Mathematics Test

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Mathematics Test - 11

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Weekly Quiz Competition

Question 1
1 / -0

State the nature of the given quadratic equation (x + 1) (x + 2) + 2 = 0

A
Real and Distinct Roots

B
Imaginary roots

C
Real and Equal rrots

D
None of the above

Solution

let event \(A=\) minimun is 3 let event \(B =\) maximum is 6
total ways \(=\cdot{ }^{8} C_{3}\) \(P(A \cap B)=\frac{2}{.^{8} C_{3}}=\frac{2}{\frac{8 !}{3 ! 5 !}}\)
\(=\frac{2 \times 3 !}{8 \times 7 \times 6}=\frac{1}{28}\)
\(P\left(\frac{A}{B}\right)=\frac{P(A \cap B)}{P(B)}\)
\(=\frac{\frac{1}{28}}{\frac{s^{5} C_{2}}{s C_{3}}}\)
\(=\frac{\frac{2}{s^{8} C_{3}}}{\frac{s C_{2}}{s C_{3}}}\)
\(=\frac{2}{5^{5} C_{2}}\)
\(=\frac{2 \times 2}{5 \times 4}=\frac{1}{5}\)
option 1 is correct
Question 2
1 / -0

If \(\mathrm{A} \& \mathrm{B}\) are two events then \(P\{(A \cap \bar{B}) \cup(\bar{A} \cap B)\}=\)

A
P(A ∪ B) − P(A ∩ B)

B
P(A ∪ B) + P(A ∩ B)

C
P(A) + P(B)

D
P(A) + P(B) + P(A ∩ B)

Solution

\(P(A \cup B)-P(A \cap B)\)
Question 3
1 / -0

The number of four digit odd numbers that can be formed using 1,2,3,4,5,6,7,8,9 (repetition of digits is allowed ) are

A
5 × 8³

B
5 × 9²

C
5 × 7³

D
5 × 9³

Solution

For a number to be odd, the units digit should be either of 1,3,5,7 or 9
\(\therefore\) Total number of ways of choosing units digit \(=5\) since, it is given that the digit may be repeated. So, we have 9 choices for ten's digit Similarly, for rest two digits we can choose any of the 9 digits
\(\therefore\) No. of ways of choosing first and second digit is also 9 for each.
\(\therefore\) Total number of ways to form odd numbers is \(5 \times 9 \times 9 \times 9=5 \times 9^{3}\)
Hence, the number of 4 digit odd numbers can be formed is \(5 \times 9^{3}\)
Question 4
1 / -0

The number of nine digit numbers that can be formed with different digits is

A
9.8!

B
8.9!

C
9.9!

D
10!

Solution

The first digit is one of (1,2,3,4,5,6,7,8,9) (i.e., all
expect 0 ). So, the no. of ways \(=9\) The remaining 8 digits are to be taken from the remaining 9 numbers. So, the no. of ways \(=9 P_{8}\) So, total no. of ways \(=9 \times^{9} P_{8}\) \(=9 \times 9 !\)
Question 5
1 / -0

What are the nature of quadratic equation x²+ 3x + 4 = 0 ?

A
Real and Distinct

B
Not Real

C
Real and Equal

D
None of these

Solution

Using \(D=(3)^{2}-(4)(1)(4)\)

\(D=9-16\)

\(D=-7\)

\(D<0\)

So the given quadratic equation have imaginary roots.
Question 6
1 / -0

If the first term of an AP is 5 and the common difference is −2, then the sum of the first 6 terms is

A
0

B
5

C
6

D
15

Solution

Sum of an AP is:
\(\frac{n}{2} \times[2 a+(n-1) d]\)
\(a=5, d=-2, n=6\)
Sum \(=\frac{6}{2} \times[(2 \times 5)+(6-1) \times(-2)]\)
\(=3 \times(10-10)\)
\(=0\)
Question 7
1 / -0

The value of \(\frac{\sin 4 \theta}{\cos \theta}\) is

A
4 cos θ cos 2θ

B
4 sin θ cos 2θ

C
4 sin θ sin 2θ

D
4 cos θ sin 2θ

Solution

The value of \(\frac{\sin 4 \theta}{\cos \theta}\) is
\(=\frac{2 \sin 2 \theta \cos 2 \theta}{\cos \theta}\)
\(=\frac{4 \sin \theta \cos \theta \cos 2 \theta}{\cos \theta}\)
\(=4 \sin \theta \cos 2 \theta\)
Hence, answer is option B.
Question 8
1 / -0

Find the value of \(\sin ^{-1} x+\sin ^{-1} \frac{1}{x}+\cos ^{-1} x+\cos ^{-1} \frac{1}{x}\)

A
−π

B
+π

C
−2π

D
+2π

Solution

The value of \(\sin ^{-1} x+\sin ^{-1} \frac{1}{x}+\cos ^{-1} x+\cos ^{-1} \frac{1}{x}\) is defined only when
\(x, \frac{1}{x} \epsilon[-1,1]\)
which is possible only when \(x=\pm 1\) For which \(\sin ^{-1} x+\sin ^{-1} \frac{1}{x}+\cos ^{-1} x+\cos ^{-1} \frac{1}{x}=\pi\)
Question 9
1 / -0

\(\int e^{\sin x} \sin 2 x d x=\)

A
2e^{sin x}(sin x + 1) + c

B
e^{sin x}(sin x + 2) + c

C
e^{sin x}(2 sin x −2) + c

D
e^{sin x}(3 sin x − 2) + c

Solution

\(\int e^{\sin x} 2 \sin x \cos x d x\)
\(=2 \int e^{\sin x} \sin x d(\sin x)\)
Take \(\sin x=t\)
\(=2 \int e^{t} t d t=2\left(t e^{t}-e^{t}\right)\)
\(=2 e^{t}(t-1)\)
\(=2 e^{\sin x}(\sin x-1)\)
\(\Rightarrow \int e^{\sin x} \sin 2 x d x=2 e^{\sin x}(\sin x-1)+c\)
Question 10
1 / -0

\(\int(x+2) \sqrt{x+1} d x=\)

A
2/5 (x + 1)^3/2(3x + 8) + c

B
2/15 (x − 1)^3/2(3x − 8) + c

C
2/15 (x + 1)^3/2(3x + 8) + c

D
2/15 (x − 1)^3/2(3x + 8) + c

Solution

\(\int(x+2) \sqrt{x+1} d x\)
\(\int[(x+1)+1](\sqrt{x+1}) d x\)
\(\int(x+1)^{\frac{3}{2}} d x+\int \sqrt{x+1} d x\)
\(=\frac{2}{5}(x+1)^{5 / 2}+\frac{2}{3}(x+1)^{3 / 2}+c\)
\(=\frac{2}{15}\left[3(x+1)^{5 / 2}+5(x+1)^{3 / 2}\right]+c\)
\(=\frac{2}{15}(x+1)^{3 / 2}(3 x+8)+c\)

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Mathematics Test - 11

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Mathematics Test - 11

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Question 1

Real and Distinct Roots

Imaginary roots

Real and Equal rrots

None of the above

Solution

Question 2

P(A ∪ B) − P(A ∩ B)

P(A ∪ B) + P(A ∩ B)

P(A) + P(B)

P(A) + P(B) + P(A ∩ B)

Solution

Question 3

5 × 83

5 × 92

5 × 73

5 × 93

Solution

Question 4

9.8!

8.9!

9.9!

10!

Solution

Question 5

Real and Distinct

Not Real

Real and Equal

None of these

Solution

Question 6

0

5

6

15

Solution

Question 7

4 cos θ cos 2θ

4 sin θ cos 2θ

4 sin θ sin 2θ

4 cos θ sin 2θ

Solution

Question 8

−π

+π

−2π

+2π

Solution

Question 9

2esin x(sin x + 1) + c

esin x(sin x + 2) + c

esin x(2 sin x −2) + c

esin x(3 sin x − 2) + c

Solution

Question 10

2/5 (x + 1)3/2(3x + 8) + c

2/15 (x − 1)3/2(3x − 8) + c

2/15 (x + 1)3/2(3x + 8) + c

2/15 (x − 1)3/2(3x + 8) + c

Solution

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5 × 8³

5 × 9²

5 × 7³

5 × 9³

2e^{sin x}(sin x + 1) + c

e^{sin x}(sin x + 2) + c

e^{sin x}(2 sin x −2) + c

e^{sin x}(3 sin x − 2) + c

2/5 (x + 1)^3/2(3x + 8) + c

2/15 (x − 1)^3/2(3x − 8) + c

2/15 (x + 1)^3/2(3x + 8) + c

2/15 (x − 1)^3/2(3x + 8) + c