SSC CHSL Tier 1 2019 Quantitative Aptitude Questions with Solutions

Explanation

The area of a regular hexagon with side length a is given by the formula: Area = 6 × (√3/4) × a²

Given that the area of the field is 2400√3 m², we can substitute this into the formula to find the side length: 6 × (√3/4) × a² = 2400√3 (3/2) × a² = 2400 a² = 1600 a = 40 m

The perimeter of the hexagon is: Perimeter = 6 × a = 6 × 40 = 240 m

The cost of fencing the field at the rate of Rs 18.50 per metre is: Cost = Perimeter × Rate = 240 × 18.50 = 120 × 37 = 4440

Explanation

call:google:search{queries:["In a circle with centre O, AD is diameter and AC is a chord" "OBA = 60","In a circle with centre O, AD is diameter and AC is a chord. B is a point"]}

Explanation

Let the total work be 1 unit.

The combined rate of work of A and B is: A + B = 1/25 per day

Since B alone completes 33 ⅓% (which is 1/3) of the work in 15 days, the rate of work of B is: B = (1/3) ÷ 15 = 1/45 per day

The rate of work of A alone is the difference between their combined rate and B's rate: A = (A + B) - B A = 1/25 - 1/45 A = (9 - 5) / 225 = 4/225 per day

The time required for A alone to complete 4/15 of the work is calculated by dividing the required portion of work by A's daily rate: Time = (4/15) ÷ (4/225) Time = (4/15) × (225/4) = 15 days

Explanation

Correct Answer: D

Since tangents drawn from an external point to a circle are equal in length, we have AB = AC.

In triangle ABC, since AB = AC, the opposite angles are equal: ∠ABC = ∠ACB. Using the angle sum property of triangle ABC: ∠BAC + ∠ABC + ∠ACB = 180° 40° + 2 × ∠ABC = 180° 2 × ∠ABC = 140° ∠ABC = 70° Therefore, ∠ACB = 70°.

Since the chord CP is parallel to the tangent BA (BA || CP) and BC is the transversal line, the alternate interior angles are equal: ∠BCP = ∠ABC = 70°

By the alternate segment theorem, the angle between the tangent AB and the chord BC is equal to the angle subtended by the chord BC in the alternate segment: ∠BPC = ∠ABC = 70°

In triangle BCP, the sum of angles is 180°: ∠CBP + ∠BCP + ∠BPC = 180° ∠CBP + 70° + 70° = 180° ∠CBP + 140° = 180° ∠CBP = 40°

Explanation

The marked price of the article is Rs 600. After allowing a 25% discount, the discount amount is: Discount = 600 × 25/100 = Rs 150

The selling price is calculated by subtracting the discount from the marked price: Selling Price = 600 - 150 = Rs 450

Since there is a loss of Rs 30, the cost price of the article is: Cost Price = Selling Price + Loss = 450 + 30 = Rs 480

The loss percentage is given by the formula: Loss Percentage = (Loss / Cost Price) × 100 Loss Percentage = (30 / 480) × 100 = (1 / 16) × 100 = 6.25%

Explanation

To find the simple interest, use the formula: Simple Interest (SI) = P × R × T / 100

Here, Principal (P) = 12,000, Rate (R) = 12% per annum, and Time (T) = 1¼ years = 5/4 years. SI = 12000 × 12 × (5/4) / 100 = 1800

For compound interest compounded 5-monthly: Total time period = 1¼ years = 15 months. Number of compounding periods (n) = 15 / 5 = 3. Rate of interest per 5-month period (r) = 12% × 5/12 = 5%.

Using the compound interest formula: Amount (A) = P × (1 + r/100)ⁿ A = 12000 × (1 + 5/100)³ A = 12000 × (21/20)³ A = 12000 × 9261 / 8000 A = 1.5 × 9261 = 13891.50

Compound Interest (CI) = A - P CI = 13891.50 - 12000 = 1891.50

The difference between the compound interest and the simple interest is: Difference = CI - SI Difference = 1891.50 - 1800 = 91.50

Explanation

Average speed for a round trip with speeds v₁ and v₂ is given by the formula: Average Speed = 2v₁v₂ / (v₁ + v₂)

Substituting v₁ = 12 km/h and v₂ = 9 km/h: Average Speed = 2 × 12 × 9 / (12 + 9) = 216 / 21 = 72/7 km/h

The total time taken is 2⅓ hours, which is 7/3 hours.

Total distance covered is the average speed multiplied by the total time: Total Distance = (72/7) × (7/3) = 24 km

Explanation

We simplify the denominator of the given expression: (1 - sin θ)(sec θ + tan θ)

Since sec θ = 1/cos θ and tan θ = sin θ/cos θ, the term (sec θ + tan θ) can be written as (1 + sin θ)/cos θ.

Substituting this back into the denominator: (1 - sin θ) × (1 + sin θ)/cos θ = (1 - sin² θ)/cos θ

Using the trigonometric identity 1 - sin² θ = cos² θ, the denominator simplifies to: cos² θ/cos θ = cos θ

Now, substituting the simplified denominator back into the original expression: cot θ/cos θ = (cos θ/sin θ)/cos θ = 1/sin θ = cosec θ

Explanation

Let the three numbers be a, b, and c. According to the problem:

(a + b)/2 + c = 177 which simplifies to a + b + 2c = 354 (b + c)/2 + a = 163 which simplifies to 2a + b + c = 326 (c + a)/2 + b = 138 which simplifies to a + 2b + c = 276

Adding these three simplified equations together gives:

4a + 4b + 4c = 354 + 326 + 276 4(a + b + c) = 956 a + b + c = 239

To find the value of each individual number, we subtract this sum from each of the three simplified equations:

c = 354 - 239 = 115 a = 326 - 239 = 87 b = 276 - 239 = 37

The largest number is 115 and the smallest number is 37. The average of the largest and smallest numbers is:

(115 + 37) / 2 = 152 / 2 = 76

Explanation

The given algebraic expressions are x + y = 7 and xy = 10.

To find the value of 1/x³ + 1/y³, simplify the expression by finding a common denominator: 1/x³ + 1/y³ = (x³ + y³) / (xy)³

Use the algebraic identity for the sum of cubes to evaluate the numerator: x³ + y³ = (x + y)³ - 3xy(x + y)

Substitute the given values of x + y and xy into this identity: x³ + y³ = 7³ - 3 × 10 × 7 x³ + y³ = 343 - 210 x³ + y³ = 133

Next, calculate the denominator using the value of xy: (xy)³ = 10³ = 1000

Substitute the values of the numerator and denominator back into the simplified expression: 1/x³ + 1/y³ = 133 / 1000 = 0.133

Explanation

Let the initial income of the man be 100.

Since he spends 72% of his income, his initial expenditure is 72, and his initial savings is 100 - 72 = 28.

When his income increases by 28%, the new income becomes 128. When his expenditure increases by 25%, the new expenditure becomes 72 + (25% of 72) = 72 + 18 = 90.

The new savings of the man is the difference between his new income and new expenditure: New savings = 128 - 90 = 38.

The increase in his savings is 38 - 28 = 10.

To find the percentage increase in savings: Percentage increase = (Increase in savings / Initial savings) × 100 Percentage increase = (10 / 28) × 100 = 250 / 7 ≈ 35.7%

Explanation

Using the BODMAS rule, the expression 3/4 ÷ 3/4 of 3/4 × 4/3 + 5/2 ÷ 2/5 of 5/4 - (2/3 + 2/3 of 5/6) is simplified in order of operations: Brackets, Of, Division, Multiplication, Addition, and Subtraction.

First, solve the bracket: (2/3 + 2/3 of 5/6)

Evaluating the of operation inside the bracket: 2/3 of 5/6 = 2/3 × 5/6 = 5/9

Now, add inside the bracket: 2/3 + 5/9 = 6/9 + 5/9 = 11/9

Next, evaluate the remaining of terms in the expression: 3/4 of 3/4 = 9/16 2/5 of 5/4 = 1/2

Substituting these back into the expression: 3/4 ÷ 9/16 × 4/3 + 5/2 ÷ 1/2 - 11/9

Now perform the division operations: 3/4 ÷ 9/16 = 3/4 × 16/9 = 4/3 5/2 ÷ 1/2 = 5/2 × 2 = 5

Substituting these values: 4/3 × 4/3 + 5 - 11/9

Now, evaluate the multiplication: 4/3 × 4/3 = 16/9

Finally, solve addition and subtraction: 16/9 + 5 - 11/9 = 5/9 + 5 = 50/9

Explanation

For the nine-digit number 43x1145y2 to be divisible by 88, it must be divisible by both 8 and 11.

Divisibility by 8 requires the number formed by the last three digits, 5y2, to be divisible by 8. Testing single-digit values of y starting from 0, we find that for y = 1, the number 512 is divisible by 8. Thus, the smallest value of y is 1.

Divisibility by 11 requires the difference between the sum of digits at odd positions and the sum of digits at even positions to be a multiple of 11 or 0. Sum of digits at odd positions = 4 + x + 1 + 5 + 2 = 12 + x Sum of digits at even positions = 3 + 1 + 4 + 1 = 9 Difference = (12 + x) - 9 = 3 + x

For 3 + x to be a multiple of 11, the single-digit value for x must be 8 (since 3 + 8 = 11).

Substituting the values of x = 8 and y = 1 into the required expression: 3x - 2y = 3(8) - 2(1) = 24 - 2 = 22

Explanation

In the correct distribution, the sum is divided in the ratio 3 : 4 : 5 : 8. Total parts = 3 + 4 + 5 + 8 = 20 parts. D's share in this correct distribution = (8 / 20) × 4360 = 1744.

In the mistaken distribution, the sum is divided in the ratio 1/3 : 1/4 : 1/5 : 1/8. Multiplying the ratio terms by the LCM of denominators (120) simplifies this ratio to 40 : 30 : 24 : 15. Total parts in this mistaken ratio = 40 + 30 + 24 + 15 = 109 parts. D's share in this mistaken distribution = (15 / 109) × 4360 = 15 × 40 = 600.

The difference in D's share = 1744 - 600 = 1144. As a result of the mistake, D received Rs. 1,144 less.

Explanation

Let the selling price of one article be x.

The selling price of 72 articles is 72x.

The loss incurred is equal to the selling price of 8 articles, which is 8x.

The cost price of the articles is given by: Cost Price = Selling Price + Loss Cost Price = 72x + 8x = 80x

The loss percentage is calculated as: Loss % = (Loss / Cost Price) × 100 Loss % = (8x / 80x) × 100 = 10%

Explanation

Let cos θ = x. The given quadratic equation is: 2x² - 5x + 2 = 0

Factoring the quadratic equation: 2x² - 4x - x + 2 = 0 2x(x - 2) - 1(x - 2) = 0 (2x - 1)(x - 2) = 0

This gives x = ½ or x = 2. Since the value of cos θ must lie between -1 and 1, we discard x = 2. Thus, cos θ = ½. Since 0° < θ < 90°, we get θ = 60°.

Now, substitute θ = 60° into the given expression: cosec θ + cot θ = cosec 60° + cot 60° cosec 60° + cot 60° = 2/√3 + 1/3 Wait, cot 60° is 1/√3, so: cosec 60° + cot 60° = 2/3 + 1/3 No, let's write it clearly: cosec 60° + cot 60° = 2/√3 + 1/3 No, 1/√3: cosec 60° + cot 60° = 2/√3 + 1/√3 = 3/√3 = √3.

Explanation

Rearranging the terms of the given quadratic equation: 8x² - 4xy + y² - 12x + 9 = 0

We can split 8x² into 4x² + 4x² to group the terms into perfect squares: (4x² - 4xy + y²) + (4x² - 12x + 9) = 0

This simplifies to: (2x - y)² + (2x - 3)² = 0

Since the sum of two non-negative squared terms is zero, each term must individually equal zero: 2x - 3 = 0 → x = 3/2

2x - y = 0 → y = 2x = 2(3/2) = 3

Substituting the values of x and y into the expression 14x - 5y: 14(3/2) - 5(3) = 21 - 15 = 6

Explanation

We use standard algebraic identities to solve for the required expression.

First, we find the sum of squares using the identity: x² + y² + z² = (x + y + z)² - 2(xy + yz + zx)

Substituting the given values: x² + y² + z² = 19² - 2(114) = 361 - 228 = 133

Next, we find the sum of cubes using the identity: x³ + y³ + z³ - 3xyz = (x + y + z)(x² + y² + z² - (xy + yz + zx))

Substituting our values into the identity: x³ + y³ + z³ - 3(216) = 19 × (133 - 114) x³ + y³ + z³ - 648 = 19 × 19 = 361 x³ + y³ + z³ = 361 + 648 = 1009

Now, we calculate the required value: √(x³ + y³ + z³ + xyz) = √(1009 + 216) = √1225 = 35

Explanation

The given expression can be simplified by evaluating the numerator and the denominator separately.

For the numerator: tan 13° tan 37° tan 45° tan 53° tan 77°

Using the complementary angle identity tan(90° - θ) = cot θ: tan 77° = tan(90° - 13°) = cot 13° tan 53° = tan(90° - 37°) = cot 37° tan 45° = 1

This simplifies the numerator to: (tan 13° × cot 13°) × (tan 37° × cot 37°) × tan 45° = 1 × 1 × 1 = 1

For the denominator, recognizing the typographic representation of the expression as 2 csc² 60° (cos² 60° - 3 cos 60° + 2): csc 60° = 2/√3, which gives csc² 60° = 4/3 cos 60° = ½, which gives cos² 60° = ¼

Substituting these values: 2 × (4/3) × (¼ - 3(½) + 2) = (8/3) × (¼ - 6/4 + 8/4) = (8/3) × (3/4) = 2

Dividing the simplified numerator by the simplified denominator yields: 1 / 2

Explanation

In triangle ABC, since DE is parallel to AC, triangle BDE is similar to triangle BAC.

The ratio of the sides of these similar triangles is BD/BA. Since BD = 2 cm and DA = 3 cm, we have: BA = BD + DA = 2 + 3 = 5 cm

The ratio of the side lengths of triangle BDE and triangle BAC is: BD/BA = 2/5

The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides: Area(ΔBDE)/Area(ΔBAC) = (BD/BA)² = (2/5)² = 4/25

Let the area of triangle BAC be 25 units and the area of triangle BDE be 4 units.

The area of the trapezium ACED is the difference between the areas of the two triangles: Area(trapezium ACED) = Area(ΔBAC) - Area(ΔBDE) = 25 - 4 = 21 units

The ratio of the area of triangle BDE to the area of trapezium ACED is: Area(ΔBDE) : Area(trapezium ACED) = 4:21

Explanation

To solve the given trigonometric equation: cos²θ - sin²θ - 3cosθ + 2 = 0

We substitute sin²θ = 1 - cos²θ into the equation: cos²θ - (1 - cos²θ) - 3cosθ + 2 = 0 2cos²θ - 3cosθ + 1 = 0

Factorizing this quadratic equation in terms of cosθ: (2cosθ - 1)(cosθ - 1) = 0

Since the angle lies in the interval 0 < θ < 90°, cosθ cannot be 1. Therefore: 2cosθ - 1 = 0 cosθ = 1/2 θ = 60°

Using θ = 60°, we evaluate the expression 4 cosec θ + cot θ: 4 cosec 60° + cot 60° = 4 × (2/√3) + 1/√3 = 8/√3 + 1/√3 = 9/√3 = 3√3

Alternatively, if evaluating the expression 4 cos θ + cot θ at θ = 30°: 4 cos 30° + cot 30° = 4 × (√3/2) + √3 = 2√3 + √3 = 3√3

Explanation

To solve the given equation: a² + 4b² + 49c² + 18 = 2(2b + 28c - a)

First, expand and rearrange all the terms to one side: a² + 4b² + 49c² + 18 = 4b + 56c - 2a a² + 2a + 4b² - 4b + 49c² - 56c + 18 = 0

Next, group the terms to form perfect squares. The constant 18 can be split into 1 + 1 + 16: (a² + 2a + 1) + (4b² - 4b + 1) + (49c² - 56c + 16) = 0 (a + 1)² + (2b - 1)² + (7c - 4)² = 0

Since the sum of squares of real numbers must be zero, each individual squared term must be equal to zero: a + 1 = 0 → a = -1 2b - 1 = 0 → 2b = 1 7c - 4 = 0 → 7c = 4

Now, substitute these values into the expression 3a + 2b + 7c: 3a + 2b + 7c = 3(-1) + 1 + 4 = -3 + 1 + 4 = 2

Explanation

When two trains of equal length, say L meters, cross each other while running in the same direction, the total distance to be covered is the sum of their individual lengths, which is 2L meters.

The relative speed of the trains running in the same direction is the difference between their speeds: Relative speed = 54 km/h - 42 km/h = 12 km/h

Converting the relative speed from km/h to m/s by multiplying with 5/18: Relative speed = 12 × 5/18 = 10/3 m/s

Using the standard relationship Distance = Speed × Time: 2L = (10/3) × 63 2L = 210 L = 105 meters

Explanation

Let the radius of the larger circle be R = 15 cm and the radius of the smaller circle be r = 9 cm.

A chord of the larger circle that is tangent to the smaller circle is perpendicular to the radius of the smaller circle at the point of contact. This perpendicular line from the center of the concentric circles bisects the chord.

By applying Pythagoras' theorem in the right-angled triangle formed by the radius of the smaller circle (r), half of the chord (x), and the radius of the larger circle (R):

R² = r² + x²

Substituting the given values:

15² = 9² + x² 225 = 81 + x² x² = 144 x = 12 cm

The total length of the chord is 2x:

2 × 12 = 24 cm

Explanation

Using the simple interest formula: SI = (P × R × T) / 100

Substitute the given values into the formula to find the principal (P): 2940 = (P × 10 × 3.5) / 100 2940 = 0.35 × P P = 2940 / 0.35 P = 8400

The principal sum is Rs. 8,400.

To calculate the compound interest for 2½ years at 10% per annum compounded yearly, the rate of interest for the first 2 years is 10% per annum, and for the remaining ½ year, the rate of interest is half of the annual rate: R = 10% × ½ = 5%

The total amount (A) at the end of 2½ years is: A = P × (1 + 10/100) × (1 + 10/100) × (1 + 5/100) A = 8400 × 1.1 × 1.1 × 1.05 A = 8400 × 1.21 × 1.05 A = 10164 × 1.05 A = 10672.20

The compound interest (CI) is the difference between the amount and the principal: CI = A - P CI = 10672.20 - 8400 = 2272.20

Rounding to the nearest rupee, the compound interest is Rs. 2,272.

Explanation

We simplify the numerator and denominator of the expression separately.

Numerator: (sec θ + tan θ)(1 - sin θ) = (1/cos θ + sin θ/cos θ)(1 - sin θ) = ((1 + sin θ)/cos θ)(1 - sin θ) = (1 - sin² θ)/cos θ = cos² θ/cos θ = cos θ

Denominator: cosec θ (1 + cos θ)(cosec θ - cot θ) = (1/sin θ)(1 + cos θ)(1/sin θ - cos θ/sin θ) = (1/sin θ)(1 + cos θ)((1 - cos θ)/sin θ) = (1 - cos² θ)/sin² θ = sin² θ/sin² θ = 1

Dividing the simplified numerator by the simplified denominator: cos θ / 1 = cos θ

Explanation

Let the marked price of the article be MP and the cost price be CP.

The article is sold for Rs. 360 after a 10% discount on MP. SP = MP × (1 - Discount%) 360 = MP × 0.90 MP = 360 / 0.90 = Rs. 400

If no discount is given, the selling price becomes the marked price of Rs. 400. Under this condition, the profit is 25%. MP = CP × (1 + Profit%) 400 = CP × 1.25 CP = 400 / 1.25 = Rs. 320

Explanation

Let the incomes of A and B be 2I and 3I respectively, and let their expenditures be E and 2E respectively.

The problem states that 90% of B's expenditure is equal to A's income: 0.9 × 2E = 2I 1.8E = 2I I = 0.9E

Savings is calculated as Income minus Expenditure: Savings of A = 2I - E = 2(0.9E) - E = 1.8E - E = 0.8E Savings of B = 3I - 2E = 3(0.9E) - 2E = 2.7E - 2E = 0.7E

The ratio of the savings of A and B is: 0.8E : 0.7E = 8 : 7

Explanation

Let the cost price of the article be CP.

The initial transaction results in an 8% loss, meaning the selling price is 92% of the cost price. The second scenario results in a 10.5% gain, meaning the selling price is 110.5% of the cost price.

The difference between these two selling prices is: 110.5% - 92% = 18.5% of CP

According to the given information, this difference is equal to Rs. 37: 18.5% of CP = 37

Using the direct formula to find the cost price: CP = (Difference in Selling Price × 100) / (Loss% + Gain%) CP = (37 × 100) / (8 + 10.5) CP = 3700 / 18.5 CP = 200

Explanation

Let the value of D be 100.

Since C is 60% less than D, we have: C = 100 - 60% of 100 = 40

Since B is 25% more than C, we have: B = 40 + 25% of 40 = 40 + 10 = 50

Since A is 20% more than B, we have: A = 50 + 20% of 50 = 50 + 10 = 60

Comparing A and D: A = 60 and D = 100. The difference is 100 - 60 = 40. The percentage by which A is less than D is (40 / 100) × 100% = 40%.

Explanation

In triangle ABC, because DE || BC, triangle ADE is similar to triangle ABC. For similar triangles, the ratio of their areas is equal to the square of the ratio of their corresponding sides:

Area(triangle ADE) / Area(triangle ABC) = (DE / BC)² = (3 / 5)² = 9 / 25

This means if the area of triangle ADE is 9 units, the area of triangle ABC is 25 units.

The area of the quadrilateral DECB is the difference between the area of the entire triangle ABC and the smaller triangle ADE:

Area(quadrilateral DECB) = Area(triangle ABC) - Area(triangle ADE) = 25 - 9 = 16 units

The required ratio of the area of triangle ADE to the area of quadrilateral DECB is 9 : 16.

Explanation

Since the side lengths of the triangle ABC are AB = 7 cm, BC = 24 cm, and AC = 25 cm, we observe that 7² + 24² = 49 + 576 = 625 = 25². This satisfies the Pythagorean theorem, which means triangle ABC is a right-angled triangle with the right angle at vertex B, and AC is the hypotenuse.

Let BM be the median from vertex B to the hypotenuse AC. In any right-angled triangle, the length of the median drawn to the hypotenuse is exactly half the length of the hypotenuse: BM = AC / 2 = 25 / 2 = 12.5 cm

The centroid G of a triangle lies on the median BM and divides it in the ratio 2:1 from the vertex B. Thus, the length of BG is two-thirds of the total length of the median BM: BG = (2 / 3) × BM = (2 / 3) × (25 / 2) = 25 / 3 = 8⅓ cm

Explanation

The relation between men, days, and working hours for a constant amount of work is given by the formula:

M₁ × D₁ × H₁ = M₂ × D₂ × H₂

Here, M₁ = 30 persons, D₁ = 10 days, H₁ = 8 hours per day, M₂ = 40 persons, and D₂ = 6 days.

Substituting these values into the formula:

30 × 10 × 8 = 40 × 6 × H₂

2400 = 240 × H₂

H₂ = 2400 ÷ 240

H₂ = 10

Explanation

The volume V of a right circular cone is given by the formula V = ⅓ × Base Area × h, where h is the height.

Given that the volume V is 924 cm³ and the height h is 18 cm, substituting these values into the formula gives:

924 = ⅓ × Base Area × 18

924 = Base Area × 6

Base Area = 924 ÷ 6 = 154 cm²

Explanation

For a number to be divisible by 72, it must be divisible by both 8 and 9, as 8 and 9 are co-prime.

According to the divisibility rule of 8, the last three digits of the number, 8y6, must be divisible by 8. Since 800 is divisible by 8, the two-digit number y6 must also be divisible by 8. The possible values of y that make y6 divisible by 8 are 1, 5, and 9 because 16, 56, and 96 are divisible by 8.

To find the value for the largest value of y, we choose y = 9.

According to the divisibility rule of 9, the sum of all digits in the number 342x1896 must be divisible by 9.

Sum of digits = 3 + 4 + 2 + x + 1 + 8 + 9 + 6 = 33 + x

For 33 + x to be divisible by 9, the value of x must be 3 because the next multiple of 9 after 33 is 36.

Now, substituting x = 3 and y = 9 into the expression √(9x + y):

√(9 × 3 + 9) = √(27 + 9) = √36 = 6

Explanation

Given cot θ = 1/√3, for an acute angle θ, we have θ = 60°.

The corresponding trigonometric values at θ = 60° are: sin² θ = 3/4 cos² θ = 1/4 cosec² θ = 4/3 sec θ = 2

Substituting these values into the first term of the expression: (2 - sin² θ) / (1 - cos² θ) = (2 - 3/4) / (1 - 1/4) = (5/4) / (3/4) = 5/3

Substituting these values into the second term of the expression: cosec² θ + sec θ = 4/3 + 2 = 10/3

Adding the two terms together: 5/3 + 10/3 = 15/3 = 5

Explanation

Let O₁ and O₂ be the centers of the two circles with radii r₁ = 15 cm and r₂ = 12 cm, respectively. Let AB be the common chord of length 18 cm.

The line segment joining the centers of two intersecting circles is the perpendicular bisector of their common chord. Therefore, if O₁O₂ intersects AB at M, then AM = MB = 18/2 = 9 cm, and ∠O₁MA = ∠O₂MA = 90°.

In the right-angled triangle O₁MA: O₁M = √(O₁A² - AM²) = √(15² - 9²) = √(225 - 81) = √144 = 12 cm

In the right-angled triangle O₂MA: O₂M = √(O₂A² - AM²) = √(12² - 9²) = √(144 - 81) = √63 = 3√7 cm

The distance between the centers O₁ and O₂ is the sum of these segments: O₁O₂ = O₁M + O₂M = 12 + 3√7 cm

Explanation

To find the value of the expression, we evaluate the numerator and denominator separately using the BODMAS rule.

Numerator evaluation: The numerator is 3 ÷ {5 - 5 ÷ (6 - 7) × 8 + 9}. First, solve the inner bracket: 6 - 7 = -1

Next, solve the division and multiplication inside the curly braces from left to right: 5 ÷ (-1) = -5 -5 × 8 = -40

Substituting this back into the curly braces: 5 - (-40) + 9 = 5 + 40 + 9 = 54

Now, evaluate the outer division: 3 ÷ 54 = 3/54 = 1/18

Denominator evaluation: The denominator is 4 + 4 × 4 ÷ 4 of 4 (where "of" represents "का" in the original question paper). According to BODMAS, the "of" operation has higher priority than division: 4 of 4 = 16

Substituting this back into the denominator expression: 4 + 4 × 4 ÷ 16

Now, perform the division: 4 ÷ 16 = 1/4

Perform the multiplication: 4 × 1/4 = 1

Finally, perform the addition: 4 + 1 = 5

Combining the numerator and denominator: (1/18) ÷ 5 = 1 / (18 × 5) = 1/90

Explanation

Use the algebraic identity: (a + b + c)² = a² + b² + c² + 2(ab + bc + ca)

Substitute the given values into the equation: 5² = 27 + 2(ab + bc + ca) 25 = 27 + 2(ab + bc + ca) 2(ab + bc + ca) = -2 ab + bc + ca = -1

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

Substitute the values: 125 - 3abc = 5 × (27 - (-1)) 125 - 3abc = 5 × 28 125 - 3abc = 140 -3abc = 15 abc = -5

Find the value of 4abc: 4abc = 4 × (-5) = -20

Explanation

The algebraic identity for the difference of two cubes is: a³ - b³ = (a - b)(a² + ab + b²)

By representing the given terms in this form: (√3 x)³ - (√2 y)³ = (√3 x - √2 y)((√3 x)² + (√3 x)(√2 y) + (√2 y)²) 3√3 x³ - 2√2 y³ = (√3 x - √2 y)(3x² + √6 xy + 2y²)

Comparing this result with the given expression (√3 x - √2 y)(Ax² - Bxy + Cy²): Ax² - Bxy + Cy² = 3x² + √6 xy + 2y²

By comparing the coefficients: A = 3 -B = √6, which gives B = -√6 C = 2

Substituting these values into the required expression: A² - B² + C² = 3² - (-√6)² + 2² A² - B² + C² = 9 - 6 + 4 A² - B² + C² = 7

Explanation

Let Anu's initial income be 100. Since she spends 90% of her income: Initial Expenditure = 90 Initial Savings = 100 - 90 = 10

The ratio of her expenditure to savings is 90 : 10 or 9 : 1.

Using the weighted average formula to find the overall percentage increase in income (salary): Percentage increase in income = (Expenditure Ratio × Expenditure Increase + Savings Ratio × Savings Increase) / (Expenditure Ratio + Savings Ratio) Percentage increase in income = (9 × 25% + 1 × 30%) / (9 + 1) Percentage increase in income = (225% + 30%) / 10 Percentage increase in income = 255% / 10 = 25.5%

Alternatively, using the basic method: New Expenditure = 90 × 1.25 = 112.5 New Savings = 10 × 1.30 = 13 New Income = 112.5 + 13 = 125.5 Percentage increase in income = (125.5 - 100) / 100 × 100% = 25.5%

Explanation

For similar triangles, the ratio of their areas is equal to the square of the ratio of their corresponding sides.

Since ΔABC ~ ΔQPR, the corresponding side to AB is QP. This gives the relation: ar(ΔABC) / ar(ΔQPR) = (AB / QP)²

Given that ar(ΔABC) / ar(ΔPQR) = 9/16, we have: (AB / QP)² = 9/16

Taking the square root on both sides: AB / QP = 3/4

Substituting the given value of AB = 12 cm: 12 / QP = 3/4 QP = (12 × 4) / 3 QP = 16 cm

Explanation

According to the tangent properties of a circle, the lengths of tangents drawn from an external point to a circle are equal. Applying this to each vertex of the quadrilateral ABCD:

From vertex A, AP = AS = 8 cm. From vertex C, CQ = CR = 5 cm.

Since BC is made of segments BQ and CQ, we have BC = BQ + CQ. Substituting the given values: 11 = BQ + 5 BQ = 6 cm.

From vertex B, BP = BQ = 6 cm.

The length of side AB is the sum of segments AP and BP: AB = AP + BP = 8 + 6 = 14 cm.

Explanation

The given equation is: 3cos²A + 6sin²A = 3

Dividing the entire equation by 3: cos²A + 2sin²A = 1

Using the trigonometric identity cos²A = 1 - sin²A, the equation becomes: 1 - sin²A + 2sin²A = 1 1 + sin²A = 1 sin²A = 0 sin A = 0

Since A is constrained to the interval 0° ≤ A ≤ 90°, the only value satisfying this condition is A = 0°.

Explanation

For any triangle ABC where AD is the altitude to BC and AE is the angle bisector of angle A, the angle DAE between the altitude and the angle bisector is given by the standard formula:

angle DAE = ½ |angle B - angle C|

Substituting the given values, angle B = 72° and angle C = 26°:

angle DAE = ½ |72° - 26°|

angle DAE = ½ × 46°

angle DAE = 23°

This direct formula yields the measure of angle DAE swiftly without needing to calculate all intermediate angles.

Explanation

Using complementary angle relations, we can simplify both the numerator and the denominator of the given expression.

First, simplify the numerator: (cos 9° + sin 81°)(sec 9° + cosec 81°)

Since sin 81° = sin(90° - 9°) = cos 9° and cosec 81° = cosec(90° - 9°) = sec 9°, we can substitute these into the expression: (cos 9° + cos 9°)(sec 9° + sec 9°) = (2 cos 9°)(2 sec 9°) = 4 × cos 9° × sec 9°

Since cos θ × sec θ = 1, the numerator simplifies to: 4 × 1 = 4

Next, simplify the denominator: sin 56° sec 34° + cos 25° cosec 65°

Since sec 34° = sec(90° - 56°) = cosec 56° and cosec 65° = cosec(90° - 25°) = sec 25°, we can substitute these into the expression: sin 56° cosec 56° + cos 25° sec 25°

Since sin θ × cosec θ = 1 and cos θ × sec θ = 1, the denominator simplifies to: 1 + 1 = 2

Dividing the numerator by the denominator gives: 4 ÷ 2 = 2

Explanation

To simplify the given expression, it can be broken down into two parts.

The first part is: (sin θ / (1 + cos θ)) + ((1 + cos θ) / sin θ)

Taking the common denominator, this becomes: (sin² θ + (1 + cos θ)²) / (sin θ (1 + cos θ)) = (sin² θ + 1 + 2cos θ + cos² θ) / (sin θ (1 + cos θ))

Using the fundamental trigonometric identity sin² θ + cos² θ = 1, the numerator simplifies to: (1 + 1 + 2cos θ) / (sin θ (1 + cos θ)) = (2 + 2cos θ) / (sin θ (1 + cos θ)) = 2(1 + cos θ) / (sin θ (1 + cos θ)) = 2 / sin θ

The second part is: 1 / (tan θ + cot θ)

Expressing tan θ and cot θ in terms of sin θ and cos θ gives: 1 / ((sin θ / cos θ) + (cos θ / sin θ)) = 1 / ((sin² θ + cos² θ) / (sin θ cos θ))

Using the identity sin² θ + cos² θ = 1: = 1 / (1 / (sin θ cos θ)) = sin θ cos θ

Multiplying the two simplified parts together yields: (2 / sin θ) × (sin θ cos θ) = 2 cos θ

Explanation

To solve the given equation: 3 - 2 sin² θ - 3 cos θ = 0

Substitute sin² θ = 1 - cos² θ: 3 - 2(1 - cos² θ) - 3 cos θ = 0 3 - 2 + 2 cos² θ - 3 cos θ = 0 2 cos² θ - 3 cos θ + 1 = 0

Factor the quadratic equation: (2 cos θ - 1)(cos θ - 1) = 0

This gives: cos θ = 1 or cos θ = ½

Since 0° < θ < 90°, cos θ cannot be 1. Therefore: cos θ = ½ θ = 60°

Evaluating the expression 2 cosec θ + tan θ (where cosec θ is represented as cos θ in the question text) at θ = 60°: 2 cosec 60° + tan 60° = 2(2/√3) + √3 = 4/√3 + √3 = 7/√3 = 7√3/3

Explanation

The selling price of each article is 5,220.

For the first article sold at a gain of 16%: Cost Price₁ = Selling Price / (1 + Gain%) = 5220 / 1.16 = 4500 Profit on this article = 5220 - 4500 = 720

For the second article sold at a loss of 10%: Cost Price₂ = Selling Price / (1 - Loss%) = 5220 / 0.90 = 5800 Loss on this article = 5800 - 5220 = 580

Net profit or loss = Profit on the first article - Loss on the second article = 720 - 580 = 140

Since the result is positive, there is an overall profit of140.

Explanation

Let the total work be the Least Common Multiple (LCM) of 16, 32, and 24, which is 96 units.

Efficiency of (A + B) = 96 / 16 = 6 units per day Efficiency of (C + A) = 96 / 32 = 3 units per day Efficiency of (B + C) = 96 / 24 = 4 units per day

Adding these efficiencies gives: 2(A + B + C) = 6 + 3 + 4 = 13 units per day Efficiency of (A + B + C) = 13 / 2 = 6.5 units per day

Work completed by A, B, and C working together for 12 days: 12 × 6.5 = 78 units

Remaining work: 96 - 78 = 18 units

Efficiency of C alone: Efficiency of (A + B + C) - Efficiency of (A + B) = 6.5 - 6 = 0.5 units per day

Time taken by C alone to complete the remaining work: 18 / 0.5 = 36 days