Mathematics Test 238

Result

Mathematics Test 238

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Weekly Quiz Competition

Question 1
4 / -1

A

B

C

D

Solution
Question 2
4 / -1

A
AP

B
GP

C
HP

D
none of these

Solution
Question 3
4 / -1

If α, β be roots of x² – x – 1 = 0 and A_n = αⁿ + βⁿ, then Arithmetic Mean of A_n–1 and A_n is

A
2A_n+1

B
(1/2)A_n+1

C
2A_n–2

D
none of these

Solution
Question 4
4 / -1

In the value of 100! the number of zeros at the end is

A
11

B
22

C
23

D
24

Solution

= 20 + 4 + 0

= 24

Now, power of 5 is less than that of 2.

⇒ Number of power of 5 will decide the number of 10 formed

⇒ Number of 10 formed = 24

∴ Number of zeros in the end of 100! is 24.

The correct answer is Option 4.
Question 5
4 / -1

If n integers taken at random are multiplied together, then the probability that the last digit of the product is 1, 3, 7 or 9 is

A
2ⁿ/5ⁿ

B
4ⁿ – 2ⁿ/5ⁿ

C
4ⁿ/5ⁿ

D
none of these

Solution

Calculation:

Given, n integers taken at random are multiplied together

Now, last digit can be 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.

∴ Number of ways of choosing last digit = 10ⁿ

According to the question, last digit be 1, 3, 7, 9.

∴ Favourable number of ways = 4ⁿ

∴ Required Probability = 4ⁿ/10ⁿ = 2ⁿ/5ⁿ.

∴ The probability that the last digit of the product is 1, 3, 7 or 9 is 2ⁿ/5ⁿ.

The correct answer is Option 1.
Question 6
4 / -1

The distance of the lines 2x – 3y = 4 from the point (1, 1) measured parallel to the line x + y = 1 is

A
√2

B

C

D
6

Solution

Calculation:

Equation of line parallel to x + y = 1 is x + y + c = 0

Let it passes through (1, 1) so the line is along direction of x + y = 1

⇒ 1 + 1 + c = 0

⇒ c = − 2

⇒ x + y − 2 = 0 … (i)

Given line is 2x − 3y − 4 = 0 … (ii)

Solving (i) and (ii), we get:

x = 2 and y = 0

∴ The point of intersection is (2, 0)

∴ Distance of (1, 1) from 2x − 3y − 4 = 0 along x + y + 1 = 0

= Distance of (1, 1) from (2, 0)

∴ The distance of the lines 2x – 3y = 4 from the point (1, 1) measured parallel to the line x + y = 1 is √2.

The correct answer is Option 1.
Question 7
4 / -1

In a triangle ABC, if A is (2, –1), and 7x – 10y + 1 = 0 and 3x – 2y + 5 = 0 are equations of an altitude and an angle bisector, respectively, drawn from B, then equation of BC is

A
x + y + 1 = 0

B
5x + y + 17 = 0

C
4x + 9y + 30 = 0

D
x – 5y – 7 = 0

Solution

Calculation:

Given, A is (2, –1), and

Equation of BD and BE are 7x – 10y + 1 = 0 and 3x – 2y + 5 = 0 repectively.

Solving, BD and BE we get,

Coordinates of B = (- 3, - 2)
Question 8
4 / -1

Let f (x) = (x – 1)^p (x – 2)^q where p > 1, q > 1, each critical point of f (x) is a point of extremum when

A
p = 3, q = 4

B
p = 4, q = 2

C
p = 2, q = 3

D
none of these

Solution

For it to be point of extremum, p - 1 should be odd and q - 1 should be odd

⇒ p should be even and q shoudl be even.

Possible combination is p = 4 and q = 2.

∴ The value of p and q are p = 4 and q = 2.

The correct answer is Option 2.
Question 9
4 / -1

A variable chord of circle x² + y² = 4 is drawn from the point P(3, 5) meeting the circle at the points A and B. A point Q is taken on this chord such that 2PQ = PA + PB. Locus of Q is

A
x² + y² + 3x + 4y = 0

B
x² + y² = 36

C
x² + y² = 16

D
x² + y² – 3x – 5y = 0

Solution

Calcualtion:

Given, 2PQ = PA + PB

⇒ PQ − PA = PB − PQ

⇒ AQ = QB

⇒ Q is midpoint of AB.

Let Q has coordinates (h, k)

∴ Equation of chord AB is given by T = S₁

⇒ hx + ky − 4 = h² + k² − 4

Now, chord passes through the point P(3, 5)

⇒ 3h + 5k = h² + k²

⇒ h² + k² − 3x − 5y = 0

Replacing h by x and k by y, we get:

⇒ x² + y² − 3x − 5y = 0

∴ The equation of chord is x² + y² − 3x − 5y = 0.

The correct answer is Option 4.
Question 10
4 / -1

The angle between the tangents to the parabola y² = 4ax at the points where it intersects with the line x – y – a = 0 is

A

B

C

D

Solution

Calculation:

Given, y² = 4ax

The coordinates of the focus of the parabola y² = 4ax are (a, 0).

Now, the line x – y – a = 0 passes through this point.

⇒ It is a focal chord of the parabola.

⇒ Tangent intersects at right angle.

∴ The angle between the tangents to the parabola y² = 4ax at the points where it intersects with the line x – y – a = 0 is .

The correct answer is Option 4.