Mathematics Test 251

Result

Mathematics Test 251

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Question 1
4 / -1

Two parabolas with a common vertex and with axes along x-axis and y-axis intersect each other in the first quadrant. If the length of the latus rectum of each parabola is 3, then the equation of the common tangent to the two parabolas is

A
3(x + y) + 4 = 0

B
8(2x + y) + 3 = 0

C
x + 2y + 3 = 0

D
4(x + y) + 3 = 0

Solution
Question 2
4 / -1

The letters of the word OUGHT are written in all possible orders and these words are written out as in a dictionary. Find the rank of the word TOUGH in this dictionary.

A
89

B
98

C
65

D
58

Solution

The word TOUGH will appear after all the words that start with G, H and O. Then we look at the second letter of the words starting with T and then third.

Hence, the rank of the word TOUGH will be one more than the sum of all the possibilities just mentioned.

Total number of letters in the word OUGHT is 5 and all the five letters are different, the alphabetical order of these letters is G, H, O, T, U.

Number of words beginning with G=4 !=24, Number of words beginning with H=4 !=24, Number of words beginning with O=4 !=24, Number of words beginning with TG=3 !=6, Number of words beginning with TH=3 !=6, Number of words beginning with T O G=2 !=2, Number of words beginning with TOH=2 !=2

Next come the words beginning with TOU and TOUH is the first word beginning with TOU.

∴ Rank of 'TOUGH' in the dictionary =24 + 24 + 24 + 6 + 6 + 2 + 2 + 1 = 89
Question 3
4 / -1

Let P be a point on the parabola x² = 4y. If the distance of P from the centre of the circle x² + y² + 6x + 8 = 0 is minimum, then the equation of the tangent to the parabola at P is

A
x + y + 1 = 0

B
x + 4y - 2 = 0

C
x + 2y = 0

D
x - y + 3 = 0

Solution

Let P = (2t, t²)

For the distance between point P and centre of circle to be minimum, line drawn from the centre of circle to point P must be normal to the parabola at P.

The equation normal at P to x² = 4y:

y - t² = -1/t (x - 2t)

It passes through (-3, 0).

0 - t² = -1/t (-3 - 2t)

t³ + 2t + 3 = 0

(t + 1)(t² - t + 3) = 0

⇒t = -1

Point P is (-2, 1).

The equation of tangent to x² = 4y at (x', y') is:

xx' = 2(y + y')

So, at P(-2, 1),

x(-2) = 2(y + 1)

x + y + 1 = 0
Question 4
4 / -1

Let the population of rabbits surviving at a time t be governed by the differential equation If p(0) = 100, then p(t) equals

A
400 - 300 e^t/2

B
300 - 200 e^t/2

C
600 - 500 e^t/2

D
300 - 400 e^t/2

Solution
Question 5
4 / -1

A bag contains 4 red and 6 black balls. A ball is drawn at random from the bag. Its colour is seen and this ball, along with two additional balls of the same colour, is returned to the bag. If now, a ball is drawn at random from the bag, then the probability that this drawn ball is red is:

A
3/10

B
2/5

C
1/5

D
3/4

Solution
Question 6
4 / -1

The length of projection of the line segment joining the points (5, -1, 4) and (4, -1, 3) on the plane x + y + z = 7 is

A
2/√3

B
2/3

C
1/3

D
√2/3

Solution
Question 7
4 / -1

O is the origin and A is the point (a, b, c). Deduce the equation of the plane through A at right angles to OA.

A
ax + by + cz = a² - b² - c²

B
ax + by + cz = a² + b² + c²

C
ax - by - cz = a² + b² + c²

D
ax - by - cz - b² - c²

Solution

Given, D.R.'s of OA are a 0, b - 0, c - 0 or a, b, c.

Equation of any plane through A, i.e. (a, b, c) is: A(x – a) + B(y - b) + C(z - c) = 0

But plane is at right angles to OA which therefore is normal and whose D.R.'s are a, b, c.

So, the plane is, a(x - a) + b(y - b) + c(z - c) = 0

ax + by + cz = a² + b² + c²
Question 8
4 / -1

Tangents drawn from the point (-8, 0) to the parabola y² = 8x touch the parabola at P and Q. If F is the focus of the parabola, then the area of the triangle PFQ (in sq. units) is equal to

A
24

B
32

C
48

D
64

Solution
Question 9
4 / -1

The sides of a rhombus ABCD are parallel to the lines x - y + 2 = 0 and 7x - y + 3 = 0. If the diagonals of the rhombus intersect at P(1, 2) and the vertex A (different from the origin) is on the y-axis, then the ordinate of A is:

A
7/2

B
5/2

C
2

D
7/4

Solution
Question 10
4 / -1

If f(x) is a quadratic expression such that f(1) + f(2) = 0, and -1 is a root of f(x) = 0, then the other root of f(x) = 0 is:

A
-8/5

B
-5/8

C
5/8

D
8/5

Solution