List of top Questions asked in JEE Main- 2021

C6H6 freezes at 5.5°C. The temperature at which a solution of 10 g of C4H10 in 200 g of C6H6 freeze is ________ °C. (The molal freezing point depression constant of C6H6 is 5.12°C/m.)
The solubility product of PbI2 is \( 8.0 \times 10^{-9} \). The solubility of lead iodide in 0.1 molar solution of lead nitrate is \( x \times 10^{-6} \) mol/L. The value of \( x \) is __________. (Rounded off to the nearest integer) [Given \( \sqrt{2} = 1.41 \)]
The magnitude of the change in oxidising power of the MnO4− / Mn2+ couple is \( x \times 10^{-4} \) V, if the H+ concentration is decreased from 1 M to \( 10^{-4} \) M at 25 °C. (Assume concentration of MnO4− and Mn2+ to be same on change in H+ concentration). The value of \( x \) is __________ .
Sucrose hydrolyses in acid solution into glucose and fructose following first order rate law with a half-life of 3.33 h at 25 °C. After 9 h, the fraction of sucrose remaining is \( f \). The value of \( \log_{10}(1/f) \) is _______ × 10−2. (Rounded off to the nearest integer)
Among the following allotropic forms of sulphur, the number of allotropic forms, which will show paramagnetism is __________
The total number of amines among the following which can be synthesized by Gabriel synthesis is _________. (A) ethylamine, (B) n-propylamine, (C) benzylamine, (D) aniline
1.86 g of aniline completely reacts to form acetanilide. 10% of the product is lost during purification. Amount of acetanilide obtained after purification (in g) is _________ × 10−2.
Let a, b ∈ R. If the mirror image of the point P(a, 6, 9) with respect to the line (x-3)/7 = (y-2)/5 = (z-1)/(-9) is (20, b, -a - 9), then |a + b| is equal to :
Let a, b, c be in arithmetic progression. Let the centroid of the triangle with vertices (a, c), (2, b) and (a, b) be (10/3, 7/3). If α, β are the roots of the equation ax² + bx + 1 = 0, then the value of α² + β² - αβ is :
If P is a point on the parabola y = x² + 4 which is closest to the straight line y = 4x - 1, then the co-ordinates of P are :
For which of the following curves, the line x + √3 y = 2√3 is the tangent at the point (3√3 / 2, 1/2) ?
The value of the integral, ∫ from 1 to 3 of [x² - 2x - 2] dx, where [x] denotes the greatest integer less than or equal to x, is :
The negation of the statement ~p ∧ (p ∨ q) is :
Let A and B be 3 × 3 real matrices such that A is symmetric matrix and B is skew-symmetric matrix. Then the system of linear equations (A²B² - B²A²) X = O, where X is a 3 × 1 column matrix of unknown variables and O is a 3 × 1 null matrix, has :
Let f(x) be a differentiable function defined on [0, 2] such that f'(x) = f'(2-x) for all x ∈ (0, 2), f(0) = 1 and f(2) = e². Then the value of ∫ from 0 to 2 f(x) dx is :
Let f be a twice differentiable function defined on ℝ such that f(0) = 1, f'(0) = 2 and f'(x) ≠ 0 for all x ∈ ℝ. If $| f(x) \ f'(x) | \ | f'(x) \ f''(x) | = 0$, then the value of f(1) lies in the interval :
If the curve y = ax² + bx + c passes through (1, 2) and the tangent at origin is y = x, then a, b, c are :
If n ≥ 2 is a positive integer, then the sum of the series $^{n+1}C_2 + 2( ^2C_2 + ^3C_2 + ^4C_2 + ....... + ^nC_2 )$ is :
For the system of linear equations : x - 2y = 1, x - y + kz = -2, ky + 4z = 6, k ∈ ℝ, consider the following statements : (A) Unique solution if k ≠ 2, k ≠ -2. (B) Unique solution if k = -2. (C) Unique solution if k = 2. (D) No-solution if k = 2. (E) Infinite solutions if k ≠ -2. Which of the following statements are correct ?
Let f(x) = -55x (x<-5), 2x³ - 3x² - 120x (-5 ≤ x ≤ 4), 2x³ - 3x² - 36x - 336 (x>4). Let A = {x ∈ ℝ : f is increasing}. Then A is equal to :
A possible value of \( \tan\!\left(\frac{1}{4}\sin^{-1}\!\left(\frac{\sqrt{63}}{8}\right)\right) \) is :
The angle of elevation of a jet plane from a point A on the ground is 60°. After 20 s at speed 432 km/h, the angle changes to 30°. If the jet plane is at constant height, then its height is :
The vector equation of the plane passing through the intersection of r · (i + j + k) = 1 and r · (i - 2j) = -2, and the point (1, 0, 2) is :
For p and q, consider: (a) (~ q ∧ (p → q)) → ~ p, (b) ((p ∨ q) ∧ ~ p) → q. Which is correct ?
If \( y = f(x) \) passes through \( (1, 2) \) and \( x \frac{dy}{dx} + y = b x^4 \), then for what value of \( b \), \( \displaystyle \int_{1}^{2} f(x)\,dx = \frac{62}{5} \) ?