CDS Elementary Mathematics 8 Nov 2020 Official Paper (Previous Year Paper)

Given :

x3 + x2 + 16 is exactly divisible by x, where x is a positive integer

Calculations :

Let a number N be equal to x3 + x2 + 16. 

N = x3 + x2 + 16 

Now divide N/x 

N/x = (x3 + x2 + 16)/x 

⇒ x² + x + (16/x)

As N is divisible by x so 16 must be divisible by x 

Values of x that can divide 16 are 1, 2, 4, 8 and 16 

So there are total 5 values of x that can divide N without leaving the remainder.

∴ Option 3 is the correct answer.

Given :

There are 3 number a, b and c where abc = 30.

Calculations :

All divisors of 30 are 1, 2, 3, 5, 6, 10, 15 and 30. 

Selecting any 3 numbers from all divisors so that we get multiplication as 30 

1. 1 × 1 × 30 = 30 

Number of ways to arrange 1, 1 and 30 = 3!/2 = 3 

2. 1 × 3 × 10 = 30 

Number of ways to arrange 1, 3 and 10 = 3! = 6 

3. 2 × 3 × 5 = 30 

Number of ways to arrange 2, 3 and 5 = 3! = 6

4. 1 × 5 × 6 = 30 

Number of ways to arrange 1, 5 and 6 = 3! = 6 

5. 1 × 2 × 15 = 30 

Number of ways to arrange 1, 2 and 15 = 3! = 6   

Total solutions = 6 + 6 + 6 + 6 + 3

⇒ 27 ways 

∴ Option 2 will be the correct choice.

Alternate MethodNumber of the integral solution for abc = 30 are:

1 × 3 × 10 ⇒ Permutation = 3 !

15 × 2 × 1  ⇒  Permutation = 3 !

5 × 3 × 2 ⇒ Permutation = 3 !

5 × 6 × 1 ⇒ Permutation = 3 !

30 × 1 × 1 ⇒ Permutation = 3! / 2!

 

Total solutions = (3! × 4) + 3 = 27

Given:

The quadratic equation x2 - 4x - log10 N = 0 

Concept Used:

The quadratic equation ax² + bx + c = 0,

For minimum value of b² - 4ac ≥ 0

If log_a b = c 

Then b = a^c

Calculation:

The quadratic equation x2 - 4x - log10 N = 0 

Compare the above equation ax² + bx + c = 0 

Then a = 1, b = - 4 and c = - log₁₀ N

Putting the value of b² - 4ac ≥ 0

⇒ (- 4)² - 4⋅1⋅(- log₁₀ N) ≥ 0

⇒ 16 = - 4⋅log₁₀ N

⇒ log₁₀ N = - 4

⇒ N = (10)^-4

⇒ N = $\rm 1\over 10000$

∴The minimum value of N = $\rm 1\over 10000$

Given : 

x,y and z is a positive integer and x + y + z = 12 

Concept used :

When we have n positive integrals and we have to distribute them to r different things then 

No of ways = ^{n - 1}c_{r - 1} 

Calculations :

x, y and z can be 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 and 12 

Total number to select are 3 

Total number of solutions = ^{12 - 1}c_{3 - 1} = ¹¹c₂

⇒ 11!/[(11 - 2)!.2!]

⇒ 11!/(9! × 2!) 

⇒ 55 

∴ Option 3 will be the correct answer.

Given :

I = a2 + b2  + c2, where a and b are consecutive integers and c = ab.

Calculations :

Let a and b be 1 and 2 respectively    (a and b are consecutive numbers)

So, c = 1 × 2 = 2 

Now 

⇒ I = 1² + 2² + 2²

⇒ I = 9 

Now let the values of a and b be 2 and 3 respectively 

So, c = 2 × 3 = 6 

Now, 

⇒ I = 2² + 3² + 6² 

⇒ I = 49

In both cases, it is coming out to be the square of an odd integer.

∴ Option 4 will be the correct choice.

Given :

An eleven digit number 23P62971335.

Concept used :

A number is divisible by 9 when the sum of all its digits is divisible by 9. 

Calculations :

23P62971335 is divisible by 9 (the smallest odd composite number is 9) 

So sum of digits of  23P62971335 must be divisible by 9 

Sum of digits = 2 + 3 + P + 6 + 2 + 9 + 7 + 1 + 3 + 3 + 5 

⇒ 41 + P 

At P = 4 we get 45 which is divisible by 9 

∴ Option 1 will be the correct answer.

Shortcut Trick 

{1⁵ + (2²)² × 2 + (3²)² × 3 + 4⁵ + (5²)² × 5} / 4

{1 + 0 + (1 × 3) + 0 + (1 × 1)} / 4 = 5/4 = 1

∴ Option 2 will be the correct answer.

Alternate Method 

Concept Used:

Divisibility rule of 4:

A number is divisible by 4 if its last two digits are divisible by 4

Calculations :

Remainder when 1⁵  divided by 4 = 1

Remainder when 2⁵ or 32 divided by 4 = 0 

Remainder when 3⁵ or 243 divided by 4 = 3 

Remainder when 4⁵ or 1024 divided by 4 = 0 

Remainder when 5⁵ or 3125 divided by 4 = 1 

Remainder = 1 + 3 + 1 = 5 

Remainder when 5 divided by 4 = 1 

∴ Option 2 will be the correct answer.

Given:

3⁹⁹

Concept Used:

Unit digit of 3ⁿ

3¹ unit digit 3

3² unit digit 9

3³ unit digit 7

3⁴ unit digit 1

3⁵ unit digit 3... 

After 4 power unit digits repeat again.

Calculation:

⇒ 3⁹⁹ = (3⁴)²⁴ × (3³)

⇒ 3⁹⁹ = (1)²⁴ × 7 = (1) × 7

Unit place = 1 × 7 = 7

∴ The place of 3⁹⁹ is 7

Shortcut Trick 

X + 28X = 1740

⇒ x = 60

⇒ 60 × (28 × 60) = 240 × y

⇒ y = 420

∴ The other number is 420.

Alternate Method 

Given:

LCM of two number = 28 × HCF

First number = 240

HCF + LCM = 1740

Formula used:

LCM × HCF = first number × second number

Calculation:

Let HCF be x and the second number be y

HCF = x

LCM = 28x

According to the question,

The Sum of the HCF and the LCM is 1740

⇒ HCF + LCM = 1740

⇒ x + 28x = 1740

⇒ 29x = 1740

⇒ x = 60

∴ HCF = 60, LCM = 28 × 60 = 1680

By using formula,

LCM × HCF = first number × second number

⇒ 1680 × 60 = 240 × y

⇒ y = 420

∴ The other number is 420.

Calculations :

(xn - an) is divisible by (x-a) where n can be any natural number. 

∴ Option 1 will be the correct answer.

Additional InformationNatural Number:

The numbers that we use when we are counting or ordering {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 …}.

Odd Number:

A whole number that is not able to be divided by two into two equal whole numbers The numbers 1, 3, 5, and 7 are odd numbers.

Even Number:

A whole number that is able to be divided by two into two equal whole numbers The numbers 0, 2, 4, 6, and 8 are even numbers.

Prime Number:

Prime numbers are numbers that have only 2 factors: 1 and themselves. For example, the first 5 prime numbers are 2, 3, 5, 7, and 11.

Shortcut Trick

172020 is divided by 18 then

$\rm = {17^{2020}} = {{(18\ -\ 1)^{2020}}\over 18} = (-\ 1)^{2020} = 1$

Alternate Method

Concept used:

Divisibility rule of 18:

A number that is divisible by 18 must be divisible by both 2 and 9. 

Formula used:

(xⁿ - aⁿ) → divisible by (x + a) if n is even number.

Calculation:

By using the formula, we get

(xn - an) → divisible by (x + a) if n is even number.

Where,

x = 17, n = 2020, a = 1

⇒ (17²⁰²⁰ - 1²⁰²⁰)/(17 + 1) = (-1)²⁰²⁰/18

When 17²⁰²⁰ is divided by 18 getting the remainder 1

∴ The required answer is 1.

Additional InformationDivisibility rule of 2:

If a number ends in a 0, 2, 4, 6 or 8, it is divisible by 2. 

Divisibility rule of 9:

If the sum of digits of the number is divisible by 9, then the number itself is divisible by 9.

Given:

$\frac{1}{{1 + √ 2 }} + \frac{1}{{√ 2 + √ 3 }} + \frac{1}{{√ 3 + √ 4 }} + \ldots + \frac{1}{{√ {99} + √ {100} }}$

Formula used:

(a² - b²) = (a + b) (a - b)

Calculation:

Rationalizing the denominator, we get

⇒ $\frac{1}{{\left( {1 + √ 2 } \right)}} \times \frac{{\left( {√ 2 - 1} \right)}}{{\left( {√ 2 - 1} \right)}} + \frac{1}{{\left( {√ 2 + √ 3 } \right)}} \times \frac{{(√ 3 - √ {2)} }}{{\left( {√ 3 - √ 2 } \right)}} + \dots\frac{1}{{(√ {99} + √ {100)} }} \times \frac{{\left( {√ {100} - √ {99} } \right)}}{{\left( {100 - √ {99} } \right)}}$

On canceling the negative and positive term, we get

⇒ $\frac{{\left( {√ 2 - 1} \right)}}{1} + \frac{{\left( {√ 3 - √ 2 } \right)}}{1} + \frac{{\left( {√ 4 - √ 3 } \right)}}{1} + \ldots \frac{{\left( {√ {100} - √ {99} } \right)}}{1}$

⇒ $√ 2 - 1 + √ 3 - √ 2 + √ 4 - \surd 3 \ldots + √ {100} - √ {99} ;$

⇒ √100 - 1 = 9

∴ The required value is 9

Given: 

$\mathbf{x}^{\mathbf{m}}=\sqrt[14]{\mathbf{x} \sqrt{\mathbf{x} \sqrt{\mathbf{x}}}}$

Concept:

x^a⋅x^b = x^{(a + b)}

(x^a)^b = x^a⋅b

Calculation:

$\mathbf{x}^{\mathbf{m}}=\sqrt[14]{\mathbf{x} \sqrt{\mathbf{x} \sqrt{\mathbf{x}}}}$

⇒ $\rm \mathbf{x}^{\mathbf{m}}=[{\mathbf{x} \sqrt{\mathbf{x} ({\mathbf{x})^{1\over 2}}}}]^{1\over 14}$

⇒ $\rm \mathbf{x}^{\mathbf{m}}=[{\mathbf{x} {\mathbf({x})^{1\over 2} ({\mathbf{x})^{1\over 4}}}}]^{1\over 14}$

⇒ $\rm x^m = (x^{1 + {1\over 2} +{1\over 4}})^{1\over 14}$

⇒ $\rm x^m = [x^{(4 + 2 + 1)\over 4}]^{1\over 14}$

⇒ $\rm x^m = (x^{7\over 4})^{1\over 14}$

⇒ $\rm x^m = x ^{1\over 8}$

∴ The value of m = $\rm 1\over 8$

Given: 

1, 2, 3, 4, 5, - 1, - 2, - 3, - 4 and - 5

Formula used:

(a + b)² = a² + 2ab +b²

(a + b + c)² = a² + b² + c² + 2(ab + bc + ca) 

(a + b + c + d + e + ....................)2 = (a2 + b2 + c2 + d2 + e2 +......) + 2(ab + bc + ca +......) 

1² + 2² + 3² + ...... + n² = $\rm {{n\cdot(n\ +\ 1)\cdot(2n\ +\ 1)}\over 6}$

Calculation:

Let, sum of all possible products taken two at a time = X

(1 + 2 + 3 + 4 + 5 - 1 - 2 - 3 - 4 - 5)² = (1)² + (2)² + (3)² + (4)² + (5)² + (- 1)² + (- 2)²  (- 3)² + (- 4)² + ( - 5)² + 2(X)

⇒ 0 = 2[(1)2 + (2)2 + (3)2 + (4)2 + (5)2]] + 2X

⇒ 2X = - 2[(1)2 + (2)2 + (3)2 + (4)2 + (5)2]]

⇒ X = $\rm -\ {{5\ \times\ 6\ \times\ 11}\over{6}}$

⇒ X = - 55

∴ The sum of all possible products = - 55

Given:

Length of train = 110 m

Speed of train = 132 km/hr

Length of bridge = 165 m

Formula used:

Distance = Speed × Time

1 km/hr = (5/18) m/sec

Calculation:

Length of train = 110 m

Speed of train = 132 km/hr = 132 × (5/18) = 110/3 m/sec

Length of bridge = 165 m 

Total distance = 110 + 165 = 275 m

Time = Distance/Speed

Time = 275/(110/3)

⇒ Time = 15/2 = 7.5 seconds

∴ The required time is 7.5 seconds.

Shortcut Trick

When the simple interest is a fraction of the principal and Rate = Time:

Time = √((SI ÷ P) × 100)

Given SI ÷ P = 1 ÷ 4

⇒ Time = √((1 ÷ 4) × 100) = √25 = 5 years

∴ The correct answer is 5 years.

Alternate Method

Sum = P

R = T

SI = P/4

SI = (P × R × T) ÷ 100

Simple Interest Analysis

Given:

Simple Interest (SI) = P ÷ 4

Rate of Interest (R) = Time in years (T)

Formula Used:

SI = (P × R × T) ÷ 100

Calculations:

⇒ P ÷ 4 = (P × T × T) ÷ 100

⇒ 1 ÷ 4 = T² ÷ 100

⇒ T² = 100 ÷ 4

⇒ T² = 25

⇒ T = √25 = 5

∴ The correct answer is 5 years.

Additional Information

Direct Formula for Equal Rate and Time

When Simple Interest is 1/n of the principal and the rate percentage equals the time in years, directly apply T = √(100 ÷ n).

Simple Interest vs Compound Interest

For the first year, both SI and CI are equal for a given principal and rate of interest. After the first year, CI calculates interest on the accumulated sum.

Rule of 72

A mental math shortcut to estimate the doubling time of an investment under compound interest: Time ≈ 72 ÷ Annual Interest Rate.

Given :

A 60-page book has n lines per page, when the number of lines were reduced by 3 on each page, then the number of pages will increase by 10. 

Calculations :

Total number of lines = Page × Lines per page 

⇒ 60 × n = 60n 

Now the number of lines when 3 lines are reduced = n - 3 

Number of pages when lines are reduced = 70 

Total number of lines = 70 × (n - 3) = 70n - 210 

So, 

60n = 70n - 210 

⇒ 10n = 210 

⇒ n = 21  

∴ Option 2 is the correct answer.

Given : 

x men working x hours per day can do x units of work in x days

Shortcut Trick

Formula Used:

M1D1H1/W₁ = M2D2H2/W2

Where, M = men, D = days, H = hours, w = works

Calculations:

x.x.x/x = y.y.y/k

⇒ x² = y³/k

⇒ k =  y3/x² = y³x^-2

∴ The correct answer is option (4).

Alternate Method

Concept used :

Total work = Number of persons × Total time × Each person's work per unit time 

Calculations :

Let the work done by each person per hour be 'w' units 

According to the questions 

x = x × x × x × w         (total time - total hours × total days)

⇒ w = 1/x²

Now work done by y men in y hours and y days 

Total work(k) = y × y × y × (1/x²) 

⇒ y³/x²

⇒ y³x^-2 

∴ Option 4 will be the correct choice.

Given :

d(n) is the number of positive divisors of a positive number n. 

Calculations :

d(5) = 2       (positive divisors of 5 are 1 and 5) 

d(11) = 2     (positive divisors of 11 are 1 and 11) 

d(16) = 5     (positive divisors of 16 are 1, 2, 4, 8 and 16) 

d(55) = 4     (positive divisors of 55 are 1, 5, 11 and 55)

Checking Choices 

1. d(5) = d(11)

⇒ 2 = 2 

2. d(5).d(11) = d(55) 

⇒ 2 × 2 = 4 

⇒ 4 = 4 

3. d(5) + d(11) = d(16) 

⇒ 2 + 2 = 5 

We can conclude that choice 1 and 2 are correct. 

∴ Option 2 is the correct answer.

Given: 

An = Pn + 1, where $P_n$ is the product of the first n prime numbers

Concept:

Odd number + odd number = always an even number 

Odd number + even number = always an odd number 

Even number + enev number = always an even number

Calculation:

Here P_n = 2, 3, 5, 7, 11, 13,............

We  know that when we add one in the product of first prime numbers it becomes also a prime number

So, A_n is a prime number, not a composite number

All the prime numbers (except 2) are an odd number 

∵ Odd number + even number = always an odd number 

So, A_n + 2 is always an odd number. 

 ∵ Odd number + odd number = always even number    

So, A_n +  1 is always an even number.       

∴ Option (4) is correct.

Alternate Method

Prime  numbers are 2, 3, 5, 7, 11, ......

Let n = 1

⇒ P1 = 2 (i.e. product of 1 prime number)

⇒ A1 = P1 + 1 = 2 + 1 = 3  (i.e. odd prime number)

Hence, A1 is a prime number

A1 + 2 = 3 + 2 = 5  (i.e. odd number)

Hence, statement 2 is correct.

A1 + 2 = 3 + 1 = 4  (i.e. even number)

Hence, statement 3 is also correct.

∴ Option (4) is correct.

Shortcut Trick

For faulty weight problems where the article is sold at cost price:

Profit % = [(Stated Weight − Actual Weight) ÷ Actual Weight] × 100

Here, Stated Weight = 1000 gm and Actual Weight = 800 gm.

Profit % = [(1000 − 800) ÷ 800] × 100 = [200 ÷ 800] × 100 = 1/4 × 100 = 25%

∴ The correct answer is 25%.

Alternate Method

Given:

Reading on balance = 1000 gm (Quantity for which customer pays)

Actual weight delivered = 800 gm (Quantity that actually goes out of stock)

Selling price = Cost price (per unit weight)

Formula Used:

Profit = Selling Price (SP) − Cost Price (CP)

Profit % = (Profit ÷ CP) × 100

Let the Cost Price (CP) of 1 gm of the article be &inr;1.

⇒ Shopkeeper pays for 800 gm = &inr;800 (This is the effective CP).

⇒ Shopkeeper charges the customer for 1000 gm = &inr;1000 (Since SP = CP, this is the effective SP).

⇒ Total Profit = &inr;1000 − &inr;800 = &inr;200.

⇒ Profit % = (Profit ÷ Effective CP) × 100

⇒ Profit % = (200 ÷ 800) × 100

⇒ Profit % = 1/4 × 100 = 25%

∴ The correct answer is 25%.

Additional Information

Dishonest Dealer Formula

Profit % = [Error ÷ (True Value − Error)] × 100, where Error is the difference between the weight shown and the actual weight used.

Profit with Mark-up

If a dealer marks up price by x% and uses y% less weight, Net Profit % = [(x + y) ÷ (100 − y)] × 100.

Effective Ratio Method

Profit ratio can be found by multiplying individual ratios: (Stated Weight/Actual Weight) × (SP/CP). If SP=CP, the ratio is simply Stated Weight : Actual Weight.

Shortcut Trick

Area of river = depth × width

⇒ Area = 3 × 40 = 120 m²

Amount of water that falls into the sea = Area of river × flow rate

⇒ 120 m² × 2 km/h × 1000           (∵ 1 m³ = 1000 l)

⇒ 120 × (2000/60) m/min × 1000

⇒ 40,00,000 l

Given:

Depth of a river = 3 m

Width of a river = 40 m

Water flowing at the rate of 2 km/hr

Concept used:

Area of cross section of the river = L × B

Volume = Area × Rate of flow

1 m³ = 1000 litre

Calculation:

Depth of a river = 3 m

Width of a river = 40 m

Water flowing at the rate of 2 km/hr

2 km/hr =  (2000/60) m/min

Area of cross section of the river = L × B

Area of cross section of the river = 3 × 40 = 120 m

Volume of water flowing through this cross section in every minute = Area × Rate of flow

⇒ 120 × (2000/60)

⇒ 4000 m³

Now,

1 m3 = 1000 litres

4000 × 1000 = 4000000 litres

∴ The amount of water flow in one minute is 4000000 litres.

Shortcut TrickLet the cost price = C

y = (C × 112)/100

x = (C × 72)/100

y/x = 112/72 = 14/9

∴ The correct answer is 14:9.

Solution:

Given:

SP = Rs. x

Loss = 28%

SP = Rs. y

Profit = 12%

Concept used:

Loss% = (Loss/CP) × 100 

Calculation:

Let the cost price be Rs. 100

If television is sold at Rs. x, then

Loss % = 28%

Loss % = (Loss/CP) × 100 

28 = (Loss/100) × 100

⇒ Loss = 28

SP = CP - Loss

SP = 100 - 28 = Rs. 72

⇒ x = Rs. 72

Now,

If television is sold at Rs. y, then

Profit % = 12%

12 = (Profit/100) × 100

Profit = Rs. 12

SP = CP + Profit

SP = 100 + 12 = Rs. 112

⇒ y = Rs. 112

We have to find y/x 

y/x = 112/72 = 14/9

∴ The ratio of y to x is 14 : 9.

Alternate Method

Let CP be Rs. 100

Loss = 28%

SP(x) = 100 - 28 = Rs. 72

Profit = 12%

SP(y) = 100 + 12 = Rs. 112

y/x = 112/72 = 14/9

∴ The ratio of y to x is 14 : 9.

Shortcut TrickWe know that Time = Distance / Speed

Let the original speed be x km/hr

As per the question,

⇒ (300/x) - [300/(x + 15)] = 1

Now, we will check through the options.

Check for those values which will divide 300 completely and give us an integral value.

As we can understand easily when x = 60 km/hr,

⇒ (300/60) - [300/(60 + 15)] = 1

∴ The original speed of the car is 60 km/hr.

Given:

Distance = 300 km

Speed increased by car = 15 km/hr

Time difference = 1 hour

Formula Used:

Distance = {(S₁ ×  S₂)/(S₂ - S₁)} × Time difference

Where,

S₁ = Original speed

S₂ = Increased speed

Calculation:

Let the original speed of the car be x km/hr

⇒ Increased speed of car = (x + 15) km/hr

According to the question, we have

300 = [{x × (x + 15)}/(x + 15 - x)} × 1

⇒ 300 = (x² + 15x)/15

⇒ 4500 = x2 + 15x

⇒ x2 + 15x - 4500 = 0

⇒ x² + 75x - 60x - 4500 = 0

⇒ x(x + 75) - 60(x + 75) = 0

⇒ (x + 75)(x + 60) = 0

⇒ x = 60 and (- 75)

Now, the negative distance is not possible

 x = 60 km/hr

∴ The original speed of car is 60 km/hr.

Shortcut Trick 

                   A                :          B              :          C

{(20 × 4) + (10 × 8)}     :      (15 × 12)       :     (12 × 12)

                  40              :          45              :         36

⇒ A's share = (96800 / 121) × 40 = 32000

∴ The share of the first person in profit is Rs. 32,000.

Alternate Method 

Given:

The ratio of capital invested by three persons = (1/3) ∶ (1/4) ∶ (1/5)

Total profit = Rs. 96,800

Calculation:

Let the three persons be A, B and C

According to the question, we have

A ∶ B ∶ C = (1/3) ∶ (1/4) ∶ (1/5)      ----(1)

LCM of 3, 4 and 5 is 60

Multiply of 60 in equation (1)

⇒ A ∶ B ∶ C = 20 ∶ 15 ∶ 12

⇒ So, the investment of A, B and C are 20x, 15x and 12x

Now, according to the question

A withdraws half of his capital after 4 months

⇒ A's investment after 12 months =(20x × 4) + (10x × 8)

⇒ 80x + 80x

⇒ 160x

B's investment after 12 months = 15x × 12

⇒ 180x

⇒ C's investment after 12 months = 12x × 12

⇒ 144x

Total investment = 160x + 180x + 144x

⇒ 484x

Now,

Total profit  = Rs. 96,800

⇒ Profit of A's = (96800 × 160x)/(484x)

⇒ 32000

∴ The share of the first person in profit is Rs. 32,000.

Given:

1. x2 + y2 varies as x2 - y2 2. $\frac{x}{y^2}$ varies inversely as y 3. $\sqrt[n]{x^2y}$ varies as $\sqrt[2n]{x^4y^2}$

Concept:

If x varies as y 

then x = ky

Calculation:

Check the options:

1. x2 + y2 varies as x2 - y2

Put x = ky

x² + y² = (ky)² + y² = y²(k² + 1)

And x² - y² = (ky)² - y² = y²(k² - 1)

So, $\rm {{x^2\ +\ y^2}\over{x^2\ -\ y^2}} = {{y^2(k^2\ +\ 1)}\over {y^2(k^2\ -\ 1)}} = {{(k^2\ +\ 1)}\over {(k^2\ -\ 1)}}$

$\rm x^2 + y^2 = {{(k^2\ +\ 1)}\over {(k^2\ -\ 1)}}(x^2 - y^2)$

Then x² + y² varies as x² - y²

2. $\frac{x}{y^2}$ varies inversely as y

Put x = ky

$\rm {x\over y^2} = {ky\over y^2} = {k\over y}$

So, $\rm x\over y^2$ is inversely proportional to y

3. $\rm \sqrt[n]{x^2y}$ varies as $\rm \sqrt[2n]{x^4y^2}$ 

Put x = ky

$\rm \sqrt[2n]{x^4y^2} = \sqrt[2n]{(x^2y)^2} = \sqrt[n]{(x^2y)}$

So,  $\rm \sqrt[n]{x^2y}$ varies with $\rm \sqrt[2n]{x^4y^2}$ 

∴ 1, 2 and 3 all are correct.

Shortcut TrickEna's age = 13 years

Ena' mother age = 13 + 24 = 37 years

Ena' father age = 37 + 3 = 40 years

The age of Ena's father at the time of marriage = 40 - (13 + 4)

⇒ (40 - 17) years = 23 years

Given:

Ena born after the parent's marriage = 4 years

Age of father - Age of mother = 3 years

Age of mother - Age of Ena = 24 years

The present age of Ena = 13 years

Calculation:

According to the question, we have

Age of mother - Age of Ena = 24 years

⇒ Age of mother - 13 = 24

⇒ Age of mother = 24 + 13

⇒ Age of mother = 37 years

Now, according to the question

Age of father - Age of mother = 3 years

⇒ Age of father - 37 = 3

⇒ Age of father = 3 + 37

⇒ Age of father = 40 years

Now,

The age of Ena's father at the time of marriage = 40 - (13 + 4)

⇒ 40 - 17

⇒ 23 years

∴ The age of Ena's father at the time of marriage is 23 years.

Shortcut TrickMahesh's age = 60 years

Ram's age = Mahesh's age - 6 years = 55 years

Raju's age = Ram's age - 4 years = 51 years

Babu's age = Raju's age - 6 years = 45 years

Difference between Mahesh and Babu age = (60 - 45) years

⇒ 15 years

Given:

Mahesh = 60 years

Mahesh - Ram = 5 years

Ram - Raju = 4 years

Raju - Babu = 6 years

Calculation:

According to the question, we have 

Mahesh - Ram = 5 years

⇒ 60 - Ram = 5

⇒ Ram = 60 - 5

⇒ Ram = 55 years

Now, according to the question

Ram - Raju = 4 years

⇒ 55 - Raju = 4

⇒ Raju = 55 - 4

⇒ Raju = 51 years

Now, according to the question

Raju - Babu = 6 years

⇒ 51 - Babu = 6

⇒ Babu = 51 - 6

⇒ Babu = 45 years

Now, the age difference between Mahesh and Babu 

Mahesh - Babu = 60 - 45

⇒ 15 years

∴ The age difference between Mahesh and Babu is 15 years.

Shortcut TrickThe formula for Successive Percentage change for 2 years

⇒ x + y + (xy/100)

As per question,

⇒ Effective change in no. of items for 2 yr = x - x - (x²/100)

⇒ Effective change in no. of items for 2 yr = -x2/100

⇒ Negative sign shows decrease.

So, after 2 years the number of items = N - x2/100

∴ The number of items will be less than N.

Concept used :

If there are N number of items and there is an increment and decrement of x% respectively in the amount then final value of N 

N = N.(100 + x)/100).(100 - x)/100) 

Calculations :

Let the value of N be 100 

Let the value of x be 10 

Final number of products = 100 × (100 + 10)/100) × (100 - 10)/100) 

⇒ 100 × (110/100) × (90/100) 

⇒ 99 

Final amount is less than the initial amount(N)

∴  Option 1 will be the correct choice.

Given :

ab + xy - xb = 0 and bc + yz - cy = 0

Calculations :

According to the question 

ab + xy - xb = 0 

⇒ ab = xb - xy 

⇒ ab = x(b -y) 

⇒ x/a = b/(b - y) 

Now 

bc + yz - cy = 0 

⇒ yz = cy - bc 

⇒ yz = c(y - b) 

⇒ c/z = y/(y - b) 

Now, 

(x/a) + (c/z) =  (b/(b - y) + (y/(y - b)

⇒ (b/(b - y) - (y/(b - y)

⇒ (b - y)/(b - y) 

⇒ 1 

∴ Option 3 will be the correct answer.

Given:

Polynomial f(x) = x6 - 3x4 + 3x2 - 1 

And polynomial g(x) = x3 + 3x2 + 3x + 1

Formula Used:

(x² - y²) = (x + 1)(x - y)

(x + y)³ = x³ + 3x²y + 3xy² + y³

(x - y)3 = x3 - 3x2y + 3xy2 - y3

Shortcut Trick

f(x) = x6 - 3x4 + 3x2 - 1

⇒ f(x) = (x²)³ - 3(x²)².1 + 3.1.(x²) - 1

⇒ f(x) = (x² - 1)³

⇒ f(x) = (x - 1)3 (x + 1)3

g(x) = x3 + 3x2 + 3x + 1

⇒ g(x) = (x + 1)3

Thus,HCF of f(x) and g(x) = (x + 1)3

∴ The correct answer is option (4).

Alternate Method

Calculation:

According to the question, we have

f(x) = x6 - 3x4 + 3x2 - 1 

⇒ x⁶ - x⁴ - 2x⁴ + 2x² + x² - 1

⇒ x⁴(x² - 1) - 2x²(x² - 1) +1(x² + 1)

⇒ (x² - 1)(x⁴ - 2x² + 1)

⇒ (x + 1)(x - 1)(x4 - 2x2 + 1)

⇒ (x + 1)(x - 1)(x² - 1)²

⇒ (x + 1) (x - 1) (x + 1)² (x - 1)²

⇒ (x + 1)³(x - 1)³      ----(1)

And,

g(x) = x3 + 3x2 + 3x + 1

⇒ x³ + x² + 2x² + 2x + x + 1

⇒ x²(x + 1) + 2x(x + 1) + 1(x + 1)

⇒ (x + 1)(x² + 2x + 1)

⇒ (x + 1) (x + 1)²

⇒ (x + 1)³      ----(2)

Now,

The HCF of f(x) and g(x) is the common factor of equations (1) and (2)

⇒ (x + 1)³

∴ The HCF of the polynomials x6 - 3x4 + 3x2 - 1 and x3 + 3x2 + 3x + 1 is (x + 1)³.

Given:

The HCF of two polynomials = 3x + 1

The LCM of two polynomials = 30x3 + 7x2 - 10x - 3

One polynomial = 6x2 + 5x + 1

Formula used:

The product of two polynomials = Product of their HCF and LCM

Calculation:

Let the two polynomials be f(x) and g(x).

Here, f(x) = 6x2 + 5x + 1

According to the question,

f(x) × g(x) = (3x + 1) × (30x3 + 7x2 - 10x - 3)

⇒ (6x2 + 5x + 1) × g(x) = (3x + 1) × (30x3 + 7x2 - 10x - 3)

⇒ (6x2 + 2x + 3x + 1) × g(x) = (3x + 1) × (30x3 + 15x2 - 8x2 - 4x - 6x - 3)

⇒ [2x(3x + 1) + 1(3x + 1)] × g(x) = (3x + 1)[15x²(2x + 1) - 4x(2x + 1) - 3(2x + 1)] 

⇒ (3x + 1) × (2x + 1) × g(x) = (3x + 1) × (2x + 1) × (15x² - 4x - 3)

⇒ g(x) = (15x2 - 4x - 3)

∴ The other polynomial is 15x2 - 4x - 3.

Given:

(p + 2)(2q - 1) = 2pq - 10 

(p - 2)(2q - 1) = 2pq - 10

Calculation:

(p + 2)(2q - 1) = 2pq - 10

⇒ 2pq - p + 4q - 2 = 2pq - 10

⇒ 4q - p + 10 - 2 = 0     

⇒ 4q - p + 8 = 0      ----(1)

(p - 2)(2q - 1) = 2pq - 10

⇒ 2pq - p - 4q + 2 = 2pq - 10

⇒ -4q - p + 10 + 2 = 0

⇒ -4q - p + 12 = 0     

⇒ 4q + p -12 = 0      ----(2)

Now, add equation (1) and (2)

4q - p + 8 + 4q + p -12 = 0

⇒ 8q - 4 = 0

⇒ 8q = 4

⇒ q = 1/2

By putting the value of q in equation (1), we get

4(1/2) - p + 8 = 0

⇒ 2 - p + 8 = 0

⇒ 10 - p = 0

⇒ p = 10

Now, pq = 10 × (1/2)

⇒ 5

∴ The value of pq is 5.

<br>

Mistake Points

We can't compare p + 2 = p - 2 as these two are the terms of the two

given equations. When we divide the given equation we get 0/0 form.

So we can't do this, and to satisfy the equations you have to put the

value of p and q separately in both the equations.

Given:

The given expression is \(\frac{a^{2}~+ac}{a^{2}c-c^{3}}-\frac{a^{2}-c^{2}}{a^{2}c+2ac^2+c^3}-\frac{2c}{a^2-c^2}+\frac{3}{a+~c}\)

Shortcut TrickLet, a  =0, c = 1

Put these values on a given expression, and we get

⇒  $0-\frac{-1}{1}-\frac{2}{-1}+\frac{3}{1}$

⇒  1 + 2 + 3 = 6

From option 4

6/(0 + 1) = 6

∴ Option 4 is correct.

 Alternate Method

\(\frac{a^{2}~+ac}{a^{2}c-c^{3}}-\frac{a^{2}-c^{2}}{a^{2}c+2ac^2+c^3}-\frac{2c}{a^2-c^2}+\frac{3}{a+~c}\)

⇒ \(\frac{a(a~+c)}{c(a^2-c^2)}-\frac{a^2-c^2}{c(a^2+2ac+c^2)}-\frac{2c}{a^2-c^2}+\frac{3}{a+~c}\)

⇒ \(\frac{a(a~+c)}{c(a-c)(a+c)}-\frac{(a-c)(a+c)}{c(a+c)^2}-\frac{2c}{a^2-c^2}+\frac{3}{a+~c}\)

⇒ \(\frac{a}{c(a~-c)}-\frac{(a-c)}{c(a+c)}-\frac{2c}{(a-c)(a+c)}+\frac{3}{a+~c}\)

⇒ \(\frac{a(a~+c)-(a-c)^2-2c^2+3c(a-c)}{c(a-c)(a+~c)}\)

⇒ \(\frac{a^2~+ac-(a^2-2ac+c^2)-2c^2+3ca-3c^2}{c(a-c)(a+~c)}\)

⇒ \(\frac{a^2~+ac-a^2+2ac-c^2-2c^2+3ca-3c^2}{c(a-c)(a+~c)}\)

⇒ \(\frac{6ac~-6c^2}{c(a-c)(a+~c)}\)

⇒ \(\frac{6c(a~-c)}{c(a-c)(a+~c)}\)

⇒ $\frac{6}{a~+~c}$

∴ The value of \(\frac{a^{2}~+ac}{a^{2}c-c^{3}}-\frac{a^{2}-c^{2}}{a^{2}c+2ac^2+c^3}-\frac{2c}{a^2-c^2}+\frac{3}{a+c}\) is \(\frac{6}{a+~c}\).

Given:

The given expression is 4x⁴ + 8x³ - 4x + 1

Formula used:

(a + b - c)² = a² + b² + c² + 2ab - 2bc - 2ca

Calculation:

4x4 + 8x3 - 4x + 1

⇒ 4x4 + 8x3 - 4x + 1 + 4x² - 4x²

⇒ 4x4 + 4x2 + 1 + 8x3 - 4x - 4x2

⇒ (2x²)² + (2x)² + (1)² + 2 × 2x2 × 2x - 2 × 2x × 1 - 2 × 2x2 × 1

⇒ (2x² + 2x - 1)²

So, $\sqrt{4x^4 + 8x^3 - 4x + 1}$

⇒ $\sqrt{(2x^2 + 2x - 1)^2}$

⇒ 2x2 + 2x - 1

∴ The square root of 4x4 + 8x3 - 4x + 1 is 2x2 + 2x - 1.

Alternate Method x = 1

4x4 + 8x3 - 4x + 1

4 + 8 - 4 + 1 = 9

From option (4)

2x2​ + 2x - 1

2 + 2 - 1 = 3

3² = 9

Shortcut Trick 

⇒ x + y = 13    ......(1)

⇒ (10y + x) - (10x + y) = 27

⇒ y - x = 3        .....(2)

⇒ x × y = 5 × 8 = 40

∴ The product of the digits of the number is 40.

Alternate Method 

Given:

The sum of the digits of a two-digit number = 13

The difference between the number and that formed by reversing the digits = 27

Calculation:

Let the digit in the 10th place and unit place are a and b respectively.

So, the two-digit number is 10a + b.

According to the question,

a + b = 13      ----(1)

Again, according to the question,

(10b + a) - (10a + b) = 27

⇒ 10b + a - 10a - b = 27

⇒ 9b - 9a = 27

⇒ 9(a - b) = 27

⇒ a - b = 3      ----(2)

By adding equ(1) and equ(2), we get

(a + b) + (a - b) = 13 + 3

⇒ 2a = 16

⇒ a = 8

From equ(1), b = 13 - a

⇒ 13 - 8 = 5

Now, the product of the digits = 8 × 5 = 40

∴ The product of the digits of the number is 40.

Given :

x/(b + c) = y/(c + a) = z/(b - a) 

Calculations :

Let, x/(b + c) = y/(c + a) = z/(b - a) = k 

⇒ x = (b + c)k = bk + ck 

⇒ y = (c + a)k = ck + ak 

⇒ z = (b - a)k = bk - ak 

Now check for each option given 

1. x + y + z = bk + ck + ck + ak + bk - ak 

⇒ 2(bk + ck) 

2. x - y - z = bk + ck - (ck + ak ) - (bk - ak) 

⇒ bk + ck - ck - ak - bk + ak

⇒  0 

3. x + y - z = bk + ck + ck + ak - (bk - ak) 

⇒ bk + ck + ck + ak - bk + ak

⇒ 2(ak + ck)

4. x + 2y + 3z = bk + ck + 2(ck + ak) + 3(bk - ak) 

⇒ bk + ck + 2ck + 2ak + 3bk - 3ak 

⇒ 4bk + 3ck - ak 

∴ Option 2 will be the correct choice.

Given:

Speed of X, Y and Z = 4 km/h, 5 km/h and 6 km/h.

Formula used:

Distance = Speed × Time

The relative speed of Y with respect to X = (S_x - S_y)

Where S_x and S_y are the speed of X and Y respectively.

Calculation:

Distance covered by X in 2 hours = 4 × 2 = 8 km

Time required for Y to cover this distance = 8/(5 - 4) = 8 hours

Let Z starts t hours after Y.

According to the question,

Distance covered by Y = Distance covered by Z 

⇒ 8 × 5 = (8 - t) × 6

⇒ 40 = 48 - 6t

⇒ 6t = 8

⇒ t = 4/3 

∴ Z starts travelling 4/3 hours after Y.

Given:

$1-x-x^{n}+x^{n+1}$

Shortcut Trick

Concept used:

aⁿ - bⁿ is divisible by (a - b); if n = odd

Calculations:

1 - x - xⁿ + xⁿ⁺¹

⇒ (1 - x) - xn​ (1 - x)

⇒ (1 - x)(1 - xn)

Now, (1 - xn) = (1ⁿ - xn) 

Here, n = odd, so it is divisible by (1 -x)

Thus, expression is divisible by 

⇒ (1 -x)(1 -x) = (1 -x)²

∴ The correct answer is option (2).

Alternate Method

Formula Used:

(1 - xn) = (1 - x)(1 + x + x2 + x3 + ...... + xn - 1)

Calculation:

Let, f(x) = $1-x-x^{n}+x^{n+1}$

= (1 - x) - xⁿ(1 - x)

= (1 - x)(1 - xⁿ)

= (1 - x)(1 - x)(1 + x + x² + x³ + .....+ x^{n - 1})             [∵ (1 - xⁿ) = (1 - x)(1 + x + x² + x³ + ...... + x^{n - 1})]

= (1 - x)²(1 + x + x2 + x3 + .....+ xn - 1)

∴  $1-x-x^{n}+x^{n+1}$ is divisible by (1 - x)²

Given:

Cost price of the article = Rs. x

Selling price of the article = Rs. 75

Profit% = x%

Formula used:

Profit% = [(S.P - C.P)/C.P] × 100

Calculation:

According to the question,

x = [(75 - x)/x] × 100

⇒ x² = 7500 - 100x 

⇒ x(x + 100) = 7500

⇒ x(x + 100) = 50(50 + 100)

On comparing both sides,

x = 50

∴ The value of x is 50.

Given:

Time taken to complete the journey = t hours

Time taken to complete the journey after increasing speed = (t - y) hours

Formula used:

Distance = Speed × Time

Calculation:

Let the distance be D and the initial speed of the car be a kmph.

According to the first condition,

D = at      ----(1)

According to the second condition,

D = (a + x)(t - y)      ----(2)

From equation (1) and equation (2),

at = (a + x)(t - y)

⇒ at = at + xt - ay - xy

⇒ ay = x(t - y)

⇒ a = x(t - y)/y

Put the value of a in equation 1,

D = x(t - y)ty^-1

∴ The distance is x(t - y)ty^-1 km.

Given:

Time α √Distance

Time required to cover 78.4-meter distance = 4 s

Concept used:

If time varies as the square root of the distance, then (t₁²/t₂²) = d₁/d₂.

Where, t₁ and t₂ is the time taken whereas d₁ and d₂ is the distance.

Calculation:

We know that,

(t12/t22) = d1/d₂

⇒ 4²/t₂² = 78.4/122.5

⇒ t₂² = (122.5/78.4) × 16

⇒ t₂ = √25

⇒ t₂ = 5

∴ The required time is 5 seconds.

Given:

63 - 4x 4x + 5 = 8 

Concept:

a^m⋅aⁿ = a^{m + n}

$\rm {a^m\over a^n}$ = a^{m - n}

(a⋅b)ⁿ = aⁿ⋅bⁿ

log a^b = b⋅log a

Calculation:

Given: 63 - 4x 4x + 5 = 8 

⇒ 3^{3 - 4x}⋅ 2^{3 - 4x}⋅ 2^{2x + 10} = 2³                                         (∵(a⋅b)n = an⋅bⁿ)

⇒ 33 - 4x⋅ 23 - 4x + 2x + 10 = 2³                                          (∵am⋅an = a^{m + n})

⇒ 33 - 4x⋅ 2^{13 - 2x} = 2³

⇒ 3^{3 - 4x} = $\rm {2^3\over {2^{13\ -\ 2x}}}$

⇒ 3^{3 - 4x} = 2^{3 - (13 - 2x)}                                                     (∵$\rm {a^m\over a^n}$ =  a^{m - n})

⇒ 3^{3 - 4x} = 2^{2x - 10}

Taking log both side

⇒ log (3^{3 - 4x}) = log (2^{2(x - 5)})

⇒ (3 - 4x)⋅log 3 = 2(x - 5)⋅log 2                                     (∵ log ab = b⋅log a)

⇒ (3 - 4x)(0.477) = 2(x - 5)(0.301)

⇒ 1.4313 - 1.9084x = 0.602x - 3.010

⇒ 1.4313 + 3.010 = 0.602x + 1.9084x

⇒ 4.4413 = 2.5104x

⇒ x = $\rm {4.4413\over 2.5104}$ 

⇒ x = 1.7692

∴ The value of x lies in between 1 < x < 2.

Concept:

Euclid's algorithm: 

• Euclid's algorithm is basically used to find the GCD of two numbers
• If a and b are two positive integers, then there exist a unique positive integers q and r such that;

a = bq + r, where 0 ≤ r < b 

If b ∣ a then r = 0

Otherwise r must satisfy stronger inequality 0 < r < b.

Explanation:

Using the above concept, we can conclude that,

A euclidean algorithm is used to calculate HCF of two integers.

Given : 

Radius of the sphere is rational 

Concept used : 

If r is the radius of the sphere then, 

Surface area of sphere = 4π r²

Volume of the sphere = (4/3)π r³

π is an irrational number 

Product of a rational and irrational number is always irrational

Calculations : 

Let the radius of the sphere be x cm 

Surface area of the sphere = 4π x = 4x²π     

Volume of the sphere = (4/3)π x = (4/3)x³π 

We can conclude that both surface area and the volume are irrational 

∴ Option 4 is the correct answer.

Given : 

cosec θ - sin θ = m 

sec θ - cos θ = n 

Calculations :

Put θ = 45° 

cosec 45°  - sin 45° = m 

⇒ m = √2 - (1/√2) 

⇒ m = 1/√2 

sec 45° - cos 45° = n 

⇒ n = √2 - (1/√2) 

⇒ n = 1/√2   

Putting value of n and m in the equation 

m^4/3 . n^2/3 + m^2/3 . n^4/3 = (1/√2)^4/3 . (1/√2)^2/3 + (1/√2)^2/3 . (1/√2)^4/3

⇒ (1/√2)² + (1/√2)² 

⇒ (1/2) + (1/2) 

⇒ 1 

∴ Option 2 will be the correct answer.

Given:

cos θ + sec θ = k

Formula used:

sec θ = 1/cos θ 

a² + b² = (a + b)² - 2ab

Calculation:

cos θ + sec θ = k

⇒ cos θ + 1/cos θ = k

On squaring both sides, we get,

⇒ cos²θ + 1/cos²θ + 2 = k²

⇒ cos2θ + 1/cos2θ = k² - 2

According to the question,

sin2θ - tan2θ = (1 - cos²θ) - (sec²θ - 1)

⇒ 2 - (cos2θ + 1/cos2θ)

⇒ 2 - (k2 - 2)

⇒ sin2θ - tan2θ = 4 - k²

∴ The value of sin2θ - tan2θ is 4 - k².

Given:

ΔABC is in a semicircle where AB is a diameter.

Calculation:

ΔABC is in a semicircle where AB is a diameter.

An angle on semicircle = 90°

∠C = 90°

In a triangle, the sum of three angles is 180°

∠A + ∠B + ∠C = 180°

⇒ ∠A + ∠B + 90° = 180°

⇒ ∠A + ∠B = 180° - 90° = 90°

Now, 

cos(A+B) + sin(A + B) 

⇒ cos90° + sin90° = 1

∴ The value of cos(A+B) + sin(A + B)  is 1

Calculations :

Staement 1

sin θ = x + (1/x) 

⇒ sin θ = (x² + 1)/x

⇒ x.sin θ = x² + 1

⇒ x² - x.sin θ + 1 = 0 

For any quadratic equation(ax² + bx + c) real values exist when b² > 4ac

So, 

(- sin θ)² > 4 × 1 × 1 

⇒ sin² θ > 4 

⇒ sin θ > 2 

We know that -1 ≤ sin θ ≤ 1

So no value exist for some real value of x.

Statement 2 

cos θ = x + (1/x) 

⇒ cos θ = (x2 + 1)/x

⇒ x.cos θ = x2 + 1

⇒ x2 - x.cos θ + 1 = 0 

For any quadratic equation(ax2 + bx + c) real values exist when b2 > 4ac

So, 

(- cos θ)2 > 4 × 1 × 1 

⇒ cos2 θ > 4 

⇒ cos θ > 2 

We know that -1 ≤ cos θ ≤ 1

So no value exist for some real value of x. 

So no statement is following. 

∴ Option 4 is the correct choice.

Formula Used:

Interior angle of regular polygon = $\rm {(n - 2)\cdot\pi\over n}$

Calculation:

In pentagon n = 5

= $\rm {(5\ -\ 2)\cdot\pi\over 5}$

= $\rm 3\pi \over 5$ = 108°

∴ The magnitude (in radian) of the interior angle is $\rm 3\pi \over 5$

Shortcut Trickα + β = 15°, α - β = 75°

α = 45° and β = 15° 

α/β = 45/15 = 3/2

α = (3/2)β, ∴ k = 3/2

Alternate Method 

Given :

The difference between the two angles is 15°

And the sum of the angles in radian is (5/12)π = 75° 

And the bigger angle is k times the smaller angle.

Calculations :

Let the bigger angle be α and smaller angle be β 

According to the questions

α - β = 15°     ....(1) 

α + β = 75°      ....(2)             (π = 180°)

Solving equation (1) and (2) we get 

α = 45° and β = 30° 

We get 

α = (3/2)β     (α = kβ, given in question) 

So, 

⇒ k = 3/2 

∴ Option 2 will be the correct answer.

Calculations :

Statement 1

2 sin2 θ - cos θ + 4 = 0

⇒ 2 (1 - cos² θ) - cos θ + 4 = 0 

⇒ 2 - 2 cos² θ - cos θ + 4 = 0 

⇒ 2 cos² θ + cos θ - 6 = 0 

⇒ 2 cos² θ + 4 cos θ - 3 cos θ - 6 = 0 

⇒ 2 cos θ (cos θ + 2) - 3 (cos θ + 2) = 0 

⇒ (2 cos θ - 3)(cos θ + 2) = 0 

⇒ cos θ = 3/2 or -2  

This statement is incorrect as cos θ lies between -1 and 1 

Now statement 2

tan θ + cot θ is more than or equal to 2 

(tan θ + cot θ)/2  ≥ √(tan θ .cot θ)       (Arithmetic mean ≥ Geometric mean)

(tan θ + cot θ)/2 ≥ 1

Tan θ + cot θ ≥ 2  

We can conclude that only statement 2 is correct. 

∴ Option 2 is the correct choice.

Shortcut Trick 

⇒ 44 = (42/360) × 2 × (22/7) × R

⇒ R = 60 m

∴ The radius to be used is 60 m.

Alternate Method 

Given:

Change in direction by angle = 42°

Distance of curve road = 44 m

Concept Used:

Arc Length: The arc length is the measure of the distance along the curved line making up the arc.

It is longer than the straight line distance between its endpoints (which would be a chord).

Formula Used:

Arc Length = 2πR(C/360°)

where:

R = Radius of the arc

C = Central angle of the arc in degrees

Calculation:

Observe the following diagram carefully,

According to the question,

Length of the curved road = Length of the arc of the circle

⇒ 44 = 2πR(C/360°)

⇒ 2πR(C/360°) = 44

⇒ 2 × (22/7) × R × (42°/360°) = 44

⇒ (44/7) × R × (42/360) = 44

⇒ R = 44 × (7/44) × (360/42)

⇒ R = 60 m

∴ The radius to be used is 60 m.

Concept used :

Maximum value of sinθ = 1 

Calculations :

Maximum value of 3 sin θ - 4 = 3 × 1 - 4                (For maximum value we must take sin θ greatest value which is 1)

⇒ 3 - 4 

⇒ -1   

∴ Option 2 will be the correct choice.

Given                       

sin θ + cos θ = √2

Formula used :

sin² θ + cos² θ = 1 

(x + y)³ = x³ + y³ + 3.x.y(x + y) 

Calculations :

(cos² θ + sin² θ)³ = cos⁶ θ + sin⁶ θ + 3.sin² θ.cos² θ(cos² θ + sin² θ) 

⇒ cos⁶ θ + sin⁶ θ + 3.sin² θ.cos² θ(cos² θ + sin² θ) = 1³ = 1 

⇒  sin6 θ + cos6 θ + 3 sin2 θ cos2 θ = 1

Now, 

sin θ + cos θ = √2 

Squarring both the sides 

Sin² θ + cos² θ + 2 sin θ cos θ = 2 

⇒ 2 sin θ cos θ = 1 

⇒ sin θ cos θ = 1/2 

⇒ sin² θ cos² θ = 1/4 

⇒ 3 sin2 θ cos2 θ  = 3/4 

Now,

sin6 θ + cos6 θ + 6 sin2 θ cos2 θ = sin6 θ + cos6 θ + 3 sin2 θ cos2 θ + 3 sin2 θ cos2 θ  

⇒ 1 + (3/4) 

⇒ 7/4 

∴ Option 4 is the correct answer. 

Shortcut Trick⇒ sin θ + cos θ = √2 

At θ = 45° satisfy the above equation.

⇒ sin6 θ + cos6 θ + 6 sin2 θ cos² θ

Put θ = 45° 

⇒  (1/√2)⁶ + (1/√2)⁶ + 6 × (1/√2)² × (1/√2)²

⇒ 1/8 + 1/8 + 6 × (1/2) × (1/2)

⇒ 7/4

∴ The correct answer is option (4).

Concept used :

sin2 θ = 1 - cos2 θ

Calculations :

9 sin2 θ + 16 cos2 θ = 9 sin2 θ + 16 ( 1 - sin2 θ) 

⇒ 9 sin2 θ + 16 - 16 sin2 θ

⇒ -7 sin² θ + 16 

⇒ -7 + 16         (for least value of the equation value of sin² θ must be equal to 1)

⇒ 9 

∴ Option 2 will be the correct choice.

Shortcut Trick Minimum value of a sin² θ + b cos² θ is b if b < a or a if a < b.

The least or minimum value will be either a or b, depending on which is lesser.

⇒ 9 sin2 θ + 16 cos2 θ 

Here, 9 < 16 so the least value is 9.

∴ The correct answer is option (2).

Given :

cos 47° + sin 47° = k

Concept used :

cos 2θ = cos² θ - sin² θ 

cos2 θ + sin2 θ = 1  

2 sin θ cos θ = sin 2θ 

Calculations :

We have

cos 47° + sin 47° = k    

Squaring on both side

(cos 47° + sin 47°)² = k²

We know that,

(a + b)² = a² + 2ab + b²

⇒ cos² 47° + sin² 47° + 2 cos 47° sin 47° = k²

1 + 2 cos 47° sin 47° = k²       (∵ cos2 θ + sin2 θ = 1)

2 cos 47° sin 47° = k² - 1

sin 94°  =  k2 - 1        (∵ 2 sin θ cos θ = sin 2θ )

We know that,

sin² θ + cos² θ = 1 

⇒ sin2 94° + cos2 94° = 1 

⇒ cos2 94° = 1 - (k² - 1)²       [∵ (a - b)2 = a2 - 2ab + b2]

⇒ cos2 94° = 1 - (k⁴ + 1 - 2k²)

⇒ cos2 94° = 1 - k⁴ - 1 + 2k² 

⇒ cos2 94°= 2k² - k⁴

⇒ cos 94° = ± k√(2 - k²),

We know that cos 94° lies on the second quadrant, and cos θ is 

negative in this quadrant. 

Therefore,  

cos 94° = - k√(2 - k2)

⇒ cos 2 × 47 = - k√(2 - k2)

Since cos 2θ = cos2 θ - sin2 θ 

⇒ cos2 47° - sin2 47° =  - k√(2 - k2)

∴ Option 2 will be the correct answer.

Given :

cosec θ - sin θ = p3 and sec θ - cos θ = q³

Concept used :

1 - sin2 θ = cos² θ 

1 - cos2 θ = sin² θ 

Calculations :

cosec θ - sin θ = 1/(sin θ) - sin θ 

⇒ (1 - sin² θ)/sin θ 

⇒  (cos² θ)/(sin θ)

So, 

(cos² θ)/(sin θ) = p³      ....(1) 

Now 

sec θ - cos θ = (1/cos θ) - cos θ 

⇒ (1 - cos² θ)/cos θ 

⇒  (sin² θ)/(cos θ)

So, 

(sin² θ)/(cos θ) = q³      ....(2) 

Divide equation (2) by (1) 

(sin² θ)/(cos θ) ÷ (cos² θ)/(sin θ) = q³/p³ 

⇒ sin³ θ/cos³ θ = q³/p³ 

⇒ tan θ = q/p 

∴ Option 2 is the correct answer.

Given :

0 ≤ α, β ≤ 90° and cos (α - β) = 1

Concept used :

cosθ = 1    at θ = 0°

Calculations :

cos (α - β) = 1

So, 

α - β = 0° 

⇒ α = β  

sin α - sin β + cos α - cos β

⇒ sin α - sin α  + cos α - cos α    (α = β)

⇒ 0 + 0 

⇒ 0 

∴ Option 2 will be the correct answer.

Concept used :

cos (90° - θ) = sin θ

tan (90° - θ) = cot θ  

Calculations :

Statement 1 

cos 61° + sin 29° = sin 29° + sin 29° 

⇒ 2 sin 29° 

⇒ 2 sin 29° < 1     (sin 29° is less than 0.5 as the value of sin θ increases from 0° to 90° and the value of sin 30° is 0.5)

Statement 2 

tan 23° - cot 67° = cot 67° - cot 67° 

⇒ 0  

Only statement 1 is correct 

∴ Option 1 is the correct choice.

Given:

In quadrilateral ABCD, 

∠B = 90°

and AB2 + BC2 + CD2 - AD2 = 0.

Concept Used:

In ΔABC, ∠B = 90° Using Pythgoras Theorem 

AB2 + BC2 = AC2

Calculation:

Observe the diagram given below carefully,

According to the question,

AB2 + BC2 + CD2 - AD2 = 0

⇒ (AB2 + BC2) + CD2 - AD2 = 0

⇒ (AC²) + CD2 - AD2 = 0      ----(∵ From Pythgoras Theorem, AB² + BC² = AC²)

⇒ AC2 + CD2 = AD2      ----(i)

Here, Equation (i) is the Equation form of Pythagoras theorem applied in Δ ACD.

For Equation (i) to hold true, the side AD must be the Hypotenuse of the Δ ACD.

So, the angle opposite to the Hypotenuse i.e. AD must be 90°.

∴ ∠ ACD = 90°.

Given:

In a Δ ABC,

AC = 12 cm,

AB = 16 cm,

AD is biesctor of ∠A. 

& BD = 4 cm.

Concept Used:

Angle Bisector Theorem: An angle bisector of an angle of a triangle divides the opposite side in two segments that are proportional to the other two sides of the triangle.

Calculation:

Observe the diagram below carefully,

From the Angle Bisector Theorem,

We know that,

AB/AC = BD/DC

⇒ 16/12 = 4/DC

⇒ DC = 4 × (12/16)

⇒ DC = 4 × (3/4)

⇒ DC = 3 cm

∴ The length of DC = 3 cm.

 Shortcut Trick

When an angle bisector of an angle of a triangle divides the opposite side in two segments in  a and b  then other two sides of the triangle will be in ratio a:b