Mathematics Test

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Mathematics Test - 6

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Weekly Quiz Competition

Question 1
1 / -0

What is the equation of straight line passing through the point (4, 3) and making equal intercepts on the coordinate axes?

A
x + y = 7

B
3x + 4y = 7

C
x – y = 1

D
None of these

Solution

Let equation of line be $\frac{x}{a} + \frac{y}{a} = 1$ or $x + y = a$
Line passing through $(4, 3),$ then, $a = 0$
Required equation is $x + y = 7$
Question 2
1 / -0

Consider the curve y = e^2x.
What is the slope of the tangent to the curve at (0 ,1)?

A
0

B
1

C
2

D
4

Solution

$y = e^{2 x}$
$\frac{d y}{d x} = 2 e^{2 x}$
$\begin{aligned} \Rightarrow & \frac{d y}{d x} | (x, y) = (0, 1) = 2 e^{0} = 2 \end{aligned}$
Question 3
1 / -0

What is $\frac{co t 224^{0} - co t 134^{0}}{co t 226^{0} + co t 316^{0}}$ equal to?

A
-cosec 88⁰

B
-cosec 2⁰

C
-cosec 44⁰

D
–cosec 46°

Solution

$\frac{co t 224^{0} - co t 134^{0}}{co t 226^{0} + co t 316^{0}}$
$\frac{co t (180^{0} + 44^{0}) - co t (180^{0} - 46^{0})}{co t (180^{0} + 46) + co t (270^{0} + 46^{0})}$
$\frac{co t 44^{0} - co t 46^{0}}{co t 46^{0} - \tan 46^{0}} = \frac{\tan 46^{0} + \tan 44^{0}}{\tan 44^{0} - \tan 46^{0}}$
$= \frac{\sin (46^{0} + 44^{0})}{\sin (44^{0} - 46^{0})} = - \cos e c 2^{0}$
Question 4
1 / -0

Let z = x + iy where x, y are real variables i = √-1. If |2z-1| = |z - 2|, then the point z describes:

A
A circle

B
An ellipse

C
A hyperbola

D
A parabola

Solution

|2z-1|=|z-2|
|2(x+iy)-1|=|x+iy-2|
|(2x-1)+2yi|= |(x-2)+iy|
$\sqrt{{(2 x - 1)}^{2} + 4 y^{2}} = \sqrt{{(x - 2)}^{2} + y^{2}}$
on squaring both sides,
4x²+1-4x+4y²=x²+4-4x+y²
3x²+3y²=3
x²+y²=1
It is the equation of circle.
So, the point z describes a circle
Question 5
1 / -0

Consider the following statement in respect of the matrix A =

1. The matrix A is skew-symmetric

2. The matrix A is symmetric.

3. The matrix A is invertible.

Which of the above statements is/are correct?

A
1 only

B
3 only

C
1 and 3

D
2 and 3

Solution
Question 6
1 / -0

Consider the following statements:

1. If 3θ is an acute angle such that sin 3θ = cos 2θ, then the measurement of θ in radian equals to π/10

2. One radian is the angle subtended at the centre of a circle by an arc of the same circle whose length is equal to the diameter of that circle.

Which of the above statements is/are correct?

A
1 only

B
2 only

C
Both 1 and 2

D
Neither 1 nor 2

Solution

$\sin 3 θ = \cos 2 θ \sin 3 θ = \sin (\frac{π}{2} - 2 θ) 3 θ = \frac{π}{2} - 2 θ 5 θ = \frac{π}{2} \Rightarrow θ = \frac{π}{10}$

One radian is the angle subtended at the center of a circle by an arc of a circle by an arc of the same circle whose length is equal to radius of that circle.

hence statement 1 is correct.
Question 7
1 / -0

The roots of the equation 2 a²x² - 2abx + b² = 0 when a<0 and b>0 are :

A
Sometimes complex

B
Always irrational

C
Always complex

D
Always real

Solution

We have, 2a²x² — 2abx + b² = 0
Discriminant, D = (-2ab)2 - 4(2a²)(b²)
= 4a²b² — 8a²b² = —4a²b² < 0
Roots are always complex.
Question 8
1 / -0

It has been found that if A and B play a game 12 times, A wins 6 times, B wins 4 times and they draw twice. A and B take part in a series of 3 games. The probability that they win alternately, is:

A
5/12

B
5/36

C
19/27

D
5/27

Solution

$P (A) = \frac{6}{12} = \frac{1}{2}, P (B) = \frac{4}{12} = \frac{1}{3}$
Req.probability= $\frac{1}{2} . \frac{1}{3} . \frac{1}{2} + \frac{1}{3} . \frac{1}{2} . \frac{1}{3} = \frac{1}{12} + \frac{1}{18} = \frac{5}{36}$
Question 9
1 / -0

If A is any matrix, then the product AA is defined only when A is a matrix of order m × n where:

A
m>n

B
m

C
m=n

D
m≤n

Solution

A A is defined only when A is a matrix of order m*n where m=n
$A \times A = (m \times n) (m \times n) = (m \times n) (n \times n) i f m = n$
$= m \times n = n \times n o r m \times m$
= A is a square matrix
Question 10
1 / -0

Let a vector $\vec{r}$ make angle $60^{0}, 30^{0}$ with x and y-axes respectively.
What angle does $\vec{r}$ make with z=axes

A
30^o

B
60^o

C
90^o

D
120^o

Solution

$\cos^{2} a + \cos^{2} β + \cos^{2} γ = 1$
$\cos^{2} 60^{0} + \cos^{2} 30^{0} + \cos^{2} γ = 1$
$\frac{1}{4} + \frac{3}{4} + \cos^{2} γ = 1$
$\cos^{2} γ = 0 \Rightarrow γ = 90^{0}$

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Mathematics Test - 6

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Mathematics Test - 6

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Question 1

x + y = 7

3x + 4y = 7

x – y = 1

None of these

Solution

Question 2

0

1

2

4

Solution

Question 3

-cosec 880

-cosec 20

-cosec 440

–cosec 46°

Solution

Question 4

A circle

An ellipse

A hyperbola

A parabola

Solution

Question 5

1 only

3 only

1 and 3

2 and 3

Solution

Question 6

1 only

2 only

Both 1 and 2

Neither 1 nor 2

Solution

Question 7

Sometimes complex

Always irrational

Always complex

Always real

Solution

Question 8

5/12

5/36

19/27

5/27

Solution

Question 9

m>n

m

m=n

m≤n

Solution

Question 10

30o

60o

90o

120o

Solution

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-cosec 88⁰

-cosec 2⁰

-cosec 44⁰

30^o

60^o

90^o

120^o