Tag: maths

Questions Related to maths

Let $ A(1,2,3), B(0,0,1), C(-1,1,1)$ are the vertices of a $\triangle ABC$. Then, the equation of internal angle bisector through A to side BC is 
maths theorems on triangles theorem of remote interior angles of a triangle use of properties of parallel lines angle sum property of a triangle

  1. $\underset{r}{\rightarrow}=\widehat{i}+2\widehat{j}+3\widehat{k}+\mu (3\widehat{i}+2\widehat{j}+3\widehat{k})$

  2. $\underset{r}{\rightarrow}=\widehat{i}+2\widehat{j}+3\widehat{k}+\mu (3\widehat{i}+4\widehat{j}+3\widehat{k})$

  3. $\underset{r}{\rightarrow}=\widehat{i}+2\widehat{j}+3\widehat{k}+\mu (3\widehat{i}+3\widehat{j}+2\widehat{k})$

  4. $\underset{r}{\rightarrow}=\widehat{i}+2\widehat{j}+3\widehat{k}+\mu (3\widehat{i}+3\widehat{j}+4\widehat{k})$

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Correct Option: B

In a  $\triangle A B C,$  side  $A B$  has the equation  $2 x + 3 y = 29$  and the side  $A C$  has the equation  $x + 2 y = 16.$  If the mid point of  $B C$  is  $( 5,6 ) ,$  then the equation of  $B C$  is

maths theorems on triangles theorem of remote interior angles of a triangle use of properties of parallel lines angle sum property of a triangle

  1. $2 x + y = 16$

  2. $x + y = 11$

  3. $2 x - y = 4$

  4. $x + y = - 11$

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Correct Option: B
Explanation:

$\cfrac { x _{ 1 }+x _{ 2 } }{ 2 } =5\Rightarrow x _{ 1 }+x _{ 2 }=10.....(1)\quad and\quad y _{ 1 }+y _{ 2 }=12.....(2)$

$Point(x _1,y _1)$ lie on line AC
then
$x _1+2y _1=16...(3)$
Similarly $2x _2+3y _2=29....(4)$
$\Rightarrow 2(x _1+x _2)+4y _1+3y _2=32+29\2\times 10+4y _1+36-3y _1=61\y _1=5\Rightarrow x _1=6\Rightarrow x _2=4\ \Rightarrow y _2=7$
now,
we take these two points and make equation,
$AC= x+y=11$

In triangle, three angles are  $x , x + 10 ^ { \circ } + x + 20 ^ { \circ }$  then the biggest is

maths theorems on triangles theorem of remote interior angles of a triangle use of properties of parallel lines angle sum property of a triangle

  1. $70 ^ { \circ }$

  2. $80 ^ { \circ }$

  3. $90 ^ { \circ }$

  4. none

Show answer
Correct Option: A

In. triangle ABC,$\angle A$ + $\angle B$ = 144 and$\angle A$ + $\angle C$ = 124.
Calculate smallest angle of the triangle.

maths theorems on triangles theorem of remote interior angles of a triangle use of properties of parallel lines angle sum property of a triangle

  1. $36^o$

  2. $56^o$

  3. $46^o$

  4. none of these

Show answer
Correct Option: A
Explanation:

$\angle A + \angle B = 144$...(I)
$\angle A + \angle C = 124$...(II)
In triangle ABC,
$\angle A  + \angle B + \angle C = 180 $
Add, I and II,
$\angle A + \angle B + \angle A + \angle C = 144+ 124$
$180 + \angle A = 268 $
$\angle A = 268 - 180 $
$\angle A = 88$
Put this value in (I)
$\angle A + \angle B = 144$
$88 + \angle B = 144$
$\angle B = 56$
Put this value in (II)
$\angle A + \angle C = 124$
$88 + \angle C = 124$
$\angle C = 36$

If every side of a triangle is doubled, then the area of the new triangle is 'K' times the area of the old one. The value of K is

maths theorems on triangles theorem of remote interior angles of a triangle use of properties of parallel lines angle sum property of a triangle

  1. 2

  2. 3

  3. $\sqrt 2$

  4. 4

Show answer
Correct Option: D
Explanation:
Let the area of the triangle be $x$.

We know that the area of the triangle
$=\dfrac{1}{2}\times Height \times Base$
$x=\dfrac{1}{2}\times Height \times Base$              $........ (1)$

According to the question,
$Kx=\dfrac{1}{2}\times 2 \times Height \times 2 \times Base$
$Kx=4\times x$
$K=4$

Hence, this is the answer.

The ratio of the areas of two similar triangles is equal to the

maths theorems on triangles theorem of remote interior angles of a triangle use of properties of parallel lines angle sum property of a triangle

  1. ratio ofcorresponding medians

  2. ratio ofcorresponding sides

  3. ratio of the squares ofcorresponding sides

  4. none of these

Show answer
Correct Option: C
Explanation:

The ratio of the areas of two similar triangles is equal to the square of ratio of their corresponding sides.

Number of ordered pairs $(x, y)$ of real numbers satisfying the equation $x^{2} + y^{2} - 24x - 26y + 313 = 0$ is equal to

decimal representation of rational numbers rational and irrational numbers maths

  1. Infinite

  2. Finite but more than one

  3. Exactly one

  4. Zero

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Correct Option: D

The number having no reciprocal is

decimal representation of rational numbers rational and irrational numbers maths

  1. $2$

  2. $1$

  3. $-1$

  4. $0$

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Correct Option: D
Explanation:

If a number is 2 then its reciprocal is $\dfrac{1}{2}$

The number '0' has no reciprocal.

The decimal part of $9.99$ is

decimal representation of rational numbers rational and irrational numbers maths

  1. $0.99$

  2. $9$

  3. $8$

  4. $0.98$

Show answer
Correct Option: A
Explanation:

Decimal part of $9.99$ is $0.99$.

So, option A is correct.

The integral part of $3.89$ is

decimal representation of rational numbers rational and irrational numbers maths

  1. $3$

  2. $0.89$

  3. $4$

  4. None of these

Show answer
Correct Option: A
Explanation:

Integral part of $3.89$ is $3$.