SSC CHSL Tier 1 2012 Quantitative Aptitude Questions with Solutions

Explanation

The sum of an interior angle and its corresponding exterior angle of a regular polygon is always 180°. Given that the interior angle is 120°, the exterior angle is:

Exterior Angle = 180° - 120° = 60°

Since the sum of all exterior angles of any polygon is always 360°, the number of sides (n) of the regular polygon can be calculated using the formula:

n = 360° / Exterior Angle

Substituting the value of the exterior angle:

n = 360° / 60° = 6

Explanation

In geometry, if S denotes the circumcentre of triangle ABC, then the angle subtended by the arc BC at the circumcentre S is twice the angle subtended by the same arc at any point on the remaining part of the circle, which is the vertex A.

By this theorem: ∠BSC = 2 × ∠BAC

Given that ∠BAC = 40°, we substitute this value into the relation: ∠BSC = 2 × 40° = 80°

Explanation

The medians of a triangle divide the triangle into six smaller triangles of equal area.

When the three medians AD, BE, and CF of triangle ABC intersect at the centroid G, the six smaller triangles formed are ΔAFG, ΔBFG, ΔBDG, ΔCDG, ΔCEG, and ΔAEG. The area of each of these triangles is equal to 1/6 of the total area of triangle ABC.

Area of each small triangle = 36 ÷ 6 = 6 cm²

The quadrilateral BDGF is composed of two of these smaller triangles, which are ΔBFG and ΔBDG.

Area of quadrilateral BDGF = Area of ΔBFG + Area of ΔBDG = 6 + 6 = 12 cm²

Explanation

Let H be the height of the cloud above the surface of the lake. Since the observer is at a height h above the lake surface, the height of the cloud above the level of the observer is H - h. Let d be the horizontal distance from the observer to the vertical line of the cloud.

From the right-angled triangle formed by the observer and the cloud: tan 30° = (H - h) / d d = (H - h) / tan 30° d = (H - h)√3

The reflection of the cloud lies at a depth H below the lake surface. Therefore, the depth of the reflection below the level of the observer is H + h.

From the right-angled triangle formed by the observer and the reflection of the cloud: tan 60° = (H + h) / d d = (H + h) / tan 60° d = (H + h) / √3

Equating the two expressions for d: (H - h)√3 = (H + h) / √3

Multiplying both sides by √3: 3(H - h) = H + h 3H - 3h = H + h 2H = 4h H = 2h

Explanation

For any right-angled triangle, the circumcentre lies at the midpoint of the hypotenuse. Therefore, the circumradius is exactly half the length of the hypotenuse.

Using the Pythagorean theorem, the length of the hypotenuse AC is: AC = √(AB² + BC²) AC = √(15² + 20²) AC = √(225 + 400) AC = √625 = 25 cm

The circumradius is: Circumradius = AC / 2 = 25 / 2 = 12.5 cm

Explanation

From the given trigonometric equations, we have: sin(x + y) = √3/2 This implies x + y = π/3

Similarly: tan(x - y) = 1/√3 This implies x - y = π/6

Adding these two equations to eliminate y: (x + y) + (x - y) = π/3 + π/6 2x = π/2 x = π/4

Explanation

Given A + B = 90°, we have B = 90° - A. Using the trigonometric co-function identity, we get sec B = sec(90° - A) = cosec A.

Substituting this into the given expression: sec²A + sec²B - sec²A · sec²B = sec²A + cosec²A - sec²A · cosec²A

We can express the sum sec²A + cosec²A in terms of sine and cosine: sec²A + cosec²A = 1/cos²A + 1/sin²A = (sin²A + cos²A) / (sin²A · cos²A) = 1 / (sin²A · cos²A) = sec²A · cosec²A

Substituting this result back into the expression yields: sec²A · cosec²A - sec²A · cosec²A = 0

Explanation

Substituting the value P = 999 into the expression P(P + 2):

P(P + 2) = 999 × (999 + 2) P(P + 2) = 999 × 1001

To simplify the calculation, express 1001 as 1000 + 1:

P(P + 2) = 999 × (1000 + 1) P(P + 2) = 999000 + 999 P(P + 2) = 999999

Explanation

The discount allowed on the book is the difference between its marked price and its selling price: Discount = Marked Price - Selling Price = 165 - 132 = 33

The percentage discount is calculated on the marked price using the formula: Discount % = (Discount / Marked Price) × 100 Discount % = (33 / 165) × 100 = (1 / 5) × 100 = 20%

Explanation

The initial mixture has a total volume of 60 litres, with milk and water in the ratio 2:1.

The quantity of milk in the mixture is: (2 / 3) × 60 = 40 litres

The quantity of water in the mixture is: (1 / 3) × 60 = 20 litres

Since only water is added to the mixture, the amount of milk remains unchanged at 40 litres. In the new mixture, the ratio of milk to water is 1:2.

Let x be the amount of water to be added. The new quantity of water becomes 20 + x.

Using the new ratio: 40 / (20 + x) = 1 / 2

Cross-multiplying to solve for x: 80 = 20 + x x = 60

Explanation

Convert the mixed numbers to improper fractions: 2 ⅓ = 7/3 3 ¼ = 13/4 1 ⅙ = 7/6

Now, express the ratio of these improper fractions: 7/3 : 13/4 : 7/6

To eliminate the denominators, find the least common multiple (LCM) of the denominators 3, 4, and 6, which is 12.

Multiply each term of the ratio by the LCM: (7/3) × 12 = 28 (13/4) × 12 = 39 (7/6) × 12 = 14

This simplifies the ratio to 28 : 39 : 14.

Explanation

The ratio of men, women, and boys employed in the factory is 5:7:10. Let the common ratio factor be x.

The sum of the ratio terms represents the total number of workers, which is given as 660: 5x + 7x + 10x = 660 22x = 660 x = 660 ÷ 22 x = 30

By multiplying the ratio terms by the common factor of 30, we find the individual counts of each category: Number of men = 5 × 30 = 150 Number of women = 7 × 30 = 210 Number of boys = 10 × 30 = 300

Explanation

The equation says the absolute value of x plus 2 equals the square root of 4, which is 2. So x plus 2 equals 2 or x plus 2 equals negative 2. Solving these gives x equal to 0 or x equal to negative 4, so the solution set is zero and negative four.

Explanation

Let the required number be x.

According to the problem, the sum of the number and 5/7 is 1: x + 5/7 = 1 x = 1 - 5/7 x = 2/7

To find what percentage of 5/7 this number represents, we calculate: (2/7 ÷ 5/7) × 100% = (2/7 × 7/5) × 100% = 2/5 × 100% = 40%

This confirms the number is 40% of 5/7.

Explanation

The total distance the train needs to cover to cross the platform is the sum of the length of the train and the length of the platform.

Total Distance = Length of Train + Length of Platform Total Distance = 250 + 520 = 770 m

To convert the speed from km/hr to m/s, multiply by 5/18.

Speed = 55 × 5/18 = 275/18 m/s

Using the standard formula for time:

Time = Distance / Speed Time = 770 / (275/18) Time = (770 × 18) / 275 Time = 13860 / 275 Time = 50.4 seconds

Explanation

Let the daily work capacity of a man be M and a woman be W.

According to the given condition, the work done by 12 men is equal to the work done by 18 women: 12 M = 18 W 2 M = 3 W 1 M = 1.5 W

We need to find the time taken by 4 men and 8 women to reap the same field. Converting men into equivalent women: 4 M + 8 W = 4 × 1.5 W + 8 W = 6 W + 8 W = 14 W

Using the work equivalence formula, if W₁ women complete the work in D₁ days, then W₂ women will complete the same work in D₂ days: W₁ × D₁ = W₂ × D₂ 18 × 7 = 14 × D₂ D₂ = (18 × 7) / 14 = 9 days

Explanation

The share of wages is distributed in proportion to the efficiency of the individuals when they work together for the same duration. Since efficiency is inversely proportional to the time taken to complete the work, the ratio of their work done is calculated as:

Ratio of time taken by A and B = 30 : 40 = 3 : 4

Ratio of efficiency of A and B = 1/30 : 1/40 = 4 : 3

The total payment of ₹ 7,000 is divided in this ratio of 4 : 3.

Share of B = (3 / (4 + 3)) × 7,000 = (3/7) × 7,000 = ₹ 3,000

Explanation

The total surface area of a cube is directly proportional to the square of its edge length, as TSA = 6a². When the edge of the cube is increased by 50%, the new edge length becomes 1.5 times the original edge length.

The new total surface area scales by the square of this factor, which is (1.5)² = 2.25 times the original surface area.

The percentage increase in the total surface area is calculated as (2.25 - 1) × 100% = 125%.

Explanation

Volume of a rectangular tank = Base Area × Depth

The volume of water is given in cubic meters and the base area is given in square centimeters. To perform the calculation in standard units, the base area is first converted into square meters:

Base Area = 6500 cm² Since 1 m² = 10000 cm²: Base Area = 6500 / 10000 = 0.65 m²

Let the depth of water be h meters. Substituting the values into the formula:

2.6 = 0.65 × h h = 2.6 / 0.65 h = 4 m

Explanation

The True Discount (TD) on a sum due (Amount) is equal to the Simple Interest (SI) on its Present Worth (PW) for the same time and rate of interest.

According to the question, the true discount on ₹ 1,800 is equal to the simple interest on ₹ 1,500 for 3 years at the same rate. This directly implies that the Present Worth (PW) is ₹ 1,500.

The relation between Amount, Present Worth, and True Discount is: Amount = Present Worth + True Discount 1,800 = 1,500 + True Discount True Discount = 1,800 - 1,500 = 300

Since True Discount is the Simple Interest on the Present Worth: True Discount = (Present Worth × Rate × Time) / 100 300 = (1,500 × R × 3) / 100 300 = 45R R = 300 / 45 = 20/3 = 6 ⅔ %

Explanation

To find the profit of the trader in this transaction, we calculate the present worth of the credit payment.

The present worth (PW) is calculated using the formula: PW = (Amount × 100) / (100 + (Rate × Time))

Given: Amount = ₹ 14,300 Rate = 10% per annum Time = 1 year

Substituting these values into the formula: PW = (14,300 × 100) / (100 + (10 × 1)) PW = 1,430,000 / 110 PW = ₹ 13,000

The profit earned by the trader is the difference between the present worth of the credit payment and the actual cash down value of the article: Profit = Present Worth - Cash Down Value Profit = ₹ 13,000 - ₹ 12,500 Profit = ₹ 500

Explanation

Under compound interest, the amount accumulated at the end of a given year acts as the principal for the succeeding year.

Amount after 2 years = £500 Amount after 3 years = £540

The interest earned during the third year is: Interest = Amount after 3 years - Amount after 2 years Interest = 540 - 500 = £40

The rate of interest per annum (R) is given by the relation: R = (Interest / Principal) × 100

Using the second-year amount of £500 as the principal for the third year: R = (40 / 500) × 100 = 8%

Explanation

Let the amount of money lent at 5% simple interest be x. The remaining amount lent at 4% simple interest is 2,000 - x.

The total annual interest received is 92. Using the simple interest formula: (x × 5)/100 + ((2,000 - x) × 4)/100 = 92

Multiply both sides of the equation by 100: 5x + 4(2,000 - x) = 9,200

Expand and simplify: 5x + 8,000 - 4x = 9,200 x + 8,000 = 9,200 x = 1,200

Explanation

For a rectangular board of width w and length l, the perimeter p is given by the formula: p = 2(l + w)

From this, the length l can be expressed as: l = p/2 - w

The area k of the rectangle is given by the formula: k = l × w

Substituting the value of l into the area formula yields: k = (p/2 - w) × w k = (pw)/2 - w²

Multiplying the entire equation by 2 to eliminate the fraction gives: 2k = pw - 2w²

Rearranging the terms to form a standard quadratic equation results in: 2w² - pw + 2k = 0

Explanation

The relationship between income, expenditure, and savings is given by the formula:

Total Income = Total Expenditure + Savings

First, calculate the total expenditure for the entire year of 12 months: Expenditure for the first 5 months = 5 × £12,000 = £60,000 Expenditure for the next 7 months = 7 × £13,000 = £91,000 Total annual expenditure = £60,000 + £91,000 = £151,000

Since the annual savings are £29,000, the total annual income is: Total annual income = £151,000 + £29,000 = £180,000

The average monthly income is obtained by dividing the total annual income by 12: Average monthly income = £180,000 ÷ 12 = £15,000

Explanation

The length of the journey can be calculated using the relation:

Distance = (S₁ × S₂ / (S₁ - S₂)) × Δt

where S₁ and S₂ are the two speeds, and Δt is the difference in time.

Given: S₁ = 40 km/hr S₂ = 35 km/hr Δt = 15 minutes = 15/60 hour = ¼ hour

Substituting these values into the formula:

Distance = (40 × 35 / (40 - 35)) × (1/4) Distance = (1400 / 5) × (1/4) Distance = 280 × (1/4) Distance = 70 km

Explanation

Three years ago, the average age of the 5-member family was 17 years. Therefore, the total age of these 5 members three years ago was 5 × 17 = 85 years.

Since three years have passed, each of the 5 members is now 3 years older. Thus, the total age of these 5 members today is 85 + (5 × 3) = 100 years.

Today, with the addition of the baby, the family has 6 members. The average age of the family remains the same as 17 years. Therefore, the total age of all 6 members today is 6 × 17 = 102 years.

The age of the baby is the difference between the total age of the 6 members today and the total age of the 5 older members today, which is 102 - 100 = 2 years.

Explanation

The ratio of materials, labour, and overheads is 4:3:2. Since the labour component represents 3 parts of the ratio and is given as £45, the value of 1 part is £45 ÷ 3 = £15.

The total cost is the sum of all components, represented by 4 + 3 + 2 = 9 parts. Multiplying 9 parts by £15 per part gives a total cost price of £135.

With a selling price of £180, the profit is computed as £180 - £135 = £45. The profit percentage is calculated by dividing the profit by the cost price and multiplying by 100, which gives (45 ÷ 135) × 100 = 33 ⅓%.

Explanation

Let the cost price of the shirt be CP.

The difference between selling the shirt at a profit of 5% and 4% is 1% of the cost price.

Given that this difference is £3: 1% of CP = 3 (1/100) × CP = 3 CP = 3 × 100 = 300

Thus, the cost price of the shirt is £300.

The two selling prices are calculated as: First selling price (at 4% profit) = 300 × 1.04 = 312 Second selling price (at 5% profit) = 300 × 1.05 = 315

So, the cost price is £300, and the two selling prices are £312 and £315.

Explanation

Let CP be the cost price of the toy.

The difference between the two profit percentages corresponds to the increase in the selling price: 20% - 16% = 4% of CP

Since this 4% difference is equal to £10: 4/100 × CP = 10 CP = (10 × 100) / 4 = 250

Explanation

To find the value of the given expression, we translate the words directly into a mathematical equation:

(2/3) × 30% × 4a

First, write 30% as a fraction:

30% = 30/100 = 3/10

Now substitute this into the expression:

(2/3) × (3/10) × 4a

Simplify the multiplication by canceling the common factor of 3 in the numerator and denominator:

(2/10) × 4a

This simplifies to:

(1/5) × 4a = 4a/5

Explanation

Let the number be x.

Four-fifth of the number represents (4/5)x, which is equivalent to 80% of the number.

According to the question, 65% of the number is 21 less than 80% of the number. This relationship can be expressed as:

80% of x - 65% of x = 21

15% of x = 21

0.15 × x = 21

x = 21 / 0.15

x = 140

Explanation

The length of the bridge is calculated using the distance formula:

Distance = Speed × Time

The speed of the man is given as 6 km/hr, and the time taken to cross the bridge is 20 minutes. First, convert the time from minutes to hours:

Time = 20 minutes = 20/60 hour = 1/3 hour

Now, substitute the speed and time into the distance formula:

Distance = 6 km/hr × 1/3 hour = 2 km

This gives the total length of the bridge as 2 kilometers.

Explanation

The relationship between distance, speed, and time is given by the formula: Distance = Speed × Time

First, convert the speed of the car from kilometers per hour (km/hr) to meters per second (m/s) by multiplying by 5/18: Speed = 36 × (5/18) m/s = 10 m/s

Using the converted speed and the given time of 32 seconds, the distance covered is calculated as: Distance = 10 m/s × 32 s = 320 meters

Explanation

The interior angle of a regular polygon with n sides is related to its exterior angle by the formula: Exterior angle = 180° - Interior angle

Substituting the given interior angle of 120°: Exterior angle = 180° - 120° = 60°

The number of sides n of a regular polygon is calculated using the formula: n = 360° ÷ Exterior angle

Substituting the calculated exterior angle: n = 360° ÷ 60° = 6

Explanation

By definition, if S represents the circumcentre of Δ ABC, then the circumcircle passes through the vertices A, B, and C with S as its centre.

According to the central angle theorem, the angle subtended by an arc at the centre of a circle is twice the angle subtended by the same arc at any point on the circumference.

Here, the arc BC subtends the central angle ∠BSC at the circumcentre S, and the angle ∠BAC at the circumference.

Therefore, ∠BSC = 2 × ∠BAC.

Substituting the given value: ∠BSC = 2 × 40° = 80°

Explanation

The three medians AD, BE, and CF of a triangle ABC intersect at the centroid G and divide the triangle into six smaller triangles of equal area: ΔAFG, ΔBFG, ΔBDG, ΔCDG, ΔCEG, and ΔAEG.

Given that the total area of triangle ABC is 36 cm², the area of each of these six smaller triangles is 36 ÷ 6 = 6 cm².

The quadrilateral BDGF is composed of two of these smaller triangles, namely ΔBFG and ΔBDG. Therefore, the area of quadrilateral BDGF is the sum of the areas of ΔBFG and ΔBDG, which is 6 + 6 = 12 cm².

Explanation

Let the height of the cloud above the surface of the lake be H and the horizontal distance from the point of observation to the cloud be d. Since the point of observation is at a height h above the lake surface, the vertical height of the cloud above this observation level is H - h.

Using the angle of elevation of 30°: tan(30°) = (H - h) / d d = (H - h) / tan(30°) = (H - h)√3

The reflection of the cloud is at a depth H below the lake surface. Therefore, the vertical depth of the reflection below the horizontal level of the point of observation is H + h.

Using the angle of depression of 60°: tan(60°) = (H + h) / d d = (H + h) / tan(60°) = (H + h) / √3

Equating the two expressions for d: (H - h)√3 = (H + h) / √3

Multiplying both sides by √3: 3(H - h) = H + h 3H - 3h = H + h 2H = 4h H = 2h

Explanation

In a right-angled triangle, the circumradius is always equal to half of the length of the hypotenuse.

Using the Pythagorean theorem for the triangle ABC right-angled at B: AC = √(AB² + BC²)

Substitute the given values: AC = √(15² + 20²) AC = √(225 + 400) AC = √625 AC = 25 cm

Therefore, the circumradius is: R = AC / 2 R = 25 / 2 R = 12.5 cm

Explanation

From the given trigonometric relations, we can find the values of the angles:

sin(x + y) = √3/2 → x + y = π/3

tan(x - y) = 1/√3 → x - y = π/6

By adding these two linear equations, we get:

(x + y) + (x - y) = π/3 + π/6

2x = π/2

x = π/4

Explanation

Given A + B = 90°, we can write B = 90° - A.

Taking the secant on both sides gives sec B = sec(90° - A) = cosec A. Squaring both sides yields sec² B = cosec² A.

Substituting sec² B = cosec² A into the given expression: sec² A + cosec² A - sec² A cosec² A

Expressing the terms in terms of sine and cosine: 1/cos² A + 1/sin² A - 1/(cos² A sin² A)

Taking the common denominator for the first two terms: (sin² A + cos² A)/(cos² A sin² A) - 1/(cos² A sin² A)

Using the fundamental trigonometric identity sin² A + cos² A = 1: 1/(cos² A sin² A) - 1/(cos² A sin² A) = 0

Explanation

Volume of a sphere is given by the formula V = (4/3) π r³, which means the volume is directly proportional to the cube of its radius (V ∝ r³).

When the radius of the sphere is increased by 50%, the new radius becomes 1.5 times the original radius (r' = 1.5r).

The new volume is proportional to the cube of the new radius: V' ∝ (1.5r)³ = 3.375r³

This shows that the new volume is 3.375 times the original volume (V' = 3.375V).

The percentage increase in the volume is calculated as: Percentage increase = ((V' - V) / V) × 100 = (3.375 - 1) × 100 = 2.375 × 100 = 237.5%

Explanation

The volume of the cylindrical container is calculated using the formula: V = π × R² × H

Substituting the given values of radius R = 6 cm and height H = 15 cm: V = π × 6² × 15 = 540π cm³

Each child receives an ice-cream cone consisting of a conical part and a hemispherical top. Let the radius of the base of the cone be r. The height of the conical portion is given as h = 4r.

The total volume of one ice-cream cone is the sum of the volume of the cone and the volume of the hemisphere: V_cone = (1/3)πr²h + (2/3)πr³ V_cone = (1/3)πr²(4r) + (2/3)πr³ V_cone = (4/3)πr³ + (2/3)πr³ V_cone = 2πr³

Since the ice-cream is distributed equally among 10 children, the volume of 10 ice-cream cones is equal to the volume of the cylindrical container: 10 × 2πr³ = 540π 20πr³ = 540π r³ = 27 r = 3 cm

Explanation

The volume of a hemisphere is given by V₁ = (2/3) × π × R³, where R is the radius of the hemisphere. Here, R = 15 cm. V₁ = (2/3) × π × 15³ = (2/3) × π × 3375 = 2250π cm³

The volume of a cylinder is given by V₂ = π × r² × h, where r is the radius and h is the height. The diameter of the cylindrical bottle is 5 cm, so its radius r = 5/2 = 2.5 cm, and height h = 6 cm. V₂ = π × (5/2)² × 6 = π × (25/4) × 6 = 37.5π cm³

The number of bottles necessary to empty the bowl is N = V₁ ÷ V₂. N = 2250π ÷ 37.5π = 2250 ÷ 37.5 = 60

Explanation

The value of an infinite nested radical of the form √(a + √(a + √(a + ...))) is given by the formula: x = (1 + √(1 + 4a)) / 2

Substituting a = 6 into the formula: x = (1 + √(1 + 4 × 6)) / 2 x = (1 + √(1 + 24)) / 2 x = (1 + √25) / 2 x = (1 + 5) / 2 x = 3

Alternatively, for any expression of this form where a can be factored into two consecutive integers n(n + 1), the value of the positive square root series is always the larger factor, n + 1. Since 6 can be factored as 2 × 3, the larger factor is 3.

Explanation

To find the value of x + 1/x, we first rationalize the expressions.

The given expression for x is: x = (√3 + √2) / (√3 - √2)

Multiplying the numerator and denominator by (√3 + √2): x = (√3 + √2)² / ((√3 - √2)(√3 + √2)) x = (3 + 2 + 2√6) / (3 - 2) x = 5 + 2√6

The reciprocal 1/x is given by: 1/x = (√3 - √2) / (√3 + √2)

Multiplying the numerator and denominator by (√3 - √2): 1/x = (√3 - √2)² / ((√3 + √2)(√3 - √2)) 1/x = (3 + 2 - 2√6) / (3 - 2) 1/x = 5 - 2√6

Adding the values of x and 1/x: x + 1/x = (5 + 2√6) + (5 - 2√6) x + 1/x = 10

Explanation

Applying the algebraic identity for the square of a binomial, (a - b)² = a² - 2ab + b², the left side of the equation is expanded as:

(c - 3)² = c² - 2(c)(3) + 3² = c² - 6c + 9

Equating this expanded form to the right side of the given equation:

c² - 6c + 9 = c² + kc + 9

By comparing the coefficients of the c term on both sides of the equation, we get:

k = -6