If the roots of the equation x^2 + x + 1 = 0 are in the ratio of m : n, then which one of the following relation holds ? -Maths 9th

If the roots of the equation x^2 + x + 1 = 0 are in the ratio of m : n, then which one of the following relation holds ? -Maths 9th
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For what value of m the ratio of the roots of the equation 12x^2 – mx + 5 = 0 is 3 : 2 ? -Maths 9th

Description : For what value of m the ratio of the roots of the equation 12x^2 – mx + 5 = 0 is 3 : 2 ? -Maths 9th

Last Answer : Given equation: 12x2+mx+5=0 The roots are in ratio 3:2 Hence,let roots of the equations are 3α and 2α. Applying condition for sum and product of the roots, 3α+2α=−12m​ and 3α×2α=125​⇒α2=725​⇒α=±62​5​​And m=−60α ⇒m=±510​Hence, A is the correct option.

If the difference in the roots of the equation x^2 – px + q = 0 is unity, then which one of the following is correct ? -Maths 9th

Description : If the difference in the roots of the equation x^2 – px + q = 0 is unity, then which one of the following is correct ? -Maths 9th

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If the roots of the equation x^2 – 2ax + a^2 + a – 3 = 0 are real and less than 3, then which one of the following is correct ? -Maths 9th

Description : If the roots of the equation x^2 – 2ax + a^2 + a – 3 = 0 are real and less than 3, then which one of the following is correct ? -Maths 9th

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If the roots of the equation a(b – c) x^2 + b(c – a)x + c(a – b) = 0 are equal, then a, b, c are in : -Maths 9th

Description : If the roots of the equation a(b – c) x^2 + b(c – a)x + c(a – b) = 0 are equal, then a, b, c are in : -Maths 9th

Last Answer : As we know that for the quadratic equation ax2+bx+c=0, roots will be equal if D=B2−4AC=0 Therefore, for the equation, a(b−c)x2+b(c−a)x+c(a−b)=0 A=a(b−c),B=b(c−a),C=c(a−b) D=0 B2−4AC=0 (b(c−a))2−4(a(b−c))(c(a−b))=0 ⇒ab+bc=2ac Hence a,b and c are in HP.

If the roots of the equation x^3 – ax^2 + bx – c = 0 are three consecutive integers, then what is the smallest possible value of b ? -Maths 9th

Description : If the roots of the equation x^3 – ax^2 + bx – c = 0 are three consecutive integers, then what is the smallest possible value of b ? -Maths 9th

Last Answer : Let the roots of the equation x3 – ax2 + bx – c = 0 be (α – 1), α, (α + 1) ∴ S2 = (α – 1)α + α(α + 1) + (α + 1) ( ... ; 1 = b ⇒ 3α2 – 1 = b ∴ Minimum value of b = – 1, when α = 0.

If the equation (a^2 + b^2) x^2 – 2 (ac + bd)x + (c^2 + d^2) = 0 has equal roots, then which one of the following is correct ? -Maths 9th

Description : If the equation (a^2 + b^2) x^2 – 2 (ac + bd)x + (c^2 + d^2) = 0 has equal roots, then which one of the following is correct ? -Maths 9th

Last Answer : The given quadratic equation is (a2 + b2)x2 − 2(ac + bd)x + (c2 + d2) = 0. If the roots of given quadratic equation are equal, then its discriminant is zero.

What is the ratio of sum of squares of roots to the product of the roots of the equation 7x^2 + 12x + 18 = 0? -Maths 9th

Description : What is the ratio of sum of squares of roots to the product of the roots of the equation 7x^2 + 12x + 18 = 0? -Maths 9th

Last Answer : Let α, β be the roots of the equation 7x2 + 12x + 18 = 0. ∴ Required ratio = α2 + β2 : αβ = ​​−10849187 = −67 = – 6 : 7.

For what value of k will the roots of the equation kx^2 – 5x + 6 = 0 be in the ratio 2 : 3 ? -Maths 9th

Description : For what value of k will the roots of the equation kx^2 – 5x + 6 = 0 be in the ratio 2 : 3 ? -Maths 9th

Last Answer : (b) 1 Let the roots of the equation kx2 - 5x + 6 = 0 be α and β. Then, α + β = \(\frac{5}{k}\) ...(i) αβ = \(\frac{6}{k}\) ...(ii) Given \(\ ... frac{9}{k}\) ⇒ 9k2 - 9k = 0 k(k - 1) = 0 ⇒ k = 0 or 1 But k = 0 does not satisfy the condition, so k = 1.

If the roots of the equation ax^2 + bx + c = 0 are equal in magnitude but opposite in sign, then which one of the following is correct ? -Maths 9th

Description : If the roots of the equation ax^2 + bx + c = 0 are equal in magnitude but opposite in sign, then which one of the following is correct ? -Maths 9th

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If the sum of the roots of the equation ax^2 + bx + c = 0 is equal to the sum of their squares, then which one of the following is correct ? -Maths 9th

Description : If the sum of the roots of the equation ax^2 + bx + c = 0 is equal to the sum of their squares, then which one of the following is correct ? -Maths 9th

Last Answer : Given equation: ax2+bx+c=0 Let α and β be the roots of given quadratic equation Sum of the roots i.e. α+β=a−b Product of roots i.e. αβ=ac It is given that, Sum of the roots = Sum of squares of the roots i ... )2−2αβ i.e. a−b =(a−b )2−a2c i.e. −ab=b2−2ac i.e. ab+b2=2ac Hence, C is the correct option.

If one of the roots of the equation x^2 + ax + 3 = 0 is 3 and one of the roots of the equation x2 + ax + b = 0 is three -Maths 9th

Description : If one of the roots of the equation x^2 + ax + 3 = 0 is 3 and one of the roots of the equation x2 + ax + b = 0 is three -Maths 9th

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What are the roots of the equation 4^x – 3.2^(x + 2) + 32 = 0 ? -Maths 9th

Description : What are the roots of the equation 4^x – 3.2^(x + 2) + 32 = 0 ? -Maths 9th

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What are the roots of the quadratic equation a^2 b^2 x^2 – (a^2 + b^2)x + 1 = 0 ? -Maths 9th

Description : What are the roots of the quadratic equation a^2 b^2 x^2 – (a^2 + b^2)x + 1 = 0 ? -Maths 9th

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If an integer P is chosen at random in the interval 0 ≤ p ≤ 5, the probability that the roots of the equation x^2 + px -Maths 9th

Description : If an integer P is chosen at random in the interval 0 ≤ p ≤ 5, the probability that the roots of the equation x^2 + px -Maths 9th

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Two students A and B solve an equation of the form x^2 + px + q = 0. A starts with a wrong value of p and obtains the roots as 2 and 6. -Maths 9th

Description : Two students A and B solve an equation of the form x^2 + px + q = 0. A starts with a wrong value of p and obtains the roots as 2 and 6. -Maths 9th

Last Answer : Let αα and ββ be the roots of the quadratic equation x2+px+q=0x2+px+q=0 Given that, A starts with a wrong value of p and obtains the roots as 2 and 6. But this time q is correct. i.e., a product of roots ... 1 Now, from Eqs. (ii) and (iii), we get α=−3 and β=−4α=−3 and β=−4 which are correct roots.

The equation whose roots are the negatives of the roots of the equation x^7 + 3x^5 + x^3 – x^2 + 7x + 2 = 0 is : -Maths 9th

Description : The equation whose roots are the negatives of the roots of the equation x^7 + 3x^5 + x^3 – x^2 + 7x + 2 = 0 is : -Maths 9th

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Let p and q be the roots of the quadratic equation x^2 – (a – 2)x – a – 1 = 0. What is the minimum possible value of p^2 + q^2 ? -Maths 9th

Description : Let p and q be the roots of the quadratic equation x^2 – (a – 2)x – a – 1 = 0. What is the minimum possible value of p^2 + q^2 ? -Maths 9th

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The equation whose roots are twice the roots of the equation x^2 – 3x + 3 = 0 is -Maths 9th

Description : The equation whose roots are twice the roots of the equation x^2 – 3x + 3 = 0 is -Maths 9th

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Let R be a relation on the set N, defined by {(x, y) : 2x – y = 10} then R is -Maths 9th

Description : Let R be a relation on the set N, defined by {(x, y) : 2x – y = 10} then R is -Maths 9th

Last Answer : (a) ReflexiveGiven, {(\(x\), y) : 2\(x\) – y = 10} Reflexive, \(x\) R \(x\) = 2\(x\) – \(x\) = 10 ⇒ \(x\) = 10 ⇒ y = 10 ∴ Point (10, 10) ∈ N ⇒ R is reflexive.

If the sum as well as the product of roots of a quadratic equation is 9, then the equation is: -Maths 9th

Description : If the sum as well as the product of roots of a quadratic equation is 9, then the equation is: -Maths 9th

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The temperature of a liquid can be measured in Kelvin units as x K or in Fahrenheit units as y°F. The relation between the two system of measurement of temperature is given by the linear equation y = 9/5 ( x - 273 ) + 32 -Maths 9th

Description : The temperature of a liquid can be measured in Kelvin units as x K or in Fahrenheit units as y°F. The relation between the two system of measurement of temperature is given by the linear equation y = 9/5 ( x - 273 ) + 32 -Maths 9th

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What are the roots of the equation (a + b + x) ^(–1) = a^(–1) + b^(–1) + x^(–1) ? -Maths 9th

Description : What are the roots of the equation (a + b + x) ^(–1) = a^(–1) + b^(–1) + x^(–1) ? -Maths 9th

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What are the roots of the equation log10 (x^2 – 6x + 45) = 2? -Maths 9th

Description : What are the roots of the equation log10 (x^2 – 6x + 45) = 2? -Maths 9th

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The sum of the roots of the equation (1/(x+a))+(1/(x+b))=1/c is zero. What is the product of the roots of the equation ? -Maths 9th

Description : The sum of the roots of the equation (1/(x+a))+(1/(x+b))=1/c is zero. What is the product of the roots of the equation ? -Maths 9th

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In quadratic equation ax^2 + bx + c = 0 , Identities the sum and product of Roots? -Maths 9th

Description : In quadratic equation ax^2 + bx + c = 0 , Identities the sum and product of Roots? -Maths 9th

Last Answer : If the two roots of the quadratic equation ax2 + bx + c = 0 obtained by the quadratic formula be denoted by a and b, then we have α = \(\frac{-b+\sqrt{b^2-4ac}}{2a},\) β = \(\frac{-b-\sqrt{b^2-4ac}}{2a}\) ∴ Sum of roots ... then α + β = - (- \(\frac{5}{6}\)) = \(\frac{5}{6}\), ab = \(\frac{7}{6}\).

Which one of the following is the equation whose roots are respectively three times the roots of the equation ax^2 + bx + c = 0 ? -Maths 9th

Description : Which one of the following is the equation whose roots are respectively three times the roots of the equation ax^2 + bx + c = 0 ? -Maths 9th

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If p, q, r are positive and are in A.P., the roots of quadratic equation px^2 + qx + r = 0 are real for : -Maths 9th

Description : If p, q, r are positive and are in A.P., the roots of quadratic equation px^2 + qx + r = 0 are real for : -Maths 9th

Last Answer : Given p,q,r are in A.P. then q=2p+r​.....(1). Now px2+qx+r=0 will have real root then q2−4pr≥0. or, 4(p+r)2​−4pr≥0 or, p2+r2−14pr≥0 or, r2−14rp+49p2≥48p2 or, (r−7p)2≥(43​p)2 or, (pr​−7)2≥(43​)2 [ Since p=0 for the given equation to be quadratic] or, ∣∣∣∣∣​pr​−7∣∣∣∣∣​≥43​.

The quadratic equation whose roots are three times the roots of 3ax^2 + 3bx + c = 0 is -Maths 9th

Description : The quadratic equation whose roots are three times the roots of 3ax^2 + 3bx + c = 0 is -Maths 9th

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On a set N of all natural numbers is defined the relation R by a R b iff the GCD of a and b is 2, then R is -Maths 9th

Description : On a set N of all natural numbers is defined the relation R by a R b iff the GCD of a and b is 2, then R is -Maths 9th

Last Answer : (c) Symmetric only Let a ∈N. Then (a, a) ∉R as the GCD of a' and a' is a' not 2. R is not reflexive Let a, b ∈N. Then, (a, b) ∉R ⇒ GCD of a' and b' is 2 ⇒ GCD of b' and a' is 2 ⇒ (b, a) ∈R ∴ R ... , let a = 4, b = 10, c = 12 GCD of (4, 10) = 2 GCD of (10, 12) = 2 But GCD of (4, 12) = 4.

If R is a relation defined on the set of natural numbers N such that (a, b) R (c, d) if and only if a + d = b + c, then R is -Maths 9th

Description : If R is a relation defined on the set of natural numbers N such that (a, b) R (c, d) if and only if a + d = b + c, then R is -Maths 9th

Last Answer : (d) An equivalence relationWe can check the given properties as follows: Reflexive: Let (a, b) ∈ N x N. Then (a, b) ∈ N ⇒ a + b = b + a (Communtative law of Addition) ⇒ (a, b) R (b, a) ⇒ (a, b) R (a, ... , f) ⇒ (a, b) R (e, f) on N x N so R is transitive.Hence R is an equivalence relation on N N.

If R is a relation on a finite set A having n elements, then the number of relations on A is -Maths 9th

Description : If R is a relation on a finite set A having n elements, then the number of relations on A is -Maths 9th

Last Answer : (d) \(2^{n^2}\)Set A has n elements ⇒ n(A) = n ⇒ A × A has n × n = n2 elements ∴ Number of relations on A = Number of subsets of A × A = \(2^{n^2}\)

If a, b, c, d are positive reals such that a + b + c + d = 2, then M = (a + b) (c + d) satisfies the relation -Maths 9th

Description : If a, b, c, d are positive reals such that a + b + c + d = 2, then M = (a + b) (c + d) satisfies the relation -Maths 9th

Last Answer : answer:

Explain Nature of Roots of a quadratic equation. -Maths 9th

Description : Explain Nature of Roots of a quadratic equation. -Maths 9th

Last Answer : answer:

Let N be the set of natural numbers. Describe the following relation in words giving its domain -Maths 9th

Description : Let N be the set of natural numbers. Describe the following relation in words giving its domain -Maths 9th

Last Answer : The given relation stated in words is R = {(x, y) : x is the fourth power of y; x ∈ N, y ∈ {1, 2, 3, 4}}.

If R is a relation in N × N defined by (a, b) R (c, d) if and only if ad = bc, show that R is an equivalence relation. -Maths 9th

Description : If R is a relation in N × N defined by (a, b) R (c, d) if and only if ad = bc, show that R is an equivalence relation. -Maths 9th

Last Answer : (i) R is reflexive. For all (a, b) ∈ N N we have (a, b) R (a, b) because ab = ba ⇒ R is reflexive. (ii) R is symmetric. Suppose (a, b) R (c, d) Then (a, b) R (c, d) ⇒ ... ) R (e, f) ⇒ R is transitive. Since R is reflexive, symmetric and transitive, therefore, R is an equivalence relation on N N.

Let R be a relation defined as a Rb if | a – b | > 0, then the relation is -Maths 9th

Description : Let R be a relation defined as a Rb if | a – b | > 0, then the relation is -Maths 9th

Last Answer : (d) Symmetric and transitive| a - a | = | 0 | = 0 so (a, a) ∉R ⇒ R is not reflexive(a, b) ∈ R ⇒ | a - b | > 0 ⇒ | b - a | > 0 ⇒ (b, a) ∈R (∵ | a - b | = | b - a |) ⇒ R is symmetric (a, b) ∈ R ... a, b, c. ∴ | a - b | > 0 and | b - c | > 0 ⇒ | a - c | > 0 ⇒ (a, c) ∈ R ⇒ R is transitive.

On the set R of all real numbers, a relation R is defined by R = {(a, b) : 1 + ab > 0}. Then R is -Maths 9th

Description : On the set R of all real numbers, a relation R is defined by R = {(a, b) : 1 + ab > 0}. Then R is -Maths 9th

Last Answer : (a) Reflexive and symmetric only(a, a) ∈ R ⇒ 1 + a . a = 1 + a2 > 0 V real numbers a ⇒ R is reflexive (a, b) ∈ R ⇒ 1 + ab > 0 ⇒ 1 + ba > 0 ⇒ (b, a) ∈ R ⇒ R is symmetricWe observe that \(\big(1,rac{1}{2}\big) ... }{2},-1\big)\) ∈ Rbut (1, - 1) ∉ R as 1 + 1 (-1) = 0 \( ot>\) 0 ⇒ R is not transitive.

In Fig. 6.10, if m|n, then find the value of x. -Maths 9th

Description : In Fig. 6.10, if m|n, then find the value of x. -Maths 9th

Last Answer : Solution :-

If (log x)/(l + m - 2n) = (log y)/(m + n - 2l) = (log z)/(n + l - 2m), then xyz is equal to : -Maths 9th

Description : If (log x)/(l + m - 2n) = (log y)/(m + n - 2l) = (log z)/(n + l - 2m), then xyz is equal to : -Maths 9th

Last Answer : Let l+m−2nlogx​=m+n−2llogy​=n+l−2mlogz​=k(say) So, we get logx=k(l+m−2n) ....... (i) logy=k(m+n−2l) ....... (ii) logz=k(n+l−2m) ....... (iii) ∴logx+logy+logz=k(l+m−2n)+k(m+n−2l)+k(n+l−2m) ⇒logx+logy+logz=kl+km−2kn+km+kn−2kl+kn+kl−2km ⇒log(xyz)=0 ⇒logxyz=log1 ⇒xyz=1

If (log x)/(l + m - 2n) = (log y)/(m + n - 2l) = (log z)/(n + l - 2m), then xyz is equal to : -Maths 9th

Description : If (log x)/(l + m - 2n) = (log y)/(m + n - 2l) = (log z)/(n + l - 2m), then xyz is equal to : -Maths 9th

Last Answer : (b) 1Let \(rac{ ext{log}\,x}{l+m-2n}\) = \(rac{ ext{log}\,y}{m+n-2l}\) = \(rac{ ext{log}\,z}{n+l-2m}\) = k. Thenlog x = k(l + m – 2n), log y = k(m + n – 2l); log z = k(n + l – 2m) ⇒ log x + log y + log z = k(l + m – 2n) + k(m + n – 2l) + k(n + l – 2m)⇒ log(xyz) = 0 ⇒ log(xyz) = log 1 ⇒ xyz = 1.

If 1/3 log3 M + 3 log3 N = 1 + log 0.008 5, then -Maths 9th

Description : If 1/3 log3 M + 3 log3 N = 1 + log 0.008 5, then -Maths 9th

Last Answer : (b) \(N^9=rac{9}{M}\)\(rac{1}{3}\) log3 M + 3 log3 N = 1 + log0.008 5⇒ log3 M1/3 + log3N3 = 1 + log0.0085 ⇒ log3 M1/3 N3 = 1 + log0.0085 ⇒ M1/3 N3 = 3(1 + log0.0085) ⇒ M1/3 N3 = 31 . 3log0.0085⇒ \(N^9=rac{27 ... ^9=rac{27}{M}\big(3^{log_{rac{1}{5}}5}\big)\) = \(rac{1}{M}\) (27)(3-1) = \(rac{9}{M}.\)

The ratio of girls and boys in a class is 1: 3. Set up an equation between the students of a class and boys and then draw its graph. Also find the number of boys in a class of 40 students from the graph. -Maths 9th

Description : The ratio of girls and boys in a class is 1: 3. Set up an equation between the students of a class and boys and then draw its graph. Also find the number of boys in a class of 40 students from the graph. -Maths 9th

Last Answer : Total number of boys and girl = 40, Ratio = 1 : 3 Number of girls be A and Number of boys be B. Ratio of number of girls and boys is 1 : 3, so Therefore 3A=B To find number of boys we ... the number 30 represents the number of girls. 40 as total on the line A = 10, which is the common equation.

If (–5, 4) divides the line segment between the co-ordinate axes in the ratio 1 : 2, then what is its equation ? -Maths 9th

Description : If (–5, 4) divides the line segment between the co-ordinate axes in the ratio 1 : 2, then what is its equation ? -Maths 9th

Last Answer : (d) x = yThe equations of the given lines are: 4x + 3y = 12 ...(i) 3x + 4y = 12 ...(ii) Solving the simultaneous equations (i) and (ii), we get\(x\) = \(rac{12}{7}\), y = \(rac{12}{7}\)∴ Point of the ... )isy - 0 = \(\bigg(rac{rac{12}{7}-0}{rac{12}{7}-0}\bigg)\) (x - 0), i.e., y = x.

If one roots of the equation `x^(2) - mx + n = 0` is twice the other root, then show that `2m^(2) = 9pi`.

Description : If one roots of the equation `x^(2) - mx + n = 0` is twice the other root, then show that `2m^(2) = 9pi`.

Last Answer : If one roots of the equation `x^(2) - mx + n = 0` is twice the other root, then show that `2m^(2) = 9pi`.

If `alpha, beta` are roots of equation `x^(2)-4x-3=0` and `s_(n)=alpha^(n)+beta^(n), n in N` then the value of `(s_(7)-4s_(6))/s_(5)` is

Description : If `alpha, beta` are roots of equation `x^(2)-4x-3=0` and `s_(n)=alpha^(n)+beta^(n), n in N` then the value of `(s_(7)-4s_(6))/s_(5)` is

Last Answer : If `alpha, beta` are roots of equation `x^(2)-4x-3=0` and `s_(n)=alpha^(n)+beta^(n), n in N` then the value ... 4s_(6))/s_(5)` is A. 4 B. 3 C. 5 D. 7

If one root of the equation ax^2 + x – 3 = 0 is –1, then what is the other root ? -Maths 9th

Description : If one root of the equation ax^2 + x – 3 = 0 is –1, then what is the other root ? -Maths 9th

Last Answer : answer:

The equation of the line bisecting the join of (3, – 4) and (5, 2) and having its intercepts on the x-axis and y-axis in the ratio 2 : 1 is -Maths 9th

Description : The equation of the line bisecting the join of (3, – 4) and (5, 2) and having its intercepts on the x-axis and y-axis in the ratio 2 : 1 is -Maths 9th

Last Answer : (d) \(\lambda\) = \(\mu\) Let the equation of the line in the intercept from be\(rac{x}{\lambda}\)+ \(rac{y}{\mu}\) = 1Since it passes through (4, 3) and (2, 5)\(rac{4}{\lambda}\) + \(rac{3}{\mu}\) = 1 ... ) = 1 - \(rac{3}{7}\) = \(rac{4}{7}\) = \(\lambda\) = 7∴ \(\lambda\) = \(\mu\) = 7.

If the roots of x^3 – 12x^2 + 12x – 28 = 0 are in A.P, their common difference is -Maths 9th

Description : If the roots of x^3 – 12x^2 + 12x – 28 = 0 are in A.P, their common difference is -Maths 9th

Last Answer : answer:

The equation of motion for a vibrating system with viscous damping is d 2 x/dt 2 + c/m X dx/dt + s/m X x = 0 If the roots of this equation are real, then the system will be A. over damped B. under damped C. critically damped D. none of the mentioned

Description : The equation of motion for a vibrating system with viscous damping is d 2 x/dt 2 + c/m X dx/dt + s/m X x = 0 If the roots of this equation are real, then the system will be A. over damped B. under damped C. critically damped D. none of the mentioned

Last Answer : A. over damped

The equation of motion for a vibrating system with viscous damping is d 2 x/dt 2 + c/m X dx/dt + k/m X x = 0 If the roots of this equation are real, then the system will be a) over damped b) under damped c) critically damped d) none of the mentioned Ans:a

Description : The equation of motion for a vibrating system with viscous damping is d 2 x/dt 2 + c/m X dx/dt + k/m X x = 0 If the roots of this equation are real, then the system will be a) over damped b) under damped c) critically damped d) none of the mentioned Ans:a

Last Answer : a) over damped