A problem in mathematics is given to three students A, B, C and their respective probability of solving the problem is 1/2, 1/3 and 1/4. Probability that the problem is solved is

- 1/3
- 1/2
- 3/4
- 2/3

Events A, B, C are mutually exclusive events such that P(A) = (3x + 1)/3, P(B) = (x - 1)/4, P(C) = (1 - 2x)/4. The set of possible values of x are in the interval

- [1/3, 1/2]
- [1/3, 13/3]
- [0, 1]
- [1/3, 2/3]

The number of solution of tan x + sec x = 2 cos x in [0, 2π) is

- 0
- 1
- 2
- 3

The negation of the statement "If I become a teacher, then I will open a school" is

- I will not become a teacher or I will open a school
- Either I will not become a teacher or I will not open a school
- Neither I will become a teacher nor I will open a school
- I will become a teacher and I will not open a school

A triangle with vertices (4, 0), (-1, -1), (3, 5) is

- right angled but not isosceles
- neither right angled nor isosceles
- isosceles and right angled
- isosceles but not right angled

If the two circles (x-1)^{2} + (y-3)^{2} = r^{2} and x^{2} + y^{2} - 8x + 2y + 8 = 0 intersect in two distinct point, then

- 2 < r < 8
- r = 2
- r > 2
- r < 2

A plane which passes through the point (3, 2, 0) and the line (x-4)/1 = (y-7)/5 = (z-4)/4 is

- 2x - y + z = 5
- x + 2y - z = 1
- x - y + z = 1
- x + y + z = 5

The number of ways of selecting 15 teams from 15 men and 15 women, such that each team consists of a man and a woman, is:

- 1880
- 1120
- 1240
- 1960

If 2+3i is one of the roots of the equation 2x^{3} – 9x^{2} + kx – 13 = 0, k ∈ R, then the real root of this equation:

- does not exist
- exists and is equal to 1/2
- exists and is equal to -1/2
- exists and is equal to 1

Let the sum of the first three terms of an A.P. be 39 and the sum of its last four terms be 178. If the first term of this A.P. is 10, then the median of the A.P. is:

- 26.5
- 28
- 29.5
- 31

If the coefficients of the three successive terms in the binomial expansion of (1+x)^{n} are in the ratio 1:7:42, then the first of these terms in the expansion is:

- 6th
- 7th
- 8th
- 9th

What is the sum of the squares of the roots of the equation x^{2} + 2x - 143 = 0?

- 170
- 180
- 190
- 290

If the difference between the roots of ax^{2} + bx + c = 0 is 1, then which one of the following is correct?

- b
^{2}= a(a + 4c) - a
^{2}= b(b + 4c) - a
^{2}= c(a + 4c) - b
^{2}= a(b + 4c)

If α and β are the roots of the equation x^{2} - q(1+x) - r = 0, then what is (1+α)(1+β) equal to?

- 1 - r
- q - r
- 1 + r
- q + r

If \( A = \begin{bmatrix}1 & 2 \\2 & 3 \end{bmatrix} \) and \( B = \begin{bmatrix}1 & 0 \\1 & 0 \end{bmatrix} \) then what is determinant of AB?

- 0
- 1
- 10
- 20

A and B are two matrices such that AB = A and BA = B then what is B^{2} equal to?

- B
- A
- I
- -I

If the 2nd, 5th and 9th terms of a non-constant A.P. are in G.P., then the common ratio of this G.P. is:

- 4/3
- 1
- 7/4
- 8/5

Let P be the point on the parabola, y^{2} = 8x which is at a minimum distance from the centre C of the circle, x^{2} + (y + 6)^{2} = 1. Then the equation of the circle, passing through C and having its centre at P is:

- x
^{2}+ y^{2}– x + 4y – 12 = 0 - x
^{2}+ y^{2}– x/4 + 2y – 24 = 0 - x
^{2}+ y^{2}– 4x + 9y + 18 = 0 - x
^{2}+ y^{2}– 4x + 8y + 12 = 0

The system of linear equations

x + λy – z = 0

λx – y – z = 0

x + y – λz = 0

has a non-trivial solution for:

- exactly one value of λ.
- exactly two values of λ.
- exactly three values of λ.
- infinitely many values of λ.

The eccentricity of the hyperbola whose length of the latus rectum is equal to 8 and the length of its conjugate axis is equal half of the distance between its foci, is:

- 4/√3
- 2/√3
- √3
- 4/3

If the standard deviation of the numbers 2, 3, a and 11 is 3.5, then which of the following is true?

- 3a
^{2}– 32a + 84 = 0 - 3a
^{2}– 34a + 91 = 0 - 3a
^{2}– 23a + 44 = 0 - 3a
^{2}– 26a + 55 = 0

The integral \( \int \dfrac{2x^{12}+5x^9}{(x^5+x^3+1)^3} dx \) is equal to:

- \( \dfrac{-x^5}{(x^5+x^3+1)^2} + C \)
- \( \dfrac{x^{10}}{2(x^5+x^3+1)^2} + C \)
- \( \dfrac{x^5}{2(x^5+x^3+1)^2} + C \)
- \( \dfrac{-x^{10}}{2(x^5+x^3+1)^2} + C \)

If the line, (x-3)/2 = (y+2)/-1 = (z+4)/3, lies in the plane, lx + my – z = 9, then l^{2} + m^{2} is equal to:

- 18
- 5
- 2
- 26

If 0 ≤ x < 2π, then the number of real values of x, which satisfy the equation

cosx + cos2x + cos3x + cos4x = 0 is:

- 5
- 7
- 9
- 3