SSC CHSL Tier 1 2012 Quantitative Aptitude Questions with Solutions

Explanation

Using the formula Speed = Distance / Time, we find the speed of the train when it passes the tree. Since the tree has negligible length, the distance covered is equal to the length of the train (150 meters):

Speed = 150 / 12 = 12.5 meters per second

To pass a 250-meter-long tunnel, the train must cover a total distance equal to the sum of its own length and the tunnel's length:

Total Distance = 150 + 250 = 400 meters

Using the formula Time = Distance / Speed, the time taken to cross the tunnel is:

Time = 400 / 12.5 = 32 seconds

Explanation

The formula for the total amount in compound interest is: A = P × (1 + R/100)ᵗ

Given the Principal P = ₹1800, Rate R = 10%, and Compound Interest CI = ₹378. The total Amount A is: A = P + CI = 1800 + 378 = 2178

Substituting these values into the formula: 2178 = 1800 × (1 + 10/100)ᵗ 2178 / 1800 = (11/10)ᵗ 1.21 = (1.1)ᵗ

Since 1.21 is the square of 1.1: (1.1)² = (1.1)ᵗ

Comparing the exponents, we get t = 2.

Explanation

The simple interest SI obtained from a principal P invested at an annual rate of interest R for T years is given by the formula SI = (P × R × T) / 100.

The simple interest earned is the difference between the final amount and the principal: SI = ₹3300 - ₹2500 = ₹800

Substituting the known values into the simple interest formula: 800 = (2500 × 8 × T) / 100 800 = 200 × T T = 800 / 200 T = 4 years

Explanation

For a standard three-dimensional cube, the geometric properties are defined by its vertices, edges, and faces.

A cube has 8 vertices, which represents the value of A. It has 12 edges, which represents the value of B. It has 6 faces, which represents the value of C.

To find the sum of these properties, we add them together:

A + B + C = 8 + 12 + 6 = 26

Explanation

The total area of the square field is calculated by squaring the length of its side: Area of square = 10² = 100 m²

Since the horse is tied at one corner of the square, the area it can graze forms a quadrant of a circle with a radius equal to the length of the rope (r = 7 m). The angle at the corner of a square is 90°.

Area grazed = (1/4) × π × r² Area grazed = (1/4) × (22/7) × 7 × 7 Area grazed = (1/4) × 22 × 7 Area grazed = 154 / 4 = 38.5 m²

The area of the square field left ungrazed is the difference between the total area of the field and the grazed area: Area left ungrazed = 100 - 38.5 = 61.5 m²

Explanation

Area of a rectangle is given by the formula: Area = length × width

Initially, for rectangle ABCD, the area (A₁) is: A₁ = AB × AD = h² × 3p = 3ph²

When h is doubled, the new value of h is 2h, making the new length: AB' = (2h)² = 4h²

When p is halved, the new value of p is p/2, making the new width: AD' = 3(p/2) = 1.5p

The new area (A₂) is: A₂ = AB' × AD' = 4h² × 3(p/2) = 6ph²

Comparing the new area with the initial area: A₂ = 2 × (3ph²) = 2A₁

This shows that the area is doubled.

Explanation

The volume of a rectangular cuboid is given by the formula: Volume = length × width × height

To find the number of bricks required, all dimensions must be converted to the same unit. Converting the dimensions of the wall from meters to centimeters: Length of the wall = 8 m = 800 cm Height of the wall = 6 m = 600 cm Thickness of the wall = 22.5 cm

The volume of the wall is: 800 cm × 600 cm × 22.5 cm = 10,800,000 cm³

The dimensions of one brick are given as 25 cm × 11.25 cm × 6 cm. The volume of one brick is: 25 cm × 11.25 cm × 6 cm = 1687.5 cm³

The number of bricks required is calculated by dividing the total volume of the wall by the volume of a single brick: Number of bricks = Volume of the wall ÷ Volume of one brick Number of bricks = 10,800,000 ÷ 1687.5 = 6400

Alternatively, this can be solved directly by simplifying the expression: (800 × 600 × 22.5) ÷ (25 × 11.25 × 6) Dividing 800 by 25 equals 32. Dividing 600 by 6 equals 100. Dividing 22.5 by 11.25 equals 2. Multiplying these quotients together gives: 32 × 100 × 2 = 6400

Explanation

To find the value of the given expression, divide both the numerator and the denominator by x:

2x / (3x² - 5x + 3) = 2 / (3x - 5 + 3/x)

Rearranging the terms in the denominator gives:

2 / (3(x + 1/x) - 5)

Substitute the given value x + 1/x = 5 into this simplified expression:

2 / (3(5) - 5) = 2 / (15 - 5) = 2/10 = 1/5

Explanation

By expanding the given expression: p(p² + 3p + 3) = p³ + 3p² + 3p

Using the algebraic identity for the cube of a binomial: (p + 1)³ = p³ + 3p² + 3p + 1

Subtracting 1 from both sides gives: (p + 1)³ - 1 = p³ + 3p² + 3p

Substituting p = 99 into this equivalent expression: (99 + 1)³ - 1 = 100³ - 1

Since 100³ is 1,000,000, the final calculation is: 1,000,000 - 1 = 999,999

Explanation

The algebraic identity for the cube of a binomial is: (3a - 4)³ = 27a³ - 108a² + 144a - 64

The expression can be rewritten by isolating this perfect cube: 27a³ - 108a² + 144a - 317 = (3a - 4)³ - 253

Substituting a = 4/3 into the equation: 3a - 4 = 3(4/3) - 4 = 4 - 4 = 0

Therefore, the value of the expression is: 0³ - 253 = -253

Explanation

The algebraic identity for the sum of cubes is: a³ + b³ + c³ - 3abc = (a + b + c)(a² + b² + c² - ab - bc - ca)

The denominator of the given expression can be rewritten by factoring out a negative sign: ab + bc + ca - (a² + b² + c²) = -(a² + b² + c² - ab - bc - ca)

Thus, the given expression simplifies to: (a + b + c)(a² + b² + c² - ab - bc - ca) / -(a² + b² + c² - ab - bc - ca) = -(a + b + c)

Substituting the values a = 1, b = 2, and c = -3: -(1 + 2 + (-3)) = 0

Explanation

The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides. Since DE is parallel to BC, ΔADE is similar to ΔABC.

We are given: AD/BD = 2/3

Since AB = AD + BD, the ratio of the side AD to AB is: AD/AB = AD / (AD + BD) = 2 / (2 + 3) = 2/5

Using the property of similar triangles, the ratio of the area of ΔADE to the area of ΔABC is: Area(ΔADE) / Area(ΔABC) = (AD/AB)² = (2/5)² = 4/25

If Area(ΔABC) is represented as 25 units, then Area(ΔADE) is 4 units.

The area of the trapezium DECB is the difference between the areas of the two triangles: Area(trapezium DECB) = Area(ΔABC) - Area(ΔADE) = 25 - 4 = 21 units

The ratio of the area of trapezium DECB to the area of ΔABC is: Area(trapezium DECB) : Area(ΔABC) = 21 : 25

Explanation

The centroid of a triangle is the point of intersection of its three medians. A fundamental property of the centroid is that it divides the triangle into three smaller triangles of equal area: ΔAGB, ΔBGC, and ΔCGA.

Therefore, the area of ΔAGB is exactly one-third of the total area of ΔABC.

Area(ΔAGB) = Area(ΔABC) ÷ 3

Substituting the given value of Area(ΔABC) = 108 cm²:

Area(ΔAGB) = 108 ÷ 3 = 36 cm²

Explanation

The centroid of a triangle divides each median in the ratio 2:1, measuring from the vertex to the midpoint of the opposite side. This means that the distance from a vertex to the centroid is two-thirds of the total length of that median.

Therefore, the length of each median is 3/2 times the distance from the corresponding vertex to the centroid.

Calculating the lengths of the medians: First median = (3/2) × 4 = 6 cm Second median = (3/2) × 6 = 9 cm Third median = (3/2) × 8 = 12 cm

Explanation

Picture the balloon's center O at height h above the observer, with the line to the center making angle β with the ground. The balloon of radius r subtends angle α at the observer's eye, so the half-angle α/2 satisfies sin(α/2) = r divided by the distance from the observer to the center, giving that distance as r cosec(α/2). The height of the center is then this distance times sin β, so h = r cosec(α/2) sin β.

Explanation

By using the trigonometric identity cosec(90° - θ) = sec θ, the given equation can be simplified as:

sec θ + sec θ = 4 2 sec θ = 4 sec θ = 2

Since θ is an acute angle (0 < θ < 90°), sec θ = 2 corresponds to cos θ = ½, which gives θ = 60°.

Substituting θ = 60° into tan θ: tan 60° = √3

Explanation

Using standard trigonometric identities, we can simplify the terms of the expression.

First, we know that 1/cosec²θ = sin²θ. Therefore: 9/cosec²θ = 9sin²θ

Second, using the fundamental identity 1 + tan²θ = sec²θ, we have: 5/(1+tan²θ) = 5/sec²θ

Since 1/sec²θ = cos²θ, this simplifies to: 5cos²θ

Substituting these back into the original expression gives: 9sin²θ + 4cos²θ + 5cos²θ

Combining the cos²θ terms yields: 9sin²θ + 9cos²θ

Factoring out 9, we get: 9(sin²θ + cos²θ)

Using the Pythagorean identity sin²θ + cos²θ = 1: 9(1) = 9

Explanation

The given equation is |x + 2| = √4.

First, simplify the right side of the equation. The principal square root of 4 is 2, so the equation becomes: |x + 2| = 2

By definition of the absolute value, this equation can be split into two linear equations: x + 2 = 2 or x + 2 = -2

Solving the first equation: x = 2 - 2 x = 0

Solving the second equation: x = -2 - 2 x = -4

The solution set consists of 0 and -4.

Explanation

Let the required number be x.

x + 5/7 = 1 x = 1 - 5/7 x = 2/7

To find what percentage of 5/7 this number is, we divide 2/7 by 5/7 and multiply by 100: Percentage = (2/7) ÷ (5/7) × 100% Percentage = 2/5 × 100% = 40%

Explanation

To find the time taken by the train to cross the platform, the total distance to be covered 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

Next, convert the speed of the train from km/hr to m/s by multiplying with 5/18: Speed = 55 × 5/18 = 275/18 m/s

Using the formula: Time = Distance ÷ Speed Time = 770 ÷ (275/18) Time = (770 × 18) ÷ 275 Time = (70 × 18) ÷ 25 Time = 1260 ÷ 25 = 50.4 seconds

Explanation

The work done by 12 men is equal to the work done by 18 women. This gives the efficiency ratio: 12 men = 18 women 1 man = 1.5 women

We need to find the time taken by 4 men and 8 women to complete the same work. Converting men to women: 4 men + 8 women = (4 × 1.5) women + 8 women = 6 women + 8 women = 14 women

Using the work-time formula, M₁ × D₁ = M₂ × D₂, where M is the number of workers and D is the number of days: 18 women × 7 days = 14 women × D days 126 = 14 × D D = 126 / 14 = 9 days

Explanation

The share of payment for work done together is distributed in the ratio of the individual efficiencies (or rates of work) of the workers.

Since efficiency is inversely proportional to the time taken to complete the work individually:

Efficiency of A : Efficiency of B = 1/30 : 1/40 = 4 : 3

The total payment of ₹7,000 is divided in the ratio 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 proportional to the square of its edge length, as represented by the formula A = 6a² where a is the edge length.

Since the surface area is a two-dimensional quantity, an increase of 50% in the edge length results in a successive percentage increase in the surface area. Using the successive percentage change formula:

Net percentage increase = x + y + (x × y) / 100

Substituting x = 50 and y = 50:

Net percentage increase = 50 + 50 + (50 × 50) / 100 = 100 + 25 = 125%

Explanation

The volume of a rectangular tank is given by the formula:

Volume = Base Area × Depth

First, convert the area of the base from square centimeters to square meters. Since 1 square meter is equal to 10000 square centimeters, we have:

Base Area = 6500 ÷ 10000 = 0.65 m²

Now, substitute the volume and the base area into the formula to find the depth:

2.6 = 0.65 × Depth

Depth = 2.6 ÷ 0.65 = 4 m

Explanation

The true discount on ₹1,860 for 3 years is equal to the simple interest on ₹1,500 for 3 years at the same rate. Let the rate of interest be R% per annum.

The formula for Simple Interest (SI) is: SI = (Principal × Rate × Time) / 100 SI = (1500 × R × 3) / 100 = 45R

The formula for True Discount (TD) is: TD = (Amount × Rate × Time) / (100 + (Rate × Time)) TD = (1860 × R × 3) / (100 + 3R) = 5580R / (100 + 3R)

Since the true discount is equal to the simple interest: 45R = 5580R / (100 + 3R)

Dividing both sides by R (since R ≠ 0): 45 = 5580 / (100 + 3R) 45 × (100 + 3R) = 5580 4500 + 135R = 5580 135R = 1080 R = 1080 / 135 R = 8%

Explanation

The present value of the credit payment due 1 year hence is calculated by discounting the amount at the given interest rate of 10% per annum.

Present Value = Amount ÷ (1 + R/100)

Substituting the given values: Present Value = 14300 ÷ (1 + 10/100) = 14300 ÷ 1.1 = 13000

The cash equivalent of the credit price is ₹13,000. Since the cash down value of the article is ₹12,500, the profit gained by the trader in terms of present value is:

Profit = Present Value - Cash Down Value Profit = 13000 - 12500 = 500

Explanation

The marked price of the book is ₹165 and the selling price is ₹132.

The discount is calculated as: Discount = Marked Price - Selling Price Discount = 165 - 132 = ₹33

The percentage discount allowed is given by the formula: Discount % = (Discount / Marked Price) × 100 Discount % = (33 / 165) × 100 Discount % = 1/5 × 100 = 20%

Explanation

Let the initial ratio of milk and water be M : W = 2 : 1. The total mixture is 60 litres, which represents 2 + 1 = 3 units. This means 1 unit is equal to 20 litres.

The initial quantity of milk is 2 × 20 = 40 litres, and the initial quantity of water is 1 × 20 = 20 litres.

Since only water is added to the mixture, the quantity of milk remains unchanged at 40 litres.

The new ratio of milk to water is 1 : 2.

If 1 unit of milk in the new mixture corresponds to 40 litres, then 2 units of water in the new mixture must correspond to 2 × 40 = 80 litres.

The quantity of water to be further added is the difference between the new and initial quantities of water: 80 litres − 20 litres = 60 litres.

Explanation

The ratio consists of the mixed fractions: 2 ⅓ : 3 ¼ : 1 ⅙

Converting these mixed fractions into improper fractions: 2 ⅓ = 7/3 3 ¼ = 13/4 1 ⅙ = 7/6

The ratio becomes: 7/3 : 13/4 : 7/6

To simplify and eliminate the fractional denominators, multiply each term by the Least Common Multiple (LCM) of the denominators 3, 4, and 6, which is 12: (7/3) × 12 = 28 (13/4) × 12 = 39 (7/6) × 12 = 14

This yields the simplified ratio 28:39:14.

Explanation

Let the number of men, women, and boys employed in the factory be 5x, 7x, and 10x respectively.

The total number of workers is given as 660: 5x + 7x + 10x = 660 22x = 660 x = 30

Substituting the value of x: Number of men = 5 × 30 = 150 Number of women = 7 × 30 = 210 Number of boys = 10 × 30 = 300

Explanation

Average Monthly Income is calculated using the relation between total income, total expenditure, and total savings over the twelve months of the year.

Total Annual Expenditure = (5 × 12,000) + (7 × 13,000) = 60,000 + 91,000 = ₹1,51,000

Total Annual Income = Total Annual Expenditure + Total Savings = 1,51,000 + 29,000 = ₹1,80,000

Average Monthly Income = Total Annual Income / 12 = 1,80,000 / 12 = ₹15,000

Explanation

Let the length of the journey be d km. Using the relation time = distance / speed, the time difference between the two speeds is:

d/35 - d/40 = 15/60

Simplifying the time difference of 15 minutes to ¼ hours:

d/35 - d/40 = ¼

Multiplying the entire equation by 280 (the least common multiple of 35 and 40) to clear the denominators:

8d - 7d = 70

d = 70

Explanation

Three years ago, the sum of the ages of the 5 family members was 5 × 17 = 85 years.

In the 3 years that have passed, each of the 5 members has grown 3 years older. Therefore, the sum of their ages today is 85 + (5 × 3) = 100 years.

With the addition of the baby, the family now has 6 members. The current average age of the family is the same as before, which is 17 years. The sum of the ages of all 6 members today is 6 × 17 = 102 years.

The age of the baby is the difference between the current sum of all 6 members and the current sum of the original 5 members, which is 102 - 100 = 2 years.

Explanation

The ratio of materials, labour, and overheads is 4:3:2. Let the costs of these three components be 4x, 3x, and 2x respectively.

Since the cost of labour is ₹45: 3x = 45 x = 15

The total cost of manufacturing the article is: 4x + 3x + 2x = 9x 9 × 15 = ₹135

The selling price of the article is ₹180. The profit earned is: Profit = Selling Price - Total Cost = 180 - 135 = ₹45

The profit percentage is: Profit % = (Profit / Total Cost) × 100 = (45 / 135) × 100 = (1/3) × 100 = 33 ⅓%

Explanation

Let the Cost Price of the shirt be x.

The Selling Price at 4% profit is 1.04x. The Selling Price at 5% profit is 1.05x.

The difference between these two selling prices is: 1.05x - 1.04x = 3 0.01x = 3 x = 3 / 0.01 = 300

Thus, the Cost Price of the shirt is ₹300.

The first Selling Price at 4% profit is: 300 × 1.04 = ₹312

The second Selling Price at 5% profit is: 300 × 1.05 = ₹315

Explanation

Let CP be the cost price of the toy.

The difference in profit percentages is: 20% - 16% = 4%

This 4% increase in the profit margin corresponds directly to the ₹10 increase in the selling price.

Using the relationship between the percentage change and the actual price change: 4% of CP = 10 0.04 × CP = 10 CP = 10 / 0.04 = 250

Explanation

We translate the statement mathematically as: (2/3) × (30/100) × 4a

First, simplify 2/3 of 30%: (2/3) × (30/100) = (2/3) × (3/10) = 2/10 = 1/5

Next, multiply this result by 4a: (1/5) × 4a = 4a/5

Explanation

Let the required number be x.

According to the given condition, 65% of the number is 21 less than four-fifth of the number.

Four-fifth of the number is written as (4/5)x, which is equivalent to 80% of x or 0.8x.

We can set up the equation: 0.8x - 0.65x = 21

Simplifying the left-hand side: 0.15x = 21

To find x, divide 21 by 0.15: x = 21 ÷ 0.15 x = 2100 ÷ 15 x = 140

Explanation

The length of the bridge can be calculated directly using the distance formula:

Distance = Speed × Time

First, convert the time from minutes to hours:

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

Now, substitute the speed of 6 km/hr and the time of 1/3 hour into the formula:

Distance = 6 km/hr × 1/3 hour = 2 km

Explanation

To find the distance covered in meters, the speed must first be converted from kilometers per hour (km/hr) to meters per second (m/s) using the conversion factor 5/18:

Speed = 36 × (5/18) m/s = 10 m/s

Using the standard distance formula:

Distance = Speed × Time

Substituting the converted speed and time into the formula:

Distance = 10 m/s × 32 s = 320 meters

Explanation

Between year 2 and year 3 the amount grows only by the interest earned in that single year, so the rate equals the increase divided by the amount at the start of that year. Here the amount rises from 500 to 540, an increase of 40, which is 40 out of 500, or 8 percent per annum.

Explanation

Using the method of allegation:

If the entire amount of ₹2,000 was lent at 5%, the interest earned in one year would be 5% of 2,000 = ₹100.

If the entire amount was lent at 4%, the interest earned would be 4% of 2,000 = ₹80.

The actual interest received is ₹92.

Applying the allegation rule, the ratio of the amount lent at 5% to the amount lent at 4% is: (92 - 80) : (100 - 92) = 12 : 8 = 3 : 2

The amount of money lent at 5% is: (3 / (3 + 2)) × 2,000 = (3/5) × 2,000 = 1,200

Explanation

Let the length of the rectangular circuit board be L cm. The perimeter p of a rectangle is given by the formula: p = 2(w + L)

Rearranging this formula to express L in terms of p and w: w + L = p/2 L = p/2 - w

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

Substituting the expression for L into the area formula: k = w × (p/2 - w) k = pw/2 - w²

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

Rearranging the terms into standard quadratic form: 2w² - pw + 2k = 0

Explanation

The volume of a sphere depends on the cube of its radius. If the radius grows by 50 percent, the new radius is 1.5 times the original, so the new volume is 1.5 cubed, or 3.375 times the original volume. That is an increase of 2.375 times, which as a percentage is 237.5 percent.

Explanation

Let R and H be the radius and height of the cylindrical container, and r and h be the radius and height of the conical portion of the ice-cream cone.

The volume of the cylindrical container is: V_cyl = π × R² × H

Substituting R = 6 cm and H = 15 cm: V_cyl = π × 6² × 15 = 540π cm³

The total volume of one ice-cream cone, consisting of a conical portion of height h = 4r and a hemispherical top of radius r, is: V_cone = (1/3) × π × r² × h + (2/3) × π × r³

Substituting h = 4r into the equation: V_cone = (1/3) × π × r² × (4r) + (2/3) × π × r³ V_cone = (4/3) × π × r³ + (2/3) × π × r³ V_cone = 2π × r³

The volume of 10 such ice-cream cones is equal to the volume of the cylindrical container: 10 × 2π × r³ = 540π 20π × r³ = 540π r³ = 27 r = 3 cm

Explanation

The volume of a hemispherical bowl of radius R is given by the formula V_bowl = (2/3) × π × R³. Given the internal radius R = 15 cm: V_bowl = (2/3) × π × 15³ = (2/3) × π × 3375 = 2250π cm³

The volume of a cylindrical bottle of radius r and height h is given by the formula V_bottle = π × r² × h. Given the diameter of the bottle is 5 cm, its radius r = 2.5 cm. The height h = 6 cm: V_bottle = π × 2.5² × 6 = π × 6.25 × 6 = 37.5π cm³

The number of bottles necessary to empty the bowl is obtained by dividing the volume of the hemispherical bowl by the volume of a single cylindrical bottle: Number of bottles = V_bowl / V_bottle = 2250π / 37.5π = 60

Explanation

If a + b + c = 0, then the algebraic identity states that a³ + b³ + c³ = 3abc.

Letting a = 1.7, b = 3.9, and c = -5.6, we find: a + b + c = 1.7 + 3.9 + (-5.6) = 5.6 - 5.6 = 0

Therefore, the given expression can be written as: (1.7)³ + (3.9)³ - (5.6)³ = (1.7)³ + (3.9)³ + (-5.6)³ = 3 × (1.7) × (3.9) × (-5.6) = 19.89 × (-5.6) = -111.384

Explanation

Let the given infinite nested radical expression be x:

x = √(6 + √(6 + √(6 + …)))

Since the pattern repeats infinitely, the nested term inside the outermost square root is also equal to x:

x = √(6 + x)

Squaring both sides of the equation to eliminate the radical gives:

x² = 6 + x

Rearranging this into a standard quadratic equation:

x² - x - 6 = 0

Factoring the quadratic equation:

(x - 3)(x + 2) = 0

This yields two possible values: x = 3 or x = -2. Since the principal square root is always non-negative, the value of the expression must be positive, which leaves x = 3.

Explanation

Given that x = (√3 + √2)/(√3 - √2)

To rationalize the denominator, we multiply the numerator and the denominator by (√3 + √2): x = (√3 + √2)² / ((√3)² - (√2)²) x = (3 + 2 + 2√6) / (3 - 2) x = 5 + 2√6

The reciprocal of x is: 1/x = 1 / (5 + 2√6)

Rationalizing the denominator of 1/x by multiplying the numerator and denominator by (5 - 2√6): 1/x = (5 - 2√6) / (5² - (2√6)²) 1/x = (5 - 2√6) / (25 - 24) 1/x = 5 - 2√6

To find the value of x + 1/x: x + 1/x = (5 + 2√6) + (5 - 2√6) x + 1/x = 10

Explanation

Using the algebraic identity (a - b)² = a² - 2ab + b², we expand the expression (c - 3)²:

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

By comparing this expanded form with the given equation:

c² + kc + 9 = c² - 6c + 9

Comparing the coefficients of c on both sides:

k = -6