Solve the followings Questions.

1. Verify that the numbers given alongside of the cubic polynomials below are their zeroes. Also verify the relationship between the zeroes and the coefficients in each case:

(i) 2x³ + x² – 5x + 2 ; 1/1,1,-2

(ii) x³ – 4x² + 5x – 2 ; 2,1,1

Answer:

(i) 2x³ + x² – 5x + 2; ¹/₂ , 1, -2

p(x) = 2x³ + x² – 5x + 2

Zeros for this polynomial are ¹/₂ , 1, -2

p(¹/₂) = 2(¹/₂)³ + (¹/₂)² – 5(¹/₂) + 2

= (¹/₄) + (¹/₄) – (⁵/₂) + 2

= 0

p(1) = 2(1)³ + (1)² – 5(1) + 2 = 0

p(-2) = 2(-2)³ + (-2)² – 5(-2) + 2

= -16 + 4 + 10 + 2

= 0

Therefore, ½ , 1, and −2 are the zeroes of the given polynomial. Comparing the given polynomial with ax³ + bx² + cx + d, we obtain a = 2, b = 1, c = −5, d = 2

Let us take α = ¹/₂ ,β = 1, γ = -2

α + β + γ = ¹/₂ + 1 + (-2) = –¹/₂ = –^b/_a

αβ + βγ + γα = ¹/₂ x 1 + 1(-2) + ¹/₂ (-2) = ^-5/₂ = ^c/_a

αβγ = ¹/₂ x 1 x (-2) = –¹/₁ = – ²/₂ = –^d/_a

Therefore, the relationship between the zeroes and the coefficients is verified.

(ii) x³ – 4x² + 5x – 2; 2,1,1

p(x) = x³ – 4x² + 5x – 2

Zeros for this polynomial are 2 , 1, 1

p(2) = 2³ – 4(2)² + 5(2) – 2

= 8 – 16 + (10) – 2

= 0

p(1) = (1)³ – 4(1)² + 5(1) – 2 = 0

Therefore, 2 , 1, and 1 are the zeroes of the given polynomial. Comparing the given polynomial with ax³ + bx² + cx + d, we obtain a = 1, b = -4, c = 5, d = -2

Verify:

Sum of zeros = 2+1+1 = 4 = – ^(-4)/₁ = –^b/_a

Multiplication of zeroes taking two at a time = (2)(1)+(1)(1)+(2)(1) = 2+1+2= 5 = ⁵/₁ = ^c/_a

Multiplication of zeroses = 2 x 1 x 1 = 2 = – ^(-2)/₁ = –^d/_a

Hence, the relationship between the zeroes and the coefficients is verified.

2. Find a cubic polynomial with the sum of the product of its zeroes taken two at a time and the product of its zeroes are 2, -7,-14respectively.

Answer:

Let the polynomial be ax³ + bx² + cx + d and the zeroes be α, β and γ.

It is given that

α + β + γ = ²/₁ = –^b/_a

αβ + βγ + γα = ^-7/₂ = ^c/_a

αβγ = – ¹⁴/₁ = –^d/_a

If a = 1, then b = −2, c = −7, d = 14 Hence, the polynomial is x³ – 2x² – 7x + 14

3. If the zeroes of the polynomialx³ – 3x² + x + 1 are a – b, a , a + b, finda and b .

Answer:

p(x) = x³ -3x²+x+1

Zeroes are a − b, a + a + b Comparing the given polynomial with px³+qx²+rx+1, we obtain p = 1, q = −3, r = 1, t = 1

Sum of zeros = a – b + a + a + b

^-q/ _p = 3a

–^(-3)/₁ = 3a

3 = 3a

a = 1

The zeroes are 1-b, 1, 1+b

Multiplication of zeroes = 1(1-b)(1+b)

^-t/_p = 1 – b²

^-1/₁= 1 – b²

1 – b² = -1

1 + 1 = b²

b = ±√2

4. If the two zeroes of the polynomial x⁴ – 6x³ – 26x² + 138x – 35 are 2 + √3, 2 – √3 find other zeroes.

Answer:

Given 2 + √3 and 2 – √3 are zeroes of the given polynomial. So, (2 + √3)(2 – √3) is a factor of polynomial.

Therefore,[x – (2 + √3)][x – (2 – √3)] = x² + 4 − 4x − 3 = x² − 4x + 1 is a factor of the given polynomial

For finding the remaining zeroes of the given polynomial, we will find

Clearly, x⁴ -6x³ -26x²+138x -35 = (x² – 4x + 1)(x² – 2x – 35)

We know, (x² – 2x – 35) = (x – 7)(x + 5)

Therefore, the value of the polynomial is also zero when (x -7 = 0) or (x + 5 = 0)

Or x = 7 or −5

therefore, 7 and −5 are also zeroes of this polynomial.

5. If the polynomial x⁴ – 6x³ + 16x² – 25x +10 is divided by another polynomial x² – 2x + k, the remainder comes out to be x+a,findkanda.

Answer:

By division algorithm we know,

Dividend = Divisor × Quotient + Remainder

Dividend − Remainder = Divisor × Quotient

x⁴ – 6x³ + 16x² – 25x + 10 – x – a = x⁴ – 6x³ + 16x² – 26x + 10 – a will be divisible by x² – 2x + k

It can be observed that (-10+2k)x +(10-a-8k+k²) will be zero.

Thus, (-10 + 2k) = 0 and (10 – a – 8k +k²) = 0

For (-10+2k) = 0 , 2k = 10 and thus k = 5

For (10 – a – 8k + k²) = 0

10 − a − 8 × 5 + 25 = 0

10 − a − 40 + 25 = 0

− 5 − a = 0

Therefore, a = −5

Hence, k = 5 and a = −5

CHAPTER NAME	OLD NCERT	NEW NCERT
Real Numbers	EXERCISE 1.1
	EXERCISE 1.2	1.1	CLICK HERE
	EXERCISE 1.3	1.2	CLICK HERE
	EXERCISE 1.4
Polynomials	EXERCISE 2.1	2.1	CLICK HERE
	EXERCISE 2.2	2.2	CLICK HERE
	EXERCISE 2.3
	EXERCISE 2.4
Pair of Linear Equations in Two Variables	EXERCISE 3.1
	EXERCISE 3.2	3.1	CLICK HERE
	EXERCISE 3.3	3.2	CLICK HERE
	EXERCISE 3.4	3.3	CLICK HERE
	EXERCISE 3.5
	EXERCISE 3.6
	EXERCISE 3.7
Quadratic Equations	EXERCISE 4.1	4.1	CLICK HERE
	EXERCISE 4.2	4.2	CLICK HERE
	EXERCISE 4.3
	EXERCISE 4.4	4.3	CLICK HERE
Arithmetic Progressions	EXERCISE 5.1	5.1	CLICK HERE
	EXERCISE 5.2	5.2	CLICK HERE
	EXERCISE 5.3	5.3	CLICK HERE
	EXERCISE 5.4	5.4 (Optional)	CLICK HERE
Triangles	EXERCISE 6.1	6.1	CLICK HERE
	EXERCISE 6.2	6.2	CLICK HERE
	EXERCISE 6.3	6.3	CLICK HERE
	EXERCISE 6.4
	EXERCISE 6.5
	EXERCISE 6.6
Coordinate Geometry	EXERCISE 7.1	7.1	CLICK HERE
	EXERCISE 7.2	7.2	CLICK HERE
	EXERCISE 7.3
	EXERCISE 7.4
Introduction to Trigonometry	EXERCISE 8.1	8.1	CLICK HERE
	EXERCISE 8.2	8.2	CLICK HERE
	EXERCISE 8.3
	EXERCISE 8.4	8.3	CLICK HERE
Some Applications of Trigonometry	EXERCISE 9.1	9.1	CLICK HERE
Circles	EXERCISE 10.1	10.1	CLICK HERE
	EXERCISE 10.2	10.2	CLICK HERE
Construction
Areas Related to Circles	EXERCISE 12.1
	EXERCISE 12.2	11.1	CLICK HERE
	EXERCISE 12.3
Surface Areas and Volumes	EXERCISE 13.1	12.1	CLICK HERE
	EXERCISE 13.2	12.2	CLICK HERE
	EXERCISE 13.3
	EXERCISE 13.4
	EXERCISE 13.5
Statistics	EXERCISE 14.1	13.1	CLICK HERE
	EXERCISE 14.2	13.2	CLICK HERE
	EXERCISE 14.3	13.3	CLICK HERE
	EXERCISE 14.4
Probability	EXERCISE 15.1	14.1	CLICK HERE
	EXERCISE 15.2