Mathematics Test

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Mathematics Test - 7

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Weekly Quiz Competition

Question 1
4 / -1

The area bounded by the curve y = x | x |, x-axis and the ordinates x = 1, x = -1 is given by

A
0

B
1/3

C
2/3

D
None of these

Solution

Required area $=\left|\int_{-1}^{1} x\right| x|d x|$
$=\left|\int_{-1}^{0} x\right| x|d x|+\int_{0}^{1} x|x| d x=\left|\int_{-1}^{0}-x^{2} d x\right|+\int_{0}^{0} \mid x^{2} d x$
$=\mid\left[-\frac{x^{3}}{3}\right]_{-1}^{0}-\left[\frac{x^{3}}{3}\right]_{0}^{1}=\frac{1}{3}+\frac{1}{3}=\frac{2}{3}$
Question 2
4 / -1

Let f : R be a differentiable function and f(1) = 4. Find the value of

$\lim _{x \rightarrow 1} \int_{4}^{f(x)} \frac{2 t}{x-1} d t,$ if $f^{\prime}(1)=2$

A
16

B
8

C
4

D
2

Solution

$\begin{aligned} \lim _{x \rightarrow 1} \frac{\int_{4}^{f(x)} 2 t}{x-1} d t & \\=& \lim _{x \rightarrow 1} \frac{2 f(x) \cdot f^{\prime}(x)}{1} \\=& 2 f(1) \cdot f^{\prime}(1)=2 \cdot 4 \cdot 2=16 \end{aligned}$
Question 3
4 / -1

Out of 40 consecutive integers, two are chosen at random. The probability that their sum is odd is

A
1/200

B
20/39

C
1/2

D
None of these

Solution

The total number of ways in which 2 integers can be chosen from the given 40 integers is $40 \mathrm{C}_{2} \cdot 780 .$ The sum of the selected numbers is odd if exactly one of them is odd and other is even. Therefore, favourable number of cases $={ }^{20} \mathrm{C}_{1} \times{ }^{20} \mathrm{C}_{1}=$ 400
Hence, the required probability $=\frac{400}{780}=\frac{20}{39}$
Question 4
4 / -1

What does y = x² represent in a 3-dimensional coordinate system?

A
A parabola

B
A cylinder

C
A paraboloid

D
None of these

Solution

Given the equation is of a parabola with a focus on origin. The graph of the parabola is as follows:

Since the graph clearly represents the equation of a cylinder. It is having the equation about the y-axis in the z-x plane having the shape of a cylinder.
Question 5
4 / -1

If

then $\frac{d y}{d x}$

A
0

B
1

C
x

D
y

Solution

$\begin{aligned}{y=\left(1+x+\frac{x^{2}}{2 !}+\frac{x^{3}}{3 !}+\ldots . . . \infty\right.}) \\ y=& e^{x} \\ \frac{d y}{d x}=e^{x}=y \end{aligned}$
Question 6
4 / -1

The probability that a man will live 10 more years is 1/4 and the probability that his wife will live 10 more years is 1/3. The probability that neither of them will be alive in 10 years is

A
11/12

B
1/2

C
7/12

D
None of these

Solution

Probability that a man will live 10 more years = 1/4

Probability that man not live 10 more years = 1 - 1/4 = 3/4

Probability that his wife will live 10 more years = 1/3

Probability that his wife not live 10 more years = 1 - 1/3 = 2/3

Then, probability that neither will be alive in 10 years = (3/4) x (2/3) = 1/2
Question 7
4 / -1

$\int_{0}^{\pi} \sqrt{1+\sin x} d x$ is equal to

A
2

B
2√2

C
4

D
None of these

Solution

$\begin{aligned} &=\int_{\pi}^{\pi} \sqrt{\cos ^{2}(x / 2)+\sin ^{2}(x / 2)+2 \sin (x / 2) \cos (x / 2)} d x \\ &=\int_{0}^{\pi}[\cos (x / 2)+\sin (x / 2)] d x \\ &=\left[\frac{\sin (x / 2)}{(1 / 2)}-\frac{\cos (x / 2)}{(1 / 2)}\right]_{0}^{\pi} \\ &=2[\sin (x / 2)-\cos (x / 2)]^{\pi}=4 \end{aligned}$
Question 8
4 / -1

If f(x) = | x |³, then f′(0) =

A
0

B
1/2

C
-1

D
None of these

Solution

$R f^{\prime}(0)=\frac{m}{1-0} \frac{f(0+h)-f(0)}{h}$
$=\lim _{h \rightarrow 0} \frac{|h|^{3}-0}{h}=\lim _{h \rightarrow 0} \frac{h^{3}}{h}=\lim _{h \rightarrow 0} h^{2}=0$
$L f^{\prime}(0)=\lim _{h \rightarrow 0} \frac{f(0-h)-f(0)}{-h}=\lim _{h \rightarrow 0} \frac{|-h|^{3}}{-h}$
$={ }_{h \rightarrow 0}^{\lim } \frac{h^{3}}{-h}=-h \rightarrow 0^{\lim } h^{2}=0$
$R f^{\prime}(0)=L f^{\prime}(0)=0,$ Hence, $f^{\prime}(0)=0$
Question 9
4 / -1

The value of sin 10° + sin 20° + sin 30° + ........+ sin 360° is

A
1

B
0

C
-1

D
None of these

Solution

Since, sin 190° = - sin 10°, sin 200° = - sin 20°,

sin 210° = - sin 30°, sin 360° = sin 180° = 0 etc.

Hence all the terms in the expression cancels out, therefore answer is 0.
Question 10
4 / -1

The area bounded by curve xy = c and x-axis between x = 1 and x = 4, is

A
c log 3 sq. units

B
2 log c sq. units

C
2c log 2 sq. units

D
2c log 5 sq. units

Solution

Required area $=\int^{4} y d x=\int_{1}^{4} \frac{c}{x} d x=2 c \log 2 s q$

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Re-Attempt Weekly Quiz Competition

Mathematics Test - 7

Result

Mathematics Test - 7

Score

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SHARING IS CARING

Question 1

0

1/3

2/3

None of these

Solution

Question 2

16

8

4

2

Solution

Question 3

1/200

20/39

1/2

None of these

Solution

Question 4

A parabola

A cylinder

A paraboloid

None of these

Solution

Question 5

0

1

x

y

Solution

Question 6

11/12

1/2

7/12

None of these

Solution

Question 7

2

2√2

4

None of these

Solution

Question 8

0

1/2

-1

None of these

Solution

Question 9

1

0

-1

None of these

Solution

Question 10

c log 3 sq. units

2 log c sq. units

2c log 2 sq. units

2c log 5 sq. units

Solution

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