Maths

A student appeared in an examination consisting of 8 true – false type questions. The student guesses the answers with equal probability. The smallest value of n, so that the probability of guessing at least 'n' correct answers is less than 1/2, is :

Which of the following is the negation of the statement “for all M > 0, there exists x ∈ S such that x ≥ M”?

Two sides of a parallelogram are along the lines 4x + 5y = 0 and 7x + 2y = 0. If the equation of one of the diagonals of the parallelogram is 11x + 7y = 9, then other diagonal passes through the point:

For real numbers α and β ≠ 0, if the point of intersection of the straight lines (x-α)/1 = (y-1)/2 = (z-1)/3 and (x-4)/β = (y-6)/3 = (z-7)/3 lies on the plane x+2y-z = 8, then α - β is equal to:

The value of limₓ→₀ x/(√1-sinx - √1+sinx) is equal to :

A possible value of 'x', for which the ninth term in the expansion of {3ˡᵒᵍ₃√⁽²⁵⁽ˣ⁻¹⁾⁺⁷⁾ + 3⁻¹/ˡᵒᵍ₃⁽⁵⁽ˣ⁻¹⁾⁺¹⁾}¹⁰ in the increasing powers of 3⁻¹/ˡᵒᵍ₃⁽⁵⁽ˣ⁻¹⁾⁺¹⁾ is equal to 180, is :

Let C be the set of all complex numbers. Let S₁ = {z∈ C:|z-2|≤1} and S₂ = {z∈ C: z(1+i)+z̄(1-i) ≥ 4}. Then, the maximum value of |z - 5/2|² for z ∈ S₁ ∩ S₂ is equal to:

Let α = maxₓ∈ᵣ{8²sin³ˣ 4⁴cos³ˣ} and β = minₓ∈ᵣ{8²sin³ˣ 4⁴cos³ˣ}. If 8x² + bx + c = 0 is a quadratic equation whose roots are α¹/⁵ and β¹/⁵, then the value of c – b is equal to :

Let A and B be two 3 x 3 real matrices such that (A² - B²) is invertible matrix. If A⁵ = B⁵ and A³B² = A²B³, then the value of the determinant of the matrix A³ + B³ is equal to :

Consider a circle C which touches the y-axis at (0, 6) and cuts off an intercept 6√5 on the x-axis. Then the radius of the circle C is equal to :