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## Tamilnadu Samacheer Kalvi 11th Maths Solutions Chapter 2 Basic Algebra Ex 2.13

Choose the correct or the most suitable questions.

Question 1.

If |x + 2| ≤ 9, then x belongs to

(a) (-∞, -7)

(b) [-11, 7]

(c) (-∞, -7) ∪ [11, ∞)

(d)(-11, 7)

Solution:

(b) [-11, 7]

Hint:

Given |x + 2| ≤ 9

– 9 ≤ (x + 2) ≤ 9

– 9 – 2 ≤ x ≤ 9 – 2

– 11 ≤ x ≤ 7

∴ x ∈ [-11, 7]

Question 2.

Given that x, y and b are real numbers x < y, b ≥ 0, then ……..

(a) xb < yb (b) xb > yb

(c) xb ≤ vb

(d) xlb ≥ ylb

Solution:

(a) xb < yb

Hint:

Question 3.

(a) [2, ∞]

(b) (2, ∞)

(c) (-∞, 2)

(d) (-2, ∞)

Solution:

(b) (2, ∞)

Hint:

Question 4.

The solution of 5x – 1 < 24 and 5x + 1 > -24 is …….

(a) (4, 5)

(b) (-5, -4)

(c) (-5, 5)

(d) (-5, 4)

Solution:

(c) (-5, 5)

Hint:

Question 5.

The solution set of the following inequality |x – 1| ≥ |x – 3| is …….

(a) [0, 2]

(b) (2, ∞)

(c) (0, 2)

(d) (-∞, 2)

Solution:

(b) (2, ∞)

Question 6.

(a) 16

(b) 18

(c) 9

(d) 12

Solution:

(b) 18

Hint:

Question 7.

(a) -2

(b) -8

(c) -4

(d) -9

Solution:

(c) -4

Hint:

Question 8.

(a) 0.5

(b) 2.5

(c) 1.5

(d) 1.25

Solution:

(a) 0.5

Hint:

Question 9.

(a) 2

(b) 1

(c) 3

(d) 4

Solution:

(b) 1

Hint:

Question 10.

If 3 is the logarithm of 343, then the base is ……

(a) 5

(b) 7

(c) 6

(d) 9

Solution:

(b) 7

Hint.

Given log_{x} 343 = 3

343 = x^{3}

7 × 7 × 7 = x^{3}

7^{3} = x^{3}

x = 7

Base of the logarithm x = 7

Question 11.

Find a so that the sum and product of the roots of the equation 2x^{2} + (a – 3)x + 3a – 5 = 0 are equal is ……..

(a) 1

(b) 2

(c) 0

(d) 4

Solution:

(b) 2

Hint:

Question 12.

If a and b are the roots of the equation x^{2} – kx + 16 = 0 and satisfy a^{2} + b^{2} = 32, then the value of k is ……

(a) 10

(b) -8

(c) (-8, 8)

(d) 6

Solution:

(c) -8, 8

Hint:

Given a and b are the roots of x^{2} – kx + 16 = 0 satisfying a^{2} + b^{2} = 32

Sum of the roots a + b = \(\frac{-(-k)}{1}\)

a + b = k

Product of the roots ab = \(\frac{16}{1}\)

ab = 16

a^{2} + b^{2} = (a + b)^{2} – 2ab

32 = k^{2} – 2 × 16

32 = k^{2} – 32

k^{2} = 32 + 32 = 64

k = ± 8

Question 13.

The number of solutions of x^{2} + |x – 1| = 1 is ………

(a) 1

(b) 0

(c) 2

(d) 3

Solution:

(c) 2

We have two solutions 0, 1

Question 14.

The equations whose roots are numerically equal but opposite in sign to the roots of 3x^{2} – 5x – 7 = 0 is ……

(a) 3x^{2} – 5x – 7 = 0

(b) 3x^{2} + 5x – 7 = 0

(c) 3x^{2} – 5x + 7 = 0

(d) 3x^{2} + x – 7 = 0

Solution:

(b) 3x^{2} + 5x – 7 = 0

Hint:

Question 15.

If 8 and 2 are the roots of x^{2} + ax + c = 0 and 3, 3 are the roots of x^{2} + ax + b = 0, then the roots of the equation x^{2} + ax + b = 0 are …….

(a) 1, 2

(b) -1, 1

(c) 9, 1

(d) -1, 2

Solution:

(c) 9, 1

Hint:

Sum = 8 + 2 = 10 = -a ⇒ a = -10

Product = 3 × 3 = 9 = b ⇒ b = 9

Now the equation x^{2} + ax + b = 0

⇒ x^{2} – 10x + 9 = 0

⇒ (x- 9) (x – 1) = 0

x = 1 or 9

Question 16.

If a and b are the real roots of the equation x^{2} – kx + c = 0, then the distance

between the points (a, 0) and (b, 0) is ……..

Solution:

Hint:

a + b = k, ab = c

Question 17.

(a) 1

(b) 2

(c) 3

(d) 4

Solution:

(c) 3

Question 18.

(a) -1/2

(b) -2/3

(c) 1/2

(d) 2/3

Solution:

(a) -1/2

Hint:

Question 19.

The number of real roots of (x + 3)^{4} + (x + 5)^{4} = 16 is ……

(a) 4

(b) 2

(c) 3

(d) 0

Solution:

(a) 4

Hint:

The given equation is (x + 3)^{4} + (x + 5)^{4} = 16

Since it is a fourth degree equation it has four roots.

∴ Number of roots = 4

Question 20.

The value of log_{3} 11 . log_{11} 13 . log_{13} 15 . log_{15} 27 . log_{27} 81 is …….

(a) 1

(b) 2

(c) 3

(d) 4

Solution:

(d) 4

Solution:

(d) 4

Hint.

log_{3} 11 . log_{11} 13 . log _{13} 15 . log _{15} 27 . log _{27} 81

= log_{3} 13 . log _{13} 15 . log _{15} 27 . log _{27} 81

= log _{3} 15 . log _{15} 27 . log _{27} 81

= log _{3} 27 . log _{27} 81

= log _{3} 81

= log _{3}3^{4}

= 4 log _{3}3

= 4 × 1

= 4