Practice Questions

Q84.Consider a set of 3n numbers having variance 4. In this set, the mean of first 2n numbers is 6 and the mean of the remaining n numbers is 3. A new set is constructed by adding 1 into each of the first 2n numbers, and subtracting 1 from each of the remaining n numbers. If the variance of the new set is k, then 9k is equal to ______. JEE Main 2021 (17 Mar Shift 2) JEE Main Previous Year Paper

Q84.The number of integral values of k for which the equation 3 sin x + 4 cos x = k + 1 has a solution, k ∈R is _______. cos x + 1, then number of solutions of the given equation when x ∈[0, π2 ] is

Q84.If lim x x = 2, then a + b + c is equal to ________. sin x→0 Q85. ⎡ −30 20 56 ⎤ ⎡ 2 7 ω2 ⎤ −1+i√3 Let P = 90 140 112 and A = −1 −ω 1 where ω = 2 , and I3 be the identity matrix ⎣ 120 60 14 ⎦ ⎣ 0 −ω −ω + 1 ⎦ of order 3. If the determinant of the matrix (P −1AP −I3)2 is αω2 , then the value of α is equal to _________.

Q84.If the function f(x) = cos(sin x)−cos x is continuous at each point in its domain and f(0) = k1 , then k is x4 _________. x is −ba loge 2 when x = 1, where a and

Q84.If the variable line 3x + 4y = α lies between the two circles (x −1)2 + (y −1)2 = 1 and (x −9)2 + (y −1)2 = 4, without intercepting a chord on either circle, then the sum of all the integral values of α is

Q84.If A = 0 and (I2 + A)(I2 −A)−1 = then 13(a2 + b2) is equal to _____ . θ [ b a ], [tan( 2 ) 0 ]

Q84.Consider the statistics of two sets of observations as follows: Size Mean Variance Observation I 10 2 2 Observation II n 3 1 If the variance of the combined set of these two observations is 17 9 , then the value of n is equal to ________.

Q84.The sum of first four terms of a geometric progression (G. P. ) is 6512 and the sum of their respective reciprocals is 65 18 . If the product of first three terms of the G. P. is 1, and the third term is α, then 2α is _________. n nCr, if n ≥r ≥0

Q84.Let E be an ellipse whose axes are parallel to the co-ordinates axes, having its centre at (3, −4), one focus at (4, −4) and one vertex at (5, −4). If mx −y = 4, m > 0 is a tangent to the ellipse E, then the value of 5m2 is equal to _____ .

Q84.If lim ax−(e4x−1) exists and is equal to b, then the value of a −2b is ___ . x→0 ax(e4x−1)

Q84.If the co-efficient of x7 and x8 in the expansion of (2 + x3 ) n are equal, then the value of x ≠2 , and f(x) = 7, x = 2 where P(x) is a polynomial such that

Q84.Let the points of intersections of the lines 𝑥- 𝑦+ 1 = 0, 𝑥- 2𝑦+ 3 = 0 and 2𝑥- 5𝑦+ 11 = 0 are the mid points of the sides of a triangle ABC . Then the area of the triangle ABC is

Q84.Let a1, a2, … , a10 be an A. P. with common difference −3 and b1, b2, … , b10 be a G. P. with common ratio 2. Let ck = ak + bk, k = 1, 2, … , 10. If c2 = 12 and c3 = 13, then ∑10k=1 ck is equal to ______.

Q84.If the minimum area of the triangle formed by a tangent to the ellipse x2 = 1 and the co-ordinate axis is + 4a2 b2 kab, then k is equal to ___________.

Q84.Let S be the sum of all solutions (in radians) of the equation sin4 θ + cos4 θ −sin θ cos θ = 0 in [0, 4π] then 8S is equal to π

Q84.If 1P 1 + 2 ⋅2P 2 + 3 ⋅3P 3 + … + 15 ⋅15P 15 = qP r −s, 0 ≤s ≤1, then q+sCr−s is equal to

Q84.The number of elements in the set {n ∈{1, 2, 3, … , 100} ∣(11)n > (10)n + (9)n} is ___________.

Q84.The ratio of the coefficient of the middle term in the expansion of ( 1 + 𝑥) 20 and the sum of the coefficients of two middle terms in expansion of ( 1 + 𝑥) 19 is . 𝑥+ 1 𝑥- 1 10

Q84.If the arithmetic mean and the geometric mean of the pth and qth terms of the sequence −16, 8, −4, 2, … satisfy the equation 4x2 −9x + 5 = 0 , then p + q is equal to _______.

Q84.Consider a triangle having vertices A(−2, 3), B(1, 9) and C(3, 8). If a line L passing through the circum- centre of triangle ABC, bisects line BC, and intersects y-axis at point (0, α2 ), then the value of real number α is _______.

Q85.Let L be a common tangent line to the curves 4x2 + 9y2 = 36 and (2x)2 + (2y)2 = 31 . Then the square of the slope of the line L is ______.

Q85.The locus of a point, which moves such that the sum of squares of its distances from the points (0, 0), (1, 0), (0, 1)(1, 1) is 18 units, is a circle of diameter d. Then d2 is equal to −1), for x ≤0 2 (x

Q85.Let 𝑀 be any 3 × 3 matrix with entries from the set 0, 1, 2. The maximum number of such matrices, for which the sum of diagonal elements of 𝑀𝑇𝑀 is seven, is______.

Q85.If √3(cos2 x) = (√3 −1) ________.

Q85.If f(x) = sin(cos−1( 1+22x1−22x )) and its first derivative with respect to b are integers, then the minimum value of a2 −b2 is _______.