Mathematics Test 199

Result

Mathematics Test 199

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Weekly Quiz Competition

Question 1
4 / -1

If area of pentagon PQRST be 7, where P(–1, –1), Q(2, 0), R(3, 1), S(2, 2) and T(–1, t), t > 0, then the value of t is

A
1

B
4/3

C
2

D
5/3

Solution
Question 2
4 / -1

The sum of all value of λ for which the lines 2x + y + 1 = 0; 3x + 2λy + 4 = 0; x + y - 3λ = 0 are concurrent, is

A
1/4

B
1/2

C
7/2

D
7/12

Solution
Question 3
4 / -1

If A and B are two independent events such that P(A' ∩ B') = 2/15, P(A ∩ B') = 1/6 then P(B) =

A
5/9

B
4/9

C
3/10

D
7/10
Question 4
4 / -1

A hyperbola has centre at origin and one focus at (6, 8). If its two directrices are 3x + 4y + 10 = 0 and 3x + 4y –10 = 0 and eccentricity is e,then the value of 4e²/5 is equal to

A
1

B
2

C
3

D
4

Solution

Distance between centre and focus = ae = 10

Distance between directrices = 2a/e = 4
Question 5
4 / -1

Number of integral values of 'k' for which the chord of the circle x² + y² = 125 passing through P(8, k) gets bisected at P (8, k) and has integral slope is

A
8

B
6

C
4

D
2

Solution

The slope of the chord is

⇒ k = ± 1, ± 2, ± 4, ± 8 but (8, k) must also lie inside the circle x² + y² = 125

⇒ 64 + k² – 125 < 0

⇒ k² < 61

⇒ k can be equal to ± 1, ± 2, ± 4

⇒ 6 values
Question 6
4 / -1

Locus of the feet of perpendiculars drawn from points (1, 2) and (3, 4) on a variable tangent to the conic | z - (1+ 2i) | - | z - (3+ 4i) | = 2 is

A
| z - (2 + 3i) | = 1

B
| z - (2 + 3i) | = 4

C
| z - (1 + i) | = 2

D
| z - (1 + i) | = 1

Solution

represents a hyperbola with foci (1, 2) and (3, 4) and length of transverse axis = 2.

∴ 2a = 2 ⇒ a = 1

∵ Feet of perpendiculars from foci on any tangent lie on auxilliary circle of the hyperbola.

∴ Locus will be auxilliary circle.

∴ Centre = mid point of foci = (2, 3)

and radius = semi transverse axis = 1

∴ Equation of auxilliary circle is |z - (2 + 3i) |= 1
Question 7
4 / -1

Number of numbers greater than a million and divisible by 5 which can be formed by using only the digits 1, 2,1, 2, 0, 5 and 2 is :

A
120

B
110

C
90

D
none

Solution
Question 8
4 / -1

The dual of statement (p v ~ q) ∧ (~ p) is

A
(p v ~ q) ∧ (~ p)

B
(p ∧ ~ q) ∧ (~ p)

C
(p v ~ q) v (~ p)

D
(p ∧ ~ q) v (~ p)

Solution

In dual statement v replace by ∧ and ∧ replace by v so answer is (p ∧ ~ q) v (~ p).
Question 9
4 / -1

A normal is drawn to the parabola y² = 9x at the point P(4, 6), S being the focus, a circle is described on the focal distance of the point P as diameter. The length of the intercept made by the circle on the normal at P is

A
4

B
15/4

C
6

D
17/4

Solution

Required intercept will be equal to the perpendicular distance from the focus on the tangent at P.

Tangent at P,
Question 10
4 / -1

Consider ellipse Let C is centre of the ellipse and P is a variable point lying on the ellipse. If the angle between CP and tangent at P is minimum, then P may be

A

B

C

D
(0, 2)

Solution