SSC CGL Tier 1 2016 Quantitative Aptitude Questions with Solutions

Explanation

To find the value of (ab + bc + ca), we use the algebraic identity for the square of a trinomial: (a + b + c)² = a² + b² + c² + 2(ab + bc + ca).

Given that a + b + c = 6 and a² + b² + c² = 14, we substitute these values into the identity:

6² = 14 + 2(ab + bc + ca)

36 = 14 + 2(ab + bc + ca)

By rearranging the equation to solve for the unknown term:

2(ab + bc + ca) = 36 - 14

2(ab + bc + ca) = 22

ab + bc + ca = ²²⁄₂

ab + bc + ca = 11

Explanation

The given equation is (a)/(b) + ᵇ⁄ₐ = 1. By taking the least common multiple of the denominators, the expression simplifies to (a² + b²)/(ab) = 1. Multiplying both sides by ab results in a² + b² = ab, which can be rearranged to a² - ab + b² = 0.

The algebraic identity for the sum of two cubes is a³ + b³ = (a + b)(a² - ab + b²). Substituting the derived value a² - ab + b² = 0 into this identity, we get a³ + b³ = (a + b) × 0 = 0.

Explanation

In a circle, the radius is perpendicular to the tangent at the point of contact. Therefore, △OQP is a right-angled triangle with the right angle at Q. Using the Pythagorean theorem for △OQP, where the radius OQ = 5 cm and the distance OP = 13 cm (the hypotenuse), the length of the tangent PQ is calculated as: PQ = √(OP² - OQ²) = √(13² - 5²) = √(169 - 25) = √(144) = 12 cm.

The quadrilateral PQOR is formed by two congruent right-angled triangles, △OQP and △ORP. The area of one triangle is ½ × base × height = ½ × 5 × 12 = 30 cm². The total area of the quadrilateral PQOR is the sum of the areas of these two triangles, which is 30 + 30 = 60 cm².

Explanation

In △ATC, the sum of angles is 180°. Given ∠ATC = 30° and ∠ACT = 48°, we find ∠CAT = 180° - (30° + 48°) = 102°. Since T, A, and B lie on a straight line, ∠CAB and ∠CAT are supplementary angles. Therefore, ∠CAB = 180° - 102° = 78°.

According to the Alternate Segment Theorem, the angle between the tangent TC and the chord AC is equal to the angle subtended by the chord in the alternate segment. Thus, ∠ABC = ∠ACT = 48°. In △ABC, the sum of angles is 180°, so ∠ACB = 180° - (∠CAB + ∠ABC) = 180° - (78° + 48°) = 54°.

The angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle. Consequently, ∠AOB = 2 × ∠ACB = 2 × 54° = 108°.

Explanation

The given equation is 3sinθ + 4cosθ = 5. Dividing the entire equation by 5, we obtain: ⅗sinθ + ⅘cosθ = 1

Comparing this result with the fundamental trigonometric identity sin²θ + cos²θ = 1, which can be written as (sinθ)sinθ + (cosθ)cosθ = 1, we can directly equate the coefficients. This yields sinθ = ⅗ and cosθ = ⅘.

This substitution is valid because the values satisfy the identity (⅗)² + (⅘)² = ⁹⁄₂₅ + ¹⁶⁄₂₅ = ²⁵⁄₂₅ = 1. For any equation of the form asinθ + bcosθ = √(a² + b²), the value of sinθ is always (a)/(√(a² + b²)). In this case, a = 3 and b = 4, so √(3² + 4²) = 5, making sinθ = ⅗.

Explanation

For a period of 2 years, the difference D between compound interest and simple interest is given by the formula D = P × ((R)/(100))², where P is the principal amount and R is the annual rate of interest.

Substituting the given values P = 5000 and R = 8:

D = 5000 × (⁸⁄₁₀₀)²

D = 5000 × ⁶⁴⁄₁₀₀₀₀

D = (5000 × 64)/(10000)

D = ⁶⁴⁄₂ = 32

The difference between compound interest and simple interest on Rs. 5000 for 2 years at 8% per annum is Rs. 32.

Explanation

The greenhouse effect describes the process by which certain gases in the atmosphere trap heat that would otherwise escape into space. Solar radiation enters the atmosphere in the form of shortwave light and warms the Earth's surface. The surface then emits this energy back as longwave infrared radiation. Greenhouse gases like carbon dioxide and methane are transparent to incoming sunlight but absorb the outgoing infrared rays, reflecting a portion of that energy back toward the ground. This mechanism regulates the Earth's temperature by retaining solar energy within the lower atmosphere, ensuring the planet remains warm enough to sustain biological life.

Explanation

Udham Singh was an Indian revolutionary who assassinated Michael O'Dwyer in London on March 13, 1940, to avenge the Jallianwala Bagh massacre. Michael O'Dwyer was the Lieutenant Governor of Punjab during the 1919 massacre and had endorsed the military actions taken at the site. Udham Singh, who was deeply affected by the tragedy in Amritsar, joined the Ghadar Party and spent years planning his retribution. After carrying out the assassination at Caxton Hall, he was arrested, tried for murder, and executed at Pentonville Prison on July 31, 1940. In the historical narrative of the Indian independence movement, his act is celebrated as a significant stand against colonial atrocities.

Explanation

The rate of work for an individual is the reciprocal of the time taken to complete the task. X completes the work in p days, so X's work rate is 1/p units per day. Similarly, Y completes the work in q days, so Y's work rate is 1/q units per day. When working together, their combined daily work rate is the sum of their individual rates:

Combined Rate = ¹⁄ₚ + (1)/(q) = (q + p)/(pq)

The number of days required to complete the entire work is the reciprocal of the combined daily work rate:

Time Taken = (1)/(Combined Rate) = (1)/(p+q){pq} = (pq)/(p+q)

Explanation

The net gain or loss percentage can be calculated using the successive percentage change formula x + y + ˣʸ⁄₁₀₀, where x represents the mark-up percentage and y represents the discount percentage. Since mark-up increases the price and discount decreases it, x = 40 and y = -25.

Gain % = 40 - 25 + (40 × (-25))/(100) Gain % = 15 - ¹⁰⁰⁰⁄₁₀₀ Gain % = 15 - 10 = 5%

Alternatively, if the cost price (CP) is assumed to be 100, a 40% mark-up results in a marked price (MP) of 140. A 25% discount on the marked price is calculated as 0.25 × 140 = 35. The selling price (SP) is then 140 - 35 = 105. The profit is SP - CP = 105 - 100 = 5, which corresponds to a 5% gain on the original cost price of 100.

Explanation

Let the current ages of the two boys be 3x and 4x. According to the problem, after 3 years, the ratio of their ages becomes 4:5. This can be expressed as:

(3x + 3)/(4x + 3) = ⅘

Cross-multiplying to solve for x:

5(3x + 3) = 4(4x + 3) 15x + 15 = 16x + 12 x = 3

Alternatively, observing the ratio change from 3:4 to 4:5, the difference in ratio units for both is 1. Since this change occurs over 3 years, 1 unit equals 3 years.

The current ages are 3 × 3 = 9 years and 4 × 3 = 12 years. After 21 years, their ages will be 9 + 21 = 30 years and 12 + 21 = 33 years. The ratio of their ages after 21 years is 30:33. Dividing both terms by their greatest common divisor, 3, we get the simplified ratio 10:11.

Explanation

Let the cost price of one book be CP and the selling price of one book be SP. The given condition states that 25 × CP = 20 × SP. By rearranging this equation, the ratio of the selling price to the cost price is determined as (SP)/(CP) = ²⁵⁄₂₀ = ⁵⁄₄.

Profit percentage is calculated using the formula: Profit % = ( (SP - CP)/(CP) ) × 100

By substituting the proportional values where SP = 5 and CP = 4: Profit % = ( (5 - 4)/(4) ) × 100 = ¼ × 100 = 25%

For problems where the cost price of x articles equals the selling price of y articles, the profit percentage can be directly calculated using the shortcut: Profit % = ( (x - y)/(y) ) × 100 = ( (25 - 20)/(20) ) × 100 = ⁵⁄₂₀ × 100 = 25%

Explanation

Let the larger number be x. Since the smaller number is 25% of the larger number, the difference between them is 75% of the larger number. The problem states this difference is 12, so we can set up the equation 75% of x = 12. Converting the percentage to a fraction, we get ¾x = 12. Solving for x by multiplying both sides by ⁴⁄₃ yields x = 12 × ⁴⁄₃ = 16.

Explanation

To find the speed of the train, we use the formula Speed = (Total Distance)/(Total Time). When a train passes a platform, the total distance covered is the sum of the length of the train and the length of the platform.

Total Distance = 500 m + 221 m = 721 m Total Time = 35 seconds

The speed in meters per second is ⁷²¹⁄₃₅. To convert this speed into kilometers per hour, we multiply by the factor ¹⁸⁄₅:

Speed (km/hr) = ⁷²¹⁄₃₅ × ¹⁸⁄₅

Simplifying the fraction ⁷²¹⁄₃₅ by dividing both the numerator and denominator by 7 gives: (721 ÷ 7)/(35 ÷ 7) = ¹⁰³⁄₅ = 20.6 m/s

Finally, converting to km/hr: 20.6 × ¹⁸⁄₅ = 20.6 × 3.6 = 74.16 km/hr

Explanation

The formula for Simple Interest is SI = (P × R × T)/(100), where P is the Principal, R is the Rate of interest per annum, and T is the Time period in years.

Given that P = 400, T = 10, and SI = 280, we can rearrange the formula to solve for the Rate R:

R = (SI × 100)/(P × T)

Substituting the values into the equation:

R = (280 × 100)/(400 × 10)

R = ²⁸⁰⁰⁰⁄₄₀₀₀

R = 7

Thus, the rate of interest per annum is 7%.

Explanation

The given condition a+b=2c can be rearranged to relate the denominators of the expression. By subtracting c from both sides and shifting terms, we get b-c = c-a, which is equivalent to b-c = -(a-c).

Substituting this relation into the required expression (a)/(a-c) + (c)/(b-c):

(a)/(a-c) + (c)/(-(a-c))

(a)/(a-c) - (c)/(a-c)

Combining the fractions over the common denominator a-c:

(a-c)/(a-c) = 1

This result can also be verified quickly by choosing values that satisfy a+b=2c. If we let c=2, a=3, and b=1, the expression becomes ³⁄₃₋₂ + ²⁄₁₋₂, which simplifies to 3 - 2 = 1.

Explanation

To find the value of (x)/(1+x+x²), divide both the numerator and the denominator by x. This manipulation simplifies the expression as follows:

(x)/(x){(1+x+x²)/(x)} = (1)/(1){x + 1 + x}

By rearranging the terms in the denominator, the expression becomes:

(1)/((x + 1){x) + 1}

Substituting the given condition x + ¹⁄ₓ = 5 into the simplified expression yields:

(1)/(5 + 1) = ⅙

Explanation

The centroid G of a triangle is the point of concurrency of its medians. A key geometric property of the centroid is that it divides each median in a 2:1 ratio, measured from the vertex to the midpoint of the opposite side.

In △ ABC, AD is the median originating from vertex A. Since G is the centroid, it divides the segment AD such that the ratio of the distance from the vertex to the centroid (AG) to the distance from the centroid to the midpoint (GD) is AG:GD = 2:1.

To find the ratio AG:AD, we express the total length of the median as the sum of its two segments: AD = AG + GD. Using the parts from the ratio, if AG = 2k and GD = k for some constant k, then AD = 2k + k = 3k. Therefore, the ratio AG:AD is calculated as 2k:3k, which simplifies to 2:3.

Explanation

The circumcentre of a triangle is defined as the point of concurrency where the perpendicular bisectors of the sides of the triangle meet. According to the perpendicular bisector theorem, any point lying on the perpendicular bisector of a line segment is equidistant from the endpoints of that segment. Since the circumcentre P lies on the perpendicular bisectors of all three sides, it must be equidistant from all three vertices of the triangle. This common distance PA = PB = PC = R is known as the circumradius. Consequently, a circle centered at P with radius R will pass through all three vertices, making P the center of the circumcircle.

Explanation

Given the equation sinθ + cosθ = 1, we square both sides to find the product of the terms:

(sinθ + cosθ)² = (1)²

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

sin²θ + cos²θ + 2sinθ = 1

Using the fundamental trigonometric identity sin²θ + cos²θ = 1, we substitute this into the equation:

1 + 2sinθ = 1

Subtracting 1 from both sides of the equation results in:

2sinθ = 1 - 1

2sinθ = 0

Dividing by 2, we find:

sinθ = 0

Explanation

The n-th term of an Arithmetic Progression is given by the formula aₙ = a + (n - 1)d, where a is the first term and d is the common difference. According to the given condition, 7 × a₇ = 11 × a₁₁. Substituting the expression for each term, we have 7(a + 6d) = 11(a + 10d). Expanding both sides gives 7a + 42d = 11a + 110d. Rearranging the terms to one side leads to 11a - 7a + 110d - 42d = 0, which simplifies to 4a + 68d = 0. Dividing the entire equation by 4, we obtain a + 17d = 0. Since the 18th term of the AP is defined as a₁₈ = a + (18 - 1)d = a + 17d, and we found that a + 17d = 0, the 18th term must be 0. Generally, in an Arithmetic Progression, if m times the m-th term is equal to n times the n-th term, then the (m + n)-th term is always zero. Given m = 7 and n = 11, the (7 + 11) = 18-th term is 0.

Explanation

The new average age can be determined by calculating the change in the total sum of ages. Since one teacher is replaced by another, the number of teachers remains constant at 8.

The net change in the sum of ages is given by the difference between the age of the teacher who joins and the age of the teacher who died: 39 - 55 = -16 years.

The change in the average age is obtained by dividing this net change by the total number of teachers: ⁻¹⁶⁄₈ = -2 years.

The new average age is the original average adjusted by this change: 40 - 2 = 38 years.

Alternatively, using the total sum method, the initial sum of ages is 8 × 40 = 320 years. After the replacement, the new sum of ages is 320 - 55 + 39 = 304 years. The new average is calculated as: ³⁰⁴⁄₈ = 38

Explanation

The given conditions are a²{b+c} = 1, b²{c+a} = 1, and c²{a+b} = 1. From the first equation, we have a² = b+c. By adding a to both sides of the equation, we get a² + a = a+b+c. Factoring the left side yields a(a+1) = a+b+c, which can be rearranged to ¹⁄ₐ₊₁ = (a)/(a+b+c). Due to the symmetry of the given equations, it follows that (1)/(b+1) = (b)/(a+b+c) and (1)/(c+1) = (c)/(a+b+c). The expression to be evaluated is 2 ( ¹⁄₁₊ₐ + (1)/(1+b) + (1)/(1+c) ). Substituting the derived values into this expression, we obtain 2 ( (a)/(a+b+c) + (b)/(a+b+c) + (c)/(a+b+c) ). Combining the fractions results in 2 ( (a+b+c)/(a+b+c) ), which simplifies to 2 × 1 = 2.

Explanation

Dividing the given equation 2x + ²⁄ₓ = 3 by 2 on both sides, we get x + ¹⁄ₓ = ³⁄₂. For any expression of the form x + ¹⁄ₓ = k, the value of x³ + (1)/(x³) is determined by the identity k³ - 3k. Substituting k = ³⁄₂ into this formula:

x³ + (1)/(x³) = (³⁄₂)³ - 3(³⁄₂) = ²⁷⁄₈ - ⁹⁄₂

To subtract these fractions, we use a common denominator of 8:

²⁷⁄₈ - ³⁶⁄₈ = -⁹⁄₈

The problem asks for the value of x³ + (1)/(x³) + 2. Adding 2 to our result:

-⁹⁄₈ + 2 = -⁹⁄₈ + ¹⁶⁄₈ = ⅞

Explanation

The length of a transverse common tangent (L) between two circles is determined by the formula L = √(d² - (r₁ + r₂)²), where d represents the distance between the centers of the circles and r₁, r₂ represent their respective radii. Given two equal circles with radii of 3 cm each (r₁ = 3, r₂ = 3) and a distance between their centers of 10 cm (d = 10), the calculation proceeds as follows:

L = √(10² - (3 + 3)²) L = √(100 - 6²) L = √(100 - 36) L = √(64) L = 8 cm

This geometric relationship is derived by applying the Pythagorean theorem to a right-angled triangle where the hypotenuse is the distance between the centers and one leg is the sum of the radii. By shifting the tangent line parallelly until it passes through one of the centers, the tangent length becomes the second leg of this triangle.

Explanation

In △ ABC, AD is the median to side BC, which implies that D is the midpoint of BC. Therefore, BD = DC = ½ BC. The problem states that AD = ½ BC, leading to the relation AD = BD = DC. This indicates that point D is equidistant from all three vertices A, B, and C. In geometry, the point equidistant from the vertices of a triangle is its circumcenter. Since the circumcenter D lies on the side BC, BC must be the diameter of the circumcircle. According to Thales's Theorem, the angle subtended by a diameter at any point on the circumference is 90°. Alternatively, considering the isosceles triangles formed, in △ ABD, ∠ BAD = ∠ ABD, and in △ ACD, ∠ CAD = ∠ ACD. The sum of the angles in △ ABC is ∠ ABD + ∠ ACD + (∠ BAD + ∠ CAD) = 180°. Substituting the equal angles gives 2(∠ BAD + ∠ CAD) = 180°, which simplifies to 2(∠ BAC) = 180°, hence ∠ BAC = 90°.

Explanation

Applying the rule of componendo and dividendo to the given equation (sinθ + cosθ)/(sinθ - cosθ) = ³⁄₁:

((sinθ + cosθ) + (sinθ - cosθ))/((sinθ + cosθ) - (sinθ - cosθ)) = (3 + 1)/(3 - 1)

(2sinθ)/(2cosθ) = ⁴⁄₂

tanθ = 2

The expression to be evaluated is sin⁴θ - cos⁴θ. Using the algebraic identity a² - b² = (a - b)(a + b), we can factorize it as:

sin⁴θ - cos⁴θ = (sin²θ - cos²θ)(sin²θ + cos²θ)

Since sin²θ + cos²θ = 1, the expression simplifies to:

sin⁴θ - cos⁴θ = sin²θ - cos²θ

We can express sin²θ and cos²θ in terms of tanθ:

cos²θ = (1)/(1 + tan²θ) = (1)/(1 + 2²) = ⅕

sin²θ = 1 - cos²θ = 1 - ⅕ = ⅘

Substituting these values into the simplified expression:

sin²θ - cos²θ = ⅘ - ⅕ = ⅗

Explanation

The slant height l of the cone is calculated using the Pythagorean theorem with its vertical height h = 40 cm and radius r = 30 cm. Using the relation l = √(h² + r²), we get l = √(40² + 30²) = √(1600 + 900) = √(2500) = 50 cm. The curved surface area of the cone is given by the formula π r l, which results in π × 30 × 50 = 1500π cm². According to the problem, the curved surface area of the sphere, 4π R², is equal to that of the cone. Setting up the equation 4π R² = 1500π and dividing both sides by 4π, we find R² = 375. Taking the square root of both sides, R = √(375) = √(25 × 15) = 5√(15) cm.

Explanation

For problems where a person moves a distance d towards a tower of height h and the angle of elevation changes from ₁ to ₂, the relationship is given by the formula d = h(cot ₁ - cot ₂).

In this case, the distance moved d = 20 metres, the initial angle ₁ = 30°, and the final angle ₂ = 60°. Substituting these values into the formula:

20 = h(cot 30° - cot 60°)

Since cot 30° = √3 and cot 60° = (1)/(√(3)), the equation becomes:

20 = h(√3 - (1)/(√(3)))

Simplifying the term in the bracket by taking the common denominator:

20 = h((3 - 1)/(√(3)))

20 = h((2)/(√(3)))

Rearranging the equation to solve for h:

h = 20 × √3{2}

h = 10√3