Tag: maths

Questions Related to maths

State true or false:

In parallelogram $ ABCD $. $ E $ is the mid-point of $ AB $ and $ AP $ is parallel to $ EC $ which meets $ DC $ at point $ O $ and $ BC $ produced at $ P $. Hence
$ BP= 2AD $


maths properties of parallel lines and their transversal how to check for similarity in triangles? criteria for similarity of triangles criteria for triangle similarity

  1. True

  2. False

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Correct Option: A
Explanation:

In $\triangle$s, APB and ECB,
$\angle ABP = \angle EBC $ (Common angle)
$\angle PAB = \angle CEB$ (Corresponding angles of parallel lines)
$\angle APB = \angle ECB $ (Third angle of the triangle)
Thus $\triangle APB \sim \triangle ECB$ 
Hence, $\frac{AB}{EB} = \frac{BP}{BC}$ (Corresponding sides of similar triangles)
$2 = \frac{BP}{BC}$
$BP = 2 BC$
$BP = 2 AD$  (BC = AD)

In quadrilateral ABCD, the diagonals AC and BD intersect each at point O. If $AO=2CO$ and $BO=2DO$; Then,

$\displaystyle \Delta AOB$ is similar to $\displaystyle \Delta COD$

maths properties of parallel lines and their transversal how to check for similarity in triangles? criteria for similarity of triangles criteria for triangle similarity

  1. True

  2. False

Show answer
Correct Option: B
Explanation:

Given: $AO = 2 CO$ or $\dfrac{AO}{CO} = 2$
Also given, $BO = 2 DO$ or $\dfrac{BO}{DO} = 2$
In $\triangle AOB$ and $\triangle COD$, we know 
$\angle AOB = \angle COD$
$\dfrac{AO}{CO} = \dfrac{BO}{DO}$
Thus, $\triangle AOB \sim \triangle COD$ (SAS rule)

$\angle BAC$ of triangle $ABC$ is obtuse and $AB=AC$. $P$ is a point in $BC$ such that $PC= 12$ cm. $ PQ $ and $PR$ are perpendiculars to sides $AB$ and $AC$ respectively. If $PQ= 15$ cm and $=9$ cm; find the length of $PB$.

maths properties of parallel lines and their transversal how to check for similarity in triangles? criteria for similarity of triangles criteria for triangle similarity

  1. $20$

  2. $24$

  3. $36$

  4. $18$

Show answer
Correct Option: A
Explanation:

Given: $AB = AC$, $PQ \perp AB$ and $PR \perp AC$
Since, $AB = AC$
$\angle ABC = \angle ACB$...(I) (Isosceles triangle property)

Now, In $\triangle PBQ$ and $\triangle PRC$
$\angle PBQ = \angle PCR$ (From I)
$\angle PQB = \angle PRC$ (Each $90^{\circ}$)
$\angle QPB = \angle RPC$ (Third angle)
Thus, $\triangle QPB \sim \triangle RPC$ (AAA rule)
Hence, $\dfrac{PQ}{PR} = \dfrac{PB}{PC}$
$\dfrac{15}{9} = \dfrac{PB}{12}$
$PB = \dfrac{15 \times 12}{9}$
$PB = 20$ cm

In standard from, the number 829030000 is written as $K\times { 10 }^{ 8 }$ where K is equal to 

maths numbers and place value many forms of ten thousand standard form of numbers number names, numerals and place values

  1. 82903

  2. 829.03

  3. 82.903

  4. 8.2903

Show answer
Correct Option: A
Explanation:

$829030000=K \times 10^8$

Thus, $K=8.2903$
Hence, option D is correct.

The sum of the powers of the prime factors in $108 \times 192$  is

maths numbers and place value many forms of ten thousand standard form of numbers number names, numerals and place values

  1. $5$

  2. $7$

  3. $8$

  4. $12$

Show answer
Correct Option: D
Explanation:

$\displaystyle 108=2\times 2\times  3\times 3\times 3=2^{2}\times 3^{3}$
$\displaystyle 192=2\times2\times2\times2\times2\times2\times3$
$\displaystyle =2^{6}\times 3^{1}$
$\displaystyle 108 \times 192=2^{2}\times 3^{3}\times 2^{6}\times 3^{1}$
$\displaystyle =2^{8}\times 3^{4}$
Sum of the powers = 8 + 4 = 12

The value of $\displaystyle (243)^{\frac{-2}{5}}$ is---

maths numbers and place value many forms of ten thousand standard form of numbers number names, numerals and place values

  1. $\displaystyle \frac{1}{9}$

  2. $\displaystyle \frac{2}{9}$

  3. 9

  4. None of these

Show answer
Correct Option: A
Explanation:

$\displaystyle (243)^{-2/5} =\displaystyle(3^5)^{-2/5}$
                $\displaystyle = 3^{-2} = \frac{1}{9}$

Find the value of $\displaystyle (64)^{-2/3}$---

maths numbers and place value many forms of ten thousand standard form of numbers number names, numerals and place values

  1. 16

  2. $\displaystyle \frac{1}{16}$

  3. $\displaystyle -\frac{1}{16}$

  4. None of these

Show answer
Correct Option: B
Explanation:

$\displaystyle (64)^{-2/3} = 4^3 \times 1^{-2/3} = 4^{-2}$
                 $\displaystyle = \frac{1}{4^2} = \frac{1}{16}$   

The value of $[(-3)^{(-2)}]^{(-3)}$ is---

maths numbers and place value many forms of ten thousand standard form of numbers number names, numerals and place values

  1. 243

  2. 27

  3. 729

  4. None of these

Show answer
Correct Option: C
Explanation:

$[(-3)^{(-2)}]^{(-3)} = (-3)^6$
                           = 729

Charge of an electron is $0.00000000000000000016$ coulomb. This number can also be written in standard form as:

maths numbers and place value many forms of ten thousand standard form of numbers number names, numerals and place values

  1. $\displaystyle 1\cdot 6\times 10^{19}$

  2. $\displaystyle 1\cdot 6\times 10^{-20}$

  3. $\displaystyle 1\cdot 6\times 10^{-19}$

  4. $\displaystyle 1\cdot 6\times 10^{18}$

Show answer
Correct Option: C
Explanation:

$0.0000000000000000000016=\displaystyle \frac{16}{100000000000000000000}$


=$\displaystyle \frac{1\cdot 6\times 10^{1}}{10^{20}}$

=$\displaystyle 1\cdot 6\times 10^{-19}$

The value of $(3^0 - 2^1) \times 4^2$ is---

maths numbers and place value many forms of ten thousand standard form of numbers number names, numerals and place values

  1. -16

  2. 32

  3. 64

  4. 0

Show answer
Correct Option: A
Explanation:

$(3^0 - 2^1) \times 4^2 = (1-2) \times 16$
                                         $= -1 \times 16 = -16$