Mathematics Test

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Mathematics Test - 2

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Weekly Quiz Competition

Question 1
1 / -0

Interval in which tan^–1 (sinx + cosx) is increasing

A

(0, π/8)

B

(0, π/2)

C

(-π/2, π/2)

D

All of these
Question 2
1 / -0

$F o r 0 < θ < \frac{π}{2}, t h e s o l u t i o n (s) o f$

$\sum_{m = 1}^{6} \cos e c (θ + \frac{(m - 1) π}{4}) \cos e c (θ + \frac{m π}{4})$

= 4 $\sqrt{2}$ is(are)

(A) $\frac{π}{4}$

(B) $\frac{π}{6}$

(C) $\frac{π}{12}$

(D) $\frac{5 π}{12}$

A

Option (C, D)

B

Option (A, B)

C

Option (A, C)

D

Option (B, C)
Question 3
1 / -0

From points on a given circle tangents are drawn to another circle. Then the locus of the mid-points of the chord of contact is will be :

A

Parabola

B

Hyperbola

C
Circle

D

Eclipse

Solution

Let (p,q) be a point on a given circle.

$x^{2} + y^{2} + 2 g x + 2 f y + c = 0$

so that

$p^{2} + q^{2} + 2 g p + 2 f q + c = 0$ ....(1)

Let tangents be drawn to the circle $x^{2} + y^{2} = a^{2}$ from P and mid-point of chord of contact AB be (h,k).

AB is px+qy=a2 as chord of contact.

AB is $h x + k y = h^{2} + k^{2}$ , by $T = S_{1}$

Comparing,

$h p = k q = h^{2} + k^{2} a^{2}$

∴ $p = h^{2} + k^{2} a^{2} h, q = h^{2} + k^{2} a^{2} k$

Put these values of p and q in (1) and generalize (p,q) and you get a circle.
Question 4
1 / -0

A

option A,B,C

B

Option A

C

Option D

D

Option B
Question 5
1 / -0

The number of seven digit integers, with sum of the digits equal to 10 and formed by using the digits 1, 2 and 3 only, is

A

77

B

55

C

66

D

88
Question 6
1 / -0

Two dice are tossed 6 times, then the probability that 7 will show an exactly four of tosses is

A

225/188442

B

116/20003

C

125/15552

D
None of the above

Solution

Two dice will show 7 if they show : $\{(1, 6), (2, 5), (6, 1), (5, 2), (4, 3), (3, 4)\}$

the probability of success (p)= $6 / (6 \times 6) = 1 / 6$

the probability of failure (q)= $1 - 1 / 6 = 5 / 6$

Probability of showing 7 exactly 4 times = ${}^{6}C_{4} \times p^{4} \times q^{2} = 15 \times {(1 / 6)}^{4} \times {(5 / 6)}^{2} = \frac{125}{15552}$
Question 7
1 / -0

The tangent $\mathrm{P} \mathrm{T}$ and the normal $\mathrm{P} \mathrm{N}$ to the parabola $\mathrm{y}^{2}=4 \mathrm{ax}$ at a point $\mathrm{P}$ on it meet its axis at points $\mathrm{T}$ and $\mathrm{N}$, respectively. The locus of the centroid of the $\triangle \mathbf{P T} \mathrm{N}$ is a parabola whose-

A

vertex $=\left(\frac{2 \mathrm{a}}{3}, 0\right)$ and focus $=(\mathrm{a}, 0)$

B

directrix is $\mathrm{x}=0$

C

latus rectum $=\left(\frac{2 \mathrm{a}}{3}\right)$

D
None of these

Solution

Equation of tangent and normal at point $\mathrm{P}\left(\mathrm{at}^{2}, 2 \mathrm{at}\right)$ is $\mathrm{ty}=\mathrm{x}+\mathrm{at}^{2}$ and $\mathrm{y}=-\mathrm{t} \mathrm{x}+\mathrm{2} \mathrm{at}+$

at $^{2}$

Let centroid of $\triangle \mathbf{P T N}$ is $\mathbf{R}(\mathbf{h}, \mathbf{k})$

$\therefore h=\frac{a t^{2}+\left(-a t^{2}\right)+2 a+a t}{3}$

and $\mathrm{k}=\frac{2 \mathrm{at}}{3} \Rightarrow 3 \mathrm{~h}=2 \mathrm{a}+\mathrm{a} \cdot\left(\frac{3 \mathrm{k}}{2 \mathrm{a}}\right)^{2}$

$\Rightarrow 3 \mathrm{~h}=2 \mathrm{a}+\frac{9 \mathrm{k}^{2}}{4 \mathrm{a}}$

$\Rightarrow 9 \mathrm{k}^{2}=4 \mathrm{a}(3 \mathrm{~h}-2 \mathrm{a})$

Therefore Locus of centroid is $\mathrm{y}^{2}=\frac{4 \mathrm{a}}{3}\left(\mathrm{x}-\frac{2 \mathrm{a}}{3}\right)$

And vertex $\left(\frac{2 \mathrm{a}}{3}, 0\right) ;$ Directrix $\mathrm{x}-\frac{2 \mathrm{a}}{3}=-\frac{\mathrm{a}}{3}$

$\Rightarrow \mathrm{x}=\frac{\mathrm{a}}{3}$

Latus rectum $=\frac{4 a}{3}$

Therefore Focus $\left(\frac{a}{3}+\frac{2 a}{3}, 0\right)$

$\Rightarrow F=(a, 0)$
Question 8
1 / -0

If the function f(x) = x^3 + e^x/2 and g(x) = f^−1 (x), then the value of g′(1) is

A

2

B

3

C

4

D

5
Question 9
1 / -0

In a triangle ABC with fixed base BC, the vertex A moves such that cosB + cosC = 4sin^2 A/2 . If a, b and c denote the lengths of the sides of the triangle opposite to the angles A, B and C, respectively, then

A

b + c = 2a and locus of point A is an ellipse

B

locus of point A is an ellipse

C

b + c = 2a

D

locus of point A is a pair of straight lines
Question 10
1 / -0

The domain of the function $f: R \rightarrow R$ defined by $\sqrt{x^{2}-x-110}$ is:

A

$(-\infty,-10] \cup[11, \infty)$

B

$(-\infty, 11] \cup[-10, \infty)$

C

$(-\infty, 10] \cup[-11, \infty)$

D

$(-\infty, 10] \cup[11, \infty)$

Solution

Given:

The function $f: R \rightarrow R$ defined by $\sqrt{x^{2}-x-110}$

We know that the domain of a function is the complete set of possible values of the independent variable.

To find the domain,

$x^{2}-x-110 \geq 0$

$\Rightarrow x^{2}-11 x+10 x-110 \geq 0$

$\Rightarrow x(x-11)+10(x-11) \geq 0$

$\Rightarrow (x+10)(x-11) \geq 0$

$\Rightarrow x \leq-10$ or $x \geq 11$

$\Rightarrow x \in(-\infty,-10] \cup[11, \infty)$

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Mathematics Test - 2

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Mathematics Test - 2

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Question 1

(0, π/8)

(0, π/2)

(-π/2, π/2)

All of these

Question 2

Option (C, D)

Option (A, B)

Option (A, C)

Option (B, C)

Question 3

Parabola

Hyperbola

Circle

Eclipse

Solution

Question 4

option A,B,C

Option A

Option D

Option B

Question 5

77

55

66

88

Question 6

225/188442

116/20003

125/15552

None of the above

Solution

Question 7

vertex \(=\left(\frac{2 \mathrm{a}}{3}, 0\right)\) and focus \(=(\mathrm{a}, 0)\)

directrix is \(\mathrm{x}=0\)

latus rectum \(=\left(\frac{2 \mathrm{a}}{3}\right)\)

None of these

Solution

Question 8

2

3

4

5

Question 9

b + c = 2a and locus of point A is an ellipse

locus of point A is an ellipse

b + c = 2a

locus of point A is a pair of straight lines

Question 10

\((-\infty,-10] \cup[11, \infty)\)

\((-\infty, 11] \cup[-10, \infty)\)

\((-\infty, 10] \cup[-11, \infty)\)

\((-\infty, 10] \cup[11, \infty)\)

Solution

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