RD Sharma solution class 7 chapter 7 Algebraic Expression Exercise 7.1

Exercise 7.1

Page-7.7

Question 1:

Identify the monomials, binomials, trinomials and quadrinomials from the following expressions:
(i) a²
(ii) a² − b²
(iii) x³ + y³ + z³
(iv) x³ + y³ + z³ + 3xyz
(v) 7 + 5
(vi) abc + 1
(vii) 3x − 2 + 5
(viii) 2x − 3x + 4
(ix) xy + yz + zx
(x) ax³ + bc² + cx + d

Answer 1:

The monomials, binomials, trinomials and quadrinomials are as follows.
(i) a² is a monomial expression as it contains one term only.
(ii) a² - b² is a binomial expression as it contains two terms.
(iii) x³ + y³ + z³ is a trinomial expression as it contains three terms.
(iv) x³ + y³ + z³ + xyz is a quadrinomial expression as it contains four terms.
(v) 7 + 5 = 12 is a monomial expression as it contains one term only.
(vi) abc +1 is a binomial expression as it contains two terms.
(vii) 3x - 2 + 5 = 3x + 3 is a binomial expression as it contains two terms.
(viii) 2x - 3y + 4 is a trinomial expression as it contains three terms.
(ix) xy + yz + zx is a trinomial expression as it contains three terms.
(x) ax³ +bx² +cx + d is a quadrinomial expression as it contains four terms.

Page-7.8

Question 2:

Write all the terms of each of the following algebraic expressions:
(i) 3x
(ii) 2x − 3
(iii) 2x² − 7
(iv) 2x² + y² − 3xy + 4

Answer 2:

The terms of each of the given algebraic expressions are as follows.
(i) 3x is the only term of the given algebraic expression.
(ii) 2x and -3 are the terms of the given algebraic expression.
(iii) 2x² and -7 are the terms of the given algebraic expression.
(iv) 2x², y², -3xy  and 4 are the terms of the given algebraic expression.

Question 3:

Identify the like terms and also mention the numerical coefficients of those terms:
(i) 4xy, −5x²y, −3yx, 2xy²
(ii) $7a^{2}bc, -3c{a}^{2}b, -\frac{5}{2}ab{c}^2, \frac{3}{2}ab{c}^2, -\frac{4}{3}cb{a}^2$

Answer 3:

       Like terms                                          Numerical coefficients
(i)   4xy, -3yx                                                4, -3
(ii) {7a²bc ,-3ca²b, $-\frac{4}{3}cb{a}^2$}                      { 7, -3,$-\frac{4}{3}$}
      {$-\frac{5}{2}ab{c}^2,\frac{3}{2}ab{c}^2$ }                                 {$-\frac{5}{2},\frac{3}{2}$}

Question 4:

Identify the like terms in the following algebraic expressions:
(i) $a^2+b^2-2a^2+c^2+4a$
(ii) $3x+4xy-2yz+\frac{5}{2}zy$
(iii) $abc+a{b}^{2}c+2ac{b}^2+3c^{2}ab+b^{2}ac-2a^{2}bc+3ca{b}^2$

Answer 4:

The like terms in the given algebraic expressions are as follows.
(i) The like terms in the given algebraic expression are a²  and -2a².

(ii) The like terms in the given algebraic expression are $-2\mathrm{yz} \mathrm{and} \frac{5}{2}\mathrm{zy}$.
(iii) The like terms in the given algebraic expression are ab²c, 2acb², b²ac and 3cab².

Question 5:

Write the coefficient of x in the following:
(i) −12x
(ii) −7xy
(iii) xyz
(iv) −7ax

Answer 5:

 The coefficients of x are as follows.
(i) The numerical coefficient of x is -12.
(ii) The numerical coefficient of x is -7y.
(iii) The numerical coefficient of x is yz.
(iv) The numerical coefficient of x is -7a.

Question 6:

Write the coefficient of x² in the following:
(i) −3x²
(ii) 5x²yx
(iii) $\frac{5}{7}{x}^{2}z$
(iv) $-\frac{3}{2}a{x}^2+yx$

Answer 6:

The coefficients of x² are as follows.
 (i) The numerical coefficient of x² is -3.
(ii) The numerical coefficient of x² is 5yz.

(iii) The numerical coefficient of x² is $\frac{5}{7}z$.
(iv) The numerical coefficient of x² is $-\frac{3}{2}a$.

Question 7:

Write the coefficient of:
(i) y in −3y
(ii) a in 2ab
(iii) z in −7xyz
(iv) p in −3pqr
(v) y² in 9xy²z
(vi) x³ in x³ + 1
(vii) x² in −x²

Answer 7:

 The coefficients are as follows.

 (i) The coefficient of  y is -3.
 (ii) The coefficient of  a is 2b.
 (iii) The coefficient of  z is -7xy.
 (iv) The coefficient of  p is -3qr.
 (v) The coefficient of  y² is 9xz.
 (vi) The coefficient of  x³ is 1.
 (vii) The coefficient of  -x² is - 1.

Question 8:

Write the numerical coefficient of each of the following:
(i) xy
(ii) −6yz
(iii) 7abc
(iv) −2x³y²z

Answer 8:

The numerical coefficient of each of the given terms is as follows.
(i) The numerical coefficient in the term xy is 1.
(ii) The numerical coefficient in the term -6yz is - 6.
(iii) The numerical coefficient in the term 7abc is 7.
(iv) The numerical coefficient in the term -2x³y²z is - 2.

Question 9:

Write the numerical coefficient of each term in the following algebraic expressions:
(i) $4x^{2}y-\frac{3}{2}xy+\frac{5}{2}x{y}^2$
(ii) $-\frac{5}{3}{x}^{2}y+\frac{7}{4}xyz+3$

Answer 9:

The numerical coefficient of each term in the given algebraic expressions is as follows.
             Term                         Coefficient

(i)       4x²y                                   4
                          
         $-\frac{3}{2}xy$                              $-\frac{3}{2}$
           $\frac{5}{2}x{y}^2$                                 $\frac{5}{2}$

(ii)    $-\frac{5}{3}{x}^{2}y$                            $-\frac{5}{3}$
          $\frac{7}{4}xyz$                                 $\frac{7}{4}$
           3                                        3
        

Question 10:

Write the constant term of each of the following algebraic expressions:
(i) x²y − xy² + 7xy − 3
(ii) a³ − 3a² + 7a + 5

Answer 10:

The constant term of each of the given algebraic expressions is as follows.
(i) The constant term in the given algebraic expression is -3.
(ii) The constant term in the given algebraic expression is 5.

Question 11:

Evaluate each of the following expressions for x = − 2, y = −1, z = 3:
(i) $\frac{x}{y}+\frac{y}{z}+\frac{z}{x}$
(ii) $x^2+y^2+z^2-xy-yz-zx$

Answer 11:

We have x = −2, y = −1 and z = 3.
Thus,
(i)
 $$\begin{gathered} \frac{x}{y}+\frac{y}{z}+\frac{z}{x} \\ =\left(\frac{-2}{-1}\right)+\left(\frac{-1}{3}\right)+\left(\frac{3}{-2}\right) \\ =2-\frac{1}{3}-\frac{3}{2} \\ =\frac{12-2-9}{6}=\frac{12-11}{6}=\frac{1}{6} \end{gathered}$$

(ii) x² + y² + z²- xy - yz - zx
     = (-2)² + (-1)² + (3)² - (-2)(-1) - (-1)(3) - (3)(-2)
     = 4 + 1 + 9 - 2 + 3 + 6
     = (4 + 1 + 9 + 3 + 6) - 2
     = 23 - 2 = 21

Question 12:

Evaluate each of the following algebraic expressions for x = 1, y = −1, z = 2, a = −2, b = 1, c = −2:
(i) ax + by + cz
(ii) ax² + by² − cz²
(iii) axy + byz + cxy

Answer 12:

We have x = 1, y = −1, z = 2, a = −2, b = 1 and c = −2.
Thus,
(i) ax + by + cz
    = (-2)(1) + (1)(-1) + (-2)(2)
    = - 2 + (- 1) + (- 4)
    = - 2 -1 - 4 = -7

(ii) ax² + by² - cz²
    = (-2)(1)² + (1)(-1)² - (-2)(2)²
    = -2 x 1 + 1 - (-2 x 4)
    = -2 + 1 - (-8)
    = -2 + 1 + 8
    = -2 + 9
    = 7

(iii) axy + byz + cxy
     = (-2)(1)(-1) + (1)(-1)(2) + (-2)(1)(-1)
     = 2 + (-2) + 2
     = 2 - 2 + 2
     = 4 - 2
     = 2