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**Determine whether the following statement pattern is a tautology, contradiction, or contingency:**

[(∼ p ∧ q) ∧ (q ∧ r)] ∧ (∼ q)

Concept: Statement Patterns and Logical Equivalence

**Choose the correct alternative :**

If p : He is intelligent.

q : He is strong

Then, symbolic form of statement “It is wrong that, he is intelligent or strong” is

Concept: Truth Value of Statement

**Choose the correct alternative:**

Negation of p → (p ˅ ~q) is

Concept: Logical Connective, Simple and Compound Statements

The dual of the statement (p ˅ q) ˄ (r ˅ s) is ______.

Concept: Truth Value of Statement

Write the negation of the statement “An angle is a right angle if and only if it is of measure 90°”

Concept: Logical Connective, Simple and Compound Statements

**Examine whether the statement pattern**

[p → (~ q ˅ r)] ↔ ~[p → (q → r)] is a tautology, contradiction or contingency.

Concept: Statement Patterns and Logical Equivalence

Using truth table prove that p ˅ (q ˄ r) ≡ (p ˅ q) ˄ (p ˅ r).

Concept: Truth Value of Statement

**Complete the truth table.**

p |
q |
r |
q → r |
r → p |
(q → r) ˅ (r → p) |

T | T | T | T | `square` | T |

T | T | F | F | `square` | `square` |

T | F | T | T | `square` | T |

T | F | F | T | `square` | `square` |

F | T | T | `square` | F | T |

F | T | F | `square` | T | `square` |

F | F | T | `square` | F | T |

F | F | F | `square` | T | `square` |

The given statement pattern is a `square`

Concept: Statement Patterns and Logical Equivalence

If p ∨ q is true, then the truth value of ∼ p ∧ ∼ q is ______.

Concept: Algebra of Statements

Write the converse, inverse, and contrapositive of the statement. "If 2 + 5 = 10, then 4 + 10 = 20."

Concept: Logical Connective, Simple and Compound Statements

**Determine whether the following statement pattern is a tautology, contradiction, or contingency:**

[(∼ p ∧ q) ∧ (q ∧ r)] ∧ (∼ q)

Concept: Statement Patterns and Logical Equivalence

The total cost of 3 T.V. and 2 V.C.R. is ₹ 35,000. The shopkeeper wants profit of ₹1000 per television and ₹ 500 per V.C.R. He can sell 2 T.V. and 1 V.C.R. and get the total revenue as ₹ 21,500. Find the cost price and the selling price of a T.V. and a V.C.R.

Concept: Application of Matrices

The sum of three numbers is 6. If we multiply third number by 3 and add it to the second number we get 11. By adding the first and third number we get a number which is double the second number. Use this information and find a system of linear equations. Find the three numbers using matrices.

Concept: Application of Matrices

Find matrices A and B, if `2"A" - "B" = [(6, -6, 0),(-4, 2, 1)] and "A" - 2"B" = [(3, 2, 8),(-2, 1, -7)]`

Concept: Algebra of Matrices

If A^{2} + 5A + 3I = 0, |A| ≠ 0, then A^{–1} = ______

Concept: Application of Matrices

**State whether the following statement is True or False:**

If `[(3, 0),(0, 2)][(x),(y)] = [(3),(2)]`, then x = 1 and y = – 1

Concept: Types of Matrices

If A = [a_{ij}]_{2×2}, where a_{ij} = i – j, then A = ______

Concept: Inverse of Matrix

Find k, if A = `[(3, -2),(4, -2)]` and A^{2} = kA – 2I, where I is identity matrix of order 2

Concept: Algebra of Matrices

If A = `[(2, 1),(0, 3),(1, -1)]` and B = `[(0, 3, 5),(1, -7, 2)]`, then verify (BA)^{T} = A^{T}B^{T}

Concept: Properties of Matrices

**Complete the following activity.**

The cost of 4 kg potato, 3kg wheat and 2kg rice is ₹ 60. The cost of 1 kg potato, 2 kg wheat and 3kg rice is ₹ 45. The cost of 6 kg potato, 3 kg rice and 2 kg wheat is ₹ 70. Find the per kg cost of each item by matrix method.

**Solution:** Let the cost of potato, wheat and rice per kg be x, y and z respectively.

Therefore by given conditions,

4x + ( )y + 2( ) = ( )

x + 2y + ( )( ) = ( )

( )x + 2y + 3z = ( )

Matrix form of above equations is,

`[("( )", 3, "( )"),(1, "( )", 3),("( )", 2, "( )")] [(x),(y),(z)] =[("( )"), (45), ("( )")]`

R_{1} ↔ R_{2}

`[(1, 2, 3),("( )", "( )", "( )"),(6, 2, 3)] [(x),(y),(z)] =[("( )"), (60), ("( )")]`

R_{2} – 4R_{1}, R_{3} – 6R_{1}

`[(1, 2, 3),("( )", -5, "( )"),(0, "( )", -15)] [(x),(y),(z)] =[(45), ("( )"), (-200)]`

`(-1)/5 "R"_2, (-1)/5 "R"_3`

`[("( )", 2, 3),(0, "( )", 2),(0, 2, "( )")] [(x),("( )"),(z)] =[(45), (24), (40)]`

R_{3} – 2R_{2}

`[(1, 2, 3),(0, 1, 2),(0, 0, -1)] [(x),(y),(z)] =[("( )"), ("( )"), ("( )")]`

By pre multiplying we get,

x + 2y + ( )z = ( ) .....(i)

y + 2z = 24 ......(ii)

–z = ( ) ......(iii)

From (iii), we get, z = ( )

From (ii), we get, y = ( )

From (i), we get, x = ( )

Therefore the cost of Potato, Wheat and Rice per kg are _______, _______ and _______ respectively.

Concept: Application of Matrices