Territorial Army Official Paper I (Conducted on 25 Sep 2022) (Previous Year Paper)

The pattern followed in the given series is:

Giant : Dwarf → The words are opposite of each other.

Similarly,

Opposite of Genius is Idiot.

Hence, the correct answer is Option 3.

The pattern followed in the given series is:

Forecast : Future → A Forecast is related to Future.

Similarly,

Regret : ? → ? will be related to regret.

Now we consider the given options:

Option 1) Present : Regret is not done for present.

Option 2) Past : Regret is done for past(if something bad happened).

Option 3) Atone : It is not related to regret.

Option 4) Sins : Regret is a reason of sins.

Hence, the correct answer is option (2).

The pattern followed in the given series is:

Planet : Orbit → Orbit is the path followed by a Planet.

Similarly,

Projectile : ? → ? will be the path followed by a projectile.

Now we consider the given options:

Option 1) Track : It is the path followed by a train

Option 2) Milky Way : It is a galaxy.

Option 3) Trajectory : It is the path followed by a projectile.

Option 4) Path : a course or direction without any context.

Hence, the correct answer is option (3).

The pattern followed in the given series is:

Pesticide : Crop → Pesticide is used to protect the crops.

Similarly,

Antiseptic : ? → Antiseptic is used to protect ?

Now we consider the given options:

Option 1) Clotting : Antiseptic is not used to protect clotting.

Option 2) Bandage : It is an object to stop bleeding.

Option 3) Wound : Antiseptic is used to protect a wound.

Option 4) Bleeding : Antiseptic is not used to protect bleeding.

Hence, the correct answer is option (3).

The pattern followed in the given series is:

Symphony : Composer → A composer uses symphony to make music.

Similarly,

Fresco : ? → Fresco will be used in a related profession.

Now we consider the given options:

Option 1) Painter : A painter uses fresco(painting on water).

Option 2) Inventor : Fresco is not related to this profession.

Option 3) Singer : Fresco is not related to this profession. 

Option 4) Writer : Fresco is not related to this profession.

Hence, the correct answer is option (1).

The pattern followed in the given series is:

    First term ÷ 2 - 1 = Second term

Now we apply the pattern on the given series:

42 : 20 :: 64 : ?

42 : 20

42 ÷ 2 - 1 = 20

21 - 1 = 20

20 = 20

64 : ?

64 ÷ 2 - 1 = ?

32 - 1 = ?

31 = Answer

Hence, the correct answer is option (1).

The pattern followed in the given series is:

(√First term) + 2 = √Second term

Now we apply the pattern on the given series:

49 : 81 :: 100 : ?

(√49) + 2 = √81

7 + 2 = 9

9 = 9

Similarly,

49 : 81 :: 100 : ?

(√100) + 2 = √?

10 + 2 = √?

12 = √Answer

Answer = 144

Hence, the correct answer is option (2).

The pattern followed in the given series is:

   First term ÷ 100 = Second term

Now we apply the pattern on the given series:

0.16 : 0.0016 :: 1.02 : ?

0.16 ÷ 100 = 0.0016

0.0016 = 0.0016

0.16 : 0.0016 :: 1.02 : ?

1.02 ÷ 100 = ?

0.0102 = Answer

Hence, the correct answer is option (1).

The pattern followed in the given series is: 

             (First term)² = Second term

Now we apply the pattern on the series:

1 : 1 :: 25 : ?

1² = 1

1 = 1

1 : 1 :: 25 : ?

25² = ?

625 = Answer

Hence, the correct answer is option (2).

We consider the given options and analyze:

Option 1) Rhea: It is not a type of fish.

Option 2) Trout: It is a type of fish.

Option 3) Lamprey: It is a type of fish.

Option 4) Salmon: It is a type of fish.

So, Rhea is the odd one.

Hence, the correct answer is option (1).

We consider the given options and analyze:

Option 1) Commander: It is a rank in Navy.

Option 2) Commodore: It is a rank in Navy.

Option 3) Brigadier: It is a rank in Army.

Option 4) Admiral: It is a rank in Navy.

So, Brigadier is the odd one.

Hence, the correct answer is option (3).

We consider the given options and analyze:

Option 1) Microscope: It is an optical instrument.

Option 2) Telescope: It is an optical instrument.

Option 3) Periscope: It is an optical instrument.

Option 4) Stethoscope: It is not an optical instrument.

So, Stethoscope is the odd one.

Hence, the correct answer is option (4).

We consider the given options and analyze:

Option 1) Thyroxin: It is a hormone.

Option 2) Adrenalin: It is a hormone.

Option 3) Iodine: It is not a hormone.

Option 4) Insulin: It is a hormone.

So, Iodine is the odd one.

Hence, the correct answer is option (3).

According to the English alphabets and their positional values:

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The pattern followed in the given codes is:

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Similarly,

Hence, the correct answer is option (3).

According to the English alphabets and their positional values:

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The pattern followed in the given codes is:

1.First four alphabets are increased to 1 positional value and placed at last four places.

2.Last four alphabets are decreased by 1 positional value and placed at first four places.

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Similarly,

Hence, the correct answer is option (2).

According to the English alphabets and their positional values:

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The pattern followed in the given codes is:

Alphabets at odd places are decreased by 1 positional value.

Alphabets at even places are increased by 1 positional value.

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Similarly,

Hence, the correct answer is option (2).

Given: Sam ranked 9th from top and 38th from the bottom in the class.

If Sam is ranked 38th from the bottom that means there are 37 students below her and 9th from the top means there are 8 students above her.

Total number of Students = Sam's rank from bottom + Sam's rank from top - 1

= 38 + 9 - 1(Sam) = 46.

Hence, the correct answer is option (2).

Given: Manoj and Sachin are ranked 7th and 11th respectively from top in a class of 31 students.

Students below Manoj = 31 - 7 =  24

Rank of Manoj from bottom = 24 + 1 = 25th

Students below Sachin  = 31 - 11 = 20

Rank of Sachin from bottom = 20 + 1 = 21st

Hence, the correct answer is option (3).

From the given information:  

Step 1: In a row of boys, A is 15th from the left and B is 4th from the right. There are three boys between A and B. 

Step 2: C is just left of A.

Now C is immediate left of A which means C is ranked 14th (15th - 1).

We can see that A is 8th (B + 4) from right. Then,

Rank of C from Right = 8 + 1 = 9th

Hence, the correct answer is Option 1.

Given: Richard is 15th from the front in a column of boys, there were thrice as many behind him as there were in front.

Boys in front of Richard = 14

Boys behind Richard = 14 × 3 = 42 

Total number of boys = 14 + 42 + 1(Richard) = 57

Number of boys between Richard & 7th boy from the end of the column:

⇒57 - (15 + 7)

⇒57 - 22

Answer = 35 boys

Hence, the correct answer is option (3).

Given: N rank 5th in the class, S is 8th from the last. If T is 6th after N and just in the middle of N and S.

Rank of N = 5th

Rank of S = 8th from last

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Rank of T = 6th after N = 5 + 6 = 11th

T is just in middle of N and S which means number of students between N and T is equal to the number of students between T and S i.e., 6.

Now, rank of S from start is 11 + 6 = 17th

And rank of S from last is 8th

Total number of students = Rank from start + Rank from last - 1

Total number of students = 17 + 8 - 1 = 24

Hence, the correct answer is option (2).

Given: In a row of 21 girls when Monica was shifted by four places towards the right, she became 12th from the left end.

(Left)

(Right)

From the figure we can see that Monica's initial from the right end is 14th.

Monica from right end = 21 - 8 + 1 = 14

Hence, the correct answer is option (4).

From the given information:

We know that, water is drawn from a well but in this case well is called island.

Hence, the correct answer is option (1).

From the given information:

We know that birds fly in air but in this case air is called blue.

Hence, the correct answer is option (2).

From the given information:

We know that fruit grows on a tree and in this case tree is called sky.

Hence, the correct answer is an option (3).

Given : Q means to +, J means to ×, T means - , K means ÷

We substitute the values in the given equation and solve it using the BODMAS rule:

Given: 30 K 2 Q 3 J 6 T 5 = ?

⇒ 30 ÷ 2 + 3 × 6 - 5 = ?

⇒ 15 + 3 × 6 - 5 = ?

⇒ 15 + 18 - 5 = ?

⇒ 33 - 5 = ?

⇒ 28 = Answer

Hence, the correct answer is option (3).

We substitute the values in the given equation and solve it using the BODMAS rule:

Given: 18 Q 12 P 4 R 5 S 6 = ?

⇒18 × 12 ÷ 4 + 5 - 6 = ?

⇒ 18 × 3 + 5 - 6 = ?

⇒ 54 + 5 - 6 = ?

⇒ 59 - 6 = ?

⇒ 53 = Answer

Hence, the correct answer is option (2).

Given : A stands for + B stands for - C stands for ×

We substitute the values in the given equation and solve it using the BODMAS rule:

Given: (10 C 4) A (4 C 4) B 6? 

⇒(10 × 4) + (4 × 4) - 6? 

⇒ 40 + 16 - 6 = ?

⇒ 56 - 6 = ?

⇒50 = Answer

Hence, the correct answer is option (1).

Given:

Thickness of paper = 1 mm = 1 × 10-6  km

The thickness of paper after 1 fold = 2 mm

The thickness of paper after 2 fold = 22 mm

The thickness of paper after 50 fold = 250 mm

x = 250 × 10-6 km

x =  250 /106

Taking log on both sides

log x = 50 × log2 - 6 × log10

⇒ log x = 50 × 0.3010 - 6 × 1

⇒ log x = 15.05 - 6

⇒ log x = 9.05

⇒ log x ≈ 9

⇒ x = anti log 9

⇒ x = 109 km

⇒ x = 109 km

⇒  x = 1 billion km

∴ The thickness of the pile after 50 folds is 1 billion km.

Given:

Age of A = 10 yrs

Age of A's mother = 10 + 20 = 30 yrs

Age of A's father = 30 + 5 = 35 yrs

Now age of A's father when A was born = 35 - 10 = 25 yrs

We have, A was born 2 years after his father’s marriage, so

the age of A's father when he got married → 25 - 2 = 23 yrs.

Hence, the correct answer is option (3).

Given: I was of your present age when you were born.

From this statement we can say that the father is double the age of his son.

Father's present age = 36 yrs

Son's present age = 36/2 = 18 yrs

Son's age 5 years back = 18 - 5 = 13 yrs

Hence, the correct answer is option (2).

From the given information:

Step 1: A father is now 3 times old as his son.

      Father = 3 × Son

Step 2: 5 years back, he was 4 times old as his son.

⇒ Father - 5 = 4(Son - 5)

⇒ Father - 5 = 4(Son) - 20 

⇒ Father - 4(Son) + 15 = 0

Now we combine step 1 and 2,

 3 × Son - 4(Son) + 15 = 0

 -1(Son) = -15

We eliminate negative sign from both sides,

 Son = 15 yrs

Hence, the correct answer is option (1).

Given:

The sum of the ages of the couple is 11 times the difference of their ages.

Calculations:

Let age of the women be 10x + y years

Age of her husband = 10y + x years

According to the question,

⇒ (10y + x) – (10x + y) = 1/11 × (10x + y + 10y + x)

⇒ 10y + x - 10x - y = 1/11 × 11x + 11

⇒ 9y – 9x = x + y

⇒ 8y = 10x

⇒ x/y = 4/5

Hence, Age of the women is 45 years.

Given : 7 read French, 8 read English while 3 of them read none of these two.

According to the formula:

Total = E + F - EF + N

where E is English, F is French , EF is both English & French and N is neither.

We substitute the given values to the formula,

15 = 7 + 8 - EF + 3

15 = 18 - EF

EF = 18 - 15

EF = 3 

So, there are 3 people who read both English and French.

Hence, the correct answer is option (2).

Given : M went to movies 9 days ago. She goes to the movies only on Thursdays.

9 days from Thursday → (Thursday to Thursday is 7 days), remaining days = 2.

So, Thursday + 2 i.e., Saturday.

Hence, the correct answer is option (2).

We have : 30th January 2003 was Thursday

Now we count the number of days:

31st January = 1 day

February 2003 = 28 days(not a leap year)

1st and 2nd march = 2 days

Total number of days = 31 days

Now we divide the total with 7 and the remainder will represents the day of the week: 31/7 → 3 is the remainder.

So, 2nd March 2003 will be Thursday + 3 i.e., Sunday.

Hence, the correct answer is option (1).

The pattern followed in the given series is:

   (First term)² + (Second term)² + (Third term)² = Fourth term

Now we apply the pattern on the given columns:

Column 1: (4)² + (2)² + (1)² = 21

16 + 4 + 1 = 21

21 = 21

Column 2: (5)2 + (3)2 + (8)2 = 98

25 + 9 + 64 = 98

98 = 98

Column 3: (6)2 + (7)2 + (3)2 = ?

36 + 49 + 9 = ?

94 = Answer

Hence, the correct answer is option (1).

The pattern followed in the given series is:

   Sum of digits of first term - Sum of digits of third term = Second term

Now we apply the pattern on the given rows:

Row 1: (9 + 6 + 3) - (8 + 4 + 4) = 2

18 - 16 = 2

2 = 2

Row 2: (4 + 6 + 4) - (9 + 0 + 3) = ?

14 - 12 = ?

2 = Answer

Hence, the correct answer is option (2).

The pattern followed in the given series is:

    (Fourth term - Second term) ÷ First term = Third term

Now we appl the pattern on the given columns:

Column 1: (29 - 8) ÷ 7 = 3

21 ÷ 7 = 3

3 = 3

Column 2: (19 - 7) ÷ 4 = 3

12 ÷ 4 = 3

3 = 3

Column 3: (31 - 6) ÷ 5 = ?

25 ÷ 5 = ?

5 = Answer

Hence, the correct answer is option (3).

The pattern followed in the given series is:

   (First term - Third term) × 11 = Second term [Taking Positive value only]

Now we apply the pattern in the given rows:

Row 1: (42 - 38) × 11 = 44

4 × 11 = 44

44 = 44

Row 2: (23 - 28) × 11 = 55  [Taking Positive value only]

5 × 11 = 55

55 = 55

Row 3: (37 - 39) × 11 = ?

2 × 11 = ?

22 = Answer

Hence, the correct answer is option (1).

From the given words: Doctors, Human Beings and Married People

We draw a Venn diagram,

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All married people are human beings & all doctors are human beings but only some married people are doctors.

Hence, the correct answer is option (4).

From the given words: Man, Worker and Garden

We draw a Venn diagram,

<br>

Some men are workers but not all workers are men also there is no relation between man & worker with garden.

Hence, the correct answer is option (4).

From the given words: Males Cousins and Nephews

We draw a Venn diagram,

<br>

All nephews are males some male and nephews are cousins. 

Hence, the correct answer is option (1).

When we complete the given figure by symmetrical pattern, we get:

            

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Now we compare the given options:

Option 1)  

The lines at the corner are positioned incorrectly.

Option 2)  

The smaller triangle is missing from the pattern.

Option 3) 

 The pattern is not symmetrical.

Option 4) 

The pattern is correct.

Hence, the correct answer is option (4).

The pattern followed in the given series is:

   The term is increased by addition of increasing natural number.

Hence, the correct answer is option (2).

The pattern followed in the given series is:

    Each term is a square of number.

Hence, the correct answer is option (1).

The pattern followed in the given series is:

    Term × 2 + 1 = Next term

Hence, the correct answer is option (3).

The pattern followed in the given series is:

   Difference between the terms is 13.

Now we apply the pattern on the given options:

Option 1) 95 - 82 = 13

Option 2) 69 - 56 = 13

Option 3) 55 - 42 = 13

Option 4) 48 - 34 = 14

Hence, the correct answer is option (4).

The pattern followed in the given series is:

   First term × (First term - 1) = Second term

Now we apply the pattern on the given options:

Option 1) 11 × (11 - 1) = 115

11 × 10 = 115

110 ≠ 115

Option 2) 10 × (10 - 1) = 90

10 × 9 = 90

90 = 90

Option 3) 9 × (9 - 1) = 72

9 × 8 = 72

72 = 72

Option 4) 8 × (8 - 1) = 56

8 × 7 = 56

56 = 56

Hence, the correct answer is option (1).

The pattern followed in the given options is:

   Difference between the terms is a multiple of 6.

Now we apply the pattern on the options:

Option 1) 81 - 63 = 18

18 is a multiple of 6.

Option 2) 24 - 48 = 24

24 is a multiple of 6.

Option 3) 21 - 15 = 6.

6 is a multiple of 6.

Option 4) 13 - 39 = 26

26 is not a multiple of 6.

Hence, the correct answer is option (4).

Given:

A number, when divided by 987, gives a remainder 59.

Concept used:

Dividend = (Divisor × Quotient) + Remainder

Calculation:

Let the quotient be k.

According to the above concept,

⇒ Dividend = (987 × k) + 59

⇒ 987k + 59

Now, when the number is divided by 21

⇒ $\frac{987k + 59}{{21}}$ = 47k + $\frac{59}{{21}}$

While solving we get quotient as 2 and remainder as 17.

∴ The required remainder is 17.

Concept used:

Positive integer: Positive integers are the numbers we use to count: 1, 2, 3, 4 and so on.

Euclid division lemma: If we have two positive integers a and b, then there exist unique integers q and r which satisfies the condition a = bq + r where 0 ≤ r < b.

Calculation:

According to the above concept,

Positive integer = 143 × k + 28  

⇒ 13 × 11 × k + 13 × 2 + 2

⇒ 13(11k + 2) + 2

So, required remainder = 2

∴ The remainder obtained on dividing the same number by 13 is 2.

Given:

(Np-1 - 1) is a multiple of p

Concept used:

Prime number: A number which is divisible by one and itself.

Co-prime number: Those number that have only one common factor, namely 1.

Integers: A whole number (not a fractional number) that can be positive, negative, or zero

Calculation:

N is prime to p, that means N and p are two co-prime numbers.

Since, (Np-1 - 1) is a multiple of p.

So, (Np-1 - 1) = (k × p)

⇒ k = (Np-1 - 1)/p

Let p = 5 and N = 2  [co-prime numbers with p being a prime number]

So, k = (2^5-1 - 1)/5

⇒ (15/5) = 3  [integers]

That means the given condition satisfies.

Again take p = 4 and N = 3  [co-prime numbers with p being a non prime number]

So, k = (3⁴-1 - 1)/4

⇒ (26/4) = 6.5  [Not a integer]

That means the given condition does not satisfies.

∴ P must be a prime number only.

Given:

LCM of two numbers is 16 times their HCF. 

The sum of LCM and HCF is 850.

If one number is 50

Concept used:

LCM = Product of the numbers/HCF

Calculation:

Let the HCF be x.

Then, LCM = 16x and another number is n.

Now, the sum of the LCM and HCF is

⇒ x + 16x = 850

⇒ 17x = 850

⇒ x = 50

Now, according to the above concept

⇒ 16x = $\frac{50 \times n}{{x}}$

⇒ 16 × 50 = $\frac{50n}{{50}}$

⇒ n = 800

∴ The other number is 800.

Given:

The HCF of two numbers = 98

LCM = 2352

Concept used:

Co-prime number: 

Coprime numbers are those numbers that do not have any common

factor other than 1.

LCM × HCF = Product of the two numbers

Calculation:

Let the two numbers be 98x and 98y respectively.

Now, according to the above concept,

⇒ (98x) × (98y) = 98 × 2352

⇒ xy = 24

Since co-prime factors of 24 are 3 and 8

Let x = 3 and y = 8

Now, the sum of the number = (98 × 3 + 98 × 8)

⇒ 98(3 + 8) = (98 × 11) = 1078

∴ The required sum of the numbers is 1078.

Given:

The numbers between 500 and 1000 are divisible by 13

Formula used:

_Tn = a + (n - 1)d

Where,

T_n = last number, a = first number,

d = common difference

Calculation:

The first number after 500 and divisible by 13 = 507

And, the number just less than 1000 and divisible by 13 = 988

Hence sequence is

⇒ 507, 520, 533, ......988

Now, according to the above concept,

Tn = a + (n - 1)d

⇒ 988 = 507 + (n - 1)13

⇒ (n - 1) = 481/13

⇒ (n - 1) = 37

⇒ n = 38

∴ Number between 500 and 1000 which is divisible by 13 is 38.

Concept used:

Non-terminating: A decimal number that continues endlessly, with no group of digits repeating endlessly.

Repeating decimal:  A number whose decimal representation eventually becomes periodic (i.e. the same sequence of digits repeats indefinitely).

Calculation:

According to the above options,

⇒ $\frac{13}{8}$ = 1.625

⇒ $\frac{3}{16}$ = 0.1875

⇒ $\frac{3}{11}$ = 0.272727

⇒ $\frac{137}{25}$ = 5.48

∴ The required non-terminating and repeating decimal is $\frac{3}{11}$.

Calculation:

According to the given options

⇒ $\frac{7}{9}$ = 0.77

⇒ $\frac{11}{14}$ = 0.78

⇒ $\frac{3}{4}$ = 0.75

⇒ $\frac{10}{13}$ = 0.76

∴ The largest of the following is $\frac{11}{14}$.

Shortcut Trick

To find the square root of a surd in the form A + 2√B, find two numbers whose sum is A and product is B.

Here, sum = 9 and product = 14.

The two numbers are 7 and 2 (since 7 + 2 = 9 and 7 × 2 = 14).

Therefore, the square root is simply √7 + √2.

∴ The correct answer is √2 + √7.

Alternate Method

Given: Expression = 9 + 2√14

Formula Used: (a + b)² = a² + b² + 2ab

⇒ Let the square root be (√x + √y).

⇒ (√x + √y)² = x + y + 2√xy

⇒ Comparing x + y + 2√xy with 9 + 2√14:

⇒ x + y = 9 and xy = 14

⇒ 9 + 2√14 = 7 + 2 + 2√(7 × 2)

⇒ 9 + 2√14 = (√7)² + (√2)² + 2(√7)(√2)

⇒ 9 + 2√14 = (√7 + √2)²

⇒ Square root = √(√7 + √2)² = √7 + √2

∴ The correct answer is √2 + √7.

Additional Information

Square Root of a Quadratic Surd

For an expression of the form a ± 2√b, the square root is √m ± √n, where m + n = a and mn = b.

Algebraic Identity for Surds

The identity (a + b)² = a² + b² + 2ab allows us to transform a binomial surd into a perfect square, making it easy to find the square root.

Rationalization

Multiplying a surd like (a + √b) by its conjugate (a − √b) results in a rational number (a² − b).

Concept used:

Perfect square: A number that can be expressed as the product of an integer by itself or as the second exponent of an integer.

Calculation:

∴ The required smallest number is 69, which is added to 1780 to make it perfect square.

Alternate MethodGiven:

The given number = 1780

Calculation:

⇒ **√**1780 = 42.190

Now, finding the square of the next number

⇒ 43² = 1849

Required number = (1849 - 1780) = 69

∴ The required smallest number is 69, which is added to 1780 to make it perfect square.

Shortcut Trick

Use the direct formula for percentage less: [r ÷ (100 + r)] × 100

Given r = 20%

⇒ Required % = [20 ÷ (100 + 20)] × 100

⇒ Required % = [20 ÷ 120] × 100 = 100 ÷ 6 = 50 ÷ 3

∴ The correct answer is 50/3 %.

Alternate Method

Given:

Let the salary of Y be 100.

Salary of X is 20% more than Y.

⇒ Salary of X = 100 + 20% of 100 = 120

Calculations:

Difference between salaries = 120 − 100 = 20

Percentage by which Y is less than X = (Difference ÷ Salary of X) × 100

⇒ Required % = (20 ÷ 120) × 100

⇒ Required % = (1 ÷ 6) × 100

⇒ Required % = 50 ÷ 3

∴ The correct answer is 50/3 %.

Additional Information

Percentage Increase/Decrease Formula

If A is r% more than B, then B is less than A by [r ÷ (100 + r)] × 100%.

Inverse Case

If A is r% less than B, then B is more than A by [r ÷ (100 − r)] × 100%.

Fraction Conversion

20% = 1/5. If X is 1/5 more than Y, then Y is 1/(5+1) = 1/6 less than X. 1/6 = 16.66% or 50/3%.

Given:

A student has to secure 40% of the marks to pass an exam. 

He gets only 45 marks and fails by 5 marks. 

Calculation:

A student gets 45 marks and failed by 5 marks.

Passing marks = (45 + 5) = 50 marks.

Now, according to the question

⇒ 40% = 50

⇒ 100% = [(50 × 100)/40]

⇒ 125 marks.

∴ The maximum marks are 125.

Given:

The number which has to be added to each term of the ratio 49 ∶ 68

Calculation:

Let the number be x.

According to the question

⇒ $\frac{49+x}{{68+x}}$ = $\frac{3}{{4}}$

⇒ 196 + 4x = 204 + 3x

⇒ x = 8

∴ The required number is 8.

Shortcut Trick

Sum of ages 5 years ago = 75 − (5 × 2) = 65 years.

Product of ages 5 years ago = 750.

Find two factors of 750 that sum to 65: 50 × 15 = 750 and 50 + 15 = 65.

Father's age 5 years ago = 50 years.

Father's present age = 50 + 5 = 55 years.

∴ The correct answer is 55.

Alternate Method

Given:

Present sum of ages = 75 years

Product of ages 5 years ago = 750

Formula Used:

(x − a) × (y − a) = Product; where x + y = Sum

5 Years Ago Present

Prod: 750

Sum: 75

• 5 Years

Let Father's present age be F years.

Then, Son's present age = (75 − F) years.

5 years ago, Father's age = (F − 5) years.

5 years ago, Son's age = (75 − F − 5) = (70 − F) years.

According to the question:

⇒ (F − 5) × (70 − F) = 750

⇒ 70F − F² − 350 + 5F = 750

⇒ −F² + 75F − 350 − 750 = 0

⇒ F² − 75F + 1100 = 0

⇒ F² − 55F − 20F + 1100 = 0

⇒ F(F − 55) − 20(F − 55) = 0

⇒ (F − 55)(F − 20) = 0

⇒ F = 55 or F = 20

Since a father must be older than his son, Father's age = 55.

∴ The correct answer is 55.

Additional Information

Constant Age Difference

The difference between the ages of two people remains constant throughout their lives, regardless of the time elapsed.

Average Age Property

If the average age of n people is X, then after k years, their average age will be X + k.

Quadratic Roots in Ages

In problems involving product of ages, the larger root of the quadratic equation typically represents the older person's age.

Shortcut Trick

To find the fourth proportional to three numbers a, b, and c, use the direct formula: (b × c) ÷ a.

Given values are a = 7, b = 11, and c = 14.

Fourth Proportional = (11 × 14) ÷ 7 = 11 × 2 = 22.

This result matches the value given in option (4).

∴ The correct answer is 22.

Alternate Method

Given:

First term (a) = 7

Second term (b) = 11

Third term (c) = 14

Formula Used:

If a, b, c, and d are in proportion, then a : b = c : d, which means Product of Extremes = Product of Means (a × d = b × c).

Calculations:

⇒ Let the fourth proportional be x.

⇒ 7 : 11 :: 14 : x

⇒ 7 / 11 = 14 / x

⇒ 7 × x = 11 × 14

⇒ x = (11 × 14) ÷ 7

⇒ x = 11 × 2

⇒ x = 22

Therefore, the value 22 confirms that option (4) is the correct choice.

∴ The correct answer is 22.

Additional Information

Mean Proportional

The mean proportional between two numbers a and b is √(a × b).

Third Proportional

If a, b, and c are in continued proportion (a:b = b:c), then the third proportional c = b² ÷ a.

Invertendo and Alternendo

If a : b = c : d, then b : a = d : c (Invertendo) and a : c = b : d (Alternendo).

Compound Ratio

The compound ratio of (a : b) and (c : d) is (a × c) : (b × d).

Given:

The average weight of a class of 15 boys and 10 girls = 38.4 kg

Concept used:

Average weight = (Total weight of girls + Total weight of boys)/(No. of boys + No. of girls)

Calculation:

Let the average weight of girls be x

Total weight of boys = (40 × 15) = 600 kg

Now, according to the above concept,

⇒ 38.4 = $\frac{600+10 \times x}{{15 +10}}$

⇒ 38.4 = $\frac{600+10x}{{25}}$

⇒ 960 = 600 + 10x

⇒ 10x = 360

⇒ x = 36 kg

∴ The average weight of the girls is 36 kg.

Shortcut Trick

Ratio of Boys to Girls in the class = 70 : 30 = 7 : 3.

Deviation of Boys' average (75) from Total average (72) = +3.

Total positive deviation for 7 parts of boys = 7 × 3 = +21.

This must be balanced by a −21 deviation for 3 parts of girls.

Average deviation per girl = −21 ÷ 3 = −7.

Average marks of Girls = 72 − 7 = 65.

∴ The correct answer is 65 marks.

Alternate Method

Given:

Total students in class = 100

Number of Boys = 70

Average marks of Boys = 75

Average marks of Class = 72

Formula Used:

Total marks = Number of students × Average marks

Calculations:

⇒ Total marks of the complete class = 100 × 72 = 7200

⇒ Total marks of the boys = 70 × 75 = 5250

⇒ Number of girls = 100 − 70 = 30

⇒ Total marks of the girls = 7200 − 5250 = 1950

⇒ Average marks of the girls = 1950 ÷ 30

⇒ Average marks of girls = 65

∴ The correct answer is 65 marks.

Additional Information

Weighted Average Formula

The average of two groups combined is A_avg = (n₁A₁ + n₂A₂) ÷ (n₁ + n₂), where n is the count and A is the average.

Alligation Rule

The ratio of the quantities of two components is inversely proportional to the difference between their averages and the mixture average: n₁ ÷ n₂ = (A₂ − A_avg) ÷ (A_avg − A₁).

Deviation Method

The sum of the deviations of all observations from the mean is always zero. This is used to find missing values when the group mean is known.

Given:

The sum of two numbers = 7

The sum of their squares = 25

Formula used:

(a + b)² = a² + b² + 2ab

Calculation:

Let the two numbers be x and y

x + y = 7           -----(1)

x² + y² = 25      -----(2)

We know that

(a + b)2 = a2 + b2 + 2ab

⇒ (x + y)² = x² + y² + 2xy

⇒ xy = [(x + y)² - (x² + y²)]/2       ----(3)

From equation (1) & (2), we get

⇒ xy = [(7)² - 25)/2]

⇒ xy = [(49 - 25)/2]

⇒ xy = (24/2) = 12

∴ The product of the two numbers is 12.

Given:

Difference between CI and SI for 2 years = Rs.2000

Rate = 4% 

Formula used:

Difference between CI and SI for 2 years = P × (R/100)²

Where, P = Principal, R = Rate 

Calculation:

⇒ P × R²/10000 = 2000

⇒ P × 16/10000 = 2000

⇒ P = 2000 × 10000/16

⇒ P = 1250000

∴ The required Sum is Rs.1250000.

Given:

An article is sold at a 20% discount, the selling price is Rs. 24

Calculation:

According to the question

⇒ 80% = 24

Then, 70% = ($\frac{70 × 24}{{80}}$)

⇒ Rs. 21

∴ The required selling price is Rs. 21

Alternate Method

Formula used:

SP = {MP × (100 - D)}/100

Calculation:

According to the above concept,

SP = {MP × (100 - D)}/100

⇒ 24 = {MP × (100 - 20)}/100

⇒ 24 = (MP × 80/100)

⇒ MP = Rs. 30

Again using the same formula and considering D = 30%

SP = {MP × (100 - D)}/100

⇒ SP = {30 × (100 - 30)}/100

⇒ SP = (30 × 70/100)

⇒ SP = Rs. 21

∴ The required selling price is Rs. 21

Shortcut Trick

Ratio of CP : MP = (100 − Discount%) : (100 + Profit%)

⇒ CP : MP = (100 − 25) : (100 + 25) = 75 : 125 = 3 : 5

Let CP = 300 and MP = 500.

New Selling Price at 10% discount = 500 − 10% of 500 = 450.

Profit% = [(450 − 300) ÷ 300] × 100 = 150/300 × 100 = 50%.

∴ The correct answer is 50%.

Alternate Method

Given: First discount = 25%, First profit = 25%, Second discount = 10%.

Formula: MP/CP = (100 + Profit%) ÷ (100 − Discount%)

⇒ Let Cost Price (CP) = x

⇒ Marked Price (MP) = CP × (100 + 25) ÷ (100 − 25)

⇒ MP = x × (125 ÷ 75) = (5/3)x

⇒ New Selling Price (SP_new) = MP × (100 − 10) ÷ 100

⇒ SP_new = (5/3)x × (90 ÷ 100) = (5/3)x × 0.9

⇒ SP_new = 1.5x

⇒ Profit = SP_new − CP = 1.5x − x = 0.5x

⇒ Profit% = (Profit ÷ CP) × 100 = (0.5x ÷ x) × 100 = 50%

∴ The correct answer is 50%.

Additional Information

CP and MP Relationship

The direct formula connecting Cost Price and Marked Price is MP/CP = (100 + Profit%) / (100 − Discount%).

Successive Discounts

If two discounts d₁ and d₂ are given, the equivalent discount is d₁ + d₂ − (d₁×d₂)/100.

Effective Profit

Profit % is always calculated on the Cost Price unless stated otherwise, while Discount % is calculated on the Marked Price.

Shortcut Trick

Cost Price (CP) of 1 orange = 1000 ÷ 200 = Rs. 5.

To gain 25% profit, Selling Price (SP) of 1 orange = 5 × 1.25 = Rs. 6.25.

Number of oranges for Rs. 100 = Total Amount ÷ SP per orange.

Number of oranges = 100 ÷ 6.25 = 16 oranges.

∴ The correct answer is 16.

Alternate Method

Given: Total CP = Rs. 1000, Total Quantity = 200, Profit% = 25%.

Formulas: SP = CP × (100 + P%) ÷ 100 | Quantity = Total SP ÷ SP per unit.

⇒ CP of 200 oranges = Rs. 1000.

⇒ CP of 1 orange = 1000 ÷ 200 = Rs. 5.

⇒ SP of 1 orange = 5 × (125 ÷ 100) = Rs. 6.25.

⇒ Number of oranges for Rs. 100 = 100 ÷ 6.25 = 16.

∴ The correct answer is 16.

Additional Information

Unitary Method in Profit and Loss

Always calculate the price per unit to simplify calculations when the quantity changes between buying and selling scenarios.

Profit Percentage

Profit is always calculated on the Cost Price (CP) unless stated otherwise. Formula: SP = CP × (1 + Profit%/100).

Given:

The speed of car = 45 km/h

Formula used:

Distance = Speed × Time

Calculation:

According to the question

Time is taken by car = 2 hours 20 min   [10:40 AM to 1 PM]

⇒ 2 + $\frac{20}{{60}}$ = $\frac{7}{{3}}$ hours

Now, Distance = Speed × Time

⇒ (45 × $\frac{7}{{3}}$) = 105 km

∴ The speed of the car is 105 km.

Given:

Downstream = 32 km

Upstream = 14 km

Time taken to cover each distance = 6 hours

Formula used:

Speed of current = $\frac{1}{{2}}$(Downstream - Upstream)

Calculation:

According to the question

Rate of downstream = ($\frac{32}{{6}}$) km/hr

Rate of Upstream = ($\frac{14}{{6}}$) km/hr

Speed of current = $\frac{1}{{2}}$(Downstream - Upstream)

⇒ $\frac{1}{{2}}$($\frac{32}{{6}}$ - $\frac{14}{{6}}$)

⇒ ($\frac{1}{{2}}$ × $\frac{18}{{6}}$) 

⇒ $\frac{3}{{2}}$ = 1.5 km/h

∴ The speed of the current is 1.5 km/h.

Given:

Speed of boy when he is going to school = 12 km/h

Speed of the boy when he returned to his house = 8 km/h

Time = 50 min

Formula used:

Speed = Distance/Time

Calculation:

Let the distance between his school and house be x.

According to the question

Total time taken = Time taken from school to house + Time taken from house to school

⇒ $\frac{50}{{60}}$ h = $\frac{x}{{12}}$ + $\frac{x}{{8}}$

⇒ $\frac{50}{{60}}$ h = ($\frac{2x+3x}{{24}}$)

⇒ $\frac{50}{{60}}$ h = $\frac{5x}{{24}}$

⇒ x = 4 km/h

Now, the total distance walked = 2x = (2 × 4) = 8 km

∴  The total distance walked is 8 km.

Mistake PointsDon't mark option 1 i.e. 4 km/h since it is the distance covered in one way only.

Shortcut Trick

Total work = LCM(15, 20, 25) = 300 units.

Efficiency ratio (A : B : C) = 300/15 : 300/20 : 300/25 = 20 : 15 : 12.

Since they worked together for the entire duration, their wages are distributed directly in the ratio of their efficiencies.

C's share = [12 ÷ (20 + 15 + 12)] × 4700 = (12 ÷ 47) × 4700 = ₹1200

∴ The correct answer is ₹1200.

Alternate Method

Given:

A can finish the work in 15 days.

B can finish the work in 20 days.

C can finish the work in 25 days.

Total wages earned = ₹4700.

Concept Used:

Wages are distributed in proportion to the total work done by each person. If they work together for the same number of days, the wages are divided strictly in the ratio of their daily efficiencies.

Calculation:

⇒ Ratio of their wages = Efficiency of A : Efficiency of B : Efficiency of C

⇒ Wage ratio = 20 : 15 : 12

⇒ Sum of ratio parts = 20 + 15 + 12 = 47 parts

⇒ Total wages = 47 parts = ₹4700

⇒ Value of 1 part = 4700 ÷ 47 = ₹100

⇒ Share of C = 12 parts

⇒ Share of C = 12 × 100 = ₹1200

∴ The correct answer is ₹1200.

Additional Information

Work and Wages

Wages are always distributed in the ratio of the total work done by each individual. If persons work for an equal number of days from start to finish, the wages are divided proportionally to their per-day efficiency.

Leaving Mid-way or Joining Later

If someone leaves the work midway or joins late, their total share is strictly proportional to the actual units of work they have personally completed (Efficiency × Number of days worked).

Pipes and Cisterns

This concept operates identically to Time & Work. Filling pipes have a positive efficiency (similar to completing a task), whereas leakages or emptying pipes have a negative efficiency (similar to destroying work).

Given:

5 tractors can plough 5 hectares of land in 5 days

Concept used:

M₁D₁/W1​ = M₂D₂/W2

Where, M = Number of tractors,

D = Number of days, W = Hectares

Calculation:

Let the number of required tractors be y

According to the above concept,

⇒ M1D₁/W₁​ = M2D₂/W₂

⇒ ($\frac{5\times 5}{{5}}$) = ($\frac{y\times 50}{{100}}$)

⇒ 5 = $\frac{y}{{2}}$

⇒ y = 10

∴ The required number of tractors is 10.

Given:

9√2 - √8 - 4√2

Calculation:

9√2 - √8 - 4√2

⇒ 9√2 - 2√2 - 4√2

⇒ 3√2

∴ The required value is 3√2.

Given:

The shadow of a pole 6 m high and 15 m long.

At the same time, the shadow of a tree is 25 m long.

Calculation:

Let the height of the tree be H.

According to the question

⇒ $\frac{6}{{H}}$ = $\frac{15}{{25}}$

⇒ 15H = 150

⇒ H = 10 m

∴ The required height of the tree is 10 m.

Shortcut Trick

Let the two-digit number be x.

Difference between 1/3 and 1/4 of the number is 8.

⇒ (1/3 − 1/4) × x = 8

⇒ (1/12) × x = 8

⇒ x = 96

Sum of digits = 9 + 6 = 15.

∴ The correct answer is 15.

Alternate Method

Given:

One-third of the number = x/3

One-fourth of the number = x/4

Calculations:

⇒ x/3 − x/4 = 8

⇒ (4x − 3x) / 12 = 8

⇒ x / 12 = 8

⇒ x = 8 × 12

⇒ x = 96

The two-digit number is 96.

⇒ Sum of digits = 9 + 6 = 15

∴ The correct answer is 15.

Additional Information

Two-Digit Number Representation

A two-digit number with tens digit 'a' and units digit 'b' is represented as 10a + b.

Sum of Digits

The sum of digits for a number 10a + b is simply a + b.

Fractional Difference

To find the whole when the difference between two fractions is given: Value = Difference ÷ (1/n₁ − 1/n₂).

Shortcut Trick

Check the options to see which satisfies both conditions: sum of digits is 8 and adding 18 reverses the number.

For option 35: 3 + 5 = 8 (Condition 1 satisfied).

Adding 18 to the number: 35 + 18 = 53.

The digits 3 and 5 are reversed to 5 and 3 (Condition 2 satisfied).

∴ The correct answer is 35.

Alternate Method

Given:

Sum of digits of a two-digit number = 8

Number + 18 = Reversed number

Formula Used:

A two-digit number with tens digit x and units digit y is represented as: 10x + y.

Calculations:

Let the tens digit be x and the units digit be y.

⇒ x + y = 8 → (Equation 1)

The original number is 10x + y and the reversed number is 10y + x.

⇒ (10x + y) + 18 = 10y + x

⇒ 10x − x + y − 10y = −18

⇒ 9x − 9y = −18

⇒ x − y = −2

⇒ y − x = 2 → (Equation 2)

Adding (Equation 1) and (Equation 2):

⇒ (x + y) + (y − x) = 8 + 2

⇒ 2y = 10

⇒ y = 5

Substituting y = 5 in Equation 1:

⇒ x + 5 = 8

⇒ x = 3

⇒ The number is 10(3) + 5 = 35.

∴ The correct answer is 35.

Additional Information

Difference between Number and its Reverse

The difference between a two-digit number (10x + y) and its reverse (10y + x) is always equal to 9 × |x − y|. In this case, 18 ÷ 9 = 2, which is the difference between the digits.

Sum of a Number and its Reverse

The sum of a two-digit number and its reverse is always a multiple of 11, specifically 11 × (x + y).

Algebraic Representation

Always represent a two-digit number as 10T + U, where T is the tens digit and U is the units digit, to solve digit-reversal problems effectively.

Concept used:

sin 30° = $\frac{1}{{2}}$

Explanation:

According to the question 

Let us suppose sin 30° in the place of sin 31° + sin 32° 

⇒ ($\frac{1}{{2}}$ + $\frac{1}{{2}}$) = $\frac{2}{{2}}$ = 1

Here, we can clearly see that if we put the value of sin 30° we get its value be 1.

So, if we find the value of sin 31° + sin 32°, we have to put the sin value greater than 1.

∴ The required answer is Its value is greater than 1.

Given:

1 + tan θ = √2

Concept used:

tanθ = 1/cotθ 

Calculation: 

1 + tan θ = √2

⇒ tanθ = (√2 - 1)

⇒ cotθ = 1/(√2 - 1)

Multiplying the numerator and denominator with(√2 + 1), we get

⇒ ($\frac{1}{{√2 - 1}}$ × $\frac{√2+1}{{√2 + 1}}$)

⇒ $\frac{√2+1}{{2 - 1}}$ = √2 + 1

⇒ cotθ = √2 + 1

⇒ cotθ  - 1 = √2

∴ The required value is √2.

Given:

tanθ = $\frac{3}{4}$

Formula used:

tanθ = Perpendicular/Base

sinθ = Perpendicular/Hypotenuse

By Pythagoras' theorem,

H² = P² + B²

Calculation:

tan θ = Perpendicular/Base = $\frac{3}{4}$

So, by Pythagoras' theorem,

H2 = P2 + B²

⇒ H² = 3² + 4²

⇒ H² = (9 + 16)

⇒ H² = 25

⇒ H = 5 cm

⇒ sin θ = Perpendicular/Hypotenuse = $\frac{3}{5}$

∴ The required value is $\frac{3}{5}$.

Given:

sin 25° sin 35° sec 65° sec 55° 

Concept used:

sec θ = 1/cos θ 

cos(90 - θ) = sinθ 

Calculation:

According to the question

sin 25° sin 35° sec 65° sec 55° 

⇒ sin 25°. sin 35°. $\frac{1}{{cos65° }}$. $\frac{1}{{cos55° }}$

⇒ sin 25°. sin 35°. $\frac{1}{{cos(90^{\circ}-25° )}}$. $\frac{1}{{cos(90^{\circ}-35°) }}$

⇒ sin 25°. sin 35°. $\frac{1}{{sin25° }}$. $\frac{1}{{sin35° }}$

⇒ 1 

∴ The required answer is 1.

Concept used:

Following condition is must in any triangle

(sum of two sides) > Third side

Calculation:

According to the given options

Option 1:

(3, 4, 5)

(sum of two sides) > Third side

⇒ (3 + 4) > 5

⇒ 7 > 5

It represent the sides of a triangle.

Option 2:

(4, 7, 10)

(sum of two sides) > Third side

⇒ (4 + 7) > 10

⇒ 11 > 10

It represent the sides of a triangle.

Option 3:

(3, 6, 8)

(sum of two sides) > Third side

⇒ (3 + 6) > 8

⇒ 9 > 8

It represent the sides of a triangle.

Option 4:

(2, 3, 6)

(sum of two sides) > Third side

⇒ (2 + 3) > 6

⇒ 5 > 6

It does not represent the sides of a triangle.

∴ The required option 4 is correct.

Given:

A round balloon of unit radius subtends an angle of 90° at the eye of an observer standing at a point, say A.

Concept used:

cos θ = Base / hypotenuse

cos 45° = $\frac{1}{{√ 2}}$

Calculation:

According to the above figure,

In ΔOAB,

sin 45° = $\frac{OB}{{OA}}$

⇒ $\frac{1}{{√ 2}}$ = $\frac{1}{{OA}}$

⇒ OA = √2 cm

∴ The distance of the center of the balloon from point A is√2.

Given:

The shadow of a pole is √3 times the length of the pole.

Concept used:

tanθ = Perpendicular/base

tan 30° = $\frac{1}{{\sqrt 3}}$

Calculation:

The height of the tower be h.

So, the height of the shadow be √3h

Now, In ΔABC,

⇒ tan ∠ACB = $\frac{h}{{\sqrt 3h}}$

⇒ tan ∠ACB = $\frac{1}{{\sqrt 3}}$

⇒ tan ∠ACB = tan 30° 

⇒ ∠ACB = 30° 

∴ The angle of elevation is 30°

Concept used:

Right angle = 90° 

We know that the sum of all the sides of the triangle is 180°

tan 60° = √3

Calculation:

According to the above concept,

The sum of all the sides of the triangle is 180°

α  + β + 90° = 180 

⇒ α = 180° - (90° + β)

⇒ α = 180° - (90° + 30°)

⇒ α = (180° - 120°)

⇒ α = 60° 

So, the value of tan α = tan 60° = √3

∴ The required value of tan α is √3.

Given:

The top of a building 90 m high.

The angle of depression from the top and the bottom of a tree are 30° and 45° respectively. 

Concept used:

tan 45° = 1

tan 30° = $\frac{1}{{√ 3}}$

Calculation:

Let the height of the tree = h m.

According to the above figure,

In ΔADB,

tan 45° = $\frac{AB}{{DB}}$

⇒ 1 = $\frac{90}{{DB}}$

⇒ DB = 90 m

So, CE = DB = 90 m

In ΔCAE,

tan 30° = $\frac{AE}{{CE}}$

⇒ $\frac{1}{{√ 3}}$ = $\frac{AE}{{90}}$

⇒ AE = $\frac{90}{{√ 3}}$ = 30√3 m.

So, h = 90 - AE = (90 - 30√3) m

∴ The required height of the tree is 90 - 30√3 m.

Given:

The hypotenuse of the right-angled isosceles triangle = 6√2 cm

Concept used:

Isosceles right-angled triangle:

• The two sides on the right angle are equal (let a).
• One angle is 90°
• Hypotenuse = a√2

 

Area of right triangle = $\frac{1}{{2}}$ × base × height

Calculation:

According to the above concept,

Hypotenuse of the triangle = 6√2 cm

⇒ a√2 = 6√2

⇒ a = 6 cm

Now, the area of a right triangle

⇒ $\frac{1}{{2}}$ × base × height 

⇒ ($\frac{1}{{2}}$ × 6 × 6) cm²  = 18 cm²

∴ The required area is 18 cm².

Given:

The diameter of a circle circumscribing a square = 15√2 cm

Concept used:

By Pythagoras theorem

AC² = AB² + BC²

Calculation:

According to the above figure,

AC = 15√2 cm

Let the side of a square be x.

⇒ x² + x² = (15√2)²

⇒  2x² = 450

⇒ x² = 225

⇒ x = 15 cm

∴ The length of the side of the square is 15 cm.

Concept used:

1 mm = 0.1 cm

1 m = 100 cm

Calculation:

According to the above concept,

1 mm = 0.1 cm

⇒ 200 mm = 20 cm

Now, 1 m = 100 cm

⇒ 10 m = 1000 cm

Now, according to the question

⇒ (1000/20) = 50

∴ The required answer is 50.

Given:

The ratio of outer and inner perimeters of a circular path = 23 ∶ 22

The path is 5 m wide.

Concept used:

Circumference of circle = 2πr

Diameter = 2 × Radius

Calculation:

Let the radius of the outer circle be R.

And, the radius of the inner circle is r.

Now, according to the question

⇒ $\frac{2\pi R}{{2\pi r}}$ = $\frac{23}{{22}}$

⇒ $\frac{R}{{r}}$ = $\frac{23}{{22}}$

So, R = 23x and r = 22x    (x is common ratio)

According to the question

R - r = 5 

⇒ 23x - 22x = 5

⇒ x = 5 m

Now, the inner radius of the circle = 22x = (22 × 5) = 110 m

And, the diameter of the inner circle = (2 × 110) m = 220 m

∴  The diameter of the inner circle is 220 m.

Given:

The product of the lengths of the diagonals of a square = 50 square units.

Concept used:

Diagonal of square = a√2

Calculation:

Let the side of the square be a.

Diagonal of square = √2 a

Both the diagonals have the same length,

Product of length of diagonals = (a√2)² = 2a²

⇒ 50 = 2 × a²

⇒ a = 5 units

∴ The length of a side of the square is 5 units.

Concept used:

Volume of cylinder = πr²h

Volume of sphere = $\frac{4}{{3}}$πr³

Calculation:

The volume of the cylindrical log = πr²h

But h = 2r

⇒ Volume of the cylindrical log = 2πr³

The radius of the greatest possible sphere = r

So, the volume of this sphere = $\frac{4\pi r^3}{{3}}$

⇒ The required fraction = ($\frac{4\pi r^3}{{3}}$/2πr³) = 2/3

∴ The fraction of the original log which is cut away is 2/3.

Given:

The diagonal of a cube = 4√3 cm

Formula used:

Diagonal of cube = √3a

Volume of cube = a³

Calculation:

According to the question

The diagonal of a cube = 4√3 cm

⇒ √3a = 4√3

⇒ a = 4 cm

Now, volume of cube = a³

⇒ (4)³ = 64 cm³

∴ The required volume is 64 cu cm.

Given:

The size of a cube is increased by 100%

Concept used:

Total surface area of cube = 6a²

Calculation:

Let edge of a smaller cube be 1 m.

Then, edge of larger cube be 2 m.

Surface area of smaller cube = 6 × (1)² = 6 m²

And, surface area of larger cube = 6 × (2)² = 24 m²

% increase in surface area = ($\frac{24-6}{{6}}$ × 100) = 300%

∴ The surface area of the cube increased is 300%.

Concept used:

Bounded plane figure: Simple plane closed figure bounded by three line segments is called a triangle

Explanation:

According to the above concept,

If a plane figure is bounded by straight lines only, Here at least three lines are required because the triangle is only a closed figure.

The least number of straight lines for a bounded plane figure is 3.

∴ The required answer is 3.

Given:

There are five lines in a plane, no two of which are parallel.

Concept used:

Max number of points of intersection = Number of ways of selecting two lines out of the given line

Calculation:

By using the above concept, 

⇒ Number of points of intersection = 5C2

⇒ Number of points of intersection = 

⇒ Number of points of intersection =  = 10

∴ The required value is 10.