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Polynomials
(Q.1) Simplify : .
(1 Mark)
(A) x + y + z (B) (C) 1(D) –1
(Q.2) If and then thevalue of is
(1 Mark)
(A) x + y (B) xy(C) x – y (D) .
(Q.3) If (x + 1) is a factor of f(x) = a0.xn + a1. x
n – 1 + a2x
n – 2 + ………+an = 0, then (1 Mark)
(A) a0 + a1 + a2 + ……….+ an = 0 (B) a0 + a2 + a4 +………. = 0(C) a0 + a3 + a5 + ………= 0
(D) a0 + a2 + a4 +……= a0 + a1 + a2 + ……
(Q.4) Simplify : .
(1 Mark)
(A) 3(a + b) (b + c) (c + a) (B) 2(a + b) (b + c) (c + a)(C) (a + b) (b + c) (c + a) (D) 1
(Q.5) If (x – 1) & (x + 3) are the factors of then the other factor is _________. (1 Mark)
(A) x + 1(B) x –1(C) x – 3 (D) x + 2
(Q.6) If is divisible by (x +a) then ________ (1 Mark)
(A) a = b(B) (C) either a = b or (D) neither a = b nor .
(Q.7) is divisible by (x – y) when “n” is __________. (1 Mark)
(A) an even number(B) an odd number(C) a prime number (D) a natural number
(Q.8) is divisible by (x – y) when ‘n’ is____________. (1 Mark)
(A) an even number (B) an odd number(C) a prime number(D) a natural number.
(Q.9) If a + c +e = 0 and b + d = 0 then ax4 + bx3 + cx2 + dx + e is exactly divisible by (1 Mark)
(A) x + 1 (B) x – 1(C) (x + 1)and (x-1)(D) (x - 2 )
(Q.10) If ‘a’ and ‘b’ are unequal and x2 + ax + b and x2 + bx + a have a common factor, then a + b is equal to (1 Mark)
(A) –1 (B) 0(C) 1 (D) –2
(Q.11) If and have a common factor, then (p – q)2 = _________ .
(1 Mark)
(A) 2(5p – 3q)(B) 2(3p-5q)(C) 3p – 5q (D) 5p – 3q.
(Q.12) Find the quadratic polynomial which is exactly divided by (2x – 3) and (x + 1). (1 Mark)
(A) (B) (C) (D)
(Q.13) The value of 6a + 11 b , if x3 – 6x2 + ax + b is exactly divisible by (x2 – 3x + 2) is (1 Mark)
(A) 0(B) 132(C) 66(D) –66
(Q.14) If is exactly divisible by (x – 3), then m = ___________ . (1 Mark)
(A) – 4 (B) – 40(C) (D) .
(Q.15) Express in lowest terms.
Top
(1 Mark)
(A) x + y – z (B) x + y + z(C) x – y + z (D) x – y – z
(Q.16) What is the reminder if we divide 3x2 – x3 –3x + 5 by x – 1 – x2 (1 Mark)
(A) 1 (B) 2(C) 3 (D) 5
(Q.17) Find the zeroes of the quadratic polynomial ?
(1 Mark)
(A) 2, 5 (B) –2, –5(C) –2, 5 (D) 2, –5
(Q.18) Find the zeroes of the polynomial x2 – 3. (1 Mark)
(A) , (B) (C) (D) 3,0
(Q.19) The graph of y = P(x) is given in fig below. Find the number of zeroes of P(x) .
(1 Mark)
(A) 4(B) 3(C) 2(D) 5.
(Q.20) The graph of y = P(x) is given in fig below, find the nature of zeroes of P(x) .
(1 Mark)
(A) 2(B) 3(C) 4(D) 5
(Q.21) If are the zeroes of the cubic polynomial x3 – 3x
2 +5x – 3, then sum of the roots (1 Mark)
(A) (B) (C) (D) or 7
(Q.22) If are the zeroes of the cubic polynomial ax3 + bx2 + cx + d = 0 ,then find the value of (1 Mark)
(A) (B) = a/d (C) = 1 (D) = a/b
(Q.23) If are the zeroes of the cubic polynomial , then is equal to (1 Mark)
(A) (B) (C) (D) a/b
(Q.24) Find the sum and product of zeroes of the quadratic polynomials .
(1 Mark)
(A) (B) (C) (D) 3
(Q.25) Find all the zeroes of , if you know that two of its zeroes are . (1 Mark)
(A) , 1(B) (C) D)
(Q.26) (5x2+14x+2)
2-(4x
2-5x+7)
2is divided by (x
2+x+1), then quotient Q and reminder given by (1 Mark)
(A) (B) (C) (D)
(Q.27) The value of (1 Mark)
(A) (B) (C) (D)
(Q.28) The factor of are (1 Mark)
(A) (x+y+z)(xy+yz+zx)(B) (x-y)(y-z)(z-x)(C) (x-y-z)(xy-yz-zx)(D) (x-y)(y-z)(x-z)
(Q.29) Solve for x: 12abx2 — (9a
2 — 8b
2) x - 6ab = 0 (3 Marks)
[CBSE-DELHI 2006]
(Q.30) Find the values of k so that (x - 1) is a factor of k2x
2 - 2kx - 3.
(
3
M
a
r
k
s
)
[
C
B
S
E
-
D
E
L
H
I
2
0
0
3
]
(Q.31) Find the zeroes of the quadratic polynomial 6x2 – 3 – 7x and verify the relationship between the zeroes and the co-efficients of the polynomial.
(
2
M
a
r
k
s
)
[
C
B
S
E
-
D
E
L
H
I
2
0
0
8
]
(Q.32) Write the zeroes of the polynomial x2 – x – 6.
(
1
M
a
r
k
)
[
C
B
S
E
-
D
E
L
H
I
2
0
0
8
]
(Q.33) If one zeros of polynomial is reciprocal of the other, find the value of a. (2 Marks)[CBSE-Outside Delhi 2008]
(Q.34) Is x = -3 a solution of the equation 2x2
+ 5x + 3 = 0 ? (1 Mark)[CBSE-Outside Delhi 2008]
(Q.35) Write the number of zeros of the polynomials Y = f(x), whose graph is given in the figure.
(1 Mark)[CBSE-Outside
Delhi 2008]
(Q.36) Find the zeroes of the polynomial , given that the product of its 2 zeroes is 12. (5 Marks)
(Q.37) If and are the zeroes of the quadratic polynomial such that , find the value of . (3 Marks)
(Q.38) If and are the zeroes of the quadratic polynomial 2x2 - 5x +7, find a polynomial whose zeroes are 2 + 3 and 3 + 2 . (6 Marks)
(Q.39) If the zeroes of the polynomial are a – b, a and a + b find a, b. (2 Marks)
(Q.40) If the polynomial f(x) = x4 - 6x
3 + 16x
2 - 25x + 10 is divided by another polynomial x
2 - 2x + k , the remainder comes out to be x + a , find k and a. (6 Marks)
(Q.41) If two zeroes of the polynomial f(x) = x 4 - 6x
3 - 26x
2 +138x - 35 are , find other zeroes.
(6 Marks)
(Q.42) Find the zeroes of the polynomial f(x) = x3 - 5x
2 - 2x + 24, given that the product of its two zeroes is 12. (6 Marks)
(Q.43) Divide by and verify the division algorithm. (3 Marks)
(Q.44) (2 Marks)
(Q.45) Find a quadratic polynomial, the sum and product of whose zeroes are -3 and 2 respectively. (3 Marks)
(Q.46) Find the zeroes of the polynomial and verify the relationship between the zeroes and its coefficients. (5 Marks)
(Q.47) Find the values of a and b so that x4 + x
3 + 8x
2 + ax + b is divisible by x
2 + 1. (3 Marks)
(Q.48) If f(x) = ax3 + bx
2 + cx + d, a 0, then what will be the sum of zeroes? (1 Mark)
(Q.49) Find the condition that the zeroes of the polynomial f(x) = x3 - px
2 + qx - r may be in arithmetic progression. (3 Marks)
(Q.50) If the sum of the zeroes of the polynomial f(x) = 2x3 - 3kx
2 + 4x - 5 is 6, then find the value of k. (1 Mark)
(Q.51) Find a quadratic polynomial, if the sum and the product of the zeroes are 4 and 4 respectively. (1 Mark)
(Q.52) The product of two zeroes of the polynomial f(x) = 2x3 + 6x
2 -4x + 9 is 3, then find its third zero. (1 Mark)
(Q.53) If (x + 2)(2x - 1)(3x - 2) = 0, find the zeroes of the polynomial. (1 Mark)
(Q.54) If f(x) is divisble by q(x), what will be the value of r(x), where f(x) = g(x)q(x) + r(x)? (1 Mark)
(Q.55)
(2 Marks)
(Q.56) Find a quadratic polynomial, the sum of whose roots is -1 and the sum of their reciprocals is .
(2 Marks)
(Q.57) (2 Marks)
(Q.58)
(2 Marks)
(Q.59) (2 Marks)
(Q.60) (2 Marks)
(Q.61) Find a cubic polynomial with the sum, sum of the products of its zeroes taken two at a time, and the product of its zeroes as 2, - 7 and –14 respectively. (2 Marks)
(Q.62) (2 Marks)
(Q.63) If the zeroes of the polynomial f(x) = x3 - 3x
2 +x + 1 are a – b, a and a + b, find the values of a and b. (2 Marks)
(Q.64)
(2 Marks)
(Q.65) Find the zeroes of the polynomial f(x) = x3 - 5x
2 – 16x + 80, if its two zeroes are equal in magnitude but opposite in sign. (3 Marks)
(Q.66)
(3 Marks)
(Q.67) Find the zeroes of the quadratic polynomial 8x2 – 21 - 22x and verify the relationship between the zeroes and the coefficients of the polynomial. (3 Marks)
(Q.68)
(3 Marks)
(Q.69)
(3 Marks)
(Q.70)
(2 Marks)
(Q.71) If the sum of the squares of the zeroes of a quadratic polynomial x2 –18x + p is 180, find the value of p. (2 Marks)
(Q.72) If the square of the difference of the zeroes of the polynomial f(x) = x2 + kx + 85 is equal to 144, evaluate the value of k. (3 Marks)
(Q.73) Evaluate the values of ‘b’ if m and n are the zeroes of the polynomial q(y) = by2 – 35y + 12 and m
2 + n
2 = 1. (3 Marks)
(Q.74) Check whether the polynomial l(x) = x2 – 5x +1 is a factor of the polynomial m(x) =4 x4 –13x3 – 31x2 + 35x – 10 by dividing m(x) by l(x). (3 Marks)
(Q.75) The zeroes of a quadratic polynomial p(x) = 2x2 + x + n are and . Find the value of n if
(3 Marks)
(Q.76) If If If If and and and and are the zeroes of the polynomial f (x) = 6xare the zeroes of the polynomial f (x) = 6xare the zeroes of the polynomial f (x) = 6xare the zeroes of the polynomial f (x) = 6x2222 + x + x + x + x ––––12, then find a quadratic polynomial whose zeroes are 12, then find a quadratic polynomial whose zeroes are 12, then find a quadratic polynomial whose zeroes are 12, then find a quadratic polynomial whose zeroes are and and and and .... (3 Marks)
(Q.77) Find a polynomial whose zeroes are the reciprocals of the zeroes of the polynomial f(y) = 9yf(y) = 9yf(y) = 9yf(y) = 9y2222 –––– 18y + 8. 18y + 8. 18y + 8. 18y + 8. (3 Marks)
(Q.78) If the zeroes of the polynomial x3 - 3x2 + x + 1 are (a – b), a and (a + b), find a and b. (3 Marks)
(Q.79) Find the zeroes of the quadratic polynomial x2 – 4x +3. (2 Marks)
(Q.80) Find the zeroes of the quadratic polynomial f(x) = x2 + 5x + 6 and verify the relationship between the zeroes and the coefficients of the polynomial. (3 Marks)
(Q.81) Find the zeroes of the quadratic polynomial f(x) = abx2+(b2 – ac)x - bc and verify the relationship between the zeroes and the coefficients of the polynomial.
(3 Marks)
(Q.82)
(3 Marks)
(Q.83) Find the zeroes of the polynomial 6x2 – 3. (1 Mark)
(Q.84) Find the zeroes of the quadratic polynomial x2 – 4x + 3. (1 Mark)
(Q.85) Find the zeroes of the quadratic polynomial x2 + 7x + 10. (1 Mark)
(Q.86) Find a quadratic polynomial, the sum and product of its zeroes are 1 and –6 respectively. (1 Mark)
(Q.87) Find a quadratic equation, whose roots are and . (1 Mark)
(Q.88) Find a quadratic polynomial, the sum and product of its zeroes are 8 and 15 respectively. (1 Mark)
(Q.89)
(Q.90) What are the quotient and the remainder when 3x4 +5x
3 – 7x
2 +2x +2 is divided by x
2 + 3x +1?
(Q.91)
(3 Marks)
(Q.92)
(3 Marks)
(2 Marks)
(Q.93)
(2 Marks)
(Q.94) Use the division algorithm to find the quotient q(y) and the remainder r(y) when f(y) = 12 y 3 + 17 y2 – 20y – 10 is divided by g(y) = 3y2 +
2y-5. (3 Marks)
(Q.95) Given a polynomial p(x). The graph of y = p(x) intersects the x-axis at three points. Find the number of zeroes of p(x). (1 Mark)
(Q.96) If (Z –3) is a factor of Z3 +aZ
2 + b Z + 18 and a + b = -7, find a and b. (3 Marks)
(Q.97) Find a quadratic polynomial whose zeroes are (2 +1) and (2 + 1), if and are the
zeroes of the polynomial P(x) = 2x2 –7 x + 6.
(3 Marks)
(Q.98)
(6 Marks)
(Q.99) Find a quadratic polynomial, the sum and product of whose zeroes are -7 and 7 respectively. (1 Mark)
(Q.100) If one root of the quadratic equation 2x2 + px + 4 = 0 is ‘2’, then find the other root and also find the value of ‘p’. (2 Marks)
(Q.101)
(2 Marks)
(Q.102) Find a quadratic polynomial, the sum and product of whose zeroes are -5 and 4 respectively.
(1 Mark)
(Q.103) Find a quadratic polynomial, the sum and product of whose zeroes are –3 and 2 respectively. (1 Mark)
(Q.104) Find a quadratic polynomial, whose zeroes are (2 Marks)
(Q.105) Find a quadratic polynomial, the sum and product of whose zeroes are 4 and 1 respectively. (1 Mark)
(Q.106) (2 Marks)
(Q.107)
(1 Mark)
(Q.108)
(2 Marks)
(Q.109) Form a quadratic polynomial whose one zero is 4+ 7 and the sum of zeroes is 8. (2 Marks)
(Q.110) (2 Marks)
(Q.111)
(3 Marks)
(Q.112)
(2 Marks)
(3 Marks)
(Q.113)
(2 Marks)
(Q.114) (1 Mark)
(Q.115) Find the zeroes of quadratic polynomial f(x) = x2 – 3x – 28 and verify the relationship between the zeroes and the coefficients of the polynomial.
(3 Marks)
(Q.116) Find the zeroes of the quadratic polynomial f(x) = x2+7x+12 and verify the relationship between the zeroes and the coefficients of
the polynomial. (3 Marks)
(Q.117) Find the zeroes of the quadratic polynomial f(t)= t2– 15 and verify the relationship between the
zeroes and the coefficient of the polynomial. (3 Marks)
(Q.118)
(2 Marks)
(Q.119) (2 Marks)
(Q.120)
Top
(3 Marks)
(Q.121) If sum of the squares of zeroes of the polynomial f(t) = t2 – 8t + p is 40, find the value of p. (2 Marks)
View Answer
(Q.122)
(2 Marks)
View Answer
(Q.123)
(3 Marks)
View Answer
(Q.124) (3 Marks)
View Answer
(Q.125)
(3 Marks)
View Answer
(Q.126) Find a cubic polynomial whose zeroes are –3, 8 and –1. (2 Marks)
View Answer
(Q.127) Find a cubic polynomial whose zeroes are m, n and r such that (m + n + r) = - 9, (mn + nr + rm) = 6 and mnr = 56. (2 Marks)
View Answer
(Q.128) Find a cubic polynomial with the sum, sum of the product of it zeroes taken two at a time, and product of its zeroes as 2, -41 and 42 respectively. (2 Marks)
View Answer
(Q.129) Find the zeroes of the polynomial f(t) = t3 – 3t2 - 25t + 75, if its two zeroes are equal in magnitude but opposite in sign. (3 Marks)
View Answer
(Q.130) The product of two of the zeroes of the polynomial g(t) = t3 + 3 t2 – 10t –24 is - 6 . Find the zeroes of g(t) (3 Marks)
View Answer
(Q.131) The zeroes of the cubic polynomial f(t) = t3 – 6t2 – 13t +42 are in A.P. Find its zeroes. (3 Marks)
View Answer
(Q.132) If the zeroes of the cubic polynomial g(x) = x3 + 3x2 – 13x – 15 are (m–n), m and (m+n), then find the values of m and n. (3 Marks)
View Answer
(Q.133) Divide the polynomial P(t) = 6t3 + 10t2 – 13t +1 by the polynomial g(t) = 3t – 1. Find the quotient and the remainder. (3 Marks)
View Answer
(Q.134) Divide the polynomial P(t) =2t3 – 11t2 + 16t – 4 by the polynomial g(t) = t2 – 2t +1 and verify the division algorithm. (3 Marks)
View Answer
(Q.135) Use the division algorithm to find the quotient q(t) and the remainder r(t) when f(t) = 8t3 – 38t2 + 36t +5 is divided by g(t) = 4t – 3. Top
(6 Marks)
View Answer
(Q.136) Apply the division algorithm to find the quotient and remainder on dividing f(y) by g(y) as given below:
f(y) = y4 – 3y2 +4y + 5, g(y) = y2 + 1 – y (6 Marks)
View Answer
(Q.137) Find the zeroes of the cubic polynomial m(x) = x3 – 3x
2 – 13x + 15, it being given that 1 is one of the zeroes of m(x). (6 Marks)
View Answer
(Q.138) Use the division algorithm to find the quotient q(y) and remainder r(y), when f(y) = 8y4 – 12y
3 – 2y
2 + 15y – 4 is divided by g(y) = 2y
2 – 3y + 1. (2 Marks)
View Answer
(Q.139) By applying division algorithm prove that the polynomial g(x) = x2 + 3x + 1 is a factor of the polynomial f(x) = 3x4 + 5x3 - 7 x2 + 2x + 2. (6 Marks)
View Answer
(Q.140) It is given that and are two zeroes of the polynomial f(y) = 2y4 - 3y
3 - 3y
2 + 6y - 2, find all the zeroes of f(y). (6 Marks)
View Answer
(Q.141) Use division algorithm to find all the zeroes of the polynomial p(x) = 3x4 + 6x
3 – 2x
2 - 10x - 5, it being given that and are two of its zeroes.
(6 Marks)
View Answer
(Q.142) (6 Marks)
View Answer
(Q.143) Find the polynomial g(x), if q(x) = x – 2 is the quotient and r(x) = -2x + 4 is the remainder when f(x) = x3 – 3x
2 + x + 2 is divided by g(x). (6 Marks)
View Answer
(Q.144) What must be subtracted from the polynomial f(y)=8y4 + 14y
3 – 2y
2 + 7y – 8 so as to make it exactly divisible by g(y) = 4y
2 + 3y – 2? (6 Marks)
View Answer
(Q.145) A remainder r(x) = (x + a) is obtained when the polynomial f(x) = x4 – 6x3 + 16x2 –25x + 10 is divided by the polynomial g(x) = x2 – 2x + k. Find
the values of k and a. (6 Marks)
View Answer
Most Important Questions
(Q.1) If sum of roots of a equation is 1 and their product is –6. Write the equation.
View Answer
(Q.2) Form the equation whose roots are 6 and -1. View Answer
(Q.3) Find the quadratic equation whose roots are View Answer
(Q.4) View Answer
(Q.5)
View Answer
(Q.6) View Answer
(Q.7)
View Answer
(Q.8)
View Answer
(Q.9) Find all the Zeroes of if two of its Zeroes are . View Answer
(Q.10) Divide by and verify the division algorithm. View Answer
(Q.11) Verify that 1,-1,-2 are the zeroes of the cubic polynomial p(x)= x3 + 2x2 - x - 2, and verify the relationship between the zeroes and the coefficients. View Answer
(Q.12) Find the Zeroes of the polynomial = x3 – 5x2 – 2x + 24 if give that the product of its two zeroes is 12. View Answer
(Q.13) Find a quadratic Polynomial, the sum and product of Zeroes are -3 and 2 respectively. View Answer
(Q.14) What will be the remainder of
.
View Answer
(Q.15) The G.C.D. of two polynomials (x2 + ax - 28) (x +5)
and (x2 + 8x + b)(x - 4) is (x - 4)(x + 5). Find the value of a and b. View Answer
(Q.16) Find the G.C.D. of the polynomials (2x2 - 2x- 4) and 4(x3 - 8). View Answer
(Q.17)
Find the G.C.D. and L.C.M. of the following polynomials –
p(x) = 6(x - 2)(x2 + x - 6) And, q(x) = 3(x2 + 4x - 12).
View Answer
(Q.18) Find the G.C.D. of the polynomials (x2 - 1) and (x2 - 2x + 1). View Answer
(Q.19)
Find the G.C.D. of the polynomials (x2 - 4) and (x – 2)(x + 1). View Answer
(Q.20)
Find the L.C.M. of the given polynomials 8(x3 – x2 + x) and 28(x3 + 1). View Answer
(Q.21) Write the discriminate of the quadratic equation 4x2 – ax + 2 = 0. View Answer
(Q.22)
Polynomials of degree n having ______ numbers of Zeros. View Answer
(Q.23)
is a polynomials of variable _____ and of degree ______.
View Answer
(Q.24) What will be the coefficient of in .
View Answer
(Q.25) Factorize
View Answer
(Q.26) Factorise the polynomial x2 + 2x - 6 into two linear factor. View Answer
(Q.27) Factorise the polynomial x2 + 6x - 10.
View Answer
(Q.28) Find the value of the quadratic equation 2x2 - 3x -2
at x = 1 and x = -2. View Answer
(Q.29)
Find the value of the cubic polynomial equation x3 - 6x2 + 11x - 6 at x = 1,2 and 3. View Answer
(Q.30) Show that x = 1 is a root of the polynomial 3x3 – 4x2 + 8x – 7. View Answer
(Q.31) Find zero of the polynomial 2x2 – 8. View Answer
(Q.32) Find the zeroes of the quadratic polynomial
?
(a) 2, 5 (b) –2, –5
(c) –2, 5 (d) 2, –5
View Answer
(Q.33) Find the zeroes of the polynomial x2 – 3
(a) , (b)
(c) (d) None of these
View Answer
(Q.34) Write the degrees of each of the following
polynomials.
(i)
(ii)
(iii) (iv) 7 View Answer
