CDS Elementary Mathematics 16 April 2023 Official Paper (Previous Year Paper)

Concept:

HCF: It is the product of the smallest power of the common factors involved in

the prime factorization.

HCF of (am - 1) & (an - 1) is aHCF (m, n) -1

Calculation:

We have 235 - 1 &  291 - 1

By using the above concept

HCF = [2HCF (35, 91) - 1]

Let's find the HCF of (35, 91) first

35 = 5 × 7

91 = 13 × 7

HCF of (35, 91) = 7

Now, the required value

HCF = (2⁷ - 1) = 128 - 1 = 127.

Calculation:

We need to count the number of trailing zeros in that product.

It is obtained by multiplying a factor of 10, which is equivalent to multiplying

by 2 and 5.

Since every even number contributes a factor of 2 and every multiple of 5

contributes a factor of 5, we need to determine the number of multiples of 5

among the numbers from 1 to 29.

The numbers 5, 10, 15, 20, and 25 have one factor of 5 each, while 25 contributes

an additional factor of 5. Hence, there are 6 factors of 5 in total.

Therefore, the largest power of 10 that divides the product of the numbers

from 29 to 1 is 10⁶.

∴ The largest power of 10 that divides the given product is 6.

Shortcut Trick

When dividing 65 by 11, the nearest multiple of 11 is 66, which gives a negative remainder of −1.

Substitute this negative remainder into the given expression: (−1)⁹⁹.

Since 99 is an odd power, (−1)⁹⁹ = −1.

To convert the negative remainder into a positive one, add the divisor: 11 + (−1) = 10.

∴ The correct answer is 10.

Alternate Method

Given:

Expression = 65⁹⁹ ÷ 11

Concept Used:

If an expression is of the form (ax − 1)ⁿ ÷ a, the remainder is (−1)ⁿ.

If n is an odd number, remainder is −1, which means the actual positive remainder is (a − 1).

Calculation:

⇒ 65⁹⁹ ÷ 11

⇒ Expressing 65 in terms of a multiple of 11: 65 = 66 − 1 = 11 × 6 − 1

⇒ (11 × 6 − 1)⁹⁹ ÷ 11

⇒ Applying the concept, the remainder depends only on the (−1) term.

⇒ Remainder = (−1)⁹⁹

⇒ Remainder = −1

⇒ As the remainder cannot be negative, we add the divisor (11) to it.

⇒ Final positive remainder = 11 − 1 = 10

∴ The correct answer is 10.

Additional Information

Fermat's Little Theorem

If p is a prime number and a is not divisible by p, then a^p−1 leaves a remainder of 1 when divided by p.

Wilson's Theorem

For any prime number p, the expression (p − 1)! leaves a remainder of −1 when divided by p.

Negative Remainder Concept

A negative remainder −R indicates that the number is R less than a multiple of the divisor D. The true positive remainder is calculated as D − R.

Shortcut Trick

For a quadratic equation ax² + bx + c = 0, the difference of roots is given by |α − β| = √D / |a|.

Rewrite the equation: x² − bx + (c − 5) = 0, where a = 1, b = −b, and constant term = (c − 5).

Difference = 5 ⇒ 5 = √((−b)² − 4 × 1 × (c − 5)) / 1

Squaring both sides: 25 = b² − 4c + 20 ⇒ b² = 4c + 5.

∴ The correct answer is b² = 4c + 5.

Alternate Method

Given:

Equation: x² − bx + c = 5

Difference of roots (|α − β|) = 5

Formula Used:

1. Sum of roots (α + β) = − (coefficient of x) / (coefficient of x²)
2. Product of roots (αβ) = (constant term) / (coefficient of x²)
3. (α − β)² = (α + β)² − 4αβ

Calculations:

The standard form of the equation is x² − bx + (c − 5) = 0.

⇒ Sum of roots (α + β) = −(−b) / 1 = b

⇒ Product of roots (αβ) = (c − 5) / 1 = c − 5

⇒ Given |α − β| = 5

⇒ (α − β)² = 5² = 25

Substituting sum and product values:

⇒ b² − 4(c − 5) = 25

⇒ b² − 4c + 20 = 25

⇒ b² − 4c = 25 − 20

⇒ b² − 4c = 5

⇒ b² = 4c + 5

∴ The correct answer is b² = 4c + 5.

Additional Information

Discriminant and Difference of Roots

The discriminant (D) is b² − 4ac. The difference of roots is directly related to D by the formula |α − β| = √D / |a|.

Nature of Roots

If D > 0, roots are real and distinct. If D = 0, roots are real and equal. If D < 0, roots are imaginary/complex conjugates.

Vieta's Formulas

For any polynomial equation, the relations between coefficients and roots are established by Vieta's formulas (Sum = −b/a, Product = c/a).

Calculation:

Let, x number of persons taking milk only.

According to the question

60 + 15 + (x - 5) + (50 - x) + x = 150

120 + x = 150

x = 150 - 120 = 30

∴ The required value is 30.

Concept:

 Total investment = Investment × Time  of Investment 

Calculation:

A's total investment = 2 × 6 = 12 unit

B's total investment = 3 × 5 = 15 unit

C's total investment = 4 × 4 = 16 unit

D's total investment = 5 × 3 = 15 unit

E's total investment = 6 × 2 = 12 unit

C has the highest total investment, which is 16 unit.

Since, the profit is directly proportional to the total investment.

Therefore, C would receive the highest amount of profit.

Concept:

Prime Number:

• A prime number is a positive integer greater than 1 that has no positive integer divisors other than 1 and itself.
• In other words, a prime number is only divisible by 1 and itself.
• To determine whether a number is prime or not: Test its divisibility by all the integers from 2 to the square root of the number.
• If none of these integers divide the number, then the number is prime.

Calculation:

437 = 19 × 23    

It is not a prime number 

1073 = 29 × 37 

It is not a prime number 

Check for 797:   √797 ≈ 28.2.

The prime numbers below 28 are

2, 3, 5, 7, 11, 13, 17, 19, 23

797 is not divisible by any of the given numbers 

Therefore, we can conclude that 797 is a prime number.

∴ Out of the three given numbers, only 1 i.e. 797 number is prime.

Calculation:

Let's assume that C can finish the work in x days.

Since B can do the work 1.5 times as fast as C,

B can finish the work in ($\frac{x}{1.5}$) i.e. $\frac{2x}{3}$ days.

Similarly,

A can do the work twice as fast as B, 

A can finish the work in $\frac{\frac{2x}{3}}{2}=\frac{x}{3}$ days.

Now, let's consider the combined rate of work of all three persons:

One day work of A = 3/x

One day work of B = 3/2x

One day work of C = 1/x

According to the question

3/x + 3/(2x) + 1/x = 1/12

Multiplying both sides by 12x gives:

36 + 18 + 12 = x

Therefore, x = 66.

Hence, C can finish the work alone in 66 days.

Shortcut TrickRatio of Speed is also known as the ratio of efficiency.

Ratio of efficiency of A : B : C = 6 : 3 : 2

Total work = efficiency × days = 11 × 12

Time taken by C = Total work/Efficiency of C

⇒ (11 × 12)/2 = 66 days

∴ The correct answer is 66 days.

Calculation:

Let's assume that the 2-digit number is represented as 10x + y,

where x and y are the tens and units digits respectively.

We are given that the sum of the digits is 12, so we can write:

x + y = 12

Digits are reversed, the new number is represented as 10y + x.

According to the question

10y + x = 10x + y + 18

Simplifying this equation, we get:

9y - 9x = 18

Dividing both sides by 9, we get:

y - x = 2

∴ The difference between the digits in the number is 2.

Concept:

Relative speed:

If two trains are traveling with speeds of magnitude u & v respectively, then the relative speed will be

u + v,  if both trains are traveling in the opposite direction.

u - v,  if both trains are traveling in the same direction.

Formula used:

Speed = distance/ times

Calculation:

Let the speed of train 1 (of length 200 m) is 'u' m/s & speed of train 2

(of length 150 m) be 'v' m/s.

Given that, the man is sitting in the second train & a train of length 200 meters across a man (here, 150 m is irrelevant data).

By using the above concept,

When both trains are traveling in the same direction,

u - v = 200/20

u - v = 10           -----(1)

When both trains are traveling in the opposite direction,

u + v = 200/10 

u + v =  20         -----(2)

Adding both equations, we will get

2u = 30 or u =15 m/s

Since, 1 m/s = 3.6 km/hr

u = 15 × 3.6 = 54 km/hr

∴ The correct answer is 54 km/hr.

Calculation:

2a = 3b = 6c = 9d = 12e = 18f = k (let)

a = k/2, b = k/3, c = k/6, d = k/9, e = k/12, f =k/18

LCM of (2, 3, 6, 9, 12, 18) is 36.

Multiply the a, b, c, d, e & f by 36, 

a = 18k, b = 12k, c = 6k, 

d = 4k, e = 3k, f = 2k

Hence, the required value

(a + b)/(c + d + e + f)

(18k + 12k)/(6k + 4k + 3k + 2k)

30k/15k = 2

∴ (a + b)/(c + d + e + f) = 2

Formula used:

(a + b)² = a² + 2ab + b²

(a - b)² = a² - 2ab + b²

Solution of quadratic equation ax² + bx + c = 0 is given by

$x = {-b \pm \sqrt{b^2-4ac} \over 2a}$

Calculation:

Given that 

a + b + c = 0

b = - (a + c)              ------(1)

Root of quadratic equation ax2 + bx + c = 0  are

$x = {-b \pm \sqrt{b^2-4ac} \over 2a}$

$x = {(a+c) \pm \sqrt{[-(a+c)]^2-4ac} \over 2a}$

$x = {(a+c) \pm \sqrt{a^2+2ac+c^2-4ac} \over 2a}$

$x = {(a+c) \pm \sqrt{a^2-2ac+c^2} \over 2a}$

$x = {(a+c) \pm \sqrt{(a-c)^2} \over 2a}$

$x = {(a+c) \pm {(a-c)} \over 2a}$

Considering +ve & -ve signs alternatively,

$x = {(a+c) +{(a-c)} \over 2a}$ & $x = {(a+c) -{(a-c)} \over 2a}$ 

$x = \frac{{2a}}{ 2a}$ & $x = \frac{{2c}}{ 2a}$

 $x = 1, \ \ x = \frac{c}{a}$

Formula used:

Profit % = [(SP - CP)/CP] × 100

Calculation:

The vendor initially bought 100 bananas at a cost of 1 Rs each, so the total cost is 100 Rs.

12% i.e. 12 bananas lost during transportation.

Remaing bananas are 100 - 12 = 88.

Since the remaining bananas are sold at a 4 % profit

Selling Price of  88 bananas = 100 (1 + 4/100) = 104

Selling Price of 1 banana = 104/88 = 13/11

Selling Price of 100 bananas = 100 × (13/11) = 118.18

As the cost price of 100 bananas was 100 Rs.

Profit % = [(118.18 - 100)/100] × 100

∴ Profit % = 18.18% or $18 \frac{2}{11} \%$

Formula used:

(a - b) (a + b) = a² - b²

(a² + 2ab + b²) = (a + b)²

Calculation:

 Given that,

α  = √[(5 + 3√2) (5 - 3√2)]

Since, (a - b) (a + b) = a2 - b2

α = √[(5² - (3√2)²]

α = √(25 - 18) = √7

Let the required square root is β 

β = √(8 + 2√7) = √(7 + 1 + 2√7)

β = √[(√7)² + 2.1.√7 +  (1)²]

β = √(√7 + 1)² 

Since, (a2 + 2ab + b2) = (a + b)2

β = √7 + 1

Concept:

Even Number: Any number that can be exactly divided by 2 is called an even number. Even numbers always end up with the last digit as 0, 2, 4, 6, or 8. 

Calculation:

Let's assume the odd number = 7

Even power of 7 = 7² = 49

49 = 8 × 6 + 1

Similarly, we can also check for another value.

∴ The correct answer is 1.

Calculation:

Statement 1:

Let n be 1

So, $\frac{n\left(n^2+2\right)}{3} = \frac{1\left(1^2+2\right)}{3}$

⇒ $\frac{1\left(1+2\right)}{3}$

⇒ $\frac{3}{3}$

⇒ 1

So, statement 1 is correct.

Statement 2: 

Let m be 3

So, $\frac{\mathrm{m}^4+4 \mathrm{~m}^2+11}{16} = \frac{\mathrm{3}^4+4 (3)^2+11}{16}$

⇒ $\frac{81+4 \times9+11}{16}$

⇒ $\frac{81+36+11}{16}$

⇒ $\frac{128}{16}$

⇒ 8

So, statement 2 is correct.

∴ The required answer is Option 3.

Concept:

Divisibility rule of 5: A number is divisible by 5 only if the unit digit is 0 or 5.

Calculation:

Let n - 1, n & n + 1 be 6, 7 & 8.

⇒ n² + 1 = 7² + 1 = 50

⇒ n2 - 1 = 72 - 1 = 48

⇒ n2 + n = 72 + 7 =  56

⇒ n2 - 7 = 72 - 7 = 42

∴ 5 divides n² + 1

Concept:

Divisibility rule of 11: The given number can only be completely divided by 11 if the difference between the sum of digits at the odd position and the sum of digits at the even position in a number is 0 or 11.

Calculation:

Let the required number be n.

According to the question, 

n the remainder 7 when divided by 11. Therefore, n - 7 will be completely divisible by 11.

Now, let's check the options

Option:1   99981 - 7 = 99974 which is not divisible by 11.

Option:2   99988 - 7 = 99981 which is not divisible by 11.

Option:3   99997 - 7 = 99990 which is divisible by 11.

Option:4   99999 - 7 = 99992 which is not divisible by 11.

∴ The required value is 99997.

Formula used:

Present Value = Future Value /$[1+\frac{R× T}{100}]$ 

Calculation:

Given that 

Future Value = Rs. 20,000

Interest Rate = 5%, Time = 5 years

Present Value of A's debt = 20,000 /[1 + (0.05 ×  5)]

⇒ 20,000 / (1 + 0.25) = Rs. 16,000

Again, 

Future Value = Rs. 12,000, 

Interest Rate = 5% , Time = 4 years

Present Value of B's debt = 12,000 /[1 + (0.05 × 4)]

⇒ 12,000 / (1 + 0.20) = Rs. 10,000

Therefore, In the settlement,

A needs to give B (16,000 - 10,000) = Rs. 6,000.

Given: 

The average mark in Mathematics of Section A comprising 30 students is 65

The average mark in Section B comprising 35 students is 70.

The entry of 47 marks was wrongly made as 74 

Formula used:

Average = Sum of all observations / Number of observation

Calculation:

By using the Average formula,

Total marks of students in Section A = 30 × 65 = 1950

Total marks of students in section B = 35 × 70 = 2450

Total marks of students in section A & B = 1950 + 2450 = 4400

But, the entry of 47 marks was wrongly made as 74

Actual marks =  4400 + 47 - 74 = 4373

Again, using the above formula

Average = 4373 / (30 + 35) = 67.28

∴ The average mark for both sections is 67.28.

Shortcut Trick

For the quadratic equation x² − 7x + 1 = 0, the sum of roots (α + β) = 7 and product (αβ) = 1.

Using the property: α² + β² = (α + β)² − 2αβ = 7² − 2(1) = 47.

Then, α⁴ + β⁴ = (α² + β²)² − 2(αβ)² = 47² − 2(1)² = 2209 − 2 = 2207.

∴ The correct answer is 2207.

Alternate Method

Given: The quadratic equation is x² − 7x + 1 = 0.

Formula Used:

1. Sum of roots (α + β) = −b/a
2. Product of roots (αβ) = c/a
3. α² + β² = (α + β)² − 2αβ
4. α⁴ + β⁴ = (α² + β²)² − 2(αβ)²

Calculations:

⇒ Comparing with ax² + bx + c = 0, we have a = 1, b = −7, c = 1.

⇒ α + β = −(−7)/1 = 7

⇒ αβ = 1/1 = 1

⇒ α² + β² = 7² − 2(1) = 49 − 2 = 47

⇒ α⁴ + β⁴ = 47² − 2(1)²

⇒ α⁴ + β⁴ = 2209 − 2 = 2207

∴ The correct answer is 2207.

Additional Information

Roots and Coefficients

For a quadratic equation ax² + bx + c = 0, the sum of roots is −b/a and the product is c/a.

Algebraic Identity for Fourth Power

The sum of fourth powers can be derived iteratively from the sum of squares: x⁴ + y⁴ = (x² + y²)² − 2x²y².

Reciprocal Roots Case

If the product of roots is 1 (c = a), the roots are reciprocals of each other, satisfying the relation x + 1/x = −b/a.

Concept:

For the number "a", which can be written as,

 $a=p_1^{\alpha_1} \cdot p_2^{\alpha_2}....$ 

The total number of positive factors is given as,

T(a) = (a1+1)(a2+1)

Here, p1 and p2 are the prime numbers.

Sum of factors = $\frac{P_1^{\alpha_1+1} - 1}{P_1 - 1} × \frac{P_2^{\alpha_2+1} - 1}{P_2 - 1} × \dots$

Calculation:

360 = 2³ × 3² × 5¹

Here, 2, 3 & 5 are the only prime number.

Total number of factor = (3 + 1) × (2 + 1) × (1 + 1) = 24

Sum of factors = $\frac{2^{3+1}-1}{2-1}× \frac{3^{2+1}-1}{3-1} × \frac{5^{1+1}-1}{5-1}$

Sum of factors = (2⁴ - 1) × [(3³ - 1)/2] × [(5² - 1)/4]

Sum of factors = (15) × (13) × 6

Sum of factors = 1170

∴ The number of factors of 360 is 24 and the sum of all factors is 1170.

Calculation:

XYXYXY = 100000X + 10000Y + 1000X + 100Y + 10X + Y

XYXYXY = (100000X + 1000X + 10X) + (10000Y + 100Y + Y)

XYXYXY = 101010X + 10101Y

XYXYXY = 10101 × (10X) + 10101(Y)

XYXYXY = 10101.[10X + Y]

Prime factors of 10101 are  3 x 7 x 13 x 37

XYXYXY =  3 x 7 x 13 x 37 (10X + Y)

∴ XYXYXY is divisible by 3, 7, 13 & 37.

Concept:

HCF: It is the product of the smallest power of the common factors involved in

the prime factorization.

HCF of (a^m - 1) & (aⁿ - 1) is [a^{HCF (m, n)} - 1]

Calculation:

329 - 9 = 3² (3²⁷ - 1)

338 - 9  = 3² (3³⁶ - 1)

By using the above concept

HCF = 3² [3^{HCF (27, 36)} - 1]

Let's find the HCF of (27, 36) first

27 = 3³

36 = 2² × 3²

HCF of (27, 36) = 3²

Now, the required value

HCF = 3² (3⁹ - 1) = 3¹¹ -  9

Mistake Points Do not forget to multiply with 3² as this will also be the common prime factor.

Calculation:

Given that 

x =$√{4 √{4 √{4 √{4 \ldots}}}}$

This can be written as 

⇒ x = √4x

⇒ x² = 4x

⇒ x² - 4x = 0

⇒ x(x - 4) = 0

⇒ x = 0 & 4

∴ The required value is 4.

Shortcut Trick If y =  $\sqrt{p \sqrt{p \sqrt{p \sqrt{p \ldots}}}}$ 

than always answer is p

∴  x = $\sqrt{4 \sqrt{4 \sqrt{4 \sqrt{4 \ldots}}}}$ 

∵ x  = 4

Shortcut Trick

We are given the expression 33m + 22n, which can be factored as 11(3m + 2n).

For this to be divisible by 121 (which is 11 × 11), the term (3m + 2n) must be a multiple of 11.

To minimize the sum (m + n), we evaluate the smallest positive multiple, setting 3m + 2n = 11.

Testing natural numbers: if m = 3 and n = 1, then 3(3) + 2(1) = 11, giving m + n = 4.

Alternatively, if m = 1 and n = 4, then 3(1) + 2(4) = 11, giving m + n = 5.

∴ The correct answer is 4.

Alternate Method

Given:

Expression: 33m + 22n

Divisor: 121

m and n are natural numbers (m, n ≥ 1).

Formula Used:

If a number N is divisible by D, then N = k × D for some positive integer k.

Calculations:

⇒ 33m + 22n = 121k

⇒ 11(3m + 2n) = 11 × 11k

⇒ 3m + 2n = 11k

To minimize (m + n), we should take the smallest possible positive integer value for k. Let k = 1.

⇒ 3m + 2n = 11

⇒ 2n = 11 − 3m

Since n must be a natural number, 2n is an even positive integer. Therefore, (11 − 3m) must be an even positive integer.

Let's check possible natural number values for m:

⇒ If m = 1: 2n = 11 − 3(1) = 8 ⇒ n = 4. The sum is m + n = 1 + 4 = 5.

⇒ If m = 2: 2n = 11 − 3(2) = 5 (Not possible, since n would not be an integer).

⇒ If m = 3: 2n = 11 − 3(3) = 2 ⇒ n = 1. The sum is m + n = 3 + 1 = 4.

⇒ If m ≥ 4: 11 − 3m will be negative or zero, which cannot yield a natural number for n.

Thus, the possible pairs of (m, n) for k = 1 are (1, 4) and (3, 1).

The sums are 5 and 4 respectively. The minimum value of (m + n) is 4.

∴ The correct answer is 4.

Additional Information

Divisibility Rule of 11

A number is divisible by 11 if the difference between the sum of its digits at odd positions and the sum of its digits at even positions is either 0 or a multiple of 11.

Linear Diophantine Equations

An equation of the form ax + by = c has integer solutions if and only if the greatest common divisor of a and b perfectly divides c.

Natural Numbers

Natural numbers are the set of positive integers starting from 1 (i.e., 1, 2, 3, 4, ...). Zero, fractions, decimals, and negative numbers are not considered natural numbers.

Shortcut Trick

Using the identity: (Sum)² = Sum of squares + 2 × Product

⇒ (Sum)² = 4896 + 2 × 2160 = 4896 + 4320 = 9216

⇒ Sum of the two numbers = √9216 = 96

⇒ Mean = Sum ÷ 2 = 96 ÷ 2 = 48

(Note: The given HCF = 12 is redundant and not needed for this method)

∴ The correct answer is 48.

Alternate Method

Given:

Product of two numbers = 2160

Sum of their squares = 4896

HCF of the numbers = 12

Sum of two numbers = 96 Mean = 96 ÷ 2 = 48 Mean = 48 Mean = 48

Calculations:

⇒ Let the two numbers be 12x and 12y (where x and y are co-prime).

⇒ Product = 12x × 12y = 144xy = 2160

⇒ xy = 2160 ÷ 144 = 15

⇒ Sum of squares = (12x)² + (12y)² = 4896

⇒ 144x² + 144y² = 4896

⇒ x² + y² = 4896 ÷ 144 = 34

⇒ Using the algebraic identity: (x + y)² = x² + y² + 2xy

⇒ (x + y)² = 34 + 2 × 15 = 34 + 30 = 64

⇒ x + y = √64 = 8

⇒ Sum of the numbers = 12x + 12y = 12(x + y) = 12 × 8 = 96

⇒ Mean of the numbers = Sum ÷ 2 = 96 ÷ 2 = 48

∴ The correct answer is 48.

Additional Information

Algebraic Identity for Sum of Squares

The standard formula (a + b)² = a² + b² + 2ab directly connects the sum, sum of squares, and product of two variables without finding the individual numbers.

Properties of HCF

Any two numbers can always be expressed as (HCF × x) and (HCF × y), where x and y are co-prime integers relative to each other.

Arithmetic Mean

The arithmetic mean or average of exactly two numbers a and b is given by (a + b) ÷ 2, representing their exact central balanced value.

Calculation:

Given that

Q = P + 3

Q = 2S

We can equate these and solve for P and S.

P + 3 = 2S

⇒ P = 2S - 3            ----(1)

We also know that

R = 2P and R - S = 30,

We can equate these and solve for P and S.

2P = S + 30

⇒ P = (S + 30)/2     ------(2)

2S - 3 = (S + 30) / 2

⇒ 4S - 6 = S + 30

⇒ 3S = 36

⇒ S = 12

⇒ P =  2 × 12 - 3            [From equation (1)

⇒  P = 24 - 3 = 21

⇒ Q = P + 3 = 21 + 3 = 24

Now, the sum of the ages of P and Q

P + Q = 21 + 24 = 45 years.

Concept:

If A > B then √A > √B 

Triangle inequality theorem:

For the triangle with sides a, b & c

a + b > c, b + c > a, c + a > b

Explanation:

By using the triangle property

a + b > c

a + b - c > 0

Since, a > √a, b > √b, c > √c

∴ √a + √b - √c > 0

Concept:

LCM: LCM is the product of the greatest power of each prime factor, involved in the numbers.

Calculation:

LCM of 15, 25, 35, and 45.

15 = 3 × 5

25 = 5 × 5

35 = 5 × 7

45 = 3 × 3 × 5

LCM = 3² × 5² × 7 = 1575 min = 26.25 hr

It menas, bell ring at every 26.25 hr

Since, 72/26.25 = 2.74 

Therefore, In 72 hr, bell rings 2 times.

Concept:

For the sum of four numbers to be even, you either need to have

• All four numbers be even
• All four numbers be odd
• Two of them be even and two of them be odd

Explanation:

a + b + c + d = 200   (even)

As discussed above

Case:1 All four numbers be even

S = 1 + 1 + 1 + 1 = 4

Case:2 All four numbers be odd

S  = - 1 - 1 - 1 - 1 = - 4

Case:3 Two of them be even and two of them be odd

S = 1 + 1 - 1 - 1 = 0

∴ There will be three possible values of S.

Concept:

aⁿ - bⁿ, (where n is even number) is always divisible by (a - b) & (a + b)

Calculation:

We have 97³⁰ - 14³⁰

By using the above concept, 

9730 - 1430 is divisible by

(97 - 14) = 83

And

(97 + 14) = 111 = 37 × 3

∴ 9730 - 1430 is divisible by both 37 and 83.

Concept:

• For any base b which is a transcendental number the logarithm log_bn of every integer n ≥ 2 is an irrational number.
• A rational number is a number, which can be written as p/q where p & q are integers and q ≠ 0.

Formula used:

• log mn = log m + log n
• log_aa = 1
• log_aⁿb = (1/n) log_ab

Calculation:

Statement:1 log₁₀50 is a rational number.

log1050 = log1050 =  log10(5 × 10)

log1050 = log105 + log1010   

log1050 = log105 + 1

As discussed, log105 is an irrational number, log1050 will be also irrational.

Statement:2  log100 10 is an irrational number.

log₁₀₀10 = log_10²10 = (1/2)log₁₀10 = 1/2

Which is a rational number.

∴ Both statements are incorrect.

Calculation:

Let's say the capacity of a man is m and the capacity of a woman is w. Then,

we can set up the following equations:

5(17w + 24m) = 6(12w + 23m)

85w + 120m = 72w + 138m

13w = 18m

∴ Efficiency of 13 women = Efficiency of 18 men

Shortcut Trick

Total capacity of the tank = LCM(6, 12) = 12 units.

Efficiency of (A+B+C) = 12 ÷ 6 = 2 units/hr; Efficiency of C = 12 ÷ 12 = 1 unit/hr.

Efficiency of (A+B) = 2 − 1 = 1 unit/hr.

Work done by (A+B) in the last 8 hours = 1 × 8 = 8 units.

Remaining work done by (A+B+C) in t hours = 12 − 8 = 4 units.

t = Remaining work ÷ Efficiency(A+B+C) = 4 ÷ 2 = 2.

∴ The correct answer is 2.

Alternate Method

Given:

Time taken by (A+B+C) = 6 hr

Time taken by C = 12 hr

Time for which (A+B) worked alone after C closed = 8 hr

Formula Used:

Total Work = Efficiency × Time

Calculations:

⇒ Let the total work be 12 units (LCM of 6 and 12).

⇒ Efficiency of (A+B+C) = 12 ÷ 6 = 2 units/hr.

⇒ Efficiency of C = 12 ÷ 12 = 1 unit/hr.

⇒ Efficiency of (A+B) = Efficiency of (A+B+C) − Efficiency of C

⇒ Efficiency of (A+B) = 2 − 1 = 1 unit/hr.

⇒ Work done by all three taps in t hours = 2 × t.

⇒ Work done by (A+B) in the remaining 8 hours = 1 × 8 = 8 units.

⇒ Total Work = (Work in first t hr) + (Work in last 8 hr)

⇒ 12 = 2t + 8

⇒ 2t = 12 − 8

⇒ 2t = 4

⇒ t = 2

∴ The correct answer is 2.

Additional Information

Pipes and Cisterns Efficiency

The efficiency of multiple taps working together is the algebraic sum of their individual efficiencies. If a pipe fills the tank, its efficiency is positive; if it empties, its efficiency is negative.

Total Work Concept

Total work can be assumed as the LCM of the individual times taken to fill or empty the tank, which simplifies the calculation of efficiencies as integers.

Work Equality

Total Work = (Efficiency₁ × Time₁) + (Efficiency₂ × Time₂) + ... for different phases of the task.

Shortcut Trick

Ratio of Time taken by A : B : C = x : 1.5x : 2x = 1 : 1.5 : 2 = 2 : 3 : 4.

When persons work together for the same duration, the Payment Ratio is equal to the Efficiency Ratio.

Efficiency Ratio is the inverse of the Time Ratio = 1/2 : 1/3 : 1/4.

To simplify, multiply by the LCM of denominators (12): (12 ÷ 2) : (12 ÷ 3) : (12 ÷ 4) = 6 : 4 : 3.

∴ The correct answer is 6 : 4 : 3.

Alternate Method

Given:

Time taken by A = x days

Time taken by B = 1.5x days

Time taken by C = 2x days

Formula Used:

Efficiency = Total Work ÷ Time Taken

Wage Ratio = Efficiency Ratio (when time worked is identical for all)

⇒ The Efficiency Ratio of A, B, and C is calculated as 6 : 4 : 3.

⇒ Since they work together to complete the work, they work for the same number of days.

⇒ Ratio of Payment = Efficiency of A : Efficiency of B : Efficiency of C = 6 : 4 : 3.

∴ The correct answer is 6 : 4 : 3.

Additional Information

Relationship between Wages and Efficiency

When multiple people work for the same duration, their wages are directly proportional to their efficiencies. If they work for different durations, wages are proportional to the total work done (Efficiency × Time).

Inverse Variation

For a constant amount of work, Efficiency is inversely proportional to Time (E ∝ 1/T). If time ratio is a : b, efficiency ratio is b : a.

Total Work Calculation

Total Work is usually assumed as the LCM of the individual times taken to simplify the calculation of efficiencies as whole numbers.

Calculation:

Statement:1  n3 - n is divisible by 6.

n³ - n = n(n² - 1) = n(n - 1)(n + 1)

This product is the product of three consecutive integers which is always divisible by 6

Statement:2  n5 - n is divisible by 5

n⁵ - n = n(n⁴ - 1)

We know that (a² - b²) = (a - b)(a + b)

n5 - n = n (n² - 1)(n² + 1)

n5 - n = n (n2 - 1)[(n2 - 4) + 5)]

n5 - n = n (n2 - 1)(n2 - 4) + 5 n (n2 - 1)

n5 - n = n(n - 1)(n + 1)(n - 2)(n + 2) + 5 n (n - 1)(n + 1)

n5 - n = [(n - 2)(n - 1) n (n + 1)(n + 2)] + 5 n (n - 1)(n + 1)

Since the first part is the product of five consecutive numbers which will always be divisible by 5. Therefore, 

(n² - n) is divisible by 5.

Statement:3  n5 - 5n3 + 4n is divisible by 120.

n5 - 5n3 + 4n

n(n⁴ - 5n² + 4)

n(n² - 4)(n² - 1)

n(n - 2)(n + 2)(n - 1)(n + 1)

(n - 2)(n - 1) n (n + 1)(n + 2)

Which is always divisible by 120.

∴ All three statements are correct.

Concept:

The last digit of a number that is being raised to a power often follows a cycle.

For example, the last digit of 9ⁿ repeats every 2 numbers (9, 1, 9, 1, ...).

Similarly, 7ⁿ repeats every 4 numbers (7, 9, 3, 1, 7, 9, 3, 1, ...).

Calculation:

For 9ⁿ, the last digit is 9 if n is odd and 1 if n is even.

In this case, 9²⁷ has a last digit of 9 because 27 is odd.

For 27⁹, we only need to consider the last digit of 27, which is 7.

To find the last digit of 27⁹, we need to find the remainder when 9 is divided

by 4 (because the cycle length is 4). That remainder is 1.

 So the last digit of 27⁹ is the first digit in the cycle, which is 7.

Now

9 (from 9²⁷) + 7 (from 27⁹) = 16.

∴ The last digit of the sum S = 9²⁷ + 27⁹ is 6.

Formula used:

1. (a³ - b³) = (a - b)³ + 3ab(a -b)
2. (a - b)² = a² - 2ab + b²
3. (a + b)2 = a2 + 2ab +b2
4. (a - b)(a + b) = a2 - b2

Calculation:

Given that

x = $\frac{√{3}+1}{√{3}-1}$ and y = $\frac{√{3}-1}{√{3}+1}$

⇒ xy = $\frac{√{3}+1}{√{3}-1}$ × $\frac{√{3}-1}{√{3}+1}$ = 1       -----(1)

⇒ x - y = $\frac{√{3}+1}{√{3}-1}$ - $\frac{√{3}-1}{√{3}+1}$

⇒ x - y = $\frac{(√{3}+1)^2 - (√{3}-1)^2}{(√{3}-1)(√{3}+1)}$

By using the  formula (2), (3) & (4)

⇒ x - y = $\frac{({3}+1+2√3) - ({3}+1-2√3)}{({3}-1)}=\frac{4√ 3}{2}$

⇒ x - y = 2√3                     -----(3)

We know that 

(x3 - y3) = (x - y)3 + 3xy(x -y)

⇒ (x3 - y3) = (2√3)³ + 3 × 1 (2√3) 

⇒ (x3 - y3) = (2√3)3 + 3 × 1 (2√3) 

⇒ (x3 - y3) = 8 × 3√3 + 6√3 

∴  (x3 - y3) = 30√3

Shortcut Trick

Since time is constant, the ratio of speeds is equal to the ratio of distances covered.

Downstream Speed (S_d) : Upstream Speed (S_u) = 42 : 28 = 3 : 2

Ratio of Still Water Speed (u) to Stream Speed (v) = (3 + 2) : (3 − 2) = 5 : 1

Given 5 units = 15 km/hr ⇒ 1 unit = 3 km/hr.

∴ The correct answer is 3 km/hr.

Alternate Method

Given:

Speed of boat in still water (u) = 15 km/hr

Let the speed of the stream be v km/hr.

Downstream speed = (15 + v) km/hr

Upstream speed = (15 − v) km/hr

Formula Used:

Time = Distance ÷ Speed

Calculations:

⇒ Time downstream = Time upstream

⇒ 42 ÷ (15 + v) = 28 ÷ (15 − v)

⇒ 3 ÷ (15 + v) = 2 ÷ (15 − v) [Dividing both sides by 14]

⇒ 3 × (15 − v) = 2 × (15 + v)

⇒ 45 − 3v = 30 + 2v

⇒ 45 − 30 = 2v + 3v

⇒ 15 = 5v

⇒ v = 15 ÷ 5

⇒ v = 3 km/hr

∴ The correct answer is 3 km/hr.

Additional Information

Speed in Still Water

It is the average of downstream and upstream speeds: u = (S_d + S_u) ÷ 2.

Speed of Stream

It is half the difference between downstream and upstream speeds: v = (S_d − S_u) ÷ 2.

Time, Distance, and Speed Relationship

For a constant time, Distance₁ ÷ Distance₂ = Speed₁ ÷ Speed₂.

Formula used:

When interest is compounded half-yearly

Time = 2 × original time

Rate = (1/2) × original rate.

Calculation:

Rate = 20/2 = 10%

Time = 2 × 2 = 4 half-year

S.I = 1000 + 1000 + 1000 + 1000 = 4000

C.I = 4000 + 600 + 40 + 1 = 4641

∴ C.I - S.I = 4641 - 4000 = 641

Calculation:

We have

a(b - c) (x - b) (x - c) + b(c - a) (x - c) (x - a) + c(a - b) (x - a) (x - b)

Let's solve the only first term

a(b - c) (x - b) (x - c)

(ab - ac)[x² - x(b + c) + bc]

(ab - ac)x² - x(ab - ac)(b + c) + (ab -ac)bc

As we need the coefficient of x² & x only. Therefore, ignore the constant term.

(ab - ac)x2 - x(ab - ac)(b + c)

(ab - ac)x2 - x(ab² + abc - abc - ac²)

(ab - ac)x2 - x(ab2 - ac2)

Therefore, a simplified form of the whole expression will be as below

[(ab - ac)x2 - x(ab2 - ac2)] +

[(bc - ba)x2 - x(bc2 - ba2)] +

[(ca - cb)x2 - x(ca2 - cb2)]

Statement:1 Coefficient of x²

⇒ x² (ab - ac + bc - ba + ca - cb) = 0

Statement:1 Coefficient of x

x(ab² - ac² + bc² - ba² + ca² - cb²) 

∴ Option (1) is correct.

Concept:

The factor of (aⁿ - bⁿ) is always divisible by (a - b) & (a + b).

Calculation:

1 - x - xn + xn+1

(1 - x) - xn(1 - x)

(1 - x)(1 - xⁿ)

Since (1 - xⁿ) is the factor. Therefore

Statement 2 is correct.

By using the above concept

(1 - xⁿ) = (1 - x)(1 + x). k

Where k is the remaining factor.

1 - x - xn + xn+1 = (1 - x)(1 - x)(1 + x).k 

1 - x - xn + xn+1 = (1 - 2x + x²)(1 + x).k 

∴ Statement 1 is also correct.

Concept:

A quadratic to be a perfect square, its discriminant must be zero, as this

implies it has exactly one root (counting multiplicity).

In other words, when b² - 4ac = 0.

Calculation:

We have

mx² + (m + 8)x + 9

Here, a = m, b = m + 8, and c = 9

So now we can set up the equation b² - 4ac = 0

(m + 8)² - 4m9 = 0

But, (a + b)² = a² + 2ab + b²

⇒ m² + 16m + 64 = 36m

⇒ m² - 20m + 64 = 0

⇒ m² - 16m - 4m + 64 = 0

⇒ m (m - 16) - 4(m - 16) =

⇒ (m - 4)(m - 16) = 0

∴ m = 4, 16

Concept used:

In question there are 4 variables (a, b, x and y) and only 2 equation. Hence we put any value to a and b to get value of x and y

Calculation:

Let a = 0 and b = 1

$x=a+b+\frac{(a-b)^2}{4 a+4 b}$

⇒ $x=0+1+\frac{(0-1)^2}{0+4 \times1}$

⇒ $x=\frac{5}{4}$

Again,

$y=\frac{a+b}{4}+\frac{a b}{a+b}$

⇒ $y=\frac{0+1}{4}+0$

⇒ $y=\frac{1}{4}$

Now,

(x - a)2 - (y - b)2 = $(\frac{5}{4}-0)^2 - (\frac{1}{4}-1)^2$

⇒ $(\frac{25}{16})-(\frac{9}{16}) =\frac{25-9}{16} = 1$

Also, b² = (1)² = 1

∴ The required value is b2.

Formula used:

cos 2θ = cos² θ - sin²θ = $\frac{1-\tan ^2 θ}{1+\tan ^2 }$

cosec²θ - cot²θ = 1

(a² - b²) = (a - b)(a + b)

Calculation:

Statement: 1

$\cos ^4 θ-\sin ^4 θ$

(cos²θ + sin²θ) (cos2θ - sin2θ)

(cos2θ - sin2θ) = $\frac{2\tan θ}{1-\tan ^2θ }$

cos 2θ = tan 2θ

which is incorrect.

Statement: 2

cosec²θ - cot²θ = 1

(cosecθ - cotθ)(cosecθ + cotθ) = 1

$\operatorname{cosec} θ+\cot θ=\frac{1}{\operatorname{cosec} θ-\cot θ}$

Statement: 3

As discussed in the formula used section

$\cos ^2 θ-\sin ^2 θ=\frac{1-\tan ^2 θ}{1+\tan ^2 θ} = cos2θ$

∴ Option (2) is correct.

Shortcut Trick

In a right-angled triangle with sin θ = 12/13, the Pythagorean triplet is (5, 12, 13).

Here, Perpendicular (P) = 12, Hypotenuse (H) = 13, and Base (B) = 5.

(tan θ + sec θ) = (P + H) / B = (12 + 13) / 5 = 5.

(cosec θ − cot θ) = (H − B) / P = (13 − 5) / 12 = 8/12 = 2/3.

Expression = 5² × (2/3)⁻² = 25 × (3/2)² = 25 × 9/4 = 225/4.

∴ The correct answer is 225/4.

Alternate Method

Given: sin θ = 12/13 and 0 < θ < π/2 (First Quadrant).

Formula Used: sin θ = P/H, cos θ = B/H, tan θ = P/B, sec θ = H/B, cosec θ = H/P, cot θ = B/P.

⇒ Base (B) = √(H² − P²) = √(13² − 12²)

⇒ B = √(169 − 144) = √25 = 5

⇒ tan θ = 12/5 and sec θ = 13/5

⇒ (tan θ + sec θ) = 12/5 + 13/5 = 25/5 = 5

⇒ cosec θ = 13/12 and cot θ = 5/12

⇒ (cosec θ − cot θ) = 13/12 − 5/12 = 8/12 = 2/3

⇒ Result = (5)² × (2/3)⁻²

⇒ Result = 25 × (3/2)² = 25 × 9/4 = 225/4

∴ The correct answer is 225/4.

Additional Information

Pythagorean Triplets

Common integer sets satisfying a² + b² = c² include (3, 4, 5), (5, 12, 13), (8, 15, 17), and (7, 24, 25).

Reciprocal Trigonometric Identities

sec θ = 1/cos θ, cosec θ = 1/sin θ, and cot θ = 1/tan θ = cos θ/sin θ.

Negative Exponents

Any expression a⁻ⁿ is equivalent to 1/aⁿ. Thus (x/y)⁻² = (y/x)².

Fundamental Trigonometric Identity

sin² θ + cos² θ = 1, which helps find any ratio if one is given.

Formula used:

(x + $\frac{1}{x}$)² = (x² + $\frac{1}{x^2}$) + 2

(x + $\frac{1}{x}$) = $\sqrt{[(x^2 + \frac{1}{x^2}) + 2]}$

Calculation:

Given that 

tan8 θ + cot8 θ = m

Since, cot θ = 1/tan θ. Therefore,

tan⁸θ + (1/tan²θ) = m

Let tan θ = n then 

n⁸ + 1/n⁸ = m      ---(1)

We know that , (x + $\frac{1}{x}$) = $\sqrt{[(x^2 + \frac{1}{x^2}) + 2]}$

⇒ (n⁴ + 1/n⁴) = $\sqrt{[(n^8 + \frac{1}{n^8}) + 2]}$

⇒ (n4 + 1/n4) = $\sqrt{[m + 2]}$

Again applying the same identity

⇒ (n² + 1/n²) = $\sqrt{(\sqrt{(m + 2)}+2)}$

Again applying the same identity

⇒ (n + 1/n) = $\sqrt {\sqrt{(\sqrt{(m + 2)}+2)}+2}$

But, n = tan θ 

⇒ tan θ + 1/tan θ = $\sqrt {\sqrt{(\sqrt{(m + 2)}+2)}+2}$

∴ tan θ + cot θ = $\sqrt{\sqrt{\sqrt{m+2}+2}+2}$

Concept:

-1 ≤ sin θ ≤ 1 When θ ∈ [0, π/2]

Calculation:

f(θ) = 6- 4 sin θ 

To make it minimum, (4 sinθ) should be maximum.

Since, θ ∈ [0 ,π/2]

Maximum value of sin θ = 1 

f(θ)_min = 6 - 4 × 1 = 2

∴ Minimum value of 6- 4 sin θ is 2.

Formula used:

cos(30°) = √3/2

sin(30°) = 1/2

cot(30°) = √3

tan (90° - θ) = cot θ 

tan θ × cot θ = 1

Calculation:

4 cos2 30° + 2x sin 30° - cot2 30° - 6 tan 15° tan 75° = 0

Since, tan (90° - θ) = cot θ 

⇒ 4 cos2 30° + 2x sin 30° - cot2 30° - 6 tan (90° - 75°) tan 75° = 0

⇒ 4 cos2 30° + 2x sin 30° - cot2 30° - 6 cot 75° tan 75° = 0

But, tan θ × cot θ = 1

⇒ 4 cos2 30° + 2x sin 30° - cot2 30° - 6 = 0

Using the trigonometric values

⇒ 4(√3/2)² + 2x(1/2) - (√3)² - 6= 0

⇒ 3 + x - 3 - 6 = 0

∴ x = 6

Given:

$\frac{\cos ^2 32^{\circ}+\cos ^2 58^{\circ}}{\sec ^2 50^{\circ}-\cot ^2 40^{\circ}}$ +  4 tan 13° tan 37° tan 53° tan 77°

Concept used:

1. cos(90° - x) =  sinx
2. tan(90° - x) = cotx
3. tanx . cotx = 1
4. sin²x + cos²x = 1
5. sec²x - tan²x = 1

Calculation:

$\frac{\cos ^2 32^{\circ}+\cos ^2 58^{\circ}}{\sec ^2 50^{\circ}-\cot ^2 40^{\circ}}$ + 4 tan 13° tan 37° tan 53° tan 77°

​$\frac{\cos ^2 32^{\circ}+\cos ^2 (90^{\circ}-32^{\circ})}{\sec ^2 50^{\circ}-\tan ^2 (90^{\circ}-50^{\circ})}$ + 4 tan(90° - 77°) tan (90° - 53°) tan 53° tan 77°

​$\frac{\cos ^2 32^{\circ}+\sin ^2 32^{\circ}}{\sec ^2 50^{\circ}-\tan ^2 50^{\circ}}$ + 4 cot 77° cot 53° tan 53° tan 77°

By using the above formula,

⇒ 1/1 + 4 × 1 × 1

⇒ 1 + 4 = 5

∴ The required value is 5.

Formula used:

1 + cot2 θ =  cosec² θ 

1 + tan² θ = sec² θ 

cosec θ = 1/sin θ and sec θ = 1/cos θ 

Calculation:

(1 + cot2 θ) (1 + cos θ) (1 - cos θ) - (1 + tan2 θ) (1 + sin θ) (1 - sin θ)

⇒ cosec2 θ (1 - cos² θ) - sec² θ (1 - sin² θ) (∵ (a+b)(a-b) = a² - b²)

⇒ cosec2 θ . sin² θ - sec2θ . cos² θ   

⇒ 1 - 1 = 0

Shortcut Trick 

Put θ = π/6 from option

⇒ 2 × (√3/2)² + (1/2) - 2 = 0

⇒ 3/2 + (1/2) - 2 = 0

⇒ 0 = 0

⇒ LHS = RHS

∴ The correct answer is θ = $\frac{π}{6}$.

Alternate Method 

Formula used:

sin²θ + cos²θ = 1

Calculation:

⇒ 2cos²θ + sinθ - 2 = 0

⇒ 2(1 - sin²θ) + sinθ - 2 = 0

⇒ 2 - 2sin²θ + sinθ - 2 = 0

⇒ -2sin²θ + sinθ = 0

⇒ sinθ(-2sinθ + 1) = 0

⇒ sinθ = 0 or -2sinθ + 1 = 0

⇒ sinθ = 0, sinθ = 1/2

But  0 < θ ≤ $\frac{π}{2}$, Therefore,

⇒ sin θ = 1/2

∴ The correct answer is θ = $\frac{π}{6}$.

Shortcut Trick

Distance from tower base for 45° = h and for 30° = h√3.

Distance covered in 6 min = h√3 − h = h(√3 − 1).

At uniform speed, Time is proportional to Distance.

Time for distance h(√3 − 1) is 6 min; so time for distance h is t.

t = 6 ÷ (√3 − 1) = 3(√3 + 1) ≈ 3 × 2.732 = 8.196 minutes.

∴ The correct answer is 8.196 minutes.

Alternate Method

Given:

Initial angle of depression = 30°

Final angle of depression = 45°

Time taken for this change = 6 minutes

Speed of the car is uniform.

Formula Used:

tan θ = Perpendicular ÷ Base

Distance = Speed × Time

Calculations:

⇒ Let height of tower AB = h.

⇒ In right ΔABD, tan 45° = AB ÷ BD ⇒ 1 = h ÷ BD ⇒ BD = h.

⇒ In right ΔABC, tan 30° = AB ÷ BC ⇒ 1/√3 = h ÷ BC ⇒ BC = h√3.

⇒ Distance covered CD = BC − BD = h√3 − h = h(√3 − 1).

⇒ Speed of car (v) = Distance ÷ Time = h(√3 − 1) ÷ 6.

⇒ Time t to cover distance BD = BD ÷ v = h ÷ [h(√3 − 1) ÷ 6].

⇒ t = 6 ÷ (√3 − 1).

⇒ Rationalizing: t = [6 × (√3 + 1)] ÷ [(√3 − 1)(√3 + 1)] = 6(√3 + 1) ÷ (3 − 1).

⇒ t = 3(√3 + 1) = 3(1.732 + 1) = 3 × 2.732 = 8.196 minutes.

⇒ Since 8.196 lies between 8 and 8.3, option 2 is correct.

∴ The correct answer is 8.196 minutes.

Additional Information

Angle of Depression and Elevation

The angle of depression from the top of the tower is numerically equal to the angle of elevation from the car to the top due to alternate interior angles.

Trigonometric Ratio Shortcut

For a tower of height h, distance at 30° is h√3, at 45° is h, and at 60° is h/√3.

Uniform Speed and Time

When speed is constant, the ratio of times taken to cover two distances is equal to the ratio of the distances themselves (t₁/t₂ = d₁/d₂).

Formula used:

tan θ = Perpendicular / Base

Calculation:

From the figure,

AB = DE  & BD = AE

∠EAD = ∠ADB = 30° & ∠CAE = 60°

In an Δ ABD

tan 30° = h/BD = 1/√3

⇒ BD = h√3 = AE

In an ΔAEC

tan 60° =  CE/AE = CE/h√3

⇒ √3 = CE/h√3

⇒ CE =  3h

Therefore, the height of the tower

CD = CE + ED = 3h + h = 4h

∴ The height of the tower is 4h.

Formula used:

Area of triangle = (1/2) × base × height

Caculation:

Area of ΔABC = Area of ΔABC 

⇒ (1/2) × AC × AB = (1/2) × p × BC

⇒ AC × AB = p × BC

⇒ 5 × 12 = p × 13

∴ 13p = 60

Formula used:

Area of equilateral triangle = (√3/4) (side)²

Calculation:

OA = OB = OC = radius (r)

Since OABC is a rhombus

BC = AB = r

Therefore, ΔOAB & ΔOBC are equilateral trainge 

According to the question

Area of rhombus = 32√3

Area of ΔOAB + Area of ΔOBC = 32√3

(√3/4)r² + (√3/4)r² = 32√3

2 [(√3/4) × r²] = 32√3

r² = 64

r = 8 cm

∴ The radius of the circle is 8 cm²

Formula used:

The area of each surface of the cube = (side)²  [each surface of a cube is a square]

Calculation:

Let the initial surface area of the cube = 600

then the initial area of each surface = 100 

Initial length of side = √100 = 10 unit

Final Surface area of the cube = 600 × 125/100 = 750

Then the final area of each surface = 125

(Final length)² = 125

(Final length) = √125 

Therefore, according to the question

P = [(√125 - 10)/10] × 100 

P = 10(5√5 - 10) 

P = 10(5 × 2.236 - 10) = 11.8

∴ 10 < P < 12

Formula used:

Volume of cube of side a = a³

Total Surface area of cube = 6a²

Volume of cuobid of dimensione L, B & H = LBH 

Total surface area = 2 (LB + BH + HL)

Calculation:

Side of the cube = a.

Volume of the cube = a³

Volume of each cuboid = a³/2

Dimensions of each cuboid = a × a × a/2

Total surface area of the cube = 6 × a²

Total surface area of one cuboid

⇒ 2[a × a + a × (a/2) + a × (a/2)] = 4a² 

∴ Required ratio = 6a² : 4a² = 3 : 2

Shortcut Trick

For any cuboid, the sum of its dimensions can be found directly using the formula: Sum = √(Diagonal² + Surface Area)

⇒ Sum = √(11² + 240)

⇒ Sum = √(121 + 240)

⇒ Sum = √361 = 19 cm

∴ The correct answer is 19 cm.

Alternate Method

Given:

Length of diagonal (d) = 11 cm

Total Surface Area (S) = 240 cm²

Let the length, breadth, and height of the cuboid be l, b, and h respectively.

        l h b d = 11 cm Area = 240 cm²

Formula Used:

Diagonal of a cuboid = √(l² + b² + h²)

Total Surface Area = 2(lb + bh + hl)

Algebraic Identity: (l + b + h)² = l² + b² + h² + 2(lb + bh + hl)

Calculations:

⇒ √(l² + b² + h²) = 11

Squaring both sides:

⇒ l² + b² + h² = 121

We are given the Surface Area:

⇒ 2(lb + bh + hl) = 240

Substituting these values into the standard algebraic identity:

⇒ (l + b + h)² = 121 + 240

⇒ (l + b + h)² = 361

⇒ l + b + h = √361 = 19

∴ The correct answer is 19 cm.

Additional Information

Volume of a Cuboid

The volume of a cuboid is the product of its length, breadth, and height: V = l × b × h.

Sum of Length of All Edges

Since a cuboid has exactly 12 edges (4 each of length, breadth, and height), the sum of their lengths is calculated as 4(l + b + h).

Cube Properties

For a cube with an edge side equal to 'a', the body diagonal is given by a√3, and the total surface area is 6a².

Formula used:

The Inradius of a right-angled triangle can be given as

r = (P + B - H)/2

Area of circle = πr2

Calculation:

By using the above formula 

r = (12 + 16 - 20)/2 = 8/2 = 4 cm

Area of circle = πr²

Area of circle = π (4)² ≈ 50.27 cm² or 50 cm²