Mathematics Test

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Mathematics Test - 16

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Weekly Quiz Competition

Question 1
4 / -1

If no husband and wife play in the same game, then the number of ways in which a mixed double game can be arranged from among nine married couples is-

A
756

B
1512

C
3024

D
1024

Solution

Out of 9 men, 2 men can be chosen in ⁹C₂ways.
Since no husband and wife are to play in the same game, we have to select 2 women from the remaining 7 women.
This can be done in ⁷C₂ ways.
If M₁, M₂ and W₁, W₂ are chosen, then a team can be constituted in 2 ways, viz. M₁ W₁ or M₁ W₂.
Thus, number of ways of arranging the game = ⁹C₂ x⁷C₂ x 2 = 1512
Question 2
4 / -1

The number of ways of dividing 20 persons into 10 couples is

A
20!/2¹⁰

B
²⁰C₁₀

C
²⁰P₂

D
none of these

Solution

Here, the order of the couples is not important. So, required number of ways is 20!/2¹⁰10!
Hence option D is correct.
Question 3
4 / -1

The solution of the differential equation dy/dx + y/x = x² is

A
4xy = x⁴ + 4c

B
xy = x⁴ + c

C
1/4xy = x⁴ + c

D
None of these

Solution

The given equation \(\frac{d y}{d x}+\frac{y}{x}=x^{2}\) is of the form \(\frac{d y}{d x}+P y=Q\), so \(I F=e^{I \frac{1}{x} d x}=e^{\log x}=x\)
Hence, required solution \(x y=\int x \cdot x^{2} d x+c\) \(\Rightarrow x y=\frac{x^{4}}{4}+c\)
\(\Rightarrow 4 x y=x^{4}+4 c\)
Question 4
4 / -1

The smallest interval [a, b], such that \(\int_{0}^{1} \frac{1}{\sqrt{1+x^{4}}} d x \in[a, b]\) is given by

A
[1/√2, 1]

B
[0, 1]

C
[1/2, 1]

D
[3/4, 1]

Solution

\(1-\int_{0}^{1} \frac{d x}{\sqrt{1+x^{4}}},\) Here, \(0 \leq x \leq 1 \Rightarrow 1 \leq 1+x^{4} \leq 2\)
\(\Rightarrow 1 \leq \sqrt{1+x^{4}} \leq \sqrt{2} \Rightarrow \frac{1}{\sqrt{2}} \leq \frac{1}{\sqrt{1+x^{4}}} \leq 1\)
\(\Rightarrow \frac{1}{\sqrt{2}} \leq \int_{0}^{1} \frac{d x}{\sqrt{1+x^{4}}} \leq 1\)
Hence, \(|1 / \sqrt{2}, 1|\) is the smallest interval, such that \(\mid \in[1 / \sqrt{2}, 1]\) Note: - If \(\mathrm{m}\) - least value of \((\mathrm{x})=\mathrm{M}\) - greatest value of \(\mathrm{f}(\mathrm{x})\) is \(\mathrm{Ia}\), bl. then \(m(b-a) \leq \int_{\frac{1}{3}}^{b} f(x) d x \leq M(b-a)\)
Question 5
4 / -1

If A² - A + I = 0, then inverse of A is

A
I - A

B
A - I

C
A

D
A + I

Solution

A² – A + I = 0
I = A – (A × A)
IA^-1 = AA^-1 – A(AA^-1)
A^-1 = I - A
Question 6
4 / -1

What is the number of arrangements that can be made using all the letters of the word 'LAUGH', keeping the vowels adjacent?

A
120

B
24

C
48

D
None of these

Solution

Considering AU as one letter, we have 4 letters: L, AU, G, H, which can be permuted in 4! ways. But, AU can be put together in 2! ways. Thus, the required number of arrangements is 4! × 2! = 48.
Question 7
4 / -1

Assuming that f is continuous everywhere, (1/c) \(\int_{a c}^{b c} f(x / c) d x\)is equal to

A
\((1 / c) \int_{a}^{b} f(x) d x\)

B
\(\int_{a}^{b} f(x) d x\)

C
\(c \int_{2}^{b} f(x) d x\)

D
\(\int_{a c^{2}}^{b c^{2}} f(x) d x\)

Solution

We have \(\mid-(1 / c) \int_{0}^{b c} f(x / c) d x\)
Putting \(x / c=t, d x=c\) dt, we get, \(\mid-(1 / c) \int_{\frac{1}{2}}^{b} f(t) c d t-\int_{\frac{1}{2}}^{b} f(t) d t-\int_{\frac{1}{2}}^{b} f(x) d x\)
Question 8
4 / -1

The minimum value of √(e^x² - 1) is

A
1

B
e

C
0

D
none of these

Solution

Note that \(e^{x^{4}}-1 \geq 0\) for each \(x^{e} R\) and \(e^{0^{4}}-1=0\) \(\sqrt{e^{x^{4}}-1}\) acquires the minimum value 0 at \(x=0\)
Question 9
4 / -1

If (√8+i)⁵⁰ = 3⁴⁹ (a + ib), then a² + b² is

A
3

B
9

C
16

D
√8

Solution

\((\sqrt{8}+i)^{50}=3^{49}(a+i b)\)
Taking magnitude and squaring on both sides, we get \((8+1)^{50}=3^{98}\left(a^{2}+b^{2}\right)\)
\(9^{50}=3^{98}\left(a^{2}+b^{2}\right)\)
\(3^{100}=3^{98}\left(a^{2}+b^{2}\right)\)
\(\Rightarrow a^{2}+b^{2}=9\)
Question 10
4 / -1

Cos^-1 (cos7π/6) =

A
7π/6

B
5π/6

C
π/6

D
None of these

Solution

\(\cos ^{-1}\left(\cos \frac{7 \pi}{6}\right)=\cos ^{-1}\left(\cos \left(\pi+\frac{\pi}{6}\right)\right.\)
\(=\cos ^{-1}\left(-\cos \frac{\pi}{6}\right)=\pi-\cos ^{-1}\left(\cos \frac{\pi}{6}\right)\)
\(=\pi-\frac{\pi}{6}=\frac{5 \pi}{6}\)

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Mathematics Test - 16

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Mathematics Test - 16

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SHARING IS CARING

Question 1

756

1512

3024

1024

Solution

Question 2

20!/210

20C10

20P2

none of these

Solution

Question 3

4xy = x4 + 4c

xy = x4 + c

1/4xy = x4 + c

None of these

Solution

Question 4

[1/√2, 1]

[0, 1]

[1/2, 1]

[3/4, 1]

Solution

Question 5

I - A

A - I

A

A + I

Solution

Question 6

120

24

48

None of these

Solution

Question 7

\((1 / c) \int_{a}^{b} f(x) d x\)

\(\int_{a}^{b} f(x) d x\)

\(c \int_{2}^{b} f(x) d x\)

\(\int_{a c^{2}}^{b c^{2}} f(x) d x\)

Solution

Question 8

1

e

0

none of these

Solution

Question 9

3

9

16

√8

Solution

Question 10

7π/6

5π/6

π/6

None of these

Solution

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20!/2¹⁰

²⁰C₁₀

²⁰P₂

4xy = x⁴ + 4c

xy = x⁴ + c

1/4xy = x⁴ + c