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#### Question 1:

Solving the following quadratic equations by factorization method:
(i) ${x}^{2}+10ix-21=0$
(ii)
(iii)
(iv) $6{x}^{2}-17ix-12=0$

#### Question 2:

(i)
(ii)
(iii)
(iv)
(v)
(vi) ${x}^{2}+4ix-4=0$
(vii) $2{x}^{2}+\sqrt{15}ix-i=0$
(viii) ${x}^{2}-x+\left(1+i\right)=0$
(ix) $i{x}^{2}-x+12i=0$
(x)
(xi)
(xii)

#### Question 1:

The complete set of values of k, for which the quadratic equation ${x}^{2}-kx+k+2=0$ has equal roots, consists of
(a) $2+\sqrt{12}$
(b) $2±\sqrt{12}$
(c) $2-\sqrt{12}$
(d) $-2-\sqrt{12}$

(b) $2±\sqrt{12}$

#### Question 2:

For the equation , the sum of the real roots is
(a) 1
(b) 0
(c) 2
(d) none of these

(b) 0

#### Question 3:

If a, b are the roots of the equation
(a) 1
(b) 2
(c) −1
(d) 3

(c) −1
Given equation:
Also, $a$ and $b$ are the roots of the given equation.
Sum of the roots =

Product of the roots =

#### Question 4:

If α, β are roots of the equation  is equal to
(a) 7/3
(b) −7/3
(c) 3/7
(d) −3/7

(d) −3/7

Given equation:
Also, $\alpha$ and $\beta$ are the roots of the equation.

Sum of the roots =

Product of the roots =

∴

#### Question 5:

The values of x satisfying log3 $\left({x}^{2}+4x+12\right)=2$ are
(a) 2, −4
(b) 1, −3
(c) −1, 3
(d) −1, −3

(d) −1, −3

The given equation is ${\mathrm{log}}_{3}\left({x}^{2}+4x+12\right)=2$.

#### Question 6:

The number of real roots of the equation $\left({x}^{2}+2x{\right)}^{2}-\left(x+1{\right)}^{2}-55=0$ is
(a) 2
(b) 1
(c) 4
(d) none of these

(a) 2

#### Question 7:

If α, β are the roots of the equation
(a) c / ab
(b) a / bc
(c) b / ac
(d) none of these.

(c) b / ac
Given equation:
Also, $\alpha$ and $\beta$ are the roots of the given equation.

Then, sum of the roots =
Product of the roots =

#### Question 8:

If α, β are the roots of the equation ${x}^{2}+px+1=0;\gamma ,\delta$ the roots of the equation
(a) ${q}^{2}-{p}^{2}$
(b) ${p}^{2}-{q}^{2}$
(c) ${p}^{2}+{q}^{2}$
(d) none of these.

(a) ${q}^{2}-{p}^{2}$
Given:  are the roots of the equation .
Also,   are the roots of the equation .
Then, the sum and the product of the roots of the given equation are as follows:

#### Question 9:

The number of real solutions of $\left|2x-{x}^{2}-3\right|=1$ is
(a) 0
(b) 2
(c) 3
(d) 4

(b) 2

Given equation:

(ii)

Hence, the real solutions are 2, 2.

#### Question 10:

The number of solutions of ${x}^{2}+\left|x-1\right|=1$ is
(a) 0
(b) 1
(c) 2
(d) 3

(c) 2

(i)

Since $-$2 does not satisfy the condition $x\ge 1$

(ii)

x = 1 does not satisfy the condition x < 1

So, there are two solutions.

#### Question 11:

If x is real and $k=\frac{{x}^{2}-x+1}{{x}^{2}+x+1}$, then
(a) k ∈ [1/3,3]
(b) k ≥ 3
(c) k ≤ 1/3
(d) none of these

(a) k ∈ [1/3,3]

$k=\frac{{x}^{2}-x+1}{{x}^{2}+x+1}\phantom{\rule{0ex}{0ex}}⇒k{x}^{2}+kx+k={x}^{2}-x+1\phantom{\rule{0ex}{0ex}}⇒\left(k-1\right){x}^{2}+\left(k+1\right)x+k-1=0$

For real values of x, the discriminant of $\left(k-1\right){x}^{2}+\left(k+1\right)x+k-1=0$ should be greater than or equal to zero.

And if k=1, then,
x=0, which is real       ...(ii)
So, from (i) and (ii), we get,
$k\in \left[\frac{1}{3},3\right]$

#### Question 12:

If the roots of ${x}^{2}-bx+c=0$ are two consecutive integers, then b2 − 4 c is
(a) 0
(b) 1
(c) 2
(d) none of these.

(b) 1

Given equation:
Let  be the two consecutive roots of the equation.

Sum of the roots =
Product of the roots =

#### Question 13:

The value of a such that  may have a common root is
(a) 0
(b) 12
(c) 24
(d) 32

(a) and (c)

Let $\mathrm{\alpha }$ be the common roots of the equations .

Therefore,

... (1)

... (2)

Solving (1) and (2) by cross multiplication, we get,

Disclaimer: The solution given in the book is incomplete. The solution is created according to the question given in the book and both the options are correct.

#### Question 14:

The values of k for which the quadratic equation $k{x}^{2}+1=kx+3x-11{x}^{2}$ has real and equal roots are
(a) −11, −3
(b) 5, 7
(c) 5, −7
(d) none of these

(c) 5, −7

The given equation is $k{x}^{2}+1=kx+3x-11{x}^{2}$ which can be written as.

For equal and real roots, the discriminant of $\left(k+11\right){x}^{2}-\left(k+3\right)x+1=0$.

Hence, the equation has real and equal roots when

#### Question 15:

If the equations  have a non-zero common roots, then λ =
(a) 1
(b) −1
(c) 3
(d) none of these.

(b) −1

Let $\mathrm{\alpha }$ be the common roots of the equations,   and

Therefore,

... (1)

... (2)

Solving (1) and (2) by cross multiplication, we get

#### Question 16:

If one root of the equation ${x}^{2}+px+12=0$ is 4, while the equation ${x}^{2}+px+q=0$ has equal roots, the value of q is
(a) 49/4
(b) 4/49
(c) 4
(d) none of these

(a) 49/4

It is given that, 4 is the root of the equation ${x}^{2}+px+12=0$.

$\therefore 16+4p+12=0\phantom{\rule{0ex}{0ex}}⇒p=-7$

It is also given that, the equation ${x}^{2}+px+q=0$ has equal roots. So, the discriminant of
${x}^{2}+px+q=0$ will be zero.

$\therefore {p}^{2}-4q=0\phantom{\rule{0ex}{0ex}}⇒4q={\left(-7\right)}^{2}=49\phantom{\rule{0ex}{0ex}}⇒q=\frac{49}{4}\phantom{\rule{0ex}{0ex}}$

#### Question 17:

The value of p and q (p ≠ 0, q ≠ 0) for which p, q are the roots of the equation ${x}^{2}+px+q=0$ are
(a) p = 1, q = −2
(b) p = −1, q = −2
(c) p = −1, q = 2
(d) p = 1, q = 2

(a) p = 1, q = −2

It is given that, p and q (p ≠ 0, q ≠ 0) are the roots of the equation ${x}^{2}+px+q=0$.

Now, substituting p = 1 in (1), we get,

$2+q=0\phantom{\rule{0ex}{0ex}}⇒q=-2$

Disclaimer: The solution given in the book is incorrect. The solution here is created according to the question given in the book.

#### Question 18:

The set of all values of m for which both the roots of the equation ${x}^{2}-\left(m+1\right)x+m+4=0$ are real and negative, is
(a) $\left(-\infty ,-3\right]\cup \left[5,\infty \right)$
(b) [−3, 5]
(c) (−4, −3]
(d) (−3, −1]

(c) $m\in \left(-4,-3\right]$

The roots of the quadratic equation ${x}^{2}-\left(m+1\right)x+m+4=0$ will be real, if its discriminant is greater than or equal to zero.

It is also given that, the roots of ${x}^{2}-\left(m+1\right)x+m+4=0$ are negative.
So, the sum of the roots will be negative.

$\therefore$ Sum of the roots < 0

and product of zeros >0

From (1), (2) and (3), we get,

$m\in \left(-4,-3\right]$

Disclaimer: The solution given in the book is incorrect. The solution here is created according to the question given in the book.

#### Question 19:

The number of roots of the equation $\frac{\left(x+2\right)\left(x-5\right)}{\left(x-3\right)\left(x+6\right)}=\frac{x-2}{x+4}$ is
(a) 0
(b) 1
(c) 2
(d) 3

(b) 1

Hence, the equation has only 1 root.

#### Question 20:

If α and β are the roots of $4{x}^{2}+3x+7=0$, then the value of $\frac{1}{\alpha }+\frac{1}{\beta }$ is
(a) $\frac{4}{7}$
(b) $-\frac{3}{7}$
(c) $\frac{3}{7}$
(d) $-\frac{3}{4}$

(b) −3/7

Given equation:
Also, $\alpha$ and $\beta$ are the roots of the equation.

Then, sum of the roots =

Product of the roots =

#### Question 21:

If α, β are the roots of the equation  are the roots of the equation
(a) ${x}^{2}-px+q=0$
(b) ${x}^{2}+px+q=0$
(c) $q{x}^{2}+px+1=0$
(d) $q{x}^{2}-px+1=0$

(d) $q{x}^{2}-px+1=0$
Given equation:
Also, $\alpha$ and $\beta$ are the roots of the given equation.
Then, sum of the roots =
Product of the roots =
Now, for roots , we have:
Sum of the roots =
Product of the roots =
Hence, the equation involving the roots $-\frac{1}{\alpha },-\frac{1}{\beta }$ is as follows:
${x}^{2}-\left(\alpha +\beta \right)x+\alpha \beta =0$

#### Question 22:

If the difference of the roots of ${x}^{2}-px+q=0$ is unity, then
(a) ${p}^{2}+4q=1$
(b) ${p}^{2}-4q=1$
(c) ${p}^{2}+4{q}^{2}=\left(1+2q{\right)}^{2}$
(d) $4{p}^{2}+{q}^{2}=\left(1+2p{\right)}^{2}$

(b) ${p}^{2}-4q=1$
Given equation:
Also  are the roots of the equation such that .
Sum of the roots =

Product of the roots =

#### Question 23:

If α, β are the roots of the equation
(a) c
(b) c − 1
(c) 1 − c
(d) none of these

(c) 1 − c
Given equation:
Also   are the roots of the equation.
Sum of the roots =
Product of the roots =

#### Question 24:

The least value of k which makes the roots of the equation ${x}^{2}+5x+k=0$ imaginary is
(a) 4
(b) 5
(c) 6
(d) 7

(d) 7

The roots of the quadratic equation ${x}^{2}+5x+k=0$ will be imaginary if its discriminant is less than zero.

$\therefore 25-4k<0\phantom{\rule{0ex}{0ex}}⇒k>\frac{25}{4}$

Thus, the minimum integral value of k for which the roots are imaginary is 7.

#### Question 25:

The equation of the smallest degree with real coefficients having 1 + i as one of the roots is
(a) ${x}^{2}+x+1=0$
(b) ${x}^{2}-2x+2=0$
(c) ${x}^{2}+2x+2=0$
(d) ${x}^{2}+2x-2=0$

(b) ${x}^{2}-2x+2=0$

We know that, imaginary roots of a quadratic equation occur in conjugate pair.

It is given that, 1 + i is one of the roots.

So, the other root will be $1-i$.

Thus, the quadratic equation having roots 1 + i and 1 - i is,

${x}^{2}-\left(1+i+1-i\right)x+\left(1+i\right)\left(1-i\right)=0\phantom{\rule{0ex}{0ex}}⇒{x}^{2}-2x+2=0$

#### Question 1:

If 1 – i is a root of the equation x2 + ax + b = 0, where a, bR, then the values of a and b are ____________.

#### Question 2:

If the difference of the roots of the equation x2Px + 8 = 0 is 2, then P =___________.

#### Question 3:

If the equation 2x2kx + x + 8 = 0 has real and equal roots, then k = __________.

#### Question 4:

The number of real roots of the equation x2 + 5 |x| + 4 = 0 is _________.

#### Question 5:

If one root of the equation x2 + px + 12 = 0 is 4, then the sum of the roots is ____________.

#### Question 6:

If α, β are roots of the equation x2 + x + 1 = 0, then the equation whose roots are α19 and β7 is ____________.

#### Question 7:

The value of is ____________.

#### Question 8:

If the equations px2 + 2qx + r = 0 and $q{x}^{2}-2\sqrt{pr}x+q=0$ have real roots, then q2 =____________.

#### Question 9:

If the roots of the equation x2 – 8x + a2 – 6a = 0 are real, then 'a' lies in the interval ____________. for a < –2, (a + 2) (a – 8) ≥ 0 and for a > 8, (a + 2) (a – 8) ≥ 0

#### Question 10:

If the equations x2 + x + a = 0 and x2 + ax + 1 = 0, a ≠ 1, have a common root, then a = ____________.

#### Question 11:

If the quadratic equation 2x2 – (a3 + 8a – 1) x + a2 – 4a = 0 possesses roots of opposite signs, then a lies in the interval ____________. Since a(a – 4) < 0 is true for 0 < a < 4
Hence a ∈ (0, 4)

#### Question 1:

Write the number of real roots of the equation $\left(x-1{\right)}^{2}+\left(x-2{\right)}^{2}+\left(x-3{\right)}^{2}=0$.

Comparing the given equation with the general form of the quadratic equation , we get  and

#### Question 2:

If a and b are roots of the equation ${x}^{2}-px+q=0$, than write the value of $\frac{1}{a}+\frac{1}{b}$.

Given:
Also, $a$ and  $b$ are the roots of the given equation.
Sum of the roots =         ...(1)
Product of the roots =        ...(2)

Now,          [Using equation (1) and (2)]

Hence, the value of  is $\frac{p}{q}.$

#### Question 3:

If roots α, β of the equation ${x}^{2}-px+16=0$ satisfy the relation α2 + β2 = 9, then write the value p.

Given equation:
Also,  $\alpha$ and $\beta$ are the roots of the equation satisfying
From the equation, we have:
Sum of the roots =

Product of the roots = $\alpha \beta$ =

Hence, the value of

#### Question 4:

If $2+\sqrt{3}$ is root of the equation ${x}^{2}+px+q=0$, than write the values of p and q.

Irrational roots always occur in conjugate pairs.
If

#### Question 5:

If the difference between the roots of the equation ${x}^{2}+ax+8=0$ is 2, write the values of a.

Given:
Let  are the roots of the equation.
Sum of the roots = .

Product of the roots =
Given:

∴

#### Question 6:

Write roots of the equation $\left(a-b\right){x}^{2}+\left(b-c\right)x+\left(c-a\right)=0$.

Now,
$\mathrm{\alpha }+\mathrm{\beta }=-\frac{b-c}{a-b}\phantom{\rule{0ex}{0ex}}⇒1+\mathrm{\beta }=-\frac{b-c}{a-b}\phantom{\rule{0ex}{0ex}}⇒\mathrm{\beta }=-\frac{b-c}{a-b}-1=\frac{c-a}{a-b}$

#### Question 7:

If a and b are roots of the equation ${x}^{2}-x+1=0$, then write the value of a2 + b2.

Given:
Also,  are the roots of the equation.
Then, sum of the roots =

Product of the roots =

#### Question 8:

Write the number of quadratic equations, with real roots, which do not change by squaring their roots.

Let  be the real roots of the quadratic equation

On squaring these roots, we get:

and
and  $\mathrm{\beta }\left(1-\mathrm{\beta }\right)=0$
and

Three cases arise:

So, the corresponding quadratic equation is,

So, the corresponding quadratic equation is,

So, the corresponding quadratic equation is,

Hence, we can construct 3 quadratic equations.

#### Question 9:

If α, β are roots of the equation ${x}^{2}+lx+m=0$, write an equation whose roots are $-\frac{1}{\alpha }\mathrm{and}-\frac{1}{\beta }$.

Given equation:
Also,  are the roots of the equation.
Sum of the roots =

Product of the roots =
Now, sum of the roots =
Product of the roots =

Hence, this is the equation whose roots are

#### Question 10:

If α, β are roots of the equation ${x}^{2}-a\left(x+1\right)-c=0$, then write the value of (1 + α) (1 + β).

Given:
Also, $\alpha$ and $\beta$ are the roots of the equation.
Sum of the roots =

Product of the roots =

#### Question 1:

x2 + 1 = 0

Given:

or
or
Hence, the roots of the equation are .

#### Question 2:

9x2 + 4 = 0

Given:

or,
or
or
Hence, the roots of the equation are

#### Question 3:

x2 + 2x + 5 = 0

Given:

or,
or,
Hence, the roots of the equation are .

#### Question 4:

4x2 − 12x + 25 = 0

We have:

or,
or,
or,
Hence, the roots of the equation are .

#### Question 5:

x2 + x + 1 = 0

We have:

or,

$⇒$   or,

Hence, the roots of the equation are .

#### Question 6:

$4{x}^{2}+1=0$

We have:

or
or
$⇒$     or
Hence, the roots of the equation are .

#### Question 7:

${x}^{2}-4x+7=0$

We have:

or,
or,

Hence, the roots of the equation are .

#### Question 8:

${x}^{2}+2x+5=0$

We have:

or
or

Hence, the roots of the equation are .

#### Question 9:

$5{x}^{2}-6x+2=0$

Given:
Comparing the given equation with general form of the quadratic equation , we get  and .

Substituting these values in and , we get:

and

and

and

and

and

and

Hence, the roots of the equation are

#### Question 10:

$21{x}^{2}+9x+1=0$

Given:
Comparing the given equation with  the general form of the quadratic equation , we get and .

Substituting these values in   and , we get:

and

and

and

and

Hence, the roots of the equation are

#### Question 11:

${x}^{2}-x+1=0$

We have:

or

or
Hence, the roots of the equation are

#### Question 12:

${x}^{2}+x+1=0$

We have:

or

or

Hence, the roots of the equation are

#### Question 13:

$17{x}^{2}-8x+1=0$

Given:
Comparing the given equation with the general form of the quadratic equation , we get   and .

Substituting these values in   and  , we get:

and

and

and

and

and

and

and

Hence, the roots of the equation are .

#### Question 14:

$27{x}^{2}-10+1=0$

Given:
Comparing the given equation with the general form of the quadratic equation , we get  and  .

Substituting these values in and , we get:

and

and

and

and

and

and

and

Hence, the roots of the equation are .

#### Question 15:

$17{x}^{2}+28x+12=0$

Given:
Comparing the given equation with the general form of the quadratic equation , we get   and .

Substituting these values in  and , we get:

and

and

and

and

and

and

Hence, the roots of the equation are

#### Question 16:

$21{x}^{2}-28x+10=0$

Given:
Comparing the given equation with the general form of the quadratic equation , we get   and   .

Substituting these values in and  , we get:

and

and

and

and

and

Hence, the roots of the equation are .

#### Question 17:

$8{x}^{2}-9x+3=0$

Given:
Comparing the given equation with the general form of the quadratic equation , we get   and  .

Substituting these values in    and , we get:

and

and

and

and

and

and

Hence, the roots of the equation are .

#### Question 18:

$13{x}^{2}+7x+1=0$

Given:
Comparing the given equation with the general form of the quadratic equation , we get    and .

Substituting these values in    and  , we get:

and

and

and

and

and

Hence, the roots of the equation are

#### Question 19:

$2{x}^{2}+x+1=0$

Given:
Comparing the given equation with the general form of the quadratic equation , we get  and  .

Substituting these values in  and  , we get:

and

and

and

and

Hence, the roots of the equation are .

#### Question 20:

$\sqrt{3}{x}^{2}-\sqrt{2}x+3\sqrt{3}=0$

Given:
Comparing the given equation with the general form of the quadratic equation , we get    and   .

Substituting these values in  and  , we get:

and

and

and

Hence, the roots of the equation are .

#### Question 21:

$\sqrt{2}{x}^{2}+x+\sqrt{2}=0$

Given:
Comparing the given equation with the general form of the quadratic equation , we get  and .

Substituting these values in and  , we get:

and

and

and

Hence, the roots of the equation are .

#### Question 22:

${x}^{2}+x+\frac{1}{\sqrt{2}}=0$

Given equation:
Comparing the given equation with the general form of the quadratic equation , we get  and .

Substituting these values in and , we get:

and

and

and

Hence, the roots of the equation are .

#### Question 23:

${x}^{2}+\frac{x}{\sqrt{2}}+1=0$

Given equation:
Comparing the given equation with  the general form of the quadratic equation , we get  and .

Substituting these values in    and  , we get:

and

and

and

and

Hence, the roots of the equation are .

#### Question 24:

$\sqrt{5}{x}^{2}+x+\sqrt{5}=0$

Given:
Comparing the given equation with the general form of the quadratic equation , we get  and  .

Substituting these values in   and  , we get:

and

and

and

Hence, the roots of the equation are .

#### Question 25:

$-{x}^{2}+x-2=0$

or,

or,

Hence, the roots of the equation are .

#### Question 26:

${x}^{2}-2x+\frac{3}{2}=0$

or,

or,

Hence, the roots of the equation are  .

#### Question 27:

$3{x}^{2}-4x+\frac{20}{3}=0$

Given:
Comparing the given equation with the general form of the quadratic equation , we get  and .

Substituting these values in and  , we get:

and

and

and

and

Hence, the roots of the equation are  .

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