Mathematics Questions

Browse 1,240 Mathematics questions with expert-verified, step-by-step solutions — NCERT, exemplar and previous-year questions, organised by chapter.

Q301
Are the following statements 'True' or 'False'? Justify your answers.
1. (i)If the zeroes of a quadratic polynomial $a x^{2} + b x + c$ are both positive, then $a$ , $b$ and $c$ all have the same sign.
2. (ii)If the graph of a polynomial intersects the $x$ -axis at only one point, it cannot be a quadratic polynomial.
3. (iii)If the graph of a polynomial intersects the $x$ -axis at exactly two points, it need not be a quadratic polynomial.
4. (iv)If two of the zeroes of a cubic polynomial are zero, then it does not have linear and constant terms.
5. (v)If all the zeroes of a cubic polynomial are negative, then all the coefficients and the constant term of the polynomial have the same sign.
6. (vi)If all three zeroes of a cubic polynomial $x^{3} + a x^{2} - b x + c$ are positive, then at least one of $a$ , $b$ and $c$ is non-negative.
7. (vii)The only value of $k$ for which the quadratic polynomial $k x^{2} + x + k$ has equal zeros is $\frac{1}{2}$ .
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Q302

Find the zeroes of the following polynomials by factorisation method and verify the relations between the zeroes and the coefficients of the polynomials: 1. $4 x^{2} - 3 x - 1$ 2. $3 x^{2} + 4 x - 4$ 3. $5 t^{2} + 12 t + 7$ 4. $t^{3} - 2 t^{2} - 15 t$ 5. $2 x^{2} + \frac{7}{2} x + \frac{3}{4}$ 6. $4 x^{2} + 52 x - 3$ 7. $2 s^{2} - (1 + 22) s + 2$ 8. $v^{2} + 43 v - 15$ 9. $y^{2} + \frac{3}{2} 5 y - 5$ 10. $7 y^{2} - \frac{11}{3} y - \frac{2}{3}$

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Q303

Find the zeroes of the following polynomials by factorisation method and verify the relations between the zeroes and the coefficients of the polynomials: $3 x^{2} + 4 x - 4$

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Q304

Find the zeroes of the following polynomials by factorisation method and verify the relations between the zeroes and the coefficients of the polynomials: $5 t^{2} + 12 t + 7$

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Q305

Find the zeroes of the following polynomials by factorisation method and verify the relations between the zeroes and the coefficients of the polynomials: $t^{3} - 2 t^{2} - 15 t$

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Q306

Find the zeroes of the following polynomials by factorisation method and verify the relations between the zeroes and the coefficients of the polynomials: $2 x^{2} + \frac{7}{2} x + \frac{3}{4}$

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Q307

Find the zeroes of the following polynomials by factorisation method and verify the relations between the zeroes and the coefficients of the polynomials: $4 x^{2} + 52 x - 3$

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Q308

Find the zeroes of the following polynomials by factorisation method and verify the relations between the zeroes and the coefficients of the polynomials: $2 s^{2} - (1 + 22) s + 2$

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Q309

Find the zeroes of the following polynomials by factorisation method and verify the relations between the zeroes and the coefficients of the polynomials: $v^{2} + 43 v - 15$

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Q310

Find the zeroes of the following polynomials by factorisation method and verify the relations between the zeroes and the coefficients of the polynomials: $y^{2} + \frac{3}{2} 5 y - 5$

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Q311

Find the zeroes of the following polynomials by factorisation method and verify the relations between the zeroes and the coefficients of the polynomials: $7 y^{2} - \frac{11}{3} y - \frac{2}{3}$

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Q312

If alpha, beta and gamma are the zeroes of a cubic polynomial $a x^{3} + b x^{2} + c x + d$ , then $α + β + γ = - \frac{b}{a}$ , $α β + β γ + γ α = \frac{c}{a}$ and $α β γ = \frac{- d}{a}$ . Verify these relations.

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Q313

State whether the following statement is true or false and justify: The division algorithm states that given any polynomial $p (x)$ and any non-zero polynomial $g (x)$ , there are polynomials $q (x)$ and $r (x)$ such that $p (x) = g (x) q (x) + r (x)$ , where $r (x) = 0$ or degree $r (x) <$ degree $g (x)$ .

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Q314
Graphically, the pair of equations $6 x - 3 y + 10 = 0$ and $2 x - y + 9 = 0$ represents two lines which are
1. (a)intersecting at exactly one point.
2. (b)intersecting at exactly two points.
3. (c)coincident.
4. (d)parallel.
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Q315
The pair of equations $x + 2 y + 5 = 0$ and $- 3 x - 6 y + 1 = 0$ have
1. (a)a unique solution
2. (b)exactly two solutions
3. (c)infinitely many solutions
4. (d)no solution
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Q316
If a pair of linear equations is consistent, then the lines will be
1. (a)parallel
2. (b)always coincident
3. (c)intersecting or coincident
4. (d)always intersecting
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Q317
The pair of equations $y = 0$ and $y = - 7$ has
1. (a)one solution
2. (b)two solutions
3. (c)infinitely many solutions
4. (d)no solution
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Q318
The pair of equations $x = a$ and $y = b$ graphically represents lines which are
1. (a)parallel
2. (b)intersecting at $(b, a)$
3. (c)coincident
4. (d)intersecting at $(a, b)$
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Q319
For what value of $k$ , do the equations $3 x - y + 8 = 0$ and $6 x - k y = - 16$ represent coincident lines?
1. (a) $\frac{1}{2}$
2. (b) $- \frac{1}{2}$
3. (c)2
4. (d) $- 2$
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Q320
If the lines given by $3 x + 2 k y = 2$ and $2 x + 5 y + 1 = 0$ are parallel, then the value of $k$ is
1. (a) $\frac{- 5}{4}$
2. (b) $\frac{2}{5}$
3. (c) $\frac{15}{4}$
4. (d) $\frac{3}{2}$
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