HSC Commerce (English Medium) इयत्ता १२ वी - Maharashtra State Board Important Questions for Mathematics and Statistics

For 50 students of a class, the regression equation of marks in statistics (X) on the marks in accountancy (Y) is 3y – 5x + 180 = 0. The mean marks in accountancy is 44 and the variance of marks in statistics is `(9/16)^(th)` of the variance of marks in accountancy. Find the mean marks in statistics and the correlation coefficient between marks in the two subjects.

For a bivariate data, 

`bar x = 53 , bar y = 28 , "b"_"xy" = - 0.2` , `"b"_"yx" = -1.5` Find 

 Estimate of Y , When X = 50. 

The probability that a person who undergoes kidney operation will recover is 0.5. Find the probability that of the six patients who undergo similar operations,

(a) None will recover

(b) Half of them will recover.

Using the truth table verify that p ∨ (q ∧ r) ≡ (p ∨ q) ∧ (p ∨ r).

The following data gives the marks of 20 students in mathematics (X) and statistics (Y) each out of 10, expressed as (x, y). construct ungrouped frequency distribution considering single number as a class.
Also prepare marginal distributions :

(2, 7) (3, 8) (4, 9) (2, 8) (2, 8) (5, 6) (5 , 7) (4, 9) (3, 8) (4, 8) (2, 9) (3, 8) (4, 8) (5, 6) (4, 7) (4, 7) (4, 6 ) (5, 6) (5, 7 ) (4, 6 )

If a random variable X follows Poisson distribution sucli that P(X = 1) = P(X = 2), find:
(a) the mean and
(b) P(X = 0). [Given: e^-2 = 0.1353)

A random variable X has the following probability distribution:

Find:

1. k
2. P(X < 3)
3. P(X > 4)

The negation of p ∧ (q → r) is ______________.

Examine whether the following logical statement pattern is a tautology, contradiction, or contingency.

[(p→q) ∧ q]→p

Write the converse and contrapositive of the statement -
“If two triangles are congruent, then their areas are equal.”

Express the following statement in symbolic form and write its truth value.

"If 4 is an odd number, then 6 is divisible by 3 "

Prove that the following statement pattern is equivalent :

(p ∨ q) →  r and (p → r) ∧ (q → r)

Express the truth of each of the following statements by Venn diagram:

(a) Some hardworking students are obedient.

(b) No circles are polygons.

(c) All teachers are scholars and scholars are teachers. 

Write down the following statements in symbolic form :

(A) A triangle is equilateral if and only if it is equiangular.
(B) Price increases and demand falls

Without using the truth table show that P ↔ q ≡ (p ∧ q) ∨ (~ p ∧ ~ q)

Using truth table examine whether the following statement pattern is tautology, contradiction or contingency `(p^^~q) harr (p->q)`

Write the dual of the following statements: (p ∨ q) ∧ T

If A = {2, 3, 4, 5, 6}, then which of the following is not true?

(A) ∃ x ∈ A such that x + 3 = 8

(B) ∃ x ∈ A such that x + 2 < 5

(C) ∃ x ∈ A such that x + 2 < 9

(D) ∀ x ∈ A such that x + 6 ≥ 9

Write the dual of the following statements:

Madhuri has curly hair and brown eyes.

Draw a Venn diagram for the truth of the following statement : 

All rational number are real numbers.