Differentiation Test 41

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Differentiation Test 41

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Question 1
1 / -0

How many integers are there such that $$2 \le n \le 100$$ and the highest common factor of $$n$$ and $$36$$ is $$1$$?

A
$$166$$

B
$$332$$

C
$$331$$

D
$$416$$

Solution

$$36.2^{2}.3^{2}$$
Since HCF (36,n) =1
Therefore, n sholud not be a multiple of 2 or 3.
$$ 2 \leq n\leq 100$$
Total numbers = T= 1000.
Number of numbers divisible by 2= $$N_{2}$$
Number of numbers divisible by 3= $$N_{3}$$
Number of numbers divisible by 6= $$N_{6}$$
Therefore, there are $$T-N_{2}-N_{3}+N_{6}$$ total integers
a=2
l=1000
d=2
We know, $$l=a+(N_{2}-1)d.$$
$$N_{2}=\frac{1000-2}{2}+1$$
$$N_{2}=\frac{998}{2}+1$$
$$N_{2}=499+1=500$$
Similarly,
$$N_{3}=\frac{999-3}{3}+1=333$$
$$N_{6}=\frac{999-6}{3}+1=166$$
Answer
= 999-500-333+166
= 332
Question 2
1 / -0

If area of a triangle is $$35$$ square units with vertices $$\left( {2, - 6} \right),\,\,\left( {5,\,\,4} \right)$$ and $$({k},\,\,4)$$ then $${k}$$ is :

A
$$ -2 \space \text{or} \space 12$$

B
$$12$$

C
$$-2$$

D
$$ 2 \space \text{or} \space 12$$

Solution
Question 3
1 / -0

The number of ways in which $$9$$ persons can be divided into three equal groups, is

A
$$1680$$

B
$$840$$

C
$$560$$

D
$$280$$

Solution
Question 4
1 / -0

What is the next number in the series $$2,12,36,80,150?$$

A
$$194$$

B
$$210$$

C
$$252$$

D
$$258$$

Solution

$$12 - 2 = 10$$.
$$36 - 12 = 24$$.
$$80 - 36 = 44$$.
$$150 - 80 = 70$$.
Now,
$$24 - 10 = 14$$.
$$44 - 24 = 20$$.
$$70 - 44 = 26$$.
Then next difference of the difference will be $$26 + 6 = 32$$.
Hence answer is $$150 + 70 + 32 = 252$$.
Question 5
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Let $$f:[0,2]\rightarrow R$$ be a twice differentiable function such that $$f"(x)>0$$, for all $$x\in (0,2)$$ If $$\phi (x)=f(x)+f(2-x)$$, then $$\phi$$is:

A
decreasing on $$(0,2)$$

B
decreasing on $$(0,1)$$ and increasing on $$(1,2)$$

C
increasing on $$(0,2)$$

D
increasing on $$(0,1)$$ and decreasing on $$(1,2)$$

Solution

$$\phi(x)=f(x)+f(2-x)$$
$$\phi '(x)=f'(x)-f'(2-x)$$...(1)
Since $$f"(x)>0$$
$$\Rightarrow f'(x) is increasing \forall x\in (0,2)$$
case I when $$x>2-x \Rightarrow x>1$$
$$\Rightarrow \phi'(x) >0\forall x \in (1,2)$$
$$\therefore \phi (x) is increasing on (1,2)$$
case II: when $$x<2-x \Rightarrow x<1$$
$$\Rightarrow \phi '(x) <0\forall x\in (0,1)$$
$$\therefore \phi (x) $$ is decreasing on $$(0,1)$$
Question 6
1 / -0

The number of permutations which can be formed out of the letters of the word "SERIES" three letters together, is:

A
120

B
60

C
42

D
none

Solution
Question 7
1 / -0

The coefficient of $$x^{18}$$ in the expansion of $$(1+x)(1-x)^{10}\{(1+x+x^2)^9\}$$ is?

A
$$84$$

B
$$126$$

C
$$-42$$

D
$$42$$

Solution

Coefficient of $$x^{18}$$ in $$(1+x)(1-x)^{10}(1+x+x^2)^9$$
$$=$$Coefficient of $$x^{18}$$ in $$(1-x)^2\{(1-x)(1+x+x^2)\}^9$$
$$=$$Coefficient of $$x^{18}$$ in $$(1-x^2)(1-x^3)^9$$
$$={^9C_6}=0=84$$.
Question 8
1 / -0

Differentiate the following function with respect to x.
$$\dfrac{x^3}{3}-2\sqrt{x}+\dfrac{5}{x^2}$$.

A
$$x^2-x^{-1/2}-5x^{-3}$$.

B
$$x^2-x^{-1/2}-10x^{-3}$$.

C
$$x^2-x^{-1/2}+5x^{-3}$$.

D
$$-x^2+x^{-1/2}+10x^{-3}$$.

Solution
Question 9
1 / -0

Differentiate the following function with respect to x.
$$3^x+x^3+3^3$$.

A
$$3^xlog 3+3x^2$$.

B
$$3^xlog 3+3x$$.

C
$$3^xlog 3+x^2$$.

D
$$xlog 3+3x^2$$.

Solution
Question 10
1 / -0

Let $$ u(x) $$ and $$ v(x) $$ be differentiable functions such that $$ \dfrac{u(x)}{v(x)}=7 . $$ If $$ \dfrac{u^{\prime}(x)}{v^{\prime}(x)}=p $$ and $$ \left(\dfrac{u(x)}{v(x)}\right)^{\prime}=q, $$ then $$ \dfrac{p+q}{p-q} $$ has the value equal to

A
$$1$$

B
$$0$$

C
$$7$$

D
$$-7$$

Solution

$$ u(x)=7 v(x) \Rightarrow u^{\prime}(x)=7 v^{\prime}(x) \Rightarrow p=7 $$ (given)

$$\operatorname{Again} \dfrac{u(x)}{v(x)}=7 \Rightarrow\left(\dfrac{u(x)}{v(x)}\right)^{\prime}=0 \Rightarrow q=0 $$

$$\text { Now } \dfrac{p+q}{p-q}=\dfrac{7+0}{7-0}=1$$