JEE 2017 :: Advanced 2017 :: Mathematics 2017

• Advanced 2017 - Physics 2017
• Advanced 2017 - Chemistry 2017
• Advanced 2017 - Mathematics 2017

Let  S={1,2,3.......,9} For k=1,2,...5 , Let N_k be the number of subsets of S, each containing five elements out of which exactly k are odd. Then N₁+N₂+N₃+N₄+N₅ =

Answer:

Explanation:

$N_{1}=^{5}C_{k}\times^{4}C_{5-k}$

  $N_{1}=5\times1$

$N_{2}=10\times4$

$N_{3}=10\times6$

$N_{4}=5\times4$

$N_{5}=1$

N₁+N₂+N₃+N₄+N₅ =126

Three randomly choosen non negative intergers x,y and z are found to satisfy the equation x+y+z=10 , Then the probability that z is even is 

Answer:

Explanation:

Sample space  $\rightarrow^{12}C_{2}$

Number of possibilities for z is even

   $z=0\Rightarrow^{11}C_{1}$

  $z=2\Rightarrow^{9}C_{1}$

  $z=4\Rightarrow^{7}C_{1}$

$z=6\Rightarrow^{5}C_{1}$

$z=8\Rightarrow^{3}C_{1}$

$z=10\Rightarrow^{1}C_{1}$

 Total=36

  $\therefore Probability=\frac{36}{66}=\frac{6}{11}$

How many 3x3 matrices M with entries from {0,1,2} are there, for which the sum of the diagonal entries of M^TM is 5?

Answer:

Explanation:

Sum of diagonal entries of M^TM is Σ a₁²

          $\sum_1^9a_1^2=5$

Possibilities

I.  2,1,0,0,0,0,0,0,0 which gives $\frac{9!}{7!}$ matrices

II. 1,1,1,1,1,0,0,0,0 which gives $\frac{9!}{4!\times5!}$ matrices

Total matrices= $9\times8+9\times7\times2=198$

if y=y(x) satisfies the differential equation

$8\sqrt{x}(\sqrt{9+\sqrt{x}})dy=(\sqrt{4+\sqrt{9+\sqrt{x}}})^{-1}$

dx,x >0 and $y(0)=\sqrt{7}$ , then y(256)=

Answer:

Explanation:

$\frac{dy}{dx}= \frac{1}{8\sqrt{x}\sqrt{9+\sqrt{x}}\sqrt{4+\sqrt{9+\sqrt{x}}}}$

$\Rightarrow y=\sqrt{4+\sqrt{9+\sqrt{x}}}+c$

Now, y(0)=$\sqrt{7}+c$

  c=0

$y(256)=\sqrt{4+\sqrt{9+16}}=\sqrt{4+5}=3$

If $f:R\rightarrow R$  is twice differentablr function such that f''(x)>0, for all xε R, and $f(\frac{1}{2})=\frac{1}{2}$ , f(1)=1, then 

Answer:

Explanation:

f'(x)  is increasing

 For some x in ($\frac{1}{2},1$)

f'(x)=1

f'(1)>1

• Advanced 2017 - Physics 2017
• Advanced 2017 - Chemistry 2017
• Advanced 2017 - Mathematics 2017