Search Previous Year Questions

Given:

sin θ + cos θ = $\sqrt5$ sin(90 - θ)

Concept used:

1. cot θ = cos θ ÷ sin θ
2. a² - b² = (a + b)(a - b)

Calculation:

sin θ + cos θ = $\sqrt5$ sin(90 - θ)

⇒ sin θ + cos θ = $\sqrt5$ cosθ

⇒ cos θ($\sqrt5$ - 1) = sinθ

⇒ cot θ = $\frac {1}{\sqrt5 - 1}$

⇒ cot θ = $\frac {\sqrt5 + 1}{(\sqrt5 - 1)(\sqrt5 + 1)}$

⇒ cot θ = $\frac {\sqrt5 + 1}{5 - 1}$

⇒ cot θ = $\frac {\sqrt5 + 1}{4}$

∴ The value of cot θ is $\frac {\sqrt5 + 1}{4}$.

Concept used:

B = Bar/Bracket

O = Of

D = Division

M = Multiplication

A = Addition

S = Subtraction

Calculation:

\frac{{9 \div \frac{3}{7}\ \text{of} \ \left( {9 + 6 \times \overline {4 - 2} } \right) + \left[ {\frac{1}{5} \div \frac{7}{{25}} - \left{ {\frac{5}{8} + \frac{6}{{16}}} \right}} \right]}}{{24 \div \overline {16 - 10} + 36 \div \left( {5 + 20 \div 4 - 1} \right)}}

⇒ \frac{{9 \div \frac{3}{7}\ \text{of} \ \left( {9 + 6 \times 2 } \right) + \left[ {\frac{1}{5} \div \frac{7}{{25}} - \left{ {\frac{5}{8} + \frac{6}{{16}}} \right}} \right]}}{{24 \div 6 + 36 \div \left( {5 + 20 \div 4 - 1} \right)}}

⇒ \frac{{9 \div 9 + \left[ {\frac{1}{5} \div \frac{7}{{25}} - 1 }} \right]}}{{24 \div 6 + 36 \div \left( {5 + 5 - 1} \right)}}

⇒ \frac{{1+ \left[ {\frac{1}{5} \times \frac{25}{{7}} - 1 }} \right]}}{{4 + 36 \div 9 }}

⇒ $1 -{2\over 7}\over 8$

⇒ $5\over56$

The answer is $5\over56$.

Shortcut Trick

Net difference in percentage = Gain % − Loss % = 14% − (−16%) = 30%.

Given that 30% of CP = Rs. 33.

Target SP to gain 25% = 125% of CP.

Value of 125% = (33 ÷ 30) × 125 = 1.1 × 125 = 137.5.

∴ The correct answer is Rs. 137.5.

Alternate Method

Given:

Initial Loss = 16%

Potential Gain = 14%

Difference in Selling Price = Rs. 33

Target Gain = 25%

Formula Used:

SP = CP × (100 ± Profit/Loss %)/100

Calculations:

Let the Cost Price (CP) be 100 units.

⇒ Initial Selling Price (SP₁) = 100 − 16 = 84 units

⇒ Second Selling Price (SP₂) = 100 + 14 = 114 units

⇒ Difference in SP = 114 − 84 = 30 units

⇒ According to question, 30 units = Rs. 33

⇒ 1 unit = 33 ÷ 30 = Rs. 1.1

⇒ Required Selling Price (to gain 25%) = 100 + 25 = 125 units

⇒ New Selling Price = 125 × 1.1 = Rs. 137.5

∴ The correct answer is Rs. 137.5.

Additional Information

Profit and Loss Percentage

Profit or Loss is always calculated on the Cost Price (CP) unless stated otherwise. Profit % = (Profit/CP) × 100.

Marked Price and Discount

Discount is always allowed on the Marked Price (MP). SP = MP − Discount.

Relation between CP and MP

CP/MP = (100 − Discount%)/(100 + Profit%). This ratio is useful for solving complex markup and discount problems directly.

Shortcut Trick

Total spend on pens and erasers = (10 × 8) + (10 × 5) = ₹130.

Remaining budget for sharpeners = 150 − 130 = ₹20.

Quantity of sharpeners = 20 ÷ 4 = 5 items.

Total items = 10 (pens) + 10 (erasers) + 5 (sharpeners) = 25 items.

Average price = Total Budget ÷ Total Quantity = 150 ÷ 25 = ₹6.

∴ The correct answer is ₹6.

Alternate Method

Given:

Total spending = ₹150

Pens = 10 @ ₹8 each | Erasers = 10 @ ₹5 each | Sharpeners = n @ ₹4 each

Formula: Average Price = Total Cost ÷ Total Quantity

Calculations:

⇒ Cost of pens = 10 × 8 = ₹80

⇒ Cost of erasers = 10 × 5 = ₹50

⇒ Remaining amount for sharpeners = 150 − (80 + 50)

⇒ Remaining amount = 150 − 130 = ₹20

⇒ Number of sharpeners = 20 ÷ 4 = 5

⇒ Total quantity of items = 10 + 10 + 5 = 25

⇒ Average price per item = 150 ÷ 25

⇒ Average price = ₹6

∴ The correct answer is ₹6.

Additional Information

Weighted Average Formula

When different quantities (n₁, n₂...) have different prices (p₁, p₂...), the weighted average = (n₁p₁ + n₂p₂ + ...) ÷ (n₁ + n₂ + ...).

Average Price Concept

The average price represents the uniform cost if all items were bought at the same price while maintaining the same total expenditure.

Unit Price Formula

Unit Price = Total Cost of specific items ÷ Number of units of those items.

Given:

10 men can do work = 25 days

Formula used:

Total work = efficiency × time

Calculation:

Let the efficiency of a man = M

According to the question:

⇒ 10 M × 25 = 10 M × 12 + 13 M × D

⇒ 250 M - 120 M = 13 M × D

⇒ 13 M × D = 130 M

⇒ D = 130/13 = 10 days

∴ The correct answer is 10 days.

Shortcut Trick

For 3 years, the ratio of (Amount / Principal)^1/3 gives (1 + r/100).

⇒ √³(3515.20 / 3125) = √³(35152 / 31250) = √³(17576 / 15625) = 26/25.

Rate x = (1 / 25) × 100 = 4%.

New rate = 4% + 2% = 6%.

Simple Interest = 3125 × 6% × 3 = 3125 × 0.18 = ₹562.50.

∴ The correct answer is ₹562.50.

Alternate Method

Given: Principal (P) = ₹3125, Amount (A) = ₹3515.20, Time (T) = 3 years.

Formula Used: A = P(1 + r/100)ⁿ and SI = (P × R × T) / 100.

Step 1: Calculate the value of x

⇒ 3515.20 = 3125(1 + x/100)³

⇒ 3515.20 / 3125 = (1 + x/100)³

⇒ 351520 / 312500 = (1 + x/100)³

⇒ 17576 / 15625 = (1 + x/100)³

⇒ (26/25)³ = (1 + x/100)³

⇒ 1 + x/100 = 26/25

⇒ x/100 = 26/25 − 1 = 1/25

⇒ x = 100 / 25 = 4%

Step 2: Calculate Simple Interest

⇒ New Rate = (x + 2)% = 4 + 2 = 6% p.a.

⇒ SI = (3125 × 6 × 3) / 100

⇒ SI = (3125 × 18) / 100

⇒ SI = 56250 / 100 = 562.50

∴ The correct answer is ₹562.50.

Additional Information

Compound Interest Formula

The total amount A = P(1 + r/100)ⁿ, where P is Principal, r is Rate, and n is Time in years.

Simple Interest vs Compound Interest Difference

For 2 years, the difference is D = P(r/100)². For 3 years, D = P(r/100)² × (3 + r/100).

Effective Rate of Interest

In CI, the effective rate for 2 years is x + y + (xy/100). For SI, the effective rate is simply Rate × Time.

Shortcut Trick

For expressions of the form a³ + b³ + c³ − 3abc, use the direct formula:

Value = (a + b + c) ÷ 2 × [3(a² + b² + c²) − (a + b + c)²]

⇒ Value = (2 ÷ 2) × [3(26) − (2)²]

⇒ Value = 1 × [78 − 4] = 74

∴ The correct answer is 74.

Alternate Method

Given:

Sum of terms: a + b + c = 2

Sum of squares: a² + b² + c² = 26

Formula Used:

1. (a + b + c)² = a² + b² + c² + 2(ab + bc + ca)
2. a³ + b³ + c³ − 3abc = (a + b + c)(a² + b² + c² − (ab + bc + ca))

Calculations:

First, find the value of (ab + bc + ca):

⇒ (2)² = 26 + 2(ab + bc + ca)

⇒ 4 = 26 + 2(ab + bc + ca)

⇒ 2(ab + bc + ca) = 4 − 26

⇒ 2(ab + bc + ca) = −22

⇒ ab + bc + ca = −11

Now, substitute the values into the cubic identity:

⇒ a³ + b³ + c³ − 3abc = 2 × [26 − (−11)]

⇒ a³ + b³ + c³ − 3abc = 2 × [26 + 11]

⇒ a³ + b³ + c³ − 3abc = 2 × 37

⇒ a³ + b³ + c³ − 3abc = 74

∴ The correct answer is 74.

Additional Information

Special Case (a + b + c = 0)

If the sum of the three variables is zero, then a³ + b³ + c³ = 3abc.

Alternative Form of Cubic Identity

a³ + b³ + c³ − 3abc = 1/2 × (a + b + c) × [(a − b)² + (b − c)² + (c − a)²].

Condition for Zero Result

The expression a³ + b³ + c³ − 3abc equals zero if either a + b + c = 0 or a = b = c.

Given:

The length, breadth and height of a cuboid are in the ratio 1 : 2 : 3.

Formula used:

Volume of cuboid = Length(L) × Breadth(B) × Height(H)

Calculation:

Let the Length, breadth and height of the cuboid be 100, 200, and 300 respectively.

Volume of cuboid = 100 × 200 × 300 = 6000000

Length increased by 200%, Breadth increase by 300% and height by 300%

Now, new length = 300, breadth = 800 and height = 1200 respectively

⇒ new volume of cuboid = 300 × 800 × 1200

= 288000000

Increase in volume = (new volume of the cuboid – the volume of the cuboid)/volume of the cuboid

= (288000000 – 6000000)/6000000

= 282000000/6000000

= 47 times

∴ The increase in the volume of the cuboid will be 47 times.

Shortcut Trick

Given: b = 2% of a ⇒ b = 0.02a

Required: b% of 10 = (b ÷ 100) × 10 = b ÷ 10

Substitute b: (0.02a) ÷ 10 = 0.002a

Check Option (2): 20% of a/100 = (20 ÷ 100) × (a ÷ 100) = 0.2 × 0.01a = 0.002a

∴ The correct answer is 20% of a/100.

Alternate Method

Given: 2% of a = b

Formula Used: x% of y = (x × y) ÷ 100

⇒ b = (2 ÷ 100) × a

⇒ b = 2a ÷ 100

⇒ Required value = b% of 10 = (b ÷ 100) × 10

⇒ Required value = (b × 10) ÷ 100 = b ÷ 10

⇒ Substituting the value of b: (2a ÷ 100) ÷ 10 = 2a ÷ 1000

⇒ Checking Option (2): 20% of a/100 = (20 ÷ 100) × (a ÷ 100) = 20a ÷ 10000 = 2a ÷ 1000

⇒ Since both expressions result in 2a/1000, Option (2) is correct.

∴ The correct answer is 20% of a/100.

Additional Information

Commutative Property of Percentages

x% of y is always equal to y% of x. Formula: (x × y) ÷ 100 = (y × x) ÷ 100.

Percentage to Fraction Conversion

To convert a percentage into a fraction, divide by 100. Example: 2% = 2/100 = 1/50.

Successive Percentages

a% of b% of c = (a × b × c) ÷ 10000. This is useful for multi-step percentage problems.

Shortcut Trick

According to the Angle Bisector Theorem, the ratio of the two segments BD and DC is equal to the ratio of the corresponding sides AB and AC.

Ratio BD : DC = AB : AC

⇒ Ratio = 10 : 14

⇒ Ratio = 5 : 7

∴ The correct answer is 5 : 7.

Alternate Method

Given:

Side AB = 10 cm

Side AC = 14 cm

AD is the internal angle bisector of ∠BAC.

Formula Used:

Internal Angle Bisector Theorem: BD/DC = AB/AC

Calculations:

⇒ BD/DC = 10/14

⇒ BD/DC = (2 × 5) / (2 × 7)

⇒ BD/DC = 5/7

⇒ BD : DC = 5 : 7

∴ The correct answer is 5 : 7.

Additional Information

Internal Angle Bisector Theorem

In a triangle, the angle bisector of an angle divides the opposite side into two segments that are proportional to the other two sides of the triangle.

External Angle Bisector Theorem

For an external angle bisector at A meeting BC produced at D, the ratio is still BD/CD = AB/AC.

Length of Angle Bisector

The length of the bisector AD can be calculated using the formula: AD² = AB × AC − BD × DC.