SSC CHSL Tier 1 2023 Quantitative Aptitude Questions with Solutions

Explanation

In the first alloy, the ratio of Aluminium to Zinc is 5 : 6, giving a total of 11 parts. Quantity of Aluminium = 242 × 5/11 = 110 kg Quantity of Zinc = 242 × 6/11 = 132 kg

In the second alloy, the ratio of Aluminium to Zinc is 3 : 5, giving a total of 8 parts. Quantity of Aluminium = 144 × 3/8 = 54 kg Quantity of Zinc = 144 × 5/8 = 90 kg

By mixing these together, the total quantities in the new alloy are: Total Aluminium = 110 + 54 = 164 kg Total Zinc = 132 + 90 = 222 kg

The ratio of Aluminium to Zinc in the new alloy is: 164 : 222 = 82 : 111

Explanation

The time taken by two individuals to complete a work together is given by the relation: 1/T_together = 1/T_Sita + 1/T_Gita

Given that Sita and Gita together can do the job in 4 days (T_together = 4) and Sita alone can do it in 6 days (T_Sita = 6), we substitute these values into the formula to find the time taken by Gita alone (T_Gita): 1/4 = 1/6 + 1/T_Gita

Rearranging the terms to solve for Gita's rate: 1/T_Gita = 1/4 - 1/6

Using the least common multiple of 4 and 6, which is 12: 1/T_Gita = 3/12 - 2/12 1/T_Gita = 1/12

This yields: T_Gita = 12 days

Explanation

The formula to calculate the Selling Price (SP) from the Cost Price (CP) and profit percentage is: SP = CP × (1 + Profit%/100)

Substituting the given values CP = ₹24,000 and Profit% = 9%: SP = 24,000 × (1 + 9/100) = 24,000 × 1.09 = ₹26,160

The formula to calculate the discount percentage from the Marked Price (MP) and Selling Price (SP) is: Discount% = ((MP - SP) / MP) × 100

Substituting MP = ₹30,000 and SP = ₹26,160: Discount% = ((30,000 - 26,160) / 30,000) × 100 Discount% = (3,840 / 30,000) × 100 = 12.8%

Explanation

The volume of a cuboidal tank is calculated using the formula: Volume = length × width × depth

Given the dimensions of the tank: Length = 12 m Width = 9 m Depth = 7.5 m

Substituting these values into the formula: Volume = 12 × 9 × 7.5 = 810 m³

Since 1 cubic metre is equal to 1000 litres: Capacity = 810 × 1000 = 810000 litres

Explanation

The average of three numbers A, B, and C is given by the formula: Average = (A + B + C) / 3

Substituting the given values into the formula: 1/13 = (1/7 + 1/8 + C) / 3

Multiplying both sides by 3: 3/13 = 1/7 + 1/8 + C

First, simplify the sum of A and B by finding a common denominator: 1/7 + 1/8 = (8 + 7) / 56 = 15/56

Now, substitute this back to solve for C: 3/13 = 15/56 + C C = 3/13 - 15/56

To subtract these fractions, find the least common multiple of 13 and 56, which is 13 × 56 = 728: C = (3 × 56 - 15 × 13) / 728 C = (168 - 195) / 728 C = -27/728

Explanation

According to the Basic Proportionality Theorem, if a line is drawn parallel to one side of a triangle intersecting the other two sides, then it divides those two sides in the same ratio.

In triangle MNO, since AB is parallel to NO, the relation is: MA/AN = MB/BO

Substituting the given values into this ratio: 2.5/7.5 = 2.2/BO

Simplifying the fraction on the left side: 1/3 = 2.2/BO

Solving for BO: BO = 2.2 × 3 = 6.6

Explanation

Correct Answer: C

Using the trigonometric identity: tan²θ = sec²θ - 1

First, convert the mixed fraction of sec θ into an improper fraction: sec θ = 2 4/23 = (2 × 23 + 4)/23 = 50/23

Now, substitute this value into the identity: tan²θ = (50/23)² - 1 tan²θ = 2500/529 - 1 tan²θ = (2500 - 529)/529 tan²θ = 1971/529

Convert the improper fraction 1971/529 back into a mixed fraction: 1971 ÷ 529 = 3 with a remainder of 384 tan²θ = 3 384/529

Explanation

The speed of the current is calculated using the formula:

Speed of current = (Downstream speed - Upstream speed) / 2

Given: Downstream speed = 35 km/h Upstream speed = 25 km/h

Substituting these values:

Speed of current = (35 - 25) / 2 Speed of current = 10 / 2 Speed of current = 5 km/h

Explanation

To determine which set contains only divisors of 2520, we find the prime factorization of 2520:

2520 = 2³ × 3² × 5¹ × 7¹

For any number to be a divisor of 2520, its prime factorization must not contain any prime factors other than 2, 3, 5, and 7, and the exponents of these prime factors must be less than or equal to their respective exponents in the factorization of 2520 (which are 3, 2, 1, and 1).

Analyzing the elements of the correct set: The number 8 can be written as 2³. Since the exponent of 2 is 3, which is less than or equal to 3, 8 is a divisor of 2520. The number 9 can be written as 3². Since the exponent of 3 is 2, which is less than or equal to 2, 9 is a divisor of 2520. The number 7 can be written as 7¹. Since the exponent of 7 is 1, which is less than or equal to 1, 7 is a divisor of 2520.

Since 8, 9, and 7 are all divisors of 2520, this set satisfies the given condition.

Explanation

Let each of the two equal angles of the isosceles triangle be x.

The sum of these two equal angles is x + x = 2x.

The unequal angle is five times this sum, which is 5 × 2x = 10x.

Using the angle sum property of a triangle, the sum of all three interior angles is 180°: x + x + 10x = 180° 12x = 180° x = 15°

Explanation

Correct Answer: D

Let the cost price of the goods be CP.

The difference between a loss of 10% and a gain of 8% represents the total percentage change in the selling price, which is 10% + 8% = 18% of the cost price.

Using the relation: Difference in Selling Price = CP × (Gain% + Loss%) / 100

Substituting the given values: 270 = CP × 18 / 100

Solving for CP: CP = 270 × 100 / 18 CP = 15 × 100 CP = 1500

Explanation

Using complementary angle relationships in trigonometry, cos θ = sin(90° - θ).

Applying this to the terms in the expression: cos 65° = sin(90° - 65°) = sin 25° cos 55° = sin(90° - 55°) = sin 35°

Additionally, using the reciprocal identity for trigonometric functions: tan 50° cot 50° = 1

Substituting these values back into the given expression: (sin 25° / sin 25°) + (sin 35° / sin 35°) - 1 = 1 + 1 - 1 = 1

Explanation

The rate of the inlet pipe is 1/51 per hour and the rate of the outlet pipe is 1/76.5 (or 2/153) per hour. When both are opened together, the net rate of filling is 1/51 - 2/153 = 1/153 per hour, meaning the tank will be completely filled in 153 hours.

Explanation

Correct Answer: D

The relation between the Cost Price (CP) and the Marked Price (MP) when given a gain percent (g) and a discount percent (d) is given by the formula:

MP / CP = (100 + g) / (100 - d)

Substituting the given values, g = 5 and d = 25:

MP / CP = (100 + 5) / (100 - 25) MP / CP = 105 / 75 MP / CP = 7 / 5

The markup percentage is:

Markup % = ((MP - CP) / CP) × 100 Markup % = ((7 - 5) / 5) × 100 Markup % = (2 / 5) × 100 Markup % = 40%

Explanation

Using the co-function trigonometric identity, sin(90° - θ) = cos θ.

For θ = 25°: sin 65° = sin(90° - 25°) = cos 25°

Substituting sin 65° = cos 25° into the given expression: (cos²25° - sin²65°) / (cos²25° + sin²65°) = (cos²25° - cos²25°) / (cos²25° + cos²25°) = 0 / (2 cos²25°) = 0

Explanation

In a circle, the line segment drawn from the center perpendicular to a chord bisects the chord. This forms a right-angled triangle where the hypotenuse is the radius of the circle, one leg is the perpendicular distance from the center to the chord, and the other leg is half the length of the chord.

Applying the Pythagorean theorem to this right-angled triangle: Radius² = (Perpendicular distance)² + (Half of the chord length)²

Substituting the given values: Radius² = 5² + 12² Radius² = 25 + 144 Radius² = 169 Radius = √169 = 13 cm

Explanation

The total debt A discharged by t equal annual instalments of ₹I each at simple interest rate r% per annum is given by the formula:

A = t × I + I × r × t × (t - 1) / 200

In this problem, the instalment amount I is ₹800, the rate of interest r is 5% per annum, and the number of instalments t is 6. Substituting these values into the formula:

A = 6 × 800 + 800 × 5 × 6 × 5 / 200

A = 4800 + 40 × 15

A = 4800 + 600

A = 5400

Thus, the amount of debt that will be discharged is ₹5,400.

Explanation

From the given equation x² - 11x + 1 = 0, dividing both sides by x gives: x + 1/x = 11

Squaring both sides of this equation: (x + 1/x)² = 11² x² + 2 + 1/x² = 121 x² + 1/x² = 119

Squaring both sides again: (x² + 1/x²)² = 119² x⁴ + 2 + 1/x⁴ = 14161 x⁴ + 1/x⁴ = 14159

Multiplying this equation by x⁴ yields: x⁸ + 1 = 14159x⁴ x⁸ - 14159x⁴ = -1

Substituting this value into the required expression x⁸ - 14159x⁴ + 11: -1 + 11 = 10

Explanation

Using the algebraic identity for the square of a trinomial: (a + b + c)² = a² + b² + c² + 2(ab + bc + ac)

We can express the required value of a² + b² + c² as: a² + b² + c² = (a + b + c)² - 2(ab + bc + ac)

Substitute the given values into the equation: a + b + c = 16 ab + bc + ac = 81

Now, calculate the values: a² + b² + c² = 16² - 2(81) a² + b² + c² = 256 - 162 a² + b² + c² = 94

Explanation

The mean proportional (or geometric mean) between two numbers a and b is defined as the value x that satisfies the ratio a/x = x/b. This relation simplifies to the formula:

x = √(a × b)

Substituting the given values, a = 3 and b = 27, into the formula:

x = √(3 × 27) = √81 = 9

Explanation

The length of the direct common tangent to two circles touching each other externally is given by the formula:

PQ = 2√(r₁ × r₂)

where r₁ and r₂ are the radii of the two circles.

Given: r₁ = 9 cm r₂ = 4 cm

Substituting the values into the formula:

PQ = 2 × √(9 × 4) PQ = 2 × √36 PQ = 2 × 6 PQ = 12 cm

Explanation

Using the algebraic identity for the difference of squares, x² - y² = (x - y)(x + y), let x = a + b + c and y = a - b + c.

First, calculate the sum and the difference of the terms: x + y = (a + b + c) + (a - b + c) = 2a + 2c x - y = (a + b + c) - (a - b + c) = 2b

Substitute these back into the difference of squares formula: (a + b + c)² - (a - b + c)² = (2b)(2a + 2c) = 4ab + 4bc

Now, add the remaining term 4ac from the original expression: 4ab + 4bc + 4ac = 4(ab + bc + ac)

Explanation

To find the best discount for the customer, we maximize the equivalent single discount. The formula for the equivalent discount of two successive discounts d₁ and d₂ is d₁ + d₂ - (d₁ × d₂)/100.

When the sum of the two discounts is constant (d₁ + d₂ = 40% for all offers), the maximum discount is achieved when the product d₁ × d₂ is minimized.

For the third offer of 35% and 5%, the sum is 40% and the product is 35 × 5 = 175. This gives the smallest product among all the combinations, resulting in the highest equivalent discount of 40% - 1.75% = 38.25%. Therefore, the third offer is the most beneficial for the customer.

Explanation

Under the monthly reducing balance installment scheme, the relationship between the remaining principal P, the monthly installment E, and the monthly interest rate r is given by the formula:

P = E × [1 - (1 + r)⁻ⁿ] / r

Here, the cash price of the computer is ₹39,000. Under the plan, the cash down payment of ₹19,000 is made, leaving the balance principal to be financed under the monthly installment scheme as P = ₹19,000.

The monthly installment E is ₹4,200, and the number of installments n is 5.

Substituting these values into the formula:

19,000 = 4,200 × [1 - (1 + r)⁻⁵] / r

19,000 / 4,200 = [1 - (1 + r)⁻⁵] / r

4.5238 = [1 - (1 + r)⁻⁵] / r

Solving this equation for the monthly interest rate r yields:

r ≈ 0.031425 or 3.1425% per month

To find the annual rate of interest R:

R = r × 12 ≈ 3.1425% × 12 ≈ 37.71% per annum

Explanation

The Present Value (PV) of a future cash flow is calculated using the formula:

PV = FV / (1 + r)ⁿ

where: FV is the Future Value (₹2.1 lakh) r is the interest rate (5% or 0.05) n is the number of years (2)

Substituting these values into the formula:

PV = 2.1 / (1 + 0.05)² PV = 2.1 / (1.05)² PV = 2.1 / 1.1025 PV ≈ 1.90476 lakh

Rounding to three decimal places yields approximately 1.905 lakh.

Explanation

The net selling price (SP) after successive discounts can be calculated using the formula:

SP = MP × (1 - d₁/100) × (1 - d₂/100) × (1 - d₃/100)

where MP is the marked price of 36,000, and d₁, d₂, and d₃ are the successive discount percentages of 18%, 12%, and 5% respectively.

Substituting the values into the formula:

SP = 36000 × (1 - 18/100) × (1 - 12/100) × (1 - 5/100)

SP = 36000 × 0.82 × 0.88 × 0.95

SP = 34200 × 0.82 × 0.88

SP = 28044 × 0.88

SP = 24678.72

Explanation

Dividing the given quadratic equation x² - 5√5 x + 1 = 0 by x, we get: x + 1/x = 5√5

Using the algebraic relation (x - 1/x)² = (x + 1/x)² - 4, we substitute the value of x + 1/x: (x - 1/x)² = (5√5)² - 4 (x - 1/x)² = 125 - 4 = 121 x - 1/x = 11 (taking the positive value as the options are positive)

Now, using the algebraic identity for x³ - 1/x³: x³ - 1/x³ = (x - 1/x)³ + 3(x - 1/x)

Substituting x - 1/x = 11 into this formula: x³ - 1/x³ = 11³ + 3 × 11 x³ - 1/x³ = 1331 + 33 = 1364

Explanation

The volume of a cube is given by the formula V = a³, where a is the side length.

Given that the volume is 729 cm³, we calculate the side length of the cube: a = ∛729 = 9 cm

The total surface area of a cube is calculated using the formula TSA = 6a².

Substituting the side length into the surface area formula: TSA = 6 × 9² = 6 × 81 = 486 cm²

Explanation

To find the company that sold the most shirts, we calculate the sum of sales for each company over the five weeks:

Sales of C1 = 190 + 225 + 240 + 215 + 205 = 1075 Sales of C2 = 226 + 244 + 175 + 235 + 210 = 1090 Sales of C3 = 184 + 214 + 235 + 260 + 265 = 1158 Sales of C4 = 202 + 202 + 275 + 200 + 245 = 1124

Company C3 has the highest total sales with 1158 shirts sold.

Explanation

The time taken by A alone to complete the work can be found using the direct work formula:

Time taken by A = (T_both × T_B) ÷ (T_B - T_both)

Here, T_both is the time taken by A and B together (15 days), and T_B is the time taken by B alone (21 days).

Substituting these values into the formula:

Time taken by A = (15 × 21) ÷ (21 - 15) Time taken by A = 315 ÷ 6 Time taken by A = 52.5 days

In fractional form, 52.5 days is equal to 52½ days.

Explanation

The circumference of a circle is given by the formula C = 2πr, where r is the radius of the circle.

Substituting the given values into the formula: 132 = 2 × (22/7) × r 132 = (44/7) × r r = (132 × 7) / 44 r = 3 × 7 = 21 cm

The area of a circle is given by the formula A = πr².

Substituting the radius r = 21 cm into the area formula: A = (22/7) × 21² A = (22/7) × 21 × 21 A = 22 × 3 × 21 A = 1386 cm²

Explanation

The average speed of an object is defined as the total distance traveled divided by the total time taken.

Average Speed = Total Distance / Total Time

The total distance traveled by the bus is: Total Distance = (60 × 3) + (50 × 4) + (50 × 5) = 180 + 200 + 250 = 630 km

The total time taken is: Total Time = 3 + 4 + 5 = 12 hours

Substituting these values into the formula: Average Speed = 630 / 12 = 52.5 km/h

Explanation

Using the inverse proportion formula for food provision, the total amount of food available is constant, which gives:

M₁ × D₁ = M₂ × D₂

Here, the initial number of boys M₁ is 70, and the initial number of days D₁ is 36.

When 50 more boys join the camp, the new number of boys M₂ becomes:

M₂ = 70 + 50 = 120

Substituting these values into the formula to find the new number of days D₂:

70 × 36 = 120 × D₂

D₂ = (70 × 36) / 120

D₂ = 2520 / 120

D₂ = 21 days

Explanation

By Fermat's Little Theorem, for any prime number p and an integer a not divisible by p, a⁽p-1) ≡ 1 (mod p).

Here, p = 7 and a = 3.

3⁶ ≡ 1 (mod 7)

We can express 3⁸ as:

3⁸ = 3⁶ × 3²

Taking modulo 7 on both sides:

3⁸ ≡ 1 × 3² (mod 7) 3⁸ ≡ 9 (mod 7)

Since 9 divided by 7 leaves a remainder of 2:

3⁸ ≡ 2 (mod 7)

Explanation

The area of a sector of a circle is given by the formula: Area = (θ / 360°) × π × r²

Given that the central angle θ = 10° and π = 22/7, we can set up the equation using the area of 616 cm²: 616 = (10 / 360) × (22 / 7) × r²

Simplify the terms: 616 = (1 / 36) × (22 / 7) × r² 616 = (11 / 126) × r²

Solve for r²: r² = (616 × 126) / 11 r² = 56 × 126 r² = 7056

Taking the square root of both sides: r = √7056 r = 84 cm

Explanation

Given the equation: √a = 3b

Squaring both sides of the equation to eliminate the square root: (√a)² = (3b)² a = 9b²

Dividing both sides by b²: a/b² = 9

Explanation

In a proportion relationship of the form a:b = c:d, the product of the extremes equals the product of the means, which is represented by the formula a × d = b × c.

Applying this formula to the given ratio 2:9 = 4:x: 2 × x = 9 × 4 2x = 36 x = 18

Explanation

The given expression to simplify is (cot³ A - 1) / (cot A - 1).

Using the algebraic identity for the difference of cubes, x³ - y³ = (x - y)(x² + xy + y²), we can expand the numerator by setting x = cot A and y = 1: cot³ A - 1 = (cot A - 1)(cot² A + cot A + 1)

Substituting this expansion into the original expression: (cot³ A - 1) / (cot A - 1) = (cot A - 1)(cot² A + cot A + 1) / (cot A - 1) = cot² A + cot A + 1

Using the standard trigonometric identity cosec² A = 1 + cot² A, we substitute cosec² A for cot² A + 1: cot² A + cot A + 1 = (cot² A + 1) + cot A = cosec² A + cot A

Explanation

Let x = 0.25. The given expression can be written in terms of x as:

(x⁴ + 2x² + 1 - x²) / (x² + x + 1)

Simplifying the numerator: x⁴ + x² + 1

Using the algebraic identity: x⁴ + x² + 1 = (x² + x + 1)(x² - x + 1)

Substitute this into the expression: (x² + x + 1)(x² - x + 1) / (x² + x + 1) = x² - x + 1

Now substitute x = 0.25 back into the simplified expression: (0.25)² - 0.25 + 1 = 0.0625 - 0.25 + 1 = 0.8125

Explanation

When two items are sold at the same selling price, with one yielding a gain of x% and the other a loss of x%, the overall transaction always results in a loss. The net loss percentage is given by the formula:

Loss % = x² / 100

Substituting the given value of x = 20:

Loss % = 20² / 100 = 400 / 100 = 4%

Explanation

Using the fundamental trigonometric identity: sec² θ - tan² θ = 1

This can be factored as: (sec θ - tan θ)(sec θ + tan θ) = 1

Given that sec θ + tan θ = a, substituting this into the equation yields: (sec θ - tan θ) × a = 1 sec θ - tan θ = 1/a

Adding these two equations: (sec θ + tan θ) + (sec θ - tan θ) = a + 1/a 2 sec θ = (a² + 1)/a sec θ = (a² + 1)/(2a)

Explanation

The selling price of the mixture is ₹30 per kg, and the gain is 10%.

The cost price of the mixture is calculated using the formula: Cost Price = (Selling Price × 100) / (100 + Gain %) Cost Price = (30 × 100) / (100 + 10) = 300/11 per kg

Let the cost price of the cheaper tea be CP₁ = ₹26 per kg and that of the dearer tea be CP₂ = ₹32 per kg.

By the rule of alligation, the ratio in which the two varieties must be mixed is: Ratio of cheaper to dearer = (CP₂ - Cost Price of mixture) / (Cost Price of mixture - CP₁) Ratio = (32 - 300/11) / (300/11 - 26) Ratio = ((352 - 300) / 11) / ((300 - 286) / 11) Ratio = 52 / 14

Simplifying the fraction by dividing both terms by 2 yields 26:7.

Explanation

The distance travelled is calculated using the formula:

Distance = Speed × Time

First, convert the speed of the car from kilometers per hour (km/h) to meters per second (m/s) by multiplying with 5/18:

Speed = 63 × 5/18 = 17.5 m/s

Now, substitute the speed and the given time of 10 seconds into the distance formula:

Distance = 17.5 m/s × 10 s = 175 m

Explanation

Let the cost price of one pen be ₹1. Since Shyam paid for 16 pens, his total cost price is 16 × ₹1 = ₹16.

He received 4 pens free, giving him a total of 20 pens. He marked the price of each pen 20% above its cost price.

Marked price of one pen = ₹1 × 1.20 = ₹1.20

Since he sold all 20 pens at this marked price without any discount, his total selling price is 20 × ₹1.20 = ₹24.

The profit percentage is calculated using the formula: Profit % = ((Total Selling Price - Total Cost Price) / Total Cost Price) × 100%

Substituting the values: Profit % = ((24 - 16) / 16) × 100% = (8 / 16) × 100% = 50%

Explanation

The length of a direct common tangent (L) to two circles with radii R and r, whose centres are at a distance d apart, is given by the formula:

L = √(d² - (R - r)²)

Given: Radius of the first circle, R = 25 cm Radius of the second circle, r = 20 cm Distance between centres, d = 35 cm

Substituting these values into the formula:

L = √(35² - (25 - 20)²) L = √(35² - 5²) L = √((35 - 5)(35 + 5)) L = √(30 × 40) L = √1200 L = √(400 × 3) L = 20√3 cm

Explanation

Using the trigonometric identity: sin² θ + cos² θ = 1

For an acute angle θ, sin θ is positive: sin θ = √(1 - cos² θ) sin θ = √(1 - (11/17)²) sin θ = √(1 - 121/289) sin θ = √(168/289) sin θ = (2√42) / 17

Now, the value of tan θ is: tan θ = sin θ / cos θ tan θ = ((2√42) / 17) / (11/17) tan θ = 2√42 / 11

Explanation

Correct Answer: D

Using the standard algebraic identity for the difference of cubes: a³ - b³ = (a - b)³ + 3ab(a - b)

Substitute the given values (a - b) = 9 and (a³ - b³) = 4401 into the formula: 4401 = 9³ + 3ab(9)

Since 9³ = 729: 4401 = 729 + 27ab

Subtract 729 from both sides: 27ab = 4401 - 729 27ab = 3672

Divide by 27 to find the value of ab: ab = 3672 / 27 ab = 136

Explanation

To find the combined average weight of both groups, the weighted average formula is used:

Combined Average = (n₁ × W₁ + n₂ × W₂) / (n₁ + n₂)

Here, the ratio of the number of members in group A to group B is 1:2, so we can assume n₁ = 1 and n₂ = 2. The average weight of group A is W₁ = 12 kg, and the average weight of group B is W₂ = 24 kg.

Substituting these values into the formula:

Combined Average = (1 × 12 + 2 × 24) / (1 + 2) Combined Average = (12 + 48) / 3 Combined Average = 60 / 3 Combined Average = 20

This gives a combined average weight of 20 kg.

Explanation

Using the trigonometric identity for the sine of the sum of two angles: sin(A + B) = sin A cos B + cos A sin B

Substituting A = 60° and B = 30° into this formula: sin(60° + 30°) = sin 90°

Since sin 90° = 1, the value of the expression is 1.

Explanation

Using the algebraic identity (x + 1/x)² = x² + 1/x² + 2, we substitute x = α.

Given that α + 1/α = 6.

Squaring both sides: (α + 1/α)² = 6² α² + 1/α² + 2 = 36 α² + 1/α² = 34

Now, substituting this value into the required expression: ¾(α² + 1/α²) = ¾ × 34 = 102/4 = 25.5