Tag: maths

Questions Related to maths

Identify the function for the following sequence $4, 10, 18, 28...$

maths binomial theorem, sequence and series series introduction to series introduction to sequences and series

  1. $2n(n+3)$

  2. $n(n+3)$

  3. $n(n-3)$

  4. $n^2(n+3)$

Show answer
Correct Option: B
Explanation:

When $n = 1$, $1(1+3)=4$ i.e. $1(1+3)$
When $n = 2$, $2(2+3)=10$ i.e. $2(2+3)$
When $n = 3$, $3(3+3)=18$ i.e. $3(3+3)$
When $n = 4$, $4(4+3)=28$ i.e. $4(4+3)$
So, $4, 10, 27, 28..$ is the function for the sequence is $n(n+3)$.

Identify the sequence for the following function $n(n+3)$.

maths binomial theorem, sequence and series series introduction to series introduction to sequences and series

  1. $4, 10, 18, 28..$

  2. $4, 12, 18, 28..$

  3. $2, 10, 18, 28..$

  4. $4, 10, 18, 38..$

Show answer
Correct Option: A
Explanation:

Given function is $n(n+3)$
When $n = 1$, $1(1+3)=4$
When $n = 2$, $2(2+3)=10$
When $n = 3$, $3(3+3)=27$
When $n = 4$, $4(4+3)=28$....
So, $4, 10, 27, 28..$ is the function for the sequence is $n(n+3)$.

What is the next number in the sequence $2, 15, 41, 80, ?$

maths binomial theorem, sequence and series series introduction to series introduction to sequences and series

  1. $111$

  2. $120$

  3. $121$

  4. $132$

Show answer
Correct Option: D
Explanation:

Differences are $13, 26, 39$
$\Rightarrow 4^{th} $ difference $= 52$,
$\Rightarrow$ required number $= 80 + 52 = 132$.


$\therefore$ The solution is $132$.


A, B, C, D are four points in a straight line. Distance from A to  B is 10, B to C is 5, C to D  is 4 and A to D is 1. Which  one of the following is the correct sequence of the  points ?

maths binomial theorem, sequence and series series introduction to series introduction to sequences and series

  1. A- B - C - D

  2. A - C - B - D

  3. A - D - C - B

  4. A - C - D - B

Show answer
Correct Option: C
Explanation:
Given:$A,B,C,D$ are four points in a staraight line
Distance from $A$ to $B$ is $AB=10$ units
Distance from $B$ to $C$ is $BC=5$ units
Distance from $C$ to $D$ is $CD=4$ units
Distance from $A$ to $D$ is $AD=1$ unit
Here $10>5>4>1\Rightarrow\,AB>BC>CD>DA$
$AB$ is the longest line,the sequence is $A-B$
$\Rightarrow\,BC<CD\Rightarrow\,C$ lies between $A$ and $B$
$\therefore\,$the sequence is $A-C-B$
$AC=5$ units then $CB=5$ units
Now,$D$ may lie between $AC$ or $AB$
$\because \,CD<AC,\,CD<BC$
Since $AD$ starts from $A$ and lies between $A$ and $C$
and $AD=1$ unit
$\therefore\,D$ cannot lie between $B$ and $C$
Hence $D$ lies between $A$ and $C$
The correct sequence is $A-D-C-B$

Which of the following statements is false?

maths binomial theorem, sequence and series series introduction to series introduction to sequences and series

  1. $\left {a _{ij}\right } _{= 1}^{-\infty}$ can be viewed as a function $g : N\rightarrow R$ defined by $g(k) = a _{k}, \forall k\epsilon N$

  2. A sequence may have infinitely many terms

  3. A function is not necessarily a sequence

  4. The function $f : R\rightarrow R$ given by $f(x) = 2x + 1, x\epsilon R$ is a sequence

Show answer
Correct Option: A
Explanation:
(A) Both are absolutely different things and can't be viewed as same.
This statement is false.

(B) A sequence may have infinitely many terms. for ex. function $f$ : N $\rightarrow $ N given by $f$($x$) = $x$ will have many terms. As our domain is infinity, so putting infinite values of $x$ will result in infinite no. of terms $f(x)$ as $f(x)$ is $x$.
This statement is true.

(C) There are examples of functions which do not necessarily forms a sequence. You will study things related to this in higher standard or definitely in college.  
This statement is true.

(D) The domain should be natural number, as domain represents the position of that particular term whether it is $1^{st}$ or $2^{nd}$ term, and it can't be $(2.5)^{th}$ term (NOTE : 2.5 is real no.), so there should be natural no. in place of real no. in domain. "A sequence may be regarded as a function whose argument can take on only positive integral values—that is, a functiondefined on the set of natural numbers"
So this statement is also false.

Find the first five terms of the sequence specified by the recursion formula
${a} _{k+1}={a} _{k}+3$, if ${a} _{1}=7$.

maths binomial theorem, sequence and series series introduction to series introduction to sequences and series

  1. $7,10,13,16,19$

  2. $6,9,12,15,18$

  3. $8,11,14,17,20$

  4. $5,8,11,14,17$

Show answer
Correct Option: A
Explanation:

Given ${a} _{k+1} = {a} _{k}+3$ and ${a} _{1}=7$

So ${a} _{2}={a} _{1}+3 = 7+3=10$
${a} _{3}={a} _{2}+3 = 10+3=13$
${a} _{4}={a} _{3}+3 = 13+3=16$
${a} _{5}={a} _{4}+3=16+3=19$

The equation $3x^4-5x^3+3x^2-4x+5=0$ is of the type

reciprocal equations theory of equations maths

  1. Quadratic

  2. Linear

  3. Reiprocal

  4. None of the above

Show answer
Correct Option: D
Explanation:

Given equation is $3x^4-5x^3+3x^2-4x+5=0$ .... $(i)$

The maximum power of $x$ in this equation is $ 4$, so this is $4th$ degree equation  
So, $(i)$ is neither linear nor quadratic equation.
In reciprocal equation ($ ax^4 +bx^3 +cx^2 +dx +e = 0 $)  the multiplication of the roots i.e ($\dfrac{e}{a}$) should be  $1$

In the given equation $(i)$, $\dfrac{e}{a}  = \dfrac{5}{3}$  
So the multiplication of roots is not equal to one
Therefore, equation $(i)$ is not a reciprocal equation.

Hence, option D is correct.

$x+\dfrac{1}{x}=2, x^{1680}+\dfrac{1}{x^{1680}}$

reciprocal equations theory of equations maths

  1. $1$

  2. $-1$

  3. $2$

  4. $-2$

Show answer
Correct Option: C
Explanation:

We have,

$x+\dfrac{1}{x}=2$
We know that if $x+\dfrac{1}{x}=2$, then
$x^n+\dfrac{1}{x^n}=2$
Therefore,
$x^{1680}+\dfrac{1}{x^{1680}}=2$

Simplify the reciprocal equation $\dfrac{3}{12}=\dfrac{3}{2x}$

reciprocal equations theory of equations maths

  1. $0$

  2. $3$

  3. $6$

  4. $1$

Show answer
Correct Option: C
Explanation:

Given equation is $\dfrac {3}{12}=\dfrac {3}{2x}$

$\Rightarrow 6x=36$
$\Rightarrow x=\dfrac {36}{6}$
$\Rightarrow x=6$
Therefore, the reciprocal of the given function is $6$.

Simplify $\sqrt { 1+{ \left( \cfrac { { x }^{ 4 } }{ -2{ x }^{ 2 } }  \right)  }^{ 2 } } $

reciprocal equations theory of equations maths

  1. $\cfrac { { x }^{ 4 }+1 }{ 2{ x }^{ 2 } } $

  2. $\cfrac{\sqrt{{x}^{2}+1}}{2}$

  3. $\cfrac{{x}^{4}+2{x}^{2}-1}{2{x}^{2}}$

  4. $\cfrac { { x }^{ 4 }-1 }{ 2{ x }^{ 4 } } $

Show answer
Correct Option: B
Explanation:

$\sqrt{1+(\dfrac{x^4}{-2x^2})^2}$


$\Rightarrow \sqrt{1+\dfrac{x^8}{4x^4}}$

$\Rightarrow \sqrt{\dfrac{4x^4+x^8}{4x^4}}$

$\Rightarrow \dfrac{\sqrt{4x^4+x^8}}{2x^2}$

$\Rightarrow\cfrac{\sqrt{{x}^{2}+1}}{2}$