Mathematics Test 222

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Mathematics Test 222

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Question 1
4 / -1

If a₁, a₂, a₃,....,a₂₀ are the arithmetic means between 13 and 67, then the maximum value of the product a₁⋅a₂⋅a₃⋅....a₂₀ is

A
(20)²⁰

B
(60)²⁰

C
(80)²⁰

D
(40)²⁰

Solution

∴ The maximum value of a₁⋅a₂⋅a₃⋅....a₂₀ is (40)²⁰
Question 2
4 / -1

The locus of the mid point of the line segment joining the focus to a moving point on the parabola y² = 4ax is another parabola with directrix

A
x = −a

B
x = −a/2

C
x = 0

D
x = a/2

Solution

Let p(h,k) be the mid point of the line segment joining the focus (a,0) and a general point Q(x, y) on the parabola. Then,

⇒ x = 2h − a, y = 2k

Put these values of x and y in y²= 4ax to get

4k² = 4a(2h − a)

⇒ 4k² = 8ah − 4a²

⇒ k² = 2ah − a²

So, the locus of P(h, k) is y² = 2ax − a²

Its directrix is x − a/2 = −a/2

⇒ x = 0.
Question 3
4 / -1

The length of the diameter of the circle which touches the X-axis at the point (1, 0) and passes through the point (2, 3) is

A
6/5

B
5/3

C
10/3

D
3/5

Solution

Let, the centre of the required circle is (1, k) and radius |k|

So, the equation of the required circles is

(x−1)² +(y − k)² = k²

Given, it passes through (2, 3)

Thus diameter = 10/3
Question 4
4 / -1

The vertex of a parabola is (1,2) and its axis is parallel to y-axis. If parabola passes through (0,6), then the length of its latus rectum is _____

A
1/2

B
2

C
1/4

D
4

Solution

The equation will be of the form

(y − 2) = λ(x − 1)²

Passing through (0,6)

⇒ 4 = λ(0 − 1)²

⇒ λ = 4

⇒ (y − 2) = 4(x−1)²

Comparing with x² = 4ay, we get

The length of latus rectum = 4a = 1/4
Question 5
4 / -1

There is an unlimited number of identical balls of four different colours. How many arrangements of at most 88 balls in a row can be made by using them?

A
97380

B
87380

C
87370

D
87480

Solution

The number of arrangements of one ball = 4, because there are only four different balls.

The number of arrangements of two balls = 4 × 4 = 4²

∴ The required number of arrangements = 4 + 4²+ 4³+....+4⁸
Question 6
4 / -1

The value of the determinant is equal to

A
2 (10! 11!)

B
2 (10! 13!)

C
2 (10! 11! 12!)

D
2 (11! 12! 13!)

Solution
Question 7
4 / -1

If f(x) = a − (x − 3)^8/9, then the maximum value of f(x) is

A
3

B
a - 3

C
a

D
None of these

Solution
Question 8
4 / -1

The volume of the largest possible right circular cylinder that can be inscribed in a sphere of radius √3 is

A
4π

B
4/3√3π

C
8/3√3π

D
2π

Solution

Let r be the radius of the cylinder and 2h be the height.
Question 9
4 / -1

If p, q ∈ {1, 2, 3, 4} the number of equations of the form px² + qx + 1 = 0 having real roots is

A
15

B
9

C
7

D
8

Solution

For real roots, we must have

Disc ≥ 0 ⇒ q² − 4p ≥ 0 ⇒ q² ≥ 4p

If p=1, then

q² ≥ 4p ⇒ q² ≥ 4 ⇒ q = 2, 3, 4

If p = 2, then q² ≥ 4p ⇒ q² ≥ 8 ⇒ q = 3, 4

If p = 3, then q² ≥ 4p ⇒ q² ≥ 12 ⇒ q = 4

If p = 4, then q² ≥ 4p ⇒ q² ≥ 16 ⇒ q = 4

Thus, there are 7 values of (p, q)

⇒7 such equations are possible.
Question 10
4 / -1

sin²θcos³θ − sin⁴θcosθ is equal to

A
1/2sin²2θ

B
1/4sinθsin4θ

C
1/2cosθsin4θ

D
1/4cosθsin4θ

Solution