Page 1 1. Which of the following are quadratic equation in x? (i) 2 3 0 x x (ii) 2 5 2 3 0 2 x x (iii) 2 2 7 5 2 x x (iv) 2 1 1 2 0 3 5 x x (v) 2 3 4 0 x x x (vi) 6 3 x x (vii) 2 2 2 x x x (viii) 2 2 1 5 x x (ix) 3 3 2 8 x x (x) 2 3 3 2 6 1 2 x x x x (xi) 2 1 1 2 3 x x x x Sol: (i) 2 3 x x is a quadratic polynomial 2 3 0 x x is a quadratic equation. (ii) Clearly, 2 5 2 3 2 x x is a quadratic polynomial. 2 5 2 3 0 2 x x is a quadratic equation. (iii) Clearly, 2 2 7 5 2 x x is a quadratic polynomial. 2 2 7 5 2 0 x x is a quadratic equation. (iv) Clearly, 2 1 1 2 3 5 x x is a quadratic polynomial. 2 1 1 2 0 3 5 x x is a quadratic equation. (v) 2 3 4 x x x contains a term with , x i.e., 1 2 , x where 1 2 is not a ineger. Page 2 1. Which of the following are quadratic equation in x? (i) 2 3 0 x x (ii) 2 5 2 3 0 2 x x (iii) 2 2 7 5 2 x x (iv) 2 1 1 2 0 3 5 x x (v) 2 3 4 0 x x x (vi) 6 3 x x (vii) 2 2 2 x x x (viii) 2 2 1 5 x x (ix) 3 3 2 8 x x (x) 2 3 3 2 6 1 2 x x x x (xi) 2 1 1 2 3 x x x x Sol: (i) 2 3 x x is a quadratic polynomial 2 3 0 x x is a quadratic equation. (ii) Clearly, 2 5 2 3 2 x x is a quadratic polynomial. 2 5 2 3 0 2 x x is a quadratic equation. (iii) Clearly, 2 2 7 5 2 x x is a quadratic polynomial. 2 2 7 5 2 0 x x is a quadratic equation. (iv) Clearly, 2 1 1 2 3 5 x x is a quadratic polynomial. 2 1 1 2 0 3 5 x x is a quadratic equation. (v) 2 3 4 x x x contains a term with , x i.e., 1 2 , x where 1 2 is not a ineger. Therefore, it is not a quadratic polynomial. 2 3 4 0 x x x is not a quadratic equation. (vi) 6 3 x x 2 2 6 3 3 6 0 x x x x 3 3 6 x x is not quadratic polynomial; therefore, the given equation is quadratic. (vii) 2 2 2 x x x 2 3 3 2 2 2 0 x x x x 3 2 2 x x is not a quadratic polynomial. 3 2 2 0 x x is not a quadratic equation. (viii) 2 2 1 5 x x 4 2 1 5 x x 4 2 5 1 0 x x 4 2 5 1 x x is a polynomial with degree 4. 4 2 5 1 0 x x is not a quadratic equation. (ix) 3 3 2 8 x x 3 2 3 2 6 12 8 8 6 12 16 0 x x x x x x This is of the form 2 0 a x b x c Hence, the given equation is a quadratic equation. (x) 2 3 3 2 6 1 2 x x x x 2 2 2 2 6 4 9 6 6 3 2 6 13 6 6 18 12 31 6 0 x x x x x x x x x x This is of the form 2 0 a x b x c Hence, the given equation is not a quadratic equation. (xi) 2 1 1 2 3 x x x x Page 3 1. Which of the following are quadratic equation in x? (i) 2 3 0 x x (ii) 2 5 2 3 0 2 x x (iii) 2 2 7 5 2 x x (iv) 2 1 1 2 0 3 5 x x (v) 2 3 4 0 x x x (vi) 6 3 x x (vii) 2 2 2 x x x (viii) 2 2 1 5 x x (ix) 3 3 2 8 x x (x) 2 3 3 2 6 1 2 x x x x (xi) 2 1 1 2 3 x x x x Sol: (i) 2 3 x x is a quadratic polynomial 2 3 0 x x is a quadratic equation. (ii) Clearly, 2 5 2 3 2 x x is a quadratic polynomial. 2 5 2 3 0 2 x x is a quadratic equation. (iii) Clearly, 2 2 7 5 2 x x is a quadratic polynomial. 2 2 7 5 2 0 x x is a quadratic equation. (iv) Clearly, 2 1 1 2 3 5 x x is a quadratic polynomial. 2 1 1 2 0 3 5 x x is a quadratic equation. (v) 2 3 4 x x x contains a term with , x i.e., 1 2 , x where 1 2 is not a ineger. Therefore, it is not a quadratic polynomial. 2 3 4 0 x x x is not a quadratic equation. (vi) 6 3 x x 2 2 6 3 3 6 0 x x x x 3 3 6 x x is not quadratic polynomial; therefore, the given equation is quadratic. (vii) 2 2 2 x x x 2 3 3 2 2 2 0 x x x x 3 2 2 x x is not a quadratic polynomial. 3 2 2 0 x x is not a quadratic equation. (viii) 2 2 1 5 x x 4 2 1 5 x x 4 2 5 1 0 x x 4 2 5 1 x x is a polynomial with degree 4. 4 2 5 1 0 x x is not a quadratic equation. (ix) 3 3 2 8 x x 3 2 3 2 6 12 8 8 6 12 16 0 x x x x x x This is of the form 2 0 a x b x c Hence, the given equation is a quadratic equation. (x) 2 3 3 2 6 1 2 x x x x 2 2 2 2 6 4 9 6 6 3 2 6 13 6 6 18 12 31 6 0 x x x x x x x x x x This is of the form 2 0 a x b x c Hence, the given equation is not a quadratic equation. (xi) 2 1 1 2 3 x x x x 2 2 2 1 1 2 3 x x x x 2 2 2 2 4 2 3 2 4 3 2 1 2 1 3 2 1 2 2 3 2 2 1 0 x x x x x x x x x x x x x This is not of the form 2 0 a x b x c Hence, the given equation is not a quadratic equation. 2. Which of the following are the roots of 2 3 2 1 0? x x (i) 1 (ii) 1 3 (iii) 1 2 Sol: The given equation is 2 3 2 1 0 . x x (i) 1 x L.H.S. 2 2 1 x x 2 3 1 2 1 1 3 2 1 0 . . . R H S Thus, 1 is a root of 2 3 2 1 0 . x x (ii) On subtracting 1 3 x in the given equation, we get: L.H.S. 2 3 2 1 x x 2 1 1 3 2 1 3 3 1 2 3 1 9 3 1 2 3 3 0 3 0 . . . R H S Thus, 1 3 is a root of 2 3 2 1 0 x x Page 4 1. Which of the following are quadratic equation in x? (i) 2 3 0 x x (ii) 2 5 2 3 0 2 x x (iii) 2 2 7 5 2 x x (iv) 2 1 1 2 0 3 5 x x (v) 2 3 4 0 x x x (vi) 6 3 x x (vii) 2 2 2 x x x (viii) 2 2 1 5 x x (ix) 3 3 2 8 x x (x) 2 3 3 2 6 1 2 x x x x (xi) 2 1 1 2 3 x x x x Sol: (i) 2 3 x x is a quadratic polynomial 2 3 0 x x is a quadratic equation. (ii) Clearly, 2 5 2 3 2 x x is a quadratic polynomial. 2 5 2 3 0 2 x x is a quadratic equation. (iii) Clearly, 2 2 7 5 2 x x is a quadratic polynomial. 2 2 7 5 2 0 x x is a quadratic equation. (iv) Clearly, 2 1 1 2 3 5 x x is a quadratic polynomial. 2 1 1 2 0 3 5 x x is a quadratic equation. (v) 2 3 4 x x x contains a term with , x i.e., 1 2 , x where 1 2 is not a ineger. Therefore, it is not a quadratic polynomial. 2 3 4 0 x x x is not a quadratic equation. (vi) 6 3 x x 2 2 6 3 3 6 0 x x x x 3 3 6 x x is not quadratic polynomial; therefore, the given equation is quadratic. (vii) 2 2 2 x x x 2 3 3 2 2 2 0 x x x x 3 2 2 x x is not a quadratic polynomial. 3 2 2 0 x x is not a quadratic equation. (viii) 2 2 1 5 x x 4 2 1 5 x x 4 2 5 1 0 x x 4 2 5 1 x x is a polynomial with degree 4. 4 2 5 1 0 x x is not a quadratic equation. (ix) 3 3 2 8 x x 3 2 3 2 6 12 8 8 6 12 16 0 x x x x x x This is of the form 2 0 a x b x c Hence, the given equation is a quadratic equation. (x) 2 3 3 2 6 1 2 x x x x 2 2 2 2 6 4 9 6 6 3 2 6 13 6 6 18 12 31 6 0 x x x x x x x x x x This is of the form 2 0 a x b x c Hence, the given equation is not a quadratic equation. (xi) 2 1 1 2 3 x x x x 2 2 2 1 1 2 3 x x x x 2 2 2 2 4 2 3 2 4 3 2 1 2 1 3 2 1 2 2 3 2 2 1 0 x x x x x x x x x x x x x This is not of the form 2 0 a x b x c Hence, the given equation is not a quadratic equation. 2. Which of the following are the roots of 2 3 2 1 0? x x (i) 1 (ii) 1 3 (iii) 1 2 Sol: The given equation is 2 3 2 1 0 . x x (i) 1 x L.H.S. 2 2 1 x x 2 3 1 2 1 1 3 2 1 0 . . . R H S Thus, 1 is a root of 2 3 2 1 0 . x x (ii) On subtracting 1 3 x in the given equation, we get: L.H.S. 2 3 2 1 x x 2 1 1 3 2 1 3 3 1 2 3 1 9 3 1 2 3 3 0 3 0 . . . R H S Thus, 1 3 is a root of 2 3 2 1 0 x x (iii) On subtracting 1 2 x in the given equation, we get L.H.S. 2 3 2 1 x x 2 1 1 3 2 1 2 2 1 3 1 1 4 3 2 4 3 8 4 5 0 4 Thus, . . . . . L H S R H S Hence, 1 2 is a solution of 2 3 2 1 0 . x x 3. Find the value of k for which 1 x is a root of the equation 2 3 0. x k x Sol: It is given that 1 x is a root of 2 3 0 . x k x Therefore, 1 x must satisfy the equation. 2 1 1 3 0 4 0 4 k k k Hence, the required value of k is 4. 4. Find the value of a and b for which 3 4 x and 2 x are the roots of the equation 2 6 0 a x b x Sol: It is given that 3 4 is a root of 2 6 0; a x b x therefore, we have: 2 3 3 6 0 4 4 a b Page 5 1. Which of the following are quadratic equation in x? (i) 2 3 0 x x (ii) 2 5 2 3 0 2 x x (iii) 2 2 7 5 2 x x (iv) 2 1 1 2 0 3 5 x x (v) 2 3 4 0 x x x (vi) 6 3 x x (vii) 2 2 2 x x x (viii) 2 2 1 5 x x (ix) 3 3 2 8 x x (x) 2 3 3 2 6 1 2 x x x x (xi) 2 1 1 2 3 x x x x Sol: (i) 2 3 x x is a quadratic polynomial 2 3 0 x x is a quadratic equation. (ii) Clearly, 2 5 2 3 2 x x is a quadratic polynomial. 2 5 2 3 0 2 x x is a quadratic equation. (iii) Clearly, 2 2 7 5 2 x x is a quadratic polynomial. 2 2 7 5 2 0 x x is a quadratic equation. (iv) Clearly, 2 1 1 2 3 5 x x is a quadratic polynomial. 2 1 1 2 0 3 5 x x is a quadratic equation. (v) 2 3 4 x x x contains a term with , x i.e., 1 2 , x where 1 2 is not a ineger. Therefore, it is not a quadratic polynomial. 2 3 4 0 x x x is not a quadratic equation. (vi) 6 3 x x 2 2 6 3 3 6 0 x x x x 3 3 6 x x is not quadratic polynomial; therefore, the given equation is quadratic. (vii) 2 2 2 x x x 2 3 3 2 2 2 0 x x x x 3 2 2 x x is not a quadratic polynomial. 3 2 2 0 x x is not a quadratic equation. (viii) 2 2 1 5 x x 4 2 1 5 x x 4 2 5 1 0 x x 4 2 5 1 x x is a polynomial with degree 4. 4 2 5 1 0 x x is not a quadratic equation. (ix) 3 3 2 8 x x 3 2 3 2 6 12 8 8 6 12 16 0 x x x x x x This is of the form 2 0 a x b x c Hence, the given equation is a quadratic equation. (x) 2 3 3 2 6 1 2 x x x x 2 2 2 2 6 4 9 6 6 3 2 6 13 6 6 18 12 31 6 0 x x x x x x x x x x This is of the form 2 0 a x b x c Hence, the given equation is not a quadratic equation. (xi) 2 1 1 2 3 x x x x 2 2 2 1 1 2 3 x x x x 2 2 2 2 4 2 3 2 4 3 2 1 2 1 3 2 1 2 2 3 2 2 1 0 x x x x x x x x x x x x x This is not of the form 2 0 a x b x c Hence, the given equation is not a quadratic equation. 2. Which of the following are the roots of 2 3 2 1 0? x x (i) 1 (ii) 1 3 (iii) 1 2 Sol: The given equation is 2 3 2 1 0 . x x (i) 1 x L.H.S. 2 2 1 x x 2 3 1 2 1 1 3 2 1 0 . . . R H S Thus, 1 is a root of 2 3 2 1 0 . x x (ii) On subtracting 1 3 x in the given equation, we get: L.H.S. 2 3 2 1 x x 2 1 1 3 2 1 3 3 1 2 3 1 9 3 1 2 3 3 0 3 0 . . . R H S Thus, 1 3 is a root of 2 3 2 1 0 x x (iii) On subtracting 1 2 x in the given equation, we get L.H.S. 2 3 2 1 x x 2 1 1 3 2 1 2 2 1 3 1 1 4 3 2 4 3 8 4 5 0 4 Thus, . . . . . L H S R H S Hence, 1 2 is a solution of 2 3 2 1 0 . x x 3. Find the value of k for which 1 x is a root of the equation 2 3 0. x k x Sol: It is given that 1 x is a root of 2 3 0 . x k x Therefore, 1 x must satisfy the equation. 2 1 1 3 0 4 0 4 k k k Hence, the required value of k is 4. 4. Find the value of a and b for which 3 4 x and 2 x are the roots of the equation 2 6 0 a x b x Sol: It is given that 3 4 is a root of 2 6 0; a x b x therefore, we have: 2 3 3 6 0 4 4 a b 9 3 6 16 4 9 12 6 16 9 12 96 0 a b a b a b 3 4 32 .......... a b i Again, 2 is a root of 2 6 0; a x b x therefore, we have: 2 2 2 6 0 4 2 6 2 3 ......... a b a b a b i i On multiplying (ii) by 4 and adding the result with (i), we get: 3 4 8 4 32 12 11 44 a b a b a 4 a Putting the value of a in (ii), we get: 2 4 3 8 3 5 b b b Hence, the required values of a and b are 4 and 5, respectively. 5. 2 3 3 1 0 x x Sol: 2 3 3 1 0 x x 2 3 0 3 1 0 2 3 3 1 3 1 2 3 x or x x or x x or x Hence the roots of the given equation are 3 1 . 2 3 a n d 6. 2 4 5 0 x x Sol: 2 4 5 0 x x 4 5 0 x x 0 4 5 0 x o r xRead More

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