Mathematics Test

Question 1
1 / -0

What is the least value of 25 cosec² x + 36 sec² x?

A
1

B
11

C
120

D
121

Question 2
1 / -0

Using the principal of mathematical induction, prove that \(1+\frac{1}{1+2}+\frac{1}{1+2+3}+\ldots+\frac{1}{1+2+3+\ldots+n}=\frac{2 n}{n+1}\) for all:

A
\(n \in R\)

B
\(n \in N\)

C
\(n \in I\)

D
\(n \in\) Z

Question 3
1 / -0

Consider the matrix \(f(x)=\left[\begin{array}{ccc}\cos x & -\sin x & 0 \\ \sin x & \cos x & 0 \\ 0 & 0 & 1\end{array}\right]\).

Given below are two statements :

Statement I: \(\mathrm{f}(-\mathrm{x})\) is the inverse of the matrix \(\mathrm{f}(\mathrm{x})\).

Statement II: \(\mathrm{f}(\mathrm{x}) \mathrm{f}(\mathrm{y})=\mathrm{f}(\mathrm{x}+\mathrm{y})\).

In the light of the above statements, choose the correct answer from the options given below

A
Statement I is false but Statement II is true

B
Both Statement I and Statement II are false

C
Statement I is true but Statement II is false

D
Both Statement I and Statement II are true

Question 4
1 / -0

The angle between the straight lines, whose direction cosines are given by the equations \(2 I+2 m-n=0\) and \(m n+n I+\operatorname{Im}=0\), is

A
\(\frac{\pi}{2}\)

B
\(\pi-\cos ^{-1}\left(\frac{4}{9}\right)\)

C
\(\cos ^{-1}\left(\frac{8}{9}\right)\)

D
\(\frac{\pi}{3}\)

Question 5
1 / -0

If \(p\) and \(q\) are the lengths of the perpendiculars from the origin on the lines, \(\mathrm{x} \operatorname{cosec} \alpha-\mathrm{y} \sec \alpha=\mathrm{k} \cot 2 \alpha\) and \(\mathrm{x} \sin \alpha+\mathrm{y} \cos \alpha=\mathrm{k} \sin 2 \alpha\) respectively, then \(\mathrm{k}^2\) is equal to

A
\(4 p^2+q^2\)

B
\(2 p^2+q^2\)

C
\(p^2+2 q^2\)

D
\(2 p^2+q^2 p^2+4 q^2\)

Question 6
1 / -0

If the direction cosines of a line are \(\left(\frac{1}{k}, \frac{2}{k},-\frac{2}{k}\right)\), then \(k\) is:

A
\(\pm\left(\frac{1}{\sqrt{3}}\right)\)

B
\(\frac{1}{3}\)

C
\(\pm \sqrt{3}\)

D
\(\pm 3\)

Question 7
1 / -0

\(\tan 54^{\circ}\) can be expressed as:

A
\(\frac{\sin 9^{\circ}+\cos 9^{\circ}}{\sin 9^{\circ}-\cos 9^{\circ}}\)

B
\(\frac{\sin 9^{\circ}-\cos 9^{\circ}}{\sin 9^{\circ}+\cos 9^{\circ}}\)

C
\(\frac{\cos 9^{\circ}+\sin 9^{\circ}}{\cos 9^{\circ}-\sin 9^{\circ}}\)

D
\(\frac{\sin 36^{\circ}}{\cos 36^{\circ}}\)

Question 8
1 / -0

If sin A, sin B, and cos A are in GP, then the roots of x² + 2x cot B + 1 = 0 are always:

A
Real

B
Imaginary

C
Greater than 1

D
Equal

Question 9
1 / -0

If \(O A B\) is an equilateral triangle inscribed in the parabola \(y^{2}=4 a x\) with \(O\) as the vertex, then the length of the side of the \(\Delta O A B\) is:

A
\(8 a \sqrt{3}\)

B
\(4 a \sqrt{3}\)

C
\(2 a \sqrt{3}\)

D
\(a \sqrt{3}\)

Question 10
1 / -0

\(x+i y=\sqrt{\frac{a+i b}{c+i d}}\), then the value of \(x^{2}+y^{2}\) is:

A
\(\sqrt{\frac{a^{2}+b^{2}}{c^{2}+d^{2}}}\)

B
\(\frac{a^{2}+b^{2}}{c^{2}+d^{2}}\)

C
\(a d-b c\)

D
\(\sqrt{\frac{a-i b}{a+i b}}\)

Submit Test

Mathematics Tes...

Question 1

1

11

120

121

Question 2

\(n \in R\)

\(n \in N\)

\(n \in I\)

\(n \in\) Z

Question 3

Statement I is false but Statement II is true

Both Statement I and Statement II are false

Statement I is true but Statement II is false

Both Statement I and Statement II are true

Question 4

\(\frac{\pi}{2}\)

\(\pi-\cos ^{-1}\left(\frac{4}{9}\right)\)

\(\cos ^{-1}\left(\frac{8}{9}\right)\)

\(\frac{\pi}{3}\)

Question 5

\(4 p^2+q^2\)

\(2 p^2+q^2\)

\(p^2+2 q^2\)

\(2 p^2+q^2 p^2+4 q^2\)

Question 6

\(\pm\left(\frac{1}{\sqrt{3}}\right)\)

\(\frac{1}{3}\)

\(\pm \sqrt{3}\)

\(\pm 3\)

Question 7

\(\frac{\sin 9^{\circ}+\cos 9^{\circ}}{\sin 9^{\circ}-\cos 9^{\circ}}\)

\(\frac{\sin 9^{\circ}-\cos 9^{\circ}}{\sin 9^{\circ}+\cos 9^{\circ}}\)

\(\frac{\cos 9^{\circ}+\sin 9^{\circ}}{\cos 9^{\circ}-\sin 9^{\circ}}\)

\(\frac{\sin 36^{\circ}}{\cos 36^{\circ}}\)

Question 8

Real

Imaginary

Greater than 1

Equal

Question 9

\(8 a \sqrt{3}\)

\(4 a \sqrt{3}\)

\(2 a \sqrt{3}\)

\(a \sqrt{3}\)

Question 10

\(\sqrt{\frac{a^{2}+b^{2}}{c^{2}+d^{2}}}\)

\(\frac{a^{2}+b^{2}}{c^{2}+d^{2}}\)

\(a d-b c\)

\(\sqrt{\frac{a-i b}{a+i b}}\)

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