JEE Main entrance exam Question Bank Solutions

Let A and B be 3 × 3 real matrices such that A is symmetric matrix and B is skew-symmetric matrix. Then the systems 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 ______.

If A = `[(0, -tan  θ/2),(tan  θ/2, 0)]` and (I₂ + A) (I₂ – A)^–1 = `[(a, -b),(b, a)]` then 13(a² + b²) is equal to ______. 

Consider three observations a, band c such that b = a + c. If the standard deviation of a + 2, b + 2, c + 2 is d, then which of the following is true?

Let the position vectors of two points P and Q be `3hat"i" - hat"j" + 2hat"k"` and `hat"i" + 2hat"j" - 4hat"k"`, respectively. Let R and S be two points such that the direction ratios of lines PR and QS are (4, –1, 2) and (–2, 1, –2) respectively. Let lines PR and QS intersect at T. If the vector `vec("TA")` is perpendicular to both `vec("PR")` and `vec("QS")` and the length of vector `vec("TA")` is `sqrt(5)` units, then the modulus of a position vector of A is ______.

The mean and standard deviation of 20 observations were calculated as 10 and 2.5 respectively. It was found that by mistake one data value was taken as 25 instead of 35. If α and `sqrtβ` are the mean and standard deviation respectively for correct data, then (α, β) is ______.

If the mean of the data 7, 7, 9, 7, 8, 7, λ, 8 is 7, then the variance of this data is ______.

Let `A(hati + 2hatj + 3hatk), B(hatj + hatk)` and `C(3hati + 2hatj + 2hatk)` are position vectors of vertices of ΔABC and the circumradius of ΔABC is R then `(4R^2)/11` is ______.

The position vectors of points A, B, C are respectively `vec"a", vec"b", vec"c"`. If L divides AB in 3:4 and M divides BC in 2:1 both externally, then `vec("LM")` is ______.

The mean of a data set consisting of 20 observations is 40. If one observation 53 was wrongly recorded as 33, then the correct mean will be ______.

`int (a^x - b^x)^2/(a^xb^x)dx` equals ______.

If seven women and seven men are to be seated around a circular table such that there is a man on either side of every woman, then the number of seating arrangements is ______.

The mean and variance of 10 observations were calculated as 15 and 15 respectively by a student who took by mistake 25 instead of 15 for one observation. Then, the correct standard deviation is ______.

The mean and standard deviation of 50 observations are 15 and 2 respectively. It was found that one incorrect observation was taken such that the sum of correct and incorrect observations is 70. If the correct mean is 16, then the correct variance is equal to ______.

If ω(≠1) is a cube root of unity and (1 + ω)⁷ = A + Bω, then A and B are respectively the numbers ______.

If the constant term in the expansion of `(3x^3 - 2x^2 + 5/x^5)^10` is 2^k. l, where l is an odd integer, then the value of k is equal to ______.

Let c, k ∈ R. If f(x) = (c + 1)x² + (1 – c²)x + 2k and f(x + y) = f(x) + f(y) – xy, for all x, y ∈ R, then the value of |2(f(1) + f(2) + f(3) + ... + f(20))| is equal to ______.

Let α be a root of the equation 1 + x² + x⁴ = 0. Then the value of α¹⁰¹¹ + α²⁰²² – α³⁰³³ is equal to ______.

Let n ≥ be an integer. If 9ⁿ – 8n–1 = 64α and 6ⁿ – 5n = 25β, then α – β is ______.

The distance of the point (1, 1, 9) from the point of intersection of the line `(x - 3)/1 = ("y" - 4)/2 = ("z" - 5)/2` and the plane x + y + z = 17 is ______.

Let f: (0,2)→R be defined as f(x) = `log_2(1 + tan((πx)/4))`. Then, `lim_(n→∞) 2/n(f(1/n) + f(2/n) + ... + f(1))` is equal to ______.