#### Chapters

Chapter 2: Relations

Chapter 3: Functions

Chapter 4: Measurement of Angles

Chapter 5: Trigonometric Functions

Chapter 6: Graphs of Trigonometric Functions

Chapter 7: Values of Trigonometric function at sum or difference of angles

Chapter 8: Transformation formulae

Chapter 9: Values of Trigonometric function at multiples and submultiples of an angle

Chapter 10: Sine and cosine formulae and their applications

Chapter 11: Trigonometric equations

Chapter 12: Mathematical Induction

Chapter 13: Complex Numbers

Chapter 14: Quadratic Equations

Chapter 15: Linear Inequations

Chapter 16: Permutations

Chapter 17: Combinations

Chapter 18: Binomial Theorem

Chapter 19: Arithmetic Progression

Chapter 20: Geometric Progression

Chapter 21: Some special series

Chapter 22: Brief review of cartesian system of rectangular co-ordinates

Chapter 23: The straight lines

Chapter 24: The circle

Chapter 25: Parabola

Chapter 26: Ellipse

Chapter 27: Hyperbola

Chapter 28: Introduction to three dimensional coordinate geometry

Chapter 29: Limits

Chapter 30: Derivatives

Chapter 31: Mathematical reasoning

Chapter 32: Statistics

Chapter 33: Probability

#### RD Sharma Mathematics Class 11

## Chapter 14 : Quadratic Equations

#### Pages 5 - 6

*x*^{2} + 1 = 0

9*x*^{2} + 4 = 0

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

4*x*^{2} − 12*x* + 25 = 0

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

\[4 x^2 + 1 = 0\]

\[x^2 - 4x + 7 = 0\]

\[x^2 + 2x + 5 = 0\]

\[5 x^2 - 6x + 2 = 0\]

\[21 x^2 + 9x + 1 = 0\]

\[x^2 - x + 1 = 0\]

\[x^2 + x + 1 = 0\]

\[17 x^2 - 8x + 1 = 0\]

\[27 x^2 - 10 + 1 = 0\]

\[17 x^2 + 28x + 12 = 0\]

\[21 x^2 - 28x + 10 = 0\]

\[8 x^2 - 9x + 3 = 0\]

\[13 x^2 + 7x + 1 = 0\]

\[2 x^2 + x + 1 = 0\]

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

\[\sqrt{2} x^2 + x + \sqrt{2} = 0\]

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

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

\[\sqrt{5} x^2 + x + \sqrt{5} = 0\]

\[- x^2 + x - 2 = 0\]

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

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

#### Page 13

Solving the following quadratic equation by factorization method:

\[x^2 + 10ix - 21 = 0\]

Solving the following quadratic equation by factorization method:

\[x^2 + \left( 1 - 2i \right) x - 2i = 0\]

Solving the following quadratic equation by factorization method:

\[x^2 - \left( 2\sqrt{3} + 3i \right) x + 6\sqrt{3}i = 0\]

Solving the following quadratic equation by factorization method:

\[6 x^2 - 17ix - 12 = 0\]

Solve the following quadratic equation:

\[x^2 - \left( 3\sqrt{2} + 2i \right) x + 6\sqrt{2i} = 0\]

Solve the following quadratic equation:

\[x^2 - \left( 5 - i \right) x + \left( 18 + i \right) = 0\]

Solve the following quadratic equation:

\[\left( 2 + i \right) x^2 - \left( 5 - i \right) x + 2 \left( 1 - i \right) = 0\]

Solve the following quadratic equation:

\[x^2 - \left( 2 + i \right) x - \left( 1 - 7i \right) = 0\]

Solve the following quadratic equation:

\[i x^2 - 4 x - 4i = 0\]

Solve the following quadratic equation:

\[x^2 + 4ix - 4 = 0\]

Solve the following quadratic equation:

\[2 x^2 + \sqrt{15}ix - i = 0\]

Solve the following quadratic equation:

\[x^2 - x + \left( 1 + i \right) = 0\]

Solve the following quadratic equation:

\[i x^2 - x + 12i = 0\]

Solve the following quadratic equation:

\[x^2 - \left( 3\sqrt{2} - 2i \right) x - \sqrt{2} i = 0\]

Solve the following quadratic equation:

\[x^2 - \left( \sqrt{2} + i \right) x + \sqrt{2}i = 0\]

Solve the following quadratic equation:

\[2 x^2 - \left( 3 + 7i \right) x + \left( 9i - 3 \right) = 0\]

#### Pages 15 - 16

Write the number of real roots of the equation \[(x - 1 )^2 + (x - 2 )^2 + (x - 3 )^2 = 0\].

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

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

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

If the difference between the roots of the equation \[x^2 + ax + 8 = 0\] is 2, write the values of a.

Write roots of the equation \[(a - b) x^2 + (b - c)x + (c - a) = 0\] .

If *a* and *b* are roots of the equation \[x^2 - x + 1 = 0\], then write the value of a^{2} + b^{2}.

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

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

If α, β are roots of the equation \[x^2 - a(x + 1) - c = 0\] then write the value of (1 + α) (1 + β).

#### Pages 16 - 18

The complete set of values of *k*, for which the quadratic equation \[x^2 - kx + k + 2 = 0\] has equal roots, consists of

\[2 + \sqrt{12}\]

\[2 \pm \sqrt{12}\]

\[2 - \sqrt{12}\]

\[- 2 - \sqrt{12}\]

For the equation \[\left| x \right|^2 + \left| x \right| - 6 = 0\] ,the sum of the real roots is

1

0

2

none of these

If a, b are the roots of the equation \[x^2 + x + 1 = 0, \text { then } a^2 + b^2 =\]

1

2

-1

3

If α, β are roots of the equation \[4 x^2 + 3x + 7 = 0, \text { then } 1/\alpha + 1/\beta\] is equal to

7/3

−7/3

3/7

-3/7

The number of real roots of the equation \[( x^2 + 2x )^2 - (x + 1 )^2 - 55 = 0\] is

2

1

4

none of these

If α, β are the roots of the equation \[a x^2 + bx + c = 0, \text { then } \frac{1}{a\alpha + b} + \frac{1}{a\beta + b} =\]

c / ab

a / bc

b / ac

none of these.

If α, β are the roots of the equation \[x^2 + px + 1 = 0; \gamma, \delta\] the roots of the equation \[x^2 + qx + 1 = 0, \text { then } (\alpha - \gamma)(\alpha + \delta)(\beta - \gamma)(\beta + \delta) =\]

\[q^2 - p^2\]

\[p^2 - q^2\]

\[p^2 + q^2\]

none of these.

If *x* is real and \[k = \frac{x^2 - x + 1}{x^2 + x + 1}\], then

*k* ∈ [1/3,3]

*k* ≥ 3

*k* ≤ 1/3

none of these

If the roots of \[x^2 - bx + c = 0\] are two consecutive integers, then *b*^{2} − 4 *c* is

0

1

2

none of these.

The value of a such that \[x^2 - 11x + a = 0 \text { and } x^2 - 14x + 2a = 0\] may have a common root is

0

12

24

32

The values of *k* for which the quadratic equation \[k x^2 + 1 = kx + 3x - 11 x^2\] has real and equal roots are

−11, −3

5, 7

5, −7

none of these

If the equations \[x^2 + 2x + 3\lambda = 0 \text { and } 2 x^2 + 3x + 5\lambda = 0\] have a non-zero common roots, then λ =

1

-1

3

none of these.

If one root of the equation \[x^2 + px + 12 = 0\] while the equation \[x^2 + px + q = 0\] has equal roots, the value of q is

49/4

4/49

4

none of these

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

p = 1, q = −2

p = −1, q = −2

p = −1, q = 2

p = 1, q = 2

The set of all values of *m* for which both the roots of the equation \[x^2 - (m + 1)x + m + 4 = 0\] are real and negative, is

\[( - \infty , - 3] \cup [5, \infty )\]

[−3, 5]

(−4, −3]

(−3, −1]

The number of roots of the equation \[\frac{(x + 2)(x - 5)}{(x - 3)(x + 6)} = \frac{x - 2}{x + 4}\] is

0

1

2

3

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

\[\frac{4}{7}\]

\[- \frac{3}{7}\]

\[\frac{3}{7}\]

\[- \frac{3}{4}\]

If α, β are the roots of the equation \[x^2 + px + q = 0 \text { then } - \frac{1}{\alpha} + \frac{1}{\beta}\] are the roots of the equation

\[x^2 - px + q = 0\]

\[x^2 + px + q = 0\]

\[q x^2 + px + 1 = 0\]

\[q x^2 - px + 1 = 0\]

If the difference of the roots of \[x^2 - px + q = 0\] is unity, then

\[p^2 + 4q = 1\]

\[p^2 - 4q = 1\]

\[p^2 + 4 q^2 = (1 + 2q )^2\]

\[4 p^2 + q^2 = (1 + 2p )^2\]

If α, β are the roots of the equation \[x^2 - p(x + 1) - c = 0, \text { then } (\alpha + 1)(\beta + 1) =\]

c

*c* − 1

1 − *c*

none of these

The least value of *k *which makes the roots of the equation \[x^2 + 5x + k = 0\] imaginary is

4

5

6

7

The equation of the smallest degree with real coefficients having 1 + *i* as one of the roots is

\[x^2 + x + 1 = 0\]

\[x^2 - 2x + 2 = 0\]

\[x^2 + 2x + 2 = 0\]

\[x^2 + 2x - 2 = 0\]

#### RD Sharma Mathematics Class 11

#### Textbook solutions for Class 11

## RD Sharma solutions for Class 11 Mathematics chapter 14 - Quadratic Equations

RD Sharma solutions for Class 11 Maths chapter 14 (Quadratic Equations) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the CBSE Mathematics Class 11 solutions in a manner that help students grasp basic concepts better and faster.

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Concepts covered in Class 11 Mathematics chapter 14 Quadratic Equations are The Modulus and the Conjugate of a Complex Number, Algebra of Complex Numbers, Complex Numbers, Square Root of a Complex Number, Need for Complex Numbers, Algebraic Properties of Complex Numbers, Algebra of Complex Numbers - Equality, Quadratic Equations, Argand Plane and Polar Representation.

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