Tag: maths

Questions Related to maths

Two cards are drawn simultaneously from a well shuffled pack of $52$ cards. The expected number of aces is?

maths introduction of probability theory types of events some more terms in probability events and its algebra

  1. $\dfrac{1}{221}$

  2. $\dfrac{3}{131}$

  3. $\dfrac{2}{113}$

  4. $\dfrac{1}{131}$

Show answer
Correct Option: A
Explanation:

$\begin{matrix} Two\, \, cards\, \, are\, \, drawn\, \, from\, \, a\, \, well\, \, shuffered\, \, pack\, \, of\, \, 52\, \, card. \ Then,\, \, number\, \, access\, \, is\, \,  \ \Rightarrow number\, \, of\, \, sample\, \, space=52/2 \ \Rightarrow There\, \, are\, \, four\, \, ace=\dfrac { { 4/2 } }{ { 52/2 } } =\frac { { 4\times 3 } }{ { 52\times 51 } }  \ =\dfrac { 1 }{ { 13\times 17 } } \, \, \, \, \, Ans. \  \end{matrix}$

Probability of any event $x$ lies

maths introduction of probability theory types of events some more terms in probability events and its algebra

  1. $0 < x < 1$

  2. $0\leq x < 1$

  3. $0\leq x \leq 1$

  4. $1 < x < 2$

Show answer
Correct Option: C
Explanation:

$P\in \left [ 0,1 \right ]$ 


Option $(C)$ is correct.

Probability of impossible event is

maths introduction of probability theory types of events some more terms in probability events and its algebra

  1. $1$

  2. $0$

  3. $\dfrac {1}{2}$

  4. $-1$

Show answer
Correct Option: B
Explanation:

(P) Of possible event is $0$

Which one can represent a probability of an event

maths introduction of probability theory types of events some more terms in probability events and its algebra

  1. $\dfrac {7}{4}$

  2. $-1$

  3. $-\dfrac {2}{3}$

  4. $\dfrac {2}{3}$

Show answer
Correct Option: D
Explanation:

     $P <  1$


 Optoion (D) is correct $2/3$

Probability of sure event is

maths introduction of probability theory types of events some more terms in probability events and its algebra

  1. $1$

  2. $0$

  3. $\dfrac {1}{2}$

  4. $2$

Show answer
Correct Option: A
Explanation:

Probability of sure event is always $1$

If P(A) = P(B), then

maths introduction of probability theory types of events some more terms in probability events and its algebra

  1. A and B are the same events

  2. A and B must be same events

  3. A and B may be different events

  4. A and B are mutually exclusive events.

Show answer
Correct Option: C
Explanation:
Given $P(A) = P(B)$
Then $A$ can be different event from $B$ because $A\;\xi\;B$ are Mutually exclusive.
i.e, $A\;\xi\;B$ may be different events.
Hence, the answer is $A$ and $B$ may be different events.

Sum of roots is $-1$ and sum of their reciprocals is $\dfrac{1}{6}$, then equation is?

maths theory of equations forming quadratic equation vieta’s formula for quadratic equations properties of roots of a quadratic equations

  1. $x^2+x-6=0$

  2. $x^2-x+6=0$

  3. $6x^2+x+1=0$

  4. $x^2-6x+1=0$

Show answer
Correct Option: A
Explanation:

$\Rightarrow$  Let $\alpha$ and $\beta$ are roots of the equation.

According to the given condition,
$\Rightarrow$  $\alpha+\beta=-1$                             ------ ( 1 )
Again according to the given condition,
$\Rightarrow$  $\dfrac{1}{\alpha}+\dfrac{1}{\beta}=\dfrac{1}{6}$

$\Rightarrow$  $\dfrac{\beta+\alpha}{\alpha\beta}=\dfrac{1}{6}$

$\Rightarrow$  $6(\alpha+\beta)=\alpha\beta$
$\Rightarrow$  $6(-1)=\alpha\beta$                          [ From ( 1 ) ]
$\therefore$  $\alpha\beta=-6$               ----  ( 2 )
Now, required equation,
$\Rightarrow$  $x^2-(\alpha+\beta)x+(\alpha\beta)=0$
Using ( 1 ) and ( 2 ) we get,
$\Rightarrow$  $x^2-(-1)x+(-6)=0$
$\therefore$  $x^2+x-6=0$

If the roots of $x^{3}-kx^{2}+14x-8=0$ are in geometric progression ,then $k=$

maths theory of equations forming quadratic equation vieta’s formula for quadratic equations properties of roots of a quadratic equations

  1. -3

  2. 7

  3. 4

  4. 0

Show answer
Correct Option: B
Explanation:

Let $\dfrac{a}{r}.a.ar $ be the roots 

$\Rightarrow \dfrac{a}{r}.a.ar=8$

$\Rightarrow a^{3}=8$

$\Rightarrow a=2$

$a=2 $ is a root given equation

$\Rightarrow 8-4k+28-8=0$

$\Rightarrow K=7$

The quadratic equation whose roots are twice the roots of  $2 x ^ { 2 } - 5 x + 2 = 0$  is:

maths theory of equations forming quadratic equation vieta’s formula for quadratic equations properties of roots of a quadratic equations

  1. $8 x ^ { 2 } - 10 x + 2 = 0$

  2. $x ^ { 2 } - 5 x + 4 = 0$

  3. $2 x ^ { 2 } - 5 x + 2 = 0$

  4. $x ^ { 2 } - 10 x + 6 = 0$

Show answer
Correct Option: B
Explanation:

$\begin{array}{l} Let\, \alpha \, and\, \beta \, be\, the\, root\, of\, the\, given\, equation. \ Then,\, \alpha +\beta =\frac { 5 }{ 2 } and\, \alpha \beta =\frac { 2 }{ 2 } =1 \ \therefore 2\alpha +2\beta  \ \therefore \left( { \alpha +\beta  } \right)  \ \therefore 5\left( { 2\alpha  } \right) \left( { 2\beta  } \right) =4 \ So\, the\, required\, equation\, is: \ { x^{ 2 } }-5x+4=0 \end{array}$


So, option $B$ is correct answer.

The sum and the product of the zeroes of a quadratic polynomial are $ \dfrac{-1}{2} $ and $ \dfrac{1}{2}$ respectively, then the polynomial is :

maths theory of equations forming quadratic equation vieta’s formula for quadratic equations properties of roots of a quadratic equations

  1. $2x^{2}+x+1$

  2. $2x^{2}-x+1$

  3. $2x^{2}-x-1$

  4. $2x^{2}+x-1$

Show answer
Correct Option: A
Explanation:

Given: Sum of zeroes $=-\dfrac 12$ and product of zeroes $=\dfrac 12$

We know,
$x^2-(\text{sum of zeroes})x+(\text{product of zeroes})=0$
$\Rightarrow x^2-\left(-\dfrac 12\right)x+\dfrac 12=0$
$\Rightarrow 2x^2+x+1=0$
is the required polynomial.