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Here's a detailed explanation of how each solution was derived, step-by-step, as you requested:

1. Identify positive and negative integers:

To identify positive and negative integers, we first understand that integers are whole numbers (no fractions or decimals). Then, we classify them based on whether they are greater than, less than, or equal to zero.

• 1(a) $4, \frac{3}{2}, (+2), +7, 0, -5$

  • Step 1: Identify integers. Integers are whole numbers. From the given list, $\frac{3}{2}$ is a fraction, so it is not an integer. The integers are $4, (+2), +7, 0, -5$.
  • Step 2: Classify based on sign.
    • Positive integers are numbers greater than 0.
    • Negative integers are numbers less than 0.
    • 0 is neither positive nor negative.
  • Answer:
    • Positive integers: $4, (+2), +7$
    • Negative integers: $-5$
    • Neither positive nor negative: 0

• 1(b) $-11, 51, 4.5, \frac{6}{7}, -7, \frac{5}{8}$

  • Step 1: Identify integers. $4.5, \frac{6}{7}, \frac{5}{8}$ are decimals or fractions, so they are not integers. The integers are $-11, 51, -7$.
  • Step 2: Classify based on sign.
  • Answer:
    • Positive integers: 51
    • Negative integers: $-11, -7$

2. Write the opposite of the following:

To find the opposite, we identify the key concept or direction in the statement and replace it with its inverse.

• 2(a) 10 kg वजन में वृद्धि (Increase in weight by 10 kg)

  • Step 1: The opposite of "increase" is "decrease".
  • Answer: Decrease in weight by 10 kg

• 2(b) पश्चिम की ओर जाना (Going towards West)

  • Step 1: The opposite direction of "West" is "East".
  • Answer: Going towards East

• 2(c) $-80$

  • Step 1: The opposite of a number is the same numerical value with the opposite sign.
  • Answer: $*+80*$

• 2(d) समुद्र तल से 100 m ऊपर (100 m above sea level)

  • Step 1: The opposite of "above" is "below".
  • Answer: 100 m below sea level

• 2(e) ₹500 का नुकसान (Loss of ₹500)

  • Step 1: The opposite of "loss" is "profit" or "gain".
  • Answer: Profit of ₹500

• 2(f) देश में हिंसा (Violence in the country)

  • Step 1: The opposite of "violence" is "peace".
  • Answer: Peace in the country

3. Write the numbers used in the following using the sign (+ or -):

To represent these situations using signs, we assign '+' for increases, above, deposits, or positive changes, and '-' for decreases, below, losses, or negative changes.

• 3(a) एक हवाई जहाज जमीन से दो हजार पाँच सौ मीटर की ऊँचाई पर उड़ रहा है। (An aeroplane is flying at a height of two thousand five hundred meters above the ground.)

  • Step 1: "Height" or "above the ground" indicates a positive value.
  • Answer: $\boxed{+2500 \text{ m}}$

• 3(b) एक पनडुब्बी समुद्र तल से 800 m की गहराई पर चल रही है। (A submarine is moving at a depth of 800 m below sea level.)

  • Step 1: "Depth" or "below sea level" indicates a negative value.
  • Answer: $\boxed{-800 \text{ m}}$

• 3(c) तापमान $10^\circ C$ ऊपर से $0^\circ C$ तक। (Temperature $10^\circ C$ above $0^\circ C$.)

  • Step 1: "Above $0^\circ C$" indicates a positive temperature.
  • Answer: $\boxed{+10^\circ C}$

• 3(d) बैंक में ₹100 जमा करना। (Depositing ₹100 in the bank.)

  • Step 1: "Depositing" money means an increase in the bank balance, which is positive.
  • Answer: $\boxed{+₹100}$

• 3(e) ₹100 का नुकसान। (Loss of ₹100.)

  • Step 1: "Loss" indicates a decrease in value, which is negative.
  • Answer: $\boxed{-₹100}$

4. निम्नलिखित से क्या दर्शाया जाता है? (What is represented by the following?)

To interpret the meaning of a signed number, we use the given context or the standard convention for positive and negative values.

• 4(a) $+10^\circ C$ (यदि $-10^\circ C$ तापमान को नीचे दर्शाता है) ($+10^\circ C$ (if $-10^\circ C$ represents temperature below $0^\circ C$))
  • Step 1: If a negative sign (-) represents "below $0^\circ C$", then a positive sign (+) must represent the opposite, which is "above $0^\circ C$".
  • Answer: This represents a temperature of $10^\circ C$ above $0^\circ C$.