15 multiple-choice questions and 2 case-based questions on the current (rationalised) syllabus, in the style of the board paper's Sections A and E. Try each one first; the answer, a one-line reason and a worked solution open on a tap.
Choose one option. Section A of the board paper has 18 MCQs of 1 mark each, spread over all chapters.
The zeroes of the polynomial \(x^2 - 9x + 14\) are
Why: Split −9x as −2x − 7x: x² − 9x + 14 = (x − 2)(x − 7).
\(x^2 - 9x + 14 = (x - 2)(x - 7)\), which is \(0\) when \(x = 2\) or \(x = 7\).
The sum and the product of the zeroes of \(4x^2 + 7x - 2\) are respectively
Why: Sum = −b/a and product = c/a with a = 4, b = 7, c = −2.
Sum \(= -\dfrac{b}{a} = -\dfrac74\); product \(= \dfrac{c}{a} = -\dfrac24 = -\dfrac12\). (The zeroes are \(\tfrac14\) and \(-2\).)
A quadratic polynomial whose zeroes are \(3 + \sqrt2\) and \(3 - \sqrt2\) is
Why: Sum of zeroes = 6, product = 9 − 2 = 7, so x² − 6x + 7.
Sum \(= 6\), product \(= (3+\sqrt2)(3-\sqrt2) = 9 - 2 = 7\). Polynomial: \(x^2 - (\text{sum})x + \text{product} = x^2 - 6x + 7\).
If \(-4\) is a zero of \(x^2 + kx - 20\), then \(k\) equals
Why: Substitute x = −4 and set the value to 0.
\((-4)^2 + k(-4) - 20 = 0 \Rightarrow 16 - 4k - 20 = 0 \Rightarrow k = -1\). (The other zero is \(5\).)
The graph of a quadratic polynomial \(y = p(x)\) touches the \(x\)-axis at exactly one point and does not cross it. The number of distinct zeroes of \(p(x)\) is
Why: The zeroes are the x-coordinates of the points where the graph meets the x-axis.
Each point where the graph meets the \(x\)-axis gives one zero. One point of contact gives exactly one zero (e.g. \(y = (x-2)^2\) has only the zero \(2\)).
If \(\alpha\) and \(\beta\) are the zeroes of \(x^2 - 6x + 5\), then \(\alpha^2\beta + \alpha\beta^2\) equals
Why: α²β + αβ² = αβ(α + β) = 5 × 6.
\(\alpha + \beta = 6\), \(\alpha\beta = 5\). So \(\alpha^2\beta + \alpha\beta^2 = \alpha\beta(\alpha + \beta) = 5 \times 6 = 30\).
The zeroes of \(x^2 - 12x + k\) are in the ratio \(1 : 5\). The value of \(k\) is
Why: Let the zeroes be a and 5a: 6a = 12 gives a = 2, so k = 2 × 10.
Zeroes \(a\) and \(5a\): \(a + 5a = 12 \Rightarrow a = 2\). Zeroes \(2\) and \(10\); \(k = \) product \(= 20\).
If one zero of \(5x^2 + 13x + k\) is the reciprocal of the other, then \(k\) equals
Why: Zeroes α and 1/α have product 1, and product = k/5.
Product of zeroes \(= \alpha \cdot \dfrac1\alpha = 1 = \dfrac{k}{5}\), so \(k = 5\). (Then \(5x^2 + 13x + 5\) has real zeroes since \(169 - 100 \gt 0\).)
The zeroes of \(x^2 - 3\) are
Why: x² = 3 gives x = ±√3.
\(x^2 - 3 = (x - \sqrt3)(x + \sqrt3)\), so the zeroes are \(\sqrt3\) and \(-\sqrt3\).
For the quadratic polynomial \(ax^2 + bx + c\) (\(a \ne 0\)), if \(c = 0\) then
Why: c = 0 makes the product of zeroes c/a = 0, so one zero is 0.
With \(c = 0\): \(ax^2 + bx = x(ax + b)\), whose zeroes are \(0\) and \(-\dfrac{b}{a}\). So one zero is \(0\) (both only if also \(b = 0\)).
For some \(b \gt 0\), the graph of \(y = x^2 + bx + 9\) touches the \(x\)-axis at exactly one point. That point is
Why: Touching at one point means one repeated zero: x² + bx + 9 = (x + 3)² with b = 6.
One point of contact means the polynomial is a perfect square: \(x^2 + bx + 9 = (x + 3)^2\) or \((x - 3)^2\). With \(b \gt 0\), it is \((x + 3)^2\) (\(b = 6\)), whose only zero is \(-3\). The graph touches the \(x\)-axis at \((-3, 0)\). (\((0, 9)\) is where it meets the \(y\)-axis.)
A quadratic polynomial whose zeroes have sum \(-4\) and product \(-21\) is
Why: Use x² − (sum)x + (product).
\(x^2 - (-4)x + (-21) = x^2 + 4x - 21\). Check: \((x + 7)(x - 3)\) has zeroes \(-7, 3\) with sum \(-4\), product \(-21\).
If \(\alpha\) and \(\beta\) are the zeroes of \(2x^2 - 7x + 3\), then \((\alpha - \beta)^2\) equals
Why: (α − β)² = (α + β)² − 4αβ = 49/4 − 6.
\(\alpha + \beta = \dfrac72\), \(\alpha\beta = \dfrac32\). \((\alpha - \beta)^2 = \dfrac{49}{4} - 4 \times \dfrac32 = \dfrac{49}{4} - 6 = \dfrac{25}{4}\).
The zeroes of \(x^2 - 5x + 2\) are \(\alpha\) and \(\beta\). The value of \(\dfrac{1}{\alpha^2} + \dfrac{1}{\beta^2}\) is
Why: 1/α² + 1/β² = (α² + β²)/(αβ)² = ((α + β)² − 2αβ)/(αβ)².
\(\alpha + \beta = 5\), \(\alpha\beta = 2\). \(\alpha^2 + \beta^2 = 25 - 4 = 21\), so \(\dfrac{1}{\alpha^2} + \dfrac{1}{\beta^2} = \dfrac{\alpha^2 + \beta^2}{(\alpha\beta)^2} = \dfrac{21}{4}\).
A quadratic polynomial \(p(x)\) has zeroes \(\dfrac12\) and \(-3\), and \(p(0) = -6\). Then \(p(x)\) is
Why: p(x) = k(x − 1/2)(x + 3); p(0) = −3k/2 = −6 gives k = 4.
\(p(x) = k\left(x - \tfrac12\right)(x + 3)\); \(p(0) = -\tfrac32 k = -6 \Rightarrow k = 4\). So \(p(x) = (2x - 1)(2x + 6) = 4x^2 + 10x - 6\).
Section E of the board paper has three case-based questions of 4 marks: a real-life passage, then parts of 1, 1 and 2 marks, with a choice (OR) on the 2-mark part.
The owner of a home bakery models her weekly profit (in hundreds of rupees) from selling \(x\) dozen cupcakes by the quadratic polynomial \(P(x) = -x^2 + 14x - 40\). The graph of \(y = P(x)\) is a parabola opening downwards; she makes a profit only where the graph lies above the \(x\)-axis.
Meera plans a rectangular kitchen garden on her terrace. Its length is \((x + 5)\) m and its width is \((x - 2)\) m, so its area is \(A(x) = (x + 5)(x - 2) = x^2 + 3x - 10\) square metres.
This free set is separate from the chapter's question bank. On the Polynomials chapter page the revision notes and the first step of the Route to 95 are free for everyone. CBSE Essentials adds all 39 questions in the chapter bank (short and long answers, assertion–reason and more case studies) with their full step-marking schemes, the Route to 95 checkpoints with your progress saved, Skill Builders and the full common-mistakes library. There is no AI marking on CBSE Math Revision: you check your work against the marking scheme.
Original questions written by CBSE Math Revision for the CBSE 2026-27 syllabus; not taken from NCERT, NCERT Exemplar or CBSE papers. Every answer was re-checked by computer algebra and by an independent reviewer. CBSE Math Revision is independent and not affiliated with CBSE or NCERT. Spotted a slip? Tell us and it goes in the corrections log.