Mathematics Test

Result

Mathematics Test - 11

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Question 1/10
1 / -0

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

Correct

Wrong

Skipped
A
Real and Distinct Roots

Correct

Wrong

Skipped
B
Imaginary roots

Correct

Wrong

Skipped
C
Real and Equal rrots

Correct

Wrong

Skipped
D
None of the above

Solutions

let event $A =$ minimun is 3 let event $B =$ maximum is 6
total ways $= \cdot^{8} C_{3}$ $P (A \cap B) = \frac{2}{.^{8} C_{3}} = \frac{2}{\frac{8!}{3! 5!}}$
$= \frac{2 \times 3!}{8 \times 7 \times 6} = \frac{1}{28}$
$P (\frac{A}{B}) = \frac{P (A \cap B)}{P (B)}$
$= \frac{\frac{1}{28}}{\frac{s^{5} C_{2}}{s C_{3}}}$
$= \frac{\frac{2}{s^{8} C_{3}}}{\frac{s C_{2}}{s C_{3}}}$
$= \frac{2}{5^{5} C_{2}}$
$= \frac{2 \times 2}{5 \times 4} = \frac{1}{5}$
option 1 is correct
Question 2/10
1 / -0

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

Correct

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Skipped
A
P(A ∪ B) − P(A ∩ B)

Correct

Wrong

Skipped
B
P(A ∪ B) + P(A ∩ B)

Correct

Wrong

Skipped
C
P(A) + P(B)

Correct

Wrong

Skipped
D
P(A) + P(B) + P(A ∩ B)

Solutions

$P (A \cup B) - P (A \cap B)$
Question 3/10
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

Correct

Wrong

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A
5 × 8³

Correct

Wrong

Skipped
B
5 × 9²

Correct

Wrong

Skipped
C
5 × 7³

Correct

Wrong

Skipped
D
5 × 9³

Solutions

For a number to be odd, the units digit should be either of 1,3,5,7 or 9
$∴$ 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
$∴$ No. of ways of choosing first and second digit is also 9 for each.
$∴$ 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 4/10
1 / -0

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

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Wrong

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A
9.8!

Correct

Wrong

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B
8.9!

Correct

Wrong

Skipped
C
9.9!

Correct

Wrong

Skipped
D
10!

Solutions

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 5/10
1 / -0

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

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Wrong

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A
Real and Distinct

Correct

Wrong

Skipped
B
Not Real

Correct

Wrong

Skipped
C
Real and Equal

Correct

Wrong

Skipped
D
None of these

Solutions

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

$D = 9 - 16$

$D = - 7$

$D < 0$

So the given quadratic equation have imaginary roots.
Question 6/10
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

Correct

Wrong

Skipped
A
0

Correct

Wrong

Skipped
B
5

Correct

Wrong

Skipped
C
6

Correct

Wrong

Skipped
D
15

Solutions

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 7/10
1 / -0

The value of $\frac{\sin 4 θ}{\cos θ}$ is

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A
4 cos θ cos 2θ

Correct

Wrong

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B
4 sin θ cos 2θ

Correct

Wrong

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C
4 sin θ sin 2θ

Correct

Wrong

Skipped
D
4 cos θ sin 2θ

Solutions
The value of $\frac{\sin 4 θ}{\cos θ}$ is
$= \frac{2 \sin 2 θ \cos 2 θ}{\cos θ}$
$= \frac{4 \sin θ \cos θ \cos 2 θ}{\cos θ}$
$= 4 \sin θ \cos 2 θ$
Hence, answer is option B.
Question 8/10
1 / -0

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

Correct

Wrong

Skipped
A
−π

Correct

Wrong

Skipped
B
+π

Correct

Wrong

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C
−2π

Correct

Wrong

Skipped
D
+2π

Solutions

The value of $\sin^{- 1} x + \sin^{- 1} \frac{1}{x} + \cos^{- 1} x + \cos^{- 1} \frac{1}{x}$ is defined only when
$x, \frac{1}{x} ϵ [- 1, 1]$
which is possible only when $x = \pm 1$ For which $\sin^{- 1} x + \sin^{- 1} \frac{1}{x} + \cos^{- 1} x + \cos^{- 1} \frac{1}{x} = π$
Question 9/10
1 / -0

$\int e^{\sin x} \sin 2 x d x =$

Correct

Wrong

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

Correct

Wrong

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

Correct

Wrong

Skipped
C
e^{sin x}(2 sin x −2) + c

Correct

Wrong

Skipped
D
e^{sin x}(3 sin x − 2) + c

Solutions

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

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

Correct

Wrong

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

Correct

Wrong

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

Correct

Wrong

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

Correct

Wrong

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

Solutions

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

Question 1/10

Real and Distinct Roots

Imaginary roots

Real and Equal rrots

None of the above

Question 2/10

P(A ∪ B) − P(A ∩ B)

P(A ∪ B) + P(A ∩ B)

P(A) + P(B)

P(A) + P(B) + P(A ∩ B)

Question 3/10

5 × 83

5 × 92

5 × 73

5 × 93

Question 4/10

9.8!

8.9!

9.9!

10!

Question 5/10

Real and Distinct

Not Real

Real and Equal

None of these

Question 6/10

0

5

6

15

Question 7/10

4 cos θ cos 2θ

4 sin θ cos 2θ

4 sin θ sin 2θ

4 cos θ sin 2θ

Question 8/10

−π

+π

−2π

+2π

Question 9/10

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

Question 10/10

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

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