SSC CHSL Tier 1 2023 Quantitative Aptitude Questions with Solutions

Explanation

Applying the componendo and dividendo rule to the given equation: (sin θ - cos θ) / (sin θ + cos θ) = 4/5

((sin θ - cos θ) + (sin θ + cos θ)) / ((sin θ - cos θ) - (sin θ + cos θ)) = (4 + 5) / (4 - 5) 2 sin θ / (-2 cos θ) = 9 / -1 tan θ = 9

Since tan θ = 9, we have cot θ = 1/9.

Using the trigonometric identity cosec² θ = 1 + cot² θ, we get: cosec² θ = 1 + (1/9)² = 1 + 1/81 = 82/81

Now, substitute the value of cosec² θ into the required expression: cosec² θ / (2 - cosec² θ) = (82/81) / (2 - 82/81) = (82/81) / ((162 - 82) / 81) = (82/81) / (80/81) = 82/80 = 41/40

Explanation

Average speed is calculated using the formula: Average Speed = Total Distance / Total Time

Let the total distance of the journey be D km.

The times taken to cover each part of the journey are: Time₁ = 0.35D / 70 = D / 200 hours Time₂ = 0.40D / 80 = D / 200 hours Time₃ = (1 - 0.35 - 0.40)D / 50 = 0.25D / 50 = D / 200 hours

Total time taken for the entire journey: Total Time = D/200 + D/200 + D/200 = 3D/200 hours

Substituting these values into the average speed formula: Average Speed = D / (3D/200) = 200 / 3 = 66⅔ km/h

Explanation

In right-angled triangle PQR, right-angled at Q, we use Pythagoras theorem to find the length of the side QR:

PR² = PQ² + QR²

Substituting the given values:

13² = 12² + QR² 169 = 144 + QR² QR² = 25 QR = 5 cm

For angle P: tan P = opposite side / adjacent side = QR / PQ = 5/12

For angle R: cot R = adjacent side / opposite side = QR / PQ = 5/12

Adding the two values: tan P + cot R = 5/12 + 5/12 = 10/12

Explanation

The selling price of an item after a discount is calculated using the formula:

Selling Price = Marked Price × (100 - Discount%)/100

Given: Marked Price = ₹1,700 Discount = 15%

Substituting these values into the formula:

Selling Price = 1700 × (100 - 15)/100 Selling Price = 1700 × 85/100 Selling Price = 17 × 85 Selling Price = 1445

Explanation

The net profit percentage when there is a markup of a% followed by a discount of b% is given by the successive percentage formula:

Net Profit % = a - b - ab/100

Substituting the given values, markup percentage a = 25 and discount percentage b = 12:

Net Profit % = 25 - 12 - (25 × 12)/100 Net Profit % = 13 - 300/100 Net Profit % = 13 - 3 Net Profit % = 10%

Explanation

Correct Answer: D

Using modular arithmetic, the statement that dividing n by 7 leaves a remainder of 2 is expressed as: n ≡ 2 (mod 7)

To find the remainder when 9n is divided by 7, multiply both sides by 9: 9n ≡ 9 × 2 (mod 7) 9n ≡ 18 (mod 7)

Reducing 18 modulo 7 yields: 18 = 2 × 7 + 4 9n ≡ 4 (mod 7)

Explanation

To find the aerated drink with the highest manufacturing number from 2019 to 2022, we calculate the total production for each drink across all the given years:

Total production of A = 250 + 200 + 260 + 275 = 985 thousand Total production of B = 200 + 230 + 210 + 175 = 815 thousand Total production of C = 140 + 170 + 195 + 190 = 695 thousand Total production of D = 115 + 120 + 135 + 155 = 525 thousand

Comparing these totals, aerated drink A has the highest overall manufacturing number of 985 thousand. This corresponds to option C.

Explanation

The length of a chord in a circle of radius r is given by the formula L = 2r sin(θ/2), where θ is the central angle subtended by the chord. Since the maximum value of the sine function is 1, the maximum length of a chord occurs when θ = 180°, which gives L = 2r. A chord that subtends a central angle of 180° passes through the center of the circle, which is the definition of the diameter. Therefore, the diameter represents the longest possible chord of a circle.

Explanation

The difference between compound interest and simple interest for a period of two years is given by the formula:

Difference = P × (R/100)²

where P is the principal amount and R is the rate of interest per annum.

Substituting the given values, Difference = ₹162 and R = 9%, into the formula:

162 = P × (9/100)²

162 = P × (81/10000)

P = (162 × 10000) / 81

P = 2 × 10000

P = 20000

Thus, the principal amount is ₹20000.

Explanation

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

PQ = 2√(r₁ × r₂)

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

Substituting the given radii r₁ = 36 cm and r₂ = 16 cm into the formula:

PQ = 2√(36 × 16)

PQ = 2 × 6 × 4

PQ = 48 cm

Explanation

By the intersecting chords theorem, when two chords AB and CD intersect inside a circle at point F, the products of their segments are equal:

AF × BF = CF × DF

Using the segment lengths AF = 2.4 cm and BF = 1.8 cm:

AF × BF = 2.4 × 1.8 = 4.32

Let the lengths of CF and DF be x and y respectively. We are given the total length of chord CD:

x + y = 5.7 x × y = 4.32

We can set up a quadratic equation to find the lengths:

t² - 5.7t + 4.32 = 0

Solving this equation:

(t - 4.8)(t - 0.9) = 0

This gives the two segment lengths as 4.8 cm and 0.9 cm. The length of the longer segment is 4.8 cm.

Explanation

The algebraic expression can be factored by grouping and using algebraic identities.

First, group the terms to isolate the perfect square in terms of y: 4x² - y² + 6y - 9 = 4x² - (y² - 6y + 9)

The expression inside the parentheses, y² - 6y + 9, is a perfect square trinomial of the form (a - b)² = a² - 2ab + b², where a = y and b = 3: y² - 6y + 9 = (y - 3)²

Substitute this back into the expression: 4x² - (y - 3)²

Express 4x² as (2x)² to set up a difference of squares: (2x)² - (y - 3)²

Apply the difference of squares identity, a² - b² = (a + b)(a - b), where a = 2x and b = y - 3: (2x + (y - 3))(2x - (y - 3)) = (2x + y - 3)(2x - y + 3)

Explanation

Income is the sum of expenditure and saving: Income = Expenditure + Saving

Let the initial monthly income be 100. Since A spends 68% of his income, the initial expenditure is 68, and the initial saving is: 100 - 68 = 32

After the changes: New monthly income increases by 25%: 100 × 1.25 = 125

New monthly saving increases by 15%: 32 × 1.15 = 36.8

The new monthly expenditure is the difference between the new income and the new saving: 125 - 36.8 = 88.2

The increase in monthly expenditure is: 88.2 - 68 = 20.2

The percentage increase in monthly expenditure is: (20.2 / 68) × 100% = (2020 / 68)% = (505 / 17)% = 29 12/17%

Explanation

The trigonometric identity for the sum of two angles is: tan(A + B) = (tan A + tan B) / (1 - tan A × tan B)

Substituting the given values tan A = ½ and tan B = ⅓ into the formula: tan(A + B) = (½ + ⅓) / (1 - ½ × ⅓)

Evaluating the numerator: ½ + ⅓ = 5/6

Evaluating the denominator: 1 - 1/6 = 5/6

Therefore: tan(A + B) = (5/6) / (5/6) = 1

Since tan 45° = 1, it follows that A + B = 45°.

Explanation

Let the cost price of the good be CP.

The difference between selling the good at a 10% profit instead of a 5% profit represents a difference of 5% of the cost price.

This 5% difference is equal to ₹175: 5% of CP = 175 0.05 × CP = 175 CP = 175 ÷ 0.05 CP = 3500

The actual selling price of the good is at a 5% profit: Selling Price = 105% of CP Selling Price = 1.05 × 3500 Selling Price = 3675

Explanation

Given: x = (√6 - 1)⁰.⁵

Squaring both sides: x² = √6 - 1

The reciprocal of x² is: 1/x² = 1 / (√6 - 1)

Rationalizing the denominator by multiplying the numerator and denominator by the conjugate (√6 + 1): 1/x² = (√6 + 1) / ((√6 - 1)(√6 + 1)) 1/x² = (√6 + 1) / (6 - 1) 1/x² = (√6 + 1) / 5

Evaluating the difference x² - 1/x²: x² - 1/x² = (√6 - 1) - (√6 + 1) / 5 x² - 1/x² = (5(√6 - 1) - (√6 + 1)) / 5 x² - 1/x² = (5√6 - 5 - √6 - 1) / 5 x² - 1/x² = (4√6 - 6) / 5

Explanation

Correct Answer: B

Let the total work be the Least Common Multiple of the individual days taken by X, Y and Z. LCM of 14, 28 and 42 is 84 units.

The efficiency of each individual is calculated as total work divided by the number of days: Efficiency of X = 84 / 14 = 6 units per day Efficiency of Y = 84 / 28 = 3 units per day Efficiency of Z = 84 / 42 = 2 units per day

Combined efficiency of X, Y and Z = 6 + 3 + 2 = 11 units per day.

The time taken by X, Y and Z working together to complete the work is: Total work / Combined efficiency = 84 / 11 = 7 7/11 days.

Explanation

Correct Answer: D

First, find the Cost Price (CP) of the mixture using the formula: CP = Selling Price / (1 + Profit Percentage / 100)

Given that the Selling Price is ₹20 per kg and the profit is 25%: CP = 20 / (1 + 25/100) = 20 / 1.25 = ₹16 per kg

Using the rule of alligation to find the ratio in which the wheat at ₹23 per kg (dearer) and ₹14 per kg (cheaper) must be mixed: Ratio of dearer wheat to cheaper wheat = (Mean Price - Cheaper Price) / (Dearer Price - Mean Price) Ratio = (16 - 14) / (23 - 16) = 2 / 7

Thus, the wheat should be mixed in the ratio 2:7.

Explanation

Let Ravi's initial number of chocolates be R. Bhanu's initial number of chocolates is 80% of R, which is 0.8R.

Bhanu received an additional 35% of his initial chocolates from his uncle, so his final number of chocolates is: 0.8R × (1 + 0.35) = 1.08R

Since Bhanu finally had 54 chocolates, we set up the equation: 1.08R = 54 R = 54 / 1.08 R = 50

Ravi gave 10% of his chocolates to his sister. The number of chocolates given is: 10% of 50 = 5

Explanation

The school has 6 working days in the donation week, which are Monday, Tuesday, Wednesday, Thursday, Friday, and Saturday.

To achieve an average donation of ₹8,000 per working day over these 6 days, the total target collection is calculated as: Total target collection = 6 × ₹8,000 = ₹48,000

The sum of the collections from Monday to Friday is: ₹7,230 + ₹6,435 + ₹6,927 + ₹6,855 + ₹6,562 = ₹34,009

The amount that must be collected on Saturday to meet the target is: Saturday collection = Total target collection - Monday to Friday collection Saturday collection = ₹48,000 - ₹34,009 = ₹13,991

Explanation

The selling price after successive discounts is calculated using the formula:

Selling Price = Marked Price × (1 - d₁) × (1 - d₂)

where d₁ and d₂ are the successive discount rates.

Given: Selling Price = ₹704 d₁ = 20% = 0.20 d₂ = 12% = 0.12

Substituting these values into the formula:

704 = Marked Price × (1 - 0.20) × (1 - 0.12) 704 = Marked Price × 0.80 × 0.88 704 = Marked Price × 0.704

Solving for Marked Price:

Marked Price = 704 / 0.704 = 1000

Thus, the marked price of the shirt is ₹1,000.

Explanation

To find the total calories burnt in the whole week, first calculate the total distance jogged during the week by summing the distances for each day:

Total distance = 2.5 + 4.0 + 2.5 + 3.5 + 1.5 + 2.5 + 1.5 = 18 km

The rate of calories burnt per kilometer is: 300 calories ÷ 5 km = 60 calories/km

Multiplying the total distance by this rate gives the total calories burnt for the week: 18 km × 60 calories/km = 1080 calories

Explanation

Let the direct common tangent AB touch the two circles at A and B, and let the circles touch each other externally at P. Draw a common tangent at the point of contact P, which intersects the tangent AB at point T.

The lengths of tangents drawn from an external point to a circle are equal. Therefore, for the first circle, TA = TP, and for the second circle, TB = TP. This gives TA = TB = TP.

In triangle TAP, since TA = TP, the opposite angles are equal, so ∠TAP = ∠TPA. In triangle TBP, since TB = TP, the opposite angles are equal, so ∠TBP = ∠TPB.

In triangle APB, the sum of the interior angles is 180°: ∠PAB + ∠PBA + ∠APB = 180° ∠TAP + ∠TBP + ∠APB = 180°

Substituting ∠TAP = ∠TPA and ∠TBP = ∠TPB: ∠TPA + ∠TPB + ∠APB = 180°

Since ∠TPA + ∠TPB = ∠APB, we get: ∠APB + ∠APB = 180° 2 ∠APB = 180° ∠APB = 90°

Explanation

The given expression can be simplified using the algebraic identity: a³ - b³ = (a - b)(a² + ab + b²)

Let a = 0.24 and b = 0.2.

The numerator of the expression is: 0.24 × 0.24 × 0.24 - 0.008 = a³ - b³

The denominator of the expression is: 0.24 × 0.24 + 0.048 + 0.04 = a² + ab + b² since ab = 0.24 × 0.2 = 0.048 and b² = 0.2² = 0.04.

Dividing the numerator by the denominator: (a³ - b³) / (a² + ab + b²) = a - b

Substituting the values of a and b: 0.24 - 0.2 = 0.04

Explanation

Using the relation between profit percentage, markup percentage, and discount percentage: Profit % = Markup % - Discount % - (Markup % × Discount %) / 100

Substituting the given values of profit percentage as 12% and markup percentage as 40%: 12 = 40 - d - (40 × d) / 100 12 = 40 - 1.4d 1.4d = 28 d = 28 / 1.4 = 20%

Explanation

Using the algebraic identity x² + y² = (x + y)² - 2xy, we can rewrite the numerator by letting x = sin² A and y = cos² A.

sin⁴ A + cos⁴ A = (sin² A + cos² A)² - 2 sin² A cos² A

Since sin² A + cos² A = 1, the numerator simplifies to:

1² - 2 sin² A cos² A = 1 - 2 sin² A cos² A

Substituting this back into the original expression gives:

(1 - 2 sin² A cos² A) / (1 - 2 sin² A cos² A) = 1

Explanation

In a right-angled triangle, the relation between the sides is given by the Pythagorean theorem: leg₁² + leg₂² = hypotenuse².

For the right-angled triangle PQR, the angle P is 90°, making QR the hypotenuse and PQ and PR the legs. Since triangle ABC is congruent to triangle PQR, their corresponding sides are equal. This means the hypotenuse QR is equal to BC, which is 13 cm. The given side PR is 12 cm.

Using the Pythagorean theorem for triangle PQR: PQ = √(QR² - PR²) PQ = √(13² - 12²) PQ = √(169 - 144) PQ = √25 PQ = 5 cm

Due to the congruence of triangles ABC and PQR, the side AB corresponds to PQ, which gives AB = 5 cm.

Explanation

When interest is calculated and added to the principal every six months, the compounding occurs semi-annually. For a nominal interest rate of 8% per annum, the rate for each six-month period is 8% ÷ 2 = 4%.

The effective annual rate of interest for two successive half-yearly compounding periods of r% each is calculated using the successive percentage change formula:

Effective Rate = r + r + (r × r)/100 %

Substituting r = 4% into the formula:

Effective Rate = 4 + 4 + (4 × 4)/100 % Effective Rate = 8 + 16/100 % Effective Rate = 8 + 0.16 % Effective Rate = 8.16%

Explanation

Using the fundamental trigonometric identities, we know that 1 - cos² A = sin² A and 1 - sin² A = cos² A.

Substituting these identities into the given expression simplifies it as follows:

5(1 - cos² A) / 6(1 - sin² A) = 5 sin² A / 6 cos² A = (5/6) tan² A

We are given that cot A = 5/12. Since tan A is the reciprocal of cot A, we have:

tan A = 12/5

Squaring both sides gives:

tan² A = (12/5)² = 144/25

Now, substitute this value back into the simplified expression:

(5/6) × (144/25) = (5 × 144) / (6 × 25) = 144 / (6 × 5) = 24/5

Explanation

Average speed is calculated using the formula:

Average Speed = Total Distance / Total Time

The total distance covered by the train is: Total Distance = 420 + 195 = 615 km

The time taken for each part of the journey is: Time for the first part = 420 / 60 = 7 hours Time for the second part = 195 / 65 = 3 hours

Total Time = 7 + 3 = 10 hours

Substituting these values into the average speed formula: Average Speed = 615 / 10 = 61.5 km/h

Explanation

The discount percentage is calculated using the formula: Discount % = ((Marked Price - Selling Price) / Marked Price) × 100

Given: Marked Price = ₹3,800 Selling Price = ₹3,230

Discount Amount = ₹3,800 - ₹3,230 = ₹570

Discount % = (570 / 3800) × 100 = 15%

Explanation

The mean proportion between two numbers a and b is defined by the formula √(a × b).

Substituting the values a = 6 and b = 54:

Mean proportion = √(6 × 54) = √324 = 18

Explanation

The values of the angles are given as X = 60° and Y = 30°.

The difference between the two angles is calculated as: X - Y = 60° - 30° = 30°

Substituting this value into the expression gives: sin(X - Y) = sin(30°)

From the standard trigonometric ratio values, sin(30°) is equal to ½.

Explanation

A number is divisible by 3 if the sum of its digits is divisible by 3, and it is divisible by 9 if the sum of its digits is divisible by 9. For the number 329784, the sum of the digits is 3 + 2 + 9 + 7 + 8 + 4 = 33. Because 33 is divisible by 3 but not divisible by 9, the number 329784 is divisible by 3 but not by 9. This number represents the greatest configuration because the thousands digit a is maximized at 9, and the unit digit b is 4, keeping the digit sum at 33 which avoids divisibility by 9 while maintaining divisibility by 3.

Explanation

The total volume of material remains conserved when the solid cuboid is melted to form smaller cubes.

Volume of the cuboid = Length × Breadth × Height = 20 cm × 16 cm × 8 cm = 2560 cm³

Volume of each small cube = Side³ = 4 cm × 4 cm × 4 cm = 64 cm³

Number of cubes = Volume of cuboid ÷ Volume of each cube Number of cubes = 2560 ÷ 64 = 40

Explanation

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

We are given that the cost price of 240 articles is equal to the selling price of x articles: 240 × CP = x × SP SP / CP = 240 / x

The profit percentage is 20%, which means: SP = CP × (1 + Profit%/100) SP = CP × (1 + 20/100) SP = CP × 1.2 SP / CP = 6/5

Equating the two ratios for SP / CP: 240 / x = 6/5 x = (240 × 5) / 6 x = 40 × 5 x = 200

Explanation

Using the algebraic identity (a + b + c)² = a² + b² + c² + 2(ab + bc + ca), we substitute the given values:

17² = 101 + 2(ab + bc + ca) 289 = 101 + 2(ab + bc + ca) 2(ab + bc + ca) = 289 - 101 2(ab + bc + ca) = 188

The expression to be evaluated is: (a - b)² + (b - c)² + (c - a)² = a² - 2ab + b² + b² - 2bc + c² + c² - 2ca + a² = 2(a² + b² + c²) - 2(ab + bc + ca)

Substituting the values into this expression: 2(101) - 188 = 202 - 188 = 14

Explanation

The remaining quantity of milk after n operations of drawing out and replacing a liquid of initial quantity a with b quantity of another liquid is given by the formula:

Remaining Milk = a × (1 - b/a)ⁿ

Here, the initial quantity of milk (a) is 64 litres, the quantity drawn out and replaced (b) is 8 litres, and the process is performed 3 times in total (n = 3).

Remaining Milk = 64 × (1 - 8/64)³ = 64 × (1 - 1/8)³ = 64 × (7/8)³ = 64 × 343/512 = 343/8 litres

Since the total volume of the mixture remains 64 litres, the quantity of water in the mixture is:

Quantity of Water = Total Volume - Remaining Milk = 64 - 343/8 = (512 - 343)/8 = 169/8 litres

The proportion of milk to water in the resulting mixture is:

Milk : Water = (343/8) : (169/8) = 343 : 169

Explanation

Given the equation (x² + 1)/x = 5, we can simplify it by dividing each term in the numerator by x to get x + 1/x = 5.

Squaring both sides of this equation using the algebraic identity (a + b)² = a² + 2ab + b² gives: (x + 1/x)² = 5² x² + 2 + 1/x² = 25 x² + 1/x² = 23

Squaring this result again to find the fourth power: (x² + 1/x²)² = 23² x⁴ + 2 + 1/x⁴ = 529 x⁴ + 1/x⁴ = 527

Substituting this value into the required expression x⁴ + 1/x⁴ - 36 yields: 527 - 36 = 491

Explanation

To find the performance category with the highest average, we calculate the sum and average percentage of students for each category across all five subjects.

For the Below Basic category, the sum of the percentages across the subjects MIL, Maths, Science, Social Science, and English is 40 + 19 + 41 + 34 + 8 = 142. Dividing this sum by the 5 subjects gives an average of 142 / 5 = 28.4%.

For the Basic category, the sum is 35 + 45 + 29 + 21 + 5 = 135, which gives an average of 135 / 5 = 27%.

For the Proficient category, the sum is 20 + 25 + 25 + 33 + 34 = 137, which gives an average of 137 / 5 = 27.4%.

For the Advance category, the sum is 5 + 11 + 5 + 12 + 52 = 85, which gives an average of 85 / 5 = 17%.

The Below Basic category has the highest average value of 28.4%.

Explanation

A perpendicular line drawn from the centre of a circle to a chord bisects the chord. For a chord of length 8 cm, the perpendicular from the centre divides it into two equal segments of 4 cm each.

The radius of the circle, the perpendicular distance from the centre to the chord, and the half-chord form a right-angled triangle. Applying Pythagoras' theorem:

d² + (L/2)² = R²

Here, R is the radius of 5 cm, and L/2 is the half-chord of 4 cm. Substituting these values into the formula:

d² + 4² = 5²

d² + 16 = 25

d² = 9

d = √9 = 3 cm

Explanation

A perpendicular drawn from the centre of a circle to a chord bisects the chord. Therefore, the length of half the chord is 8 / 2 = 4 cm.

This perpendicular, half of the chord, and the radius of the circle form a right-angled triangle. By applying the Pythagorean theorem, the radius r is calculated as:

r = √(3² + 4²) r = √(9 + 16) r = √25 r = 5 cm

The diameter of the circle is twice the radius:

Diameter = 2 × r = 2 × 5 = 10 cm

Explanation

The algebraic identity states that: a² + b² + c² - ab - bc - ca = ½((a - b)² + (b - c)² + (c - a)²)

Since we are given a² + b² + c² = ab + bc + ca, it follows that: ½((a - b)² + (b - c)² + (c - a)²) = 0

For the sum of squares of real numbers to equal zero, each term must individually be zero: a - b = 0, b - c = 0, c - a = 0

This gives: a = b = c

Substituting a = b = c into the expression (where the numerator term 11a³ is 11a⁴ to maintain homogeneity of degree 4): (11a⁴ + 13b⁴ + 15c⁴) / (16a²b² + 19b²c² + 17c²a²) = (11a⁴ + 13a⁴ + 15a⁴) / (16a⁴ + 19a⁴ + 17a⁴) = 39a⁴ / 52a⁴ = 39/52 = 3/4

Explanation

Correct Answer: C

A tangent to a circle is perpendicular to the radius at the point of contact. This relationship forms a right-angled triangle where the hypotenuse is the distance from the external point to the centre of the circle (d = 17 cm), one leg is the radius of the circle (r = 8 cm), and the other leg is the length of the tangent (t).

Applying Pythagoras' theorem: d² = r² + t²

Substituting the given values into the formula: 17² = 8² + t² 289 = 64 + t² t² = 289 - 64 t² = 225 t = √225 = 15 cm

Explanation

The correct average can be directly computed using the formula:

Correct Average = Incorrect Average + (Correct Value - Incorrect Value) / Number of Observations

Substituting the given values into the formula:

Correct Average = 41 + (127 - 341) / 18 Correct Average = 41 - 214 / 18 Correct Average = 41 - 107 / 9 Correct Average = 41 - 11 8/9 Correct Average = 29 1/9

Explanation

Applying the BODMAS rule, we simplify the expression by evaluating the innermost brackets first.

For the first term: ((1/5) of (35 - 5) + 4)² = ((1/5) of 30 + 4)² = (6 + 4)² = 10² = 100

For the second term: (9 - {7 + (2 - 3)} + 5)² = (9 - {7 - 1} + 5)² = (9 - 6 + 5)² = (3 + 5)² = 8² = 64

Subtracting the second term from the first term yields: 100 - 64 = 36

Explanation

The total discount percentage required to reduce the marked price from 1,600 to the selling price of 1,056 is calculated as:

Total Discount = (1,600 - 1,056) ÷ 1,600 × 100 = 544 ÷ 1,600 × 100 = 34%

The formula for the net equivalent discount of two successive discounts of y% and z% is:

Net Discount = y + z - (y × z) ÷ 100

Substituting the values y = 22 and z = 15.5 into the formula:

Net Discount = 22 + 15.5 - (22 × 15.5) ÷ 100 Net Discount = 37.5 - 341 ÷ 100 Net Discount = 37.5 - 3.41 = 34.09%

Since the resulting net discount is 34.09% instead of the required 34%, these values cannot be the successive discounts offered on the item.

Explanation

The double-angle formula for cosine is: cos 2θ = cos²θ - sin²θ

Using the fundamental trigonometric identity: sin²θ + cos²θ = 1 sin²θ = 1 - cos²θ

Substituting sin²θ into the double-angle formula gives: cos 2θ = cos²θ - (1 - cos²θ) cos 2θ = 2cos²θ - 1

This directly derives the identity.

Explanation

Using the standard algebraic identity for the difference of squares, a² - b² = (a + b)(a - b), let a = c + d and b = c - d.

Substituting these values into the identity:

((c + d) + (c - d))((c + d) - (c - d))

Simplifying the terms inside the parentheses:

(c + d + c - d)(c + d - c + d) = (2c)(2d) = 4cd

Alternatively, expanding each squared term directly:

(c + d)² = c² + 2cd + d²

(c - d)² = c² - 2cd + d²

Subtracting the second expanded expression from the first yields:

(c² + 2cd + d²) - (c² - 2cd + d²) = c² + 2cd + d² - c² + 2cd - d² = 4cd

Explanation

Let the total income of Aastha be I.

She spends 15% of her total income on travelling. Remaining income = I - 0.15I = 0.85I

She spends 20% of this remaining income on food. Remaining income after food = 0.85I × (1 - 0.20) = 0.85I × 0.80 = 0.68I

From this remaining amount, she donates ₹380 and saves ₹2,000. 0.68I - 380 = 2000 0.68I = 2380 I = 2380 / 0.68 I = 3500

The total income of Aastha is ₹3,500.