Tag: maths

Questions Related to maths

A circular wire of radius $7$ cm is cut and bend again into an arc of a circle of radius $12$ cm. The angle subtended by the arc at the centre is

maths circle measures length of an arc area of a sector of a circle sector and arc of a circle

  1. $50^\circ$

  2. $210^\circ$

  3. $100^\circ$

  4. $60^\circ$

Show answer
Correct Option: B
Explanation:

Given, radius of circular wire $= 7$ cm

Circumference of wire $= 2 \pi r = 2 \pi (7) = 14 \pi$
Radius of arc $= 12$ cm

Angle subtended by the arc $= \dfrac{\text{arc}}{\text{radius}} = \dfrac{14 \pi}{12} = \dfrac{7 \pi}{6}$

Angle subtended by arc $=\cfrac{7\pi}{6}\times \cfrac{180}{\pi}= 210^{\circ}$

In $\triangle ABC,$ if $AB=7 \ cm$ and $BC=8 \ cm$, then which of the following cannot be the length of $AC$?

maths basic geometrical concepts and shapes construction of line segment and circle of given radius construction related to lines constructing line segment

  1. $17\ cm$

  2. $16\ cm$

  3. $14\ cm$

  4. $None\ of\ these$

Show answer
Correct Option: D

Four distinct points $(2K , 3K), (1 , 0), (0 , 1)$ and $(0 , 0)$ lie on a circle when

maths basic geometrical concepts and shapes construction of line segment and circle of given radius construction related to lines constructing line segment

  1. all values of $K$ are integral

  2. $0 < K < 1$

  3. $K < 0$

  4. For one values of $K$

Show answer
Correct Option: D
Explanation:

Let A = $(0,0)$ , B$(0,1)$ , C$(2k,3k)$ , D$(1,0)$

As we can see

$AD \perp AB$

$\angle A = 90$

$\angle C = 90$

$\implies m _{BC} \times m _{DC} = -1$

$\dfrac{3k - 1}{2k} \times {3k}{2k - 1} = -1$

$\implies k(9k - 3) = -(4k - 2)k$

$k = 0 , 9k + 4k = 2+ 3 \implies 13k = 5 \implies k = \dfrac{5}{13}$

But if $k = 0$, C will be $(0,0)$ which is A

$k = \dfrac{5}{13} $ only one point.

A ray of light passing through the point A$(1, 2, 3)$, strikes the plane $x + y + z = 12$ at B on reflection passes through point C$(3, 5, 9)$. The coordinates of point B are 

maths basic geometrical concepts and shapes construction of line segment and circle of given radius construction related to lines constructing line segment

  1. $(2, 5, 5)$

  2. $(-4, 6, 10)$

  3. $(-7, 0, 19)$

  4. $(0, -5, 17)$

Show answer
Correct Option: C
Explanation:
Let image of point $A(1,2,3)$ about $x+y+z=12$ be $D(p,q,r)$ then,
$\cfrac{p-1}{1}=\cfrac{q-2}{1}=\cfrac{r-3}{1}=-2\cfrac{1+2+3-12}{1^2+1^2+1^2} \\ D(p,q,r)=D(5,6,7) $

Line joining $CD$
$\cfrac{x-5}{2}=\cfrac{y-6}{1}=\cfrac{z-7}{-2}=\lambda$
Coordinate of $B(2\lambda+5,\lambda+6,-2\lambda+7)$
This lies on plane $2\lambda+5+\lambda+6-2\lambda+7=12 \Rightarrow \lambda=-6$
$B(2\lambda+5,\lambda+6,-2\lambda+7)=B(-7,0,19)$

Thirty one crore fifty-five lakh thirty-two thousand eight hundred in numerals is written as

maths numbers international place value system international system of numeration use of commas and comparison and number systems

  1. $31,55,32,800$

  2. $3,01,55,32,800$

  3. $3,15,05,32,800$

  4. $3,15,53,02,800$

Show answer
Correct Option: A
Explanation:

In Indian System, the numeral is Thirty one crore, fifty-five lakh thirty-two thousand eight hundred can be written as $31,55,32,800$.

1 billion = __________ crores.

maths numbers international place value system international system of numeration use of commas and comparison and number systems

  1. 1

  2. 10

  3. 100

  4. 1000

Show answer
Correct Option: C
Explanation:

correct option is C..

__________ is equivalent to ten lakh in International System of numeration.

maths numbers international place value system international system of numeration use of commas and comparison and number systems

  1. Hundred thousand

  2. Ten million

  3. One million

  4. Ten thousand

Show answer
Correct Option: C
Explanation:

$1$ million is equal to $10$ lakhs.


Hence the correct answer is option C.

Kiran had $85$ currency notes in all, some of which were of Rs. $100$ denomination and the remaining of Rs. $50$ denomination. The total amount of all these currency notes was Rs. $5000$. How much amount did she have in the denomination of Rs. $50$?

maths numbers international place value system international system of numeration use of commas and comparison and number systems

  1. Rs. $2725$

  2. Rs. $3500$

  3. Rs. $1500$

  4. Rs. $2800$

Show answer
Correct Option: B
Explanation:

Let $100$ Rs note be $x$ and $50$ Rs note be $y$

Total number of notes =$x+y=85$................(1)
Total amount of these currency notes + $5000=100x+50y$.............(2)
Multiplying equation 1 with $50$
therefore we get $50x+50y=4250$.......(3)
Subtracting equation 3 from equation 2, we get
$50x=750$
$x=15$
Substituting $x=15$ in equation 1, weget $y=70$
Amount in dominnation of $50$ Rs=$70\times 50=3500$

If the line y = mx is one of the bisector of the lines $x^2 + 4xy - y^2 = 0$, then the value of no ___________.

maths pair of straight lines bisection of angle pair of bisectors of angles bisector of angle between lines

  1. $\frac{\sqrt{5} - 1}{2}$

  2. $\frac{\sqrt{5} + 1}{2}$

  3. $-(\frac{\sqrt{5} + 1}{2})$

  4. $-(\frac{\sqrt{5} -1}{2})$

Show answer
Correct Option: A

The Straight lines represented by the equation $135{ x }^{ 2 }-136xy+33{ y }^{ 2 }=0$ are equally inclined to the line 

maths pair of straight lines bisection of angle pair of bisectors of angles bisector of angle between lines

  1. $x-2y=7$

  2. $x+2y=7$

  3. $x-2y=4$

  4. $3x+2y=4$

Show answer
Correct Option: B
Explanation:

Give pair of lines is $135{ x }^{ 2 }-136xy+33{ y }^{ 2 }=0$   ...(1)


The equation of bisector of angles between pair of lines (1) is

$\displaystyle \frac { { x }^{ 2 }-{ y }^{ 2 } }{ a-b } =\frac { xy }{ h } \Rightarrow \frac { { x }^{ 2 }-{ y }^{ 2 } }{ 135-33 } =\frac { xy }{ -68 } $

$\Rightarrow 2{ x }^{ 2 }+3xy-2{ y }^{ 2 }=0\Rightarrow \left( x+2y \right) \left( 2x-y \right) =0$

One of the bisectors is $x+2y=0$ which is parallel to the line $x+2y=7$.

Hence, the line $x+2y=7$ is equally inclined to the given lines.