JEE Main entrance exam Question Bank Solutions

Let g : (0, ∞) `rightarrow` R be a differentiable function such that `int((x(cosx - sinx))/(e^x + 1) + (g(x)(e^x + 1 - xe^x))/(e^x + 1)^2)dx = (xg(x))/(e^x + 1) + c`, for all x > 0, where c is an arbitrary constant. Then ______.

If a curve y = f(x), passing through the point (1, 2), is the solution of the differential equation, 2x²dy = (2xy + y²)dx, then `f(1/2)` is equal to ______.

The largest value of a, for which the perpendicular distance of the plane containing the lines `vec"r" = (hat"i" + hat"j") + λ(hat"i" + "a"hat"j" - hat"k")` and `vec"r" = (hat"i" + hat"j") + μ(-hat"i" + hat"j" - "a"hat"k")` from the point (2, 1, 4) is `sqrt(3)`, is ______.

If the shortest distance between the lines `(x - 1)/2 = (y - 2)/3 = (z - 3)/λ` and `(x - 2)/1 = (y - 4)/4 = (z - 5)/5` is `1/sqrt(3)`, then the sum of all possible values of λ is ______.

The shortest distance between the z-axis and the line x + y + 2z – 3 = 0 = 2x + 3y + 4z – 4, is ______.

Let [x] denote greatest integer less than or equal to x. If for n ∈ N, (1 – x + x³)ⁿ = `sum_("j" = 0)^(3"n")"a"_"j"x^"j"`, then `sum_("j" = 0)^([(3"n")/2]) "a"_(2"j") + 4sum_("j" = 0)^([(3"n" - 1)/2])"a"_(2"j" + 1)` is equal to ______.

Let ABC be a triangle with A(–3, 1) and ∠ACB = θ, 0 < θ < `π/2`. If the equation of the median through B is 2x + y – 3 = 0 and the equation of angle bisector of C is 7x – 4y – 1 = 0, then tan θ is equal to ______.

Let A be a 2 × 2 real matrix with entries from {0, 1} and |A| ≠ 0. Consider the following two statements:

(P) If A¹I₂, then |A| = –1

(Q) If |A| = 1, then tr(A) = 2,

where I₂ denotes 2 × 2 identity matrix and tr(A) denotes the sum of the diagonal entries of A. Then ______.

The contrapositive of the statement "If I reach the station in time, then I will catch the train" is ______.

The contrapositive of the statement "If you are born in India, then you are a citizen of India", is ______.

If the sides a, b, c of ΔABC satisfy the equation 4x³ – 24x² + 47x – 30 = 0 and `|(a^2, (s - a)^2, (s - a)^2),((s - b)^2, b^2, (s - b)^2),((s - c)^2, (s - c)^2, c^2)| = p^2/q` where p and q are co-prime and s is semiperimeter of ΔABC, then the value of (p – q) is ______.

If D = `[(0, aα^2, aβ^2),(bα + c, 0, aγ^2),(bβ + c, (bγ + c), 0)]` is a skew-symmetric matrix (where α, β, γ are distinct) and the value of `|((a + 1)^2, (1 - a), (2 - c)),((3 + c), (b + 2)^2, (b + 1)^2),((3 - b)^2, b^2, (c + 3))|` is λ then the value of |10λ| is ______.

The minimum number of zeros in an upper triangular matrix will be ______.

How many matrices can be obtained by using one or more numbers from four given numbers?

If A and B are square matrices of order 3 × 3 and |A| = –1, |B| = 3, then |3AB| equals ______.

Let S = {1, 2, 3, 4, 5, 6, 9}. Then the number of elements in the set T = {A ⊆ S : A ≠ `phi` and the sum of all the elements of A is not a multiple of 3} is ______.

The contrapositive of the statement “If you will work, you will earn money” is ______.

The contrapositive of the statement “if I am not feeling well, then I will go to the doctor” is ______.

The statement that is TRUE among the following?

Let A = `[(0, -2),(2, 0)]`. If M and N are two matrices given by M = `sum_(k = 1)^10 A^(2k)` and N = `sum_(k = 1)^10 A^(2k - 1)` then MN² is ______.