SSC CGL Tier 1 2024 Quantitative Aptitude Questions with Solutions

Explanation

Since the policeman and the thief start running at the same time and the chase ends when the policeman catches the thief, the time of travel t is constant for both. Using the formula Distance = Speed × Time, we know that when time is constant, the distance traveled is directly proportional to the speed (D ∝ V).

The ratio of the speed of the policeman to the thief is Vₚ : Vₜ = 5 : 4. Therefore, the ratio of the distances covered by them is also Dₚ : Dₜ = 5 : 4. Let the distance covered by the policeman be 5x and the distance covered by the thief be 4x.

The difference in the distances covered represents the initial gap of 150 m that the policeman must close. 5x - 4x = 150 x = 150

The distance traveled by the thief before being caught is 4x: Dₜ = 4 × 150 = 600 m

Explanation

From the given production data for the six months, the highest production value is 108 units, which occurred in November. The least production value is 45 units, which occurred in May.

To find the required ratio of the highest production to the least production, we set up the fraction:

Ratio = (Highest production)/(Least production) = ¹⁰⁸⁄₄₅

Both 108 and 45 are divisible by their greatest common divisor, which is 9. Dividing both the numerator and the denominator by 9 yields:

108 ÷ 9 = 12 45 ÷ 9 = 5

This results in a simplified ratio of 12 : 5.

Explanation

Following the VBODMAS rule, we first solve the operations inside the parentheses, followed by of, division, multiplication, addition, and subtraction.

The expression is [(2.5 + 1.5)² ÷ 4 of ½ × 8 - 1].

Step 1: Solve the addition inside the bracket: (2.5 + 1.5) = 4

Step 2: Square the result: 4² = 16

Step 3: Solve the of operator: 4 of ½ = 4 × ½ = 2

The expression now becomes: [16 ÷ 2 × 8 - 1]

Step 4: Perform the division: 16 ÷ 2 = 8

Step 5: Perform the multiplication: 8 × 8 = 64

Step 6: Perform the final subtraction: 64 - 1 = 63

Explanation

The arc length s of a sector is given by the formula s = (θ)/(360) · 2π r, where θ is the central angle and r is the radius. Given s = 44 cm and θ = 30°, we first find the radius: 44 = ³⁰⁄₃₆₀ · 2 · ²²⁄₇ · r 44 = ¹⁄₁₂ · ⁴⁴⁄₇ · r r = (44 · 84)/(44) = 84 cm.

The area of a sector A can be directly calculated using the relationship between arc length and radius, A = ½ · s · r. Substituting the known values: A = ½ · 44 · 84 A = 22 · 84 = 1848 cm².

Explanation

The curved surface area of a cylinder is calculated using the formula CSA = 2 π r h, where r represents the radius of the base and h represents the height. When comparing two cylinders, the ratio of their curved surface areas is the product of the ratio of their radii and the ratio of their heights because the constant term 2 π cancels out:

CSA₁{CSA₂} = 2 π r₁ h₁{2 π r₂ h₂} = ( r₁{r₂} ) × ( h₁{h₂} )

Given that the ratio of the radii r₁:r₂ is 4:5 and the ratio of the heights h₁:h₂ is 5:7, we substitute these values into the relation:

CSA₁{CSA₂} = ⅘ × ⁵⁄₇

The factor of 5 in the numerator and denominator cancels out, leaving the ratio as 4/7. Thus, the ratio of the curved surface areas is 4:7.

Explanation

To simplify the expression (1)/(1+cos θ) + (1)/(1-cos θ), we first find a common denominator by multiplying the two individual denominators: (1 + cos θ)(1 - cos θ). Applying the algebraic identity (a + b)(a - b) = a² - b², the denominator becomes 1 - cos² θ.

The numerator is found by cross-multiplying: (1 - cos θ) + (1 + cos θ). Simplifying the numerator yields 1 - cos θ + 1 + cos θ = 2. This results in the combined fraction:

(2)/(1 - cos² θ)

According to the fundamental trigonometric identity sin² θ + cos² θ = 1, the expression 1 - cos² θ is equal to sin² θ. Substituting this into the fraction, we get:

(2)/(sin² θ)

Using the reciprocal trigonometric identity where (1)/(sin θ) = csc θ, it follows that (1)/(sin² θ) = csc² θ. Therefore, the simplified value of the original expression is 2 csc² θ.

Explanation

To find when pipe B should be closed, we determine the portion of the tank filled by pipe A during the total duration. Since the tank is filled in 32 minutes and pipe A is never closed, pipe A works for the entire 32 minutes. The rates of filling for pipes A and B are 1/48 and 1/66 of the tank per minute, respectively.

Work done by pipe A in 32 minutes is calculated as: Work_A = 32 × ¹⁄₄₈ = ⅔

The remaining work to be completed by pipe B is: Work_B = 1 - ⅔ = ⅓

Let x be the number of minutes pipe B remains open. The work done by pipe B in x minutes is: ˣ⁄₆₆ = ⅓

Solving for x: x = ⁶⁶⁄₃ = 22

Thus, pipe B must be closed after 22 minutes to ensure the tank is filled in exactly 32 minutes.

Explanation

Given the equation 7 tan θ = 3, we find that tan θ = ³⁄₇. To evaluate the expression (7 sin θ + 5 cos θ)/(7 sin θ - 2 cos θ), we can divide both the numerator and the denominator by cos θ to express the terms in tan θ :

(7 sin θ)/(cos θ) + (5 cos θ)/(cos θ){(7 sin θ)/(cos θ) - (2 cos θ)/(cos θ)} = (7 tan θ + 5)/(7 tan θ - 2)

Substituting the given value 7 tan θ = 3 directly into this simplified form:

(3 + 5)/(3 - 2) = ⁸⁄₁ = 8

Alternatively, since tan θ = ³⁄₇ = (sin θ)/(cos θ), we can substitute sin θ with 3k and cos θ with 7k. The expression becomes:

(7(3k) + 5(7k))/(7(3k) - 2(7k)) = (21k + 35k)/(21k - 14k) = ⁵⁶ᵏ⁄₇ₖ = 8

Explanation

Let the length of side AB be x cm. Based on the relationships provided, the length of side BC is 2x - 2 cm and the length of side AC is x + 10 cm. The perimeter of the triangle is the sum of these three sides, which is given as 40 cm. We can establish the following algebraic equation:

x + (2x - 2) + (x + 10) = 40

Simplifying the equation by combining like terms results in:

4x + 8 = 40 4x = 32 x = 8

Substituting the value of x back into the expressions for the side lengths, we find: AB = 8 cm BC = 2(8) - 2 = 14 cm AC = 8 + 10 = 18 cm

Comparing the three side lengths (8 cm, 14 cm, and 18 cm), the length of the smallest side is 8 cm.

Explanation

Given the equation cot⁴φ - cot²φ = 1, we rearrange it as: cot⁴φ = 1 + cot²φ

Using the fundamental trigonometric identity 1 + cot²φ = csc²φ, we substitute: cot⁴φ = csc²φ

Expressing the terms in terms of sine and cosine: (cos⁴φ)/(sin⁴φ) = (1)/(sin²φ) cos⁴φ = (sin⁴φ)/(sin²φ) cos⁴φ = sin²φ

Using the identity sin²φ = 1 - cos²φ, we substitute this into the equation: cos⁴φ = 1 - cos²φ cos⁴φ + cos²φ = 1

Explanation

The sum of the squares of the first n natural numbers is given by the formula Sₙ = ⁿ⁽ⁿ⁺¹⁾⁽²ⁿ⁺¹⁾⁄₆. To find the average of these squares, we divide the total sum by the number of terms n, which simplifies to the formula:

Average = ⁽ⁿ⁺¹⁾⁽²ⁿ⁺¹⁾⁄₆

For the first 46 natural numbers, we substitute n = 46 into the formula:

Average = ((46+1)(2 × 46 + 1))/(6) Average = (47 × (92 + 1))/(6) Average = (47 × 93)/(6)

Dividing both 93 and 6 by their greatest common divisor 3, we get:

Average = (47 × 31)/(2) Average = ¹⁴⁵⁷⁄₂ Average = 728.5

Explanation

Let the selling price (SP) of the scooter be x. According to the problem, the gain is ⅙th of the selling price, which is ˣ⁄₆.

The cost price (CP) is calculated by subtracting the gain from the selling price: CP = SP - Gain = x - ˣ⁄₆ = ⁵ˣ⁄₆

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

Substituting the values into the formula: Gain % = (x)/(6){⁵ˣ⁄₆} × 100 = ⅕ × 100 = 20%

The numerical value of the selling price (24,318) does not affect the calculation of the percentage since the gain is defined as a ratio of the selling price itself. By assuming the selling price is 6 units, the gain is 1 unit, making the cost price 5 units. The gain percentage is thus ⅕ × 100, which equals 20%.

Explanation

The divisibility rule for 11 states that a number is divisible by 11 if the absolute difference between the sum of the digits at odd positions and the sum of the digits at even positions is either 0 or a multiple of 11. For the number 128041, the digits at odd positions (1st, 3rd, and 5th) are 1, 8, and 4, while the digits at even positions (2nd, 4th, and 6th) are 2, 0, and 1. Calculating the sums gives Sₒ_d_d = 1 + 8 + 4 = 13 and Sₑᵥₑₙ = 2 + 0 + 1 = 3. The absolute difference between these sums is |13 - 3| = 10. Since 10 is not divisible by 11, the number 128041 is not divisible by 11.

Explanation

Let x = m - 16 and y = n - 8. To find the sum of these two variables, we add the expressions: x + y = (m - 16) + (n - 8) = m + n - 24. Substituting the given condition m + n = 24 into this equation, we get x + y = 24 - 24 = 0.

According to the algebraic property of cubes, if the sum of two numbers is zero, then the sum of their cubes is also zero. Specifically, if x + y = 0, then x = -y. Cubing both sides gives x³ = (-y)³ = -y³, which leads to x³ + y³ = 0. Substituting the original expressions back into this identity, we have (m - 16)³ + (n - 8)³ = 0.

Explanation

According to the Angle Bisector Theorem, the internal bisector of an angle in a triangle divides the opposite side into segments whose lengths are in the same ratio as the adjacent sides. In Δ DEF, since DN is the bisector of ∠ D, it divides the side EF such that:

(EN)/(NF) = (DE)/(DF)

Substituting the given values DE = 36 cm and DF = 40 cm:

(EN)/(NF) = ³⁶⁄₄₀ = ⁹⁄₁₀

This means the side EF is divided into two parts, EN and NF, in the ratio 9:10. Given that the total length of EF is 38 cm, the length of NF can be calculated as:

NF = (10)/(9 + 10) × EF

NF = ¹⁰⁄₁₉ × 38

NF = 10 × 2 = 20 cm

Explanation

Let the initial number of male and female members be 2x and 3x respectively. When the number of male members is increased by 200, the new number of male members becomes 2x + 200. According to the problem, the new ratio of males to females is 5 : 6.

(2x + 200)/(3x) = ⅚

Cross-multiplying gives 6(2x + 200) = 5(3x), which simplifies to 12x + 1200 = 15x. Subtracting 12x from both sides results in 3x = 1200. Since the number of female members is defined as 3x, the total number of female members in the club is 1200. Alternatively, solving for x gives x = 400, and multiplying by 3 confirms the female count is 1200.

Explanation

The discount amount is calculated by applying the discount percentage directly to the marked price (MP) of the article. Using the standard formula:

Discount (Rs.) = (Discount %)/(100) × Marked Price

Given that the Marked Price is Rs. 180 and the discount rate is 15%, the calculation is:

Discount = ¹⁵⁄₁₀₀ × 180

Discount = 0.15 × 180

To solve this quickly, 10% of 180 is 18, and 5% (half of 10%) is 9. Adding these together gives 18 + 9 = 27. Alternatively, simplifying the fraction:

Discount = ³⁄₂₀ × 180 = 3 × 9 = 27

The total discount offered on the article is Rs. 27.

Explanation

Let n represent the number of years that pass after the first year. In the first year, there are 6 daisy plants and 26 jasmine plants. Since the gardener adds 3 daisy plants each year, the number of daisy plants after n additional years is given by the expression 6 + 3n. Similarly, since the gardener loses 2 jasmine plants each year, the number of jasmine plants after n additional years is given by the expression 26 - 2n.

To find when the number of plants will be equal, set these two expressions equal to each other: 6 + 3n = 26 - 2n

By grouping the terms involving n on one side and the constant values on the other, we get: 3n + 2n = 26 - 6 5n = 20 n = 4

Thus, the number of daisy plants will equal the number of jasmine plants 4 years after the first year. At this point, both will have 6 + 3(4) = 18 plants.

Explanation

In a circle, the radius is always perpendicular to the tangent at the point of contact. Let O be the center of the circle and P be the point of tangency for the tangent segment from point T. This forms a right-angled triangle OPT, where the hypotenuse is the distance from the center to point T (OT = 40 cm) and one side is the length of the tangent (PT = 32 cm).

Using the Pythagorean theorem: OP² + PT² = OT² r² + 32² = 40² r² = 1600 - 1024 r² = 576 r = √(576) = 24 cm

The diameter of the circle is twice the radius: d = 2r = 2 × 24 = 48 cm