If the remainder of the polynomial a0 + a1x + a2x^2 + ....... + anx^n when divided by (x – 1) is 1, then which one of the following is correct ? -Maths 9th

If the remainder of the polynomial a0 + a1x + a2x^2 + ....... + anx^n when divided by (x – 1) is 1, then which one of the following is correct ? -Maths 9th
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Assertion (A) : The value of k for which the system of linear equations kx+2y+1=0 and 6x+4y-5=0 has a unique solution is 3. Reason (R):The system of linear equations a1x + b1y + c1= 0 and a2x + b2y + c2 = 0 has a unique solution if (a) Both A and R are correct; R is the correct explanation of A. (b) Both A and R are correct; R is not the correct explanation of A. (c) A is correct; R is incorrect. (d) R is correct; A is incorrect.

Description : Assertion (A) : The value of k for which the system of linear equations kx+2y+1=0 and 6x+4y-5=0 has a unique solution is 3. Reason (R):The system of linear equations a1x + b1y + c1= 0 and a2x + ... not the correct explanation of A. (c) A is correct; R is incorrect. (d) R is correct; A is incorrect.

Last Answer : (d) R is correct; A is incorrect.

If the polynomial x^19 + x^17 + x^13 + x^11 + x^7 + x^5 + x^3 is divided by (x^2 + 1), then the remainder is : -Maths 9th

Description : If the polynomial x^19 + x^17 + x^13 + x^11 + x^7 + x^5 + x^3 is divided by (x^2 + 1), then the remainder is : -Maths 9th

Last Answer : answer:

Determine the remainder when polynomial p(x) is divided by x - 2 . -Maths 9th

Description : Determine the remainder when polynomial p(x) is divided by x - 2 . -Maths 9th

Last Answer : p(x) = x4 - 3x2 + 2x - 5 According to remainder theorem, the required remainder will be = p(2) p(x) = x4 - 3x2 + 2x - 5 ∴ p(2) = 24 - 3(2)2 + 2(2) - 5 =16 - 12 + 4 - 5 = 3

By actual division, find the quotient and the remainder when the first polynomial is divided by the second polynomial x4 + 1 and x-1. -Maths 9th

Description : By actual division, find the quotient and the remainder when the first polynomial is divided by the second polynomial x4 + 1 and x-1. -Maths 9th

Last Answer : Actual division method

The polynomial p{x = x4 -2x3 + 3x2 -ax+3a-7 when divided by x+1 leaves the remainder 19. -Maths 9th

Description : The polynomial p{x = x4 -2x3 + 3x2 -ax+3a-7 when divided by x+1 leaves the remainder 19. -Maths 9th

Last Answer : p(x) is divided by x+ 2 =

Determine the remainder when polynomial p(x) is divided by x - 2 . -Maths 9th

Description : Determine the remainder when polynomial p(x) is divided by x - 2 . -Maths 9th

Last Answer : p(x) = x4 - 3x2 + 2x - 5 According to remainder theorem, the required remainder will be = p(2) p(x) = x4 - 3x2 + 2x - 5 ∴ p(2) = 24 - 3(2)2 + 2(2) - 5 =16 - 12 + 4 - 5 = 3

By actual division, find the quotient and the remainder when the first polynomial is divided by the second polynomial x4 + 1 and x-1. -Maths 9th

Description : By actual division, find the quotient and the remainder when the first polynomial is divided by the second polynomial x4 + 1 and x-1. -Maths 9th

Last Answer : Actual division method

The polynomial p{x = x4 -2x3 + 3x2 -ax+3a-7 when divided by x+1 leaves the remainder 19. -Maths 9th

Description : The polynomial p{x = x4 -2x3 + 3x2 -ax+3a-7 when divided by x+1 leaves the remainder 19. -Maths 9th

Last Answer : p(x) is divided by x+ 2 =

When a polynomial f(x) is divided by (x – 3) and (x + 6), the respective remainders are 7 and 22. What is the remainder when f(x) is -Maths 9th

Description : When a polynomial f(x) is divided by (x – 3) and (x + 6), the respective remainders are 7 and 22. What is the remainder when f(x) is -Maths 9th

Last Answer : answer:

If x51 + 51 is divided by x + 1, then the remainder is -Maths 9th

Description : If x51 + 51 is divided by x + 1, then the remainder is -Maths 9th

Last Answer : (d) Let p(x) = x51 + 51 . …(i) When we divide p(x) by x+1, we get the remainder p(-1) On putting x= -1 in Eq. (i), we get p(-1) = (-1)51 + 51 = -1 + 51 = 50 Hence, the remainder is 50.

If x51 + 51 is divided by x + 1, then the remainder is -Maths 9th

Description : If x51 + 51 is divided by x + 1, then the remainder is -Maths 9th

Last Answer : (d) Let p(x) = x51 + 51 . …(i) When we divide p(x) by x+1, we get the remainder p(-1) On putting x= -1 in Eq. (i), we get p(-1) = (-1)51 + 51 = -1 + 51 = 50 Hence, the remainder is 50.

If the expressions (px^3 + 3x^2 – 3) and (2x^3 – 5x + p) when divided by (x – 4) leave the same remainder, then what is the value of p ? -Maths 9th

Description : If the expressions (px^3 + 3x^2 – 3) and (2x^3 – 5x + p) when divided by (x – 4) leave the same remainder, then what is the value of p ? -Maths 9th

Last Answer : Given that the following polynomials leave the same remainder when divided by (x - 4) : We are to find the value of a. Remainder theorem: When (x - b) divides a polynomial p(x), then the remainder is p(b). So, from (i) and (ii), we get Thus, the required value of a is 1.

If x^3 + 5x^2 + 10k leaves remainder – 2x when divided by x^2 + 2, then what is the value of k ? -Maths 9th

Description : If x^3 + 5x^2 + 10k leaves remainder – 2x when divided by x^2 + 2, then what is the value of k ? -Maths 9th

Last Answer : x3+5x2+10k =(x2+2)(x+5)+10k−2x−10 ⇒10k−2x−10=−2x ⇒10k−10=0 or k=1.

f(x) = x^4 – 2x^3 + 3x^2 – ax + b is a polynomial such that when it is divided by (x – 1) and (x + 1), the remainders are respectively 5 and 19. -Maths 9th

Description : f(x) = x^4 – 2x^3 + 3x^2 – ax + b is a polynomial such that when it is divided by (x – 1) and (x + 1), the remainders are respectively 5 and 19. -Maths 9th

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By remainder theorem, find the remainder when p(x) is divided by g(x) -Maths 9th

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Last Answer : Find the remainder when p(x) is divided by g(x)

By remainder theorem, find the remainder when p(x) is divided by g(x) -Maths 9th

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Description : If x to the power 11 + 101 is divided by x+1,what is the remainder? -Maths 9th

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Description : When x^13 + 1 is divided by x –1, the remainder is : -Maths 9th

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Description : When (x^3 – 2x^2 + px – q) is divided by (x^2 – 2x – 3), the remainder is (x – 6), What are the values of p and q respectively ? -Maths 9th

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If the polynomials ax^3 + 4x^2 + 3x – 4 and x^3 – 4x + a leave the same remainder when divided by (x – 3), the value of a is : -Maths 9th

Description : If the polynomials ax^3 + 4x^2 + 3x – 4 and x^3 – 4x + a leave the same remainder when divided by (x – 3), the value of a is : -Maths 9th

Last Answer : Given ax^3 + 4x^2 + 3x - 4 and x^3 - 4x + a leave the same remainder when divided by x - 3. Let p(x) = ax^3 + 4x^2 + 3x - 4 and g(x) = x^3 - 4x + a By remainder theorem, if f(x) is divided by (x − a) then ... 4 27a+41 g(3)=27-4(3)+a 15+a f(3)=G(3) 27a+41=15+a 26a=15-41 a=15-41/26 a=-26/26 a=-1

If the expression ax^2 + bx + c is equal to 4, when x = 0, leaves a remainder 4 when divided by x + 1 and leaves a -Maths 9th

Description : If the expression ax^2 + bx + c is equal to 4, when x = 0, leaves a remainder 4 when divided by x + 1 and leaves a -Maths 9th

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Description : When x^40 + 2 is divided by x^4 + 1, what is the remainder ? -Maths 9th

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Description : The remainder when 1 + x + x^2 + x^3 + ........ + x^(1007) is divided by (x – 1) is -Maths 9th

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The remainder, when x^(200) is divided by x^2 – 3^x + 2 is -Maths 9th

Description : The remainder, when x^(200) is divided by x^2 – 3^x + 2 is -Maths 9th

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When x^3 + 2x^2 + 4x + b is divided by (x + 1), the quotient is x^2 + ax + 3 and the remainder -Maths 9th

Description : When x^3 + 2x^2 + 4x + b is divided by (x + 1), the quotient is x^2 + ax + 3 and the remainder -Maths 9th

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If ax^3 + bx^2 + x – 6 has (x + 2) as a factor and leaves a remainder 4, when divided by (x – 2), the value of a and b respectively are : -Maths 9th

Description : If ax^3 + bx^2 + x – 6 has (x + 2) as a factor and leaves a remainder 4, when divided by (x – 2), the value of a and b respectively are : -Maths 9th

Last Answer : Let p(x) = ax³ + bx² + x - 6 A/C to question, (x + 2) is the factor of p(x) , and we know this is possible only when p(-2) = 0 So, p(2) = a(-2)³ + b(-2)² - 2 - 6 = 0 ⇒ ... --(2) solve equations (1) and (2), 4a = 0 ⇒a = 0 and b = 2 Then, equation will be 2x² + x - 6

If x + 1 is a factor of the polynomial 2x2 + kx, then the value of k is -Maths 9th

Description : If x + 1 is a factor of the polynomial 2x2 + kx, then the value of k is -Maths 9th

Last Answer : (c) Let p(x) = 2x2 + kx Since, (x + 1) is a factor of p(x), then p(-1)=0 2(-1)2 + k(-1) = 0 ⇒ 2-k = 0 ⇒ k= 2 Hence, the value of k is 2.

If x + 1 is a factor of the polynomial 2x2 + kx, then the value of k is -Maths 9th

Description : If x + 1 is a factor of the polynomial 2x2 + kx, then the value of k is -Maths 9th

Last Answer : (c) Let p(x) = 2x2 + kx Since, (x + 1) is a factor of p(x), then p(-1)=0 2(-1)2 + k(-1) = 0 ⇒ 2-k = 0 ⇒ k= 2 Hence, the value of k is 2.

If x+1 is a factor of the polynomial 3x(square) - kx,then find the value of k. -Maths 9th

Description : If x+1 is a factor of the polynomial 3x(square) - kx,then find the value of k. -Maths 9th

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A cubic polynomial f(x) is such that f(1) = 1, f(2) = 2, f(3) = 3 and f(4) = 5, then f(6) equals : -Maths 9th

Description : A cubic polynomial f(x) is such that f(1) = 1, f(2) = 2, f(3) = 3 and f(4) = 5, then f(6) equals : -Maths 9th

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If the polynomial x^6 + px^5 + qx^4 – x^2 – x – 3 is divisible by x^4 – 1, then the value of p^2 + q^2 is : -Maths 9th

Description : If the polynomial x^6 + px^5 + qx^4 – x^2 – x – 3 is divisible by x^4 – 1, then the value of p^2 + q^2 is : -Maths 9th

Last Answer : The divisor is x4−1=(x−1)(x+1)(x2+1) By factor theorem, f(1)=f(−1)=0 Thus, 1+p+q−1−1−3=0 and 1+q−1−3=p−1 i.e., p+q=4 and p−q=−2 Adding the two, 2p=2 i.e. p=1 and ∴ q=3. ∴ p2+q2=1+9=10

If the sum of the zeroes of the polynomial p(x) = (k2 – 14) x2 – 2x – 12 is 1, then find the value of k. -Maths 9th

Description : If the sum of the zeroes of the polynomial p(x) = (k2 – 14) x2 – 2x – 12 is 1, then find the value of k. -Maths 9th

Last Answer : p(x) = (k2 – 14) x2 – 2x – 12 Here a = k2 – 14, b = -2, c = -12 Sum of the zeroes, (α + β) = 1 …[Given] ⇒ − = 1 ⇒ −(−2)2−14 = 1 ⇒ k2 – 14 = 2 ⇒ k2 = 16 ⇒ k = ±4

Find the remainder when y3 + y2 - 2y + 5 is divided by y - 5. -Maths 9th

Description : Find the remainder when y3 + y2 - 2y + 5 is divided by y - 5. -Maths 9th

Last Answer : Remainder = 145 Again, we should evaluate p(5) Let p(y) = y3 + y2 - 2y + 5 ∴ p(5) = 53 + 52 - 2 x 5 + 5 = 125 + 25 - 10 + 5 = 145 Thus , we find that p(5) is the remainder when p(y) is divided by y - 5 .

If the polynomials az3 +4z2 + 3z-4 and z3-4z + 0 leave the same remainder when divided by z – 3, -Maths 9th

Description : If the polynomials az3 +4z2 + 3z-4 and z3-4z + 0 leave the same remainder when divided by z – 3, -Maths 9th

Last Answer : Let p1(z) = az3 +4z2 + 3z-4 and p2(z) = z3-4z + o When we divide p1(z) by z - 3, then we get the remainder p,(3). Now, p1(3) = a(3)3 + 4(3)2 + 3(3) - 4 = 27a+ 36+ 9-4= 27a+ 41 When we ... to' the question, both the remainders are same. p1(3)= p2(3) 27a+41 = 15+a 27a-a = 15 - 41 . 26a = 26 a = -1

Find the remainder when y3 + y2 - 2y + 5 is divided by y - 5. -Maths 9th

Description : Find the remainder when y3 + y2 - 2y + 5 is divided by y - 5. -Maths 9th

Last Answer : Remainder = 145 Again, we should evaluate p(5) Let p(y) = y3 + y2 - 2y + 5 ∴ p(5) = 53 + 52 - 2 x 5 + 5 = 125 + 25 - 10 + 5 = 145 Thus , we find that p(5) is the remainder when p(y) is divided by y - 5 .

If the polynomials az3 +4z2 + 3z-4 and z3-4z + 0 leave the same remainder when divided by z – 3, -Maths 9th

Description : If the polynomials az3 +4z2 + 3z-4 and z3-4z + 0 leave the same remainder when divided by z – 3, -Maths 9th

Last Answer : Let p1(z) = az3 +4z2 + 3z-4 and p2(z) = z3-4z + o When we divide p1(z) by z - 3, then we get the remainder p,(3). Now, p1(3) = a(3)3 + 4(3)2 + 3(3) - 4 = 27a+ 36+ 9-4= 27a+ 41 When we ... to' the question, both the remainders are same. p1(3)= p2(3) 27a+41 = 15+a 27a-a = 15 - 41 . 26a = 26 a = -1

If the polynomials az3 + 42z2 + 3z – 4 and z3 - 4z + a leave the same remainder when divided by z – 3, find the value of a. -Maths 9th

Description : If the polynomials az3 + 42z2 + 3z – 4 and z3 - 4z + a leave the same remainder when divided by z – 3, find the value of a. -Maths 9th

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When a + b + c + 3a^(1/3) b^(2/3) + 3a(2/3) b^(1/3) is divided by a^(1/3) + b^(1/3) + c^(1/3), what is the remainder ? -Maths 9th

Description : When a + b + c + 3a^(1/3) b^(2/3) + 3a(2/3) b^(1/3) is divided by a^(1/3) + b^(1/3) + c^(1/3), what is the remainder ? -Maths 9th

Last Answer : answer:

If 9x^2 + 3px + 6q when divided by (3x + 1) leaves a remainder (-3/4) -Maths 9th

Description : If 9x^2 + 3px + 6q when divided by (3x + 1) leaves a remainder (-3/4) -Maths 9th

Last Answer : Given, (9x2 + 3px + 6q), when divided by (3x + 1) leaves a remainder −34−34 ∴ f(x) = 9x2 + 3px + 6q – (−34)(−34) = (9x2+3px+6q+34)(9x2+3px+6q+34) is exactly divisible by (3x + 1) ... ...(ii) Solving equations (i) and (ii), we get q = 0, p = 7474.

If the sum of zeroes of the quadratic polynomial 3x2 – kx + 6 is 3, then find the value of k. -Maths 9th

Description : If the sum of zeroes of the quadratic polynomial 3x2 – kx + 6 is 3, then find the value of k. -Maths 9th

Last Answer : Here a = 3, b = -k, c = 6 Sum of the zeroes, (α + β) = − = 3 …..(given) ⇒ −(−)3 = 3 ⇒ k = 9

The value of the polynomial 5x – 4x2 + 3, when x = -1 is -Maths 9th

Description : The value of the polynomial 5x – 4x2 + 3, when x = -1 is -Maths 9th

Last Answer : (a) Let p (x) = 5x – 4x2 + 3 …(i) On putting x = -1 in Eq. (i), we get p(-1) = 5(-1) -4(-1)2 + 3= - 5 - 4 + 3 = -6

Zero of the polynomial p(x)=2x+5 is -Maths 9th

Description : Zero of the polynomial p(x)=2x+5 is -Maths 9th

Last Answer : (b) Given, p(x) = 2x+5 For zero of the polynomial, put p(x) = 0 ∴ 2x + 5 = 0 ⇒ -5/2 Hence, zero of the polynomial p(x) is -5/2.

x + 1 is a factor of the polynomial -Maths 9th

Description : x + 1 is a factor of the polynomial -Maths 9th

Last Answer : (b) Let assume (x + 1) is a factor of x3 + x2 + x+1. So, x = -1 is zero of x3 + x2 + x+1 (-1)3 + (-1)2 + (-1) + 1 = 0 ⇒ -1+1-1 + 1 = 0 ⇒ 0 = 0 Hence, our assumption is true.

Find the value of the polynomial 3x3 – 4x2 + 7x – 5, when x = 3 and also when x = -3. -Maths 9th

Description : Find the value of the polynomial 3x3 – 4x2 + 7x – 5, when x = 3 and also when x = -3. -Maths 9th

Last Answer : Let p(x) =3x3 – 4x2 + 7x – 5 At x= 3, p(3) = 3(3)3 – 4(3)2 + 7(3) – 5 = 3×27-4×9 + 21-5 = 81-36+21-5 P( 3) =61 At x = -3, p(-3)= 3(-3)3 – 4(-3)2 + 7(-3)- 5 = 3(-27)-4×9-21-5 = -81-36-21-5 = -143 p(-3) = -143 Hence, the value of the given polynomial at x = 3 and x = -3 are 61 and -143, respectively.

Find the zeroes of the polynomial p(x)= (x – 2)2 – (x+ 2)2. -Maths 9th

Description : Find the zeroes of the polynomial p(x)= (x – 2)2 – (x+ 2)2. -Maths 9th

Last Answer : Given, polynomial is p(x) = (x – 2)2 – (x+ 2)2 For zeroes of polynomial, put p(x) = 0 (x – 2)2 – (x+ 2)2 = 0 (x-2 + x+2)(x-2-x-2) = 0 [using identity, a2-b2 =(a-b)(a + b)] ⇒ (2x)(-4) = 0

Determine which of the following polynomial has x – 2 a factor -Maths 9th

Description : Determine which of the following polynomial has x – 2 a factor -Maths 9th

Last Answer : first option is the correct answer for the given question solution is as follows:- let x-2=0 then, x=2 put x in (i) 3(2)(2)+6(2)-24=0 12+12-24=0 {use BODMAS rule for solution}... 24-24=0 0=0 this verifies our answer

The value of the polynomial 5x – 4x2 + 3, when x = -1 is -Maths 9th

Description : The value of the polynomial 5x – 4x2 + 3, when x = -1 is -Maths 9th

Last Answer : (a) Let p (x) = 5x – 4x2 + 3 …(i) On putting x = -1 in Eq. (i), we get p(-1) = 5(-1) -4(-1)2 + 3= - 5 - 4 + 3 = -6

Zero of the polynomial p(x)=2x+5 is -Maths 9th

Description : Zero of the polynomial p(x)=2x+5 is -Maths 9th

Last Answer : (b) Given, p(x) = 2x+5 For zero of the polynomial, put p(x) = 0 ∴ 2x + 5 = 0 ⇒ -5/2 Hence, zero of the polynomial p(x) is -5/2.