Polynomials

If one of the zeros of the polynomial \(p(x) = x^{2} - 4x + k\) is 1, then the value of \(k\) is:

1. a\(-3\)
2. b\(-5\)
3. c3
4. d5

If \(\alpha\) and \(\beta\) are zeros of \(x^{2} - 5x + 6\), then \(\dfrac{1}{\alpha} + \dfrac{1}{\beta}\) equals:

1. a\(\dfrac{5}{6}\)
2. b\(\dfrac{6}{5}\)
3. c\(-\dfrac{5}{6}\)
4. d\(-\dfrac{1}{6}\)

If \(\alpha\) and \(\beta\) are zeros of \(x^{2} + px + q\), then \(\alpha^{2} + \beta^{2}\) is:

1. a\(p^{2} + 2q\)
2. b\(p^{2} - 2q\)
3. c\(p^{2} - q\)
4. d\(2p^{2} - q\)

A quadratic polynomial whose zeros are \(5 + \sqrt{3}\) and \(5 - \sqrt{3}\) is:

1. a\(x^{2} - 10x - 22\)
2. b\(x^{2} + 10x + 22\)
3. c\(x^{2} - 10x + 22\)
4. d\(x^{2} + 10x - 22\)

If \(\alpha\) and \(\beta\) are zeros of \(2x^{2} - 5x + 7\), the value of \(\dfrac{\alpha}{\beta} + \dfrac{\beta}{\alpha}\) is:

1. a\(\dfrac{6}{7}\)
2. b\(\dfrac{3}{14}\)
3. c\(-\dfrac{6}{7}\)
4. d\(-\dfrac{3}{14}\)

If the polynomial \(x^{3} - 3x^{2} + 4x - 12\) is divided by \((x - 3)\), the remainder is:

1. a0
2. b12
3. c\(-12\)
4. d6

If \(\alpha\) and \(\beta\) are the zeroes of the quadratic polynomial \(x^{2} - 6x + k\) and \(3\alpha + 2\beta = 20\), then the value of \(k\) is:

1. a16
2. b8
3. c\(-12\)
4. d\(-16\)

If \(\alpha, \beta, \gamma\) are zeros of cubic polynomial \(p(x) = x^{3} - 6x^{2} + 11x - 6\), then \(\dfrac{1}{\alpha} + \dfrac{1}{\beta} + \dfrac{1}{\gamma}\) equals:

1. a\(\dfrac{6}{11}\)
2. b\(-\dfrac{11}{6}\)
3. c\(\dfrac{11}{6}\)
4. d\(-\dfrac{6}{11}\)

Assertion (A): If \(\alpha\) and \(\beta\) are the zeros of \(x^{2} - 7x + 12\), then \(\alpha + \beta = 7\) and \(\alpha\beta = 12\).

Reason (R): If \(\alpha\) and \(\beta\) are zeros of \(ax^{2} + bx + c\), then \(\alpha + \beta = -\dfrac{b}{a}\) and \(\alpha\beta = \dfrac{c}{a}\).

1. aBoth A and R are true, and R is the correct explanation for A
2. bBoth A and R are true, but R is NOT the correct explanation for A
3. cA is true, but R is false
4. dA is false, but R is true

Assertion (A): The graph of \(y = x^{2} - 4x + 5\) does not intersect the \(x\)-axis.

Reason (R): A quadratic polynomial can have at most two zeros, so it intersects the x-axis at most twice.

1. aBoth A and R are true, and R is the correct explanation for A
2. bBoth A and R are true, but R is NOT the correct explanation for A
3. cA is true, but R is false
4. dA is false, but R is true

Assertion (A): The polynomial \(x^{2} + 4\) has two real zeros.

Reason (R): The discriminant \(b^{2} - 4ac\) of a quadratic with no real zeros is negative.

1. aBoth A and R are true, and R is the correct explanation for A
2. bBoth A and R are true, but R is NOT the correct explanation for A
3. cA is true, but R is false
4. dA is false, but R is true

1. I. A quadratic polynomial can have at most 2 zeros.
2. II. A cubic polynomial always has 3 real zeros.
3. III. A linear polynomial has exactly 1 zero.

Examine the above statements about zeros of polynomials. Which are correct?

1. aOnly I and II
2. bOnly I and III
3. cOnly II and III
4. dAll three

1. I. If 1 is a zero of \(ax^{2} + bx + c\), then \(a + b + c = 0\).
2. II. If \(-1\) is a zero of \(ax^{2} + bx + c\), then \(a - b + c = 0\).
3. III. If 0 is a zero of \(ax^{2} + bx + c\), then \(c\) must equal 0.

From the statements above about coefficients, identify which are correct:

1. aAll three
2. bOnly II and III
3. cOnly I and III
4. dOnly I and II

1. I. For \(a < 0\), the graph of \(y = ax^{2} + bx + c\) is a parabola that opens downwards.
2. II. The graph of \(y = x^{2} - 3x - 4\) cuts the \(x\)-axis at \((1, 0)\) and \((-4, 0)\).
3. III. If the graph of \(y = p(x)\) for a quadratic polynomial \(p(x)\) lies completely above the \(x\)-axis, then \(p(x)\) has no zero.

Which of the statements above about the graphs of polynomials are correct?

1. aOnly I and II
2. bOnly II and III
3. cAll three
4. dOnly I and III

Match each polynomial with its number of distinct real zeros:

1. a(i)-P, (ii)-Q, (iii)-R, (iv)-S
2. b(i)-Q, (ii)-R, (iii)-P, (iv)-S
3. c(i)-Q, (ii)-P, (iii)-R, (iv)-S
4. d(i)-Q, (ii)-P, (iii)-S, (iv)-R

Match each pair of zeros with the correct quadratic polynomial:

1. a(i)-Q, (ii)-S, (iii)-P, (iv)-R
2. b(i)-Q, (ii)-P, (iii)-S, (iv)-R
3. c(i)-R, (ii)-P, (iii)-S, (iv)-Q
4. d(i)-P, (ii)-Q, (iii)-S, (iv)-R

If the sum of zeros of \(p(x) = kx^{2} + 2x + 3k\) equals their product, the value of \(k\) is:

1. a\(-\dfrac{2}{3}\)
2. b\(-\dfrac{1}{3}\)
3. c\(\dfrac{2}{3}\)
4. d\(\dfrac{1}{3}\)

The number of distinct quadratic polynomials of the form \(x^{2} + bx + 6\) (with integer \(b\)) whose zeros are both integers is:

1. a2
2. b8
3. c6
4. d4

If \(x = 2\) is a zero of polynomial \(2x^{2} + 3x + a\), then the value of \(a\) is:

1. a\(-10\)
2. b14
3. c\(-14\)
4. d10

Find a quadratic polynomial whose zeros are 4 and \(-3\).

1. a\(x^{2} + x - 12\)
2. b\(x^{2} - x - 12\)
3. c\(x^{2} + x + 12\)
4. d\(x^{2} - x + 12\)