ = 4 units. In the second utensil, the milk is 28 × ⁵⁄₇ = 20 units and the water is 28 × ²⁄₇ = 8 units. In the third utensil, the milk is 28 × ¾ = 21 units and the water is 28 × ¼ = 7 units.

Combining these, the total milk in the final mixture is 24 + 20 + 21 = 65 units. The total water in the final mixture is 4 + 8 + 7 = 19 units. Therefore, the ratio of milk to water in the final mixture is 65:19.

Explanation

For any sequence of consecutive natural numbers, the average is the arithmetic mean of the first and last terms, represented by the formula (a + l)/(2). When n additional consecutive natural numbers are added to the end of such a sequence, the new average increases by ⁿ⁄₂. In this problem, 2 more consecutive natural numbers are included. Applying the property, the increase in the average is ²⁄₂ = 1. Therefore, the average of the 8 numbers is K + 1, which is 1 more than the original average K.

Explanation

To find A's income, we use the ratio balancing method to reflect that both individuals save the same amount. The income ratio is 3:2 (difference of 1 unit) and the expenditure ratio is 5:3 (difference of 2 units). We multiply the income ratio by 2 to equalize the differences, resulting in a new income ratio of 6:4.

Now, for both A and B, the difference between the income units and expenditure units represents their savings. For A, the savings are 6 - 5 = 1 unit, and for B, the savings are 4 - 3 = 1 unit. Since each saves Rs. 1000, we equate 1 unit to Rs. 1000. A's income corresponds to 6 units in our balanced ratio, so A's income is 6 × 1000 = 6000. Alternatively, using the algebraic approach, if income is 3x and 2x, then (3x - 1000)/(2x - 1000) = ⁵⁄₃. Cross-multiplying yields 9x - 3000 = 10x - 5000, which gives x = 2000. Thus, A's income is 3 × 2000 = 6000.

Explanation

Let the Cost Price (CP) of one article be x. The CP of 10 articles is 10x. According to the problem, the Selling Price (SP) of 9 articles is equal to the CP of 10 articles, so the SP of 9 articles is 10x.

The SP of one article is ¹⁰ˣ⁄₉. Since the selling price per article is greater than the cost price per article, there is a gain. The gain per article is calculated as SP - CP = ¹⁰ˣ⁄₉ - x = ˣ⁄₉.

The gain percentage is determined by the formula: Gain % = (Gain)/(CP) × 100

Substituting the values into the formula: Gain % = (x/9)/(x) × 100 = ¹⁰⁰⁄₉ = 11 ⅑ %

Explanation

The overall effect on the value of monthly sales, which is the product of price and quantity sold, can be determined using the successive percentage change formula x + y + ˣʸ⁄₁₀₀. In this case, the price increases by 10%, so x = +10, and the quantity sold decreases by 20%, so y = -20.

Substituting these values into the formula: Net Change = 10 + (-20) + (10 × (-20))/(100) Net Change = -10 - ²⁰⁰⁄₁₀₀ Net Change = -10 - 2 = -12%

The negative result indicates a decrease. Therefore, the total value of monthly sales experienced a 12% decrease. This can also be verified by considering initial values: if the original price was 100 and the quantity sold was 100, the initial revenue was 10,000. With the changes, the new price is 110 and the new quantity is 80, making the new revenue 110 × 80 = 8,800. The drop from 10,000 to 8,800 is a reduction of 1,200, which is 12% of the original value.

Explanation

The mean of the cubes of the first n natural numbers is calculated by dividing the sum of these cubes by n. The sum of the cubes of the first n natural numbers is given by the formula Sₙ = [ⁿ⁽ⁿ⁺¹⁾⁄₂]². Here, the sequence consists of the cubes of the first 7 natural numbers, so n = 7.

Substituting n = 7 into the sum formula: S₇ = [⁷⁽⁷⁺¹⁾⁄₂]² = [(7 × 8)/(2)]² = (7 × 4)² = 28² = 784

The mean is defined as the sum divided by the number of observations: Mean = (S₇)/(n) = ⁷⁸⁴⁄₇ = 112

Alternatively, the formula for the mean of the cubes of the first n natural numbers can be simplified to (n(n+1)²)/(4). Substituting n = 7 gives: (7(7+1)²)/(4) = (7 × 64)/(4) = 7 × 16 = 112