SSC CHSL Tier 1 2023 Quantitative Aptitude Questions with Solutions

Explanation

In any triangle PQR, the sum of the interior angles is 180°: P + Q + R = 180°

The given relation is: 2R - P = Q - R

By rearranging the terms to group R on one side, we get: 3R = P + Q

Substituting this expression for P + Q into the angle sum equation: 3R + R = 180° 4R = 180° R = 45°

In the given options, the degree symbol is represented as a zero, making the correct measure of angle R equal to 45° (represented as 450).

Explanation

The formula for the length of a direct common tangent to two circles is:

L = √(d² - (R - r)²)

where L is the length of the direct common tangent, d is the distance between the centres of the circles, and R and r are the radii of the two circles.

Substituting the given values: L = 24√7 cm R = 32 cm r = 24 cm

24√7 = √(d² - (32 - 24)²)

24√7 = √(d² - 8²)

Squaring both sides: (24√7)² = d² - 64 576 × 7 = d² - 64 4032 = d² - 64 d² = 4032 + 64 d² = 4096 d = √4096 d = 64 cm

Explanation

The fundamental trigonometric identity relates cosecant and cotangent functions as: cosec² B - cot² B = 1

By rearranging the terms of this identity, we get: cot² B - cosec² B = -1

Since 0 < B < 90°, both cot B and cosec B are real and well-defined, making the identity valid for all values in this range.

Explanation

The present value P of an amount A due after n years at a compound interest rate of r% per annum is given by the formula:

A = P(1 + r/100)ⁿ

Given: Amount (A) = 13,230 Rate of interest (r) = 5% per annum Time (n) = 2 years

Substituting these values into the formula:

13,230 = P(1 + 5/100)² 13,230 = P(1 + 1/20)² 13,230 = P(21/20)² 13,230 = P(441/400)

Solving for P:

P = 13,230 × 400 / 441 P = 30 × 400 P = 12,000

Thus, the present value is 12,000.

Explanation

The relationship between Marked Price (MP) and Cost Price (CP) is given by the formula:

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

Substituting the given values into the formula:

MP / 3420 = (100 + 15) / (100 - 31)

MP / 3420 = 115 / 69

Simplifying the fraction on the right side by dividing both the numerator and the denominator by their greatest common divisor, 23:

115 / 69 = 5 / 3

Now, solving for MP:

MP = 3420 × (5 / 3)

MP = 1140 × 5

MP = 5700

Explanation

The total investment of ₹44,000 is divided into three shares in the ratio 1:4:5.

The sum of the ratio terms is 1 + 4 + 5 = 10.

Using the weighted average method, the overall dividend percentage on the total investment is calculated as:

Weighted dividend rate = (1 × 10% + 4 × 15% + 5 × 25%) ÷ 10 Weighted dividend rate = (10% + 60% + 125%) ÷ 10 Weighted dividend rate = 195% ÷ 10 = 19.5%

To find the total dividend income, calculate 19.5% of the total investment:

Total dividend income = 19.5% of 44,000 Total dividend income = (19.5 ÷ 100) × 44,000 Total dividend income = 19.5 × 440 Total dividend income = 8,580

Explanation

Correct Answer: B

The given expression is written in terms of mixed fractions. Converting these mixed fractions into improper fractions:

1 2/5 = 7/5 2 1/3 = 7/3 4 1/3 = 13/3 2 1/2 = 5/2

The expression is: 7/5 ÷ [7/3 ÷ (13/3 - (5/2 + 2/3))]

Applying the BODMAS rule, simplify the innermost parentheses first: 5/2 + 2/3 = (15 + 4)/6 = 19/6

Next, simplify the middle brackets: 13/3 - 19/6 = 26/6 - 19/6 = 7/6

Now, simplify the expression inside the square brackets: 7/3 ÷ 7/6 = 7/3 × 6/7 = 2

Finally, perform the remaining division: 7/5 ÷ 2 = 7/10

Explanation

The relationship between two selling prices for the same cost price under different profit percentages is given by the formula:

SP₂ / SP₁ = (100 + Profit₂%) / (100 + Profit₁%)

Substituting the given values into the relation:

SP₂ / 2744 = 105 / 112

Simplifying the fraction 105/112 by dividing both the numerator and the denominator by 7:

SP₂ / 2744 = 15 / 16

SP₂ = 2744 × 15 / 16

SP₂ = 171.5 × 15 = 2572.50

Explanation

Total distance is 12 km. The initial two-thirds of this distance is (2/3) × 12 = 8 km.

At a speed of 10 km/h, the time taken to cover this first part is 8 / 10 = 0.8 hours, which is equal to 0.8 × 60 = 48 minutes.

The total planned time to cover the entire distance is 78 minutes. Therefore, the remaining time for the final part of the journey is 78 - 48 = 30 minutes, or 0.5 hours.

The remaining distance to be covered is the last one-third of the total distance, which is (1/3) × 12 = 4 km.

To find the required speed for this last leg, divide the remaining distance by the remaining time: 4 / 0.5 = 8 km/h.

Explanation

Given the equation x + 1/x = 2, multiplying both sides by x gives: x² + 1 = 2x x² - 2x + 1 = 0 (x - 1)² = 0

Solving this quadratic equation gives x = 1.

Substituting x = 1 into the expression x¹⁸²³ + 1/x¹⁹²⁹: 1¹⁸²³ + 1/1¹⁹²⁹ = 1 + 1 = 2.

Explanation

The largest chord of any circle is its diameter.

The relation between the diameter d and the radius r of a circle is given by the formula: d = 2r

Substituting the given value of the radius, r = 8 cm, into the formula: d = 2 × 8 = 16 cm

Explanation

Using the standard values of the trigonometric functions: sin 60° = √3 / 2 sec 30° = 2 / √3 tan 45° = 1 cos 0° = 1 sec 60° = 2 sin 30° = ½

Substitute these values into the expression: sin 60° sec 30° + tan 45° cos 0° + sec 60° sin 30° = (√3/2 × 2/√3) + (1 × 1) + (2 × ½) = 1 + 1 + 1 = 3

Explanation

Let I be Rumi's initial income.

Initial savings = 20% of I = 0.20 I Initial expenditure = 80% of I = 0.80 I

New income after a 30% increase = 1.30 I New expenditure after a 40% increase = 1.40 × 0.80 I = 1.12 I

New savings = New income - New expenditure New savings = 1.30 I - 1.12 I = 0.18 I

The change in nominal savings is the difference between initial savings and new savings: 0.20 I - 0.18 I = 0.02 I

Given that nominal savings decreased by ¥ 750: 0.02 I = 750 I = 750 ÷ 0.02 = 37,500

Explanation

Using the standard values of trigonometric ratios, sin 45° = 1/√2 and sin 30° = ½.

Substituting these values into the given equation: sin θ × sin 45° = sin 30° sin θ × (1/√2) = ½ sin θ = √2/2 sin θ = 1/√2

For an acute angle, sin θ = 1/√(2) implies θ = 45°.

Calculating the value of 2θ: 2θ = 2 × 45° = 90°

Explanation

Correct Answer: C

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

Given: A = 82000 R = 6 n = 2

Substituting these values into the formula: 82000 = P(1 + 6/100)² 82000 = P(1.06)² 82000 = P × 1.1236 P = 82000 ÷ 1.1236 P ≈ 72980.15

Rounding off to the nearest integer, the sum invested by Akhilesh is 72,980.

Explanation

By the rule of alligation, the ratio of the quantities of the two mixtures to be mixed to get the desired concentration can be calculated directly.

Let the quantity of the first mixture containing 35% syrup be Q₁ and the quantity of the second mixture containing 20% syrup be Q₂. The concentration of syrup in the final mixture is 24%.

Using the alligation formula: Q₁ / Q₂ = (Mean concentration - Second concentration) / (First concentration - Mean concentration)

Substituting the values: Q₁ / Q₂ = (24 - 20) / (35 - 24) Q₁ / Q₂ = 4 / 11

This gives the ratio of the mixtures as 4:11.

Explanation

The area of a sector of a circle with radius r and central angle θ is given by the formula:

Area = (θ / 360°) × π × r²

Given that the radius r = 21 cm, the central angle θ = 22.5°, and π = 22/7, we can substitute these values into the formula:

Area = (22.5 / 360) × (22 / 7) × 21 × 21

Simplifying the fraction 22.5 / 360 gives 1/16:

Area = (1 / 16) × (22 / 7) × 21 × 21

Area = (1 / 16) × 22 × 3 × 21

Area = (11 / 8) × 63

Area = 693 / 8

Area = 86.625 cm²

This can be written with trailing zeros as 86.6250 cm².

Explanation

For a quadratic equation ax² + bx + c = 0 to have real and equal roots, its discriminant must be equal to zero.

The discriminant formula is: D = b² - 4ac = 0

Comparing the given equation 3x² + 2kx + 3 = 0 with the standard form, the coefficients are: a = 3 b = 2k c = 3

Substituting these values into the discriminant formula: (2k)² - 4 × 3 × 3 = 0 4k² - 36 = 0 4k² = 36 k² = 9 k = ±3

Since the question specifies the positive value of k, we get k = 3.

Explanation

Let the cost price of one article be CP and the selling price of one article be SP.

According to the question, the selling price of 34 articles is equal to the cost price of 38 articles: 34 × SP = 38 × CP

Rearranging the terms gives the ratio of the selling price to the cost price: SP / CP = 38 / 34 = 19 / 17

This means if the cost price is 17 units, the selling price is 19 units, yielding a gain of: 19 - 17 = 2 units

The gain percentage is calculated as: Gain % = (Gain / CP) × 100 Gain % = (2 / 17) × 100 = 200 / 17 %

Converting 200 / 17 into a mixed fraction: 200 ÷ 17 = 11 with a remainder of 13, which is 11 13/17 %

Explanation

Factorize the given algebraic expression by first taking out the greatest common divisor, which is 2:

2x² - 32 = 2(x² - 16)

Next, apply the difference of squares formula, a² - b² = (a + b)(a - b), where a = x and b = 4:

x² - 16 = x² - 4² = (x + 4)(x - 4)

Substituting this back into the expression yields:

2(x + 4)(x - 4)

Explanation

Let the side length of the square ABCD be s. Since ABCD is a square, the side lengths are AB = BC = CD = DA = s, and the internal angles are ∠ADC = ∠BCD = 90°.

Because triangle DEC is equilateral, we have DE = EC = CD = s, and the internal angles are ∠EDC = ∠ECD = ∠DEC = 60°.

Now, consider triangle ADE. The side AD = s and the side DE = s, making triangle ADE an isosceles triangle with AD = DE. The angle ∠ADE is calculated as: ∠ADE = ∠ADC - ∠EDC = 90° - 60° = 30°

Since triangle ADE is isosceles, the angles ∠DAE and ∠AED are equal. Using the angle sum property of triangle ADE: ∠ADE + ∠DAE + ∠AED = 180° 30° + 2 × ∠AED = 180° 2 × ∠AED = 150° ∠AED = 75°

By symmetry, in the isosceles triangle BCE where BC = CE = s, the angle ∠BCE is: ∠BCE = ∠BCD - ∠ECD = 90° - 60° = 30° This gives ∠BEC = 75°.

The sum of the angles around the point E is 360°: ∠DEC + ∠AED + ∠BEC + ∠AEB = 360° 60° + 75° + 75° + ∠AEB = 360° 210° + ∠AEB = 360° ∠AEB = 150°

Explanation

Using the algebraic identity (a - b)² = a² + b² - 2ab with a = x and b = 1/x, we have:

(x - 1/x)² = x² + 1/x² - 2(x)(1/x) (x - 1/x)² = x² + 1/x² - 2

Substitute the given value x - 1/x = 0.4 into the equation:

(0.4)² = x² + 1/x² - 2 0.16 = x² + 1/x² - 2

Add 2 to both sides of the equation to isolate the target term:

x² + 1/x² = 2 + 0.16 x² + 1/x² = 2.16

Explanation

A perpendicular drawn from the center of a circle to a chord bisects the chord. This forms a right-angled triangle where the hypotenuse is the radius (r = 13 cm), one leg is the perpendicular distance from the center (d = 5 cm), and the other leg is half the length of the chord (L/2).

Using Pythagoras' theorem: (L/2)² = r² - d²

Substituting the given values: (L/2)² = 13² - 5² (L/2)² = 169 - 25 (L/2)² = 144 L/2 = √144 = 12

Solving for the total length of the chord: L = 12 × 2 = 24 cm

Explanation

Total work is the Least Common Multiple (LCM) of 6 and 4, which is 12 units.

Efficiency of A = 12 ÷ 6 = 2 units per day. Efficiency of B = 12 ÷ 4 = 3 units per day.

Work done by A alone in the first 2 days = 2 days × 2 units per day = 4 units. Remaining work = 12 - 4 = 8 units.

After 2 days, B joins A, making their combined efficiency 2 + 3 = 5 units per day. Time taken by both to finish the remaining work = 8 ÷ 5 = 1 ⅗ days.

Total time taken to finish the entire work = 2 days + 1 ⅗ days = 3 ⅗ days.

Explanation

Let the present age of the son be s. Since the present age of the father is twice that of his son, the father's present age is 2s.

After 8.5 years, the ratio of their ages becomes 7 : 4. This relation can be written as:

(2s + 8.5) / (s + 8.5) = 7 / 4

Cross-multiplying to solve for s:

4 × (2s + 8.5) = 7 × (s + 8.5)

8s + 34 = 7s + 59.5

s = 59.5 - 34

s = 25.5

The present age of the son is 25.5 years.

Explanation

Given the relation: x + 1/x = 5

We can find the required sum of fifth powers using the standard algebraic identities for second and third powers.

For the second power: x² + 1/x² = (x + 1/x)² - 2 x² + 1/x² = 5² - 2 = 23

For the third power: x³ + 1/x³ = (x + 1/x)³ - 3(x + 1/x) x³ + 1/x³ = 5³ - 3(5) = 125 - 15 = 110

The fifth power is obtained using the product of the second and third powers: x⁵ + 1/x⁵ = (x² + 1/x²)(x³ + 1/x³) - (x + 1/x)

Substituting the values: x⁵ + 1/x⁵ = 23 × 110 - 5 x⁵ + 1/x⁵ = 2530 - 5 = 2525

Explanation

The greatest 6-digit number is 999999.

To find the least number that must be added to 999999 to make it exactly divisible by 294, perform division of 999999 by 294:

999999 ÷ 294 = 3401 with a remainder of 105

The required number to be added is the difference between the divisor and the remainder:

Required number = 294 - 105 = 189

Explanation

To find the scheme with the maximum discount, we calculate the effective discount percentage for each:

For scheme (i), the effective discount of two successive discounts of 20% and 10% is 20 + 10 - (20 × 10)/100 = 28%.

For scheme (ii), the discount is 10%.

For scheme (iii), the buy 8 get 4 free offer means 4 items are free out of a total of 12 items, giving a discount of (4/12) × 100 = 33.33%.

For scheme (iv), the buy 9 get 6 free offer means 6 items are free out of a total of 15 items, giving a discount of (6/15) × 100 = 40%.

Comparing these values, scheme (iv) provides the maximum discount of 40%.

Explanation

When a solid rectangular block is melted to form a solid sphere, the volume of the metal remains constant.

Volume of Cuboid = Volume of Sphere L × B × H = 4/3 × π × r³

Substituting the dimensions of the block (with length 88 cm, width 21 cm, and height 21 cm) and π = 22/7 into the equation:

88 × 21 × 21 = 4/3 × 22/7 × r³ 88 × 21² = 88/21 × r³

Dividing both sides by 88:

21² = r³ / 21 r³ = 21³ r = 21 cm

Explanation

The relationship between Marked Price (MP), Cost Price (CP), Profit percentage (P), and Discount percentage (D) is given by the formula:

MP / CP = (100 + P) / (100 - D)

Substitute the given values: CP = ₹5,600 P = 15 D = 20

MP / 5600 = (100 + 15) / (100 - 20) MP / 5600 = 115 / 80 MP = 5600 × 115 ÷ 80 MP = 70 × 115 MP = 8,050

Explanation

To find the number of schools where the tuition fee is less than four times the activity fee, we compare the tuition fee with four times the activity fee for each school.

For school P, the tuition fee is 120 and the activity fee is 24. Four times the activity fee is 4 × 24 = 96. Since 120 is not less than 96, school P does not satisfy the condition.

For school Q, the tuition fee is 60 and the activity fee is 12. Four times the activity fee is 4 × 12 = 48. Since 60 is not less than 48, school Q does not satisfy the condition.

For school R, the tuition fee is 210 and the activity fee is 45. Four times the activity fee is 4 × 45 = 180. Since 210 is not less than 180, school R does not satisfy the condition.

For school S, the tuition fee is 90 and the activity fee is 24. Four times the activity fee is 4 × 24 = 96. Since 90 < 96, school S satisfies the condition.

For school T, the tuition fee is 120 and the activity fee is 30. Four times the activity fee is 4 × 30 = 120. Since 120 is equal to and not less than 120, school T does not satisfy the condition.

Thus, only one school (S) meets the required condition.

Explanation

When the train passes the man walking in the opposite direction, their relative speed is the sum of their individual speeds.

v_rel = 26.5 km/h + 5 km/h = 31.5 km/h

Convert this relative speed to meters per second by multiplying by 5/18:

v_rel = 31.5 × 5/18 = 8.75 m/s

The distance covered to pass the man is equal to the length of the train, L.

L = relative speed × time = 8.75 m/s × 12 s = 105 m

When the train passes the platform, the total distance covered is the sum of the length of the train and the length of the platform, P. The speed of the train in meters per second is:

v_train = 26.5 × 5/18 = 132.5/18 m/s

Using the distance formula for passing the platform:

L + P = v_train × time

105 + P = (132.5/18) × 18

105 + P = 132.5

P = 132.5 - 105 = 27.5 m

Explanation

Let the initial income of Harsha be 100.

Since her expenditure is 45% of her income: Initial Expenditure = 45 Initial Savings = Income - Expenditure = 100 - 45 = 55

When her income increases by 15%: New Income = 100 × 1.15 = 115

Since the amount saved remains the same, her new savings are still 55. New Expenditure = New Income - Savings = 115 - 55 = 60

Increase in expenditure = 60 - 45 = 15

Percentage increase in expenditure = (Increase in expenditure / Initial expenditure) × 100 Percentage increase = (15 / 45) × 100 = (1/3) × 100 = 33.33%

Explanation

Using the algebraic identity for the sum of cubes:

a³ + b³ = (a + b)(a² - ab + b²)

By rearranging this identity, we get:

(a³ + b³) / (a² - ab + b²) = a + b

In the given expression, let a = 0.96 and b = 0.04. The expression matches this form exactly:

Numerator = 0.96³ + 0.04³ Denominator = 0.96² - 0.96 × 0.04 + 0.04²

Substituting these values into the simplified identity yields:

0.96 + 0.04 = 1

Explanation

Using the standard algebraic identity, the numerator can be expanded as: x³ + y³ + z³ - 3xyz = (x + y + z)(x² + y² + z² - xy - yz - zx)

Similarly, expanding the terms in the denominator gives: (x - y)² + (y - z)² + (z - x)² = x² - 2xy + y² + y² - 2yz + z² + z² - 2zx + x² = 2x² + 2y² + 2z² - 2xy - 2yz - 2zx = 2(x² + y² + z² - xy - yz - zx)

Substituting these expansions into the given fraction: [(x + y + z)(x² + y² + z² - xy - yz - zx)] / [2(x² + y² + z² - xy - yz - zx)]

Canceling the common non-zero term (x² + y² + z² - xy - yz - zx) from both the numerator and the denominator simplifies the expression to: ½ (x + y + z)

Explanation

By the division algorithm formula: Dividend = Divisor × Quotient + Remainder

Given that the remainder is 44: Divisor = 5 × Remainder = 5 × 44 = 220

Using the relation between the divisor and the quotient: Divisor = 11 × Quotient 220 = 11 × Quotient Quotient = 220 ÷ 11 = 20

Substituting these values back into the division algorithm formula: Dividend = 220 × 20 + 44 Dividend = 4400 + 44 Dividend = 4444

Explanation

The mean proportional of two numbers a and b is given by the formula √(ab).

For the numbers 7 and 63: Mean proportional = √(7 × 63) Mean proportional = √(7 × 7 × 9) Mean proportional = 7 × 3 = 21

The difference between 24 and this mean proportional is: Difference = 24 - 21 = 3

Explanation

Using the complementary angle identities sin(90° - θ) = cos θ and cos(90° - θ) = sin θ:

For the numerator of the fraction: sin² 58° = sin²(90° - 32°) = cos² 32° So, sin² 32° + sin² 58° = sin² 32° + cos² 32° = 1

For the denominator of the fraction: cos² 58° = cos²(90° - 32°) = sin² 32° So, cos² 32° + cos² 58° = cos² 32° + sin² 32° = 1

Thus, the fraction term simplifies to: 1 / 1 = 1

For the remaining terms: sin 37° = sin(90° - 53°) = cos 53° So, sin² 53° + cos 53° sin 37° = sin² 53° + cos² 53° = 1

Combining both parts: 1 + 1 = 2

Explanation

Since the pillar and the tower cast shadows at the same time, the ratio of the height of each object to the length of its shadow remains constant because the angle of elevation of the sun is identical.

Height of pillar / Length of pillar shadow = Height of tower / Length of tower shadow

Substituting the given values into the relation:

42 / 35 = Height of tower / 25

Simplifying the fraction on the left side:

6 / 5 = Height of tower / 25

Height of tower = (6 / 5) × 25 = 30 m

Explanation

In quadrilateral OMTN, TM and TN are tangents to the circle at points M and N respectively, with O being the centre of the circle. Since the radius from the centre of a circle to the point of contact of a tangent is always perpendicular to the tangent, the angles ∠OMT and ∠ONT are both equal to 90°.

The sum of all interior angles of a quadrilateral is 360°. Applying this to the quadrilateral OMTN:

∠MTN + ∠OMT + ∠MON + ∠ONT = 360°

Substituting the given values:

∠MTN + 90° + 105° + 90° = 360°

∠MTN + 285° = 360°

∠MTN = 360° - 285° = 75°

Explanation

Total Cost Price is the sum of the purchase price of the boxes and the transport charges: Total Cost Price = (10 boxes × ₹2,000/box) + ₹1,000 = ₹21,000

Total weight of grapes purchased = 10 × 20 kg = 200 kg Remaining weight after damage = 200 kg - 10 kg = 190 kg

The selling price of ₹2,500 per box of 20 kg translates to a rate per kilogram of: Selling price per kg = ₹2,500 / 20 kg = ₹125/kg

Total Selling Price for the remaining grapes: Total Selling Price = 190 kg × ₹125/kg = ₹23,750

Gain = Total Selling Price - Total Cost Price Gain = ₹23,750 - ₹21,000 = ₹2,750

Explanation

The speed of a ship in still water is given by the average of its downstream speed and its upstream speed.

Speed in still water = (Downstream speed + Upstream speed) / 2

Substituting the given values:

Speed in still water = (18 + 16) / 2 = 34 / 2 = 17 km/h

Explanation

When interest is compounded semi-annually, the annual interest rate is halved and the number of conversion periods is doubled.

Half-yearly interest rate R = 10% ÷ 2 = 5% Number of half-yearly periods n = 2 × 2 = 4

The formula for compound interest is: CI = P × ((1 + R/100)ⁿ - 1)

Substituting the values into the formula: CI = 1,60,000 × ((1 + 5/100)⁴ - 1) CI = 1,60,000 × ((21/20)⁴ - 1) CI = 1,60,000 × (194,481/160,000 - 1) CI = 1,60,000 × (34,481/160,000) CI = 34,481

Explanation

The rate of discount is calculated using the formula:

Discount Percentage = (Marked Price - Selling Price) ÷ Marked Price × 100%

Given: Marked Price = 90 Selling Price = 78

Discount = 90 - 78 = 12

Discount Percentage = (12 ÷ 90) × 100% = 13.33%

Explanation

Using the standard trigonometric identity: cosec²θ - cot²θ = 1 cot²θ = cosec²θ - 1

The term cosec²θ - 1 can be factored as a difference of squares: cot²θ = (cosec θ - 1)(cosec θ + 1)

We can express 1/sinθ as cosec θ. Substituting this and the factored form of cot²θ into the expression: cosec θ - cot²θ / (1 + cosec θ) = cosec θ - (cosec θ - 1)(cosec θ + 1) / (cosec θ + 1)

Canceling the common factor (cosec θ + 1) from both the numerator and the denominator: = cosec θ - (cosec θ - 1) = cosec θ - cosec θ + 1 = 1

Explanation

The given expression can be simplified using complementary angle trigonometric identities: sin(90° - θ) = cos(θ), sec(90° - θ) = cosec(θ), and tan(90° - θ) = cot(θ).

For the first term: sin(58°) / cos(32°) = sin(58°) / cos(90° - 58°) = sin(58°) / sin(58°) = 1

For the second term: The numerator is sin(55°) sec(35°) = sin(55°) sec(90° - 55°) = sin(55°) cosec(55°) = sin(55°) × (1 / sin(55°)) = 1

The denominator is tan(5°) tan(45°) tan(85°) = tan(5°) × 1 × tan(90° - 5°) = tan(5°) cot(5°) = tan(5°) × (1 / tan(5°)) = 1

Substituting these values back into the second term gives 1 / 1 = 1.

Adding the two simplified terms together: 1 + 1 = 2

Explanation

The edge of the cube is a = 3 ½ cm, which is equal to 7/2 cm.

For the largest right circular cone that can be cut out from this cube, the diameter of its base is equal to the edge of the cube, and its height is also equal to the edge of the cube.

Radius of the cone, r = a/2 = (7/2) / 2 = 7/4 cm Height of the cone, h = a = 7/2 cm

The formula for the volume of a cone is: V = ⅓ × π × r² × h

Substituting the values with π = 22/7: V = ⅓ × 22/7 × (7/4)² × 7/2 V = ⅓ × 22/7 × 49/16 × 7/2 V = (11 × 49) / 48 V = 539 / 48 V ≈ 11.229 cm³

Rounding to two decimal places, the volume of the largest cone is 11.23 cm³.

Explanation

Let the radius of the circle be R. Since the diameter PQ is perpendicular to the chord RS and intersects it at T, PQ bisects RS. Therefore, RT = TS = RS / 2 = 16 / 2 = 8 cm.

Since O is the center of the circle, the radius OQ = R and OR = R. The segment OT can be expressed as: OT = OQ - QT = R - 4

In the right-angled triangle OTR, applying the Pythagorean theorem: OR² = OT² + RT² R² = (R - 4)² + 8² R² = R² - 8R + 16 + 64 8R = 80 R = 10 cm

The diameter of the circle is 2R = 2 × 10 = 20 cm.

Alternatively, using the intersecting chords theorem for the perpendicular chords PQ and RS intersecting at T: PT × QT = RT × TS PT × 4 = 8 × 8 PT = 16 cm

The diameter PQ is PT + QT = 16 + 4 = 20 cm.

Explanation

To find the difference between the average production of wheat and the average production of pulses, we first calculate the average production for each crop over the four years.

Total production of wheat = 720 + 680 + 790 + 690 = 2880 tons Average production of wheat = 2880 ÷ 4 = 720 tons

Total production of pulses = 410 + 380 + 320 + 250 = 1360 tons Average production of pulses = 1360 ÷ 4 = 340 tons

Difference between the two averages = 720 - 340 = 380 tons

Explanation

To find the value of (x-2)³ - 1/(x-2)³, substitute y = x - 2, which gives x = y + 2.

Substituting x = y + 2 into the given equation: (y + 2) - 1/y = 7 y - 1/y = 5

Using the algebraic identity for the difference of cubes: y³ - 1/y³ = (y - 1/y)³ + 3(y - 1/y)

Substitute the value of y - 1/y = 5 into the formula: y³ - 1/y³ = 5³ + 3(5) y³ - 1/y³ = 125 + 15 y³ - 1/y³ = 140