Mathematics Test

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Mathematics Test - 30

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Weekly Quiz Competition

Question 1
1 / -0

Consider the following statements

I. \(\lim _{x \rightarrow 0} \sin \frac{1}{x}\) does not exist.
II. \(\lim _{x \rightarrow 0} x \sin \frac{1}{x}\) exists.
Which of the above statement(s) is/are correct?

A
Only I

B
Only II

C
Both I & II

D
Neither I nor II

Solution

I. \(\lim _{x \rightarrow 0} \sin \frac{1}{x}\)
\(\mathrm{LHL}=\mathrm{f}(0-0)=\lim _{h \rightarrow 0} f(0-h)\)
\(=\lim _{h \rightarrow 0} \sin \frac{1}{(-h)}=\lim _{h \rightarrow 0}-\sin \frac{1}{h}=-\sin (\infty)\)
\(\mathrm{RHL}=\mathrm{f}(0+0)=\lim _{h \rightarrow 0} f(0+h)\)
\(=\lim _{h \rightarrow 0} \sin \frac{1}{h}=\sin (\infty)\)
\(\because \mathrm{LHL} \neq \mathrm{RHL}\)
Hence \(\lim _{x \rightarrow 0} \sin \left(\frac{1}{x}\right)\) does not exist.
II. \(\lim _{x \rightarrow 0} x \sin \left(\frac{1}{x}\right)\)
\(\mathrm{LHL}=\mathrm{f}(0-0)=\lim _{h \rightarrow 0} f(0-h)\)

\(=\lim _{h \rightarrow 0}(-h) \sin \left(\frac{-1}{h}\right)\)
\(=\lim _{h \rightarrow 0} h \sin \frac{1}{h}=0\)
\(\operatorname{RHL}=f(0+0)=\lim _{h \rightarrow 0} f(0+h)\)
\(=\lim _{h \rightarrow 0}(h) \sin \left(\frac{1}{h}\right)\)
\(=0 \times \sin \infty=0\)
\(\because \mathrm{LHL}=\mathrm{RHL}\)
Hence, \(\lim _{h \rightarrow 0} x \sin \left(\frac{1}{x}\right)\) exists.
Question 2
1 / -0

Find the area of the region bounded by \(y^{2}=x, x=1, x=4\) and the \(x-\) axis.

A
14/3 sq units

B
7/3 sq units

C
28/3 sq units

D
21/3 sq units

Solution
Question 3
1 / -0

Prithvi spent Rs. 89,745 on his college fees, Rs. 51,291 on personality development classes and the remaining 27% of the total amount he had as cash with him .what was the total amount?

A
1,85,400

B
1,89,600

C
1,91,800

D
1,93,200

Solution

\(73 \%=89745+51291=141036\)
Total amount \(100 \%=\frac{141036}{73} \times 100=\mathrm{Rs} .193200\)
Question 4
1 / -0

Let \(n(U)=700, n(A)=200, n(B)=300, n(A \cap B)=100,\) then \(n\left(A^{\prime} \cap B^{\prime}\right)\) is equal to

A
400

B
600

C
300

D
None of these

Solution

\(\mathrm{n}\left(\mathrm{A}^{\prime} \cap \mathrm{B}^{\prime}\right)=\mathrm{n}(\mathrm{A} \cup \mathrm{B})^{\prime}\)
\(=\mathrm{n}(\cup)-n(A \cup B)\)
\(=n(\cup)-[n(A)+n(B)-n(A \cap B)]\)
\(=700-\{200+300-100\}=300\)
Question 5
1 / -0

Find the equation of the sphere having the centre (–2, 2, 3) and passing through the point (3, 4, –1).

A

\(x^{2}+y^{2}+z^{2}+4 x+4 y+6 z+28=0\)

B

\(x^{2}+y^{2}+z^{2}-4 x-4 y-6 z+28=0\)

C

\(x^{2}+y^{2}+z^{2}+4 x+4 y+6 z-28=0\)

D

None of the above

Solution

The equation of the sphere with centre (-2,2,3) is
\[
(x+2)^{2}+(y-2)^{2}+(z-3)^{2}=r^{2}
\]
Radius, \(r=\sqrt{(3+2)^{2}+(4-2)^{2}+(-1-3)^{2}}=\sqrt{45}\)
\(\therefore\) Required equation of the sphere is
\[
\begin{array}{l}
(x+2)^{2}+(y-2)^{2}+(z-3)^{2}=(\sqrt{45})^{2} \\
\Rightarrow x^{2}+y^{2}+z^{2}+4 x-4 y-6 z-28=0
\end{array}
\]
Question 6
1 / -0

What is \(\int e^{x}\left(\sqrt{x}+\frac{1}{2 \sqrt{x}}\right) d x\) is equal to?

A

\(x e^{x}+c\)

B

\(e^{x}(\sqrt{x})+c\)

C

\(2 e^{x}(\sqrt{x})+c\)

D

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

Solution

\(\int e^{x}\left(\sqrt{x}+\frac{1}{2 \sqrt{x}}\right) d x\)
\(\int e^{x} \sqrt{x} d x+\frac{1}{2} \int \frac{e^{x}}{\sqrt{x}} d x\)
\(e^{x} \sqrt{x}-\int \frac{1}{2 \sqrt{x}} e^{x} d x+\frac{1}{2} \int \frac{e^{x}}{\sqrt{x}} d x+c\)
\(=e^{x} \sqrt{x}+c\)
Question 7
1 / -0

\(\mathrm{A} \mathrm{no} \cdot \frac{129}{2^{2} \mathrm{s}^{7} 7^{5}}\) is

A
Terminating decimal expansion

B
Non terminating repeating decimal expansion

C
Neither terminating nor non terminating repeating decimal expansion

D
Can’t determine

Solution

As the prime factorization of the denominator is not of the form \(2^{\mathrm{n}} 5^{\mathrm{m}},\) where \(\mathrm{n}, \mathrm{m}\) are non negative integers. Hence \(\frac{129}{2^{2} 5^{7} 7^{5}}\) is a non terminating repeating decimal expansion.
Question 8
1 / -0

Two pillars of equal height and on either side of a road, which is 100m wide. The angles of elevation of the top of the pillars are 60° and 30° at a point on the road between the pillars. Find the height of each pillar.

A
25m

B
35√2 m

C
25√3 m

D
53√3 m

Solution

Let \(A B \& C D\) be two pillars, each of height h meters Let \(P\) be a point on the road such that \(A P=x\) meters Then \(\mathrm{CP}=(100-\mathrm{x})\) meters. It is given that \(\angle \mathrm{APB}=60^{\circ}\) and \(\angle \mathrm{CPD}=30^{\circ}\)
In \(\Delta P A B,\) we have \(\tan 60^{\circ}=\frac{A B}{A P}\)
\(\Rightarrow \sqrt{3}=\frac{h}{x} \Rightarrow h=\sqrt{3 x}\)
In \(\Delta P C D,\) we have

\(\tan 30^{\circ}=\frac{c D}{\rho c}\)
\(\Rightarrow \frac{1}{\sqrt{3}}=\frac{h}{100-x}\)
\(\Rightarrow \mathrm{h} \sqrt{3}=100-x\)
Eliminating h between equation (1)\(\&(2)\) we get \(3 x=100-x \Rightarrow 4 x=100\) \(\Rightarrow x=25\)
Substituting \(x=25\) in equation (i), we get \(h=25 \sqrt{3}\)
Question 9
1 / -0

\(\frac{\cot ^{2} 15^{\circ}-1}{\cot ^{2} 15^{\circ}+1}\) is equal to

A

\(\frac{1}{2}\)

B

\(\frac{\sqrt{3}}{2}\)

C

\(\frac{3 \sqrt{3}}{4}\)

D

\(\sqrt{3}\)

Solution

\(\frac{\cot ^{2} 15^{\circ}-1}{\cot ^{2} 15^{\circ}+1}=\frac{\frac{\cos ^{2} 15^{\circ}}{\sin ^{2} 15^{\circ}}-1}{\frac{\cos ^{2} 15^{\circ}}{\sin ^{2} 15^{\circ}}+1}=\frac{\cos ^{2} 15^{\circ}-\sin ^{2} 15^{\circ}}{\cos ^{2} 15^{\circ}+\sin ^{2} 15^{\circ}}\)

\(=\cos \left(2 \times 15^{\circ}\right)=\cos 30^{\circ}\)
\(=\frac{\sqrt{3}}{2}\)
Question 10
1 / -0

The two ends of latus rectum of a parabola are the points (3, 6) and (–5, 6), then the focus is

A
(1, 6)

B
(–1, 6)

C
(1, –6)

D
(–1, –6)

Solution

Focus is the mid-point of latus rectum.
Let the two end points of latus rectum be
\(\mathrm{L}_{1}(3,6) \& L_{2}(-5,6)\)
\(\therefore\) Focus, \(x=\frac{3+(-5)}{2}=-1\)
\(y=\frac{6+6}{2}=6\)
Focus \(=(x, y)=(-1,6)\)

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Re-Attempt Weekly Quiz Competition

Mathematics Test - 30

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Mathematics Test - 30

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SHARING IS CARING

Question 1

Only I

Only II

Both I & II

Neither I nor II

Solution

Question 2

14/3 sq units

7/3 sq units

28/3 sq units

21/3 sq units

Solution

Question 3

1,85,400

1,89,600

1,91,800

1,93,200

Solution

Question 4

400

600

300

None of these

Solution

Question 5

\(x^{2}+y^{2}+z^{2}+4 x+4 y+6 z+28=0\)

\(x^{2}+y^{2}+z^{2}-4 x-4 y-6 z+28=0\)

\(x^{2}+y^{2}+z^{2}+4 x+4 y+6 z-28=0\)

None of the above

Solution

Question 6

\(x e^{x}+c\)

\(e^{x}(\sqrt{x})+c\)

\(2 e^{x}(\sqrt{x})+c\)

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

Solution

Question 7

Terminating decimal expansion

Non terminating repeating decimal expansion

Neither terminating nor non terminating repeating decimal expansion

Can’t determine

Solution

Question 8

25m

35√2 m

25√3 m

53√3 m

Solution

Question 9

\(\frac{1}{2}\)

\(\frac{\sqrt{3}}{2}\)

\(\frac{3 \sqrt{3}}{4}\)

\(\sqrt{3}\)

Solution

Question 10

(1, 6)

(–1, 6)

(1, –6)

(–1, –6)

Solution

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