Mathematics Test

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Mathematics Test - 9

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Weekly Quiz Competition

Question 1
3 / -1

The distance between 3x+4y=9 and 6x+8y=15 is

A
1.5

B
0.3

C
6

D
3

Solution
Question 2
3 / -1

All points lying inside the triangle formed by the points (1,3), (5,0) and (1,2) satisfy

A
3x + 2y ≥ 0

B
2x + y - 13 ≥ 0

C
-2x + y ≥ 0

D
None of these

Solution

All points lying inside the triangle formed by the points given, then this implies that all the three vertices of triangle lies on or same side of the given conditions:

A

$3 x + 2 y \geq 0$

where (1,3),(5,0) and (1,2) satisfies the above condition. Therefore A is True.

B

$2 x + y - 13 \geq 0$

(1, 3), does not satisfy it. Therefore, B is false

C

$- 2 x + y \geq 0$

is not satisfied by (5,0) Therefore, C is false

Thus (a) is correct answer.
Question 3
3 / -1

The length of latus rectum of the parabola 13y² = 14x is

A
14/13

B
13/14

C
28/13

D
6/14

Solution

$y^{2} = \frac{14}{13} x$
length of the latus rectum = 4a
Question 4
3 / -1

The distance from the point (x,y) to the positive x-axis is

A
x

B
y

C
|x|

D
0

Solution

From the figure it is clear that the distance from P(x,y) to the x – axis is ‘y’

Hence option B is the correct answer.
Question 5
3 / -1

The equation of the circle, touching the axis of x at the origin and the line 3y = 4x + 24, is

A
x² + y² + 24y = 0

B
x² + y² - 8y = 0

C
x² + y² - 24y = 0

D
x² + y² + 6y = 0

Solution
Question 6
3 / -1

A rectangular parallelopiped is formed by planes drawn through (5,7,9) and (2,3,7) parallel to the coordinate planes. The length of the longest edge of the parallelopiped is

A
√29

B
5

C
4

D
16

Solution

Given :

Points on planes which formed rectangular parallelepiped are (5,7,9) and (2,3,7).

And this is parallel to the coordinate planes.

Now we have to find out the length of the longest edge of the parallelopiped.

We can draw the rectangular parallelepiped as follows :

Lengths of the edges are $\left|x_{1}-x_{2}\right|\left|y_{1}-y_{2}\right|$ and $\left|z_{1}-z_{2}\right|$
Here $x_{1}=5, y_{1}=7, z_{1}=9$ and
$x_{2}=2, y_{2}=3, z_{3}=7$
$-\left|x_{1}-x_{2}\right|=|5-2|=3$
$\left|y_{1}-y_{2}\right|=|7-3|=4$
$\left|z_{1}-z_{2}\right|=|9-7|=2$
The length of the longest edge is 4

Hence option C is the correct answer.
Question 7
3 / -1

The equation of the circle having centre at (0,0) and radius 4 units is

A
x²+ y² = 2

B
x² + y² = 1

C
x² + y² = 4

D
x² + y² = 16

Solution

Equation of the circle with centre origin and having radius r
is x²+ y² = r² , ie x² + y² = 16
Hence option D is the correct answer.
Question 8
3 / -1

A quadratic equation can contain .........

A
x

B
z²

C
z

D
All of the above

Solution

An equation in which at least on term is squared and none of the terms has a degree more than 2 is know as quadratic equation

⇒A quadratic equation can contain any terms of degree o or 1 or 2. Hence the correct option is D.

Hence option D is the correct answer.
Question 9
3 / -1

The vertices of a triangle are (3,2,5), (3,2,-1) and (7,2,5). The circumcentre of the triangle is

A
(4,3,1)

B
(5,-2,1)

C
(5,2,2)

D
(5,-2,2)

Solution
Question 10
3 / -1

The coordinates of two points on the circle x² + y²− 12x − 16y + 75 = 0, one nearest to the origin and the other farthest from it, are

A
(3, 4)

B
(3, 2)

C
(9, 4)

D
(9, -12)

Solution

Hence option A is the correct answer.
Question 11
3 / -1

A
L is common chord of S₁ and S₂

B
L is radical axis of S₁ and S₂

C
L is perpendicular to the line joining the centre of S₁ and S₂

D
All of the above

Solution

Clearly line L is radical axis of of the given pair of circle.

And we know radical axis is perpendicular to the line joining the centre of the circles.

Also $C_{1} C_{2} < r_{1} + r_{2}$ . So both the circles intersect at two point.

Thus line is also common chord.
Question 12
3 / -1

The length of the y – intercept of the circle x² + y² – 6x + 8y + 7 = 0 is

A
1

B
2

C
3

D
6

Solution

Given equation of the circle is $x^{2}+y^{2}-6 x+8 y+7=0$ We know the equation of the Length of the $y$ - intercept is $2 \sqrt{f^{2}-c}$
Now comparing the given equation with the general equation of the circle $x^{2}+y^{2}+2 g x+2 f y+c=0,$ we get $g=-3$ and $f=4$ and $c=7$ The length of the $y$ -intercept $=2 \sqrt{f^{2}-c}=2 \sqrt{4^{2}-7}=2 \sqrt{16-7}=2 \sqrt{9}=2 \times 3=6$
Hence option D is the correct answer.
Question 13
3 / -1

The points (5,-4,2), (4,-3,1),(7,-6,4) and (8, -7,5) are vertices of a

A
Square

B
Parallelogram

C
Rectangle

D
None of these

Solution
Question 14
3 / -1

The coordinates of the point of intersection of the line $\frac{x + 1}{1} = \frac{y + 3}{3} = \frac{z - 2}{2}$ with the plane 3x + 4y + 5z = 5 are

A
(-1, 2, 3)

B
(1, 3,-2)

C
(-1,-2, 3)

D
(1,-3, 2)

Solution

since the point is intersect with both line and plane, it must be in both line and plane

Equation of line is $\frac{x + 1}{1} = \frac{y + 3}{3} = \frac{z - 2}{2}$

and equation of plane is 3x + 4y + 5z = 5

From the given option the point lie on both line and plane is (1,3,-2)

If we apply this values in equation of plane

3 × 1 + 4 × 3 + 5 × (- 2) = 5

Which is true

∴ The point is (1, 3, -2)

Hence option D is the correct answer.
Question 15
3 / -1

The focus of the parabola x² = 4ay is

A
(a, 0)

B
(-a, 0)

C
(0, a)

D
(0, -a)

Solution

x² = 4ay

Hence option C is the correct answer.