SSC CHSL Tier 1 2016 Quantitative Aptitude Questions with Solutions

Explanation

The formula for compound interest is: A = P × (1 + R/100)ⁿ

Substituting the given values, where Amount A = 73205, Principal P = 50000, and Time n = 2 years: 73205 = 50000 × (1 + R/100)²

Dividing both sides by 50000: (1 + R/100)² = 73205 / 50000 (1 + R/100)² = 14641 / 10000

Taking the square root on both sides: 1 + R/100 = √(14641 / 10000) 1 + R/100 = 121 / 100 1 + R/100 = 1.21 R/100 = 0.21 R = 21%

Explanation

Let the number of 1 rupee, 50 paise, and 10 paise coins be 6x, 3x, and 2x respectively.

The value of these coins in rupees is: Value of 1 rupee coins = 6x × 1 = 6x rupees Value of 50 paise coins = 3x × 0.5 = 1.5x rupees Value of 10 paise coins = 2x × 0.1 = 0.2x rupees

The total value of all the coins in the bag is: 6x + 1.5x + 0.2x = 30.8 7.7x = 30.8 x = 30.8 ÷ 7.7 = 4

The number of 50 paise coins is 3x: 3 × 4 = 12

Explanation

The reflection of a point (x, y) in the x-axis is given by the formula (x, -y). Under this reflection, the horizontal distance from the y-axis remains unchanged, while the vertical position is mirrored across the x-axis.

For the point (-3, 6), substituting x = -3 and y = 6 into the formula yields the reflected coordinates (-3, -6).

Explanation

Using the laws of exponents where yᵐ × yⁿ = y⁽m+n) and yᵐ / yⁿ = y⁽m-n), the expression can be simplified by grouping the terms for each variable.

For a: (a⁴ × a³) / a² = a⁽4 + 3 - 2) = a⁵

For b: (b³ × b⁴) / b⁴ = b⁽3 + 4 - 4) = b³

For x: x² × x³ = x⁽2 + 3) = x⁵

For z: (z³ × z²) / z³ = z⁽3 + 2 - 3) = z²

Combining all the simplified terms results in b³x⁵a⁵z².

Explanation

Let the marks obtained by the two students be x and x - 13.

The sum of their marks is: x + (x - 13) = 2x - 13

According to the given condition, the higher mark is 76% of the sum of their marks: x = 0.76 × (2x - 13) x = 1.52x - 9.88 1.52x - x = 9.88 0.52x = 9.88 x = 9.88 ÷ 0.52 x = 19

The marks of the other student are: 19 - 13 = 6

Thus, the marks obtained by the two students are 19 and 6.

Explanation

To find the least number to be added to 1039 so that it is completely divisible by 29, divide 1039 by 29 to find the remainder.

1039 = 29 × 35 + 24

Here, the remainder is 24.

The least number that must be added to the dividend is given by the formula: Required number = Divisor - Remainder

Substituting the values: Required number = 29 - 24 = 5

Alternatively, the next multiple of 29 immediately greater than 1039 is: 29 × 36 = 1044

Subtracting the original number from this multiple gives the required value to be added: 1044 - 1039 = 5

Explanation

Let the list price of the article be MP and the cost price be CP.

The relationship between the marked price, cost price, selling price, discount, and loss is: SP = MP × (1 - discount percentage) SP = CP × (1 - loss percentage)

When a discount of 4% is offered, the merchant makes a loss of 10%: SP₁ = 0.96 × MP = 0.90 × CP CP = (0.96 / 0.90) × MP = (16/15) × MP

When a discount of 20% is offered, the new selling price SP₂ is: SP₂ = 0.80 × MP = (4/5) × MP

The net loss is the difference between CP and SP₂: CP - SP₂ = (16/15) × MP - (4/5) × MP CP - SP₂ = (16/15 - 12/15) × MP = (4/15) × MP

The loss percentage is given by the formula: Loss% = (Loss / CP) × 100 Loss% = ((4/15) × MP / ((16/15) × MP)) × 100 Loss% = (4 / 16) × 100 = 25%

Explanation

Two angles are supplementary if their sum is 180°. Let the measure of the required angle be x. Its supplementary angle is therefore 180° - x.

The given condition states that the angle is smaller than its supplementary angle by 30°: (180° - x) - x = 30°

Simplifying this equation: 180° - 2x = 30° 2x = 150° x = 75°

This gives the measure of the angle as 75°.

Explanation

Let the side of the square be s = 14 cm.

When a square is inscribed in a circle, the diagonal of the square is equal to the diameter of the circle.

The diagonal of a square is given by the formula: d = s√2

Substituting s = 14: d = 14√2 cm

Thus, the diameter of the circle is 14√2 cm. The radius r of the circle is half of the diameter: r = 7√2 cm

The area of the circle is given by: Area = πr² Area = π × (7√2)² Area = π × 98 Area = 98π sq.cm

Explanation

The internal radius of the hollow hemisphere bowl is r = 7 cm, and the thickness is t = 1 cm. Thus, the external radius is R = r + t = 7 + 1 = 8 cm.

The total surface area of a hollow hemisphere bowl consists of the outer curved surface area, the inner curved surface area, and the area of the circular rim at the top.

Outer curved surface area = 2πR² Inner curved surface area = 2πr² Area of the circular rim = π(R² - r²)

Summing these components gives the formula for the total surface area: Total Surface Area = 2πR² + 2πr² + π(R² - r²) Total Surface Area = π(3R² + r²)

Substituting the given values into the formula: Total Surface Area = (22/7) × (3 × 8² + 7²) Total Surface Area = (22/7) × (3 × 64 + 49) Total Surface Area = (22/7) × (192 + 49) Total Surface Area = (22/7) × 241 Total Surface Area = 5302/7 Total Surface Area ≈ 757.43 sq.cm

Explanation

The ratio of weight to height is calculated by dividing weight in kilograms by height in meters. For Ananya, this ratio is 59 divided by 1.63, which equals approximately 36.20. This represents the lowest value among all the individuals listed in the table, as the other calculated ratios are 38.51, 39.76, and 42.86.

Explanation

The average number of children per house is calculated using the weighted mean formula:

Average = (Sum of (Number of Children × Number of Houses)) / (Total Number of Houses)

First, calculate the total number of houses: Total Houses = 5 + 14 + 10 + 3 = 32

Next, calculate the total number of children across all houses: Total Children = (0 × 5) + (1 × 14) + (2 × 10) + (3 × 3) = 0 + 14 + 20 + 9 = 43

Now, divide the total number of children by the total number of houses: Average = 43 / 32 = 1.34375

Rounding to two decimal places yields 1.34.

Explanation

The total number of shares issued by the company is 6,00,000.

Ejaz initially holds a 45% share in the company. The number of shares initially held by Ejaz is calculated as: Initial shares = 45% of 6,00,000 = 0.45 × 6,00,000 = 2,70,000 shares

When Chinmay sells 15,000 of his shares to Ejaz, the new total number of shares held by Ejaz becomes: Total shares = 2,70,000 + 15,000 = 2,85,000 shares

Explanation

Let the cost price of the machine be CP.

The loss incurred when the machine is sold for Rs 51 lakhs is given by: Loss = CP - 51

The gain obtained when the machine is sold for Rs 60 lakhs is given by: Gain = 60 - CP

According to the given condition, the gain is 8 times the earlier loss: Gain = 8 × Loss 60 - CP = 8 × (CP - 51) 60 - CP = 8CP - 408 9CP = 468 CP = 52 lakhs

Explanation

A completes 75% (or 3/4) of the work in 50 days. A's 1 day's work = (3/4) / 50 = 3/200 of the total work.

The remaining work is 25% (or 1/4) of the total work, which A and B together finish in 10 days. Their combined 1 day's work = (1/4) / 10 = 1/40 of the total work.

To find B's rate of work: B's 1 day's work = Combined 1 day's work - A's 1 day's work B's 1 day's work = 1/40 - 3/200 B's 1 day's work = (5 - 3) / 200 = 2/200 = 1/100 of the total work.

Since B completes 1/100 of the work in one day, B alone would take 100 days to complete the entire work.

Explanation

We can find the correct factors of the polynomial 48x³ - 8x² - 93x - 45 by expanding the expression (4x + 3)(4x + 3)(3x - 5).

First, square the term (4x + 3): (4x + 3)² = 16x² + 24x + 9

Next, multiply this result by (3x - 5): (16x² + 24x + 9)(3x - 5) = 16x²(3x - 5) + 24x(3x - 5) + 9(3x - 5) = 48x³ - 80x² + 72x² - 120x + 27x - 45 = 48x³ - 8x² - 93x - 45

This is identical to the given polynomial, confirming that (4x + 3)(4x + 3)(3x - 5) are its factors.

Explanation

The mean proportion of two numbers x and y is given by the formula: √(xy) = 8

Squaring both sides: xy = 64 x = 64/y

The third proportion of x and y is given by the formula: y²/x = 512 y² = 512x

Substituting the value of x into the third proportion equation: y² = 512 × (64/y) y³ = 32768 y = 32

Substituting the value of y back to find x: x = 64/32 x = 2

This gives the values of x and y as 2 and 32.

Explanation

The difference between compound interest and simple interest for a period of 2 years is given by the formula:

Difference = P × (R / 100)²

where P is the principal sum and R is the rate of interest per annum.

Substituting the given values into the formula:

72 = P × (12 / 100)²

72 = P × 144 / 10000

P = (72 × 10000) / 144

P = 720000 / 144

P = 5000

Explanation

Total weight of Shivshankar, Gopesh and Reena = 97 × 3 = 291 kg

Total weight of Shivshankar and Gopesh = 93 × 2 = 186 kg

Total weight of Gopesh and Reena = 82 × 2 = 164 kg

By adding the weights of the two pairs: Shivshankar + 2 × Gopesh + Reena = 186 + 164 = 350 kg

Subtracting the total weight of all three individuals from this sum yields the weight of Gopesh: Weight of Gopesh = 350 - 291 = 59 kg

Explanation

The fundamental trigonometric identity relates cosecant and cotangent: cosec²A - cot²A = 1

This can be rewritten as: cosec²A = 1 + cot²A cosec²A - 1 = cot²A

The given expression is: x = cosec⁴A - cosec²A

By factoring out cosec²A: x = cosec²A (cosec²A - 1)

Substituting the trigonometric identities into this factored equation: x = (1 + cot²A) (cot²A) x = cot²A + cot⁴A x = cot⁴A + cot²A

Explanation

The formula for the measure of each interior angle of a regular polygon with n sides is ((n - 2) × 180°) / n.

For a regular octagon, the number of sides is n = 8. Substituting this value into the formula:

((8 - 2) × 180°) / 8 = (6 × 180°) / 8 = 1080° / 8 = 135°

Alternatively, the sum of the exterior angles of any polygon is always 360°. Since a regular octagon has 8 equal exterior angles, the measure of each exterior angle is 360° / 8 = 45°. Because the interior and exterior angles at any vertex form a linear pair, the measure of each interior angle is 180° - 45° = 135°.

Explanation

The compound inequality can be solved by splitting it into two parts.

First, solve the left inequality: 4(2x - 4) - 2 > 3x - 1 8x - 16 - 2 > 3x - 1 8x - 18 > 3x - 1 5x > 17 x > 17/5 x > 3.4

Next, solve the right inequality: 3x - 1 ≥ 4x - 7 -x ≥ -6 x ≤ 6

Combining both inequalities gives the range: 3.4 < x ≤ 6

The integer 6 lies within this range and satisfies both parts of the inequality.

Explanation

By the Secant-Secant Power Theorem, when two chords PQ and RS of a circle are extended externally to intersect at an external point T, the product of the lengths of the secant segments from the external point to the circle is equal for both secants. This gives the relation:

PT × QT = RT × ST

Substituting the given values, where QT = 9 cm, RT = 18 cm, and ST = 6 cm:

PT × 9 = 18 × 6

PT × 9 = 108

PT = 108 / 9

PT = 12 cm

Explanation

Using the trigonometric identity for negative angles, tan(-θ) = -tan(θ). Substituting θ = 240°:

tan(-240°) = -tan(240°)

The angle 240° lies in the third quadrant, where the tangent function is positive. Expressing 240° as 180° + 60° and using the identity tan(180° + θ) = tan(θ):

tan(240°) = tan(180° + 60°) = tan(60°)

The standard value of tan(60°) is √3, which gives:

tan(240°) = √3

Substituting this back into the first equation:

tan(-240°) = -√3

Explanation

Using the relation Distance = Speed × Time, the distance covered in the first part of the journey is 49 km/hr × 5 hours = 245 km.

Subtracting this from the total distance of 368 km gives the remaining distance as 368 km - 245 km = 123 km.

The speed for the remaining 3 hours is calculated by dividing the remaining distance by the remaining time, which is 123 km / 3 hours = 41 km/hr.

Explanation

The total surface area of a hollow cylinder open at both ends consists of the outer curved surface area, the inner curved surface area, and the area of the two annular bases.

The formula for the total surface area (TSA) is: TSA = 2πRh + 2πrh + 2π(R² - r²)

This formula can be simplified as: TSA = 2π(R + r)(h + R - r)

The given values are: Length (h) = 22 cm External radius (R) = 7 cm Thickness = 1 cm Internal radius (r) = External radius - Thickness = 7 - 1 = 6 cm

Substituting these values into the simplified formula: TSA = 2 × (22/7) × (7 + 6) × (22 + 7 - 6) TSA = 2 × (22/7) × 13 × 23 TSA = 13156 / 7 TSA ≈ 1879.43 sq cm

Explanation

The distance d between two points (x₁, y₁) and (x₂, y₂) is given by the distance formula:

d = √((x₂ - x₁)² + (y₂ - y₁)²)

Substituting the coordinates of the given points (4, 7) and (-1, -5) into the formula:

d = √((-1 - 4)² + (-5 - 7)²)

d = √((-5)² + (-12)²)

d = √(25 + 144)

d = √169

d = 13 units

Explanation

The relationship between the selling prices and their corresponding discount percentages on the same marked price can be established using the ratio formula:

SP₂ / SP₁ = (100 - d₂) / (100 - d₁)

Here, SP₁ is the initial selling price of Rs 26000, d₁ is the initial discount of 22%, d₂ is the new discount of 40%, and SP₂ is the new selling price.

Substituting the values into the formula:

SP₂ / 26000 = (100 - 40) / (100 - 22)

SP₂ / 26000 = 60 / 78

Simplifying the fraction 60/78 by dividing the numerator and denominator by 6:

SP₂ / 26000 = 10 / 13

Solving for SP₂:

SP₂ = 26000 × (10 / 13)

SP₂ = 2000 × 10

SP₂ = 20000

Explanation

Let the required number be x.

According to the percentage formula, we have: 0.4% of x = 26

Writing the percentage as a fraction: (0.4 / 100) × x = 26

Multiplying both sides by 100: 0.4 × x = 2600

Dividing both sides by 0.4: x = 2600 / 0.4 x = 26000 / 4 x = 6500

Explanation

Let the two parts of 20 be x and 20 - x.

According to the given condition, the sum of the squares of these two parts is 232: x² + (20 - x)² = 232

Expanding the equation: x² + 400 - 40x + x² = 232 2x² - 40x + 168 = 0

Dividing the entire equation by 2: x² - 20x + 84 = 0

Factoring the quadratic equation: x² - 14x - 6x + 84 = 0 x(x - 14) - 6(x - 14) = 0 (x - 14)(x - 6) = 0

This yields x = 14 or x = 6.

Explanation

The midpoint of the segment joining (5, -3) and (0, 2) is calculated using the midpoint formula: M = ((5 + 0)/2, (-3 + 2)/2) = (5/2, -1/2)

The slope of the segment joining the two points is: m = (2 - (-3)) / (0 - 5) = 5 / -5 = -1

The slope of the perpendicular bisector is the negative reciprocal of this slope: m' = -1 / m = -1 / -1 = 1

Using the point-slope form with midpoint (5/2, -1/2) and slope 1: y - (-1/2) = 1 × (x - 5/2) y + ½ = x - 5/2

Rearranging the terms: x - y = 3

Explanation

The weight to height ratio is calculated by dividing the weight in kilograms by the height in meters. For Arpita, this ratio is 57 ÷ 1.56, which equals approximately 36.54. Comparing this to the other individuals, Archita has a ratio of 67 ÷ 1.63 ≈ 41.10, Arpana has a ratio of 67 ÷ 1.51 ≈ 44.37, and Arundhati has a ratio of 73 ÷ 1.59 ≈ 45.91. Arpita clearly has the lowest ratio.

Explanation

The average number of children per house is calculated using the formula: Average = Total number of children ÷ Total number of houses

The total number of houses is: 7 + 11 + 15 + 5 = 38

The total number of children across all houses is: (7 × 0) + (11 × 1) + (15 × 2) + (5 × 3) = 0 + 11 + 30 + 15 = 56

Applying the formula: Average = 56 ÷ 38 ≈ 1.47

Explanation

The total number of shares issued by the company is 6 lakh, which is equal to 600,000 shares.

Dhiraj initially holds a 15% share of these total shares.

Initial shares held by Dhiraj = 15% of 600,000 = (15/100) × 600,000 = 90,000 shares

When Chinmay sells 15,000 of his shares to Dhiraj, Dhiraj receives these shares, which increases his total shareholding.

New number of shares held by Dhiraj = 90,000 + 15,000 = 105,000 shares

Explanation

Using the standard algebraic identity: (x + y)² = x² + y² + 2xy

Substitute the given values of x² + y² = 113 and xy = 56 into the equation: (x + y)² = 113 + 2(56) (x + y)² = 113 + 112 (x + y)² = 225

Taking the square root on both sides yields: x + y = √225 x + y = 15

Explanation

If mA = nB = pC, the ratio A : B : C is given by the inverse ratio 1/m : 1/n : 1/p.

For the relation 3A = 7B = 13C, the ratio A : B : C is: A : B : C = 1/3 : 1/7 : 1/13

Multiplying each term by the product of the denominators (3 × 7 × 13 = 273) simplifies the ratio to: A : B : C = (7 × 13) : (3 × 13) : (3 × 7) A : B : C = 91 : 39 : 21

Explanation

A number r is a root of a quadratic equation if substituting x = r into the equation satisfies it by yielding a value of zero.

Substituting x = 3 into the equation x² - 7x + 12 = 0 gives:

3² - 7(3) + 12 = 9 - 21 + 12 = 21 - 21 = 0

Since the equation is satisfied, 3 is a root of this equation.

Explanation

The ratio of efficiency of Amar and Badal is 2 : 1. Since efficiency and time taken are inversely proportional, the ratio of time taken by Amar and Badal to complete the work is 1 : 2.

Let the time taken by Amar be x days and Badal be 2x days. The difference in their time taken is given as 15 days: 2x - x = 15 x = 15

Thus, Amar takes 15 days and Badal takes 30 days to complete the work.

Using the combined work formula, the time taken by both working together is: Time = (a × b) / (a + b) Time = (15 × 30) / (15 + 30) = 450 / 45 = 10 days

Explanation

The intercept form of the equation of a line is x/a + y/b = 1, where a is the x-intercept and b is the y-intercept.

Given: a = 3/4 b = -2/3

Substituting these values into the intercept form: x / (3/4) + y / (-2/3) = 1

Simplifying the fractions: 4x/3 - 3y/2 = 1

Multiplying the entire equation by the least common multiple of the denominators 3 and 2, which is 6, to clear the fractions: 6 × (4x/3) - 6 × (3y/2) = 6 × 1 8x - 9y = 6

Explanation

The triple angle formula for tangent is derived using the compound angle identity:

tan(x + y) = (tan x + tan y) / (1 - tan x tan y)

By expressing 3A as 2A + A, we apply this identity:

tan 3A = (tan 2A + tan A) / (1 - tan 2A tan A)

Using the double angle identity, tan 2A = 2tan A / (1 - tan²A), and substituting this into the equation:

tan 3A = (2tan A / (1 - tan²A) + tan A) / (1 - (2tan A / (1 - tan²A)) × tan A)

To simplify, we multiply both the numerator and the denominator by (1 - tan²A):

tan 3A = (2tan A + tan A(1 - tan²A)) / (1 - tan²A - 2tan²A)

Expanding and combining like terms yields:

tan 3A = (2tan A + tan A - tan³A) / (1 - 3tan²A)

tan 3A = (3tan A - tan³A) / (1 - 3tan²A)

Explanation

Let the required angle be x.

Two angles are complementary if their sum is 90°. Therefore, the complement of the angle is 90° - x.

According to the given condition, the angle is 20° greater than its complement:

x - (90° - x) = 20°

2x - 90° = 20°

2x = 110°

x = 55°

Explanation

The volume of a hemisphere is given by the formula V = (2/3) × π × r³, where r is the radius of the hemisphere.

Substituting the given values r = 21 cm and π = 22/7 into the formula:

V = (2/3) × (22/7) × 21 × 21 × 21

V = 2 × 22 × 21 × 21

V = 44 × 441

V = 19404 cubic cm

Explanation

We can simplify the given trigonometric expression by writing cot²A in terms of sin²A and cos²A:

cot²A - cos²A = (cos²A / sin²A) - cos²A

Factoring out cos²A:

= cos²A (1 / sin²A - 1)

Using the reciprocal identity cosec²A = 1 / sin²A:

= cos²A (cosec²A - 1)

Using the standard trigonometric identity cosec²A - 1 = cot²A:

= cos²A cot²A

= cot²A cos²A

Explanation

In a right-angled triangle, the side opposite the right angle is the hypotenuse. The relationship between the side opposite to an angle θ and the hypotenuse is given by the trigonometric ratio:

sin θ = Opposite / Hypotenuse

Here, the angle θ is 30° and the length of the hypotenuse is 6 cm. Substituting these values into the formula:

sin 30° = Opposite / 6

Since sin 30° = ½, the equation becomes:

½ = Opposite / 6

Opposite = 6 × ½ = 3 cm

Explanation

To find the possible values of x, we solve the two given inequalities individually.

For the first inequality: 2x + 5 > 2 + 3x Subtracting 2x from both sides gives: 5 > 2 + x Subtracting 2 from both sides gives: x < 3

For the second inequality: 2x - 3 ≤ 4x - 5 Subtracting 2x from both sides gives: -3 ≤ 2x - 5 Adding 5 to both sides gives: 2 ≤ 2x Dividing by 2 gives: x ≥ 1

Combining both inequalities, the range for x is: 1 ≤ x < 3

Since 2 lies within the interval [1, 3), it satisfies both inequalities.

Explanation

The area of a circle is given by the formula A = πr², where r is the radius. When a quantity depending on the square of a variable increases by x%, the overall percentage increase in the quantity is given by the successive percentage formula:

Percentage increase = 2x + (x² / 100)

Given that the radius increases by x = 14%:

Percentage increase = 2(14) + (14² / 100) Percentage increase = 28 + (196 / 100) Percentage increase = 28 + 1.96 = 29.96%

Explanation

Using the trigonometric identity for angles in the second quadrant, tan(180° - θ) = -tan θ.

Substituting θ = 30°: tan 150° = tan(180° - 30°) = -tan 30°

Since the standard trigonometric value of tan 30° is 1/√3, we get: tan 150° = -1/√3

Explanation

Let the recurring decimal be represented as x = 0.151515...

Since there are two repeating digits in the decimal pattern, multiply both sides by 100: 100x = 15.151515...

Now, subtract the first equation from the second: 100x - x = 15.151515... - 0.151515... 99x = 15 x = 15/99

Simplifying the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 3, yields: x = 5/33

Explanation

Let the three numbers be x, y, and z.

According to the given conditions, the first number x is twice the second number y and thrice the third number z: x = 2y → y = x/2 x = 3z → z = x/3

The average of the three numbers is 429: (x + y + z) / 3 = 429 x + y + z = 1287

Substitute y and z in terms of x: x + x/2 + x/3 = 1287 (6x + 3x + 2x) / 6 = 1287 11x / 6 = 1287 x = (1287 × 6) / 11 x = 117 × 6 x = 702

The third number z is: z = x/3 = 702 / 3 = 234

The difference between the first and the third number is: x - z = 702 - 234 = 468

Explanation

To find the numerical value of (x + 1)³, we first solve the given linear equation for x:

8x - 4 = 3x + 6

Subtracting 3x from both sides of the equation:

5x - 4 = 6

Adding 4 to both sides:

5x = 10

Dividing by 5 gives:

x = 2

Substituting the value of x into the expression (x + 1)³:

(2 + 1)³ = 3³ = 27