NCERT Exemplar Solutions
Class 7 - Mathematics

Unit 10: ALGEBRAIC EXPRESSIONS

Sum or difference of two like terms is ____.

In the formula, area of circle = \(\pi r^2\), the numerical constant of the expression \(\pi r^2\) is ____.

\(3a^2b\) and \(-7ba^2\) are ____ terms.

\(-5a^2b\) and \(-5b^2a\) are ____ terms.

In the expression \(2\pi r\), the algebraic variable is ____.

Number of terms in a monomial is ____.

Like terms in the expression \(n(n + 1) + 6 (n - 1)\) are ____ and ____.

The expression 13 + 90 is a ____.

The speed of car is 55 km/hrs. The distance covered in y hours is ____.

\(x + y + z\) is an expression which is neither monomial nor ____.

If \((x^2y + y^2 + 3)\) is subtracted from \((3x^2y + 2y^2 + 5)\), then coefficient of y in the result is ____.

− a − b − c is same as − a − ( ____ ).

The unlike terms in perimeters of following figures are ____ and ____.

On adding a monomial ____ to −2x + 4y^2 + z, the resulting expression becomes a binomial.

3x + 23x^2 + 6y^2 + 2x + y^2 + ____ = 5x + 7y^2.

If Rohit has 5xy toffees and Shantanu has 20yx toffees, then Shantanu has ____ more toffees.

1 + \(\frac{x}{2}\) + x³ is a polynomial.

(3a − b + 3) − (a + b) is a binomial.

A trinomial can be a polynomial.

A polynomial with more than two terms is a trinomial.

Sum of x and y is x + y.

Sum of 2 and p is 2p.

A binomial has more than two terms.

A trinomial has exactly three terms.

In like terms, variables and their powers are the same.

The expression x + y + 5x is a trinomial.

4p is the numerical coefficient of q² in −4pq².

5a and 5b are unlike terms.

Sum of x² + x and y + y² is 2x² + 2y².

Subtracting a term from a given expression is the same as adding its additive inverse to the given expression.

The total number of planets of Sun can be denoted by the variable n.

In like terms, the numerical coefficients should also be the same.

If we add a monomial and binomial, then answer can never be a monomial.

If we subtract a monomial from a binomial, then answer is at least a binomial.

When we subtract a monomial from a trinomial, then answer can be a polynomial.

When we add a monomial and a trinomial, then answer can be a monomial.

Write the following statements in the form of algebraic expressions and write whether it is monomial, binomial or trinomial.

(a) x is multiplied by itself and then added to the product of x and y.

(b) Three times of p and two times of q are multiplied and then subtracted from r.

(c) Product of p, twice of q and thrice of r.

(d) Sum of the products of a and b, b and c and c and a.

(e) Perimeter of an equilateral triangle of side x.

(f) Perimeter of a rectangle with length p and breadth q.

(g) Area of a triangle with base m and height n.

(h) Area of a square with side x.

(i) Cube of s subtracted from cube of t.

(j) Quotient of x and 15 multiplied by x.

(k) The sum of square of x and cube of z.

(l) Two times q subtracted from cube of q.

Write the coefficient of x² in the following:

(i) x² − x + 4

(ii) x³ − 2x² + 3x + 1

(iii) 1 + 2x + 3x² + 4x³

(iv) y + y²x + y³x² + y⁴x³

Find the numerical coefficient of each of the terms:

(i) x³y²z, xy²z³, −3xy²z³, 5x³y²z, −7x²y²z²

(ii) 10xyz, −7xy²z, −9xyz, 2xy²z, 2x²y²z²

Simplify the following by combining the like terms and then write whether the expression is a monomial, a binomial or a trinomial.

(a) 3x²yz² − 3xy²z + x²y² + 7xy²z

(b) x⁴ + 3xy + 3x²y² − 3xy − 3xy³ + y⁴ − 3x²y²

(c) p³q²r + pq²r³ + 3p²qr² − 6p²qr²

(d) 2a + 2b + 2c − 2a − 2b − 2c − 2b + 2c + 2a

(e) 50x³ − 21x + 107 + 41x³ − x + 1 − 93 + 71x − 31x³

Add the following expressions:

(a) p² − 7pq − q² and −3p² − 2pq + 7q²

(b) x³ − x²y − xy² − y³ and x³ − 2x²y + 3xy² + 4y

(c) ab + bc + ca and −bc − ca − ab

(d) p² − q + r, q² − r + p and r² − p + q

(e) x³y² + x²y³ + 3y⁴ and x⁴ + 3x²y³ + 4y⁴

(f) p²qr + pq²r + pqr² and −3pq²r − 2pqr²

(g) uv − uw, uw − uv and uw − uw

(h) a² + 3ab − bc, b² + 3bc − ca and c² + 3ca − ab

(i) 5/8 p⁴ + 2p² + 5/8 ; 1/8 − 17p + 9/8 p² and p⁵ − p³ + 7

(j) t − t² − t³ − 14 ; 15t³ + 13 + 9t − 8t² ; 12t² − 19 − 24t and 4t − 9t² + 19t³

Subtract:

(a) −7p²qr from −3p²qr.

(b) −a² − ab from b² + ab.

(c) −4x²y − y³ from x³ + 3xy² − x²y.

(d) x⁴ + 3x²y³ + 5y⁴ from 2x⁴ − x³y³ + 7y⁴.

(e) ab − bc − ca from −ab + bc + ca.

(f) −2a² − 2b² from −a² − b² + 2ab.

(g) x³y² + 3x²y² − 7xy³ from x⁴ + y⁴ + 3x²y² − xy³.

(h) 2(ab + bc + ca) from −ab − bc − ca.

(i) 4.5x⁵ − 3.4x² + 5.7 from 5x⁴ − 3.2x² − 7.3x.

(j) 11 − 15y² from y³ − 15y² − y − 11.

(a) What should be added to x³ + 3x²y + 3xy² + y³ to get x³ + y³?

(b) What should be added to 3pq + 5p²q² + p³ to get p³ + 2p²q² + 4pq?

(a) What should be subtracted from 2x³ − 3x²y + 2xy² + 3y³ to get x³ − x²y − xy² − y³?

(b) What should be subtracted from −7mn + 2m² + 3n² to get m² + 2mn + n²?

How much is 21a³ − 17a² less than 89a³ − 64a² + 6a + 16?

How much is y⁴ − 12y² + y + 14 greater than 17y³ + 34y² − 51y + 68?

How much does 93p² − 55p + 4 exceed 13p³ − 5p² + 17p − 90?

To what expression must 99x³ − 33x² − 13x − 41 be added to make the sum zero?

Subtract \(9a^2 - 15a + 3\) from unity.

Find the values of the following polynomials at \(a = -2\) and \(b = 3\):

(a) \(a^2 + 2ab + b^2\)

(b) \(a^2 - 2ab + b^2\)

(c) \(a^3 + 3a^2b + 3ab^2 + b^3\)

(d) \(a^3 - 3a^2b + 3ab^2 - b^3\)

(e) \(\dfrac{a^2 + b^2}{3}\)

(f) \(\dfrac{a^2 - b^2}{3}\)

(g) \(\dfrac{a}{b} + \dfrac{b}{a}\)

(h) \(a^2 + b^2 - ab - b^2 - a^2\)

Find the values of following polynomials at \(m = 1\), \(n = -1\) and \(p = 2\):

(a) \(m + n + p\)

(b) \(m^2 + n^2 + p^2\)

(c) \(m^3 + n^3 + p^3\)

(d) \(mn + np + pm\)

(e) \(m^3 + n^3 + p^3 - 3mnp\)

(f) \(m^2n^2 + n^2p^2 + p^2m^2\)

If \(A = 3x^2 - 4x + 1\), \(B = 5x^2 + 3x - 8\) and \(C = 4x^2 - 7x + 3\), then find:

(i) \((A + B) - C\)

(ii) \(B + C - A\)

(iii) \(A + B + C\)

If \(P = -(x - 2)\), \(Q = -2(y +1)\) and \(R = -x + 2y\), find \(a\), when \(P + Q + R = ax\).

From the sum of \(x^2 - y^2 - 1\), \(y^2 - x^2 - 1\) and \(1 - x^2 - y^2\) subtract \((1 + y^2)\).

Subtract the sum of \(12ab - 10b^2 - 18a^2\) and \(9ab + 12b^2 + 14a^2\) from the sum of \(ab + 2b^2\) and \(3b^2 - a^2\).

Each symbol given below represents an algebraic expression:

△ = 2x² + 3y, ○ = 5x² + 3x, □ = 8y² - 3x² + 2x + 3y

The symbols are then represented in the expression: △ + ○ - □

Find the expression which is represented by the above symbols.

Observe the following nutritional chart carefully (per unit = 100g):

Rajma 60g

Cabbage 5g

Potato 22g

Carrot 11g

Tomato 4g

Apples 14g

Write an algebraic expression for the amount of carbohydrates in 'g' for

(a) y units of potatoes and 2 units of rajma

(b) 2x units tomatoes and y units apples

Arjun bought a rectangular plot with length x and breadth y and then sold a triangular part of it whose base is y and height is z. Find the area of the remaining part of the plot.

Amisha has a square plot of side m and another triangular plot with base and height each equal to m. What is the total area of both plots?

A taxi service charges ₹8 per km and levies a fixed charge of ₹50. Write an algebraic expression for the above situation, if the taxi is hired for x km.

Shiv works in a mall and gets paid ₹50 per hour. Last week he worked for 7 hours and this week he will work for x hours. Write an algebraic expression for the money paid to him for both the weeks.

Sonu and Raj have to collect different kinds of leaves for science project. They go to a park where Sonu collects 12 leaves and Raj collects x leaves. After some time Sonu loses 3 leaves and Raj collects 2x leaves. Write an algebraic expression to find the total number of leaves collected by both of them.

A school has a rectangular play ground with length x and breadth y and a square lawn with side x as shown in the figure given below. What is the total perimeter of both of them combined together?

The rate of planting the grass is ₹x per square metre. Find the cost of planting the grass on a triangular lawn whose base is y metres and height is z metres.

Find the perimeter of the figure given below:

(sides labeled \(5x - y\) and \(2(x + y)\) in the diagram)