Two dice are thrown p is the event that the sum of the scores on the upp...

Two dice are throuwn Find the probability of getting the sum of the num...

Two digit numbers are formed from the digits 0, 1, 2, 3, 4 where digit...

Three coins are tossed simultaneously p is the event of getting at least...

There are three boys and two girls A committee of two is to be formed

There are 3 red, 3 white and 3 green balls in a bag

There are 3 men and 2 women A 'Gram swachchhatta Abhiyan' committee of ...

A die is thrown Find the probability of getting an odd number

HSC 12th Board Exam: Mathematics Probability Solution

Question

A die is thrown. Find the probability of getting an odd number.

Detailed Solution

Step 1: Determine the Sample Space (S)

When a standard six-sided die is thrown, the possible outcomes are numbers from 1 to 6.
\[ S = \{1, 2, 3, 4, 5, 6\} \] Therefore, the total number of possible outcomes is: \[ n(S) = 6 \]

Step 2: Define Event A

Let \( A \) be the event of getting an odd number.
The odd numbers on a die are 1, 3, and 5.
\[ A = \{1, 3, 5\} \] Therefore, the number of favorable outcomes is: \[ n(A) = 3 \]

Step 3: Calculate Probability

The probability of an event is given by the formula: \[ P(A) = \frac{n(A)}{n(S)} \]

Substituting the values: \[ P(A) = \frac{3}{6} \]

Simplifying the fraction: \[ P(A) = \frac{1}{2} \]

Answer: The probability of getting an odd number is \( \frac{1}{2} \).

If two coins are tossed, then find the probability of the events

If a card is drawn from a pack of 52 cards, find the probability of getting

From two digit numers using the digits 0, 1, 2, 3, 4, 5 without repeat...

A die is throuwn p is the event of getting an odd number

A coin is tossed three times Then find the probability of getting head ...

A coin is tossed and a die is thrown simultaneously

A card is drawn at random from well shuffled pack of 52 cards

A box contains cards marked with the numbers 1 to 20 One card is drawn

LOGIC HSC MATHS

Ex. No. 1.1 

State which of the following sentences are statements. Justify your answer. 

In case of statements, Write down the truth value.

1. March is the month in English Calendar.

Ans. It is a statement in Logic. Its truth value is T.

2. May God bless you!

Ans. It is an Exclamatory sentence, hence it is not a statement in Logic.

3. The sum of interior angles of a quadrilateral is 360⁰.

Ans. It is a statement in Logic. Its truth value is T.

4. Every natural number is a real number.

Ans. It is a statement in Logic. Its truth value is T.

5. Is the car yellow in colour?

Ans. It is an Interrogative sentence, hence it is not a statement in Logic.

6. Every quadratic equation has only real roots.

Ans. It is a statement in Logic, Its truth value is F.

7. √-4 is a rational number.

Ans. It is a statement in Logic. Its truth value is F.

8. x² – 5x + 6 = 0 implies that x = - 2 or x = - 3

Ans. It is a statement in Logic. Its truth value is F.

9. The sum of cube roots of unit is one.

Ans. It is a statement in Logic. Its truth value is F.

10. Please, come here.

Ans. It is an imperative(request) sentence, hence it is not a statement in Logic.

11. He is a bad person.

Ans. It is an open sentence, hence it is not a statement in Logic.

12. Two is the only even prime number.

Ans. It is a statement in Logic. Its truth value is T.

13. sin 3θ = 3 sinθ – 4 sin³θ ,  for all θ ∈ R.

Ans. It is a statement in Logic. Its truth value is T.

14. What a horrible sight it was!

Ans. It is an exclamatory sentence, hence it is not a statement in Logic.

15. Do not come inside the room.

Ans. It in an imperative(order) sentence, hence it is not a statement in Logic.

16. x² + 5x + 6 = 0 , x = -2.

Ans. It is a statement in Logic. Its truth value is T.

17. Can you speak in English?

Ans. It is an Interrogative sentence, hence it is not a statement in Logic.

18. The square of every real number is positive.

Ans. It is a statement in Logic. Its truth value is F.

19. It is blue in colour.

Ans. It is an open sentence, hence it is not a statement.

20. Every rectangle is a square.

Ans. It is a statement in Logic. Its truth value is F. 

1. Express the following statements in symbolic form.

(i) Banana is a fruit but potato is a vegetable. [video]

(ii) Either we play kabaddi or go for cycling. [video]

(iii) Milk is White or grass is green. [video]

(iv) The drug is effective though it has side effects. [video]

(v) Shobha stays at home, while Dhanashree and Prashant go for a movie. [Video] 

2. Write the truth value of each of the following statements.

(i) √2 is a rational number or  4 - 3i is a complex number. [video]

(ii) Jupiter is a planet and Mars is a star. [video]

(iii) 5 + 3 ≠ 8 or 5 × 3 < 8 [video]

(iv) 3 × 0 = 3 and 3 + 0 = 3 [video]

(v) 16 is a perfect square but 17 is a prime number. [video]

(vi) 6 is an even number or Pune is a harbour. [video]

Ex. No. 1.3 

1. Write the negation of each of the following statements: [video]

(i) π is an irrational number.

(ii) 7 is greater than 5. 

(iii) Zero is not a complex number. 

(iv) It is not true that mangoes are inexpensive.  

2. Write the truth value of the negation of each of the following statements. [VIDEO]

(i) The sun sets in the east.  

(ii) Re (z) ≤ |z| , where z is a complex number. 

(iii) Moscow is in America. 

  

iv) cos²θ + sin²θ = 1, for all θ Є R. 

Ex. No. 1.4  

1. Write the following statements in symbolic form. [video]

(i) I like reading but not singing.

(ii) Yuvraj neither plays cricket nor tennis.

(iii) Rama and Shama are twins.

(iv) It is not true that 'i' is a real number.

(v) Either 49 is a perfect square or 39 is divisible by 11

(vi) Pinku never works hard yet she gets good marks. [video]

(vii) Even though it is not cloudy, it is still raining. [video]

(viii) Milk is white if and only if the sky is not blue. [video]

(ix) If Kutub - minar is in Delhi then Hyderabad is in Andhra Pradesh. [video]

2. Find the truth values of the following statements. 

1. SEBI office is in Mumbai or Dalal Street is in London.

Ans. Let p = SEBI office is in Mumbai. 

q = Dalal Street is in London. 

The truth value of p is T and q is F. 

Symbolic form is 

p V q = T V F = T

  the truth value of the given 

statement is T. 

(ii) Neither 21 is a prime number nor it is divisible by 3.

Ans. Let P = 21 is a prime number 

q = 21 is divisible by 3. 

Here, the truth value of p is F and q is T.

The symbolic form is 

~p ⋀ ~ q = ~F ⋀ ~T = T ⋀ F = F

  the truth value of the given 

statement is F.  

(iii) It is not true that 3 - 7i is a real number. [video]

Ans. Let p = 3 - 7i is a real number.

Here, the truth value of p is F. 

The symbolic form is 

~ p = ~F = T. 

  the truth value of the given

statement is T. 

(iv) In a credit transaction, party’s name must be given and single entry system is costly. [video]

Ans. Let P = Party’s name must be given in a 

credit transaction. 

q = Single Entry system is costly in a credit 

transaction. 

Here, the truth value of p is T and q is F. 

The symbolic form is

p ⋀ q = T ⋀ F = F

  the truth value of the given 

statement is F.   

(v) If Joint venture is a temporary partnership, then discount on purchases is credited to supplier. 

Let P = Joint venture is a temporary partnership. 

q = Discount on purchases is credited to Supplier. 

Here the truth value of p is T and q is F. 

The symbolic form is 

p -> q = T -> F = F

  the truth value of the given 

statement is F. 

(vi) Every accountant is free to apply his own accounting rules if and only if machinery is an asset. 

Ans. Let P = Every accountant is free to apply his

own accounting rules. 

q = Machinery is an asset. 

Here, the truth value of p is F and q is T. 

The symbolic form is 

p <-> q   = F <-> T = F. 

  the truth value of the given 

statement is F. 

3. If p and q are true and r and s are false statements, find the truth value of each of the following. 

1. p v (q ⋀ r) 

Ans. Truth value of p and q are T and truth value 

of r and s are F. 

  p v (q ⋀ r) = T v (T ⋀ F) 

  p v (q ⋀ r) = T v F 

  p v (q ⋀ r) = T

  The truth value of the given statement is T

(ii) (p ⋀ ~ r) ⋀ (~ q ⋀ s )

Ans. Truth value of p and q are T and truth value 

of r and s are F.

  (p ⋀ ~ r) ⋀ (~ q ⋀ s ) = (T ⋀ ~ F) ⋀ (~ T ⋀ F )

  (p ⋀ ~ r) ⋀ (~ q ⋀ s ) = (T ⋀ T) ⋀ (F ⋀ F )

  (p ⋀ ~ r) ⋀ (~ q ⋀ s ) = T ⋀ F

  (p ⋀ ~ r) ⋀ (~ q ⋀ s ) = F

  the truth value of the given statement is F. 

(iii) (p -> q) v (r <-> s) 

Ans. Truth value of p and q are T and truth value 

of r and s are F. 

  (p -> q) v (r <-> s) = (p -> q) v (r <-> s)

  (p -> q) v (r <-> s) = (T -> T) v (F <-> F)

  (p -> q) v (r <-> s) = T v T

  (p -> q) v (r <-> s) = T

  the truth value of the given statement is T. 

iv. [(p -> q) -> (q -> r)] -> (r -> s)

Ans. Truth value of p and q are T and truth value 

of r and s are F.

  [(p -> q) -> (q -> r)] -> (r -> s) = [(T -> T) -> (T -> F)] -> (F -> F)

  [(p -> q) -> (q -> r)] -> (r -> s) = (T -> F) -> T

  [(p -> q) -> (q -> r)] -> (r -> s) = F -> T

  [(p -> q) -> (q -> r)] -> (r -> s) = T

  the truth value of the given statement is T. 

v. [p v (q ⋀ r) ] v ~ [(p ⋀ q) v (r v s)]

Ans. Truth value of p and q are T and truth value 

of r and s are F.

  [pv(q⋀r)] v ~[(p⋀q) v (r v s)] = [Tv(T⋀F)] v ~ [(T⋀T) v (FvF)]

  [pv(q ⋀ r)] v ~ [(p ⋀ q) v (r v s)] = (T v F) v ~(T v F) 

  [pv(q ⋀ r)] v ~ [(p ⋀ q) v (r v s)] = T v ~T

  [pv(q ⋀ r)] v ~ [(p ⋀ q) v (r v s)] = T v F

  [pv(q ⋀ r)] v ~ [(p ⋀ q) v (r v s)] = T

  the truth value of the given statement is T.

4. Assuming that the following statements are true: 

 p: Yuvraj has sufficient money. 

q : He will  buy a car. 

Find the truth value of the following statements.

1. Yuvraj has sufficient money and he will buy a car. 

Ans. The symbolic form of the given statement is 

p ⋀ q = T ⋀ T = T

The truth value of the given statement is T. 

1. If Yuvraj has sufficient money then he will not buy a car. 

Ans. They symbolic form of the given statement is 

p -> ~ q = T -> ~T = T -> F = F. 

The truth value of the given statement is F. 

1. Yuvraj does not have a money  or he will buy a car. 

Ans. The symbolic for of the given statement is 

~ p v q = = ~ T v T = F v T = T. 

The truth value of the given statement is T. 

5. If p = It is a day time. 

q : It is warm. 

Give the verbal statements for the following symbolic statements: 

1. p ⋀ ~ q : It is a day time and it is not warm. 
2. p v q : It is a day time or it is warm. 
3. p -> q : If it is a day time then it is warm. 
4. q <-> p : It is warm if and only if it is a day time. 

EX. NO. 1.5

1. If B = { 5, 6, 8, 10}, determine the truth value of each of the following. 

1. ∃x ∈ B, such that 3x + 4 = 28. 

Ans. Clearly x = 8 ∈ B satisfies 3x + 4 = 28. 

So the given statement is true. 

Hence the truth value of the given statement is T. 

(ii) ∀x ∈ B, x + 7 < 14.

Ans. x = 8 ∈ B and x = 10 ∈ B do not satisfy x + 7 < 14. 

So the given statement is false. 

Hence the truth value of the given statement is F. 

(iii) ∀x ∈ B, 4x - 3 ≥ 17. 

Ans. Clearly all the elements x ∈ B satisfy 4x - 3 ≥ 17. 

So the given statement is True. 

Hence the truth value of the given statement is T. 

(iv) ∃x ∈ B, such that x is an even. 

Ans. Clearly, x = 6 ∈ B, x = 8 ∈ B, and x = 10 ∈ B satisfy x is an even. 

So the given statement is true. 

Hence the truth value of the given statement is T. 

(v) ∃y ∈ B, such that (y - 10) ∈ N. 

Ans. No element y ∈ B satisfy (y - 10) ∈ N. 

So the given statement is false, 

Hence its truth value is F. 

2. Use quantifiers to convert each of the following open sentences defined on N, into a true statement: 

1.  

Ans. ∃x ∈ N, such that   is a true statement. 

Note: x = 6 ∈ N satisfies  

(ii) 5x - 3 < 10. 

Ans. ∃x ∈ N, such that 5x - 3 < 10 is a true statement. 

Note: x = 1 ∈ N, x = 2 ∈ N satisfy 5x - 3 < 10. 

(iii) x - 7 = 9.

Ans. ∃x ∈ N, such that x - 7 = 9 is a true statement. 

Note: x = 16 ∈ N satisfies x - 7 = 9.

(iv)  

Ans. ∃x ∈ N, such that   is a true statement. 

Note: y = 1 ∈ N, y = 2 ∈ N satisfy  .

(v)  

Ans. ∃x ∈ N, such that   is a true statement. 

Note: y = 5 ∈ N, y = 6 ∈ N satisfy  

(vi)  

Ans. ∀x ∈ N, such that   is a true statement. 

Note: All x ∈ N satisfy  

Ex. No. 1.6

1. Prepare the truth tables of the following statement patterns: 

1. (p ⋀ q) -> (~p) 

The truth table is given by

p	q	p ⋀ q	~p	(p ⋀ q) -> (~p)
T	T	T	F	F
T	F	F	F	T
F	T	F	T	T
F	F	F	T	T

(ii) (p -> q) <-> (~p v q) 

p	q	p -> q	~p	~p v q	(p -> q) <-> (~p v q)
T	T	T	F	T	T
T	F	F	F	F	T
F	T	T	T	T	T
F	F	T	T	T	T

5 Circle

Do you know.

*DO YOU KNOW*?

1. *Your shoes are the first thing people subconsciously notice about you. Wear nice shoes.*

2. *If you sit for more than 11 hours a day, there's a 50% chance you'll die within the next 3 years.*

3. *There are at least 6 people in the world who look exactly like you. There's a 9% chance that you'll meet one of them in your lifetime.*

4. *Sleeping without a pillow reduces back pain and keeps your spine stronger.*

5. *A person's height is determined by their father, and their weight is determined by their mother.*

6. *If a part of your body "falls asleep", you can almost always "wake it up" by shaking your head.*

7. *There are three things the human brain cannot resist noticing - Food, attractive people and danger.*

8. *Right-handed people tend to chew food on their right side.*

9. *Putting dry tea bags in gym bags or smelly shoes will absorb the unpleasant odour.*

10. *According to Albert Einstein, if honey bees were to disappear from earth, humans would be dead within 4 years.*

11. *There are so many kind of apples, that if you ate a new one everyday, it would take over 20 years to try them all.*

12. *You can survive without eating for weeks, but you will only live 11 days without sleeping.*

13. *People who laugh a lot are healthier than those who don't.*

14. *Laziness and inactivity kills just as many people as smoking.*

15. *A human brain has a capacity to store 5 times as much information as Wikipedia*

16. *Our brain uses same amount power as 10-watt light bulb!!*

17. *Our body gives enough heat in 30 mins to boil 1.5 litres of water!!*

18. *Stomach acid (conc. HCl) is strong enough to dissolve razor blades!!*

19. *Take a 20-40 minute walk every day. & while you walk, SMILE. It is the ultimate antidepressant.*

20. *✋Fwd‬ this to everyone ☝on your list...a worthy information for people to be aware of.*👍

4 Circle

2 Circle

1 Circle

3 Circle

Our achievers in ssc 2017.... CONGRATULATIONS TO ALL

OMTEX CLASSES

Our achievers in ssc 2017....

CONGRATULATIONS TO ALL

1. BUSHRA .S(St.Joseph) = 92.00%
2. VINITHA(SIES) = 89.00%
3. ARIVUMATHI(CANOSSA) = 89.00%
4. ROHAN.V(AMBEDKAR) = 87.00%
5. AUGUSTIN (SIES) = 86.20%
6. SNEHA.S(CANOSSA) = 85.40%
7. SAJITH RAJA(OLGC) = 85.40%
8. SHOAIB (K.J.KHILNANI) = 83.00%
9. HARIHAR.B(SIES) = 82.00%
10. NIKHIL.K(SARASWATI) = 81.00%
11. SHIVANGINI (UMT) = 80.20%
12. VIGNESH.T (GURUNANAK) = 80.20%
13. PINKY (AMBEDKAR) = 79.00%
14. MEENAKSHI.P(SARASWA) = 79.00%
15. WINSTON(ST.MICHEAL) = 78.00%
16. PRIYA.M(AMBEDKAR) = 78.40%
17. INSHA.A(BSIAS) = 77.00%
18. ROSY.A(CANOSSA) = 77.20%
19. SOLOMON (SIES) = 77.00%
20. BIRUNDHA (CANOSSA) = 76.40%
21. ANJALI(CANOSSA) = 76.40%
22. SAMIDURAI (AMBEDKAR) = 76.00%
23. AMANDEEP(SIES) = 75.00%
24. NELSON.J(GURUNANAK) = 74.40%
25. ZUBER (SIES) = 74.20%
26. MUTHUMARI (UMT) = 74.00%
27. GANESH.N(SIWS) = 74.00%
28. MERLIN (KMEHS) = 74.00%
29. SALEEM (FR.AGNEL) = 74.00%
30. PETRINA(CANOSA) = 74.00%
31. ANUSHA (ST.JOSEPH) = 73.00%
32. BALAJI.T(SIES) = 73.00%
33. MOHD.HUSSAIN (UMT) = 73.00%
34. SOMEYA(ST.ANTHONY) = 72.60%
35. MANISHA.V(SIES) = 72.00%
36. KANMANI (P.W.S) = 72.00%
37. MEENATCHI SUNDA(SIWS) = 71.40%
38. VENCY(ST.ANTHONY) = 71.00%
39. RAMYA(CENTRAL ) = 71.00%
40. DANISH(ST.MICHEAL) = 71.00%
41. SWAHLE (O.LG.C) = 70.00%
42. AMANULLAH(GURUNANAK)= 70.00%

AND MANY MORE TO BE ADDED...................................

OMTEX CLASSES

OUR SOME OF PREVIOUS ACHIEVERS........

MISS. SHAYMA [AMBEDKAR] = 91.40%

MISS. VINDHYA [P.W.S] = 91.00%

MR. KEVIN [O.L.G.C.] = 90.40%

MISS. INDU KUMARI [S.I.W.S.] = 90.36%

MR. ISAC SUBIN [O.L.G.C.] = 90.20%