Mathematics Class XI

Question 1 of 25

1. Question
1 point(s)
If $ A(1,2,3), \quad B (0, 4,-2) \text { and } C (x,y,z) $ are three points on a line, where $ A $ divides $ \overline{BC} $ in ratio $ 1:2 $ externally, then find the coordinates of $ C $.
- $(1,6,7) $
- $(-1,-6,-7) $
- $(-1,6,-7) $
- $(1,-6,7) $
Correct

Incorrect

Hint

$A$ , divides $\overline{BC}$ in ratio $1:2$ externally i.e., $B$ , divides $\overline{AC}$ in ratio $1:1$ internally, i.e., $B$ is the midpoint of $\overline{AC}$ .
$\implies \left( \dfrac{x+1}{2}, \dfrac{y+2}{2}, \dfrac{z+3}{2} \right)=(0, 4,-2)$
$\implies (x+1,y+2,z+3)=(0, 8,-4)$
$\implies x=-1, y=6, z=-7$
$\therefore (x,y,z)=(-1, 6,-7)$
Question 2 of 25

2. Question
1 point(s)
If $A (-1,6,-7), B (x,y,z) \text { and } C (3,4,5) $ are three points on a line. If $ C $ divides $ \overline{AB} $ in $ 2:1 $ ratio externally, then find the coordinate of $ B $.
- $(1,5,-1) $
- $(1,5,1) $
- $ (-1,5,-1) $
- $(-1,-5,-1) $
Correct

Incorrect

Hint

$C$ divides $\overline{AB}$ in ratio $2:1$ externally.
$\implies \left( \dfrac{mx_2-nx_1}{m-n},\dfrac{my_2-ny_1}{m-n},\dfrac{mz_2-nz_1}{m-n} \right)=(3,4,5)$
$\implies \left( \dfrac{2x-1(-1)}{2-1}, \dfrac{2y-1\times6}{2-1},\dfrac{2z-1(-7)}{2-1} \right)=(3,4,5)$
$\implies (2x+1,\quad 2y-6,\quad 2z+7)=(3,4,5)$
$\implies 2x+1=3 \implies x=1$
$2y-6=4 \implies y=5$
$2z+7=5 \implies z=-1$
$\therefore B (x,y,z)=(1, 5,-1)$
Question 3 of 25

3. Question
1 point(s)
If $ B(2,3,4) $ be the midpoint of $ A(1,a,3) \text { and } C (b, 4,5) $, line joining $ \overline{AC} $, then find $ a $ and $ b $.
- $(3,2) $
- $(2,3) $
- $ (4,5) $
- $(3,4) $
Correct

Incorrect

Hint

$\left( \dfrac{1+b}{2},\dfrac{a+4}{2},\dfrac{3+5}{2} \right)=(2,3,4)$
$\implies \dfrac{1+b}{2}=2 \implies b=3$
$\dfrac{a+4}{2}=3 \implies a=2$
$\therefore (a,b)=(2,3)$
Question 4 of 25

4. Question
1 point(s)
If $G \left( 1,4, \dfrac{1}{3} \right) $ divides the line joining $ A (x,y,z) \text { and } B(1,5,-1) $ in ratio internally, then find the coordinate of $ A $.
- $(3,2,1) $
- $(2,1,3) $
- $(1,2,3) $
- $ (-1,-2,-3) $
Correct

Incorrect

Hint

$\left( \dfrac{mx_2+nx_1}{m+n}, \dfrac{my_2+ny_1}{m+n}, \dfrac{mz_2+nz_1}{m+n} \right)=\left( 1,4,\dfrac{1}{3} \right)$
$\implies \left( \dfrac{2\times1+1 \times x}{2+1},\dfrac{2 \times 5+1 \times y}{2+1},\dfrac{2(-1)+1 \times z}{2+1} \right)=\left( 1,4, \dfrac{1}{3} \right)$
$\implies \left( \dfrac{2+x}{3},\dfrac{10+y}{3},\dfrac{-2+z}{3} \right)=\left( 1,4, \dfrac{1}{3} \right)$
$\implies \dfrac{x+2}{3}=1 \implies x=1$
$\dfrac{10+y}{3}=4 \implies y=2$
$\dfrac{-2+z}{3}=\dfrac{1}{3}\implies z=3$
$\therefore A (x,y,z)=(1,2,3)$
Question 5 of 25

5. Question
1 point(s)
A ball is a solid region in the space enclosed by a _____
- circle
- sphere
- cube
- square
Correct

Incorrect

Hint

A ball is a solid hollow sphere.
Question 6 of 25

6. Question
1 point(s)
The planes $ x=a, \quad y=b$ represents______
- the line parallel to $ x $ -axis
- the line perpendicular to $ z $ -axis
- the line parallel to $ y $-axis
- the line parallel to $ z $ -axis
Correct

Incorrect

Hint

$x=a$ , represents the plane parallel to $yz$ -plane.
$y=b$ , represents the plane parallel to $xz$ -plane.
So, the intersection of $x=a, \quad y=b$ is the line parallel to $z$ -axis.
Question 7 of 25

7. Question
1 point(s)
The planes $ x=a, \quad z=c$ represents the line_________
- parallel to $z $-axis
- parallel to $ x $-axis
- parallel to $y $-axis
- perpendicular to $ y $-axis
Correct

Incorrect

Hint

$x=a$ , represents the plane parallel to $yz$ -plane.
$z=c$ , represents the plane parallel to $xy$ -plane.
So, the intersection of $x=a, \quad z=c$ is the line parallel to $y$ -axis.
Question 8 of 25

8. Question
1 point(s)
The planes $ y=b, \quad z=c $ represents the line_______
- parallel to $x $-axis
- perpendicular to $ x $ -axis
- perpendicular to $ y $-axis
- parallel to $ z $ -axis
Correct

Incorrect

Hint

$y=b$ , represents the plane parallel to $zx$ -plane.
$z=c$ , represents the plane parallel to $xy$ -plane.
So, the intersection of $y=b, \quad z=c$ is the line parallel to $x$ -axis.
Question 9 of 25

9. Question
1 point(s)
In a rhombus $ A (5, -9, 3) \text { and } C(3, 5,-7) $ are the endpoints on the diagonal $\overline{AC} $. Find the intersecting point of two diagonals $ \overline{AC} \text { and } \overline{BD} $.
- $(4,2,2) $
- $(4,-2,-2) $
- $(4,-2,2) $
- $(4,2,-2) $
Correct

Incorrect

Hint

Let $O$ be the intersection of $\overline{AC} \text { and } \overline{BD}$ .
Hence $\begin{aligned} O&=\left( \dfrac{5+3}{2}, \dfrac{-9+5}{2},\dfrac{3-7}{2} \right)\\&=(4,-2,-2) \end{aligned}$
Question 10 of 25

10. Question
1 point(s)
If the distance between the points $(5,7,a) \text { and } (3,9,4) $ is $ 3 $, then find $ a $.
- $ 5,-3 $
- $5,3 $
- $ -5,3 $
- $ -5,-3 $
Correct

Incorrect

Hint

$\text {d}=\sqrt{(5-3)^2+(7-9)^2+(a-4)^2}$
$\implies 3=\sqrt{2^2+(-2)^2+(a-4)^2}$
$\implies (a-4)^2+4+4=9$
$\implies a^2-8a+16+8-9=0$
$\implies a^2-8a+15=0$
$\implies a^2-5a-3a+15=0$
$\implies a(a-5)-3(a-5)=0$
$\implies (a-5)(a-3)=0$
$\implies a=5 \text { or } 3$
Question 11 of 25

11. Question
1 point(s)
If $ A(5,7,-3) \text { and } B (-3,-9,1) $ are the coordinates of the two endpoints of a diameter of a circle, then find the coordinate of the centre.
- $ (1,1,1) $
- $ (-1,1,1) $
- $ (1,-1,-1) $
- $ (-1,-1,-1) $
Correct

Incorrect

Hint

Let $O$ be the centre of the circle.
$\implies O=\left( \dfrac{5-3}{2}, \dfrac{7-9}{2},\dfrac{-3+1}{2} \right)$
$\therefore O=(1,-1,-1)$
Question 12 of 25

12. Question
1 point(s)
Let $A (7,8,-9) \text { and } B (-9,2,-1) $ are two points on a circle with centre $ O (-1,5,-5) $, then $ \overline{AB} $ is divided by $ O $ in ratio_____
- $2:9 $
- $9:2 $
- $ 1:2 $
- $1:1 $
Correct

Incorrect

Hint

$\begin{aligned} OA&=\sqrt{(7+1)^2+(8-5)^2+(-9+5)^2}\\&=\sqrt{64+9+16}\\&=\sqrt{89} \end{aligned}$
$\begin{aligned} OB&=\sqrt{(-9+1)^2+(2-5)^2+(-1+5)^2} \\&=\sqrt{64+9+16}\\&=\sqrt{89} \end{aligned}$
$\dfrac{OA}{OB}=\dfrac{\sqrt{89}}{\sqrt{89}}=\dfrac{1}{1}$
$\therefore$ ‘ $O$ ‘ divides $\overline{AB}$ in ratio $1:1$ .
Question 13 of 25

13. Question
1 point(s)
If a point $ P $ lies in $ yz $-plane, then the coordinates of a point ‘$P $’ on $yz $-plane are of the form________
- $(x, y, 0) $
- $(x,0,0) $
- $(0, y,z) $
- $(x,0,z) $
Correct

Incorrect

Hint

On $yz$ -plane, $x=0$ .
So, the coordinates of the required point is $(0,y,z)$
Question 14 of 25

14. Question
1 point(s)
If a point $ P $ lies in $ xy $-plane, then the coordinates of the point $P $ on $xy $-plane are of the form_____
- $(x,y,0) $
- $(0,0,z) $
- $ (0,y,z) $
- $ (x,0,z) $
Correct

Incorrect

Hint

On $xy$ -plane, $z=0$ .
So, the coordinates of the required point is $(x,y,0)$
Question 15 of 25

15. Question
1 point(s)
If a point $ P $ lies on $ zx $-plane, then the coordinates of point $ P $ on $ zx $-plane are of the form______
- $ (x,y,0) $
- $ (0,y,z) $
- $(x,0,z) $
- $ (0,y,0) $
Correct

Incorrect

Hint

On $zx$ -plane, $y=0$
So, the coordinates of the required point is $(x,0,z)$
Question 16 of 25

16. Question
1 point(s)
If a line is parallel to $ x$-axis then , it is ________
- parallel to $ yz $-plane
- perpendicular to $ yz $-plane
- parallel to $z $-axis
- parallel to $ y $ -axis
Correct

Incorrect

Hint

$x$ -axis is perpendicular to $yz$ -plane.
So, any line parallel to $x$ -axis is perpendicular to $yz$ -plane.
Question 17 of 25

17. Question
1 point(s)
If a liine is parallel to $y $-axis, then it is________
- parallel to $x $-axis
- parallel to $ z $-axis
- perpendicular to $ xz $-plane
- parallel to $ xz $ -plane
Correct

Incorrect

Hint

$y$ -axis is perpendicular to $xz$ -plane.
So, any line parallel to $y$ -axis is perpendicular to $xz$ -plane.
Question 18 of 25

18. Question
1 point(s)
If a line is parallel to $ z$-axis then , it is ________
- perpendicular to $ xy $-plane
- parallel to $xy $-plane
- parallel to $y $-axis
- parallel to $ x$ -axis
Correct

Incorrect

Hint

$z$ -axis is perpendicular to $xy$ -plane.
So, any line parallel to $z$ -axis is perpendicular to $xy$ -plane.
Question 19 of 25

19. Question
1 point(s)
If a line is parallel to $ x $-axis, then which of the following is false?
- It is perpendicular to $ y $ -axis
- It is perpendicular to $ z $-axis
- It is parallel to $ yz $-plane
- It is perpendicular to $ yz $-plane.
Correct

Incorrect

Hint

$x$ -axis is perpendicular to $y$ -axis, $z$ -axis and $yz$ -plane. So, any line parallel to $x$ -axis is not parallel to $yz$ -plane.
Question 20 of 25

20. Question
1 point(s)
If a line is parallel to $y $-axis, then which of the following is false?
- It is perpendicular to $ x $ -axis
- It is parallel to $ xz $-plane
- It is perpendicular to $ xz $ -plane
- It is perpendicular to $ z $ -axis
Correct

Incorrect

Hint

$y$ -axis is perpendicular to $x$ -axis, $z$ -axis and $xz$ -plane. So,any line parallel to $y$ -axis is not parallel to $xz$ -plane.
Question 21 of 25

21. Question
1 point(s)
If a line is parallel to $z $-axis then which of the following is false?
- parallel to $xy $-plane
- It is perpendicular to $ xy $-plane
- It is perpendicular to $ y $-axis
- It is perpendicular to $ x$-axis
Correct

Incorrect

Hint

$z$ -axis is perpendicular to $y$ -axis, $x$ -axis and $xy$ -plane.
So, any line parallel to $z$ -axis is not parallel to $xy$ -plane.
Question 22 of 25

22. Question
1 point(s)
If a line is perpendicular to $yz $-plane, then which of the following is false?
- It is parallel to $ x $-axis
- It is perpendicular to $ x $ -axis
- It is perpendicular to $ y $-axis
- It is perpendicular to $ z $-axis
Correct

Incorrect

Hint

Option 2 is false because any line perpendicular to $yz$ -plane is parallel to $x$ -axis.
Question 23 of 25

23. Question
1 point(s)
If a line is perpendicular to $zx $-plane, then which of the following is false?
- It is perpendicular to $ y $ -axis
- It is perpendicular to $ x $ -axis
- It is perpendicular to $ z $ -axis
- It is parallel to $ y $-axis
Correct

Incorrect

Hint

Any line perpendicular to $zx$ -plane is parallel to $y$ -axis.
Hence, Option 1 is incorrect.
Question 24 of 25

24. Question
1 point(s)
If a line is perpendicular to $ xy $-plane then which of the following is false?
- It is parallel to $ z $-axis
- It is perpendicular to $ x $ -axis
- It is perpendicular to $ y $ -axis
- It is perpendicular to $ z $ -axis
Correct

Incorrect

Hint

Any line perpendicular to $xy$ -plane is parallel to $z$ -axis.
Hence, Option 4 is false.
Question 25 of 25

25. Question
1 point(s)
If the end points of a diagonal of a square are $ (1,-2,3) \text { and } (2,-3,5) $, then the length of the side of the square is_______
- $ \sqrt{3} $ unit
- $2 \sqrt{3} $ unit
- $3 \sqrt{3} $ unit
- $ 4 \sqrt{3} $ unit
Correct

Incorrect
Hint

Let $A-(1,-2,3), \quad C= (2,-3,5)$
$\begin{aligned} AC&=\sqrt{(1-2)^2+(-2+3)^2+(3-5)^2}\\&=\sqrt{1+1+4} =\sqrt{6} \end{aligned}$ <br
```
*** QuickLaTeX cannot compile formula:
 

*** Error message:
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$AB=\dfrac{\sqrt{6}}{\sqrt{2}}=\sqrt{3}$
$\therefore$ length of the side of the square is $\sqrt{3}$ unit.

Mathematics Class XI

Practice Questions 4-3D Geometry-Class XI

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