Rd Sharma Xi 2020 _volume 1 Solutions for Class 12 Science Math Chapter 14 Quadratic Equations are provided here with simple step-by-step explanations. These solutions for Quadratic Equations are extremely popular among Class 12 Science students for Math Quadratic Equations Solutions come handy for quickly completing your homework and preparing for exams. All questions and answers from the Rd Sharma Xi 2020 _volume 1 Book of Class 12 Science Math Chapter 14 are provided here for you for free. You will also love the ad-free experience on Meritnation’s Rd Sharma Xi 2020 _volume 1 Solutions. All Rd Sharma Xi 2020 _volume 1 Solutions for class Class 12 Science Math are prepared by experts and are 100% accurate.
Page No 14.13:
Question 1:
Solving the following quadratic equations by factorization method:
(i)
(ii)
(iii)
(iv)
Answer:
Page No 14.13:
Question 2:
Solve the following quadratic equations:
(i)
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
(viii)
(ix)
(x)
(xi)
(xii)
Answer:
Page No 14.15:
Question 1:
The complete set of values of k, for which the quadratic equation has equal roots, consists of
(a)
(b)
(c)
(d)
Answer:
(b)
Page No 14.15:
Question 2:
For the equation , the sum of the real roots is
(a) 1
(b) 0
(c) 2
(d) none of these
Answer:
(b) 0
Page No 14.16:
Question 3:
If a, b are the roots of the equation
(a) 1
(b) 2
(c) −1
(d) 3
Answer:
(c) −1
Given equation:
Also, and are the roots of the given equation.
Sum of the roots =
Product of the roots =
Page No 14.16:
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
Answer:
(d) −3/7
Given equation:
Also, and are the roots of the equation.
Sum of the roots =
Product of the roots =
∴
Page No 14.16:
Question 5:
The values of x satisfying log3 are
(a) 2, −4
(b) 1, −3
(c) −1, 3
(d) −1, −3
Answer:
(d) −1, −3
The given equation is .
Page No 14.16:
Question 6:
The number of real roots of the equation is
(a) 2
(b) 1
(c) 4
(d) none of these
Answer:
(a) 2
Page No 14.16:
Question 7:
If α, β are the roots of the equation
(a) c / ab
(b) a / bc
(c) b / ac
(d) none of these.
Answer:
(c) b / ac
Given equation:
Also, and are the roots of the given equation.
Then, sum of the roots =
Product of the roots =
Page No 14.16:
Question 8:
If α, β are the roots of the equation the roots of the equation
(a)
(b)
(c)
(d) none of these.
Answer:
(a)
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:
Page No 14.16:
Question 9:
The number of real solutions of is
(a) 0
(b) 2
(c) 3
(d) 4
Answer:
(b) 2
Given equation:
(ii)
Hence, the real solutions are 2, 2.
Page No 14.16:
Question 10:
The number of solutions of is
(a) 0
(b) 1
(c) 2
(d) 3
Answer:
(c) 2
(i)
Since 2 does not satisfy the condition
(ii)
x = 1 does not satisfy the condition x < 1
So, there are two solutions.
Page No 14.16:
Question 11:
If x is real and , then
(a) k ∈ [1/3,3]
(b) k ≥ 3
(c) k ≤ 1/3
(d) none of these
Answer:
(a) k ∈ [1/3,3]
For real values of x, the discriminant of 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,
Page No 14.16:
Question 12:
If the roots of are two consecutive integers, then b2 − 4 c is
(a) 0
(b) 1
(c) 2
(d) none of these.
Answer:
(b) 1
Given equation:
Let be the two consecutive roots of the equation.
Sum of the roots =
Product of the roots =
Page No 14.16:
Question 13:
The value of a such that may have a common root is
(a) 0
(b) 12
(c) 24
(d) 32
Answer:
(a) and (c)
Let 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.
Page No 14.16:
Question 14:
The values of k for which the quadratic equation has real and equal roots are
(a) −11, −3
(b) 5, 7
(c) 5, −7
(d) none of these
Answer:
(c) 5, −7
The given equation is which can be written as.
For equal and real roots, the discriminant of .
Hence, the equation has real and equal roots when
Page No 14.16:
Question 15:
If the equations have a non-zero common roots, then λ =
(a) 1
(b) −1
(c) 3
(d) none of these.
Answer:
(b) −1
Let be the common roots of the equations, and
Therefore,
... (1)
... (2)
Solving (1) and (2) by cross multiplication, we get
Page No 14.16:
Question 16:
If one root of the equation is 4, while the equation has equal roots, the value of q is
(a) 49/4
(b) 4/49
(c) 4
(d) none of these
Answer:
(a) 49/4
It is given that, 4 is the root of the equation .
It is also given that, the equation has equal roots. So, the discriminant of
will be zero.
Page No 14.16:
Question 17:
The value of p and q (p ≠ 0, q ≠ 0) for which p, q are the roots of the equation are
(a) p = 1, q = −2
(b) p = −1, q = −2
(c) p = −1, q = 2
(d) p = 1, q = 2
Answer:
(a) p = 1, q = −2
It is given that, p and q (p ≠ 0, q ≠ 0) are the roots of the equation .
Now, substituting p = 1 in (1), we get,
Disclaimer: The solution given in the book is incorrect. The solution here is created according to the question given in the book.
Page No 14.17:
Question 18:
The set of all values of m for which both the roots of the equation are real and negative, is
(a)
(b) [−3, 5]
(c) (−4, −3]
(d) (−3, −1]
Answer:
(c)
The roots of the quadratic equation will be real, if its discriminant is greater than or equal to zero.
It is also given that, the roots of are negative.
So, the sum of the roots will be negative.
Sum of the roots < 0
and product of zeros >0
From (1), (2) and (3), we get,
Disclaimer: The solution given in the book is incorrect. The solution here is created according to the question given in the book.
Page No 14.17:
Question 19:
The number of roots of the equation is
(a) 0
(b) 1
(c) 2
(d) 3
Answer:
(b) 1
Hence, the equation has only 1 root.
Page No 14.17:
Question 20:
If α and β are the roots of , then the value of is
(a)
(b)
(c)
(d)
Answer:
(b) −3/7
Given equation:
Also, and are the roots of the equation.
Then, sum of the roots =
Product of the roots =
Page No 14.17:
Question 21:
If α, β are the roots of the equation are the roots of the equation
(a)
(b)
(c)
(d)
Answer:
(d)
Given equation:
Also, and 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 is as follows:
Page No 14.17:
Question 22:
If the difference of the roots of is unity, then
(a)
(b)
(c)
(d)
Answer:
(b)
Given equation:
Also are the roots of the equation such that .
Sum of the roots =
Product of the roots =
Page No 14.17:
Question 23:
If α, β are the roots of the equation
(a) c
(b) c − 1
(c) 1 − c
(d) none of these
Answer:
(c) 1 − c
Given equation:
Also are the roots of the equation.
Sum of the roots =
Product of the roots =
Page No 14.17:
Question 24:
The least value of k which makes the roots of the equation imaginary is
(a) 4
(b) 5
(c) 6
(d) 7
Answer:
(d) 7
The roots of the quadratic equation will be imaginary if its discriminant is less than zero.
Thus, the minimum integral value of k for which the roots are imaginary is 7.
Page No 14.17:
Question 25:
The equation of the smallest degree with real coefficients having 1 + i as one of the roots is
(a)
(b)
(c)
(d)
Answer:
(b)
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 .
Thus, the quadratic equation having roots 1 + i and 1 - i is,
Page No 14.17:
Question 1:
If 1 – i is a root of the equation x2 + ax + b = 0, where a, b ∈ R, then the values of a and b are ____________.
Answer:
Page No 14.17:
Question 2:
If the difference of the roots of the equation x2 – Px + 8 = 0 is 2, then P =___________.
Answer:
Page No 14.17:
Question 3:
If the equation 2x2 – kx + x + 8 = 0 has real and equal roots, then k = __________.
Answer:
Page No 14.17:
Question 4:
The number of real roots of the equation x2 + 5 |x| + 4 = 0 is _________.
Answer:
Page No 14.18:
Question 5:
If one root of the equation x2 + px + 12 = 0 is 4, then the sum of the roots is ____________.
Answer:
Page No 14.18:
Question 6:
If α, β are roots of the equation x2 + x + 1 = 0, then the equation whose roots are α19 and β7 is ____________.
Answer:
Page No 14.18:
Question 7:
The value of is ____________.
Answer:
Page No 14.18:
Question 8:
If the equations px2 + 2qx + r = 0 and have real roots, then q2 =____________.
Answer:
Page No 14.18:
Question 9:
If the roots of the equation x2 – 8x + a2 – 6a = 0 are real, then 'a' lies in the interval ____________.
Answer:
for a < –2, (a + 2) (a – 8) ≥ 0 and for a > 8, (a + 2) (a – 8) ≥ 0
Page No 14.18:
Question 10:
If the equations x2 + x + a = 0 and x2 + ax + 1 = 0, a ≠ 1, have a common root, then a = ____________.
Answer:
Page No 14.18:
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 ____________.
Answer:
Since a(a – 4) < 0 is true for 0 < a < 4
Hence a ∈ (0, 4)
Page No 14.18:
Question 1:
Write the number of real roots of the equation .
Answer:
Comparing the given equation with the general form of the quadratic equation , we get and
Page No 14.18:
Question 2:
If a and b are roots of the equation , than write the value of .
Answer:
Given:
Also, and 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
Page No 14.18:
Question 3:
If roots α, β of the equation satisfy the relation α2 + β2 = 9, then write the value p.
Answer:
Given equation:
Also, and are the roots of the equation satisfying
From the equation, we have:
Sum of the roots =
Product of the roots = =
Hence, the value of
Page No 14.18:
Question 4:
If is root of the equation , than write the values of p and q.
Answer:
Irrational roots always occur in conjugate pairs.
If
Page No 14.18:
Question 5:
If the difference between the roots of the equation is 2, write the values of a.
Answer:
Given:
Let are the roots of the equation.
Sum of the roots = .
Product of the roots =
Given:
∴
Page No 14.18:
Question 6:
Write roots of the equation .
Answer:
Now,
Page No 14.18:
Question 7:
If a and b are roots of the equation , then write the value of a2 + b2.
Answer:
Given:
Also, are the roots of the equation.
Then, sum of the roots =
Product of the roots =
Page No 14.18:
Question 8:
Write the number of quadratic equations, with real roots, which do not change by squaring their roots.
Answer:
Let be the real roots of the quadratic equation
On squaring these roots, we get:
and
and
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.
Page No 14.18:
Question 9:
If α, β are roots of the equation , write an equation whose roots are .
Answer:
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
Page No 14.18:
Question 10:
If α, β are roots of the equation , then write the value of (1 + α) (1 + β).
Answer:
Given:
Also, and are the roots of the equation.
Sum of the roots =
Product of the roots =
Page No 14.5:
Question 1:
x2 + 1 = 0
Answer:
Given:
or
or
Hence, the roots of the equation are .
Page No 14.5:
Question 2:
9x2 + 4 = 0
Answer:
Given:
or,
or
or
Hence, the roots of the equation are
Page No 14.5:
Question 3:
x2 + 2x + 5 = 0
Answer:
Given:
or,
or,
Hence, the roots of the equation are .
Page No 14.5:
Question 4:
4x2 − 12x + 25 = 0
Answer:
We have:
or,
or,
or,
Hence, the roots of the equation are .
Page No 14.5:
Question 5:
x2 + x + 1 = 0
Answer:
We have:
or,
or,
Hence, the roots of the equation are .
Page No 14.6:
Question 6:
Answer:
We have:
or
or
or
Hence, the roots of the equation are .
Page No 14.6:
Question 7:
Answer:
We have:
or,
or,
Hence, the roots of the equation are .
Page No 14.6:
Question 8:
Answer:
We have:
or
or
Hence, the roots of the equation are .
Page No 14.6:
Question 9:
Answer:
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
Page No 14.6:
Question 10:
Answer:
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
Page No 14.6:
Question 11:
Answer:
We have:
or
or
Hence, the roots of the equation are
Page No 14.6:
Question 12:
Answer:
We have:
or
or
Hence, the roots of the equation are
Page No 14.6:
Question 13:
Answer:
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 .
Page No 14.6:
Question 14:
Answer:
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 .
Page No 14.6:
Question 15:
Answer:
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
Page No 14.6:
Question 16:
Answer:
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 .
Page No 14.6:
Question 17:
Answer:
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 .
Page No 14.6:
Question 18:
Answer:
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
Page No 14.6:
Question 19:
Answer:
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 .
Page No 14.6:
Question 20:
Answer:
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 .
Page No 14.6:
Question 21:
Answer:
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 .
Page No 14.6:
Question 22:
Answer:
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 .
Page No 14.6:
Question 23:
Answer:
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 .
Page No 14.6:
Question 24:
Answer:
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 .
Page No 14.6:
Question 25:
Answer:
or,
or,
Hence, the roots of the equation are .
Page No 14.6:
Question 26:
Answer:
or,
or,
Hence, the roots of the equation are .
Page No 14.6:
Question 27:
Answer:
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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