IIT JEE Main 2016 Maths Question 38

IIT JEE Main 2016 Maths Question 38: 

The eccentricity of the hyperbola whose length of the latus rectum is equal to 8 and the length of its conjugate axis is equal to half of the distance between its foci, is:

(1) 2/√3
(2) √3
(3) 4/3
(4) 4/√3

Solution

Given in problem statement that for the hyperbola whose length of the latus rectum is equal to 8, i.e.
2b2/a=8
& the length of its conjugate axis is equal to half of the distance between its foci, hence, 2b =ae
=> 4b2 = a2e2
=> 4a2 (e2 -1) = a2e2
=> e = 2/√3

Answer to IIT JEE Main 2016 Maths Question 38

Correct answer to given problem No 38 of IIT JEE Main Maths paper of 2016 is option number (1) that is the eccentricity of hyperbola is 2/√3.

IIT JEE Main 2016 Maths Question 37

IIT JEE Main 2016 Maths Question 37: 

If the 2 , 5 and 9 terms of a non-constant AP (Arithematic Progression), are in GP (Geometric Progression), then the common ratio of GP (Geometric Progression) is:

(1) 1
(2) 7/4
(3) 8/5
(4) 4/3

Solution

Given terms of AP (Arithematic Progression) are
T2 = a  + d
T5 = a +  4d
T9 = a +  8d
If T2, T5 & T9 are in GP (Geometric Progression), then
 (a  + 4d)2 = (a + d)(a + 8d)
a2 + 16d2 + 8ad = a2 + ad + 8ad + 8d2
=> 8d2 = ad
=> d(8d – a) = 0
=> 8d = a (because d ≠ 0)
Now common ratio is (a+4d)/(a+d) = 12d/9d =4/3
.

Answer

Correct answer to given problem No 37 of IIT JEE Main Maths paper of 2016 is option number (4) that is the common ratio  for GP (Geometric Progression) is 4/3.

IIT JEE Main 2016 Maths Question 36

IIT JEE Main 2016 Maths Question 36: 

The sum of all real values of x satisfying the equation (x2 – 5x + 5)(x2 + 4x -60) = 1 is:

(1) 6
(2) 5
(3) 3
(4) -4

Solution

For ab = 1, three cases arise (a) when a is 1, then b ∈ R,
(b) If a is not zero, b = 0 and
(c) if a is -1, b ∈ even integer.

Based upon above three cases, we have
For (a) x2 – 5x + 5 = 1
=> x = 1, 4
For (b) x2 + 4x – 60 = 0
=> x = -10 , 6
For (c) x2 – 5x + 5 = -1 , x2 + 4x – 60 = even integer
x=2, 3
at x=2,  x2 + 4x – 60 = -48 (even integer)
at x = 3 (odd integer)
thus possible solutions are
x=1, 4, -10, 6, 2
and sum f solutions is 3.

Answer

Correct answer to given problem No 36 of IIT JEE Main Maths paper of 2016 is option number (3).

IIT JEE Main 2016 Maths Question 35

IIT JEE Main 2016 Maths Question 35: 

The centre(s) of those circles which touch the circle, x2 + y2 -8x -8y -4 = 0, externally and also touch the x-axis, lie on:

(1) a hyperbola
 (2) a parabola
(3) a circle
 (4) an ellipse which is not a circle:

Solution

Circles touch externally and required circle touches x-axis. Let the Centre be at point (h, k) and radius = | k |
C1C2 = r1 + r2
(h-4)2 + (k-4)2 = (6 + |k|)2
h2 -2.4h + 16 + k2 -8k + 16 = 36 + k2 + 12 |k|
h2 -8h -8k -12|k| =4
h2 -8x -8y -12|y|=4
It represents two-semi parabola.

Answer

Correct answer to given problem No 35 of IIT JEE Main Maths paper of 2016 is option number (2).

IIT JEE Main 2016 Maths Question 34

IIT JEE Main 2016 Maths Question 34: For x ∈ R, f
(x) = |log 2 – sin x| and g(x) = f(f(x)), then

(1) g'(0) =-cos(log 2)
(2) g is differentiable at x = 0 and g'(0)=-sin(log 2)
(3) g is not differentiable at x = 0
(4) g'(0) = cos(log 2)

Solution

f (x) = |log 2 – sin x|
g(x)=f(f(x))
g(x) =  |log 2 – sin (log 2 -sin x)|
g(x) = log 2 – sin (log 2 – sin x) (in neighbourhoodof x = 0)
g’ (0) =-cos(log 2 -sin 0) .(-cos 0)
g’ (0) = cos (log 2)

Answer

Correct answer to given problem No 34 of IIT JEE Main Maths paper of 2016 is option number (4).

IIT JEE Main 2016 Maths Question 33

IIT JEE Main 2016 Maths Question 33 requires to find if the integral
∫ [(2x12 + 5x9)/(x5 + x3 +1)3] dx is equal to:

(1) [(x5)/2(x5 + x3 + 1)2] + C (2) [(-x10)/2(x5 + x3 + 1)2] + C (3) [(-x5)/(x5 + x3 + 1)2] + C (4) [(x10)/2(x5 + x3 + 1)2] + C

where C is an arbitrary constant

Solution

Solution to the problem requiring to find out the required integral is obtained by using the substitution method as follows:
I = ∫ [(2x12 + 5x9)/(x5 + x3 +1)3] dx
=∫ [(2/x3 + 5x6)dx/(1/x5 + 1/x2 +1)3]
Let (1/x5 + 1/x2 +1) = t
=> (-2/x3 – 5x6)dx = dt
I = ∫dt/t3 = 1/2t2 + C
=> I = [1/2(1/x5 + 1/x2 +1)2] + C
=> I = [(x10)/2(x5 + x3 +1)2] + C

Answer

Correct answer to given problem No 33 of IIT JEE Main Maths paper of 2016 is option number (4).

IIT JEE Main 2016 Maths Question 32

IIT JEE Main 2016 Maths Question 32

If f(x) + 2f(1/x) = 3x, x<> 0 and S = {x ∈ R : f(x)=f(-x)}; then S :

(1) contains exactly two elements (2) contains more than two elements (3) is an empty set (4) contains exactly one element

Solution

Solution to the problem requiring to find out the set S as per the relation specified is as follows:
f(x) + 2f(1/x) = 3x     …………. (1)
Replace x->(1/x)
=>f(1/x) + 2f(x) = 3/x
=> 2f(1/x) + 4f(x) =6/x      ……(2)
=> from (1) and (2)
3f(x) = 6/x – 3x
=> f(x) = (2/x) – x
=> f(-x) = (-2/x) + x
=> f(x) = f(-x)
=> (2/x) – x = (-2/x) + x
=> 4/x = 2x
=>x2 = 2
=>x= +/- √2

Hence set S contains two elements, i.e. +√2 , -√2.

Answer

Correct answer to given problem No 32 of IIT JEE Main Maths paper of 2016 is option number (1) that is set S contains exactly two elements.

IIT JEE Main 2016 Maths Question 31

IIT JEE Main 2016 Maths Question 31

The area (in sq. units) of the region {(x, y) : y2 >=2x and x2 + y2 <= 4x, x>= 0, y >= 0} is:

(1) Π – {(4√2)/3} (2) (Π/2) – {(2√2)/3} (3) Π – (4/3) (4) Π – (8/3)

Solution

Solution to the problem requiring to find out the area under a region bounded by given curves, is obtained by integration as follows:

y2 = 2x,  x2 + y2 = 4x
=> x2 + 2x = 4x
=> x2 = 2x
=> x = 0, 2
Required area = ∫02 [√(4x – x2) – √(2x)] dx
= ∫02 [√(4x – x2) – √2√x] dx
=[{(x-2)/2}√(4x – x2) + 2 sin-1{(x-2)/2} -{(2/3)√2}x3/2]]20
= 0 + 0 – (8/3) – [0 + 2(-Π/2)]
=Π – (8/3)

Answer

Correct answer to given problem No 31 of IIT JEE Main Maths paper of 2016 is option number (4).

IIT JEE Main 2016 Physics Question 30

IIT JEE Main 2016 Physics Question 30

A pendulum clock loses 12 s a day if the temperature is 40ºC and gains 4s a day if the temperature is 20ºC. The temperature at which the clock will show correct time, and the coefficient of linear expansion (α) of the metal of the pendulum shaft are respectively:

(1) 30ºC; α = 1.85 × 10-3/ºC (2) 55ºC; α = 1.85 × 10-2/ºC (3) 25ºC; α = 1.85 × 10-5/ºC (4) 60ºC; α = 1.85 × 10-4/ºC

Solution

(ΔT/T) =(ΔL/L)
(ΔT/T) = (αΔT/2)
=> (ΔT/T) = (αΔT1/2)
=> (1/2)(αΔT1)x24x3600
=> 12 = (αΔT1) x 12 x 3600
=> 4 = (αΔT2) x 12 x 3600
=> 3 = (ΔT1/ΔT2) = (40 – T)/(T-20)
=> 3T – 60 = 40 – T
=> T = 25ºC
α = 1.85 × 10-5/ºC

Answer

Correct answer to given problem No 30 of IIT JEE Main Physics paper of 2016 is option number (3).

IIT JEE Main 2016 Physics Question 29

IIT JEE Main 2016 Physics Question 29

In an experiment for determination of refractive index of glass of a prism by i – δ, plot, it was found that a ray incident at angle 35º, suffers a deviation of 40º and that it emerges at angle 79º. In that case which of the following is closest to the maximum possible value of the refractive index?

(1) 1.7 (2) 1.8 (3) 1.5 (4) 1.6

Solution

∠i = 35º
δ = 40º
e = 79º
i = 35º
e = 79º
δ = 40º
δ + 4 = i + e
=> A = 74º
with above set of data, as refractive index increases emerging ray will have tendency to suffer TIR.
For i = 35º
μ = sin 35º/ sin r1
=> μ = sin 90º / sin (74º – r1)
Using reversibility principle
For i = 79º
μ = sin 79º/ sin r2 = sin 90º / sin (74º – r2)
From 2nd case r2 = 37º
=> μ =1.63

Answer

Correct answer to given problem No 29 of IIT JEE Main Physics paper of 2016 is option number (4).