Mathematics Test 203

Result

Mathematics Test 203

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Weekly Quiz Competition

Question 1
4 / -1

The last two digits of the number 3⁴⁰⁰ are

A
81

B
43

C
29

D
01

Solution

3⁴⁰⁰ = 81¹⁰⁰ = (1 + 80)¹⁰⁰= ¹⁰⁰C₀

+ ¹⁰⁰C₁ 80 + ....... + ¹⁰⁰C₁₀₀ 80¹⁰⁰

⇒ Last two digits are 01
Question 2
4 / -1

There exist positive integers A, B and C with no common factors greater than 1, such that A log₂₀₀5 + B log₂₀₀2 = C. The sum (A + B + C) equals

A
5

B
6

C
7

D
8

Solution

A log₂₀₀5 + B log₂₀₀2 = C

A log 5 + B log 2 = C log 200 = C log(5² 2³)

= 2C log 5 + 3 C log 2

hence, A = 2C

B = 3C

for no common factor greater than 1, C = 1

∴ A = 2; B = 3 ⇒ A + B + C = 6
Question 3
4 / -1

A
independent of both α and β

B
independent of α but dependent on β

C
independent of β but dependent on α

D
dependent on both α and β.

Solution

Let
E = sin²α + sin²β + cos²(α + β) + 2 · sin α · sin β · cos(α + β)
= sin²α + sin²β + cos² (α + β) + [cos (α - β) - cos (α + β)] · cos (α + β)
= sin²α + sin²β +(cos²α - sin²β) = 1. Ans.

Aliter: E = sin²α + sin²β + cos² (α + β) + 2sin α sin β cos (α + β)
= sin²α - sin² (α + β) + sin²β + 1 + 2sin α sin β cos (α + β)
= - sin (2α + β) · sin β + sin² β + 1 + 2sin α sin β cos (α + β)
= 1 - sin β [sin (2α + β) - sin β] + 2 sin α sin β cos (α + β)
= 1 - sin β [2 cos (α + β) sin α] + 2 sin α sin β cos (α + β)
= 1 - 2 sin α sin β cos (α + β) + 2 sin α sin β cos (α + β)
∴ E = 1 ⇒ (A). Ans.
Question 4
4 / -1

Smallest positive x satisfying the equation cos³3x + cos³5x = 8 cos³4x · cos³x is

A
15°

B
18°

C
22.5°

D
30°

Solution

cos³3x + cos³5x = (2 cos 4x cos x)³ = (cos 5x + cos 3x)³

cos³3x + cos³5x = cos³5x + cos³3x + 3 cos 5x cos 3x (cos 5x + cos 3x)

⇒ (3 cos 3x · cos 5x) (2 cos 4x · cos x) = 0

⇒ cos x · cos 3x · cos 4x · cos 5x = 0

⇒ smallest + ve values of x is π/10 i.e. 18° Ans.
Question 5
4 / -1

If the quadratic equations 3x² + ax + 1 = 0 and 2x² + bx + 1 = 0 have a common root, then the value of the expression 5ab - 2a² - 3b² is

A
0

B
1

C
-1

D
2

Solution
Question 6
4 / -1

Solution set of the inequality

A
(0, 2]

B
[–3, 2)

C
(– ∞, -3] ∪ (0, 2]

D
[- 3, 0) ∪ (2, ∞)
Question 7
4 / -1

The sum to infinity of the series

A
2

B
5/2

C
3

D
none of these

Solution
Question 8
4 / -1

Consider the sequence 8A + 2B, 6A + B, 4A, 2A – B, ........ Which term of this sequence will have a coefficient of A which is twice the coefficient of B?

A
10^th

B
14^th

C
17^th

D
none

Solution

coefficient of A in n^th term = 8 + (n – 1)(– 2)

= 10 – 2n

coefficient of B in n^th term = 2 + (n – 1)(– 1)

= 3 – n

10 – 2n = 2(3 – n) ⇒ 10 = 6

which is absurd ⇒ none
Question 9
4 / -1

A
9

B
5

C
1

D
none of these

Solution
Question 10
4 / -1

Let k₁, k₂ (k₁ < k₂) be two values of k for which the expression x² – y² +kx +1 can be factorised into two real linear factors, then (k₂ – k₁) is equal to

A
2

B
- 2

C
0

D
4

Solution