Tag: maths

Questions Related to maths

A dealer pays following VAT @ of 14% on 20th April Rs. 2500, 20th May Rs. 1200. Find his sales for March and April.

maths gst (goods and service tax) more about gst commission, brokerage and vat taxation, calculation of income tax

  1. $16428.55$

  2. $26428.55$

  3. $36428.55$

  4. $46428.55$

Show answer
Correct Option: B
Explanation:

Sales for April:
Total VAT = VAT% of sales
$2500=\cfrac{14}{100}\times sales$
Sales of April $= Rs. 17857.14$
Sales for May:
Total VAT = VAT% of sales
$1200=\cfrac{14}{100}\times sales$
Sales of April $= Rs. 8571.42$
Total sales for March and April $= 17857.14 + 8571.41 = Rs. 26428.55$

Shri. Batliwala sold shares of Rs $30350$ and purchased shares of Rs $69650$ in a day. He paid $0.1\%$ brokerage on both sale and purchase of shares. $18\%$ GST was charged on the brokerage. Find his total expenditure on brokerage and tax.

maths gst (goods and service tax) more about gst commission, brokerage and vat taxation, calculation of income tax

  1. $120$

  2. $118$

  3. $139.5$

  4. $105$

Show answer
Correct Option: B
Explanation:
Value of shares sold = $30350$
Brokerage =$0.1%$
$\therefore$ brokerage  value =$\cfrac{0.1}{100}\times30350$
= $303.5/10$=$30.35$
Value of shares purchased =$69650$
Brokerage =$0.1%$
$ \text {brokerage value} =\cfrac{0.1}{100}\times69650$
=$696.5/10$
=$69.65$
$\therefore$ total  brokerage  value =$(30.35+69.65)$
=$Rs 100$
$G. S. T$= $18%$ of brokerage  value 
$\therefore$ amount of gst = $\cfrac{18}{100}\times100$
=$18$
$\therefore$  total amount of brokerage +gst =$100$+$18$
=$118$.

The range of values of $\lambda$ for which the circles $ { x }^{ 2 }+{ y }^{ 2 }=4$ and ${ x }^{ 2 }+{ y }^{ 2 }-2\lambda y+5=0$ have two common tangents only is-

maths tangents and intersecting chords other theorems related to circles touching circles angle made by a chord and a tangent

  1. $\lambda \epsilon \left( -\sqrt { 5 } ,\sqrt { 5 } \right) $

  2. $\lambda <-\sqrt { 5 } or\quad \lambda >\sqrt { 5 }$

  3. $-\sqrt { 5 } <\lambda <1$

  4. none of these

Show answer
Correct Option: A

The range of values of x for which the circles ${ x }^{ 2 }+{ y }^{ 2 }=4$ and$ { x }^{ 2 }+{ y }^{ 2 }+2xy+5=0\quad$ have two on tangents only is= 

maths tangents and intersecting chords other theorems related to circles touching circles angle made by a chord and a tangent

  1. $\left( -\sqrt { 5 } ,\sqrt { 5 } \right)$

  2. $\lambda <=\sqrt { 5 } or\quad \lambda >\sqrt { 5 }$

  3. $-\sqrt { 5 } <\lambda <1$

  4. none of these

Show answer
Correct Option: A

Intercept of a tangent between two parallel tangents to a circle subtends a right angle at the centre.

maths tangents and intersecting chords other theorems related to circles touching circles angle made by a chord and a tangent

  1. True

  2. False

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Correct Option: A

In the given figure, $AD\ and AE$ are the tangents to a circle with centre $O\ and BC$ touches the circle at $F$. If $AE=5\ cm$ then perimeter of $\triangle ABC$ is 

maths tangents and intersecting chords other theorems related to circles touching circles angle made by a chord and a tangent

  1. $15\ cm$

  2. $10\ cm$

  3. $22.5\ cm$

  4. $20\ cm$

Show answer
Correct Option: A

$\overline { M N }$ and $\overline { M Q }$ are two tangents from a point $M$ to a circle with centre $0$ If $m \angle N O Q = 120 ^ { \circ } ,$ then ?

maths tangents and intersecting chords other theorems related to circles touching circles angle made by a chord and a tangent

  1. $N Q = M N = M Q$

  2. $N Q = O M$

  3. $O Q = O M$

  4. $O N = M N$

Show answer
Correct Option: A

If $\triangle ABC$ is isoscles with $AB=AC$ and $C(O,r)$ is the incircle of the of the $\triangle BAC=30^{o}$. The tangent at $C$ intersects $AB$ at a point $D$, then $L$ trisects $BC$.

maths tangents and intersecting chords other theorems related to circles touching circles angle made by a chord and a tangent

  1. True

  2. False

Show answer
Correct Option: B

The chord of contact of the pair of tangents to the circle $x^2+y^2=1$ drawn from any point on the line $2x+y=4$ passes through a fixed point. 

maths tangents and intersecting chords other theorems related to circles touching circles angle made by a chord and a tangent

  1. True

  2. False

Show answer
Correct Option: A
Explanation:

If chords are drawn to the circle from a fixed point $(x _1,y _1)$ and then tangents are drawn at point of contact, the point of intersection of all tangents lie on a fixed point.


The fixed point is called pole and fixed line is called polar.


Equation of polar is $T=0$.

$C:x^2+y^2-1=0$

Equation of polar is $T=0$.

$xx _1+yy _1-1=0$

The line is identical to given line $2x+y-4=0$.

By comparing coefficients, we get,
$\dfrac{x _1}{2}=\dfrac{y _1}{1}=\dfrac{-1}{-4}$

$x _1=\dfrac{1}{2},y _1=\dfrac{1}{4}$

Hence, the fixed point is $(\dfrac{1}{2}, \dfrac{1}{4})$.

From a point $P$ which is at a distance of $13$ cm from the centre $O$ of a circle of radius $5$ cm, the pair of tangents $PQ$ and $PR$ to the circle are drawn. Then the area of the quadrilateral $PQOR$ is:

maths tangents and intersecting chords other theorems related to circles touching circles angle made by a chord and a tangent

  1. $60$ cm$^{2}$

  2. $65$ cm$^{2}$

  3. $30$ cm$^{2}$

  4. $32.5$ cm$^{2}$

Show answer
Correct Option: A
Explanation:

The radius perpendicular tangent at the pt. of contact, therefore, $OQ\perp PQ$ and $OR\perp PR$
In rt. $\triangle OPQ$, we have
$PQ=\sqrt{OP^{2}-OQ^{2}}$
   $=\sqrt{169-25}=\sqrt{144}=12$ cm
$\Rightarrow $ $PR=12$ cm (Two tangents from the same external pt. to a circle are equal)
Now area of quad. $PQOR=2\times $Area of $\triangle POQ$
   $\displaystyle =\left ( 2\times \frac{1}{2}\times 12\times 5 \right )$ cm$^{2}=60$ cm$^{2}$