Mathematics Test 267

Result

Mathematics Test 267

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Question 1
4 / -1

Let f(x) be a continuous function such that f(a – x) + f(x) = 0 for all x[0,a]. Then, the value of the integral is equal to

A
a

B
a/2

C
fA

D
1/2 fA

Solution
Question 2
4 / -1

The circles which can be drawn to pass through (1,0) & (3,0) and touching the y-axis, intersect at an angle θ. The value of cos θ is equal to

A
1/2

B
1/√2

C
1/4

D
√3/2

Solution

Equation of line joining A (1,0) and B (3,0) is y = 0. Equation of family of circles passing through A and B is:

(x – 1)(x – 3) + (y – 0) (y – 0) + λy = 0

x² + y² – 4x + λy + 3 = 0

If above circle touches y-axis then x = 0 is a tangent. Substituting x = 0:

y² + λy + 3 = 0

Discriminant of above quadratic must be zero.

λ² – 12 = 0
Question 3
4 / -1

a, b, c are positive numbers and abc² has the greatest value 1/ 64. Then

A

B

C

D
none of these

Solution

Also for the greatest value of abc² the numbers have to be equal, i.e a = b = c/2

Also given that maximum value = 1/64

so, a + b + c = 1

i.e. a = b = 1/4, c = 1/2
Question 4
4 / -1

If A and B are two square matrices such that B = –A^–1 BA, then (A+B)² is equal to

A
O

B
A² + B²

C
A² + 2AB + B²

D
A+B

Solution
Question 5
4 / -1

The set of all values of the parameter a for which the points of minimum of the function y = 1 + a² x – x³Satisfy the inequality

A
an empty set

B
(-3√3, - 2√3)

C
(2√3,3√3)

D

Solution
Question 6
4 / -1

A

B

C

D

Solution
Question 7
4 / -1

A
an even function

B
an odd function

C
a periodic function

D
none of these

Solution
Question 8
4 / -1

If for a variable line the condition a^–2 + b^–2 = c^–2(c is a constant), is satisfied, then the locus of foot of the perpendicular drawn from origin to this is:

A

B
x² + y² = 2c²

C
x² + y² = c²

D
x² – y² = c²

Solution
Question 9
4 / -1

The eccentricity of the hyperbola whose latus rectum is half of its transverse axis, is

A

B

C

D

Solution

Length of latus rectum of hyperbola = Length of the transverse axis
Question 10
4 / -1

A tangent having slope of intersects the major and minor axes in points A and B respectively. If C is the center of the ellipse then the area of the triangle ABC is:

A
24 sq. units

B
12 sq. units

C
36 sq. units

D
48 sq. units

Solution