Mathematics Test 209

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Mathematics Test 209

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Weekly Quiz Competition

Question 1
4 / -1

Let α, β be such that π < α - β < 3π.

If sinα + sinβ = - 21/65 and cosα + cosβ = -27/65, then value of cos α - β / 2 equals

A
6/65

B
- 5/65

C

D
-6/65

Solution
Question 2
4 / -1

The number of ways in which 6 men and 5 women can dine at a round table, if no 2 women are to sit together, is given by

A
30

B
5! x 4!

C
7! x 5!

D
6! x 5!

Solution

Number of women = 5

Number of men = 6

Number of ways of 6 men at a round table is (n - 1)! = (6 - 1)! = 5!

Now, we are left with six places between the men and there are 5 women. These 5 women can be arranged in ⁶P5 ways.

Required number of ways = 5! x ⁶P5 = 5! x 6!
Question 3
4 / -1

A
continuous for all x, but not differentiable at x = 0

B
neither differentiable nor continuous at x = 0

C
discontinuous everywhere

D
continuous as well as differentiable for all x

Solution
Question 4
4 / -1

If y = a Inx + bx² + x has its extremum values at x = -1 and x = 2, then

A
a = 2, b = -1

B
a = 2, b = -1/2

C
a = -2, b = 1/2

D
None of these

Solution

y = a Inx + bx² + x has its extremum values at x = -1 and 2.

∴ dy/dx = 0 at x = -1 and 2

a/x + 2bx + 1 = 0

or, 2bx² + x + a = 0 has -1 and 2 as its roots.

∴ 2b - 1 + a = 0 ... (1)

8b + 2 + a = 0 ... (2)

Solving (1) and (2), we get a = 2, b = -1/2
Question 5
4 / -1

The slope of the tangent to a curve Y = f(x) at [x, f(x)] is 2x + 1. If the curve passes through the point (1, 2), then the area bounded by the curve, the x-axis and the line x = 1 is

A
5/6

B
6/5

C
1/6

D
6

Solution
Question 6
4 / -1

The graph of function cos x cos (x + 2) - cos² (x + 1) is a

A
straight line passing through (0, -sin² 1) with slope 2

B
straight line passing through (0, 0)

C
parabola with vertex (1, -sin² 1)

D
straight line passing through the point and parallel to the x-axis

Solution

Let y = cos x cos (x + 2) - cos² (x + 1)

= cos x[cos x cos 2 - sin x sin 2] - [cos x cos 1 - sin x sin 1]²

Put x = 0

y = cos 2 - cos² 1 = 1 - 2 sin² 1 - 1 + sin² 1

= -sin² 1

Put x = π/2 , y = -sin² 1

Put x = π

y = -1[-cos 2] - cos² 1

= cos 2 - cos² 1 = -sin² 1

Graph is shown.

Thus, the graph of the function is a straight line passing through the point and is parallel to the x-axis.
Question 7
4 / -1

A particle acted upon by forces is displaced from a point P to a point Q whose respective position vectors are and . The work done by the force is A particle acted upon by forces and is displaced from a point P to a point Q whose respective position vectors are The work done by the force is

A
4 units

B
-4 units

C
-3 units

D
3 units

Solution
Question 8
4 / -1

or events A, B and C, P(Exactly one of either A or B occurs) = P(Exactly one of either B or C occurs) = P(Exactly one of either C or A occurs) = p and P(All the three events occur simultaneously) = p², where 0 < p < 1/2 . The probability of at least one of A, B and C occurring is

A

B

C

D

Solution

Probability of either A or B occurring = P(A) - P(A ⌒ B) + P(B) - P(A ⌒ B) = p.....(i)

Similarly, probability of either B or C occurring = P(B) - P(B ⌒ C) + P(C) - P(B ⌒ C) = p......(ii)

Probability of either C or A occurring = P(C) - P(A ⌒ C) + P(A) - P(A ⌒ C) = p......(iii)

Adding equations (i), (ii) and (iii) and dividing by 2, we get

P(A) + P(B) + P(C) - P(A ⌒ B) - P(B ⌒ C) - P(C ⌒ A) = 3p/2

Also, given that P(A ⌒ B ⌒ C) = p2

Now, P(A B ∪ C) = P(A) + P(B) + P(C) - P(A ∪ B) - P(B ∪ C) - P(C ∪ A) + P(A ⌒ B ⌒ C) =
Question 9
4 / -1

If g(x) is a function defined on [-1, 1] and the area of the equilateral triangle with two of its vertices at (0, 0) and (x, g(x)) is then the function is

A

B

C

D

Solution
Question 10
4 / -1

Consider the following linear programming problem (LPP).

Maximize z = 2x + 3y

Subject to

2x + 3y ≤ 12

2x - 3y ≤ 0

y ≤ 2

x, y ≥ 0

It has

A
no feasible solution

B
unique optimal solution

C
alternative optimal solutions

D
unbounded solution

Solution

z = 2x + 3y

Subject to

2x + 3y ≤ 12

2x - 3y ≤ 0, y ≤ 2, x, y ≥ 0

Feasible region is shown in the figure.

z will be maximum at (3, 2) and maximum value of z is 12.

Hence, given LPP has unique optimal solution.