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Class 10

Explore popular questions from Quadratic Equations for Class 10. This collection covers Quadratic Equations previous year Class 10 questions hand picked by experienced teachers.

Quadratic Equations

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Q 1. Roots of the quadratic equation {tex} x ^ { 2 } - 5 x - 6 = 0 {/tex} are

A

equal but negative

B

unequal but of same signs

unequal but of opposite signs

D

equal but positive

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Q 2. The quadratic equation where one root is {tex} 3 + 2 \sqrt { 3 } {/tex} is

{tex} x ^ { 2 } - 6 x - 3 = 0 {/tex}

B

{tex} x ^ { 2 } + 6 x - 3 = 0 {/tex}

C

{tex} x ^ { 2 } + 6 x + 3 = 0 {/tex}

D

{tex} x ^ { 2 } - 6 x + 3 = 0 {/tex}

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Q 3. If one root of the equation {tex} a x ^ { 2 } + b x + c = 0 {/tex} is the reciprocal of other then

A

{tex} a = b {/tex}

B

{tex} b = c {/tex}

{tex} a = c {/tex}

D

{tex} a = - c {/tex}

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Q 4. Given that {tex} f ( x ) = 3 x ^ { 4 } - 5 x ^ { 3 } + 8 x ^ { 2 } - 6 x + 8 {/tex} and {tex} g ( x ) = x ^ { 2 } - 2 x + 2 . {/tex} Then how many real roots does the equation {tex} \frac { f ( x ) } { g ( x ) } = 0 {/tex} have?

0

B

2

C

3

D

4

Explanation

{tex} \frac { f ( x ) } { g ( x ) } {/tex} = {tex} \frac {3 x ^ { 4 } - 5 x ^ { 3 } + 8 x ^ { 2 } - 6 x + 8 } { x ^ { 2 } - 2 x + 2 } = 3x ^ 2 + x + 4 {/tex} {tex} D = b ^ 2 - 4ac = 1 - 4(3)(4) {/tex} {tex} D < 0 {/tex} So {tex} \frac { f ( x ) } { g ( x ) } = 0 {/tex} has no real roots

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Q 5. The number of real roots of equation
{tex} \left( a ^ { 2 } + b ^ { 2 } \right) x ^ { 2 } + 2 a ( \sqrt { b ^ { 2 } + c ^ { 2 } } ) x + a ^ { 2 } + c ^ { 2 } = 0 {/tex}
where, {tex} a , b {/tex} and {tex} c {/tex} are non zero is

0

B

1

C

2

D

4

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Q 6. If the roots of the equation {tex} x ^ { 2 } - 4 x + 1 = 0 {/tex} are in the ratio {tex} \mathrm { p }: \mathrm { q } {/tex} then the value of {tex} \sqrt { \frac { p } { q } } + \sqrt { \frac { q } { p } } {/tex} is

A

{tex}0{/tex}

{tex}4{/tex}

C

{tex} 2 \sqrt { 3 } {/tex}

D

Cannot be determined

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Q 7. How are the roots of the quadratic equations {tex} a x ^ { 2 } + b x + c = 0 {/tex} and {tex} c x ^ { 2 } + b x + a = 0 {/tex} are related?

A

No definite relation exist between the roots

B

The roots of second equation are the sum and the difference of the roots of the first equation

The roots of the one equation are the reciprocals of the roots of the other equation

D

The roots of the first and the second equations are equal in magnitude and opposite in sign

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Q 8. The condition that the equation {tex} x ^ { 2 } + { p } x + { q } = 0 {/tex} whose one root is the cube of the other root is

A

{tex} p = q ^ { 1 / 4 } \left[ 1 - q ^ { 1 / 2 } \right] {/tex}

B

{tex} - p = q ^ { 1 / 2 } \left[ 1 - q ^ { 1 / 4 } \right] {/tex}

{tex} - p = q ^ { 1 / 4 } \left[ 1 + q ^ { 1 / 2 } \right] {/tex}

D

{tex} p = q ^ { 1 / 2 } \left[ 1 + q ^ { 1 / 4 } \right] {/tex}

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Q 9. If the quadratic equation {tex} x ^ { 2 } + a x + b = 0 {/tex} and {tex} x ^ { 2 } + b x + a = 0 ( a \neq b ) {/tex} have a common root, then the numerical value of {tex} a + b {/tex} is

A

1

B

2

-1

D

None

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Q 10. Both the roots of the equation
{tex} ( x - b ) ( x - c ) + ( x - a ) ( x - c ) + ( x - a ) ( x - b ) {/tex} are always

positive

B

negative

C

real

D

None

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Q 11. The number of real solutions of the equation {tex} | \mathrm { x } | ^ { 2 } - 3 | \mathrm { x } | + 2 = 0 {/tex}

4

B

1

C

3

D

2

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Q 12. The equation has {tex} x - \frac { 2 } { x - 1 } = 1 - \frac { 2 } { x - 1 } {/tex}

A

no root

B

one root

two equal roots

D

Infinitely many solution

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Q 13. Let {tex} \alpha , \beta {/tex} be the roots of the equation {tex} ( x - a ) ( x - b ) = c\ c \neq 0 {/tex} Find the roots of the equation {tex} ( x - \alpha ) ( x - \beta ) + c = 0 {/tex} are

A

{tex}a{/tex} and {tex} c {/tex}

B

{tex}b{/tex} and {tex} c {/tex}

{tex}a{/tex} and {tex}b{/tex}

D

{tex} a + c , b + c {/tex}

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Q 14. How many solutions does the following equation have
{tex} \sqrt { x + 1 } - \sqrt { x - 1 } = \sqrt { 4 x - 1 } {/tex}

no solution

B

one solution

C

two solutions

D

None

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Q 15. The roots of the equation {tex} x ^ { 2 } - 2 a x + a ^ { 2 } + a - 3 - 0 {/tex} are real and less than {tex} 3 {/tex} then

{tex} a < 2 {/tex}

B

{tex} 2 \leq a \leq 3 {/tex}

C

{tex} 3 \leq a \leq 4 {/tex}

D

{tex} a > 4 {/tex}

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Q 16. Consider the following statements.
If the quadratic equations, {tex} x ^ { 2 } + a x + 2 = 0 {/tex} and {tex} x ^ { 2 } + x + b = 0 {/tex} have a common root {tex} x = 1 {/tex}, then

A

{tex} a + b = -5 {/tex}

B

{tex} a b = 6 {/tex}

C

{tex} \frac { a } { b } = \frac { 3 } { 2 } {/tex}

All of the above

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Q 17. Match the following (one to many)
Column-I and column-II contains four entries each. Entries of column-I are to be matched with some entries of column-II. One or more than one entries of column-I may have the matching with the same entries of column-II and one entry of column-II may have one or more than one matching with entries of column-I

A

{tex} ( \mathrm { A } ) - ( \mathrm { P } ) {/tex}

B

{tex} ( \mathrm { B } ) - ( \mathrm { R } \mathrm { } ) {/tex}

C

{tex} ( \mathrm { C } ) - ( \mathrm { S } \mathrm { } ) {/tex}

{tex} (\mathrm D ) - (\mathrm Q ) {/tex}

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Q 18. Which of the following is a quadratic equation?

A

{tex} x ^ { 2 } + 2 x + 1 = ( 4 - x ) ^ { 2 } + 3 {/tex}

B

{tex} - 2 x ^ { 2 } = ( 5 - x ) \left( 2 x - \frac { 2 } { 5 } \right) {/tex}

C

{tex} ( k + 1 ) x ^ { 2 } + \frac { 3 } { 2 } x = 7 , {/tex} where {tex} k = - 1 {/tex}

{tex} x ^ { 3 } - x ^ { 2 } = \left( x - 1 \right) ^ { 3 } {/tex}

Explanation

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Q 19. Which of the following is not a quadratic equation?

A

{tex} 2 ( x - 1 ) ^ { 2 } = 4 x ^ { 2 } - 2 x + 1 {/tex}

B

{tex} 2 x - x ^ { 2 } = x ^ { 2 } + 5 {/tex}

{tex} ( \sqrt { 2 } x + \sqrt { 3 } ) ^ { 2 } + x ^ { 2 } = 3 x ^ { 2 } - 5 x {/tex}

D

{tex} \left( x ^ { 2 } + 2 x \right) ^ { 2 } = x ^ { 4 } + 3 + 4 x ^ { 3 } {/tex}

Explanation

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Q 20. Which of the following equations has 2 as a root?

A

{tex} x ^ { 2 } - 4 x + 5 = 0 {/tex}

B

{tex} x ^ { 2 } + 3 x - 12 = 0 {/tex}

{tex} 2 x ^ { 2 } - 7 x + 6 = 0 {/tex}

D

{tex} 3 x ^ { 2 } - 6 x - 2 = 0 {/tex}

Explanation

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Q 21. If {tex} \frac { 1 } { 2 } {/tex} is a root of the equation {tex} x ^ { 2 } + k x - \frac { 5 } { 4 } = 0 {/tex}, then the value of {tex} k {/tex} is

{tex} 2 {/tex}

B

{tex} - 2 {/tex}

C

{tex} \frac { 1 } { 2 } {/tex}

D

{tex} \frac { 1 } { 2 } {/tex}

Explanation

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Q 22. Which of the following equations has the sum of its roots as {tex} 3 ? {/tex}

A

{tex} 2 x ^ { 2 } - 3 x + 6 = 0 {/tex}

{tex} - x ^ { 2 } + 3 x - 3 = 0 {/tex}

C

{tex} \sqrt { 2 } x ^ { 2 } - \frac { 3 } { \sqrt { 2 } } x + 1 = 0 {/tex}

D

{tex} 3 x ^ { 2 } - 3 x + 3 = 0 {/tex}

Explanation

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Q 23. Which of the following equations has two distinct real roots?

A

{tex} 2 x ^ { 2 } - 3 \sqrt { 2 } x + \frac { 9 } { 4 } = 0 {/tex}

{tex} x ^ { 2 } + x - 5 = 0 {/tex}

C

{tex} x ^ { 2 } + 3 x + 2 \sqrt { 2 } = 0 {/tex}

D

{tex} 5 x ^ { 2 } - 3 x + 1 = 0 {/tex}

Explanation

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Q 24. Values of {tex} k {/tex} for which the quadratic equation {tex} 2 x ^ { 2 } - k x + k = 0 {/tex} has equal roots is :

A

0

B

4

C

8

0 and 8

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Q 25. The quadratic equation {tex} 2 x ^ { 2 } - \sqrt { 5 } x + 1 = 0 {/tex} has

A

two distinct real roots

B

two equal real roots

no real roots

D

more than 2 real roots

Explanation