JEE Main 2024 January 31 Shift 2 with solutions Attempt Free MCQ Test -

Let A (a, b), B (3, 4) and (−6,−8) respectively denote the centroid, circumcentre and orthocentre of a triangle. Then, the distance of the point P(2a + 3,7b + 5) from the line 2x + 3y − 4 = 0 measured parallel to the line x − 2y − 1 = 0 is

Let z₁ and z₂ be two complex numbers such that z₁ + z₂ = 5 and . Then  equals-

Let a variable line passing through the centre of the circle x² + y² - 16x - 4y = 0, meet the positive co-ordinate axes at the point A and B. Then the minimum value of OA + OB, where O is the origin, is equal to

Let f, g: (0, ∞) → R be two functions defined by and  Then the value of   is equal to

Let (α, β, γ) be mirror image of the point (2, 3, 5) in the line Then 2α + 3β + 4γ is equal to

Let P be a parabola with vertex (2, 3) and directrix  . Let an ellipse of eccentricity pass through the focus of the parabola P. Then the square of the length of the latus rectum of E, is

The temperature T(t) of a body at time t = 0 is 160^o F and it decreases continuously as per the 
differential equation  where K is positive constant. If T(15) = 120^oF, then T(45) is equal to 

If 2^nd, 8^th, 44^th terms of A.P. are 1^st, 2^nd and 3^rd terms respectively of G.P. and first term of A.P. is 1 then the sum of first 20 terms of A.P. is 

Let f: → R → (0, ∞)  be a strictly increasing function such that  Then, the value of  is equal to

The area of the region enclosed by the parabola  is equal to

Let the mean and the variance of 6 observation a, b, 68, 44, 48, 60 be 55 and 194, respectively if  a > b, then a + 3b is 

If the function f : (-∞, -1) → (a,b) defined by  is one-one and onto, then the distance of the point P(2b + 4,a + 2) from the line x + e^-3 y = 4 is

Consider the function f : (0, ∞) → defined by  If m and n be respectively the number of points at which f is not continuous and f is not differentiable, then m+n is

The number of solutions, of the equation e^{sin x} - 2e ^{-sin x} = 2 is

, then  is equal to

If for some m, n;  and , then  is equal to

A coin is based so that a head is twice as likely to occur as a tail. If the coin is tossed 3 times, then the probability of getting two tails and one head is-

Let A be a 33 real matrix such that

The shortest distance between lines L₁ and L₂, where  and L₂ is the line passing through the points A(-4,4,3).B(-1,6,3) and perpendicular to the line  is

is equal to .

Let a, b, c be the length of three sides of a triangle satisfying the condition (a² +b²) x² - 2b (a + c)x + (b² + c²) = 0. If the set of all possible values of x is the interval (α, β), then 12 (α² + β²) is equal to

Let  A(–2, –1), B(1, 0), C(α,β) and  be the vertices of a parallelogram ABCD . If the point C lies on  and the point D lies on  , then the value of  is equal to _______ .

Let the coefficient of x^r in the expansion of 

Let A be a 3 x3 matrix and det (A) = 2. If  

Then the remainder when n is divided by 9 is equal to ___________.

Let  and  be a vector such that  and . Then  is equal to _____.

 then 16(a² + b² + c²) is equal to ______. 

A line passes through A(4, –6, –2) and B(16, –2,4). The point P(a, b, c) where a, b, c are non-negative integers, on the line AB lies at a distance of 21 units, from the point A. The distance between the points P(a, b, c) and Q(4,  –12, 3) is equal to ____. 

Let y = y(x) be the solution of the differential equation 

Then e^8α is equal to ______. 

Let A = {1, 2, 3, ………100}. Let R be a relation on A defined by (x, y) ∈R if and only if 2x = 3y. Let R₁ be a symmetric relation on A such that  and the number of elements in R1 is n. Then, the minimum value of n is ___________. 

A light string passing over a smooth light fixed pulley connects two blocks of masses and . If the acceleration of the system is , then the ratio of masses is