📝 Polynomials (Questions)
Multiple Choice Questions
1. If one of the zeroes of the quadratic polynomial `(k–1) x^2 + k x + 1` is `–3`, then the value of `k` is
a) `4/3`
b) `-4/3`
c) `2/3`
d) `-2/3`
Ans: (a)
2. A quadratic polynomial, whose zeroes are –3 and 4, is
a) `x^2 – x + 12`
b) `x^2 + x + 12`
c) `\x^2/2 - x/2 -6`
d) `2x^2 + 2x –24`
Ans: (c)
3. If the zeroes of the quadratic polynomial `x^2 + (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`
Ans: (d)
4. The number of polynomials having zeroes as –2 and 5 is
a) 1
b) 2
c) 3
d) more than 3
Ans: (d)
5. 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`
Ans: (b)
6. If one of the zeroes of the cubic polynomial `x^3 + 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`
Ans: (a)
7. The zeroes of the quadratic polynomial `x^2 + 99x + 127` are
a) both positive
b) both negative
c) one positive and one negative
d) both equal
Ans: (b)
8. The zeroes of the quadratic polynomial `x^2 + kx + k, k \ne 0`,
a) cannot both be positive
b) cannot both be negative
c) are always unequal
d) are always equal
Ans: (a)
9. If the zeroes of the quadratic polynomial `ax^2 + bx + c, c \ne 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
Ans: (c)
10. If one of the zeroes of a quadratic polynomial of the form `x^2+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.
Ans: (a)
11. Which of the following is not the graph of a quadratic polynomial?
Ans: (D)
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(POLYNOMIALS- CASE STUDY)
Case study -1
The below picture are few natural examples of parabolic shape which is represented by a quadratic polynomial. A parabolic arch is an arch in the shape of a parabola. In structures, their curve represents an efficient method of load, and so can be found in bridges and in architecture in a variety of forms.
a) All are real numbers
b) All are rational numbers.
c) ‘a’ is a non zero real number and b and c are any real numbers
d) All are integers.
2. If the roots of the quadratic polynomial are equal, where the discriminant, `D = b^2 -4ac`, then
a) D > 0
b) D < 0
c) D `\geq` 0
d) D = 0
3. If `\alpha` are `1/\alpha` are the zeroes of the quadratic polynomial `2x^2 -x + 8k` then `k` is
a) 4
b) 1/4
c) -1/4
d) 2
4. The graph of `x^2+1=0`
a) Intersects x‐axis at two distinct points
b) Touches the x‐axis at a point.
c) Neither touches nor intersects the x‐axis
d) Either touches or intersects the x-axis.
5. If the sum of the roots is `–p` and product of the roots is `-1/p` , then the quadratic polynomial is
(a) `k(-px^2 + x/p + 1)`
(b) `k(-px^2 - x/p - 1)`
(c) `k(-px^2 + px - 1/p)`
(d) `k(-px^2 - px + 1/p)`
Answers
1. c)
2. d)
3. b)
4. c)
5. c)
CASE STUDY 2:
An asana is a body posture, originally and still a general term for a sitting meditation pose, and later extended in hatha yoga and modern yoga as exercise, to any type of pose or position, adding reclining, standing, inverted, twisting, and balancing poses. In the figure, one can observe that poses can be related to the representation of quadratic polynomial.
1. The shape of the poses shown is
a) Spiral
b) Ellipse
c) Linear
d) Parabola
2. The graph of parabola opens downwards, if _______
a) `a ≥ 0`
b) `a = 0`
c) `a < 0`
d) `a > 0`
3. In the graph, how many zeroes are there for the polynomial?
a) 0
b) 1
c) 2
d) 3
4. The two zeroes in the above shown graph are
a) 2, 4
b) -2, 4
c) -8, 4
d) 2,-8
5. The zeroes of the quadratic polynomial `4sqrt3x^2 + 5x -2sqrt3`
(a) `2/sqrt3, sqrt3/4`
(b) `-2/sqrt3, sqrt3/4`
(c) `2/sqrt3, -sqrt3/4`
(d) `-2/sqrt3, -sqrt3/4`
Answers
1. d)
2. c)
3. c)
4. b)
5. b)
CASE STUDY 3:
Basketball and soccer are played with a spherical ball. Even though an athlete dribbles the ball in both sports, a basketball player uses his hands and a soccer player uses his feet. Usually, soccer is played outdoors on a large field and basketball is played indoor on a court made out of wood. The projectile (path traced) of soccer ball and basketball are in the form of parabola representing quadratic polynomial.
1. The shape of the path traced shown is
a) Spiral
b) Ellipse
c) Linear
d) Parabola
2. The graph of parabola opens upwards, if _______
a) `a = 0`
b) `a < 0`
c) `a > 0`
d) `a ≥0`
3. Observe the following graph and answer
In the above graph, how many zeroes are there for the polynomial?
a) 0
b) 1
c) 2
d) 3
4. The three zeroes in the above shown graph are
a) `2, 3,-1`
b) `-2, 3, 1`
c) `-3, -1, 2`
d) `-2, -3, -1`
5. What will be the expression of the polynomial?
a) `x^3 + 2x^2 -5x -6`
b) `x^3 + 2x^2 -5x + 6`
c) `x^3 + 2x^2 +5x -6`
d) `x^3 + 2x^2 +5x + 6`
Answers
1. d)
2. c)
3. d)
4. c)
5. a)
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Q) The graphs of `y = p(x)` are given in Fig below, for some polynomials `p(x)`. Find the number of zeroes of p(x), in each case.
Q) Find the zeroes of the polynomial `x^2 – 3` and verify the relationship between the zeroes and the coefficients.
Q) Find the zeroes of the polynomial `x^2 +1/6x – 2`, and verify the relation between the coefficients and the zeroes of the polynomial.
Q) Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients.
a) `4u^2 + 8u`
b) `t^2 – 15`
c) `2\sqrt3x^2 -5x + \sqrt3`
d) `2s^2 – (1 + 2 \sqrt2)s + \sqrt2`
e) `4x^2 + 5\sqrt2x – 3`
f) `v^2 + 4 \sqrt3v – 15`
g) `y^2 +3/2\sqrt5 y – 5`
h) `7y^2 –11/3y –2/3`
Q) Find the quadratic polynomial whose zeroes are `\2/3` and `\-1/4`. Verify the relation between the coefficients and the zeroes of the polynomial.
Q) If `(x + a)` is a factor of the polynomial `2x^2 + 2ax + 5x + 10`, find the value of `a`.
Q) If the product of the zeroes of the polynomial `(ax^2 -6x -6)` is 4, find the value of `a`.
Q) If one zero of the polynomial `(a^2 + 9)x^2 + 13x + 6a` is reciprocal of the other, find the value of `a`.
Q) If `\alpha`, `\beta` be the zeroes of the polynomial `2x^2 + 5x + k` such that `\alpha^2 + \beta^2 + \alpha \beta = \21/4`, then k = ?
Q) If `\alpha`, `\beta` are the zeroes of the polynomial `x^2 + 6x + 2` then `(\1/\alpha + 1/\beta)` = ?
Q) Verify that the numbers given alongside of the cubic polynomials below are their zeroes. Also, verify the relationship between the zeroes and the coefficients in each case:
a) `2x^3 + x^2 – 5x + 2; \1/2, 1, – 2` b) `x^3 – 4x^2 + 5x – 2; 2, 1, 1`
Q) 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.
Q) If the zeroes of the polynomial `x^3 – 3x^2 + x + 1` are `a – b, a, a + b`, find `a` and `b`.
Q) If two zeroes of the polynomial `x^4 – 6x^3 – 26x^2 + 138x – 35` are `2 ±\sqrt 3`, find other zeroes
Q) If the polynomial `x^4 – 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`.
Q) It is given that `-1` is one of the zero of the polynomial `x^3 + 2x^2 -11x -12`. Find the all zeroes of the given polynomial.
Q) If `\alpha` and `\beta` are the zeroes of the polynomial `f(x) = 6x^2 + x -2`, find the value of `(\alpha/\beta + \beta/alpha)`.
Q) Divide the polynomial `p(x)` by the polynomial `g(x)` and find the quotient and remainder in each of the following :
a) `p(x) = x^4 – 3x^2 + 4x + 5, g(x) = x^2 + 1 – x`
b) `p(x) = x^4 – 5x + 6, g(x) = 2 – x^2`
Q) Obtain all other zeroes of `3x^4 + 6x^3 – 2x^2 – 10x – 5`, if two of its zeroes are `\sqrt{5/3}` and `-\sqrt{5/3}`
Q) If two zeroes of the polynomial `x^4 -6x^3 -26x^2 + 138x -35` are `(2+ \sqrt 3)` and `(2- \sqrt 3)`, find the other zeroes.
Q) On dividing `x^3 - 3x^2 + x + 2` by a polynomial `g(x)`, the quotient and the remainder are (`x-2)` and `(- 2x + 4)` respectively. Find `g(x).`
Q) Find all the zeroes of the polynomial `2x^4 -11x^3 + 7x + 13x - 7` it is being given that two of its zeroes are `(3 + \sqrt 2)` and `(3 - \sqrt 2)`.
Q) If the polynomial `x^4 + 2x^3 + 8x^2 + 12x + 18` is divided by another polynomial `(x^2 + 5)`, the remainder comes out to be `(px + q)`. Find the values of `p` and `q`.
Q) What real number should be subtracted from the polynomial (`3x^3 + 10x^2 -14x + 9)` so that `(3x -2)` divides it exactly?
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