SSC CHSL Tier 1 2016 Quantitative Aptitude Questions with Solutions

Explanation

The total exports for the year 2015 are calculated by summing the values of all categories: 600 + 675 + 800 + 775 + 525 = 3375 million US$. The share of software exports is 675 million US$, which constitutes (675 / 3375) * 100 = 20 percent of the total exports.

Explanation

The given system of inequalities can be split into two parts.

The first inequality is: 5x - 3(4 - x) < 4x - 4 5x - 12 + 3x < 4x - 4 8x - 12 < 4x - 4 4x < 8 x < 2

The second inequality is: 4x - 4 < 4x + 2x/3 -4 < 2x/3 -12 < 2x -6 < x

Combining both results yields: -6 < x < 2

Only the value 1 lies within this interval.

Explanation

The reflection of any point (x, y) across the x-axis keeps the x-coordinate the same while negates the y-coordinate, transforming the coordinates to (x, -y). Applying this rule to the given point (-1, 4), the x-coordinate remains -1 and the y-coordinate becomes -4, yielding the reflected coordinates (-1, -4).

Explanation

Let the time taken by the Express train be T_E hours and the time taken by the Rajdhani train be T_R hours.

According to the first condition, the Express train takes 10 hours more than Rajdhani: T_E - T_R = 10

When the speed of the Express train is doubled, its time taken is halved to T_E/2. In this case, it takes 7 hours less than Rajdhani: T_R - T_E/2 = 7

Adding these two equations eliminates T_R: T_E - T_E/2 = 10 + 7 T_E/2 = 17 T_E = 34 hours

Substituting T_E = 34 back into the first equation: 34 - T_R = 10 T_R = 24 hours

The speed of Rajdhani is the total distance divided by the time taken by Rajdhani: Speed of Rajdhani = 660 / 24 = 27.5 km/hr

Explanation

Applying the laws of exponents, for any base p, pᵐ × pⁿ = pᵐ⁺ⁿ and pᵐ / pⁿ = pᵐ⁻ⁿ.

First, simplify the numerator by multiplying terms with like bases: (b⁵x²a³z⁴) × (b³x²a⁴z⁵) = a³⁺⁴b⁵⁺³x²⁺²z⁴⁺⁵ = a⁷b⁸x⁴z⁹

Next, divide this result by the denominator: (a⁷b⁸x⁴z⁹) / (a²b³z²) = a⁷⁻²b⁸⁻³x⁴z⁹⁻² = a⁵b⁵x⁴z⁷

Explanation

In a triangle, the median AD is the line segment connecting the vertex A to the midpoint D of the opposite side BC.

First, calculate the coordinates of the midpoint D of side BC, using the coordinates B(3, 0) and C(5, -2): xD = (3 + 5)/2 = 4 yD = (0 - 2)/2 = -1 So, the coordinates of D are (4, -1).

Next, find the equation of the line passing through the points A(2, -4) and D(4, -1). The equation of a line passing through two points (x₁, y₁) and (x₂, y₂) is given by the formula: y - y₁ = ((y₂ - y₁) / (x₂ - x₁)) × (x - x₁)

Substituting the coordinates of A and D into the formula: y - (-4) = ((-1 - (-4)) / (4 - 2)) × (x - 2) y + 4 = (3/2) × (x - 2)

Multiplying both sides by 2 to clear the fraction: 2(y + 4) = 3(x - 2) 2y + 8 = 3x - 6

Rearranging the terms into standard linear form: 3x - 2y = 14

Explanation

Let the marked price of the goods be MP and the cost price be CP. The retailer offers a discount of 28% on the marked price, which means the selling price SP is 72% of the marked price: SP = 0.72 × MP

Since the retailer sells the goods at the cost price, the selling price is equal to the cost price: CP = 0.72 × MP MP = CP / 0.72 ≈ 1.3888 × CP

The markup percentage is given by the relation: Markup % = (MP - CP) / CP × 100 Markup % = (1.3888 × CP - CP) / CP × 100 = 0.3888 × 100 = 38.88%

Explanation

To find the value of tan 300°, we use the trigonometric reduction formula for angles in the fourth quadrant. The angle 300° lies in the fourth quadrant, where the tangent function is negative.

Using the identity tan(360° - θ) = -tan θ:

tan 300° = tan(360° - 60°) tan 300° = -tan 60°

Substituting the standard value of tan 60° = √3 gives:

tan 300° = -√3

Explanation

Let x be the number. According to the question, when the number is increased by 38, it becomes 119% of itself. This can be represented mathematically as: x + 38 = 1.19x

Subtracting x from both sides of the equation yields: 0.19x = 38

Solving for x: x = 38 ÷ 0.19 x = 3800 ÷ 19 x = 200

Explanation

The difference (D) between compound interest and simple interest on a principal sum (P) for 2 years at an annual interest rate (R) is given by the formula:

D = P × (R / 100)²

Given that the difference is ₹40 and the interest rate is 8% per annum, we can substitute these values into the formula:

40 = P × (8 / 100)²

40 = P × (2 / 25)²

40 = P × (4 / 625)

Solving for P:

P = 40 × (625 / 4)

P = 10 × 625 = 6250

Explanation

In a rhombus ABCD, the adjacent sides AB and BC are equal in length, making triangle ABC an isosceles triangle. Since the angles opposite to equal sides in an isosceles triangle are equal, angle BCA is equal to angle CAB, which is 35°. The sum of the angles in triangle ABC must be 180°, so angle ABC is calculated as 180° - 35° - 35° = 110°.

Explanation

Using the fundamental trigonometric identity: 1 + cot²A = cosec²A

Taking the square root on both sides: √(1 + cot²A) = cosec A

For the expression x = cot A / √(1 + cot²A), substituting √(1 + cot²A) gives: x = cot A / cosec A

Converting cot A and cosec A into terms of sin A and cos A: cot A = cos A / sin A cosec A = 1 / sin A

Substituting these values into the expression for x: x = (cos A / sin A) / (1 / sin A) x = cos A

Explanation

Let CP be the cost price, MP be the marked price, D be the discount percentage, and P be the profit percentage.

The relationship between MP and CP with discount and profit is given by the formula: MP / CP = (100 + P) / (100 - D)

In the first case, when a 20% discount is given, the profit is 42%: MP / CP = (100 + 42) / (100 - 20) = 142 / 80 = 71 / 40

In the second case, when a 15% discount is given, we find the new profit percentage P: MP / CP = (100 + P) / (100 - 15) 71 / 40 = (100 + P) / 85

Solving for P: 100 + P = 85 × 71 / 40 100 + P = 17 × 71 / 8 100 + P = 1207 / 8 100 + P = 150.875 P = 50.875%

Explanation

To simplify the expression √[(1 - cos A) / (1 + cos A)], multiply both the numerator and the denominator inside the square root by (1 - cos A):

√[(1 - cos A) / (1 + cos A)] = √[((1 - cos A)(1 - cos A)) / ((1 + cos A)(1 - cos A))]

This simplifies to:

√[(1 - cos A)² / (1 - cos² A)]

Using the fundamental trigonometric identity sin² A + cos² A = 1, we can substitute 1 - cos² A with sin² A:

√[(1 - cos A)² / sin² A] = (1 - cos A) / sin A

Splitting the fraction gives:

(1 / sin A) - (cos A / sin A)

Using the reciprocal and quotient identities where 1 / sin A = cosec A and cos A / sin A = cot A, we get:

cosec A - cot A

Explanation

The division algorithm states that:

Dividend = (Divisor × Quotient) + Remainder

In this problem, the divisor is 38, the quotient is 70, and the remainder is 12.

Substituting these values into the formula:

Number = (38 × 70) + 12 Number = 2660 + 12 Number = 2672

Explanation

Initial total collection is the product of the initial price and the initial number of visitors, while the new total collection is the product of the new price and the new number of visitors.

Given that the ticket price changes in the ratio 17:19, and the number of visitors changes in the ratio 11:10:

Initial collection ratio units = 17 × 11 = 187 New collection ratio units = 19 × 10 = 190

Since 190 is greater than 187, the total collection has increased. Therefore, the total collection has increased in the ratio 187:190.

Explanation

A cyclic quadrilateral is a quadrilateral whose vertices all lie on a single circle. A key property of a cyclic quadrilateral is that its opposite angles are supplementary, meaning their sum is 180°. This is because the opposite angles subtend complementary arcs of the circle that together sum to 360°. Since an inscribed angle is equal to half the measure of its intercepted arc, the sum of the two opposite inscribed angles is half of 360°, which equals 180°.

Explanation

The algebraic identity relating the sum of two numbers, the sum of their squares, and their product is: (a + b)² = a² + b² + 2ab

Substituting the given values, a + b = -8 and a² + b² = 34, into the formula gives: (-8)² = 34 + 2ab 64 = 34 + 2ab

Rearranging the terms to solve for ab: 2ab = 64 - 34 2ab = 30 ab = 15

Explanation

For any n consecutive even or odd integers, the relationship between their average and the highest integer is given by the formula:

Highest integer = Average + (n - 1)

Here, the average is 50 and the number of integers, n, is 19.

Substituting these values into the formula yields:

Highest integer = 50 + (19 - 1) = 50 + 18 = 68

Explanation

The sum of an arithmetic progression when the first term, the last term, and the number of terms are known is given by the formula Sₙ = (n/2) × (a + l), where n is the number of terms, a is the first term, and l is the last term.

Given that a = 25, l = -52, and n = 12, substituting these values into the formula yields:

S₁₂ = (12/2) × (25 + (-52))

S₁₂ = 6 × (-27)

S₁₂ = -162

Explanation

The total surface area of a hollow hemisphere is the sum of its outer curved surface area, inner curved surface area, and the area of the top circular ring.

Formula: TSA = 2πR² + 2πr² + π(R² - r²) TSA = π(3R² + r²)

Here, the internal radius r is 9 cm. With a thickness of 1 cm, the external radius R is r + 1 = 10 cm.

Substituting these values into the formula: TSA = π(3 × 10² + 9²) TSA = π(3 × 100 + 81) TSA = 381π

Using the value of π = 22/7: TSA = 381 × 22/7 TSA = 8382/7 TSA ≈ 1197.42 sq cm

Explanation

Pankaj completes ½ of the job in 12 days, which means the total time he takes to complete the entire job alone is 12 × 2 = 24 days. His rate of work is 1/24 of the job per day.

Sainath completes the remaining ½ of the job in 6 days, which means the total time he takes to complete the entire job alone is 6 × 2 = 12 days. His rate of work is 1/12 of the job per day.

When working together, their combined rate of work per day is: 1/24 + 1/12 = 1/24 + 2/24 = 3/24 = 1/8

Since their combined rate of work is 1/8 of the job per day, the total number of days they take to complete the job together is the reciprocal of this rate, which is 8 days.

Explanation

The percentage of students who chose Engineering that are girls is calculated using the formula:

Percentage = (Number of girls in Engineering / Total students in Engineering) × 100

From the given data table, the number of girls who chose Engineering is 25 and the number of boys who chose Engineering is 65.

Total students in Engineering = 65 + 25 = 90

Percentage = (25 / 90) × 100 = 27.77%

Truncating to one decimal place yields 27.7%.

Explanation

The cumulative production at the end of a given month represents the total number of cars manufactured from the beginning of the period up to the end of that month.

To find the total number of cars manufactured in the months of April and May, we subtract the cumulative production at the end of March from the cumulative production at the end of May.

Production in April and May = Cumulative production in May - Cumulative production in March

From the table: Cumulative production in May = 2620 Cumulative production in March = 1520

Substituting these values into the formula: Production in April and May = 2620 - 1520 = 1100

Alternatively, this can be calculated by finding the individual production for each month: Production in April = Cumulative production in April - Cumulative production in March = 1970 - 1520 = 450 Production in May = Cumulative production in May - Cumulative production in April = 2620 - 1970 = 650 Total production in April and May = 450 + 650 = 1100

Explanation

To find the total calories burned in the given week, first calculate the total distance jogged by summing the distances for each day: Total distance = 3 + 3 + 2 + 2 + 3.5 + 2 + 3.5 = 19 km

The rate of calories burned is 400 calories for every 5 km jogged. This gives the calories burned per kilometer as: 400 ÷ 5 = 80 calories per km

Multiply the total distance jogged in the week by this rate to find the total calories burned: Total calories burned = 19 km × 80 calories per km = 1520 calories

Explanation

To find the percentage of total expenses spent on Raw Materials and Interest, first calculate the total yearly expenses by summing the values of all the items: Total expenses = 4 + 3 + 4 + 6 + 1 = 18 lakhs

Next, find the combined expense of Raw Materials and Interest: Raw Materials and Interest = 4 + 6 = 10 lakhs

Now, calculate the required percentage: Percentage = (10 / 18) × 100 = 55.56%

Explanation

The area of a circle is proportional to the square of its radius, as given by the formula A = πr². When a quantity depends on the square of a variable, an increase of x% in that variable results in a successive percentage increase. This can be calculated directly using the effective percentage change formula:

Net percentage increase = a + b + ab/100

Since the radius increases by 15% in both dimensions of the area calculation, we substitute a = 15 and b = 15 into the formula:

Net percentage increase = 15 + 15 + (15 × 15)/100 = 30 + 2.25 = 32.25%

Explanation

Without any scheme or discount, the total original cost for the group of 17 adults and 7 children is calculated as: Cost of 17 adult tickets = 17 × Rs 1600 = Rs 27200 Cost of 7 child tickets = 7 × Rs 600 = Rs 4200 Total original cost = Rs 27200 + Rs 4200 = Rs 31400

According to the offer, 1 child goes free with two adults. For 17 adults, the maximum number of free children allowed is floor(17 / 2) = 8. Since there are 7 children in the group, all 7 children can enter for free.

The savings (discount) obtained by the group is the cost of the 7 child tickets: Discount = 7 × Rs 600 = Rs 4200

To find the discount percentage: Discount percentage = (Discount / Total original cost) × 100 Discount percentage = (4200 / 31400) × 100 = 4200 / 314 ≈ 13.37%

Explanation

Any repeating decimal of the form 0.ababab... with a two-digit repeating pattern can be represented directly as the fraction ab/99.

In this case, the repeating block is 09. Substituting this into the formula gives:

0.090909... = 09/99 = 9/99

Dividing both the numerator and the denominator by their greatest common divisor, which is 9, simplifies the fraction to 1/11.

Explanation

The profit is divided among Aashay, Bhairav, and Chetan in the ratio of 4:3:6.

To find Chetan's share, we first calculate the total number of parts in the ratio: Total parts = 4 + 3 + 6 = 13 parts

The total profit of Rs 135200 corresponds to these 13 parts. Therefore, the value of 1 part is: Value of 1 part = 135200 ÷ 13 = 10400

Chetan receives 6 parts of the profit: Chetan's share = 6 × 10400 = 62400

Explanation

A cube has 12 equal edges. If a represents the length of one edge of the cube, the formula for the sum of the lengths of all its edges is:

Sum of edges = 12a

Given that the sum of the edges is 84 cm, we can find the edge length of the cube:

12a = 84

a = 84 / 12 = 7 cm

The volume V of a cube is calculated using the formula:

V = a³

Substituting the value of the edge length into the formula:

V = 7³ = 343 cubic cm

Explanation

Let the third number be 100.

Since the first number is 30% lesser than the third number, its value is: 100 - 30 = 70

Since the second number is 60% lesser than the third number, its value is: 100 - 60 = 40

The difference between the first and the second number is: 70 - 40 = 30

To make the second number equal to the first number, it must be increased by 30. The required percentage enhancement based on the second number is calculated as: (Difference / Second number) × 100 = (30 / 40) × 100 = 75%

Explanation

When two objects travel the same distance at different speeds, the distance can be calculated directly using the formula:

Distance = (s₁ × s₂) / (s₂ - s₁) × Δt

where s₁ and s₂ are the speeds of the two cars and Δt is the difference in time taken.

Substituting the given values into the formula:

s₁ = 30 km/hr s₂ = 45 km/hr Δt = 2.5 hours

Distance = (30 × 45) / (45 - 30) × 2.5

Distance = 1350 / 15 × 2.5

Distance = 90 × 2.5 = 225 km

Explanation

Solving the first inequality: 7x + 2 ≥ x - 2 6x ≥ -4 x ≥ -2/3

Solving the second inequality: 7 + 2x ≥ 3 + 3x 4 ≥ x x ≤ 4

Combining both conditions gives the interval: -2/3 ≤ x ≤ 4

The value 3 falls within this range of [-2/3, 4], making it a valid value for x.

Explanation

The original fare and the new fare are in the ratio of 5:11.

Let the original fare be 5x and the new fare be 11x.

Given that the original fare is Rs 275: 5x = 275 x = 55

The increase in the fare is the difference between the new fare and the original fare: Increase in fare = 11x - 5x = 6x

Substituting the value of x, the increase in fare is: 6 × 55 = 330

Explanation

Substituting cosA = 1/secA into the given expression:

x = (1 + cosA) / (1 - cosA) x = (1 + 1/secA) / (1 - 1/secA) x = (secA + 1) / (secA - 1)

Multiplying both the numerator and the denominator by (secA - 1):

x = (secA + 1)(secA - 1) / (secA - 1)² x = (sec²A - 1) / (secA - 1)²

Using the fundamental trigonometric identity sec²A - 1 = tan²A:

x = tan²A / (secA - 1)²

Explanation

First, find the rate of interest using the simple interest formula. The simple interest earned in 8 years is 40% of the principal (P).

SI = P × R × T / 100

0.4P = P × R × 8 / 100

R = 40 / 8 = 5% per annum

To find the compound interest for 2 years at 5% per annum, we can calculate the cumulative effective rate:

Effective Rate = R + R + (R × R) / 100

Effective Rate = 5 + 5 + (5 × 5) / 100 = 10.25%

Now, compute the compound interest on Rs. 36,000 at this effective rate:

Compound Interest = Principal × Effective Rate / 100

Compound Interest = 36000 × 10.25 / 100 = 360 × 10.25 = 3690

Explanation

A quadratic equation ax² + bx + c = 0 has real roots if its discriminant, D = b² - 4ac, is greater than or equal to zero (D ≥ 0).

For the equation 2x² + 7x + 6 = 0, the coefficients are a = 2, b = 7, and c = 6.

Calculating the discriminant: D = 7² - 4 × 2 × 6 D = 49 - 48 D = 1

Since D = 1, which is greater than zero, the equation has real roots.

Explanation

The effective discount percentage on an offer of buying N items and getting M items free is calculated using the formula:

Discount % = (Number of free items ÷ Total number of items received) × 100

In this scenario, the customer buys 4 dupattas and gets 1 dupatta free, which means they receive a total of 5 dupattas (4 + 1) while only paying for 4.

Substituting the values into the formula:

Discount % = (1 ÷ 5) × 100 = 20%

The individual price of each dupatta is irrelevant to the percentage calculation, as the ratio of the discount to the total value remains constant.

Explanation

An isosceles trapezium is defined as a quadrilateral with one pair of parallel sides and a pair of non-parallel sides that are equal in length. If we consider an isosceles trapezium ABCD where side AB is parallel to side CD, and side AD is equal to side BC, the base angles adjacent to the parallel sides are equal, meaning ∠ADC = ∠BCD. In triangles ACD and BDC, AD = BC, CD is a common side, and the included angles ∠ADC and ∠BCD are equal. By the Side-Angle-Side (SAS) congruence criterion, triangle ACD is congruent to triangle BDC. This congruence directly implies that their corresponding parts are equal, which means the diagonals AC and BD are equal in length, or congruent.

Explanation

Expand and simplify the given linear equation:

9x - 6(3 - x) = 12

Distribute -6 inside the parentheses:

9x - 18 + 6x = 12

Combine the terms containing x:

15x - 18 = 12

Add 18 to both sides of the equation:

15x = 12 + 18 15x = 30

Divide both sides by 15 to solve for x:

x = 30/15 x = 2

Explanation

We can solve this system of linear equations by substituting one variable into the other.

From the first equation, we can express x in terms of y: x = 2y + 2

Substituting this expression for x into the second equation: 3(2y + 2) + y = 20 6y + 6 + y = 20 7y + 6 = 20 7y = 14 y = 2

Substituting the value of y back into our expression for x: x = 2(2) + 2 x = 6

This gives the solution (6, 2).

Explanation

Let the length of the rectangle be L and the width be W.

The perimeter of a rectangle is given by the formula: Perimeter = 2(L + W)

Substituting the given perimeter of 140 cm: 140 = 2(L + W) L + W = 70 cm

The area of the rectangle is given by: Area = L × W = 1200 sq cm (accounting for a typographical print error where 120 sq cm was written instead of 1200 sq cm)

The diagonal d of a rectangle is given by: d = √(L² + W²)

Using the algebraic identity L² + W² = (L + W)² - 2LW: d² = (L + W)² - 2LW

Substituting the values: d² = 70² - 2(1200) d² = 4900 - 2400 d² = 2500

Taking the square root: d = √2500 = 50 cm

Explanation

The sum of the interior angles of any quadrilateral is 360°. Given that the angles are in the ratio 3:4:5:6, the sum of the ratio units is:

3 + 4 + 5 + 6 = 18 units

The value of one ratio unit is:

360° ÷ 18 = 20°

The largest angle corresponds to the maximum ratio unit of 6. The measure of this biggest angle is:

6 × 20° = 120°

Explanation

The correct average can be found directly by adjusting the incorrect average with the net change in marks divided by the total number of students.

Correct Average = Incorrect Average + (Correct Value - Incorrect Value) / Number of Students

Substituting the given values into the formula:

Correct Average = 65 + (48 - 84) / 45

Correct Average = 65 - 36 / 45

Correct Average = 65 - 0.8 = 64.2

Explanation

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

D = P × (R / 100)²

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

Given: D = Rs 405 R = 9%

Substituting these values into the formula:

405 = P × (9 / 100)²

405 = P × (81 / 10000)

P = (405 × 10000) / 81

Since 405 / 81 = 5:

P = 5 × 10000

P = 50000

The principal sum is Rs 50000.

Explanation

The slope of a line passing through two points (x₁, y₁) and (x₂, y₂) is given by the formula: m = (y₂ - y₁) / (x₂ - x₁)

For the first line passing through (-3, 1) and (x, 5), the slope is: m₁ = (5 - 1) / (x - (-3)) = 4 / (x + 3)

For the second line passing through (-2, -1) and (6, 3), the slope is: m₂ = (3 - (-1)) / (6 - (-2)) = 4 / 8 = ½

Since parallel lines have equal slopes: m₁ = m₂ 4 / (x + 3) = ½ x + 3 = 8 x = 5

Explanation

Let the selling price of one shoe be SP.

The total selling price of 22 shoes is 22 × SP.

The profit earned by the shopkeeper is equal to the selling price of 6 shoes, which is 6 × SP.

Using the standard relation: Cost Price = Selling Price - Profit

The cost price of 22 shoes is: CP = 22 × SP - 6 × SP = 16 × SP

Profit percentage is calculated using the formula: Profit Percentage = (Profit / Cost Price) × 100

Substituting the values: Profit Percentage = (6 × SP / (16 × SP)) × 100 = (6/16) × 100 = (3/8) × 100 = 37.5%

Explanation

Using the fundamental trigonometric identity: sin²A + cos²A = 1

Rearranging the terms to express cos²A in terms of sin²A: cos²A = 1 - sin²A

Taking the square root of both sides gives: cosA = √(1 - sin²A)

Since it is given that cosA = x, substituting x for cosA yields: x = √(1 - sin²A)

Explanation

We know the standard values of the trigonometric functions: cosec 45° = √2 sin 30° = ½

Substitute these values into the given equation: x = cosec 45° - sin 30° x = √2 - ½

By taking the common denominator of 2, we simplify the expression: x = (2√2 - 1)/2