JEE 2017 :: Advanced 2017 :: Mathematics 2017

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

Let f:R→ R be a differentable function such that f(0)=0,  $f(\frac{\pi}{2})=3$ and f ' (0)=1

  if $g(x)=\int_{0}^{\frac{\pi}{2}} f'(t) cosec$ $ t f(t)-\cot $ $ t cosec$ $ t f(t)] dt$

  for   $x\epsilon(0,\frac{\pi}{2}]$, then $\lim_{x \rightarrow0 }g(x)=$

Answer:

Explanation:

$g(x)=\int_{x}^{\frac{\pi}{2}} f'(t) cosec$ $ t f(t)-\cot $ $ t cosec$ $ t f(t)] dt$

$g(x)= f(\frac{\pi}{2}) cosec $ $\frac{\pi}{2}-f(x) cosec$ $ x$

$\Rightarrow$   $g(x)=3-\frac{f(x)}{\sin x}$

$\lim_{x \rightarrow 0}g(x)=\lim_{x \rightarrow 0}(\frac{3\sin  x-f(x)}{\sin x})$

$=\lim_{x \rightarrow 0}\frac{3\cos x-f'(x)}{\cos x}$

$\frac{3-1}{1}=2$

For how many values of p, the circle $x^{2}+y^{2}+2x+4y-p=0$ and the coordinate axes have exactly three common points?

Answer:

Explanation:

The circle and coordinate axis can have 3 common points. If it passes through the origin (p=0)

 If the circle is cutting one axis and touching another axis

 The only possibility is one of touching X-axis and cutting Y-axis [P=-1]

The sides of a right angles triangle are in arithmetic progression. If the triangle has area 24, then whats is the length of its smallest side?

Answer:

Explanation:

Let the sides are a-d, a and a+d

     Then, a(a-d)=48

and a²-2ad+d²+a²=a²+2ad+d²

$\Rightarrow$  a² =4ad

$\Rightarrow$   a=4d

 Thus, a=8,d=2

  Hence a - d=6

If a chord, which is not a tangent of the parabola y²=16x  has the equation 2x+y=p, and mid-point (h,k) then which of the following is(are) possible value (s) of p,h, and k?

Answer:

Explanation:

Equation of chord with mid point (h,k)

  T=s₁

yk - 8x - 8h = k² - 16h

$2x-\frac{yk}{4}=2h-\frac{k^{2}}{4}$

$\because$   2x + y = p

$\because$  k = -4   and p =  2h-4

Let [X] be the greatest integer less than or equals to x. Then , at which of the following points(s) the function 

$f(x)=x\cos(\pi(x+[x]))$  discontinous?

Answer:

Explanation:

$f(x)=x\cos(\pi(x+[x]))$

  At x=0

   $\lim_{x \rightarrow 0}f(x)=\lim_{x \rightarrow 0}x\cos(\pi(x+[x])=0$

   and f(x)=0

 It is continous at x=0 and clearing discontinous at other integer points

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