﻿ If A is a 3x3 non- singular matrix such that AAT = ATA, then BBT is equal to from Mathematics Class 12 JEE Year 2014 Free Solved Previous Year Papers

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# JEE Mathematics 2014 Exam Questions

#### Multiple Choice Questions

21.

If g is the inverse of a function f and f'(x) =  then g'(x) is equal to

• 1+ x6

• 5x4

• 1+{g(x)}5

D.

1+{g(x)}5

Here 'g' is the inverse of f(x)
⇒ fog (x) =x
On differentiating w.r.t x, we get
f'{g(x)} x g'(x) =1

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22.

Let A and B be two events such that  where . Then, the events A and B are

• independent but not equally likely

• independent and equally likely

• mutually exclusive and independent

• equally likely but not independent

A.

independent but not equally likely

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23.

The Integral  is equal to

• π-4

D.

By using the formula,
It breaks given integral in two parts and then integrates separately.

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24.

The integral  dx is equal to

B.

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25.

If =-1 and x =2 are extreme points of f(x) =α log|x| + βx2 +x, then

• α = -6, β = 1/2

• α = -6, β = -1/2

• α = 2, β = -1/2

• α = 2, β = 1/2

C.

α = 2, β = -1/2

Here, x =-1 and x = 2 are extreme points of f(x) = α log|x| +βx2 +x then,
f'(x) = α/x +2βx + 1
f'(-1) = -α -2β +1 = 0  .... (i)
[At extreme point f'(x) = 0]
f'(2) = α/x +4βx + 1 = 0 .. (ii)
On solving Eqs (i) and (ii), we get
α = 2 and β = -1/2

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26.

The area of the region described by A = {(x,y): x2 +y2 ≤ 1 and y2 ≤1-x} is

A.

Given, {(x,y): x2 +y2 ≤ 1 and y2 ≤1-x}
Required area =

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27.

If fk(x) = 1/k (sink x + cosk x), where x ε R and k ≥1, then f4 (x)-fo (x) equal to

• 1/6

• 1/3

• 1/4

• 1/12

D.

1/12

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28.

If f and ga re differentiable  functions in (0,1) satisfying f(0) =2= g(1), g(0) = 0 and f(1) = 6, then for some c ε] 0,1[

• 2f'(c) = g'(c)

• 2f'(c) = 3g'(c)

• f'(c) = g'(c)

• f'(c) = 2g'(c)

D.

f'(c) = 2g'(c)

Given, f(0) = 2 = g(1), g(0) and f(1) = 6
f and g are differentiable in (0,1)
Let h(x) = f(x)-2g(x)  .... (i)
h(0) = f(0)-2g(0)
h(0) = 2-0
h(0) = 2
and h(1) = f(1)-2g(1) = 6-2(2)
h(1) = 2, h(0) = h(1) = 2
Hence, using rolle's theorem
h'(c) = 0, such that cε (0,1)
Differentiating Eq. (i) at c, we get
f'(c) -2g'(c) = 0
f'(c) = 2g'(c)

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29.

Let the population of rabbits surviving at a time t be governed by the differential equation. If p(0) = 100 then p(t) is equal to

A.

Given differential equation  is a linear differential equation
Here, p(t) =

Hence, solution is
p(t), IF = ∫Q(t)IF dt

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30.

If A is a 3x3 non- singular matrix such that AAT = ATA, then BBT is equal to

• l +B
• l
• B-1

• (B-1)T

B.

l

If A is non - singular matrix then |A| ≠0
AAT = ATA and B = A-1AT
BBT = (A-1AT)(A-1AT)T
= A-1ATA(A-1)T       [∵ (AB)T= BTAT]
=A-1AAT(A-1)T        [∵ AAT = ATA]
=AT(A-1)T              [ ∵A-1A = l]
=A-1A)T                 [∵ (AB)T = BTAT]
lTl

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