SSC CHSL Tier 1 2022 Quantitative Aptitude Questions with Solutions

Explanation

The calculation is based on the algebraic identity: (sin θ + cos θ)² = (cos θ - sin θ)² + 4 sin θ cos θ

Applying the reciprocal product assumption where sin θ cos θ = 1: (sin θ + cos θ)² = (cos θ - sin θ)² + 4

Substituting the given value cos θ - sin θ = 6: (sin θ + cos θ)² = 6² + 4 (sin θ + cos θ)² = 36 + 4 = 40 sin θ + cos θ = √40

Explanation

Adding 1 to both sides of the given equation completes the cube on the left side:

a³ + 3a² + 3a + 1 = 63 + 1

Using the algebraic identity (a + 1)³ = a³ + 3a² + 3a + 1, the equation simplifies to:

(a + 1)³ = 64

Taking the cube root of both sides gives:

a + 1 = 4 a = 3

Substituting the value of a into the required expression:

a² + 2a = 3² + 2(3) = 9 + 6 = 15

Explanation

Sum of the students studying science from 2018 to 2022 is calculated as: 78 + 69 + 83 + 55 + 65 = 350

Average number of science students over the 5 years: 350 ÷ 5 = 70

Total strength of students studying in the year 2022 across all streams: 76 + 65 + 48 + 31 = 220

Ratio of the average number of science students to the total strength in 2022: 70 : 220 = 7 : 22

Explanation

To find the number of prime numbers between 100 and 120, we examine the odd numbers in this range, excluding those ending in 5. Any composite number in this range must have a prime factor less than or equal to √120 ≈ 10.95, meaning we only need to test for divisibility by 2, 3, 5, and 7.

Testing the remaining odd numbers: 101 is not divisible by 3 or 7, so it is prime. 103 is not divisible by 3 or 7, so it is prime. 107 is not divisible by 3 or 7, so it is prime. 109 is not divisible by 3 or 7, so it is prime. 111 has a digit sum of 3, so it is divisible by 3. 113 is not divisible by 3 or 7, so it is prime. 117 has a digit sum of 9, so it is divisible by 3. 119 is divisible by 7 because 7 × 17 = 119.

This leaves exactly 5 prime numbers in the given range, which are 101, 103, 107, 109, and 113.

Explanation

The sum of the original 16 numbers is calculated by multiplying the average by the count of numbers: Original sum = 16 × 35 = 560

The sum of the four numbers that were removed is: Sum of removed numbers = 18 + 17 + 24 + 35 = 94

Subtracting this sum from the original sum gives the sum of the remaining 12 numbers: New sum = 560 - 94 = 466

The new average is obtained by dividing this new sum by the remaining 12 numbers: New average = 466 / 12 ≈ 38.83

Explanation

To find the equivalent rate, we equate the effective annual rate of both schemes. For 10 percent compounded yearly, the multiplier for one year is 1.10. For x percent compounded half-yearly, if we consider the half-yearly rate to be x percent, the equation is (1 + x/100)² = 1.10. Solving for x gives 1 + x/100 = 1.0488, which yields x = 4.88 percent.

Explanation

In a cyclic quadrilateral JKLM, when sides JK and ML are produced to meet at Z, triangles ZKL and ZMJ are similar. By the property of secants from an external point, ZK * ZJ = ZL * ZM. Substituting the given values where ZJ = ZK + JK = 12 + 10 = 22 cm and ZM = 33 cm, we get 12 * 22 = ZL * 33, which gives ZL = 8 cm.

Explanation

Using the relation between distance, speed, and time, the total time T for a round trip of distance d is given by the formula: T = d/v₁ + d/v₂

Here, the speed of the train v₁ is 50 km/hr, the walking speed v₂ is 10 km/hr, and the total time T of the journey is 12 hours.

Substituting these values into the formula: 12 = d/50 + d/10

To solve for d, find a common denominator for the fractions on the right side: 12 = (d + 5d) / 50 12 = 6d / 50

Multiply both sides by 50 to simplify: 600 = 6d d = 100 km

Explanation

The relationship between Marked Price (MP) and Cost Price (CP) when discount and profit percentages are known is given by the formula:

MP / CP = (100 + Profit%) / (100 - Discount%)

Substituting the initial profit of 30% and discount of 20%:

MP / CP = (100 + 30) / (100 - 20) = 130 / 80 = 13/8

For the second case, the discount is 10%, so the new selling price SP is 90% of the marked price:

SP = 0.9 × MP

To find the new profit ratio, we divide this new selling price by the cost price:

SP / CP = 0.9 × (MP / CP) = 0.9 × (13/8) = 11.7 / 8 = 1.4625

The new profit percentage is (SP / CP - 1) × 100:

(1.4625 - 1) × 100 = 0.4625 × 100 = 46.25%

Explanation

From the given equation: (k + 1/k)² = 9

Taking the positive square root of both sides: k + 1/k = 3

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

For a = k and b = 1/k, this simplifies to: k³ + 1/k³ = (k + 1/k)³ - 3(k + 1/k)

Substituting the value of k + 1/k = 3: k³ + 1/k³ = 3³ - 3(3) k³ + 1/k³ = 27 - 9 k³ + 1/k³ = 18

Explanation

The perpendicular drawn from the centre of a circle to a chord bisects the chord. This forms a right-angled triangle where the hypotenuse is the radius of the circle r, one side is the distance of the chord from the centre d, and the other side is half the length of the chord L/2.

By Pythagoras theorem: r² = (L/2)² + d²

For the first chord: L = 40 cm, so L/2 = 20 cm d = 15 cm r² = 20² + 15² r² = 400 + 225 r² = 625 r = 25 cm

For the second chord: d = 20 cm r² = (L/2)² + d² 625 = (L/2)² + 20² 625 = (L/2)² + 400 (L/2)² = 225 L/2 = 15 L = 30 cm

Explanation

Using the order of operations (BODMAS), we first evaluate the "of" operator: 4 of 521 = 4 × 521 = 2084

Next, we perform the division: 3126 ÷ 2084 = 1.5

Evaluating the multiplication term with 5 × 3 = 15: 1.5 + 15 = 16.5

Explanation

When the cost price of two articles is equal, the overall profit or loss percentage is the simple arithmetic average of the individual profit and loss percentages.

Overall percentage = (Profit percentage + Loss percentage) / 2

Substituting the given values: Overall percentage = (+10% - 20%) / 2 = -10% / 2 = -5%

The negative sign indicates an overall loss of 5%.

Explanation

In triangle PQR, it is given that PQ = PR, which makes it an isosceles triangle with vertex angle P. For any isosceles triangle, the angle bisector of the vertex angle is also the altitude to the opposite side. Since PM is the angle bisector of angle P, it must be perpendicular to the base QR. Therefore, the angle PMQ is a right angle, which measures 90°.

Explanation

To find the value of P1/P2, we first calculate the total number of sharpeners sold by shopkeeper A (P1) and shopkeeper B (P2) across the 5 months.

P1 = 200 + 190 + 170 + 165 + 155 = 880

P2 = 155 + 160 + 210 + 180 + 175 = 880

Now, we calculate the ratio:

P1/P2 = 880/880 = 1

Explanation

The total surface area of a sphere is given by the formula:

A = 4πr²

Here, the radius r is 3.5 cm, which can be written as 7/2 cm. Substituting the value of r and π ≈ 22/7 into the formula:

A = 4 × (22/7) × (7/2)² A = 4 × (22/7) × (49/4) A = 22 × 7 A = 154 cm²

Explanation

Let the marks obtained by student A and student B be A and B respectively.

According to the first condition, student A got 25 marks more than student B: A = B + 25

According to the second condition, the marks of A were equal to 60% of the sum of their marks: A = 0.60 × (A + B)

Simplifying the second equation: A = 0.6A + 0.6B 0.4A = 0.6B A = 1.5B

Substituting A = 1.5B into the first equation: 1.5B = B + 25 0.5B = 25 B = 50

Using the value of B to find A: A = 50 + 25 = 75

Thus, the marks of A and B are 75 and 50 respectively.

Explanation

Using the algebraic identity (a + b)² = a² + b² + 2ab, let a = x² and b = 1/x².

Squaring both sides of the given equation x² + 1/x² = 4:

(x² + 1/x²)² = 4²

(x²)² + (1/x²)² + 2 × x² × (1/x²) = 16

x⁴ + 1/x⁴ + 2 = 16

Subtracting 2 from both sides gives:

x⁴ + 1/x⁴ = 16 - 2

x⁴ + 1/x⁴ = 14

Explanation

The inradius r of an equilateral triangle with side length a is given by the formula:

r = a / (2√3)

Substituting the given inradius r = 5√3 cm into the formula:

5√3 = a / (2√3)

Solving for a:

a = 5√3 × 2√3 a = 10 × 3 a = 30 cm

Explanation

For statement I, the combined sales of P in years A and B is 100 + 200 = 300. The combined sales of Q in years D and E is 100 + 50 = 150. The ratio of these sales is 300 : 150 = 2 : 1, which means statement I is incorrect.

For statement II, the sales of P in year C is 300 and the sales of Q in year D is 100. The percentage of sales of P in year C to that of Q in year D is (300 / 100) × 100% = 300%, which means statement II is correct.

Explanation

Let x = sec θ. The given equation is √2 x² - 4x + 2√2 = 0.

Dividing the entire equation by √2 yields: x² - 2√2 x + 2 = 0

This quadratic equation is a perfect square: (x - √2)² = 0

Solving for x gives: x = √2 sec θ = √2

Using the trigonometric identity tan² θ = sec² θ - 1: tan² θ = (√2)² - 1 = 2 - 1 = 1

Since sec θ = √2, the cosine value is: cos θ = 1 / √2 cos² θ = ½

Using the identity sin² θ = 1 - cos² θ: sin² θ = 1 - ½ = ½

Now, substitute these values into the required expression: sin² θ + tan² θ = ½ + 1 = 3/2

Explanation

For any triangle XYZ with incentre M, the angle subtended by the side YZ at the incentre M is related to the opposite vertex angle YXZ by the standard geometric formula: ∠YMZ = 90° + ½ ∠YXZ

Using the given relation between the vertex angle and the incentre angle: ∠YXZ + ∠YMZ = 150°

Substitute the formula of ∠YMZ into this equation: ∠YXZ + 90° + ½ ∠YXZ = 150°

Combine the terms of ∠YXZ: 3/2 ∠YXZ + 90° = 150° 3/2 ∠YXZ = 60° ∠YXZ = 40°

Explanation

For 10 consecutive numbers, the two middle numbers are the 5th and 6th terms. Let the first number be x, which means the 10 consecutive numbers are x, x+1, x+2, x+3, x+4, x+5, x+6, x+7, x+8, and x+9.

The two middle numbers are x+4 and x+5. Their average is given as 13.5:

((x + 4) + (x + 5)) / 2 = 13.5

(2x + 9) / 2 = 13.5

2x + 9 = 27

2x = 18

x = 9

The first 6 consecutive numbers are 9, 10, 11, 12, 13, and 14.

Using the arithmetic progression sum formula, Sum = n/2 × (first term + last term), the sum of these 6 numbers is:

Sum = 6/2 × (9 + 14)

Sum = 3 × 23 = 69

Explanation

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

First, find the value of (ab + bc + ca) using the square of a trinomial formula: (a + b + c)² = a² + b² + c² + 2(ab + bc + ca)

Substitute the given values, a + b + c = 6 and a² + b² + c² = 40, into this equation: 6² = 40 + 2(ab + bc + ca) 36 = 40 + 2(ab + bc + ca) 2(ab + bc + ca) = 36 - 40 2(ab + bc + ca) = -4 ab + bc + ca = -2

Now, substitute these values into the first identity to find the required expression: a³ + b³ + c³ - 3abc = 6 × (40 - (-2)) a³ + b³ + c³ - 3abc = 6 × (40 + 2) a³ + b³ + c³ - 3abc = 6 × 42 a³ + b³ + c³ - 3abc = 252

Explanation

The formula for compound interest is A = P(1 + R/100)ⁿ, where A is the amount, P is the principal, R is the rate of interest per annum, and n is the number of years.

Substituting the given values into the formula: 1060.90 = 1000(1 + R/100)²

Dividing both sides by 1000: 1.0609 = (1 + R/100)²

Taking the square root on both sides: 1.03 = 1 + R/100

Subtracting 1 from both sides: R/100 = 0.03 R = 3%

Explanation

cot B sin B = (cos B / sin B) × sin B = cos B

Since cosec B = 3/2, the sine of the angle is: sin B = 2/3

Using the fundamental trigonometric identity: cos B = √(1 - sin² B)

Substituting the value of sin B: cos B = √(1 - (2/3)²) cos B = √(1 - 4/9) cos B = √(5/9) cos B = √5/3

Explanation

Using the trigonometric complementary angle identities, we know that cos(90° - θ) = sin θ and tan(90° - θ) = cot θ.

Applying these identities to the terms in the given expression: cos 62° = cos(90° - 28°) = sin 28° tan 34° = tan(90° - 56°) = cot 56°

Substituting these equivalents into the expression: (3 cos 62° / sin 28°) - (2 tan 34° / cot 56°) = (3 sin 28° / sin 28°) - (2 cot 56° / cot 56°)

Simplifying the fractions: 3(1) - 2(1) = 3 - 2 = 1

Explanation

The sequence of composite numbers is 4, 6, 8, 9, 10, 12, 14, 15, and so on.

Three consecutive composite numbers are 10, 12, and 14, since the numbers between them, 11 and 13, are prime numbers.

The sum of these three consecutive composite numbers is: 10 + 12 + 14 = 36

The product of these three numbers is: 10 × 12 × 14 = 1680

Explanation

The work rate formula for combined work is used here.

Let the daily work done by A, B, and C together be 1/20. The daily work done by A alone is 1/40.

The daily work done by B and C together is the difference between the combined daily work of all three and the daily work of A alone:

Daily work of (B + C) = Daily work of (A + B + C) - Daily work of A Daily work of (B + C) = 1/20 - 1/40 Daily work of (B + C) = 2/40 - 1/40 = 1/40

Therefore, B and C together can finish 1/40 part of the work in one day.

Explanation

Using the relationship Expenditure = Income - Savings, let the incomes of A and B be 4x and 5x respectively.

The expenditures of A and B can be expressed as: A's expenditure = 4x - 6000 B's expenditure = 5x - 12000

Given that the ratio of their expenditures is 5:4, we set up the equation: (4x - 6000) / (5x - 12000) = 5/4

Cross-multiplying the terms gives: 4 × (4x - 6000) = 5 × (5x - 12000) 16x - 24000 = 25x - 60000 9x = 36000 x = 4000

The income of A is 4x: 4 × 4000 = 16000