Polynomials
1.The zeroes of x2–2x –8 are:
(a)(2,-4)
(b)(4,-2)
(c)(-2,-2)
(d)(-4,-4)
Answer: (b)
Explanation: x2–2x –8 = x2–4x + 2x –8
= x(x–4)+2(x–4)
= (x-4)(x+2)
Therefore, x = 4, -2.
2. What is the quadratic polynomial whose sum and the product of zeroes is √2, ⅓ respectively?
(a)3x2-3√2x+1
(b)3x2+3√2x+1
(c)3x2+3√2x-1
(d)None of the above
Answer: (a)
Explanation: Sum of zeroes = α + β =√2
Product of zeroes = α β = 1/3
∴ If α and β are zeroes of any quadratic polynomial, then the polynomial is;
x2–(α+β)x +αβ
= x2 –(√2)x + (1/3)
= 3x2-3√2x+1
3. If the zeroes of the quadratic polynomial ax2+bx+c, c≠0 are equal, then
(a)c and b have opposite signs
(b)c and a have opposite signs
(c)c and b have same signs
(d)c and a have same signs
Answer: (d)
Explanation:
For equal roots, discriminant will be equal to zero.
b2 -4ac = 0
b2 = 4ac
ac = b2/4
ac>0 (as square of any number cannot be negative)
4. The degree of the polynomial, x4 – x2 +2 is
(a)2
(b)4
(c)1
(d)0
Answer: b
Explanation: Degree is the highest power of the variable in any polynomial.
5. If one of the zeroes of cubic polynomial is x3+ax2+bx+c is -1, then product of other two zeroes is:
(a)b-a-1
(b)b-a+1
(c)a-b+1
(d)a-b-1
Answer: (b)
Explanation: Since one zero is -1, hence;
P(x) = x3+ax2+bx+c
P(-1) = (-1)3+a(-1)2+b(-1)+c
0 = -1+a-b+c
c=1-a+b
Product of zeroes, αβγ = -constant term/coefficient of x3
(-1)βγ = -c/1
c=βγ
βγ = b-a+1
6. If p(x) is a polynomial of degree one and p(a) = 0, then a is said to be:
(a)Zero of p(x)
(b)Value of p(x)
(c)Constant of p(x)
(d)None of the above
Answer: (a)
Explanation: Let p(x) = mx+n
Put x = a
p(a)=ma+n=0
So, a is zero of p(x).
7. Zeroes of a polynomial can be expressed graphically. Number of zeroes of polynomial is equal to number of points where the graph of polynomial is:
(a)Intersects x-axis
(b)Intersects y-axis
(c)Intersects y-axis or x-axis
(d)None of the above
Answer: (a)
8. A polynomial of degree n has:
(a)Only one zero
(b)At least n zeroes
(c)More than n zeroes
(d)Atmost n zeroes
Answer: (d)
Explanation: Maximum number of zeroes of a polynomial = Degree of the polynomial
9. The number of polynomials having zeroes as -2 and 5 is:
(a)1
(b)2
(c)3
(d)More than 3
Answer: (d)
Explanation: The polynomials x2-3x-10, 2x2-6x-20, (1/2)x2-(3/2)x-5, 3x2-9x-30, have zeroes as -2 and 5.
10. Zeroes of p(x) = x2-27 are:
(a)±9√3
(b)±3√3
(c)±7√3
(d)None of the above
Answer: b
Explanation: x2-27 = 0
x2=27
x=√27
x=±3√3
11. The zeroes of the quadratic polynomial x2 + 99x + 127are
(A) both positive
(C) one positive and one negative
(D) both equal
Answer: (B)
12. If the zeroes of the quadratic polynomial x2 + bx + c , c ≠ 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
Answer: (C)
13. The number of polynomials having zeroes as –2 and 5 is
(A) 1
(B) 2
(C) 3
(D) more than 3
Answer: (D)
14. The degree of the polynomial (x + 1)(x2 – x – x4 +1) is:
(A)2
(B) 3
(C) 4
(D) 5
Answer: (D)
Explanation: Since the highest degree variable in first bracket is x and in second bracket is x4 on multiplying x with x4.the highest power we obtain is 5.
15. If the zeroes of the quadratic polynomial x2 + (a + 1)x + b are 2 and –3, then
(A) a = –7, b = –1
(B) a = 5, b = –1
(C) a = 2, b = – 6
(D) a = 0, b = – 6
Answer: (D)
16. Given that one of the zeroes of the cubic polynomial ax3 + bx2 + cx + d is zero, the product of the other two zeroes is
(A) –c/a
(B) c/a
(C) 0
(D) 3
Answer: (B)
17. If one of the zeroes of the cubic polynomial x3 + ax2 + 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
Answer: (A)
18. If one of the zeroes of the quadratic polynomial (k – 1)x2 + kx + 1 is –3, then the value of k is
(A) 4/3
(B) – 4/3
(C) 2/3
(D) – 2/3
Answer: (A)
20. The value of p for which the polynomial x3 + 4x2 –px + 8 is exactly divisible by (x – 2) is:
(A) 0
(B) 3
(C) 5
(D) 16
Answer: (D)
21. If sum of the squares of zeroes of the quadratic polynomial 6x2 + x + k is 25/36, the value of k is:
(A) 4
(B) – 4
(C) 2
(D) – 2
Answer: (D)
22. If α and β are zeroes of x2 – 4x + 1, then 1/α + 1/β – αβ is
(A) 3
(B) 5
(C) –5
(D) –3
Answer: (A)
23. If (x + 1) is a factor of x2− 3ax +3a − 7, then the value of a is:
(A) 1
(B) –1
(C) 0
(D) –2
Answer: (A)
24. 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
Answer: (A)
25. If α, β are zeroes of x2 –6x + k, what is the value of k if 3α + 2β = 20?
(A)–16
(B) 8
(C) 2
(D) –8
Answer: (A)
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