SSC CHSL Tier 1 2016 Quantitative Aptitude Questions with Solutions

Explanation

The algebraic identity for the square of a binomial difference is (a - b)² = a² - 2ab + b².

By substituting a = 3x and b = 5xy into this formula, we get:

(3x - 5xy)² = (3x)² - 2(3x)(5xy) + (5xy)²

Calculating each term: The first term is (3x)² = 9x². The middle term is -2 × 3x × 5xy = -30x²y. The last term is (5xy)² = 25x²y².

Combining these terms gives 9x² - 30x²y + 25x²y².

Explanation

The first inequality is: 5 - 2x ≥ 4 - x

Solving for x: 5 - 4 ≥ 2x - x 1 ≥ x x ≤ 1

The second inequality is: 3(2 - x) > 2 - 4x

Expanding and solving for x: 6 - 3x > 2 - 4x 4x - 3x > 2 - 6 x > -4

Combining both inequalities, the acceptable range for x is: -4 < x ≤ 1

The value -3 satisfies this range because -3 is strictly greater than -4 and less than or equal to 1. Substituting x = -3 into the inequalities confirms this: 5 - 2(-3) = 11 and 4 - (-3) = 7, where 11 ≥ 7. 3(2 - (-3)) = 15 and 2 - 4(-3) = 14, where 15 > 14.

Explanation

Evaluating the exponents on the right side of the equation:

6² = 36 3² = 9

Substituting these values into the given equation:

x³ = 36 - 9 x³ = 27

Taking the cube root of both sides:

x = ∛27 x = 3

Explanation

We simplify the equation by expanding both sides: x(x + 2) - 15 = x(x - 5) + 11 x² + 2x - 15 = x² - 5x + 11

Subtracting x² from both sides of the equation yields: 2x - 15 = -5x + 11

Rearranging the terms to one side gives: 7x - 26 = 0

This is a linear equation of degree 1. Because the x² terms cancel out completely and the highest power of the variable x is 1, it does not represent a quadratic equation.

Explanation

Let the first number be x, and the sum of the remaining four numbers be S.

Since the average of the five numbers is 78, their total sum is: x + S = 5 × 78 = 390

According to the given condition, the first number is 2/11 times the sum of the other four: x = (2/11) × S S = (11/2) × x

Substituting the value of S in the total sum equation: x + (11/2) × x = 390 (13/2) × x = 390 x = (390 × 2) / 13 x = 60

Explanation

By the fundamental trigonometric identity, sin²A + cos²A = 1. Rearranging this equation to solve for sin²A yields sin²A = 1 - cos²A. Taking the principal square root on both sides gives sin A = √(1 - cos²A). Since it is given that sin A = x, it follows directly that x = √(1 - cos²A).

Explanation

Using the direct relationship between selling price and the corresponding percentage value: SP₁ / SP₂ = (100 - Loss₁%) / (100 - Loss₂%)

Substituting the given values: 1330 / 1463 = (100 - 15) / (100 - Loss₂%) 1330 / 1463 = 85 / (100 - Loss₂%)

Since 1463 - 1330 = 133, both values are multiples of 133: 1330 = 10 × 133 1463 = 11 × 133

Simplifying the ratio: 10 / 11 = 85 / (100 - Loss₂%)

Cross-multiplying to solve: 10 × (100 - Loss₂%) = 11 × 85 1000 - 10 × Loss₂% = 935 10 × Loss₂% = 65 Loss₂% = 6.5%

Explanation

Let the total work be represented by 1.

Work done by A in 1 day = 1/16 Work done by B in 1 day = 1/20

Combined work done by A and B in 1 day = 1/16 + 1/20 = (5 + 4) / 80 = 9/80

In 8 days, the total fraction of work completed by them together = 8 × 9/80 = 9/10

The fraction of work left = 1 - 9/10 = 1/10

Explanation

By the Basic Proportionality Theorem (Thales's Theorem), if a line is drawn parallel to one side of a triangle intersecting the other two sides, it divides those sides proportionally. Since DE is parallel to BC, the relation is:

AD / AB = AE / AC

The total length of side AB is the sum of AD and DB:

AB = AD + DB = 8 + 4 = 12 cm

Substituting the given values into the proportion:

8 / 12 = 12 / AC

AC = (12 × 12) / 8

AC = 144 / 8

AC = 18 cm

Explanation

The volume of water transferred from the cone to the cylindrical vessel remains constant. Therefore, the volume of the cone is equal to the volume of the water in the cylinder:

(1/3) × π × r₁² × h₁ = π × r₂² × h₂

Here, r₁ is the radius of the cone (3.5 cm), h₁ is the height of the cone (27 cm), r₂ is the radius of the cylinder (2.1 cm), and h₂ is the height of water in the cylinder. Canceling π from both sides gives:

(1/3) × r₁² × h₁ = r₂² × h₂

Substituting the given values into the equation:

(1/3) × (3.5)² × 27 = (2.1)² × h₂

9 × (7/2)² = (21/10)² × h₂

9 × (49/4) = (441/100) × h₂

441/4 = (441/100) × h₂

1/4 = h₂/100

h₂ = 25 cm

Explanation

To convert speed from kilometres per hour (km/h) to metres per second (m/s), multiply the speed by the conversion factor 5/18.

Speed in m/s = 1278 × 5/18

Dividing 1278 by 18 yields 71.

71 × 5 = 355

This calculation shows that the speed of the missile is 355 m/s, meaning it travels 355 metres in one second.

Explanation

Let the unknown number be x.

According to the given condition: (8/5) × (4/3) × x = 192

Simplify the fraction multiplication on the left side: (32/15) × x = 192

Solve for x by multiplying both sides by 15/32: x = 192 × (15/32) x = 6 × 15 x = 90

Now, calculate 5/6 of this number: (5/6) × 90 = 5 × 15 = 75

Explanation

Correct Answer: B

Cost of daily tickets for 27 days of travel is 27 × ₹110 = ₹2970.

Cost of the monthly pass is ₹2524.5.

Total savings by buying the monthly pass instead of daily tickets is ₹2970 - ₹2524.5 = ₹445.5.

Percentage savings is (Total savings / Cost of daily tickets) × 100 = (445.5 / 2970) × 100 = 15%.

Explanation

Let the radius of the circle be r and the side length of the square be a.

Since the areas of the circle and the square are equal, we have: πr² = a² r²/a² = 1/π

The diameter of the circle is d = 2r, and the diagonal of the square is D = a√2.

The ratio of the length of the diameter to the length of the diagonal is: d/D = 2r / (a√2)

Squaring this ratio gives: (d/D)² = 4r² / 2a² = 2r²/a²

Substituting r²/a² = 1/π into the equation: (d/D)² = 2/π

Using the standard approximation π = 22/7: (d/D)² = 2 / (22/7) = 2 × 7 / 22 = 14/22 = 7/11

Explanation

The equation of the given line is 3x - 6y = 4.

Converting this equation into the slope-intercept form, y = mx + c: 6y = 3x - 4 y = ½x - ⅔

From this, the slope (m₁) of the given line is ½.

For two lines to be perpendicular, the product of their slopes (m₁ and m₂) must satisfy the relation: m₁ × m₂ = -1

Substituting the value of m₁: ½ × m₂ = -1 m₂ = -2

The slope of the perpendicular line is -2.

Explanation

Let the marks of the two students be x and y, with x being the higher score.

The difference between their marks is: x - y = 26

The higher score is 75% of the sum of their marks: x = 0.75(x + y) x = ¾(x + y) 4x = 3(x + y) 4x = 3x + 3y x = 3y

Substituting x = 3y into the first equation: 3y - y = 26 2y = 26 y = 13

Substituting y = 13 into the expression for x: x = 3 × 13 = 39

The marks obtained by the two students are 39 and 13.

Explanation

The total profit is shared among Amir, Balaraj, and Charan in the ratio of 7:5:4.

To find Charan's share, the ratio formula is applied: Charan's share = (Charan's ratio / Total ratio) × Total profit

The sum of the ratio parts is: 7 + 5 + 4 = 16

Charan's share represents 4 parts out of the total 16 parts.

Charan's share = (4 / 16) × 7,28,000 Charan's share = ¼ × 7,28,000 = 1,82,000

Explanation

The product of 9b², 3a², and 4ab is calculated by multiplying their numerical coefficients and their algebraic terms.

Multiplying the numerical coefficients: 9 × 3 × 4 = 108

Multiplying the algebraic variables by adding the exponents of like bases: b² × a² × ab = a²⁺¹ × b²⁺¹ = a³b³

Combining the coefficient and the variable part yields: 108a³b³

Explanation

Using the trigonometric relation tan θ = sin θ / cos θ, the given expression can be written as:

tan²A - tan²B = sin²A/cos²A - sin²B/cos²B

Taking the common denominator, the expression becomes:

(sin²A cos²B - cos²A sin²B) / (cos²A cos²B)

Using the identity cos²θ = 1 - sin²θ to substitute for cos²B and cos²A in the numerator:

sin²A (1 - sin²B) - (1 - sin²A) sin²B

Expanding the numerator:

sin²A - sin²A sin²B - sin²B + sin²A sin²B

This simplifies to:

sin²A - sin²B

Substituting this back into the fraction gives:

x = (sin²A - sin²B) / (cos²A cos²B)

Explanation

The formula for the volume of a cuboid is:

Volume = length × breadth × height

By substituting the given dimensions:

Length = 12 cm Breadth = 8 cm Height = 6 cm

The calculation is:

Volume = 12 cm × 8 cm × 6 cm = 576 cm³

Explanation

To find the ratio A : B : C from the given equation 5A = 12B = 13C, we can express each term in terms of a common constant k. Let 5A = 12B = 13C = k, which gives A = k/5, B = k/12, and C = k/13.

The ratio A : B : C is therefore (1/5) : (1/12) : (1/13).

To convert this fractional ratio into whole numbers, we multiply each term by the product of the denominators since 5, 12, and 13 have no common factors. This product is 5 × 12 × 13 = 780.

Multiplying each fraction by 780 yields:

A = 780/5 = 156 B = 780/12 = 65 C = 780/13 = 60

This gives the ratio A : B : C as 156 : 65 : 60.

Explanation

The altitude h of an equilateral triangle with side length a is given by the formula:

h = (√3/2) × a

Given that the side length a is 12 cm, substituting this value yields:

h = (√3/2) × 12 = 6√3 cm

Explanation

The double-angle trigonometric identity states that cos(2θ) = 2 cos²(θ) - 1.

By substituting θ = A/2, the relation becomes: cos(A) = 2 cos²(A/2) - 1

Rearranging the terms to isolate cos(A/2): 2 cos²(A/2) = 1 + cos(A) cos²(A/2) = (1 + cos(A))/2 cos(A/2) = √[(1 + cos(A))/2]

Given that cos(A/2) = x, we obtain: x = √[(1 + cos(A))/2]

Explanation

Correct average = Incorrect average + (Correct value - Incorrect value) ÷ Total number of students

Substituting the given values:

Correct average = 69 + (54 - 45) ÷ 72

Correct average = 69 + 9 ÷ 72

Correct average = 69 + 1/8

Correct average = 69 + 0.125

Correct average = 69.125 marks

This direct shortcut adjusts the average by distributing the net difference of marks across all students, saving time by avoiding the calculation of the grand total sum.

Explanation

Let the cost price of the vacuum cleaner be V and the cost price of the dishwasher be D.

The total cost price of both items is: V + D = 78750

The vacuum cleaner was sold at a profit of 37%, so its selling price is 1.37 × V. The dishwasher was sold at a loss of 27%, so its selling price is 0.73 × D.

Since the selling price is the same for both items: 1.37 × V = 0.73 × D V/D = 73/137

Expressing the cost prices in terms of a constant k: V = 73k D = 137k

Summing these gives: 73k + 137k = 78750 210k = 78750 k = 78750/210 = 375

The cheaper item is the vacuum cleaner with a cost price of: V = 73 × 375 = 27375

Explanation

We evaluate the expression given in Option A:

sec A / (sec A - 1) + sec A / (sec A + 1)

By taking the common denominator (sec A - 1)(sec A + 1) which equals sec² A - 1, we get:

[sec A (sec A + 1) + sec A (sec A - 1)] / (sec² A - 1)

Expanding the numerator:

(sec² A + sec A + sec² A - sec A) / (sec² A - 1) = 2 sec² A / (sec² A - 1)

Using the trigonometric identity sec² A - 1 = tan² A:

2 sec² A / tan² A

Since sec A = 1 / cos A and tan A = sin A / cos A, this can be written as:

2 (1 / cos² A) / (sin² A / cos² A) = 2 / sin² A = 2 cosec² A

This matches the given expression for x.

Explanation

To find the ratio of boys to girls, first calculate the total number of boys and the total number of girls across all divisions and standards from the given table.

Total number of boys = 20 + 15 + 30 + 30 + 30 + 30 = 155 Total number of girls = 30 + 20 + 10 + 25 + 25 + 20 = 130

The ratio of boys to girls is: Ratio = 155 : 130

Dividing both terms by their common factor of 5 simplifies the ratio to: 31 : 26

Explanation

To find the number of students who scored marks 20 or more but less than 40, we subtract the number of students who scored 40 and above from the number of students who scored 20 and above.

From the table: Number of students who scored 20 and above = 56 Number of students who scored 40 and above = 25

Therefore, the number of students scoring 20 or more but less than 40 is: 56 - 25 = 31

Explanation

The GDP of the country at the end of 2013 is equivalent to the GDP at the beginning of 2014, which is given as 5 trillion.

To find the GDP at the beginning of 2015, we need to calculate the GDP at the end of 2014, which is when the growth rate of 2014 has been applied.

Using the growth rate for the year 2014, which is 8%:

GDP at the beginning of 2015 = GDP at the beginning of 2014 × (1 + Growth rate of 2014 / 100)

GDP at the beginning of 2015 = 5 × (1 + 8 / 100) = 5 × 1.08 = 5.4 trillion

Explanation

Let the third number be 100.

The first number is 50% lesser than the third number: 100 - 50 = 50

The second number is 90% lesser than the third number: 100 - 90 = 10

To make the second number equal to the first number, it must be increased from 10 to 50.

The required increase in value is: 50 - 10 = 40

The percentage enhancement required is calculated as: (Increase / Second number) × 100 = (40 / 10) × 100 = 400%

Explanation

To reduce the fraction 2714/5074 to its lowest terms, we divide both the numerator and the denominator by their Highest Common Factor (HCF).

First, we find the HCF of 2714 and 5074 using the division method: 5074 = 2714 × 1 + 2360 2714 = 2360 × 1 + 354 2360 = 354 × 6 + 136 354 = 136 × 2 + 82 136 = 82 × 1 + 54 82 = 54 × 1 + 28 54 = 28 × 1 + 26 28 = 26 × 1 + 2 26 = 2 × 13 + 0

Dividing the numerator and the denominator by their initial common factor of 2 simplifies the fraction to 1357/2537. Next, we find the HCF of these remaining terms: 2537 = 1357 × 1 + 1180 1357 = 1180 × 1 + 177 1180 = 177 × 6 + 118 177 = 118 × 1 + 59 118 = 59 × 2 + 0

The HCF of 1357 and 2537 is 59. This means the overall HCF of 2714 and 5074 is 2 × 59 = 118.

Dividing both the numerator and the denominator of the original fraction by 118 gives: 2714 ÷ 118 = 23 5074 ÷ 118 = 43

This results in the reduced fraction 23/43.

Explanation

To find the value of cosec 120°, we use the reciprocal trigonometric identity:

cosec θ = 1 / sin θ

Therefore:

cosec 120° = 1 / sin 120°

Using the angle subtraction formula in the second quadrant, where sine is positive:

sin 120° = sin(180° - 60°) = sin 60°

Since the standard value of sin 60° is √3/2, substituting this in gives:

cosec 120° = 1 / (√3/2) = 2/√3

Explanation

The circumference of the base of the cylinder is given by the formula C = 2πr. Given that the circumference is 22 cm: 2 × (22/7) × r = 22 r = 7/2 cm

The volume of a cylinder is given by the formula V = πr²h. Substituting the known values: (22/7) × (7/2)² × h = 770 (77/2) × h = 770 h = 20 cm

The curved surface area of a cylinder is calculated using the formula CSA = 2πrh, which is also equal to the circumference of the base multiplied by the height: CSA = C × h CSA = 22 × 20 = 440 cm²

Explanation

The student's marks in the five subjects are 70, 55, 70, 55, and 55.

To find the average of these marks, the sum of all marks is divided by the total number of subjects:

Sum of marks = 70 + 55 + 70 + 55 + 55 = 305

Number of subjects = 5

Average marks = 305 ÷ 5 = 61

Since five points are to be deducted from this average due to poor attendance, the net average marks are:

Net average marks = 61 - 5 = 56

Explanation

The sum of the measures of all interior angles of a polygon with n sides is given by the formula (n - 2) × 180°. Substituting n = 14 into this formula:

Sum = (14 - 2) × 180° Sum = 12 × 180° Sum = 2160°

Explanation

Since the policeman and the thief run for the same duration of time until the chase ends, the distance covered by each is directly proportional to their respective speeds.

Ratio of speeds of policeman to thief = 7 : 5

This means the ratio of distance covered by the policeman to the distance covered by the thief is also 7 : 5.

The difference in their distances represents the initial gap of 350 meters between them.

Difference in ratio units = 7 - 5 = 2 units

Since 2 units correspond to 350 meters, 1 unit corresponds to 175 meters.

The distance run by the thief is 5 units: Distance = 5 × 175 = 875 meters

Explanation

A completes 75% of the work in 25 days, which means A's daily rate of work is 75% ÷ 25 = 3% of the work per day.

The remaining 25% of the work is completed by A and B together in 5 days. This means their joint daily rate of work is 25% ÷ 5 = 5% of the work per day.

Subtracting A's daily work rate from their joint daily work rate gives B's daily work rate: 5% - 3% = 2% of the work per day.

Since B completes 2% of the work in one day, the number of days B alone would take to complete 100% of the work is: 100% ÷ 2% = 50 days.

Explanation

For an arithmetic progression of n consecutive even integers with an average A, the highest integer in the series is given by the formula:

Highest integer = A + (n - 1)

Substituting the given values, where the average A = 60 and the number of terms n = 29:

Highest integer = 60 + (29 - 1) = 60 + 28 = 88

This directly yields the highest integer in the sequence.

Explanation

To find the expression that must be added, subtract the first expression from the second expression.

First, simplify the target expression: 4(2x - 3y) + 5(x + 4y) = 8x - 12y + 5x + 20y Combining like terms gives: (8x + 5x) + (-12y + 20y) = 13x + 8y

Next, simplify the given expression: 5(2x - y) = 10x - 5y

Subtract the given expression from the target expression to find the required addition: (13x + 8y) - (10x - 5y) = 13x + 8y - 10x + 5y Grouping like terms: (13x - 10x) + (8y + 5y) = 3x + 13y

Explanation

The compound inequality can be split into two separate inequalities:

First: 3(2 - 3x) < 2 - 3x Second: 2 - 3x ≥ 4x - 6

Solving the first inequality: 6 - 9x < 2 - 3x 4 < 6x x > 2/3

Solving the second inequality: 2 - 3x ≥ 4x - 6 8 ≥ 7x x ≤ 8/7

Combining both inequalities, the range for x is: 2/3 < x ≤ 8/7

In decimal form, this is approximately: 0.67 < x ≤ 1.14

The value 1 satisfies this inequality as it lies strictly between 2/3 and 8/7.

Explanation

We start with the given expression: X = sec²A + cosec²A

Using the reciprocal trigonometric identities, we can rewrite the terms as: sec²A = 1/cos²A cosec²A = 1/sin²A

Substituting these into the original equation gives: X = (1/cos²A) + (1/sin²A)

By taking a common denominator, we combine the fractions: X = (sin²A + cos²A) / (sin²A cos²A)

Applying the Pythagorean identity sin²A + cos²A = 1: X = 1 / (sin²A cos²A)

This expression can be rewritten as a product of two fractions: X = (1/cos²A) × (1/sin²A)

Substituting the reciprocal definitions back into the equation yields: X = sec²A cosec²A

Explanation

The effective annual rate of interest is calculated using the formula:

E = (1 + r/n)ⁿ - 1

where r represents the nominal annual interest rate and n is the number of compounding periods per year.

Given that the nominal annual rate of interest r is 15% (or 0.15 in decimal form) and the interest is payable half-yearly, the compounding occurs twice a year, so n = 2. Substituting these values into the formula:

E = (1 + 0.15/2)² - 1

E = (1 + 0.075)² - 1

E = (1.075)² - 1

E = 1.155625 - 1

E = 0.155625

Converting this value to a percentage yields:

E = 0.155625 × 100% = 15.56%

Explanation

The effective discount for two successive discounts of d₁% and d₂% is given by the formula:

Effective discount = d₁ + d₂ - (d₁ × d₂)/100

Substituting the given values, d₁ = 25 and d₂ = 10:

Effective discount = 25 + 10 - (25 × 10)/100 = 35 - 2.5 = 32.5%

Explanation

For a quadratic equation ax² + bx + c = 0 to have real and distinct roots, its discriminant D = b² - 4ac must be greater than zero (D > 0).

Applying this formula to the equation 3x² - 6x + 2 = 0, where a = 3, b = -6, and c = 2:

D = (-6)² - 4 × 3 × 2 D = 36 - 24 D = 12

Since the discriminant is 12, which is greater than zero, the equation has real and distinct roots.

Explanation

In a 30°-60°-90° right-angled triangle, the sides opposite to the angles 30°, 60°, and 90° are always in the ratio 1 : √3 : 2. Since the side opposite the 30° angle is given as 9 cm, the side opposite the 60° angle is calculated directly as 9 × √3 = 9√3 cm.

Alternatively, using the Law of Sines for any triangle with these angles:

a/sin(A) = b/sin(B)

Substituting A = 30°, B = 60°, and a = 9 cm:

9/sin(30°) = b/sin(60°)

9/(1/2) = b/(√3/2)

18 = 2b/√3

b = 9√3 cm

Explanation

The median AD of a triangle ABC is the line segment joining the vertex A to the mid-point D of the opposite side BC.

First, the coordinates of the mid-point D of BC are calculated using the mid-point formula: D = ((x₁ + x₂)/2, (y₁ + y₂)/2)

Substituting the coordinates of B(-4, 0) and C(-2, 2): D = ((-4 + (-2))/2, (0 + 2)/2) = (-3, 1)

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

Using A(-5, 4) and D(-3, 1): y - 4 = ((1 - 4) / (-3 - (-5))) × (x - (-5)) y - 4 = (-3 / 2) × (x + 5) 2(y - 4) = -3(x + 5) 2y - 8 = -3x - 15 3x + 2y = -7

Explanation

Let the cost price of the watch to the wholesaler be C.

The wholesaler sells the watch to the retailer at a gain of 37%. Thus, the cost price for the retailer is: Cost Price for Retailer = C × (1 + 37/100) = 1.37 × C

The retailer then sells the watch to the customer at a loss of 25%. Thus, the price paid by the customer is: Price Paid by Customer = 1.37 × C × (1 - 25/100) = 1.37 × C × 0.75

We are given that the customer paid Rs 2620.125. Setting up the equation: 1.37 × C × 0.75 = 2620.125

Using the fractional representation of 0.75 as 3/4: C × 1.37 × 3/4 = 2620.125 C × 4.11 / 4 = 2620.125 C × 4.11 = 10480.5 C = 10480.5 / 4.11 C = 2550

Explanation

Let the present ages of Rasika and Shami be 7x and 5x respectively.

According to the question, after 17 years, the ratio of their ages becomes 12:11. This can be represented by the equation:

(7x + 17) / (5x + 17) = 12 / 11

By cross-multiplying the terms:

11 × (7x + 17) = 12 × (5x + 17)

77x + 187 = 60x + 204

77x - 60x = 204 - 187

17x = 17

x = 1

Substituting the value of x, Rasika's present age is 7x = 7 × 1 = 7 years.

Explanation

By trigonometric definition, the tangent of a sum of two angles is given by:

tan(A + B) = sin(A + B) / cos(A + B)

Using the sum formulas for sine and cosine:

sin(A + B) = sinA cosB + cosA sinB cos(A + B) = cosA cosB - sinA sinB

Substituting these into the tangent ratio:

tan(A + B) = (sinA cosB + cosA sinB) / (cosA cosB - sinA sinB)

Dividing both the numerator and the denominator by cosA cosB yields:

tan(A + B) = ((sinA cosB / cosA cosB) + (cosA sinB / cosA cosB)) / ((cosA cosB / cosA cosB) - (sinA sinB / cosA cosB))

tan(A + B) = (tanA + tanB) / (1 - tanA tanB)

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 given values (x₁, y₁) = (7, 7), (x₂, y₂) = (k, -5), and d = 13: 13 = √((k - 7)² + (-5 - 7)²)

Squaring both sides to eliminate the square root: 169 = (k - 7)² + (-12)² 169 = (k - 7)² + 144 (k - 7)² = 25

Taking the square root on both sides: k - 7 = ±5

This gives two possible values for k: k = 7 + 5 = 12 k = 7 - 5 = 2

Among the choices, 2 is the correct value.