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

A wire of length 22 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into an equilateral triangle. Then, the length of the side of the equilateral triangle, so that the combined area of the square and the equilateral triangle is minimum, is ______.

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 ______.