NCERT Exemplar Solutions
Class 7 - Mathematics - Unit 10: ALGEBRAIC EXPRESSIONS

Short Answer (S.A.)

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)