HSC Commerce (English Medium) 12th Standard Board Exam - Maharashtra State Board Question Bank Solutions for Mathematics and Statistics

Using truth table, examine whether the following statement pattern is tautology, contradiction or contingency: p ∨ [∼(p ∧ q)]

Find `(dy)/(dx) , "If"   x^3 + y^2 + xy = 10`

Express the following statement in symbolic form and write its truth value.
"If 4 is an odd number, then 6 is divisible by 3."

Show that the following statement pattern in contingency : 

(~p v q) → [p ∧ (q v ~ q)] 

Find `(dy)/(dx) if y = cos^-1 (√x)`

Find `(dy)/(dx)` if `y = sin^-1(sqrt(1-x^2))`

Use the quantifiers to convert the following open sentence defined on N into true statement
5x - 3 < 10

Use the quantifiers to convert the following open sentence defined on N into true statement:
x² ≥ 1

Prove that the following statement pattern is equivalent:
(p v q) → r and (p → r) ∧ (q → r)

Write the negation of the Following Statement :
∀ y ∈  N, y² + 3 ≤ 7

Write the negation of the following statement : 
If the lines are parallel then their slopes are equal.

State if the following sentence is a statement. In case of a statement, write down the truth value :
Every quadratic equation has only real roots.

State if the following sentence is a statement. In case of a statement, write down the truth value :
√-4 is a rational number.

Write converse and inverse of the following statement :
"If Ravi is good in logic then Ravi is good in Mathematics."

By constructing the truth table, determine whether the following statement pattern ls a tautology , contradiction or . contingency.  (p →  q) ∧  (p ∧ ~ q ).

Differentiate tan^-1 (cot 2x) w.r.t.x.

If x = tan^-1t and y = t³ , find `(dy)/(dx)`.

Discuss extreme values of the function f(x) = x.logx

Examine whether the following statement (p ∧ q) ∨ (∼p ∨ ∼q) is a tautology or contradiction or neither of them.

Using the truth table prove the following logical equivalence.

∼ (p ∨ q) ∨ (∼ p ∧ q) ≡ ∼ p