Mathematics Test 131

Result

Mathematics Test 131

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Weekly Quiz Competition

Question 1
4 / -1

If statement (P →q) → (q →r) is false, then truth values of statements p,q and r respectively can be

A
FTF

B
TTT

C
FFF

D
FTT

Solution

If (P →q) →(q →r) is false then (p →q) must be true and (q →r) must be false.
Now,
If P →q is true then P & q both can be true or false :
If q →r is false then q must be true and r must be false
So when both condition i.e. P →q true and q → r false satisfy then q is true, r is false and P can be true or false
Question 2
4 / -1

A mirror and a source of light are situated at the origin O and at a point on OX respectively. A ray of light from the source strikes the mirror and is reflected. If the direction ratios of the normal to the plane are proportional to 1, –1, 1 then direction cosines of the reflected ray are

A

B

C

D

Solution

Let the source of light be situated at A(a,0,0) where a≠0

Let OA be the incident ray, OB be the reflected ray and ON be the normal to the mirror at O.

Direction ratio of are proportional to a, 0, 0 and so its direction cosines are 1, 0, 0.

Direction cosines of ON are

Hence, direction cosines of the reflected ray are
Question 3
4 / -1

A
1/3

B
2

C
1/2

D
3/2

Solution
Question 4
4 / -1

Find the differential equation of the family of ellipse such that its centre is on the origin

A

B

C

D

Solution
Question 5
4 / -1

If then range of f(x) is -

A

B

C

D

Solution

so
Question 6
4 / -1

The eccentricity of an ellipse with its centre at the origin is 1/2. If one of the directrix is x = 4, then the equation of the ellipse is

A
4x² + 3y² = 1

B
3x² + 4y² = 12

C
4x² + 3y² = 12

D
3x² + 4y² = 1

Solution
Question 7
4 / -1

The greatest value of λ ≥ 0 for which both the equations 2x² + ( λ − 1)x + 8 = 0 and x² − 8x + λ + 4 = 0 have real roots is

A
9

B
15

C
12

D
16

Solution
Question 8
4 / -1

The number of pairs of solution of the system of equations x+y=2π/3, cosx+cosy=3/2 where x and y are real is

A
infinite

B
finite

C
zero

D
none of these

Solution

We write the second equation as

Since, the range of cos function is [-1,1], so no values of x and y satisfy the second equation.
x∈φ, y∈φ
Question 9
4 / -1

The mean of n terms is If first term is increased by 1, second term by 2, and so on, then new mean is

A

B

C

D

Solution
Question 10
4 / -1

The points of contact of the tangents drawn from the origin to the curve y = sin x lie on the curve

A
x² – y² = xy

B
x² + y² = x² y²

C
x² – y² = x² y²

D
none of these

Solution

1. Let (h, k) be a point of contact of the tangents drawn from the origin to y = sin x. Then, (h,k) lies on y = sin x..
∴k = sin h …(i)

Now,

The equation of the tangent at (h,k) is
y – k = (cos h) (x – h)
This passes through (0,0).