Mathematics Test 145

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Mathematics Test 145

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Question 1
4 / -1

The equations of the tangents of the parabola y²= 12x, which passes through the point (2,5).

A
2y = 3x + 4, y = x + 3

B
2y = 3x – 4, y = x + 3

C
y = 3x + 4, y = x – 3

D
2y = 3x – 4, y = x – 3

Solution

y²= 12 x

4a = 12 (∵ y²= 4ax)

⇒ a = 3

Equation of tangent in slope form is

y = mx + 3/m

It passes through (2,5)

⇒ 5=2m+3/m

⇒ 5m=2m²+3

⇒ 2m²-5m+3=0

⇒ (m-1)(2m-3)=0

⇒ m = 1,3/2

So equation of tangents are
Question 2
4 / -1

The focal chord to y² = 16xis tangent to (x - 6)² + y²= 2, then the possible values of the slope of this chord, are

A
{- 1, 1}

B
{- 2, 2}

C
{- 2, 1/2}

D
{2, - 1/2}

Solution

The focal chord of y² = 16 x is tangent to circle (x - 6)² + y² = 2 .

Focus of parabola is (a; 0) i.e., (4, 0) is the Focus

Now, tangents are drawn from (4, 0) to (x - 6)² + y² = 2 Since, PA is tangent to circle.

Since, tanθ = slope of tangent

∴ Slope of focal chord as tangent to circle

= ±1
Question 3
4 / -1

If P(1) = 0 and then value of x for which is P(x) > 0 satisfy interval

A
x > 1

B
1 < x < 10

C
x ∈ N

D
x ∈ φ

Solution
Question 4
4 / -1

If 5a + 4b + 20c = t, then the value of t for which the line ax + by + c - 1 = 0 always passes through a fixed point is

A
0

B
20

C
30

D
None of these

Solution
Question 5
4 / -1

If f(x) is a polynomial satisfying f (x) f (1/x) = f (x) + f (1/x) and f (2) > 1,

A
2

B
1

C
-1

D
None of these

Solution
Question 6
4 / -1

The equation of the pair of straight lines parallel to the y - axis and which are tangents to the circle x² + y² - 6x - 4y - 12 = 0 is :

A
x² - 4x - 21 = 0

B
x² - 5x + 6 = 0

C
x² - 6x - 16 = 0

D
None of these

Solution
Question 7
4 / -1

The number of common tangents that can be drawn to the circles x² + y² - 4x - 6y - 3 = 0 and x² + y² + 2x + 2y + 1 = 0 is :

A
1

B
2

C
3

D
4

Solution

Hence the given circles are externally touching each other Hence, number of common tangents = 3
Question 8
4 / -1

The distance between the circumcentre and orthocentre of the triangle whose vertices are (0, 0), (6, 8) and (-4, 3) is

A
125 / 8 unit

B
√5/2 unit

C
5√5 / 2 unit

D
5 √ 5 unit

Solution
Question 9
4 / -1

Set of values of x for which sin^-1x > cos^-1x is

A

B

C

D

Solution
Question 10
4 / -1

If n is a positive integer and C_k = ⁿC_k, then

A

B

C

D

Solution