CBSE Class 7 Mathematics Algebraic Expression Worksheet Set 02

Q.1. Identify in the following expressions terms which are not constant and give their numerical coefficients:
a) \(a + b + 5\)
b) \(2x^2y - 3xy^2 + 7\)
c) \(11 - p^2\)
d) \(13 - p + 5q^2\)
Answer:
a) Terms not constant: \(a, b\). Numerical coefficients: For \(a\) is 1, for \(b\) is 1.
b) Terms not constant: \(2x^2y, -3xy^2\). Numerical coefficients: For \(2x^2y\) is 2, for \(-3xy^2\) is -3.
c) Term not constant: \(-p^2\). Numerical coefficient: -1.
d) Terms not constant: \(-p, 5q^2\). Numerical coefficients: For \(-p\) is -1, for \(5q^2\) is 5.
In simple words: The numerical coefficient is the fixed number multiplied by the variable term.
Exam Tip: Remember that a single variable without a visible number has an implicit coefficient of 1 or -1.

Q.2. Write the coefficient of p in the following expressions
a) \(3p - 4q\)
b) \(7 - p + q\)
c) \(2r - 4pr\)
Answer:
a) Coefficient of \(p\) in \(3p - 4q\) is 3.
b) Coefficient of \(p\) in \(7 - p + q\) is -1.
c) Coefficient of \(p\) in \(2r - 4pr\) is \(-4r\).
In simple words: Look at the term containing \(p\) and write down everything multiplied by it.

Exam Tip: Be careful to include the negative sign when the term is subtracted.

Q.3. Which of the following pair of terms are like terms and which are unlike
a) \(6x, 11y\)
b) \(-4pq, 8qp\)
c) \(2xy, x\)
d) \(2mn^2, 5n^2m\)
Answer:
a) \(6x\) and \(11y\) are unlike terms (different variables).
b) \(-4pq\) and \(8qp\) are like terms (same variables with same exponents, since \(pq = qp\)).
c) \(2xy\) and \(x\) are unlike terms (different variable parts).
d) \(2mn^2\) and \(5n^2m\) are like terms (same variables \(m\) and \(n\) with matching exponents: \(m^1n^2\)).
In simple words: Like terms must have the exact same variables and powers.

Q.4. Classify the following expressions as monomial, binomial or trinomial
a) \(4x - 3y\)
b) \(2xy\)
c) 5
d) \(3x + 5y + 7\)
e) \(a + b\)
f) \(5x^2 - x - z\)
Answer:
a) \(4x - 3y\) is a binomial (2 terms).
b) \(2xy\) is a monomial (1 term).
c) 5 is a monomial (1 term).
d) \(3x + 5y + 7\) is a trinomial (3 terms).
e) \(a + b\) is a binomial (2 terms).
f) \(5x^2 - x - z\) is a trinomial (3 terms).
In simple words: Count the number of terms separated by plus or minus signs to classify the expression.

Exam Tip: Terms combined by multiplication or division count as a single term.

Q.5. Add the following expressions
a) \(x - 3y + 4z\), \(y - 2x - 8z\), \(5x - 2y - 3z\)
b) \(5x, 7x, -6x\)
c) \(mn + 5m - 2, mn + 3\)
d) \(5x - 2x^2 - 8, 8x^2 - 7x - 9\)
Answer:
a) \((x - 3y + 4z) + (y - 2x - 8z) + (5x - 2y - 3z)\)
\(= (1 - 2 + 5)x + (-3 + 1 - 2)y + (4 - 8 - 3)z\)
\(= 4x - 4y - 7z\)
b) \(5x + 7x + (-6x) = (5 + 7 - 6)x = 6x\)
c) \((mn + 5m - 2) + (mn + 3)\)
\(= (1 + 1)mn + 5m + (-2 + 3)\)
\(= 2mn + 5m + 1\)
d) \((5x - 2x^2 - 8) + (8x^2 - 7x - 9)\)
\(= (-2 + 8)x^2 + (5 - 7)x + (-8 - 9)\)
\(= 6x^2 - 2x - 17\)
In simple words: Group all like terms together and add their numerical coefficients.

Exam Tip: Arrange terms in standard polynomial form (highest power to lowest power) to avoid mistakes.

Q.6. Subtract:
a) -8xy from 7xy
b) \(x - y + 3z\) from \(2z - x - 3y\)
c) \(a^2 + b^2 - 2ab\) from \(a^2 + b^2 + 2ab\)
d) \(x^2 - y^2\) from \(2x^2 - 3y^2 + 6xy\)
Answer:
a) \(7xy - (-8xy) = 7xy + 8xy = 15xy\)
b) \((2z - x - 3y) - (x - y + 3z)\)
\(= 2z - x - 3y - x + y - 3z\)
\(= (-1 - 1)x + (-3 + 1)y + (2 - 3)z\)
\(= -2x - 2y - z\)
c) \((a^2 + b^2 + 2ab) - (a^2 + b^2 - 2ab)\)
\(= a^2 + b^2 + 2ab - a^2 - b^2 + 2ab\)
\(= (1 - 1)a^2 + (1 - 1)b^2 + (2 + 2)ab\)
\(= 4ab\)
d) \((2x^2 - 3y^2 + 6xy) - (x^2 - y^2)\)
\(= 2x^2 - 3y^2 + 6xy - x^2 + y^2\)
\(= (2 - 1)x^2 + (-3 + 1)y^2 + 6xy\)
\(= x^2 - 2y^2 + 6xy\)
In simple words: Always write the second expression first, change all signs of the expression being subtracted, and combine like terms.

Exam Tip: Use parentheses for the expression following the word 'from' to prevent sign errors.

Q.7. Subtract:
a) \(2a - 3b + 4c\) from the sum of \(a + 3b - 4c\), \(4a - b + 9c\) and \(-2b + 3c - a\)
Answer:
First, find the sum of the three expressions:
\(\text{Sum} = (a + 3b - 4c) + (4a - b + 9c) + (-2b + 3c - a)\)
\(= (1 + 4 - 1)a + (3 - 1 - 2)b + (-4 + 9 + 3)c\)
\(= 4a + 0b + 8c = 4a + 8c\)
Now, subtract \(2a - 3b + 4c\) from the sum:\
\((4a + 8c) - (2a - 3b + 4c)\)
\(= 4a + 8c - 2a + 3b - 4c\)
\(= (4 - 2)a + 3b + (8 - 4)c\)
\(= 2a + 3b + 4c\)
In simple words: Add the given group of expressions first, then subtract the specified expression from that total.

Exam Tip: Break multi-step problems into two clear parts: Step 1 (Addition) and Step 2 (Subtraction).

Q.8. What must be added to \(5x^3 - 2x^2 + 6x + 7\) to make the sum \(x^3 + 3x^2 - x + 1\)?
Answer:
Let the required expression be \(P\).
\((5x^3 - 2x^2 + 6x + 7) + P = x^3 + 3x^2 - x + 1\)
\(P = (x^3 + 3x^2 - x + 1) - (5x^3 - 2x^2 + 6x + 7)\)
\(P = x^3 + 3x^2 - x + 1 - 5x^3 + 2x^2 - 6x - 7\)
\(P = (1 - 5)x^3 + (3 + 2)x^2 + (-1 - 6)x + (1 - 7)\)
\(P = -4x^3 + 5x^2 - 7x - 6\)
In simple words: To find what needs to be added, subtract the given expression from the target sum.

Exam Tip: Equation setup: \(\text{Given} + \text{Required} = \text{Target} \implies \text{Required} = \text{Target} - \text{Given}\).

Q.9. What must be subtracted from \(a^3 - 4a^2 + 5a - 6\) to obtain \(a^2 - 2a + 1\)?
Answer:
Let the required expression be \(Q\).
\((a^3 - 4a^2 + 5a - 6) - Q = a^2 - 2a + 1\)
\(Q = (a^3 - 4a^2 + 5a - 6) - (a^2 - 2a + 1)\)
\(Q = a^3 - 4a^2 + 5a - 6 - a^2 + 2a - 1\)
\(Q = a^3 + (-4 - 1)a^2 + (5 + 2)a + (-6 - 1)\)
\(Q = a^3 - 5a^2 + 7a - 7\)
In simple words: Subtract the final result from the original expression to find what was taken away.

Exam Tip: \(\text{Original} - \text{Required} = \text{Result} \implies \text{Required} = \text{Original} - \text{Result}\).

Q.10. Simplify:
a) \(2p^3 - 3p^2 + 4p - 5 - 6p^3 + 2p^2 - 8p - 2 + 6p + 8\)
b) \(2x^2 - xy + 6x - 4y + 5xy - 4x + 6x^2 + 3y\)
c) \(5x^2 - 2x + 7 - 9 + 7x - 3x^2 + 4x^2 - x + 1\)
Answer:
a) Grouping like terms:
\(= (2 - 6)p^3 + (-3 + 2)p^2 + (4 - 8 + 6)p + (-5 - 2 + 8)\)
\(= -4p^3 - p^2 + 2p + 1\)
b) Grouping like terms:
\(= (2 + 6)x^2 + (-1 + 5)xy + (6 - 4)x + (-4 + 3)y\)
\(= 8x^2 + 4xy + 2x - y\)
c) Grouping like terms:
\(= (5 - 3 + 4)x^2 + (-2 + 7 - 1)x + (7 - 9 + 1)\)
\(= 6x^2 + 4x - 1\)
In simple words: Combine all terms with identical variable parts by adding their coefficients.

Exam Tip: Tick off terms as you group them to ensure none are missed.

Q.11. If \(p = -2, q = -1, r = 3\), find the value of :
a) \(p^2 + q^2 - r^2\)
b) \(2p^2 - q^2 + 3r^2\)
Answer:
a) Substituting \(p = -2, q = -1, r = 3\):
\(= (-2)^2 + (-1)^2 - (3)^2\)
\(= 4 + 1 - 9\)
\(= 5 - 9 = -4\)
b) Substituting \(p = -2, q = -1, r = 3\):
\(= 2(-2)^2 - (-1)^2 + 3(3)^2\)
\(= 2(4) - (1) + 3(9)\)
\(= 8 - 1 + 27\)
\(= 7 + 27 = 34\)
In simple words: Replace each variable with its given number and calculate using standard order of operations.

Exam Tip: Always put negative numbers inside parentheses when substituting into powers, e.g., \((-2)^2 = 4\).

Q.12. a) Write the numerical coefficient of -6abc
b) Write the constant term of \(3x^2 - 9\)
c) Write all the terms of algebraic expression \(4x^5 - 6y^4 + 7x^2y - 9\)
Answer:
a) Numerical coefficient of \(-6abc\) is -6.
b) Constant term of \(3x^2 - 9\) is -9.
c) Terms of the algebraic expression \(4x^5 - 6y^4 + 7x^2y - 9\) are: \(4x^5, -6y^4, 7x^2y, -9\).
In simple words: Identify the specific parts of algebraic expressions as asked (coefficients, constants, or individual terms).

Exam Tip: Include negative signs with terms and coefficients.

Q.13. Construct the \(\Delta ABC\) in which
a) \(BC = 6.2 \text{ cm}, AB = 5 \text{ cm}, AC = 4.3 \text{ cm}\)
b) \(AB = 5 \text{ cm}, AC = 4.3 \text{ cm}, \text{angle } A = 60^\circ\)
c) \(BC = 4.8 \text{ cm}, \text{angle } B = 60^\circ, \text{angle } C = 75^\circ\)
d) \(BC = 5.3 \text{ cm}, \text{angle } B = 45^\circ, \text{angle } A = 75^\circ\)
Answer:
Steps of Construction (General for SSS - part a):
1. Draw line segment \(BC = 6.2 \text{ cm}\).
2. With \(B\) as center and radius equal to 5 cm, draw an arc.
3. With \(C\) as center and radius equal to 4.3 cm, draw another arc to intersect the previous arc at point \(A\).
4. Join \(AB\) and \(AC\). \(\Delta ABC\) is required.
In simple words: Use a ruler and compass to draw triangles accurately based on given side lengths and angles.

Exam Tip: Keep compass needles sharp and ensure arcs clearly intersect at a single point.

Q.14. Construct a right triangle ABC in which base \(BC = 4.8 \text{ cm}\), Angle \(B = 90^\circ\) and \(AC = 6.2 \text{ cm}\)
Answer:
Steps of Construction:
1. Draw line segment \(BC = 4.8 \text{ cm}\).
2. At point \(B\), construct a right angle \(\angle XBC = 90^\circ\) using a compass or protractor.
3. With \(C\) as center and radius equal to 6.2 cm (hypotenuse), draw an arc to intersect the ray \(BX\) at point \(A\).
4. Join \(AC\). \(\Delta ABC\) is the required right-angled triangle.
In simple words: Draw the base, construct a 90 degree angle at one end, and swing an arc of the hypotenuse length to meet the perpendicular line.

Exam Tip: Verify the 90 degree angle with a protractor before marking the hypotenuse arc.

Q.15. Construct a triangle PQR in which \(PQ = 3.5 \text{ cm}\), \(QR = 4.2 \text{ cm}\) and angle \(Q = 120^\circ\)
Answer:
Steps of Construction:
1. Draw line segment \(QR = 4.2 \text{ cm}\).
2. At vertex \(Q\), construct an angle of \(120^\circ\) using a compass (an angle of \(60^\circ\) extended, or \(180^\circ - 60^\circ\)).
3. Along the ray making \(120^\circ\), cut off a length \(QP = 3.5 \text{ cm}\) using a compass.
4. Join \(PR\). \(\Delta PQR\) is the required triangle.
In simple words: Draw the base line, construct an obtuse angle at the vertex, measure off the second side length, and close the triangle.

Exam Tip: An obtuse angle of \(120^\circ\) opens wider than a right angle; ensure your construction reflects this visually.

Question. Add the like terms : (2/3) x² y, (-3/2) x² y, (2/7) x² y³
Answer : (5/21)x²y

Question. Add the unlike terms : 1x² , 2y, 3t, 4x²
Answer : (2y + 3z)

Question. Subtract 24ab – 10b + 18a from 26ab – 8b – 7a.
Answer : 2ab + 2b – 25a

Question. Evaluate x/5 - 5/5 when x = 2 y = –3
Answer : 1

Question. Write the coefficient of y² of algebraic expression.
2x²y – 15xy² + 7y²
Answer : (–15x, 7)

Question. A dd th e term s : x² and –3y²
Answer : x² – 3y²

Question. Subtract x + y from (3/2)x - (5/2)y
Answer : (x/2 - 7/2)

Question. If Z = 5 find the value of 2 – 3 (4 – Z)
Answer : 5

Question. Find the value of 3 (2a + b) if a = 2, b = –1.
Answer : 9

Question. What should be added to x² + y² to get 2x² + 3xy?
Answer : x² + 3xy + y²

Question. Add the algebraic expression :
3x + 11 and –2x + y
Answer : (x + y + 11)

Question. Find the value of the given algebraic expression when x = –2, 5x² + 4x – 2
Answer : 10

Question. What should be added into x² + 2x to obtain x2 + 9?
Answer : –2x + 9

Question. Add 2x² – 3x + 1 to the sum of 3x² – 2x and 3x + 7.
Answer : 5x² – 2x + 8

Question. Find the value of Algebric Expression when x = 2 x² – 3x + 5
Answer : 3

Question. Find the value of 7a – 4b when a = 2 b = – 2
Answer : 22

Question. Find the value of 5x + 3 if x = 9.
Answer : 48

Question. If a side of an equilateral triangle is 7x units then find the perimeter of the triangle.
Answer : 21x units

Question. If P = –6 find the value of 2P²/3 - 1
Answer : 23

Question. Find the value of the given algebraic expression when x = –2, 10x – 2
Answer : –22

Question. Find the value of the following expression for x = 2
15x – 5x²
Answer : 10

Question. If a + b = 5, find the value of (1/1)a (1/2) b
Answer : 5/2

Question. Find the perimeter of a square whose side is 4a cm.
Answer : 16a cm

Question. Find the perimeter of a regular pentagon whose side is 4x cm.
Answer : 20x cm

Question. Simplify : 2x - {5y - (x - 2y)} 
Answer : 3x – 7y

Question. Simplify :
(i) 12a²b + 3ba²
(ii) Also find the coefficient of the result.
Answer : (15a²b, 15)