JEE Main entrance exam Question Bank Solutions for Mathematics (JEE Main)

Let f: R → R be a function defined by f(x) = (x – 3)^n₁(x – 5)^n₂, n₁, n₂ ∈ N. Then, which of the following is NOT true?

The minimum value of α for which the equation `4/sinx + 1/(1 - sinx)` = α has at least one solution in `(0, π/2)` is ______.

The range of a ∈ R for which the function f(x) = `(4a - 3)(x + log_e5) + 2(a - 7)cot(x/2)sin^2(x/2), x ≠ 2nπ, n∈N` has critical points, is ______.

Let A = [a_ij] be a 3 × 3 matrix, where

a_ij = `{{:(1, "," if "i" = "j"),(-x, "," if |"i" - "j"| = 1),(2x + 1, ","    "otherwise"):}` 

Let a function f: R→R be defined as f(x) = det(A). Then the sum of maximum and minimum values of f on R is equal to ______.

A wire of length 36 m is cut into two pieces, one of the pieces is bent to form a square and the other is bent to form a circle. If the sum of the areas of the two figures is minimum, and the circumference of the circle is k (meter), then `(4/π + 1)`k is equal to ______.

Let P(h, k) be a point on the curve y = x² + 7x + 2, nearest to the line, y = 3x – 3. Then the equation of the normal to the curve at P is ______.

If p(x) be a polynomial of degree three that has a local maximum value 8 at x = 1 and a local minimum value 4 at x = 2; then p(0) is equal to ______.

If S₁ and S₂ are respectively the sets of local minimum and local maximum points of the function. f(x) = 9x⁴ + 12x³ – 36x² + 25, x ∈ R, then ______.

If y = alog|x| + bx² + x has its extremum values at x = –1 and x = 2, then ______.

If the function y = `(ax + b)/((x - 4)(x - 1))` has an extremum at P(2, –1), then the values of a and b are ______.

If the point (1, 3) serves as the point of inflection of the curve y = ax³ + bx² then the value of 'a ' and 'b' are ______.

The function g(x) = `(f(x))/x`, x ≠ 0 has an extreme value when ______.

Let x and y be real numbers satisfying the equation x² – 4x + y² + 3 = 0. If the maximum and minimum values of x² + y² are a and b respectively. Then the numerical value of a – b is ______.

Let f(x) = (x – a)ⁿg(x) , where g⁽ⁿ⁾(a) ≠ 0; n = 0, 1, 2, 3.... then ______.

The set of values of p for which the points of extremum of the function f(x) = x³ – 3px² + 3(p² – 1)x + 1 lie in the interval (–2, 4), is ______.

A cone of maximum volume is inscribed in a given sphere. Then the ratio of the height of the cone to the diameter of the sphere is ______.

The lateral edge of a regular rectangular pyramid is 'a' cm long. The lateral edge makes an angle a. with the plane of the base. The value of a for which the volume of the pyramid is greatest, is ______.

The greatest value of the function f(x) = `tan^-1x - 1/2logx` in `[1/sqrt(3), sqrt(3)]` is ______.

Let f(x) = |(x – 1)(x² – 2x – 3)| + x – 3, x ∈ R. If m and M are respectively the number of points of local minimum and local maximum of f in the interval (0, 4), then m + M is equal to ______.

The sum of all the local minimum values of the twice differentiable function f : R `rightarrow` R defined by

f(x) = `x^3 - 3x^2 - (3f^('')(2))/2 x + f^('')(1)`