GSEB Class 7 Maths Solutions Chapter 12 બીજગણિતીય પદાવલિ Exercise 12.3

Question 1. If \( m = 2 \), find the value of the following expressions:
Answer:
(i) For \( m-2 \):
\( = 2 - 2 \) (since \( m = 2 \))
\( = 0 \)
(ii) For \( 3m-5 \):
\( = 3(2) - 5 \) (since \( m = 2 \))
\( = 6 - 5 \)
\( = 1 \)
(iii) For \( 9-5m \):
\( = 9 - 5(2) \) (since \( m = 2 \))
\( = 9 - 10 \)
\( = -1 \)
(iv) For \( 3m^2 - 2m - 7 \):
\( = 3(2^2) - 2(2) - 7 \) (since \( m = 2 \))
\( = 3(4) - 4 - 7 \)
\( = 12 - 4 - 7 \)
\( = 12 - 11 \)
\( = 1 \)
(v) For \( \frac { 5m }{ 2 } - 4 \):
\( = \frac { 5(2) }{ 2 } - 4 \) (since \( m = 2 \))
\( = \frac { 10 }{ 2 } - 4 \)
\( = 5 - 4 \)
\( = 1 \)
In simple words: To get the answer, replace the letter 'm' with the number 2 in each math problem, then solve the new problem. Always remember the order of operations when calculating the final value.

Exam Tip: Remember to substitute the variable's value carefully and follow the order of operations (PEMDAS/BODMAS) when evaluating expressions. Pay close attention to negative signs.

Question 2. If \( p = -2 \), find the value of the following:
Answer:
(i) For \( 4p + 7 \):
\( = 4(-2) + 7 \)
\( = -8 + 7 \)
\( = -1 \)
(ii) For \( -3p^2 + 4p + 7 \):
\( = -3(-2)^2 + 4(-2) + 7 \)
\( = -3(4) - 8 + 7 \)
\( = -12 - 8 + 7 \)
\( = -20 + 7 \)
\( = -13 \)
(iii) For \( -2p^3 - 3p^2 + 4p + 7 \):
\( = -2(-2)^3 - 3(-2)^2 + 4(-2) + 7 \)
\( = -2(-8) - 3(4) - 8 + 7 \)
\( = 16 - 12 - 8 + 7 \)
\( = 4 - 8 + 7 \)
\( = -4 + 7 \)
\( = 3 \)
In simple words: Put -2 in place of 'p' in each expression. Be super careful with the minus signs, especially when squaring or cubing a negative number. Then work out the sums to find the final result.

Exam Tip: When dealing with negative numbers raised to a power, remember that an even power makes the result positive (\((-2)^2 = 4\)), while an odd power keeps it negative (\((-2)^3 = -8\)).

Question 3. If \( x = -1 \), find the value of the following expressions:
Answer:
(i) For \( 2x-7 \):
\( = 2(-1) - 7 \)
\( = -2 - 7 \)
\( = -9 \)
(ii) For \( -x + 2 \):
\( = -(-1) + 2 \)
\( = 1 + 2 \)
\( = 3 \)
(iii) For \( x^2 + 2x + 1 \):
\( = (-1)^2 + 2(-1) + 1 \)
\( = 1 - 2 + 1 \)
\( = 0 \)
(iv) For \( 2x^2 - x - 2 \):
\( = 2(-1)^2 - (-1) - 2 \)
\( = 2(1) + 1 - 2 \)
\( = 2 + 1 - 2 \)
\( = 3 - 2 \)
\( = 1 \)
In simple words: Substitute -1 for 'x' into each expression. Watch out for how minus signs change when they are squared or when you subtract a negative number. Finish the calculations to find the value.

Exam Tip: Be cautious with double negatives, as subtracting a negative number is equivalent to adding a positive number (e.g., \( -(-1) = 1 \)).

Question 4. If \( a = 2 \) and \( b = -2 \), find the value of the following:
Answer:
(i) For \( a^2 + b^2 \):
\( = (2)^2 + (-2)^2 \)
\( = 4 + 4 \)
\( = 8 \)
(ii) For \( a^2 + ab + b^2 \):
\( = (2)^2 + (2)(-2) + (-2)^2 \)
\( = 4 - 4 + 4 \)
\( = 4 \)
(iii) For \( a^2 - b^2 \):
\( = (2)^2 - (-2)^2 \)
\( = 4 - 4 \)
\( = 0 \)
In simple words: Replace 'a' with 2 and 'b' with -2 in each math problem. Work out each step carefully, especially when squaring negative numbers or multiplying different signs.

Exam Tip: When a variable appears multiple times, ensure you substitute its value consistently for every instance. Remember that \( (a-b)(a+b) = a^2 - b^2 \), which can sometimes simplify calculations for (iii).

Question 5. When \( a = 0 \) and \( b = -1 \), find the value of the given expressions:
Answer:
(i) For \( 2a + 2b \):
\( = 2(0) + 2(-1) \)
\( = 0 - 2 \)
\( = -2 \)
(ii) For \( 2a^2 + b^2 + 1 \):
\( = 2(0)^2 + (-1)^2 + 1 \)
\( = 2(0) + 1 + 1 \)
\( = 0 + 1 + 1 \)
\( = 2 \)
(iii) For \( 2a^2b + 2ab^2 + ab \):
\( = 2(0)^2(-1) + 2(0)(-1)^2 + (0)(-1) \)
\( = 2(0)(-1) + 2(0)(1) + (0)(-1) \)
\( = 0 + 0 + 0 \)
\( = 0 \)
(iv) For \( a^2 + ab + 2 \):
\( = (0)^2 + (0)(-1) + 2 \)
\( = 0 + 0 + 2 \)
\( = 2 \)
In simple words: Put 0 for 'a' and -1 for 'b' in each math expression. Any number multiplied by zero becomes zero. Be careful with squaring negative numbers, as they become positive.

Exam Tip: Any term multiplied by zero will always become zero. This can significantly simplify calculations when one of the variable values is zero.

Question 6. Simplify the given expressions and find the value for \( x = 2 \):
Answer:
(i) For \( x + 7 + 4(x - 5) \):
First, simplify the expression:
\( = x + 7 + 4x - 20 \)
\( = (x + 4x) + (7 - 20) \)
\( = 5x - 13 \)
Now, find the value when \( x = 2 \):
\( = 5(2) - 13 \)
\( = 10 - 13 \)
\( = -3 \)
(ii) For \( 3(x + 2) + 5x - 7 \):
First, simplify the expression:
\( = 3x + 6 + 5x - 7 \)
\( = (3x + 5x) + (6 - 7) \)
\( = 8x - 1 \)
Now, find the value when \( x = 2 \):
\( = 8(2) - 1 \)
\( = 16 - 1 \)
\( = 15 \)
(iii) For \( 6x + 5(x - 2) \):
First, simplify the expression:
\( = 6x + 5x - 10 \)
\( = (6x + 5x) - 10 \)
\( = 11x - 10 \)
Now, find the value when \( x = 2 \):
\( = 11(2) - 10 \)
\( = 22 - 10 \)
\( = 12 \)
(iv) For \( 4(2x - 1) + 3x + 11 \):
First, simplify the expression:
\( = 8x - 4 + 3x + 11 \)
\( = (8x + 3x) + (-4 + 11) \)
\( = 11x + 7 \)
Now, find the value when \( x = 2 \):
\( = 11(2) + 7 \)
\( = 22 + 7 \)
\( = 29 \)
In simple words: First, make each math problem simpler by combining similar terms and expanding brackets. After you have the simplest version, replace 'x' with the number 2 and work out the final answer.

Exam Tip: Always simplify the expression first before substituting the value of the variable. This reduces the chances of calculation errors and often makes the problem easier to solve.

Question 7. Simplify the given expressions and find the value for \( x = 3, a = -1 \) and \( b = -2 \):
Answer:
(i) For \( 3x - 5 - x + 9 \):
First, simplify the expression:
\( = (3x - x) + (-5 + 9) \)
\( = 2x + 4 \)
Now, find the value when \( x = 3 \):
\( = 2(3) + 4 \)
\( = 6 + 4 \)
\( = 10 \)
(ii) For \( 2 - 8x + 4x + 4 \):
First, simplify the expression:
\( = -8x + 4x + 2 + 4 \)
\( = (-8x + 4x) + (2 + 4) \)
\( = -4x + 6 \)
Now, find the value when \( x = 3 \):
\( = -4(3) + 6 \)
\( = -12 + 6 \)
\( = -6 \)
(iii) For \( 3a - 5 - 8a + 1 \):
First, simplify the expression:
\( = (3a - 8a) + (5 + 1) \)
\( = -5a + 6 \)
Now, find the value when \( a = -1 \):
\( = -5(-1) + 6 \)
\( = 5 + 6 \)
\( = 11 \)
(iv) For \( 10 - 3b - 4 - 5b \):
First, simplify the expression:
\( = (-3b - 5b) + (10 - 4) \)
\( = -8b + 6 \)
Now, find the value when \( b = -2 \):
\( = -8(-2) + 6 \)
\( = 16 + 6 \)
\( = 22 \)
(v) For \( 2a - 2b - 4 - 5 + a \):
First, simplify the expression:
\( = (2a + a) + (-2b) + (-4 - 5) \)
\( = 3a - 2b - 9 \)
Now, find the value when \( a = -1 \) and \( b = -2 \):
\( = 3(-1) - 2(-2) - 9 \)
\( = -3 + 4 - 9 \)
\( = 1 - 9 \)
\( = -8 \)
In simple words: First, simplify each mathematical expression by combining similar terms. Then, substitute the provided numbers for 'x', 'a', and 'b' and perform the remaining arithmetic operations to get the final outcome.

Exam Tip: When an expression involves multiple variables, simplify it by grouping like terms together first, then substitute the values for each variable. This method makes the calculation less complicated.

Question 8. Find the value of the given expressions:
Answer:
(i) If \( z = 10 \), find the value of \( z^3 - 3(z - 10) \):
First, simplify the expression:
\( = z^3 - 3z + 30 \)
Now, find the value when \( z = 10 \):
\( = (10)^3 - 3(10) + 30 \)
\( = 1000 - 30 + 30 \)
\( = 1000 \)
(ii) If \( p = -10 \), find the value of \( p^2 - 2p - 100 \):
Now, find the value when \( p = -10 \):
\( = (-10)^2 - 2(-10) - 100 \)
\( = 100 + 20 - 100 \)
\( = 20 \)
In simple words: For these problems, first, expand and simplify the expressions if needed. Then, carefully replace the letter with the given number and calculate the final result. Remember to handle negative numbers and powers correctly.

Exam Tip: Always fully expand and simplify any algebraic expression before substituting numerical values to avoid errors, especially with terms containing parentheses.

Question 9. If the value of \( 2x^2 + x - a \) is 5 for \( x = 0 \), then find the value of \( a \).
Answer:
Given that the expression \( 2x^2 + x - a \) equals 5 when \( x = 0 \).
Substitute \( x = 0 \) into the expression:
\( 2(0)^2 + 0 - a = 5 \)
\( 2(0) + 0 - a = 5 \)
\( 0 + 0 - a = 5 \)
\( -a = 5 \)
\( a = -5 \)
So, the required value for \( a \) is -5.
In simple words: We are told that when 'x' is 0, the math problem \( 2x^2 + x - a \) turns into 5. By putting 0 where 'x' is, we can figure out what 'a' must be to make the whole problem equal 5.

Exam Tip: When a problem states that an expression "equals" a certain value for a specific variable, set up an equation by substituting the variable's value and then solve for the unknown constant.

Question 10. Simplify the given expression and find its value for \( a = 5 \) and \( b = -3 \): \( 2(a^2 + ab) + 3 - ab \).
Answer:
First, simplify the expression:
\( = 2a^2 + 2ab - ab + 3 \)
\( = 2a^2 + (2ab - ab) + 3 \)
\( = 2a^2 + ab + 3 \)
Now, find the value when \( a = 5 \) and \( b = -3 \):
\( = 2(5)^2 + (5)(-3) + 3 \)
\( = 2(25) - 15 + 3 \)
\( = 50 - 15 + 3 \)
\( = 35 + 3 \)
\( = 38 \)
In simple words: First, simplify the provided math expression by combining terms like 'ab'. Then, replace 'a' with 5 and 'b' with -3 in the simplified expression and calculate the final numerical answer.

Exam Tip: When simplifying expressions with parentheses, apply the distributive property correctly. Then, combine all like terms before substituting the values of the variables to ensure accuracy.