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

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 ______.

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 ______.