Mathematics Test

Result

Mathematics Test - 5

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Weekly Quiz Competition

Question 1
2.5 / -0.83

If ƒ(x) = √(2tan(x)),then f^-1 (√(2)) =

A
0

B
1

C
√(2)

D
2

Solution

Correct Answer :- b

Explanation : ƒ(x) = √(2tan(x))

x = √(2tan(x))

x² = (2tan(x))

(x² )/2 = tan(x)

tan^-1 (x^2)/2 = x

By putting x = √2, we get

x = 1
Question 2
2.5 / -0.83

The value of cos15^º −sin15^º is

A
- 1/√2

B
1

C
0

D
1/√2

Solution
Question 3
2.5 / -0.83

If 2tan⁻¹ (cos x) = tan⁻¹ (2cosec x) , then x =

A
π/6

B
π/4

C
π/3

D
None of these.

Solution

If 2 tan^-1 (cos x) = tan ^-1 (2 cosec x),

2tan^-1 (cos x) = tan^-1 (2 cosec x)

= tan^-1 (2 cosec x)

= cot x cosec x = cosec x = x = π/4
Question 4
2.5 / -0.83

is equal to

A
a + b

B
log a + log b

C
log a –log b

D
a –b

Solution

We have:
limx →0 ln(1+ax)]/x=a
lim x →0 ln(1+ax)/ax=a
and then:
lim x →0 [ln(1+ax)-ln(1+bx)]/x
= a-b
Question 5
2.5 / -0.83

is equal to

A
1

B
0

C
1/2

D
none of these

Solution

On rationalizing,
Lt x-->0 (1-Cosx)/[√(1+x) +1]
= lt x-->0 2sin² (x/2)[√(1+x)+1] /x
= lt x-->0 [2sin(x/2)/x].sinx/2 [√(1+x) +1 ]
= 2.1/2 .0[1+1]=0
Question 6
2.5 / -0.83

Let f (x) = x sin 1/x, x ≠0, then the value of the function at x = 0, so that f is continuous at x = 0, is

A
2

B
1

C
-1

D
0

Solution

f(0) = lim(x →0) x sin (1/x)
We know ∀x ∈R,sin (1/x) ∈[−1,1]
Hence, f(0) = lim(x →0) xsin(1/x)
= 0
Question 7
2.5 / -0.83

A
3/4

B
4/3

C
1

D
12

Solution

sin3x/sin4x = sin3x/(1)⋅(1/sin4x)
= [(3x/1)sin3x/3x] ⋅[1/4x(4x/sin4x)]
= 3x/4x[sin(3x)/3x(4x/sin(4x))]
= 3/4 [sin3x/3x/(4x/sin4x)]
Now, as x →0, (3x)→0 so sin3x/3x →1.(Using θ= 3x)
And, as x →0, (4x)→0 so 4x/sin4x →1. (Using θ= 4x)
Therefore the limit is 3/4
lim x →0 sin3x/sin4x
= lim x →0 3/4sin3x/3x(4x/sin4x)
= 3/4 lim x →0 sin3x/3x
lim x →0 4x/sinx
= 3/4(1)(1)
= 3/4
Question 8
2.5 / -0.83

The value of

A
1/8

B
1/16

C
1/32

D
1/64

Solution
Question 9
2.5 / -0.83

A
1/8

B
1/4

C
1/2

D
1/6

Solution
Question 10
2.5 / -0.83

The instantaneous rate of change at t = 1 for the function f (t) = te^−t + 9 is

A
2

B
0

C
9

D
–1

Solution