RD Sharma 2022 Mcqs Solutions for Class 10 Maths Chapter 4 Quadratic Equations are provided here with simple step-by-step explanations. These solutions for Quadratic Equations are extremely popular among class 10 students for Maths Quadratic Equations Solutions come handy for quickly completing your homework and preparing for exams. All questions and answers from the RD Sharma 2022 Mcqs Book of class 10 Maths Chapter 4 are provided here for you for free. You will also love the ad-free experience on Meritnation’s RD Sharma 2022 Mcqs Solutions. All RD Sharma 2022 Mcqs Solutions for class 10 Maths are prepared by experts and are 100% accurate.
Page No 61:
Question 1:
Which of the following is a quadratic equation?
(a) x2 + 2x + 1 = (4 – x)2 + 3
(b)
(c) where k = –1
(d) x3 – x2 = (x – 1)3
Answer:
An equation is of the form , is called a quadratic equation.
(a) Given that,
, which is not a quadratic equation.
(b) Given that,
, which is not a quadratic equation.
(c) Given that,
, where k = −1
, which is not a quadratic equation.
(d) Given that,
, which is a quadratic equation.
Hence, the correct answer is option (d).
Page No 61:
Question 2:
Which of the following is not a quadratic equation?
(a) 2(x – 1)2 = 4x2 – 2x + 1
(b) 2x – x2 = x2 + 5
(c)
(d) (x2 + 2x)2 = x4 + 3 + 4x3
Answer:
An equation is of the form , is called a quadratic equation.
(a) Given that,
, which is a quadratic equation.
(b) Given that,
, which is a quadratic equation.
(c) Given that,
, which is not a quadratic equation.
(d) Given that,
, which is a quadratic equation.
Hence, the correct answer is option C.
Page No 61:
Question 3:
Which of the following equations has 2 as a root?
(a) x2 – 4x + 5 = 0
(b) x2 + 3x – 12 = 0
(c) 2x2 – 7x + 6 = 0
(d) 3x2 – 6x – 2 = 0
Answer:
If α is one of the roots of the quadratic equation , then x = α satisfies the equation the quadratic.
(a) Given,
Substituting x = 2 in , we get
So we conclude, x = 2 is not a root of .
(b) Given,
Substituting x = 2 in , we get
So we conclude, x = 2 is not a root of .
(c) Given,
Substituting x = 2 in , we get
So we conclude, x = 2 is a root of .
(d) Given,
Substituting x = 2 in , we get
So we conclude, x = 2 is not a root of .
Hence, the correct answer is option (c).
Page No 61:
Question 4:
Which of the following equations has the sum of its roots as 3?
(a) 2x2 – 3x + 6 = 0
(b) –x2 + 3x – 3 = 0
(c)
(d) 3x2 – 3x + 3 = 0
Answer:
If α and β are the roots of the quadratic equation , , then sum of roots = α + β = .
(a) Given, .
Comparing with , we get
a = 2, b = −3 and c = 6.
∴ Sum of the roots = .
So, we conclude sum of the roots of given quadratic equation is not 3.
(b) Given, .
Comparing with , we get
a = −1, b = 3 and c = −3.
∴Sum of the roots = .
So, we conclude sum of the roots of given quadratic equation is 3.
(c) Given, .
Comparing with , we get
∴Sum of the roots = .
So, we conclude sum of the roots of given quadratic equation is not 3.
(d) Given, .
Comparing with , we get
a = 3, b = −3 and c = 3.
∴Sum of the roots = .
So, we conclude sum of the roots of the given quadratic equation is not 3.
Hence, the correct answer is option (b).
Page No 61:
Question 5:
The quadratic equation has
(a) two distinct real roots
(b) two equal real roots
(c) no real roots
(d) more than 2 real roots
Answer:
Given, equation
Now, comparing with , we get
a = 2, b = and c = 1.
Thus, we conclude given quadratic equation has no real roots since the discriminant is negative.
Hence, the correct answer is option (c).
Page No 61:
Question 6:
Which of the following equations has two distinct roots?
(a)
(b) x2 + x – 5 = 0
(c) x2 + 3x + = 0
(d) 5x2 – 3x + 1 = 0
Answer:
If a quadratic equation is in the form of , then
(i) If , then its roots are distinct and real.
(ii) If , then its roots are real and equal.
(iii) If , then its roots are not real or imaginary roots.
(a) Given,
Now, comparing with , we get
a = 2, b = and c = .
Thus, the equation has real and equal roots.
(b) Given
Now, comparing with , we get
a = 1, b = 1 and c = −5.
Thus, the equation has real and distinct roots.
(c) Given
Now, comparing with , we get
a = 1, b = 3 and c =
Thus, equation has no real roots.
(d) Given
Now, comparing with , we get
a = 5, b = −3 and c =
Thus, the equation has no real roots.
Hence, the correct answer is option (b).
Page No 61:
Question 7:
Which of the following equations has no real roots?
(a) x2 – 4x + = 0
(b) x2 + 4x – = 0
(c) x2 – 4x – = 0
(d) 3x2 + + 4 = 0
Answer:
If a quadratic equation is in the form of , then
(i) If , then its roots are distinct and real.
(ii) If , then its roots are real and equal.
(iii) If , then its roots are not real or imaginary roots.
(a) Given
Now, comparing with , we get
a = 1, b = −4 and c = .
Thus, the equation has no real roots.
(b) Given
Now, comparing with , we get
a = 1, b = 4 and c = .
Thus, the equation has real and distinct roots.
(c) Given
Now, comparing with , we get
a = 1, b = −4 and c = .
Thus, the equation has real and distinct roots.
(d) Given
Now, comparing with , we get
a = 3, b = and c = 4.
Thus, the equation has real and equal roots.
Hence, the correct answer is option (a).
Page No 61:
Question 8:
The equation (x2 + 1)2 – x2 = 0 has
(a) four real roots
(b) two real roots
(c) no real roots
(d) one real root
Answer:
Given equation,
Let
∴
Now, comparing with , we get.
a = 1, b = 1 and c = 1
Since, .
Thus, we conclude given equation has no real roots.
Hence, the correct answer is option (c).
Page No 61:
Question 9:
If x = 0.2 is a root of the equation x2 – 0.4k = 0, then k =
(a) 1
(b) 10
(c) 0.1
(d) 100
Answer:
Given equation, has x = 0.2 as a root.
Now, substitute the value 0.2 in the given equation, we get
Hence, the correct answer is option (c).
Page No 61:
Question 10:
If is a root of the equation then the value of k is
(a) –2
(b) 2
(c)
(d)
Answer:
Since, is a root of equation .
Put in given equation, we get
Thus, the value of k = 2.
Hence, the correct answer is option (b).
Page No 61:
Question 11:
Which of the following is not a quadratic equation?
(a) 3(x + 1)2 = 2x2 + x + 4
(b) 5x + 2x2 = x2 + 9
(c) (x2 – 2x)2 = x4 + 3 + 4x2
(d)
Answer:
An equation is of the form , is called a quadratic equation.
(a) Given that,
3(x + 1)2 = 2x2 + x + 4
which is a quadratic equation.
(b) Given that,
5x + 2x2 = x2 + 9
, which is a quadratic equation.
(c) Given that,
, which is not a quadratic equation.
(d) Given that,
, which is not a quadratic equation.
Disclaimer: Both (c) and (d) options are correct.
Page No 62:
Question 12:
Which of the following equations has 3 as a root?
(a) x2 – 4x + 3 = 0
(b) x2 + 4x + 3 = 0
(c) x2 + 5x + 6 = 0
(d) x2 + 7x + 12 = 0
Answer:
If α is one of the root of quadratic equation , then x = α satisfies the equation
(a) Given, x2 – 4x + 3 = 0
Substituting x = 3 in , we get
So we conclude, x = 3 is a root of x2 – 4x + 3 = 0.
(b) Given, x2 + 4x + 3 = 0
Substituting x = 3 in x2 + 4x + 3, we get
So we conclude, x = 3 is not a root of x2 + 4x + 3 = 0.
(c) Given, x2 + 5x + 6 = 0
Substituting x = 3 in x2 + 5x + 6, we get
So we conclude, x = 3 is not a root of x2 + 5x + 6 = 0.
(d) Given, x2 + 7x + 12 = 0
Substituting x = 3 in x2 + 7x + 12, we get
So we conclude, x = 3 is not a root of x2 + 7x + 12 = 0.
Hence, the correct answer is option (a).
Page No 62:
Question 13:
A quadratic equation can have
(a) at least two roots
(b) at most two roots
(c) exactly two roots
(d) any number of roots
Answer:
A quadratic equation can have atmost two roots, zero, one and two.
Hence, the correct answer is option (b).
Page No 62:
Question 14:
The discriminant of the quadratic equation (x + 2)2 = 0 is
(a) –2
(b) 2
(c) 4
(d) 0
Answer:
The quadratic equation (x + 2)2 = 0 can be written as .
On comparing with , we get
Hence, the correct answer is option (d).
Page No 62:
Question 15:
The values of k for which the quadratic equation has real and equal roots are
(a)
(b) 36, −36
(c) 6, −6
(d)
Answer:
The given quadratic equation , has equal roots.
Here, .
As we know that
Putting the values of .
The given equation will have real and equal roots, if D = 0
Thus,
Therefore, the value of k is 6, −6.
Hence, the correct option is (c).
Page No 62:
Question 16:
If y = 1 is a common root of the equations , then ab equals
(a) 3
(b) −7/2
(c) 6
(d) −3
Answer:
Since, y = 1 is a root of the equations .
So, it satisfies the given equation.
Since, y = 1 is a root of the equations .
So, it satisfies the given equation.
From (1) and (2),
Thus, ab is equal to 3.
Hence, the correct option is (a).
Page No 62:
Question 17:
If one of the equation x2 + ax + 3 = 0 is 1, then its other root is
(a) 3
(b) −3
(c) 2
(d) 1
Answer:
Let be the roots of quadratic equation in such a way that
Here,
Then , according to question sum of the roots
And the product of the roots
Therefore, value of other root be
Thus, the correct answer is
Page No 62:
Question 18:
If one root the equation 2x2 + kx + 4 = 0 is 2, then the other root is
(a) 6
(b) −6
(c) −1
(d) 1
Answer:
Let be the roots of quadratic equation in such a way that
Here,
Then , according to question sum of the roots
And the product of the roots
Putting the value of in above
Putting the value of k in
Therefore, value of other root be
Thus, the correct answer is
Page No 62:
Question 19:
A quadratic equation whose one root is 2 and the sum of whose roots is zero, is
(a) x2 + 4 = 0
(b) x2 − 4 = 0
(c) 4x2 − 1 = 0
(d) x2 − 2 = 0
Answer:
Let be the roots of quadratic equation in such a way that
Then, according to question sum of the roots
And the product of the roots
As we know that the quadratic equation
Putting the value of in above
Therefore, the require equation be
Thus, the correct answer is
Page No 62:
Question 20:
If the sum and product of the roots of the equation kx2 + 6x + 4k = 0 are real, then k =
(a)
(b)
(c)
(d)
Answer:
The given quadric equation is , and roots are equal
Then find the value of c.
Let be two roots of given equation
And,
Then, as we know that sum of the roots
And the product of the roots
According to question, sum of the roots = product of the roots
Therefore, the value of
Thus, the correct answer is
Page No 62:
Question 21:
If the sum of the roots of the equation x2 − x = λ(2x − 1) is zero, then λ =
(a) −2
(b) 2
(c)
(d)
Answer:
The given quadric equation is , and roots are zero.
Then find the value of .
Here,
As we know that
Putting the value of
The given equation will have zero roots, if
Therefore, the value of
Thus, the correct answer is
Page No 62:
Question 22:
If x = 1 is a common roots of the equations ax2 + ax + 3 = 0 and x2 + x + b = 0, then ab =
(a) 3
(b) 3.5
(c) 6
(d) −3
Answer:
is the common roots given quadric equation are , and
Then find the value of q.
Here, ….. (1)
….. (2)
Putting the value of in equation (1) we get
Now, putting the value of in equation (2) we get
Then,
Thus, the correct answer is
Page No 62:
Question 23:
If the equation x2 + 4x + k = 0 has real and distinct roots, then
(a) k < 4
(b) k > 4
(c) k ≥ 4
(d) k ≤ 4
Answer:
The given quadric equation is , and roots are real and distinct.
Then find the value of k.
Here,
As we know that
Putting the value of
The given equation will have real and distinct roots, if
Therefore, the value of
Thus, the correct answer is
Page No 62:
Question 24:
If ax2 + bx + c = 0 has equal roots, then c =
(a)
(b)
(c)
(d)
Answer:
The given quadric equation is , and roots are equal
Then find the value of c.
Let be two roots of given equation
Then, as we know that sum of the roots
And the product of the roots
Putting the value of
Therefore, the value of
Thus, the correct answer is
Page No 62:
Question 25:
The value of is
(a) 4
(b) 3
(c) −2
(d) 3.5
Answer:
Let
Squaring both sides we get
The value of x cannot be negative.
Thus, the value of x = 3
Therefore, the correct answer is
Page No 62:
Question 26:
If 2 is a root of the equation x2 + bx + 12 = 0 and the equation x2 + bx + q = 0 has equal roots, then q =
(a) 8
(b) −8
(c) 16
(d) −16
Answer:
2 is the common roots given quadric equation are , and
Then find the value of q.
Here, ….. (1)
….. (2)
Putting the value of in equation (1) we get
Now, putting the value of in equation (2) we get
Then,
As we know that
Putting the value of
The given equation will have equal roots, if
Thus, the correct answer is
Page No 62:
Question 27:
If p and q are the roots of the equation x2 − px + q = 0, then
(a) p = 1, q = −2
(b) b = 0, q = 1
(c) p = −2, q = 0
(d) p = −2, q = 1
Answer:
Given that p and q be the roots of the equation
Then find the value of p and q.
Here,
p and q be the roots of the given equation
Therefore, sum of the roots
….. (1)
Product of the roots
As we know that
Putting the value of in equation (1)
Therefore, the value of
Thus, the correct answer is
Page No 62:
Question 28:
The value of c for which the equation ax2 + 2bx + c = 0 has equal roots is
(a)
(b)
(c)
(d)
Answer:
The given quadric equation is , and roots are equal
Then find the value of c.
Let be two roots of given equation
Then, as we know that sum of the roots
And the product of the roots
Putting the value of
Therefore, the value of
Thus, the correct answer is
Page No 63:
Question 29:
If has equal roots, then k =
(a)
(b)
(c)
(d)
Answer:
The given quadric equation is , and roots are equal
Then find the value of k.
Here,
As we know that
Putting the value of
The given equation will have real and distinct roots, if
Therefore, the value of
Thus, the correct answer is
Page No 63:
Question 30:
If one of the equation ax2 + bx + c = 0 is three times times the other, then b2 : ac =
(a) 3 : 1
(b) 3 : 16
(c) 16 : 3
(d) 16 : 1
Answer:
Let be the roots of quadratic equation in such a way that
Here,
Then,
according to question sum of the roots
….. (1)
And the product of the roots
….. (2)
Putting the value of in equation (2)
Thus, the correct answer is
Page No 63:
Question 31:
If the sum of the roots of the equation is equal to half of their product, then k =
(a) 6
(b) 7
(c) 1
(d) 5
Answer:
The given quadric equation is , and roots are equal
Then find the value of k.
Let be two roots of given equation
And,
Then, as we know that sum of the roots
And the product of the roots
According to question, sum of the roots product of the roots
Therefore, the value of
Thus, the correct answer is
Page No 63:
Question 32:
If one root of the equation 4x2 − 2x + (λ − 4) = 0 be the reciprocal of the other, then λ =
(a) 8
(b) −8
(c) 4
(d) −4
Answer:
Let be the roots of quadratic equation in such a way that
Here,
Then , according to question sum of the roots
And the product of the roots
Therefore, value of
Thus, the correct answer is
Page No 63:
Question 33:
If the equation x2 − ax + 1 = 0 has two distinct roots, then
(a) |a| = 2
(b) |a| < 2
(c) |a| > 2
(d) None of these
Answer:
The given quadric equation is , and roots are distinct.
Then find the value of a.
Here,
As we know that
Putting the value of
The given equation will have real and distinct roots, if
Therefore, the value of
Thus, the correct answer is
Page No 63:
Question 34:
If the equation 9x2 + 6kx + 4 = 0 has equal roots, then the roots are both equal to
(a)
(b)
(c) 0
(d) ±3
Answer:
The given quadric equation is , and roots are equal.
Then find roots of given equation.
Here,
As we know that
Putting the value of
The given equation will have equal roots, if
So, putting the value of k in quadratic equation
When then equation be and when then
Therefore, the value of
Thus, the correct answer is
Page No 63:
Question 35:
If the equation ax2 + 2x + a = 0 has two distinct roots, if
(a) a = ±1
(b) a = 0
(c) a = 0, 1
(d) a = −1, 0
Answer:
The given quadric equation is , and roots are distinct.
Then find the value of a.
Here,
As we know that
Putting the value of
The given equation will have real and distinct roots, if
Therefore, the value of
Thus, the correct answer is
Page No 63:
Question 36:
The positive value of k for which the equation x2+ kx + 64 = 0 and x2 − 8x + k = 0 will both have real roots, is
(a) 4
(b) 8
(c) 12
(d) 16
Answer:
The given quadric equation are , and roots are real.
Then find the value of a.
Here, ….. (1)
….. (2)
As we know that
Putting the value of
The given equation will have real and distinct roots, if
Therefore, putting the value of in equation (2) we get
The value of satisfying to both equations
Thus, the correct answer is
Page No 63:
Question 37:
If the equations has equal roots, then
(a) ab = cd
(b) ad = bc
(c)
(d)
Answer:
The given quadric equation is , and roots are equal.
Here,
As we know that
Putting the value of
The given equation will have equal roots, if
Thus, the correct answer is
Page No 63:
Question 38:
If the roots of the equations are equal, then
(a) 2b = a + c
(b) b2 = ac
(c)
(d) b = ac
Answer:
The given quadric equation is , and roots are equal.
Here,
As we know that
Putting the value of
The given equation will have equal roots, if
Thus, the correct answer is
Page No 63:
Question 39:
If the equation x2 − bx + 1 = 0 does not possess real roots, then
(a) −3 < b < 3
(b) −2 < b < 2
(c) b > 2
(d) b < −2
Answer:
The given quadric equation is , and does not have real roots.
Then find the value of b.
Here,
As we know that
Putting the value of
The given equation does not have real roots, if
Therefore, the value of
Thus, the correct answer is
Page No 63:
Question 40:
If a and b can take values 1, 2, 3, 4. Then the number of the equations of the form ax2 + bx + 1 = 0 having real roots is
(a) 10
(b) 7
(c) 6
(d) 12
Answer:
Given that the equation .
For given equation to have real roots, discriminant (D) ≥ 0
⇒ b2 − 4a ≥ 0
⇒ b2 ≥ 4a
⇒ b ≥ 2√a
Now, it is given that a and b can take the values of 1, 2, 3 and 4.
The above condition b ≥ 2√a can be satisfied when
i) b = 4 and a = 1, 2, 3, 4
ii) b = 3 and a = 1, 2
iii) b = 2 and a = 1
So, there will be a maximum of 7 equations for the values of (a, b) = (1, 4), (2, 4), (3, 4), (4, 4), (1, 3), (2, 3) and (1, 2).
Thus, the correct option is (b).
Page No 63:
Question 41:
The number of quadratic equations having real roots and which do not change by squaring their roots is
(a) 4
(b) 3
(c) 2
(d) 1
Answer:
As we know that the number of quadratic equations having real roots and which do not change by squaring their roots is 2.
Thus, the correct answer is
Page No 63:
Question 42:
If has no real roots, then
(a) ab = bc
(b) ab = cd
(c) ac = bd
(d) ad ≠ bc
Answer:
The given quadric equation is , and roots are equal.
Here,
As we know that
Putting the value of
The given equation will have no real roots, if
Thus, the correct answer is
Page No 63:
Question 43:
If sin α and cos α are the roots of the equations ax2+ bx + c = 0, then b2 =
(a) a2 − 2ac
(b) a2 + 2ac
(c) a2 − ac
(d) a2 + ac
Answer:
The given quadric equation is , and are roots of given equation.
And,
Then, as we know that sum of the roots
…. (1)
And the product of the roots
…. (2)
Squaring both sides of equation (1) we get
Putting the value of , we get
Putting the value of , we get
Therefore, the value of
Thus, the correct answer is
Page No 63:
Question 44:
If a and b are roots of the equation x2 + ax + b = 0, then a + b =
(a) 1
(b) 2
(c) −2
(d) −1
Answer:
The given quadric equation is , and their roots are a and b
Then find the value of
Let be two roots of given equation
And,
Then, as we know that sum of the roots
And the product of the roots
Putting the value of a in above
Therefore, the value of
Thus, the correct answer is
Page No 63:
Question 45:
India is one of the largest importers of crud oil. Oil companies produce crude oil in barrels. Suppose the maximum oil produced by company is 300 barrels and profit made from sale of these barrels is given by the function P(x) = –10x2 + 3500x – 66,000, where P(x) is profit in rupees and x is the number of barrels produced and sold.
Based on the above information answers the following questions:
(i) When no barrel is produced, then the profit or loss is
(b) Profit â¹44,000
(c) Loss â¹66,000
(d) Loss â¹88,000
(b) 20
(c) 30
(d) 40
(b) earns profit of â¹184,000
(c) is in loss of â¹185,000
(d) is in loss of â¹184,000
(b) loss of â¹266,000
(c) profit of â¹342,000
(d) loss of â¹342,000
(b) a circle
(c) a parabola
(d) an ellipse
Answer:
(i) When no barrel is produced, x = 0
P(0) = −10(0)2 + 3500(0) − 66000
⇒ P(0) = − 66000
Thus, when no barrel is produced, the company will bear a loss of â¹66,000.
Hence, the correct answer is option (c).
(ii) The break-even point is the No-profit No-loss condition, i.e., at the break-even point, the company's profit is zero.
âµ P(x) = 0
⇒ −10x2 + 3500x − 66000 = 0
⇒ x2 − 350x + 6600 = 0
â⇒ x2 − 330x − 20x + 6600 = 0
â⇒ x(x − 330) − 20(x − 330) = 0
⇒ (x − 20)(x − 330) = 0
⇒ x = 20, 330
The maximum oil produced by company is 300 barrels. So, x = 330 will be rejected.
Thus, the break-even point is x = 20
Hence, the correct answer is option (b).
(iii) On producing 100 barrels, the company, x = 100
P(100) = −10(100)2 + 3500(100) − 66000
⇒ P(100) = −100000 + 350000 − 66000
⇒ P(100) = 184000
Thus, on producing 100 barrels, the company earns profit of â¹184,000.
Hence, the correct answer is option (b).
(iv) If company produces 400 barrels, then x = 400
P(400) = −10(400)2 + 3500(400) − 66000
⇒ P(400) = −1600000 + 1400000 − 66000
⇒ P(400) = −266000
Thus, on producing 400 barrels, the company earns loss of â¹266,000.
Hence, the correct answer is option (b).
(v) The profit function is a quadratic equation.
Thus, the graph of the profit function is a parabola.
Hence, the correct answer is option (c).
Page No 64:
Question 46:
Raghav has a field with total area of 1260 m2. He uses it to grow wheat and rice. The land used to grow wheat i.e. wheatland is rectangular in shape while the riceland is in the shape of a square as shown in the following figure. The length of wheatland is 3 metre more than twice the length of riceland.
Based on the above information answers the following questions:
(i) If the length of the riceland is x metre, then total length of the field (in metres) is
(b) 3x + 3
(c) 4x + 4
(d) 3x + 5
(b) 6x + 8
(c) 3x + 4
(d) 4x + 3
(b) 15
(c) 20
(d) 25
(b) 760 m2
(c) 820 m2
(d) 860 m2
(b) 20 : 43
(c) 23 : 40
(d) 40 : 23
Answer:
(i) The length of the rice land is x metres.
The length of wheatland is 3 metres more than twice the length of riceland.
The length of the wheatland is 2x + 3.
Thus, the length of the field = x + 2x + 3 = (3x + 3) m
Hence. the correct answer is option (b).
(ii) The figure given below shows the measurements of each side of the field.
The perimeter of the field = 4x + 2(2x + 3) = (8x + 6) m
Hence, the correct answer is option (a).
(iii) The total area of the field = 1260 m2
⇒ x2 + x(2x + 3) = 1260
⇒ x2 + 2x2 + 3x = 1260
⇒ 3x2 + 3xâ − 1260 = 0
⇒ x2 + xâ − 420 = 0
⇒ x2 + 21xâ − 20x − 420 = 0
⇒ (x + 21)(xâ − 20) = 0
⇒ x = −21â, 20
As x is the length then it can not be negative.
Thus, x = 20
Hence, the correct answer is option (c)
(iv) The area of the wheatland = x(2x + 3)
= 2x2 + 3x
= 2(20)2 + 3(20)
= 860 m2
Hence, the correct answer is option (d).
(v) The area of riceland = x2 = 400
The ratio of the areas of the wheat and rice land = 860 : 400 = 43 : 20
Thus, the ratio of the areas of the wheat and rice land is 43 : 20.
Hence, the correct answer is option (a).
Page No 64:
Question 47:
Each of the following questions contains STATEMENT-1 (Assertion) and STATEMENT-2 (Reason) and has following four choices (a), (b), (c) and (d), only one of which is the correct answer. Mark the correct choice.
(a) Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
(b) Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
(c) Statement-1 is true, Statement-2 is false.
(d) Statement-1 is false, Statement-2 is true.
Statement-1 (Assertion): If is a root of a quadratic equation with rational coefficients, then its other root is
Statement-2 (Reason): Surd roots of a quadratic equation with rational coefficients occur in conjugate pairs.
Answer:
If one root of the quadratic equation is irrational, then another root is always irrational when the coefficients are real and the other root is conjugate of the first root.
Thus, statement-2 is true.
If is a root of a quadratic equation with rational coefficients, then its other root is
The roots of a quadratic equation ax2 + bx + c = 0 are given by .
So, when b = −2 and D = 3 for a quadratic equation then its roots will be and
Thus, statement-1 is true.
Thus, Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
Hence, the correct answer is option (a).
Page No 65:
Question 48:
Each of the following questions contains STATEMENT-1 (Assertion) and STATEMENT-2 (Reason) and has following four choices (a), (b), (c) and (d), only one of which is the correct answer. Mark the correct choice.
(a) Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
(b) Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
(c) Statement-1 is true, Statement-2 is false.
(d) Statement-1 is false, Statement-2 is true.
Statement-1 (Assertion): If p, q, r and s are real numbers and pr = 2(q + s), then at least one of the equations x2 + px + q = 0 and x2 + rx + s = 0 has real roots.
Statement-2 (Reason): The sum of two real numbers is positive, then both the numbers are positive.
Answer:
The sum of two real numbers is positive, then both numbers need not be positive.
For example: (−3) + 8 = 5
Thus, statement-2 is false.
We have,
x2 + px + q = 0 .....(1)
âx2 + rx + s = 0 .....(2)
Let D1 and D2 be the discriminants of equations (1) and (2), then
D1 = p2 − 4q
D2 = r2 − 4s
⇒â D1 + D2 = p2 + r2 − 4(q + s)
⇒â D1 + D2 = p2 + r2 − 2pr [âµ pr = 2(q + s)]
⇒â D1 + D2 = (p − r)2 ≥ 0
At least one of D1 and D2 is greater than or equal to zero.
So, at least one of two equations has real roots.
Thus, Statement-1 is true, Statement-2 is false.
Hence, the correct answer is option (c).
Page No 65:
Question 49:
Each of the following questions contains STATEMENT-1 (Assertion) and STATEMENT-2 (Reason) and has following four choices (a), (b), (c) and (d), only one of which is the correct answer. Mark the correct choice.
(a) Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
(b) Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
(c) Statement-1 is true, Statement-2 is false.
(d) Statement-1 is false, Statement-2 is true.
Statement-1 (Assertion): If a + b + c = 0, then ax2 + bx + c = 0 has real roots.
Statement-2 (Reason): If one root of a quadratic equation is real, then the other root is also real.
Answer:
We have, ax2 + bx + c = 0
So, discriminant D = b2 − 4ac
Given: a + b + c = 0
⇒ b = −(a + c)
∴ D = [−(a + c)]2 − 4ac
⇒ D = a2 + c2 + 2ac − 4ac
⇒ D = a2 + c2 − ac
⇒ D = (a − c)2 ≥ 0
Thus, if a + b + c = 0, then ax2 + bx + c = 0 has real roots.
Thus, statement-1 is true.
It is also true that if one root of a quadratic equation is real, then the other root is also real.
âThus, statement-2 is true.
Thus, Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
Hence, the correct answer is option (b).
Page No 65:
Question 50:
Each of the following questions contains STATEMENT-1 (Assertion) and STATEMENT-2 (Reason) and has following four choices (a), (b), (c) and (d), only one of which is the correct answer. Mark the correct choice.
(a) Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
(b) Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
(c) Statement-1 is true, Statement-2 is false.
(d) Statement-1 is false, Statement-2 is true.
Statement-1 (Assertion): If a – b + c = 0, then ax2 + bx + c = 0 has real roots.
Statement-2 (Reason): Roots of x2 – x + 1 = 0 are not real.
Answer:
We have, ax2 + bx + c = 0
So, discriminant D = b2 − 4ac
Given: a – b + c = 0
⇒ b = (a + c)
∴ D = (a + c)2 − 4ac
⇒ D = a2 + c2 + 2ac − 4ac
⇒ D = a2 + c2 − 2ac
⇒ D = (a − c)2 ≥ 0
Thus, if a – b + c = 0, then ax2 + bx + c = 0 has real roots.
So, Statement-1 is true.
We have, x2 – x + 1 = 0
So, discriminant D = (−1)2 − 4(1)(1)
⇒ D = 1 − 4 = −3 < 0
Thus, roots of x2 – x + 1 = 0 are not real.
So, Statement-2 is true.
Thus, Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
Hence, the correct answer is option (b).
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