HSC Science (General) १२ वीं कक्षा - Maharashtra State Board Question Bank Solutions

Using the rules in logic, write the negation of the following:

p ∧ (q ∨ r)

Using the rules in logic, write the negation of the following:

(p → q) ∧ r

Using the rules in logic, write the negation of the following:

(∼p ∧ q) ∨ (p ∧ ∼q)

Suppose `bar"a" = bar"0"`:

If `bar"a".bar"b" = bar"a".bar"c"`, then is `bar"b" = bar"c"` ?

Suppose `bar"a" = bar"0"`:

If `bar"a" xx bar"b" = bar"a" xx bar"c"`, then is `bar"b" = bar"c"` ?

Suppose `bar"a" = bar"0"`:

If `bar"a".bar"b" = bar"a".bar"c" and bar"a" xx bar"b" = bar"a" xx bar"c"`,  then is `bar"b" = bar"c"`?

A(2, 3), B(−1, 5), C(−1, 1) and D(−7, 5) are four points in the Cartesian plane, Check if, `bar("CD")` is parallel to `bar("AB")`

The non zero vectors `bar("a")` and `bar("b")` are not collinear find the value of `lambda` and `mu`: if `bar("a") + 3bar("b") = 2lambdabar("a") - mubar("b")`

If `bar("a") = 4hat"i" + 3hat"k"` and `bar("b") = -2hat"i" + hat"j" + 5hat"k"`, then find `2bar("a") + 5bar("b")`

If the vectors `2hat"i" - "q"hat"j" + 3hat"k"` and `4hat"i" - 5hat"j" + 6hat"k"` are collinear then find the value of q

Find `bar("a")*(bar("b") xx bar("c"))`, if `bar("a") = 3hat"i" - hat"j" + 4hat"k", bar("b") = 2hat"i" + 3hat"j" - hat"k", bar("c") = -5hat"i" + 2hat"j" + 3hat"k"`

If `bar("c") = 3bar("a") - 2bar("b")` then prove that `[(bar("a"), bar("b"), bar("c"))]` = 0

Without using truth table prove that (p ∧ q) ∨ (∼ p ∧ q) v (p∧ ∼ q) ≡ p ∨ q

`"Check whether the vectors"  2hati+2hatj+3hatk,-3hati+3hatj+2hatk  "and"  3hati+4hatk  "form a triangle or not".`

Without using truth table, prove that : [(p ∨ q) ∧ ∼p] →q is a tautology.

The simplified form of [(~ p v q) ∧ r] v [(p ∧ ~ q) ∧ r] is ______.

Without using truth table prove that

[(p ∧ q ∧ ∼ p) ∨ (∼ p ∧ q ∧ r) ∨ (p ∧ q ∧ r) ∨ (p ∧ ∼ q ∧ r) ≡ (p ∨ q) ∧ r

The statement p → (q → p) is equivalent to ______.

Show that the simplified form of (p ∧ q ∧ ∼ r) ∨ (r ∧ p ∧ q) ∨ (∼ p ∨ q) is q ∨ ∼ p.