Mathematics Test 204

Result

Mathematics Test 204

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Weekly Quiz Competition

Question 1
4 / -1

The area (in square units) bounded by the curves y² = 4x and x² = 4y in the plane is

A
8/3

B
16/3

C
32/3

D
64/3
Question 2
4 / -1

The length of the tangent from (0,0) to the circle 2x² + 2y² + x - y + 5 = 0 is

A
√5

B
√5 ∕ 2

C
√2

D
Question 3
4 / -1

If the line 3x-4y=λ touches the circle x²+y²-4x-8y-5=0, λ can have the values

A
-35,15

B
-35,-15

C
35,-15

D
35,15
Question 4
4 / -1

The differential equation which represents the family of plane curves y=exp. (cx) is

A
y'=cy

B
xy'-logy=0

C
x log y=yy'

D
y log y=xy'

Solution

y = e^cx

dy/dx = c. e^cx

y' = cy
Question 5
4 / -1

In the following question, a Statement-1 is given followed by a corresponding Statement-2 just below it. Read the statements carefully and mark the correct answer-
Tangents are drawn from the point (17,7) to the circle x²+y²=169.
Statement-1:
The tangents are mutually perpendicular.
Statement-2:
The locus of the points from which mutually perpendicular tangents can be drawn to the given circle is x²+y²=338.

A
Statement -1 is True, Statement -2 is true; Statement-2 is a correct explanation for Statement-1

B
Statement -1 is True, Statement -2 is true; Statement-2 is not a correct explanation for Statement-1

C
Statement -1 is True, Statement -2 is False

D
Statement -1 is False, Statement -2 is True

Solution

Clearly, m₁m₂ = - 1.

Hence, the two tangents arc mutually perpendicular.

Statement 1 is true.

Now, the locus of the point of intersection of two mutually perpendicular tangents to the circle x² + y² = r² is the director circle i.e. the circle x² +y²= 2r².

For the given circle r = 13. ..

Its director circle is x² + y² = 338.

Hence, statement 2 is true and a cogect explanation of statement as the point (17, 7) lies on the director circle of the circle (i).
Question 6
4 / -1

The value of a for which the system of equations

a³x+(a+1)³y+(a+2)³z = 0
ax+(a+1)y+(a+2)z = 0
x+y+z = 0

has a non-zero solution, is

A
1

B
0

C
-1

D
none of these

Solution
Question 7
4 / -1

The chance of getting a doublet with 2 dice is

A
2/3

B
1/6

C
5/6

D
5/36

Solution

Total outcomes = 36

Doublet are 6 (1,1),(2,2),(3,3),(4,4),(5,5),(6,6)

Probability of getting doublet = 6/36

= 1/6
Question 8
4 / -1

If cov. (x, y) = 0, then ρ(x, y) equals

A
0

B
1

C
-1

D
+1/2

Solution

Covariance is a quantitative measure of the extent to which the deviation of one variable from its mean matches the deviation of the other from its mean. It is a mathematical relationship that is defined as:

Cov(X,Y) = E[(X − E[X])(Y − E[Y])]

Correlation between two random variables, ρ(X,Y) is the covariance of the two variables normalized by the variance of each variable. This normalization cancels the units out and normalizes the measure so that it is always in the range [0, 1]:
Question 9
4 / -1

The two opposite vertices of a square on xy-plane are A(-1,1) and B(5,3), the equation of other diagonal (not passing through A and B) is

A
x-3y+4=0

B
2x-y+3=0

C
y+3x-8=0

D
x+2y-1=0

Solution
Question 10
4 / -1

If the normal to the curve y=f(x) at the point (3,4) makes an angle 3π/4 with the positive x-axis, then f'(3)

A
-1

B
-3/4

C
4/3

D
1

Solution

Given y = f(x)

differentiating w.r.t x

y' = f'(x) which is the slope of the tangent

Hence the slope of the normal is - 1/f'(x) = 3pi/4 = -1

therefore f'(x) = 1

Hence f'(3) = 1