Question : The cost of 2 kg apples and 1 kg grapes on a day was found to be Rs 160. After a month, the cost of 4 kg of apples and 2 kg of grapes is Rs 300. Represent the situation algebraically and graphically. Solution : Let us denote the cost of 1 kg …
Question : The coach of a cricket team buys 3 bats and 6 balls for Rs 3900. Later, she buys another bat and 2 more balls of the same kind for Rs 1300. Represent this situation algebraically and geometrically. Solution : Let us denote the set of one bat by Rs x and one ball …
Question : Aftab tells his daughter, “Seven years ago, I was seven times as old as you were then. Also, three years from now, I shall be three times as old as you will be.” (Isn’t this interesting?) Represent this situation algebraically and graphically. Solution : Let the present age of father be x years …
Solution : Let us divide \(x^4 – 6x^3 + 16x^2 – 25x + 10\) by \(x^2 – 2x + k\) \(\therefore\) Remainder = (2k – 9)x – (8 – k)k + 10 But the remainder is given as x + a, On comparing their coefficients, we have : 2k – 9 = 1 \(\implies\) k …
Solution : We have : \(2 \pm \sqrt{3}\) are two zeroes of the polynomial p(x) = \(x^4 – 6x^3 – 26x^2 + 138x – 35\) Let x = \(2 \pm \sqrt{3}\), So, x – 2 = \(\pm \sqrt{3}\) Squaring, we get \(x^2 – 4x + 4\) = 3, i.e. \(x^2 – 4x + 1\) = …
Solution : Since (a – b), a and (a + b) are the zeroes of the polynomials \(x^3 – 3x^2 + x + 1\), therefore (a – b) + a + (a + b) = \(-(-3)\over 1\) = 3 So, 3a = 3 \(\implies\) a = 1 (a – b)a + a(a + b) + …
Solution : Let the cubic polynomial be \(ax^3 + bx^2 + cx + d\), and its zeroes be \(\alpha\), \(\beta\) and \(\gamma\). Then, \(\alpha\) + \(\beta\) + \(\gamma\) = 2 = \(-(-2)\over 1\) = \(-b\over a\) \(\alpha\)\(\beta\) + \(\beta\)\(\gamma\) + \(\gamma\)\(\alpha\) = – 7 = \(-7\over 1\) = \(c\over a\) and \(\alpha\)\(\beta\)\(\gamma\) = -14 = \(-14\over …
Question : Verify that the numbers given alongside of the cubic polynomials below are their zeroes. Also verify the relationship between the zeroes and the coefficient in each case : (i) \(2x^3 + x^2 – 5x + 2\); \(1\over 2\), 1, -2 (ii) \(x^3 – 4x^2 + 5x – 2\); 2, 1, 1 Solution : …
Question : Give examples of polynomials p(x), g(x), q(x) and r(x), which satisfy the division algorithm and (i) deg p(x) = deg q(x) (ii) deg q(x) = deg r(x) (iii) deg q(x) = 0 Solution : (i) Let q(x) = \(3x^2 + 2x + 6\), Degree of q(x) …
Question : On dividing \(x^3 – 3x^2 + x + 2\) by a polynomial g(x), the quotient and the remainder were x – 2 and -2x + 4, respectively. Find g(x). p(x) = \(x^3 – 3x^2 + x + 2\) q(x) = x – 2 and r(x) = -2x + 4 Solution : By division …
