Theory of Equations Test

Result

Theory of Equations Test - 3

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Weekly Quiz Competition

Question 1
1 / -0

Number of positive integers n for which n ² + 96 is a perfect square is

A
8

B
11

C
4

D
9

Solution

Let, m be a positive integer for which n² + 96 = m ²

m ² - n ² = 96

⇒ ( m + n ) ( m - n ) = 96

( m + n ) ( m - n ) - 2n } = 96

⇒ m + n and m - n must be both even 96 = 2 x 48 or, 4 x 24 or, 6 x 16 or, 8 x 12

Number of solution = 4.
Question 2
1 / -0

Statement – 1: If x ⊂ R (2, 3) then x² – 5x + 6 > 0

Statement – 2: If α < x < β, and a have opposite signs (α , β)

A
Both Statements– 1 and Statement– 2 are true and Statement– 2 is the correct explanation of Statement– 1

B
Both Statements– 1 and Statement– 2 are true and Statement– 2 is not the correct explanation of Statement– 1

C
Statement– 1 is true and Statement– 2 is false

D
Statement– 1 is false and Statement– 2 is true

Solution

x ⊂ (2, 3) x > 2 and x < 3

( x - 2 ) ( x - 3) < 0

x² - 5x + 6 < 0

If α < x < β and (α < β) then ax2 + bx +c and will have opposite sign.

The correct answer is: Statement– 1 is false and Statement– 2 is true
Question 3
1 / -0

Statement– 1: The set of all numbers a such that a² + 2a, 2a + 3 and a² + 3a + 8 are the sides of a triangle is (5, ∞)

Statement– 2: In a triangle sum of two sides is greater than the other and also sides are always positive.

A
Both Statements– 1 and Statement– 2 are true and Statement– 2 is the correct explanation of Statement– 1

B
Both Statements– 1 and Statement– 2 are true and Statement– 2 is not the correct explanation of Statement– 1

C
Statement– 1 is true and Statement– 2 is false

D
Statement– 1 is false and Statement– 2 is true

Solution

In triangle sum of two sides greater than the other

⇒ a² + 2a + 2a + > a² + 3a + 8

⇒ a > 5. (for positive a, a² + 3a + 8 is the greatest side)
Question 4
1 / -0

Let a ≠ – 2 and ( a + 2 ) x² + 2 ( a + 1 ) x a = 0 has integral roots.

Statement – 1: a can take four distinct integral values.

Statement – 2: a/a +2 must be integer.

A
Both Statements– 1 and Statement– 2 are true and Statement– 2 is the correct explanation of Statement– 1

B
Both Statements– 1 and Statement– 2 are true and Statement– 2 is not the correct explanation of Statement– 1

C
Statement– 1 is true and Statement– 2 is false

D
Statement– 1 is false and Statement– 2 is true

Solution

The equation can be rewritten as

(x - 1) [ x (a + 2) + a ] = 0

For both roots integer a / a+2 must be integer

⇒ a = 0, - 1, - 3, - 4
Question 5
1 / -0

If a < b < c < d, then for any real non– zero λ, the quadratic equation (x – a) (x – c) + λ (x – b) (x – d) = 0 has

A
one real root between a and c

B
one real root between b and c

C
Irrational roots

D
A and C are corrected

Solution

Let f (x) = (x - a) (x - c) + λ (x - b) (x - d).

Now

f (a) = λ (a - b) (a - d ) and f (c) = λ (c - d) (c - d)

∴ f (a) f (c) = λ² (a - b) (c - d) (c - b) (c - d) < 0

[Since a < b < c < d]

Hence, f (x) = 0 has one real root between a and c.

Again f (b) f (d) = ( b - a) ( b - c) ( d- a) ( d - c) < 0.

Hence, f (x) = 0 has one real root between a and d.
Question 6
1 / -0

If x² – 2x + sin² α = 0, then x may lie in the set

A
[– 1, 1]

B
[– 2, 2]

C
[1, 2]

D
[ 0, 2]

Solution

Given x² - 2x + sin² α

⇒ sin² c = 2x - x²

We know that - 1 ≥ sin α ≤ 2x - x2 ≤ 1

If 2x - x² ≥ 0 then 0 ≤ x ≤ 2

If 2x - x² ≤ 1 then ( x - 1 )² ≥ 0, which is true

0 ≤ x ≤ 2

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Theory of Equations Test - 3

Result

Theory of Equations Test - 3

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SHARING IS CARING

Question 1

8

11

4

9

Solution

Question 2

Both Statements– 1 and Statement– 2 are true and Statement– 2 is the correct explanation of Statement– 1

Both Statements– 1 and Statement– 2 are true and Statement– 2 is not the correct explanation of Statement– 1

Statement– 1 is true and Statement– 2 is false

Statement– 1 is false and Statement– 2 is true

Solution

Question 3

Both Statements– 1 and Statement– 2 are true and Statement– 2 is the correct explanation of Statement– 1

Both Statements– 1 and Statement– 2 are true and Statement– 2 is not the correct explanation of Statement– 1

Statement– 1 is true and Statement– 2 is false

Statement– 1 is false and Statement– 2 is true

Solution

Question 4

Both Statements– 1 and Statement– 2 are true and Statement– 2 is the correct explanation of Statement– 1

Both Statements– 1 and Statement– 2 are true and Statement– 2 is not the correct explanation of Statement– 1

Statement– 1 is true and Statement– 2 is false

Statement– 1 is false and Statement– 2 is true

Solution

Question 5

one real root between a and c

one real root between b and c

Irrational roots

A and C are corrected

Solution

Question 6

[– 1, 1]

[– 2, 2]

[1, 2]

[ 0, 2]

Solution

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