Mathematics Test

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Mathematics Test - 4

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Weekly Quiz Competition

Question 1
1 / -0

What is the value of $\sin ^{-1} \frac{4}{5}+2 \tan ^{-1} \frac{1}{3} ?$

A

$\frac{\pi}{3}$

B

$\frac{\pi}{2}$

C

$\frac{\pi}{4}$

D

$\frac{\pi}{6}$

Solution

Given:

$\sin ^{-1} \frac{4}{5}+2 \tan ^{-1} \frac{1}{3} $

Let,

$\sin ^{-1} \frac{4}{5}=x$

$\Rightarrow \sin x=\frac45$

Now,

$\cos ^{2} {x}=1-\sin ^{2} {x}$

$=1-\frac{16}{25}$

$=\frac{9}{25}$

So

$=cos x=\frac35$

Now,

$\cot x=\frac{\cos x}{\sin x}=\frac34\quad\cdots(1)$

$2 \tan ^{-1} \frac{1}{3}=\tan ^{-1}\left(\frac{\frac{2}{3}}{1-\frac{1}{9}}\right) \quad \quad\left(\because 2 \tan ^{-1} {x}=\tan ^{-1}\left(\frac{2 {x}}{1-{x}^{2}}\right)\right)$

$=\tan ^{-1}\left(\frac{\frac{2}{3}}{\frac{8}{9}}\right)$

$=\tan ^{-1}\left(\frac{3}{4}\right)$

$=\tan ^{-1}(\cot x) \quad \quad(\text { from equation 1})$

$=\tan ^{-1}\left(\tan \left(\frac{\pi}{2}-{x}\right)\right)$

$=\frac{\pi}2-x \quad \quad\left(\because \tan ^{-1}(\tan x)=x\right)$

$=\frac{\pi}{2}-\sin ^{-1} \frac{4}{5}$

$\therefore \sin ^{-1} \frac{4}{5}+2 \tan ^{-1} \frac{1}{3}=\sin ^{-1} \frac{4}{5}+\frac{\pi}{2}-\sin ^{-1} \frac{4}{5}$

$=\frac{\pi}{2}$
Question 2
1 / -0

For function f(x) = x cos 1/x , x ≥ 1,

A

for all x in the interval [1, ∞), f(x + 2) – f(x) > 2 and f′(x) is strictly decreasing in the interval [1, ∞) Both.

B

for atleast one x in interval [1, ∞), f(x + 2) – f(x) < 2

C

for all x in the interval [1, ∞), f(x + 2) – f(x) > 2

D

f′(x) is strictly decreasing in the interval [1, ∞)
Question 3
1 / -0

$F i n d t h e v a l u e o f \lim_{x \to 0} \frac{\tan x - \cos x}{x} = ?$

A

0

B

1

C

2

D

-1

Solution

$\lim_{x \to 0} \frac{\tan x - \cos x}{x} = ?$

$\Rightarrow \lim_{x \to 0} \frac{\frac{\sin x}{\cos x} - \cos x}{x}$

$\Rightarrow \lim_{x \to 0} \frac{\frac{\sin x - \cos^{2} x}{\cos x}}{x}$

$\lim_{x \to 0} \frac{\sin x}{x} . \frac{1}{\cos x} - \lim_{x \to 0} \frac{\cos^{2} x}{x \cos x}$

$\lim_{x \to 0} \frac{\sin x}{x} . \frac{1}{\cos x} - \lim_{x \to 0} \frac{\cos x}{x} = 1.1 - 0 = 1$
Question 4
1 / -0

The locus of the orthocentre of the triangle formed by the lines (1 + p)x – py + p(1 + p) = 0, (1 + q)x – qy + q(1 + q) = 0 and y = 0, where p ≠ q, is

A

a straight line

B

a parabola

C

an ellipse

D

a hyperbola
Question 5
1 / -0

Root(s) of the expression 2sin^2 θ + sin^2 2θ = 2

A

π/4

B

π/6

C

π

D

π/3
Question 6
1 / -0

Find the the statements/expressions of parabola :

A

The eccentricity of the conic lies in the interval 1 ≤ x < ∞

B

Points z in the complex plane satisfying |z + 2| – |z – 2| = ± 3

C

The locus of the point (h, k) for which the line hx + ky = 1 touches the circle x^2 + y^2 = 4

D

All of these
Question 7
1 / -0

$L e t a_{1}, a_{2}, a_{3}, . . ., a_{11} b e r e a l n u m b e r s a t i s f y i n g a_{1} = 15, 27 - 2 a_{2} > 0 a n d a_{k} = 2 a_{k - 1} - a_{k - 2} f o r k = 3, 4, . . . . 11 .$ $I f = \frac{a_{1}^{2} + a_{2}^{2} + . . . . + a_{11}^{2}}{11} = 90, t h e n t h e v a l u e o f \frac{a_{1} + a_{2} + . . . . + a_{11}}{11} i s e q u a l t o :$

A

0

B

1

C

2

D

3

Solution

⇒ a_k = 2a_k-1 - a_k-2

⇒ a₁, a₂, a₃ ...., a₁₁ are in A.P. with common difference d.

a₁ = 15,a₂ = 15+d, a₃ = 15+2d ..... a₁₁ = 15+10d

$= \frac{a_{1}^{2} + a_{2}^{2} + . . . . + a_{11}^{2}}{11}$

$\frac{15^{2} \times 11 + 30 d (11 \times 5) + d^{2} 35 \times 11}{11} = 90$

⇒ 35d² + 150d + 225 = 90

⇒ 35d² + 150d + 135 = 0

⇒ 7d² + 30d + 27 = 0

⇒ $d = - 3, - \frac{9}{7}$

∵ $a_{2} < \frac{27}{2}$ ∴d≠ $- \frac{9}{7}$

∴ $\frac{a_{1} + a_{2} + . . . . + a_{11}}{11}$

$= \frac{11 [30 + 10 (- 3)]}{2 \times 11} = \frac{[30 - 30]}{2} = \frac{0}{2} = 0$
Question 8
1 / -0

An ellipse intersects the hyperbola 2x^2 – 2y^2 = 1 orthogonally. The eccentricity of the ellipse is reciprocal of that of the hyperbola. If the axes of the ellipse are along the coordinates axes, then

A

equation of ellipse is x^2 + 2y^2 = 2

B

equation of ellipse is x^2 + 2y^2 = 4

C

None of these

D

the foci of ellipse are (±√2 , 0)
Question 9
1 / -0

Let f: R $\to$ R be a continuous function

which satisfies f(x) = $\int_{0}^{x} f (t) d t .$ Then

the value of f(In5) is.

A

0

B

1

C

2

D

3
Question 10
1 / -0

The smallest value of k, for which both the roots of the equation x^2 − 8kx + 16(k^2 − k + 1 ) = 0 are real, distinct and have values at least 4, is

A

2

B

4

C

3

D

1

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Mathematics Test - 4

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Mathematics Test - 4

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Question 1

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

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

\(\frac{\pi}{4}\)

\(\frac{\pi}{6}\)

Solution

Question 2

for all x in the interval [1, ∞), f(x + 2) – f(x) > 2 and f′(x) is strictly decreasing in the interval [1, ∞) Both.

for atleast one x in interval [1, ∞), f(x + 2) – f(x) < 2

for all x in the interval [1, ∞), f(x + 2) – f(x) > 2

f′(x) is strictly decreasing in the interval [1, ∞)

Question 3

0

1

2

-1

Solution

Question 4

a straight line

a parabola

an ellipse

a hyperbola

Question 5

π/4

π/6

π

π/3

Question 6

The eccentricity of the conic lies in the interval 1 ≤ x < ∞

Points z in the complex plane satisfying |z + 2| – |z – 2| = ± 3

The locus of the point (h, k) for which the line hx + ky = 1 touches the circle x^2 + y^2 = 4

All of these

Question 7

0

1

2

3

Solution

Question 8

equation of ellipse is x^2 + 2y^2 = 2

equation of ellipse is x^2 + 2y^2 = 4

None of these

the foci of ellipse are (±√2 , 0)

Question 9

0

1

2

3

Question 10

2

4

3

1

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