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QUESTION: 1

A quadratic equation whose one root is 3 is

Solution:

x = 3 satisfies only the equation x^{2}−5x+6 = 0

As (3)^{2}−5×3+6 = 0

⇒ 9−15+6 = 0

⇒ 15−15 = 0

⇒ 0 = 0

⇒ L.H.S. = R.H.S.

QUESTION: 2

The common root of 2x^{2}+x−6 = 0 and x^{2}−3x−10 = 0 is

Solution:

Given: p (x) = 2x^{2} + x - 6 = 0 and q (x) = x^{2} - 3x — 10 = 0

∴ p (-2) = 2(-2)^{2} + (-2) - 6 = 0 = 8 - 2 - 6 = 8 - 8 = 0

∴ q (-2) = (-2)^{2} - 3 (-2) - 10 = 0 = 4 + 6 - 10 = 10 - 10 = 0

Since p (-2) = 0 and q (-2) = 0 therefore, - 2 is the common root of 2x^{2} + x - 6 = 0 and x^{2} - 3x - 10 = 0

QUESTION: 3

In a cricket matchKumble took three wickets less than twice the number of wickets taken by Srinath. The product of the number of wickets taken by these two is 20, then the number of wickets taken by Kumble is

Solution:

Let the number of wickets taken by Srinath be x then the number of wickets taken by Kumble will be 2x−3

According to question, x (2x−3) = 20

⇒ 2x^{2}−3x−20 = 0

⇒ 2x^{2}−8x+5x−20 = 0

⇒ 2x(x−4)+5(x−4) = 0

⇒ (x−4)(2x+5) = 0

⇒ x−4 = 0and 2x+5 = 0

Therefore, number of wickets taken by Srinath is 4.

Then number of wickets taken by Kumble = 2 x 4 - 3 = 5

QUESTION: 4

(x^{2}+1)^{2 }− x^{2} = 0 has

Solution:

Given: (x^{2}+1)^{2 }− x^{2 }= 0

⇒ x^{2 }+ 1 + 2x − x^{2 }= 0

⇒ 2x + 1 = 0

⇒ x = −1/2

Therefore, (x^{2}+1)^{2 }− x^{2} = 0 has no real roots.

QUESTION: 5

If the sum and product of the roots of the equation kx^{2}+6x+4k = 0 are equal, then k =

Solution:

QUESTION: 6

One of the roots of the quadratic equation a^{2}x^{2}−3abx+2b^{2} = 0 is

Solution:

Explanation here:a^{2}x^{2}−3abx+2b^{2 }= 0

⇒ a^{2}x^{2}−2abx−abx+2b^{2 }= 0

⇒ ax(ax−2b)−b(ax−2b) = 0

⇒ (ax−b)(ax−2b) = 0

⇒ x−b = 0and ax−2b = 0

QUESTION: 7

The product of two successive integral multiples of 5 is 1050. Then the numbers are

Solution:

Let one multiple of 5 be x then the next consecutive multiple of will be (x+5) According to question,

x = −35 is not possible therefore x = 30

Then the other multiple of 5 is x+5 = 30+5 = 35

Then the number are 30 and 35.

QUESTION: 8

The angry Arjun carried some arrows for fighting with Bheeshma. With half the arrows, he cut down the arrows thrown by Bheeshma on him and with six other arrows he killed the rath driver of Bheeshma. With one arrow each he knocked down respectively the rath, flag and bow of Bheeshma. Finally with one more than four times the square root of arrows he laid Bheeshma unconscious on an arrow bed. The total number of arrows that Arjun had is

Solution:

QUESTION: 9

Which of the following has no real root?

Solution:

Since b^{2} - 4ac < 0 therefore, x^{2} - 4x + 3√2 = 0 has no real roots

QUESTION: 10

If ‘sin α’ and ‘cos α’ are the roots of the equation ax^{2}+bx+c = 0 a then b^{2} =

Solution:

QUESTION: 11

The quadratic equation whose roots are 7+√3 and 7−√3 is

Solution:

QUESTION: 12

The roots of the quadratic equation 9a^{2}b^{2}x^{2}−16abcdx−25c^{2}d^{2} = 0 are

Solution:

QUESTION: 13

The constant that must be added and subtracted to solve the quadratic equation by the method of completing the square is

Solution:

QUESTION: 14

Which of the following has two distinct roots?

Solution:

In equation x^{2}+x−5 = 0

a = 1,b = 1,c = −5

∴ b^{2}−4ac = (1)^{2}−4×1×(−5) = 1 + 20 = 21

Since b^{2}−4ac > 0 therefore, x^{2}+x−5 = 0 has two distict roots.

QUESTION: 15

If ‘a’ and ‘b’ are the roots of the equation x^{2}+ax+b = 0, then a+b =

Solution:

Since sum of the roots = -b/a ∴ a+ b = -a/1 = -a

QUESTION: 16

If x = 2 is a root of the quadratic equation 3x^{2} – px – 2 = 0, then the value of ‘p’ is

Solution:

Given: p(x) = 3x^{2}−px−2 = 0

∴ p(2) = 3(2)^{2}−p(2)−2 = 0

⇒ 12−2p−2 = 0

⇒ −2p = −10

⇒ p = 5

QUESTION: 17

The sum of reciprocals of Sharma’s age 3 years ago and 5 years from now is 1/3, then his present age is

Solution:

Let Sharma’s present age be x years then his age 3 years ago is (x−3) years and 5 years from now is (x+5) years. According to question,

But x = 3 does not satisfy the given condition.

Therefore, Sharma's present age is 7 years.

QUESTION: 18

If ax^{2}+bx+c = 0 has equal roots, then c is equal to

Solution:

If ax^{2}+bx+c = 0 has equal roots, then

b^{2}−4ac = 0

⇒ 4ac = b^{2}

⇒ c =

QUESTION: 19

If the quadratic equation kx(x – 2) + 6 = 0 has equal roots, then the value of ‘k’ is

Solution:

If the quadratic equation 6x2 - 2kx + 6 = 0 has equal roots, then

QUESTION: 20

If one root of the equation ax^{2}+bx+c = 0 is three times the other, then b^{2} : ac =

Solution:

Let one root be α then other root will be 3α.

∴ Sum of the roots

And Product of the roots

Equating the coefficients of α^{2} and subtracing e.q. (ii) from e.q. (i), we get

QUESTION: 21

5x^{2}+8x+4 = 2x^{2}+4x+6 is a

Solution:

Given: 5x^{2}+8x+4 = 2x^{2}+4x+6

⇒ 5x^{2}−2x^{2}+8x−4x+4−6 = 0

⇒ 3x^{2}+4x−2 = 0

Here, the degree is 2, therefore it is a quadratic equation.

QUESTION: 22

A train travels 360km at a uniform speed. If the speed had been 5 km/hr more, it would have taken 1 hour less for the same journey, then the actual speed of the train is

Solution:

QUESTION: 23

Let b = a + c. Then the equation ax^{2}+bx+c = 0 has equal roots if

Solution:

QUESTION: 24

The discriminant of the quadratic equation 3√3x^{2}+10x+√3 = 0 is

Solution:

Given: 3√3x^{2}+10x+√3 = 0 Discriminant = b^{2}−4ac = (10)^{2}−4×3√3×√3 = 100−36 = 64

QUESTION: 25

If one root of the equation 4x^{2}−2x+(λ−4) = 0 be the reciprocal of the other, then the value of is

Solution:

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