Chapter 1: Polynomials Chapter1: Polynomials Level – I (QUESTIONS EXACTLY COPIED FROM NCERT) 1. Which of the following expressions is/are polynomials in one variable? 2 II. y + 2 2 I. 4 x 2 − 3x + 7 III. 3 t + t 2 IV. y + y (a) Only I (b) I & II (c) III & IV (d) All 2. The coefficients of x in 2 x − 1 is 2 (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. x2 + x II. x − x3 III. 3t IV. 7x3 (a) Only I (b) Only III (c) II & IV (d) I, III & IV 5. The zeroes of the polynomial x + 2 is (a) 2 (b) –2 (c) 2 (d) 3 6. Find the value of polynomial 5x − 4 x 2 + 3 at x = −1 (a) 6 (b) –6 (c) 2 (d) –2 7. The value of p ( 2 ) if p ( x ) = ( x − 1)( x + 1) is (a) 0 (b) 1 (c) 2 (d) 3 8. The zero of the polynomial p ( x ) = cx + d ,c 0,c, d are real numbers is (a) − d (b) − c (c) c (d) d c 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 Old ICC building, near Firoz Hospital, Sir Syed Nagar, Medical Road, Aligarh: 9997607607, 9997394458 Chapter 1: Polynomials 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) x 2 − 5x + 5 (b) x 2 + 5x + 5 (c) x 2 − 5 x (d) x 2 + 5 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 zero of the quadratic polynomial 3x2 – x – 4 is 5 (a) (b) –1 (c) –2 (d) 1 3 16. The quadratic polynomial, the sum and product of whose zeroes are – 3 and 2, respectively is ( (a) k x 2 + 3 x + 2 ) ( (b) k x 2 − 3 x + 2 ) (c) x 2 − x + 6 (d) x 2 + x − 6 17. Other two zeroes of 2x4 – 3x3 – 3x2 + 6x – 2, if you know that two of its zeroes are 2 and – 2. 1 1 1 1 (a) − & −1 (b) − & 1 (c) &1 (d) & −1 2 2 2 2 5 5 18. Obtain all other zeroes of 3x4 + 6x3 – 2x2 – 10x – 5, if two of its zeroes are and – ⋅ 3 3 (a) 1 & −2 (b) −2 & −1 (c) −1& −1 (d) 1 & −1 19. On dividing x3 – 3x2 + x + 2 by a polynomial g(x), the quotient and remainder were x – 2 and –2x + 4, respectively. Find g(x). (a) x2 − x + 1 (b) x2 + x + 1 (c) x2 + x − 1 (d) x2 − x − 1 Old ICC building, near Firoz Hospital, Sir Syed Nagar, Medical Road, Aligarh: 9997607607, 9997394458 Chapter 1: Polynomials 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) x 2 + 2 x 2 − 7 x + 14 (b) x3 + 2 x 2 + 7 x + 14 (c) x3 + 2 x 2 − 7 x − 14 (d) x3 − 2 x 2 − 7 x + 14 21. If the zeroes of the polynomial x3 – 3x2 + 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 x4 – 6x3 – 26x2 + 138x – 35 are 2 3 , find other zeroes. (a) −5,5 (b) −5,7 (c) −5,−7 (d) 5, 7 23. If the polynomial x4 – 6x3 + 16x2 – 25x + 10 is divided by another polynomial x2 – 2x + 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 LEVEL – II (QUESTIONS BASED ON NCERT) 1. x − 5 is a factor of the cubic polynomial x3 − 3 5x 2 + 13x − 3 5 , , 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 x + 3x + k is 2, then the value of k is: 2 (a) 10 (b) –10 (c) 5 (d) –5 3. If and are the zeroes of the quadratic polynomial x 2 – 2 x – 8, then + + is: (a) 6 (b) –6 (c) –10 (d) 10 4. Given that two of the zeroes of the cubic polynomial ax + bx + cx + d are 0, the third zero is. 3 2 (a) –b/a (b) a/c (c) a/b (d) –d/a 5. The quadratic polynomial formed by the reciprocal of zeroes of the quadratic polynomial x2 – 3x + 2 is: (a) –3x2 + x + 2 (b) 2 x2 – 3x + 1 (c) x2 + 2 x – 3 (d) 2 x2 + 3x –1 6. If the zeroes of the quadratic polynomial ax2 + bx + c,a 0 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 a ( b – c ) x + b ( c – a ) x + c ( a – b ) = 0 are 1 and: 2 c (a − b ) b(c − a ) a (b − c ) c (a − b ) (a) (b) (c) (d) b(c − a ) a (b − c ) b(c − a ) a (b − c ) 8. One of the quadratic polynomial, whose zeroes are –3 and 4, is (a) x 2 − x + 12 (b) x 2 + x + 12 (c) x 2 + x − 12 (d) 2 x 2 + 2 x − 24 Old ICC building, near Firoz Hospital, Sir Syed Nagar, Medical Road, Aligarh: 9997607607, 9997394458 Chapter 1: Polynomials 9. If the zeroes of a quadratic polynomial ax2 + bx + c are both negative, then a, b and c all have the same sign. (a) c and a have 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 sign 10. If the zeroes of the quadratic polynomial x − ( a + 1) x + b are 2 and –3 then. 2 (a) a = – 7, b = –1 (b) a = 5, b = –1 (c) a = 2, b = – 6 (d) a = −2, b = – 6 11. If one of the zeroes of a quadratic polynomial of the form x2 + ax + b is the negative of the other, 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 x3 – 4 x2 + 5x – 2, 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 expression? 3x − 4 x2 − 2 x +1 x2 − 2 (a) (b) (c) (d) x2 + 1 2x2 −1 x −4 x −5 14. The ratio of the sum and the product of the roots of 7 x2 –12 x + 18 = 0 is: (a) 7:12 (b) 3:2 (c) 2:3 (d) 7:18 15. If one root of 3x2 + 11x + k = 0 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 x2 + 99 x + 127 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 ( k − 1) x 2 + kx + 1 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 ax 3 + bx 2 + 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 x3 + ax 2 + bx + c is –1, then the product of the other two zeroes is. (a) b – a + 1 (b) b – a – 1 (c) a – b + 1 (d) a – b –1 20. If , , are the zeros of the cubic polynomial x3 – mx2 + nx –1, then: (a) + + = – m, + + = n, = 1 (b) + + = –m, + + = n, = –1 (c) + + = m, + += n, = 1 (d) + + = m, + + = n, = –1 Old ICC building, near Firoz Hospital, Sir Syed Nagar, Medical Road, Aligarh: 9997607607, 9997394458 Chapter 1: Polynomials 21. If zeroes of the polynomial x3 – 3x + x + 1 are a – b, a, a + b, find a and b: (a) a = 2, b = 3 (b) a = 1, b = 2 (c) a = 3, b = 0 (d) a = 2, b= 3 22. The zeroes of the quadratic polynomial x 2 + kx + k , k 0 (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. 3 3 (a) x 2 + 2 x − (b) x 2 − + 2 2 x x 3 3 (c) x + − (d) x 2 − 2 x − 2 2 2 2 24. If the remainder on division of x + 2 x + kx + 3 by x – 3 is 21, the quotient and the value 3 2 of k. And the zeroes of the cubic polynomial x3 + 2 x2 + kx – 18 are. (a) k = –9, x2 – 5x + 6, Zeroes 3, –2, –3 (b) k = –9, x2 + 5x – 6, Zeroes 3, –2, –3 (c) k = –9, x2 + 5x + 6 Zeroes 3, –2, –3 (d) k = –9, x2 – 5x – 6, Zeroes 3 3, –2, –3 25. The zeroes of the cubic polynomial x3 – 6 x2 + 3x + 10 are of the form a, a + b, a + 2b for some real numbers a and b, the values of a and b are (a) a = 1,3 b = 3, –3 (b) a = 1,5 b = 3, –3 (c) a = –1,3 b = 3, –3 (d) a = –1,5 b = 3, –3 26. 2 is a zero of the cubic polynomial 6 x3 + 2 x 2 − 10 x − 4 2 ,its other two zeroes are −3 −1 −1 −3 (a) − 2 , (b) 2 , (c) − 2 , (d) 2 , 2 2 3 2 27. Are the following statements ‘True’ or ‘False’? (i) If the zeroes of a quadratic polynomial ax2 + bx + c are both positive, then a, b and c all have the same sign. (ii) If the graph of a polynomial intersects the xaxis at only one point, it cannot be a quadratic Polynomial. (iii) If the graph of a polynomial intersects the xaxis at exactly two points, It need not be a quadratic Polynomial. (iv) If two of the zeroes of a cubic polynomial are zero, then it does not have constant Term. (v) The only value of k for which the quadratic polynomial kx2 + x + k has equal zero is 1/2. Old ICC building, near Firoz Hospital, Sir Syed Nagar, Medical Road, Aligarh: 9997607607, 9997394458 Chapter 1: Polynomials ANSWERS 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 Old ICC building, near Firoz Hospital, Sir Syed Nagar, Medical Road, Aligarh: 9997607607, 9997394458
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