If ?, ? are the roots of the equation 2x² - 3x + 1 = 0, form an

equation whose roots are	?	and	?
	?		?

Correct Answer: 11, 13

Explanation:

According to question ,
Let, the two consecutive odd positive integers be 2x + 1 and 2x + 3 where x is a whole number.
Now, ( 2x + 1 )² + ( 2x + 3 )² = 290
? 4x² + 4x + 1 + 4x² + 12x + 9 = 290
? 8x² + 16x - 280 = 0
? x² + 2x - 35 = 0
? ( x + 7 ) ( x - 5 ) = 0
? x = 7, - 5
But, x = ?7 is not possible, since ?7 is not a whole number .
? x = 5.
We get , 2x + 1 = 2 x 5 + 1 = 11 and 2x + 3 = 2 x 5 + 3 = 13
Thus , two consecutive positive odd integers are 11 and 13 .

Correct Answer: p2 = 4q

Explanation:

On comparing it with ax² + bx + c = 0 , we get a = 1, b = p, c = q
The roots of the equation x² + px + q = 0 are equal if
b² - 4ac = 0
? p² - 4q = 0 ? p² = 4q.
Thus , required answer is option B .

Correct Answer: k ? 25 12

Explanation:

Correct Answer: b ( 3ac - b2 ) a 3

Explanation:

Correct Answer: 2b , b a a

Explanation:

Correct Answer: (a ? 2b ? 4) (a ? 2b + 2)

Explanation:

Given that :- The factors of a² + 4b² + 4b ? 4ab ? 2a ? 8
= a² + 4b² ? 4ab ? 2a + 4b ? 8
= (a ? 2b) ² ? 2(a ? 2b) ? 8
Let, (a ? 2b) = x
? The given expression = x² ? 2x ? 8
= x² ? 4x + 2x ? 8
= x(x ? 4) + 2(x ? 4)
= (x ? 4) (x + 2)
Putting the value of x , we get
? a² + 4b² + 4b ? 4ab ? 2a ? 8 = (a ? 2b ? 4) (a ? 2b + 2)
Therefore , required answer will be (a ? 2b ? 4) (a ? 2b + 2) .

Correct Answer: 6

Explanation:

According to question ,we can say that
Let , x = ?30 + ?30 + ?30 +........
On squaring both sides, we have
x² = 30 + ?30 + ?30 + ?30 +........
? x² = 30 + x ? x² - x - 30 = 0
? x² - 6x + 5x - 30 = 0
? x( x - 6 ) + 5( x - 6 ) = 0
? ( x - 6 ) ( x + 5 ) = 0
? x = 6 because x ? - 5
Hence , required answer is 6 .

Correct Answer: 28

Explanation:

Let, the two natural consecutive odd numbers be n and (n + 2)
Now, according to the question,
? n² + ( n + 2 )² = 394
? n² + n² + 4 + 4n = 394
? 2n² + 4n - 390= 0
? n² + 2n - 195 = 0
? n² + 15n - 13n - 195 = 0
? n( n + 15 ) - 13( n + 15 ) = 0
? ( n + 15 ) ( n - 13 ) = 0
? n = 13 and n ? - 15
? The two natural consecutive odd numbers be 13 and (13 + 2) = 15 .
? the sum of the numbers = 13 + 15 = 28
Quicker Approach:
By mental operation, 13² + 15² = 169 + 225 = 394
? Required sum = 13 + 15 = 28

Correct Answer: c = a

Explanation:

Correct Answer: 1

Explanation:

According to question ,we have
From equation ?. x² ? 4 = 0
? x = ± 2,
From equation ?. y² + 6y + 9 = 0
? ( y + 3 )² = 0
? ( y + 3 )( y + 3 ) = 0
? y + 3 = 0 ? y = -3 , -3
Clearly , x > y is required answer .