Mathematics Test 274

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Mathematics Test 274

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Question 1
4 / -1

In the given figure, co-ordinates of the midpoint of AB are?

A
(0, 2)

B
(0, 3)

C
(1, 2)

D
(3, 1)

Solution

Here we are given two points A(0, 0) and B (6, 2).

Since the midpoint falls midway between A and B, it has x-coordinates of A and B.

x-coordinates of midpoint of AB = 0+6/2 = 3.

and y-coordinates of the midpoint of AB = 0 + 2/2 = 1.

Thus the midpoint of AB has co-ordinates (3, 1).
Question 2
4 / -1

The coordinate of any point, which lies in xy plane, is?

A
(x,0,y)

B
(x,y,0)

C
(x,0,x)

D
(y,0,x)

Solution

Given that the point lies in xy plane

In xy plane, the coordinate of z will be zero

So (x,y,0) represents a point which lies in xy plane

therefore option b is correct.
Question 3
4 / -1

The point (3,0,−4) lies on the?

A
Y-axis

B
Z-axis

C
XY-plane

D
XZ-plane

Solution

(3,0,−4) → Given point

Clearly, y=0 and x and z have non-zero value.

If the point lies on x−z plane, this condition is possible.

Hence, the answer is XZ- plane.
Question 4
4 / -1

Equation of the plane perpendicular to the plane x – 2y + 5z + 1 = 0 which passes through the points (2, –3, 1) and (–1, 1, –7) is given by?

A
4x – 4y + z + 7 = 0

B
4x + 7y + 2z + 11 = 0

C
2x + y – z = 0

D
2x + y – 3z = 0

Solution

Equation of the plane through (2, –3, 1) is

a(x – 2) + b(y + 3) + c(z – 1) = 0

It passes through (–1, 1, –7) and perpendicular to the plane

x – 2y + 5z + 1 = 0

∴ –3a+ 4b – 8c = 0 and a – 2b + 5c = 0

∴ a/4 = a/7 = c/2

Hence, equation of required plane is

4x + 7y + 2z + 11 = 0
Question 5
4 / -1

If a line OP of length r (Where 'O' is the origin) makes an angle α with x-axis and lies on the xz-plane, then what are the coordinates of P?

A
(rcosα,0,rsinα)

B
(0,rsinα)

C
(rcosα,0)

D
(0,rcosα)

Solution

As OP lies on XZ plane so y co-ordinate is 0.

QP = r cos α

PR = r sin α

Co-ordinate of P = (rcosα,0,rsinα)

Option a is correct.
Question 6
4 / -1

If the projections of a line on the axes are 9, 12 and 8. Then the length of the line is?

A
7

B
17

C
21

D
25

Solution
Question 7
4 / -1

The plane meets the coordinate axes at A, B, C respectively. D and E are the mid-points of AB and AC respectively. Coordinates of the mid-point of DE are?

A

B

C

D

Solution
Question 8
4 / -1

The foot of the perpendicular from (0, 2, 3) to the line

A
(3, 2, –1)

B
(–2, 3, 4)

C
(2, –1, 3)

D
(2, 3, –1)

Solution

Let P (5λ - 3, 2λ + 1, 3λ - 4) be the foot of perpendicular of A (0, 2, 3). Then AP and given line are perpendicular to each other.

∴ (5λ - 3) 5 + (2λ - 1) 2 + (3λ - 7) 3 = 0

⇒ 38λ - 38 = 0 ⇒ λ = 1

Hence, P = (2, 3, - 1)
Question 9
4 / -1

In a single throw of two dice, find the probability that neither a doublet nor a total of 10 will appear.

A
7/9

B
2/9

C
8/9

D
9/7

Solution

Total outcomes = 36

Favourable outcomes for getting a doublet or a total of 10 = [(1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6), (4, 6), (6, 4)]

Probability = 8/36

Probability that neither a doublet nor a total of 10 will appear = 1 - 8/36 = 28/36 = 7/9
Question 10
4 / -1

A letter is known to have come from either TATANAGAR or CALCUTTA. On the envelope, just two consecutive letters, TA, are visible. The probability that the letter has come from CALCUTTA is

A
4/11

B
1/3

C
5/12

D
7/19

Solution

Let E₁ denote the event that the letter has come from TATANAGAR and E₂ the event that the letter has come from CALCUTTA.

Let B denote the event that the two consecutive alphabets visible on the envelope are TA.

We have P (E₁) = 1/2, P (E₂) = 1/2, P (B | E₁) = 2/8, P (B|E₂) = 1/7

Therefore, by the Bayes' theorem we have