MATHS EDUCATION : April 2024

Saturday, 27 April 2024

12.Application of derivatives

.Application of derivatives Basic concept Class-12

(इस chapter के अंदर हम निम्नलिखित टॉपिक पढ़ने वाले हैं)

1. Rate of change of quantities

2. Increasing and decreasing functions

3. Tangent and Normal

4. Approximations

5. Maxima and Minima

Exercise	Topic
6.1	Introduction
6.2	Rate of Change of Quantities
6.3	Increasing and Decreasing Functions
6.4	Tangents and Normals
6.5	Approximations
6.6	Maxima and Minima
Others	Miscellaneous Q&A

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1. Rate of change of quantities:

⬇️

Area/Volume/C.S.A/T.S.A/Radius/Length / Breath. etc..

⬇️

Variable---------------------------- ---Constant

↙️ ↘️.

1. Dependent. 1. Arbitrary

And ( ax+by+c=0)

2. Independent. 2. Absolute

( 2x+3y=8)

If the change in one variable y with respect to x then dy/dx = f'(x) denoted the rate of change with respect to x

lim x➡️0 ∆y/∆x = dy/dx

A = π r²

↙️. ↘️

Dep Independent

(बनने वाला) (बनाने वाला)

C = 2π r

↙️. ↘️

Dep Independent

(बनने वाला) (बनाने वाला)

(r में change होने से circle के Area और circumference दोनों change होंगे )

------------------------------------------------------------ A. ➡️r. ➡️ t

dA/dr × dr/dt = dA/dt

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2. Increasing and decreasing. functions.

(जब x का मान बढ़ाने पर y का मान भी बढ़ता हो तो function, Increasing होगा, और जब x का मान बढ़ाने पर y का मान घटता हो तो function decreasing होगा )

कोई function Increasing है या फिर decreasing है यह पता करने के निम्नलिखित दो तरीके हैं l

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1 . By graph. 2. By f'(x)

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1 . By graph.

Increasing function. ⬇️

. ⬇️

. .

Decreasing function.

⬇️

Neither Increasing nor decreasing function.

⬇️

By Table

(i) y =x²

(A) x ≥ 0

X	0	1	2	3	4
y	0	1	4	9	16

x➡️ and y➡️ Increasing

(B) x ≤ 0

x	0	-4	-3	-2	-1
y	0	16	9	4	1

x➡️ and y⬅️ decreasing

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(ii) y = 7

Neither Increasing nor Decreasing

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(iii) y = 2x + 1. Increasing

-------------------------------------------------------------

(iv) y = -3x + 2. decreasin

----------------------------------------------------------- 2. By f'(x)-------

CONCEPT

(A). f is Increasing in [a b] if f'(x) ≥ 0 for each X ∈ [a b ]

(B). f is Decreasing in [a b] if f'(x) ≤ 0 for each X ∈ [a b ]

(C). f is contant [a b ] if f'(x) = 0 for X ∈ [a b ]

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WORKING RULE FOR INCREASING AND DECREASING

(i) Function

(ii) Differentiation

(iii) Linear Factor

(iv) f'(x) = 0

(v) Find the values of x (turning point/Extreme point)

(vi) Show the value of x on number line

(vii) Write Intervals

(viii) Find f'(x) at any simple x in Interval

f'(x) ≥ 0 , Increasing

f'(x) > 0, Strictly Increasing

f'(x) ≤ 0, Decreasing

f'(x) < 0, Strictly Decreasing

Example f(x) = x² - x - 6

f'(x) = 2x -1

f'(x) = 0

2x -1 = 0

x = 1/2

- ∞ -------------------- 1/2 ------------- ∞.

Intervals	Sign of f’(x)	Nature
( -∞ 1/2 )	- ( negative)	Strictly Decreasing
( 1/2 ∞)	+ ( Positive)	Strictly Increasing

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(Out of syllabus)

3. Tangent and Normal.

Equation of tangent. y-y1 = dy/dx( x-x1)

Equation of Normal. y-y1 = - dx/dy( x-x1)

When tangent is parallel to x-axis then:

dy/dx =0

When tangent is perpendicular to x-axis then:

dx/dy = 0

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(Out of syllabu

4.Approximations

f (x+∆x ) = f(x) + ∆x.f'(x)

Ex: √(25.1) = √ (25+ 0.1)

↙️ ↘️

x ∆x

f(x) = √x f'(x) = 1/2√x

= f(x) + ∆x. f'(x)

= √x + ∆x• 1/2√x

= √25 + 0.1 × 1/2√25

= 5. +0.1/10

=. 5 + 0.01

=. 5.01. Ans

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5. Maxima and Minima.

Application of derivatives (A.O.D) का सबसे Important topics है ,नाम से ही पता लग रहा है कि किसके विषय में बात होने वाली है l

अधिकतम और न्यूनतम

Maxima and Minima का पता हम निम्नलिखित तीन तरीकों से लगा सकते हैंl

1. Graph देख कर ,

2. By First derivatives test.

3. By Second derivatives test

1. Graph देख कर ,:. ........

जिस domain पर function defined है उसके अंदर सबसे छोटी value को function की minimum तथा सबसे बड़ी value को function की maximum value कहते हैं l

(A)

(B)

(C)

(D)

(E)

(F)
(G)

(H)

(I)

(J)

(K)

(L)

एक function बहुत सारी maxima या minima values सकती हैं l इन सभी को Local Maxima या Local minima जाता है इन सभी में जो सबसे बड़ा होगा उसे Absolute Maxima तथा जो सबसे छोटा होगा उसे Absolute Minima जाता है l

Absolute Maxima ⊂ Local Maxima

Absolute Minima ⊂ Local Minima

(M)

()

2. By First derivatives test.

(i) Function

(ii) Differentiation

(iii) Linear Factor

(iv) f'(x) = 0

(v) Find the values of x (turning point/Extreme point/ critical point)

f'(x) का sign: + ➡️ - ( maximum)

- ➡️+ ( Minimum)

Graph(D)

Example : f(x) = x² - x - 6

f'(x) = 2x -1

at f'(x) = 0

2x -1 = 0

x = 1/2

-∞. -------------------- 1/2 ------------- ∞.

At value	Sign of f’(x)
( 1/2- h)	- ( negative)	(- to +) MINIMUM
( 1/2+h)	+ ( Positive)

Example: f(x) = x³ - 6x² + 9x + 15

f'(x)= 3x² - 12x + 9

= 3(x-1) (x-3)

at f'(x) =0 , x = 1 ,3

-------------------⬇️--------------------------⬇️-------------- f'(x)=.

(+) ⬇️.(-) (-) ⬇️(+)

(1-h)----•1•---(1+h)-----(3-h)----•3•------(3+h)

at x=1,f(x) will be maximum.

Maximum vale= f(1)=19

at x=3, f(x) will be minimum

Minimum value=f((3)= 15

Note: When the sign of f'(x) doesn't change then it's called point of inflation

Ex: f(x)= x³

f'(x)= 3x²

at f'(x) =0. x=0

--------------- ⬇️---------------------------------------

f'(x)=. (+) ⬇️.(+) (No change)

(0-h)----•0•---(0+h)----

f'(0-h)= (+)

f'(0+h)= (+) so x=0 is point of inflation

3. By Second derivatives test.

i) Function

(ii) Differentiation

(iii) Linear Factor

(iv) f'(x) = 0

(v) Find the values of x =a,b,c..

(turning point/Extreme point/ critical point)

(vi) f''(x) at a,b,c ..

(vii) If f''(x) = (+) then f(x) is Minimum

If f''(x) = (-) then f(x) is Maximum

f''(x)= 0 ( go to first derivatives test)

EXAMPLE: f(x) = 3x⁴ + 4x³- 12x² + 12

f'(x) = 12x³ + 12x² - 24x

= 12x( x+2) (x -1)

at f'(x) = 0 , x = 0 ,1 , -2

f''(x) = 36x² + 24x - 24

at x=0, f"(x)= -24(Neg) [ MAXIMUM]

⬇️

Point of Maxima

at x=1, f"(x)= 36 (Positive) [ MINIMUM]

⬇️

Point of Minima

at x= -2, f"(x)= 72 (Positive) [MINIMUM]

⬇️

Point of Minima

f(0) =12

f(1)= 7

f(-2)= -20

. MODULUS FUNCTION AND TRIGONOMETRY FUNCTION .

(i).

f(x)= |x+2| -1↗️ ((x+2)-1

↘️-(x+2)-1

f'(x)= ↗️ 1

↘️-1.

(Critical point not possible )

Then

|x+2| ≥ 0

|x+2| -1 ≥ -1

f(x) ≥ -1

So -1 is minimum value .

for point |x+2| -1 = 0

x - 2 = 0

x= -2 (point of Minima)

(ii).

g(x) = - |x+1| + 3

we know that. :|x+1| ≥ 0

- |x+1|≤ 0 ,

( Mode or Neg mei sign change krte hai)

- |x+1| + 3 ≤ 0 + 3

g(x) ≤ 3 (Maximum)

for point |x+1| = 0

x +1 = 0

x= -1 (point of Maxima)

(iii)

h(x) = sin2x + 5

-1 ≤ sin2x ≤ 1

-1 +5 ≤ sin2x ≤ 1+5

4 ≤ h(x). ≤ 6

Mini. Maxi

Point sin2x = 0. x ,=nπ +(-1)ñalpha

®️®️®️®️®️®️®️®️®️ ®️®️

Thursday, 25 April 2024

WORKSHEET OF POLYNOMIAL CLASS-10

CLASS-10

100 IMPORTANT QUESTIONS OF POLYNOMIAL

••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••

1. A quadratic polynomial, whose zeroes are –3 and 4, is :

(a) x² – x + 12

(b) x² + x + 12

(c) x²/2 - x/ 2 – 6

(d) 2x² + 2x – 24

2. The zeroes of the quadratic polynomial x² + 99x + 127 are :

(a) both positive

(b) both negative

(c) one positive and one negative

(d) both equal

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3. If the zeroes of the quadratic polynomial ax² + bx + c, c ≠ 0 are equal, then :

(a) c and a have opposite signs

(b) c and b have opposite signs

(c) c and a have the same sign

(d) c and b have the same sign

4. If the sum of the zeroes of the polynomial p(x) = 2x³ – 3kx² + 4x – 5 is 6, then the value of k is :

(a) 2

(b) 4

(c) –2

(d) –4

5. If the product of the zeroes of the polynomial p(x) = ax³ – 6x² + 11x – 6 is 4, then the value of a is :

(a) 1/2

(b) 5/3

(c) 3/2

(d) −1/5

6. If α and β are the zeroes of the polynomial p(x) = x² + px + q, then a polynomial having 1/α and 1/β as its zeroes is :

(a) x² + px/q + 1/q

(b) x² - px/q + 1/q

(c) x² - px/q - 1/q

(d) x² + px/q + 1/q

7. If α and β are the zeroes of p(x) = x² – 5x + b and α - β =1, then b is :

(a) 1

(b) 6

(c) 4

(d) 0

8. Given that one of the zeroes of a cubic polynomial 3x³ + 5x² – 9x + a is zero. The value of a is :

(a) −5/3

(b) –3

(c) 3

(d) 0

9. If a and b are zeroes of x² + 4x + 4 then

(a) a=b, a + b = –4

(b) a = –b, a + b = 4

(c) a = b, a + b = 4

(d) a = –b, a + b = –4

10. If the two zeroes of the cubic polynomial x³ + x² are zero each, then the third zero is :

(a) 0

(b) 1

(c) 2

(d) –1

11. If the graph of y = p(x), where p(x) is a quadratic polynomial cuts x-axis at different points, the p(x) has :

(a) no zero

(b) exactly one zero

(c) exactly two zeroes

(d) none of these

12. If α , β and γ are zeroes of a cubic polynomial ax³ + bx² + cx + d, then αβ +βγ + γα is :

(a) d/a

(b) −c/a

(c) c/a

(d) −a/b

13. If two zeroes of the polynomial x³ – 5x² – 16x + 80 are equal in magnitude but opposite in sign, then its zeroes are :

(a) 4, –4, 5

(b) 3, –3, 5

(c) 2, –2, 5

(d) 1, –1, 5

14. If α and β are the zeroes of the quadratic polynomial p(x) = x² – 5x + 4, then

1/α + 1/β - 2 α β equals:

(a) 27/4

(b) −27/4

(c) 4/27

(d) −4/27

15. If two zeroes of the polynomial x³ + x² – 9x – 9 are 3 and –3, then its third zero is :

(a) –1

(b) 2

(c) –2

(d) 3

16. If √5 and − √5 are two zeroes of the polynomial p(x) = x³ + 3x² – 5x – 15, then its other zero is :

(a) 2

(b) –3

(c) 4

(d) –4

17. If the polynomial p(x) = 3x⁴ + 5x³ – 7x² + 2x + 2 is divided by g(x) = x² + 3x + 1, then the value of quotient is :

(a) 2x² – 4x + 2

(b) 3x² – 4x + 2

(c) 3x² + 4x + 2

(d) 3x² – 4x – 2

18. On dividing 2x² + 3x – 2 by x + 2, the quotient is :

(a) 3x + 1

(b) 3x – 1

(c) 2x – 1

(d) 2x + 1

19. On dividing 3x³ + x² + 2x + 5 by 1 + 2x + x², the remainder is :

(a) 9x – 10

(b) 9x + 10

(c) 8x + 5

(d) 8x – 5

20. If zeroes of a polynomial p(x) are √2 and − √2 , then the polynomial which is a factor of p(x) is :

(a) x² + 2

(b) x² – 2

(c) x – 2

(d) x + 2

21. The graph of y = p(x), where p(x) is a polynomial in variable

x is as follows. The number of zeroes of p(x) is :

(a) 2

(b) 3.

(c) 5.

(d) 0.

22. The sum of the zeroes of the polynomial 2x² – 8x + 6 is :

(a) –3

(b) 3

(c) –4

(d) 4

23. If α and β are the zeroes of the polynomial px² – 2x + 3p and α + β = αβ, then p =

(a) 3/2

(b) 2/3

(c) 3

(d) 2

24. If α and β are the zeroes of ax² – bx + c = 0 (a ≠ 0), then α + β is :

(a) −b/a

(b) a/b

(c) b/a

(d) −c/a

25. If α and β are the zeroes of the polynomial x² + x + 1, then

1/α + 1/β is :

(a) -1

(b) 1

(c) 1/2

(d) -1/2

26. The graph of y = p(x), where p(x) is a polynomial, is shown.

The number of zeroes of the polynomial p(x) is :

(a) 3

(b) 2

(c) 1

(d) no zero

27. If α and β are the zeroes of the polynomial x² + 6x + 2,

then 1/α + 1/β is :

(a) 3

(b) –3

(c) 12

(d) –12

28. A quadratic polynomial whose zeroes are

3/ 5 and −1/2 is :

(a) 10x² + x + 3

(b) 10x² + x – 3

(c) 10x² – x + 3

(d) 10x² – x – 3

29. If 1 is a zero of the polynomial p(x) = ax² – 3(a – 1), then the value of a is :

(a) 3

(b) 2

(c) -2/3

(d) 3/2

30. If one zero of the quadratic polynomial 2x² – 3x + p is 3, then its other zero is :

(a) −3/2

(b) 3/2

(c) 1/2

(d) – 1/2

31. If α, β are the zeroes of a polynomial such that α + β = 6 and αβ = 4, then the polynomial is :

(a) x² – 6x + 4

(b) –x² + 6x – 4

(c) x² – 6x – 4

(d) none of these

32. If two zeroes of a quadratic polynomial are 5 – 3√2 and 5 + 3√2 , then the quadratic polynomial is :

(a) x² – 10x – 7

(b) x² – 10x + 6

(c) x² – 10x + 14

(d) x² – 10x + 7

33. If α and β are the zeroes of the quadratic polynomial p(x) = ax² + bx + c, then 1/α + 1/β is

(a) −b/a

(b) b)c

(c) −b/c

(d) −a/b

34. If α and β are the zeroes of the quadratic polynomial p(x) = ax² + bx + c, then 1/α + 1/β – 2αβ is :

(a) (−ab -2c²) /ac

(b) (ab+2c²) /ac

(c) (ab+b²) /2c

(d) (−c² +ab)/2ac

35. If α and β are the zeroes of the quadratic polynomial p(x) = ax² + bx + c, then α³β² + α²β³ =

(a) −bc/a³

(b) −bc² / a³

(c) −ac/b³

(d) −a²c /b³

36. If α and β are the zeroes of the quadratic polynomial x² – 5x + k such that α – β = 1, then k =

(a) 2

(b) 3

(c) 4

(d) 6

37. If α, β and γ are the zeroes of the polynomial px³ – 5x + 9 and α³ + β³ + γ³ = 27, then :

(a) p = 0

(b) p = 1

(c) p = 2

(d) p = –1

38. If α and β are the zeroes of the polynomial p(x) = 2x² + 5x + k satisfying the relation α² + β² + αβ = 21/4 , then the value of k is :

(a) 1

(b) 2

(c) 3

(d) 4

39. The value of k such that 3x² + 2kx – k – 5 has the sum of the zeores as half of their product is :

(a) 2/3

(b) 5/3

(c) 7/3

(d) 8/3

40. If the sum and the product of the zeroes of the polynomial p(x) = 4x² – 27x + 3k² are equal, then the value of k is :

(a) ±2

(b) ±3

(c) ±5

(d) ±1

41. If α, β are the zeroes of x² – 6x + k, then the value of k when 3α + 2β = 20 is :

(a) 13

(b) –14

(c). 15

(d) –16

42. If the degree of polynomial p(x) is n, then the maximum number of zeroes it can have

is :

(a) n

(b) n²

(c) n³

(d) none of these

43. The value of k, if –4 is a zero of polynomial x² – x – (2k + 2) is :

(a) 5

(b) 6

(c) 7

(d) 9

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44. Zeroes of the polynomial 4x² – 9 are :

(a) ±2 /3

(b) ± 3/2

(c) ±5/2

(d) none of these

45. Sum of the zeroes of the polynomial x³ – 4x is :

(a) 1/2

(b) – 1/2

(c) 0

(d) – 3/2

46. If p and q are the zeroes of ax² – bx + c, a ≠ 0, then the value of p + q is :

(a) b/a

(b) c/a

(c) d/a

(d) – b/a

47. If 2 and –3 are the zeroes of the quadratic polynomial x² + (a + 1)x + b, then the value of a + b is :

(a) –5

(b) 0

(c) 6

(d) –6

48. On dividing x³ – 3x² + x + 2 by a polynomial g(x), the quotient and remainder were x – 2

and –2x + 4 respectively, then g(x) is :

(a) x² + x – 1

(b) x² – x + 1

(c) x² + 2x – 2

(d) x² + 2x + 3

49. If p(x) = 3x⁴ + 5x³ – 7x² + 2x + 2 is divided by g(x) = x² + 3x + 1, then reminder is :

(a) 1

(b) 0

(c) –1

(d) 2

50. If 1 is a zero of the polynomial 7x – x3 – 6, then other zeroes are :

(a) 1 and 2

(b) 2 and –3

(c) –3 and 2

(d) –2 and –3

FILL IN THE BLANKS.

51. If one of the zeroes of polynomial x² + ax + 5 is –1, then find the value of a._____________.

52. If two zeroes of the polynomial are √3 and -√3, then find the polynomial _____________.

53. Find the sum of zeroes of the polynomial 2x² - x - 1.___________.

54. If x² +4x + 4 is diveded by x+2, find remiander ____________.

55. .If sum and product of zeroes of the polynomial are 5 and -5, then find the polynomial.

56. If a and b are two zeroes of a polynomial x² -x -6, then find the value of (a+b) / ab

57. If zeroes are 1 ,2 ,3 of the polynomial then find polynomial__________.

60. If the zeroes of the polynomial x² + px + q are double in value to the zeroes of 2x² – 5x – 3, then the value of p and q are respectively ___________ and ____________.

61. If α and β are the zeroes of the polynomial 2x² + 7x + 5, then find α + β + αβ _________.

62. Find the degree of 5x³ + 2 x³y⁴ + 3√2.

63. If α and β are the zeroes of the quadratic polynomial p(x) = kx² + 4x + 4 such that

α² + β² = 24, then the value of k = ____________.

64. If α and β are the zeroes of the quadratic polynomial p(x) = x² – 5x + k such that α – β = 1, then the value of k = ____________.

65. If α and β are the zeroes of the quadratic polynomial p(x) = x² – px + q, then

α² + β² = ____________.

III. VERY SHORT ANSWER QUESTIONS.

66.: Find the quadratic polynomial if its zeroes are 0, √5.

Ans .x2 – √5x

67. How many zeros does the polynomial (x – 3)2 – 4 have? Also, find its zeroes.

Solution: the roots are 1 and 5.

68. What is the zero of a polynomial?

69. What is the zero of the linear polynomial ax – b?

70. Graph of a quadratic polynomial may be a straight line. Is it true?

71. A cubic polynomial can have at most how many zeroes?

72. Find the sum of the zeroes of the polynomial 2x2 – x + 4.

73. What is the product of zeroes of the polynomial y² – 5y – 3?

74. Write the polynomial whose zeroes are – 1 and 2

75.the zeroes of the polynomial x³ – 3x² + x + 1 are a – b, a, a + b, then find the value of a and b.

Solution: a=1 ,b=√2

76. The zeroes of x2–2x –8 are:

(a) (2,-4)
(b) (4,-2)
(c) (-2,-2)
(d) (-4,-4)
Answer: (b) (4,-2)

77. What is the quadratic polynomial whose sum and the product of zeroes is √2, 1/3 respectively?

(a) 3x2-3√2x+1

(b) 3x2+3√2x+1

(c) 3x2+3√2x-1

(d) None of the above

Answer: (a) 3x2-3√2x+1•

78.

If the zeroes of the quadratic polynomial ax2+bx+c, c≠0 are equal, then

(a) c and b have opposite signs

(b) c and a have opposite signs

(c) c and b have same signs

(d) c and a have same signs

Answer: (d) c and a have same signs

79.

The degree of the polynomial, x4 – x2 +2 is
(a) 2
(b) 4
(c) 1
(d) 0
Answer: (b) 4

80.

If one of the zeroes of cubic polynomial is x3+ax2+bx+c is -1, then product of other two zeroes is:

(a) b-a-1

(b) b-a+1

(c) a-b+1

(d) a-b-1

Answer: (b) b-a+1

Explanation: Since one zero is -1, hence;

P(x) = x3+ax2+bx+c

P(-1) = (-1)3+a(-1)2+b(-1)+c

0 = -1+a-b+c

c=1-a+b

Product of zeroes, αβγ = -constant term/coefficient of x3

(-1)βγ = -c/1

c=βγ

βγ = b-a+1

81.

If p(x) is a polynomial of degree one and p(a) = 0, then a is said to be:

(a) Zero of p(x)

(b) Value of p(x)

(c) Constant of p(x)

(d) None of the above

Answer: (a) Zero of p(x)

82.

Zeroes of a polynomial can be expressed graphically. Number of zeroes of polynomial is equal to number of points where the graph of polynomial is:

(a) Intersects x-axis

(b) Intersects y-axis

(c) Intersects y-axis or x-axis

(d) None of the above

Answer: (a) Intersects x-axis

83•

A polynomial of degree n has:

(a) Only one zero

(b) At least n zeroes

(c) More than n zeroes

(d) At most n zeroes

Answer: (d) At most n zeroes

Explanation: Maximum number of zeroes of a polynomial = Degree of the polynomial

84.

The number of polynomials having zeroes as -2 and 5 is:

(a) 1

(b) 2

(c) 3

(d) More than 3

Answer: (d) More than 3

Explanation: The polynomials x2-3x-10, 2x2-6x-20, (1/2)x2-(3/2)x-5, 3x2-9x-30, have zeroes as -2 and 5.

.Zeroes of p(x) = x2-27 are:

(a) ±9√3

(b) ±3√3

(c) ±7√3

(d) None of the above

Answer: (b) ±3√3

86.

Given that two of the zeroes of the cubic polynomial ax3 + bx2 + cx + d are 0, the third zero is
(a) -b/a
(b) b/a
(c) c/a
(d) -d/a
Answer: (a) -b/a

87.If one zero of the quadratic polynomial x2 + 3x + k is 2, then the value of k is

(a) 10

(b) –10

(c) 5

(d) –5

Answer: (b) -10

.A quadratic polynomial, whose zeroes are –3 and 4, is

(a) x² – x + 12

(b) x² + x + 12

(c) (x²/2) – (x/2) – 6

(d) 2x² + 2x – 24

Answer: (c) (x²/2) – (x/2) – 6

89.

The zeroes of the quadratic polynomial x2 + 99x + 127 are

(a) both positive

(b) both negative

(c) one positive and one negative

(d) both equal

Answer: (b) both negative

90.

The zeroes of the quadratic polynomial x2 + 7x + 10 are

(a) -4, -3

(b) 2, 5

(c) -2, -5

(d) -2, 5

Answer: (c) -2, -5

91.

If the discriminant of a quadratic polynomial, D > 0, then the polynomial has

(a) two real and equal roots

(b) two real and unequal roots

(c) imaginary roots

(d) no roots

Answer: (b) two real and unequal roots

92.

The product of the zeroes of the cubic polynomial ax3 + bx2 + cx + d is
(a) -b/a
(b) c/a
(c) -d/a
(d) -c/a
Answer: (c) -d/a

.If the graph of a polynomial intersects the x-axis at three points, then it contains ____ zeroes.

(a) Three

(b) Two

(c) Four

(d) More than three

Answer: (a) Three

94.. Which of the following is not the graph of quadratic polynomial?

Answer: d

95.If x3 + 1 is divided by x² + 5, then the possible degree of quotient is

(a) 0
(b) 1
(c) 2
(d) 3

Ans b

96. If x3 + 11 is divided by x² – 3, then the possible degree of remainder is

(a) 0
(b) 1
(c) 2
(d) less than 2

Ans d

97.. A quadratic polynomial whose one zero is 6 and sum of the zeroes is 0, is

(a) x2 – 6x + 2

(b) x2 – 36

(c) x2 – 6

(d) x2 – 3

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98. If x¹⁰¹ +1001 is divided by x + 1

Then reminder is:

(a) 0

(b) 1

(c) 1001

(d) 1000

99.

A quadratic polynomial whose one zero is 6 and sum of the zeroes is 0, is

(a) x2 – 6x + 2

(b) x2 – 36

(c) x2 – 6

(d) x2 – 3

100.

If p(x) = ax2 + bx + c, then is equal to

(a) 0

(b) 1

(c) sum of zeroes

(d) product of zeroes

101. Find the zeros of the polynomial x² + x - p(p-1)

102. Find zeros of the polynomial x² - 3x - m ( m + 3)

102. If 3 is a zero of the polynomial kx² + x+ k, then find the value of k.

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WORKSHEET OF POLYNOMIAL CLASS-10

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