Mathematics Test

Result

Mathematics Test - 15

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Weekly Quiz Competition

Question 1
1 / -0

A
x²+ y² = 4/p²

B
x² + y² = 4p²

C
1/x²+ 1/y²= 2/p²

D
1/x² + 1/y²= 4/p²

Solution
Question 2
1 / -0

Find the equation of the line passing through the intersection of the lines x + 2y - 3 = 0 and 4x – y + 7 = 0 and which is parallel to y – x + 10 = 0.

A
3x + 3y + 10 = 0

B
- 3x + 3y + 10 = 0

C
- 3x + 3y - 10 = 0

D
None of these

Solution

Let the equation be x + 2y - 3 + k (4x - y + 7) = 0 ............ (i)
i.e., (1 + 4k) x + (2 - k) y + 7k - 3 = 0
i.e., y = (-1 - 4k)/(2-k) + 3 - 7k
Since it is parallel to y - x + 10 = 0 (i.e., y = x - 10), slopes are equal.
So, (- 1 - 4k)/(2 - k) = 1
- 1 - 4k = 2 - k
k = - 1
Substituting this in eq (i), we get:
The required line is x + 2y - 3 + (-1)(4x - y + 7) = 0
i.e. - 3x + 3y - 10 = 0
Question 3
1 / -0

A
2/3

B
-2/3

C
3/2

D
-3/2

Solution
Question 4
1 / -0

A line which is parallel to the x-axis and crosses the curve y =√x at an angle of 45^ois

A
x = 1/4

B
y = 1/4

C
y = 1/2

D
y = 1

Solution
Question 5
1 / -0

The locus of the point of intersection of the lines x sin θ + (1 – cos θ) y = a sin θ and x sin – (1 + cos θ ) y + a sin θ = 0 is

A
x² – y²= a²

B
x²+ y²= a²

C
y² = ax

D
None of these

Solution
Question 6
1 / -0

The equation of a circle touching the coordinate axes and the line x cos α + y sin α = 2 is x² + y² – 2gx + 2gy + g² = 0, where g is equal to

A
2 (cosα + sinα +1)^-1

B
2 (cos α - sin α +1)^-1

C
2 (cos α + sin α -1)^-1

D
-2 (cos α - sin α -1)^-1

Solution
Question 7
1 / -0

C₁and C₂ are circles of unit radii with centres at (0, 0) and (1, 0), respectively. C₃ is a circle of unit radius, which passes through the centres of the circles C₁ and C₂ and has its centre above the x-axis. The equation of the common tangent to C₁ and C₃, which does not pass through C₂, is

A
x -√3 y + 2 = 0

B
√3 x - y + 2 = 0

C
√3 x - y - 2 = 0

D
x +√3 y + 2 = 0

Solution
Question 8
1 / -0

A chord of the circle x² + y² – 4x – 6y = 0 passing through the origin subtends an angle tan^–1(7/4) at the point where the circle meets the positive y-axis. The equation of the chord is

A
2x + 3y = 0

B
x + 2y = 0

C
x – 2y = 0

D
2x – 3y = 0

Solution
Question 9
1 / -0

If a circle passes through (a, b) and cuts the circle x² + y² = 4 orthogonally, then the locus of its centre is

A
2ax – 2by + (a²+ b² + 4) = 0

B
2ax + 2by – (a² + b² + 4) = 0

C
2ax + 2by + (a² + b² + 4) = 0

D
2ax – 2by – (a² + b² + 4) = 0

Solution
Question 10
1 / -0

The circles x² + y² – 10x + 16 = 0 and x² + y² = r² intersect each other at two distinct points if

A
r < 2

B
r > 8

C
2 < r < 8

D
2≤ r≤ 8

Solution

Centres of the given circles are (5, 0) and (0, 0) and their radii are 3 and r, respectively. The two circles will intersect at two distinct points if the distance between their centres is greater than the difference of their radii and less than the sum of radii.
⇒ |3 – r| < 5 < 3 + r⇒ 2 < r < 8
Question 11
1 / -0

A
y/x

B
x/y

C
x²/y²

D
y²/x²

Solution
Question 12
1 / -0

Let f(x) = sin x + cos x. Then,

A
x = 17π/4 is a point of minima

B
x = 13π/4 is a point of maxima

C
x = 21π/4 is a point of minima

D
x = 29π/4 is a point of maxima

Solution
Question 13
1 / -0

A
e⁴

B
e⁵

C
e³

D
None of these

Solution
Question 14
1 / -0

A

B

C

D

Solution
Question 15
1 / -0

A
g(x) = x + 2

B
g(x) = x² + 4

C
g(x) = xh(x), where h(x) is a polynomial

D
g(x) is a constant function

Solution