Chapter 1: Polynomials Old ICC building, near Firoz Hospital, Sir Syed N agar, Medical Road , Aligarh : 9997607607, 9997394458 Chapter - 1 : Polynomials 1. Which of the following expressions is/are polynomials in one variable ? I. 2 4 3 7 x x − + II. 2 2 y + III. 3 2 t t + IV. 2 y y + (a) Only I (b) I & II (c) I II & IV (d) All 2. The coefficients of 2 x in 2 1 x − is (a) 2 (b) 0 (c) – 1 (d) Not defined 3. The degree of polynomial ‘3’ is (a) 1 (b) 0 (c) not defined (d) constant 4. Classify the following as cubic polynomials: I 2 x x + II 3 x x − III 3 t IV 3 7 x (a) Only I (b) Only III (c) II & IV (d) I, III & IV 5. The zeroes of the polynomial 2 x + is (a) 2 (b) – 2 (c) 2 (d) 3 6. Find the value of polynomial 2 5 4 3 x x − + at 1 x = − (a) 6 (b) – 6 (c) 2 (d) – 2 7. The value of ( ) 2 p i f ( ) ( )( ) 1 1 p x x x = − + is (a) 0 (b) 1 (c) 2 (d) 3 8. The zero of the polynomial ( ) 0 p x cx d ,c ,c, d = + are real numbers is (a ) d c − (b) c d − (c) c d (d) d c 9. The graphs of ( ) y p x = are given in Fig. below, for some polynomials ( ) p x . Then the number of zeroes of ( ) p x is/are (a) 0 (b) 1 (c) 2 (d) 3 10. The graphs of ( ) y p x = are given in Fig. below, for some polynomials ( ) p x . Then the number of zeroes of ( ) p x is/are (a) 0 (b) 1 (c) 2 (d) 3 Level – I ( QUESTIONS EXACTLY COPIED FROM NCERT) Chapter 1: Polynomials Old ICC building, near Firoz Hospital, Sir Syed N agar, Medical Road , Aligarh : 9997607607, 9997394458 11. The graphs of ( ) y p x = are given in Fig. below, for some polynomials ( ) p x . Then the number of zeroes of ( ) p x is/are (a) 1 (b) 2 (c) 3 (d) 4 12. The graphs of ( ) y p x = are given in Fig. below, for some polynomials ( ) p x . Then the number of zeroes of ( ) p x is/are (a) 0 (b) 1 (c) 5 (d) 3 13. The quadratic polynomial with the given numbers 0 as sum and 5 as the product of its zeroes is (a) 2 5 5 x x − + (b) 2 5 5 x x + + (c) 2 5 x x − (d) 2 5 x + 14. The graphs of ( ) y p x = are given in Fig. below, for some polynomials ( ) p x . Then the number of zeroes of ( ) p x is/are (a ) 0 (b) 1 (c) 2 (d) 3 15. One of the z ero of the quadratic polynomial 3 x 2 – x – 4 is (a) 5 3 (b) – 1 (c) – 2 (d) 1 16. The quadratic polynomial, the sum and product of whose zeroes are – 3 and 2, respectively is (a) ( ) 2 3 2 k x x + + (b) ( ) 2 3 2 k x x − + (c) 2 6 x x − + (d) 2 6 x x + − 17. Other two zeroes of 2 x 4 – 3 x 3 – 3 x 2 + 6 x – 2, if you know that two of its zeroes are 2 and – 2 ( a) 1 1 2 & − − (b) 1 1 2 & − (c) 1 1 2 & (d) 1 1 2 & − 18. Obtain all other zeroes of 3 x 4 + 6 x 3 – 2 x 2 – 10 x – 5, if two of its zeroes are 5 3 and – 5 3 ⋅ (a) 1 2 & − (b) 2 1 & − − (c) 1 1 & − − (d) 1 1 & − 19. On dividing x 3 – 3 x 2 + x + 2 by a polynomial g ( x ), the quotient and remainder were x – 2 and – 2 x + 4, respectively. Find g ( x ). (a) 2 1 x x − + (b) 2 1 x x + + (c) 2 1 x x + − (d) 2 1 x x − − Chapter 1: Polynomials Old ICC building, near Firoz Hospital, Sir Syed N agar, Medical Road , Aligarh : 9997607607, 9997394458 20. Find a cubic polynomial with the sum, sum of the product of its zeroes taken two at a time, and the product of its zeroes as 2, – 7, – 14 respectively. (a) 2 2 2 7 14 x x x + − + (b) 3 2 2 7 14 x x x + + + (c) 3 2 2 7 14 x x x + − − (d) 3 2 2 7 14 x x x − − + 21. If the zeroes of the polynomial x 3 – 3 x 2 + x + 1 are a – b , a , a + b , find a and b ? (a) 1 2 , (b) 1 2 , (c) 1 2 , (d) 2 3 , 22. If two zeroes of the polynomial x 4 – 6 x 3 – 26 x 2 + 138 x – 35 are 2 3 , find other zeroes. (a) 5 5 , − (b) 5 7 , − (c) 5 7 , − − (d) 5 7 , 23. If the polynomial x 4 – 6 x 3 + 16 x 2 – 25 x + 10 is divided by another polynomial x 2 – 2 x + k , the remainder comes out to be x + a , the value of k and a , respectively are (a) 5 5 , (b) 5 5 , − (c) 5 5 , − (d) 5 5 , − − 1. 5 x − is a factor of the cubic polynomial 3 2 3 5 13 3 5 x x x , − + − , then the zeroes of the Polynomial are. (a) 5 3 5 , (b) 5 3 5 , − (c) 3 5 , (d) 5 2 5 , 2. If one zero of the quadratic polynomial 2 3 x x k + + is 2, then the value of k is : (a) 10 (b) – 10 (c) 5 (d ) – 5 3. If and are the zeroes of the quadratic polynomial 2 – 2 – 8, x x then + + is: (a) 6 (b) – 6 (c) – 10 (d) 10 4. Given that two of the zeroes of the cubic polynomial 3 2 ax bx cx d + + + are 0, the third zero is. (a) – b / a (b) a / c (c) a / b (d) – d / a 5. The quadratic polynomial formed by the reciprocal of zeroes of the quadrat ic polynomial 2 – 3 2 x x + is: (a) 2 –3 2 x x + + (b) 2 2 – 3 1 x x + (c) 2 2 – 3 x x + (d) 2 2 3 –1 x x + 6. If the zeroes of the quadratic polynomial 2 0 ax bx c,a + + are equal, then (a) c and a have opposite signs (b) c and b have opposite signs (c) c and a have the same sign (d) c and b have the same sign 7. The two roots of the equations ( ) ( ) ( ) 2 – – – 0 a b c x b c a x c a b + + = are 1 and: (a) ( ) ( ) a c b b a c − − (b) ( ) ( ) c b a a c b − − (c) ( ) ( ) a c b c b a − − (d) ( ) ( ) c b a b a c − − 8. One of the quadratic polynomial, whose zeroes are – 3 and 4, is (a) 2 12 x x − + (b) 2 12 x x + + (c) 2 12 x x + − (d) 2 2 2 24 x x + − LEVEL – II (QUESTIONS BASED ON NCERT) Chapter 1: Polynomials Old ICC building, near Firoz Hospital, Sir Syed N agar, Medical Road , Aligarh : 9997607607, 9997394458 9. If the zeroes of a quadratic polynomial 2 ax bx c + + are both negative, then a , b and c all have the same sign. (a) c and a h ave opposite signs (b) c and b have opposite signs (c) c and a have the same sign (d) a, c and b have the same sig n 10. If the zeroes of th e quadratic polynomial ( ) 2 1 x a x b − + + are 2 and – 3 then (a) – 7, –1 a b = = (b) 5, –1 a b = = (c) 2, – 6 a b = = (d) 2, – 6 = − = a b 11. If one of the zeroes of a quadratic polynomial of the form 2 x ax b + + is the negative of the ot her, then it (a) has no linear term and the constant term is negative (b) has no linear term and the constant term is positive. (c) can have a linear term but the constant term is negative (d) can have a linear term but the constant term is positive. 12. If 2 is one of the zeroes of the polynomial 3 2 – 4 5 – 2, x x x + then the other two zeros are: (a) 1, 1 (b) 1, – 1 (c) – 1, 2 (d) – 1, – 1 13. Which one of the following is an irrational expre ssion? (a ) 2 3 4 1 x x − + (b) 1 2 2 2 2 − − x x (c) 4 1 − + x x (d) 2 2 5 x x − − 14. The ratio of the sum and the product of the roots of 2 7 –12 18 0 x x + = is: (a) 7:12 (b) 3:2 (c) 2:3 (d) 7:18 15. If one root of 2 3 11 0 x x k + + = be reciprocal of the other, then the value of k is: (a) l (b) 2 (c) 3 (d) 4 16. The zeroes of the quadratic polynomial 2 99 127 x x + + are (a) Both positive (b) both negative (c) one positive and one negative ( d) both equal 17. If one of the zeroes of the quadratic polynomial ( ) 2 1 1 k x kx − + + is – 3, then the value of k is (a) 4/3 (b) – 4/3 (c) 2/3 (d) – 2/3 18. Given that one of the zeroes of the cubic polynomial 3 2 ax bx cx d + + + is zero, the product of the other two zeroes is (a) – c/a (b) c / a (c) 0 (d) – b / a 19. If one of the zeroes of the cubic polynomial 3 2 x ax bx c + + + is – 1, then the product of the other two zeroes is. (a) – 1 b a + (b) – – 1 b a (c) – 1 a b + (d) – –1 a b 20. If , , are th e zeros of the cubic polynomial 3 2 – –1, x mx nx + then: (a) + + = – m, + + = n, = 1 (b) + + = – m, + + = n, = – 1 (c) + + = m, + + = n, = 1 (d) + + = m, + + = n, = – 1 Chapter 1: Polynomials Old ICC building, near Firoz Hospital, Sir Syed N agar, Medical Road , Aligarh : 9997607607, 9997394458 21. If zeroes of the polynomial 3 – 3 1 x x x + + are – , , , a b a a b + find a and b : (a) 2, 3 a b = = (b) 1, a b = = 2 (c) 3, 0 a b = = (d) 2 a = , 3 b = 22. The zeroes of the quadratic polynomial 2 , 0 x kx k k + + (a) Cannot be both positive (b) cannot be both negative (c) Are always unequal (d) are always equal 23. Quadratic polynomial, the sum and product of whose zeroes are 2 and – 3/2, respectively is. (a) 2 3 2 2 x x + − (b) 2 3 2 x x − + (c) 2 3 2 2 x x + − (d) 2 3 2 2 x x − − 24. If the remainder on division of 3 2 2 3 x x kx + + + by – 3 x is 21, the quotient and the value of k . And the zeroes of the cubic polynomial 3 2 2 – 18 x x kx + + are. (a) 2 –9, – 5 6, k x x = + Zeroes 3, –2, –3 (b) 2 –9, 5 – 6, k x x = + Zeroes 3, –2, –3 (c) 2 –9, 5 6 k x x = + + Zeroes 3, –2, –3 (d ) 2 –9, – 5 – 6, k x x = Zeroes 3 3, –2, –3 25. The zeroes of the cubic polynomial 3 2 – 6 3 10 x x x + + are of the form , , 2 a a b a b + + for some rea l n umbers a and b , the values of a and b are (a) 1,3 a = 3, –3 b = (b) 1,5 a = 3, –3 b = (c) –1,3 a = 3, –3 b = (d) –1,5 a = 3, –3 b = 26. 2 is a zero of the cubic polynomial 3 2 6 2 10 4 2 x x x , + − − its other two zeroes are ( a ) 3 2 2 , − − (b) 1 2 2 , − (c) 1 2 3 , − − (d) 3 2 2 , − 27. Are the following statements ‘True’ or ‘False’? (i) If the zeroes of a quadratic polynomial ax 2 + bx + c are both positive, then a , b and c all have the same sign. (ii) If the graph of a polynomial intersects the x - axis at only one point, it cannot be a quadratic Polynomial. (iii) If the graph of a polynomial intersects the x - axis at exactly two points, It need not be a quadratic Polynomial. (iv) If two of the zeroes of a cubic poly nomial are zero, then it doe s not have constant Term (v) The only value of k for which the quadratic polynomial kx 2 + x + k has equal zero is 1/2 Chapter 1: Polynomials Old ICC building, near Firoz Hospital, Sir Syed N agar, Medical Road , Aligarh : 9997607607, 9997394458 LEVEL – I 1. b 2. b 3. b 4. c 5. b 6. b 7. d 8. a 9. b 10. a 11. d 12. d 13. d 14. b 15. b 16. a 17. c 18. c 19. a 20. d 21. a 22. b 23. c LEVEL – II 1. d 2. b 3. b 4. a 5. b 6. c 7. d 8. d 9. d 10. d 11. a 12. a 13. c 14. c 15. c 16. b 17. b 18. b 19. a 20. c 21. b 22. c 23. d 24. c 25. d 26. c 27. i) F ii) F iii) F iv) T v) F ANSWERS