Mathematics Test

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Mathematics Test - 12

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Weekly Quiz Competition

Question 1
4 / -1

If \(x+\frac{1}{x}=4\), then find the value of \(x^{4}+\frac{1}{x^{4}}\).

A
\(194\)

B
\(136\)

C
\(128\)

D
\(161\)

Solution

Given,

\(x+\frac{1}{x}=4\)

By squaring,

\(\Rightarrow x^{2}+\frac{1}{x^{2}}+2=16\)

\(\Rightarrow x^{2}+\frac{1}{x^{2}}=14\)

By squaring again,

\(\Rightarrow x^{4}+\frac{1}{x^{4}}+2=196\)

\(\Rightarrow x^{4}+\frac{1}{x^{4}}=196-2=194\)
Question 2
4 / -1

If f(x) = x^a + a^x and g(x) = x^a – a^x, find the value of f(x) + g(x).

A
2x^a

B
5x^a

C
(x^a)²

D
x^a

Solution

f(x) + g(x) = 2x^a

Hence option A is correct.
Question 3
4 / -1

The order of a differential equation whose solution is y = a cos x + b sin x, where a and b are arbitrary constants, is

A
1

B
2

C
3

D
Cannot be determined

Solution

The order of differential equation = number of arbitrary constants = 2

Hence option B is correct.
Question 4
4 / -1

\(a=\frac{\sqrt{7}+\sqrt{11}}{\sqrt{11}-\sqrt{7}} ;\) and \(b=\frac{1}{a}\);then the value of\(\left(6 a^{2}-35 a b+6 b^{2}\right)\) is:

A
\(1\)

B
\(81\)

C
\(18\)

D
\(439\)

Solution

\(\because b=\frac{1}{a} \)

\(\Rightarrow a b=1\)

\(a^{2}+b^{2}=a^{2}+\frac{1}{a^{2}}\)

\(=\left(a+\frac{1}{a}\right)^{2}-2\)

\(=\left(\frac{\sqrt{7}+\sqrt{11}}{\sqrt{11}-\sqrt{7}}+\frac{\sqrt{11}-\sqrt{7}}{\sqrt{11}+\sqrt{7}}\right)^{2}-2\)

\(=\left(\frac{11+7+2 \sqrt{77}+11+7-2 \sqrt{11}}{11-7}\right)^{2}-2\)

\(=\left(\frac{36}{4}\right)^{2}-2=81-2=79\)

\(6 a^{2}-35 a b+6 b^{2}=6\left(a^{2}+b^{2}\right)-35 \times 1\)

\(=6 \times 79-35\)

\(=474-35=439\)
Question 5
4 / -1

\(\int \cos \sqrt{x} d x=?\)

A
2[√x sin √x + cos √x] + C

B
[√ x sin √x - cos √x] + C

C
2[√x sin √x - cos √x] + C

D
None of these

Solution

Putting \(x=t^{2}\) and \(d x=2 t\) dt. we get. \(1=2 \int t \cos t d t=2 \cdot\left[t \sin t-\int \sin t d t\right]+C\)
\(=2 t \sin t+2 \cos t+C\)
\(-2[\sqrt{x} \sin \sqrt{x}+\cos \sqrt{x}]+C\)
Question 6
4 / -1

The order and degree of the differential equation \(r^{2} \frac{d^{2} y}{d x^{2}}=\) \(\left[1+\left(\frac{d y}{d x}\right)^{2}\right]^{3 / 2}\) are respectively

A
2,6

B
6,2

C
2,2

D
2,3

Solution

Square the differential equation to remove the fraction,

We get, order = degree = 2
Question 7
4 / -1

The diagonals of a parallelogram PQRS are along the lines x + 3y = 4 and 6x - 2y = 7. Then PQRS must be a

A
rectangle

B
square

C
cyclic quadrilateral

D
rhombus

Solution

The slope of the first line: - 1/3

Slope of second line is: 3

Hence the diagonals are perpendicular, which is the property of a Rhombus.
Question 8
4 / -1

If \(y=\tan ^{-1}\left(\frac{\sin x}{1+\cos x}\right),\) then \(\frac{d y}{d x}=\)

A
1/

B
2

C
1/2

D
3/2

Solution

\(y=\tan ^{-1}\left(\frac{\sin x}{1+\cos x}\right)\)
\(=\tan ^{-1}\left\{\frac{\sin (2 \alpha)}{1+\cos (2 \alpha)}\right\}\)
where \(x=2 a\)
\(=\tan ^{-1}\left(\frac{2 \sin \alpha \cos \alpha}{2 \cos ^{2} \alpha}\right)\)
\(=\tan ^{-1}(\tan \alpha)\)
\(\Rightarrow y=\alpha=\frac{1}{2} x\)
\(\Rightarrow y=\frac{x}{2}\)
\(\frac{d y}{d x}=\frac{1}{2}\)
Question 9
4 / -1

\(\int e 3 \log x\left(x^{4}+1\right)^{-1} d x\) is equal to

A
log (x⁴ + 1) + C

B
1/4 log (x⁴ + 1) + C

C
- log (x⁴ + 1)

D
none of these

Solution

\(\int e^{3} \log x\left(x^{4}+1\right)^{-1} d x=\int e^{\log x^{3}} \frac{1}{x^{4}+1} d x\)
\(=\int \frac{x^{3}}{x^{4}+1} d x=1 / 4 \log \left(x^{4}+1\right)+C\)
Question 10
4 / -1

If the sets A and B are defined as

A = {(x, y) : y =1/x, 0 ≠ x ∈ R}

B = {(x, y) : y = -x, x ∈ R}, then

A
A ∩ B = A

B
A ∩ B = B

C
A ∩ B = ∅

D
None of these

Solution

since \(y=\frac{1}{x} y=-x\) meet when \(-x=\frac{1}{x} \Rightarrow x^{2}=-1\)
which does not give any real value of \(x\).
Hence, \(A \cap B=\emptyset\).

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Re-Attempt Weekly Quiz Competition

Mathematics Test - 12

Result

Mathematics Test - 12

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SHARING IS CARING

Question 1

\(194\)

\(136\)

\(128\)

\(161\)

Solution

Question 2

2xa

5xa

(xa)2

xa

Solution

Question 3

1

2

3

Cannot be determined

Solution

Question 4

\(1\)

\(81\)

\(18\)

\(439\)

Solution

Question 5

2[√x sin √x + cos √x] + C

[√ x sin √x - cos √x] + C

2[√x sin √x - cos √x] + C

None of these

Solution

Question 6

2,6

6,2

2,2

2,3

Solution

Question 7

rectangle

square

cyclic quadrilateral

rhombus

Solution

Question 8

1/

2

1/2

3/2

Solution

Question 9

log (x4 + 1) + C

1/4 log (x4 + 1) + C

- log (x4 + 1)

none of these

Solution

Question 10

A ∩ B = A

A ∩ B = B

A ∩ B = ∅

None of these

Solution

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2x^a

5x^a

(x^a)²

x^a

log (x⁴ + 1) + C

1/4 log (x⁴ + 1) + C

- log (x⁴ + 1)