Practice Questions

Q84.The number of solutions of the equation 2θ −cos2 θ + √2 = 0 in R is equal to ______.

Q84.Let 𝑥1, 𝑥2, 𝑥3, … . . , 𝑥20 be in geometric progression with 𝑥1 = 3 and the common ration 12. A new data is constructed replacing each 𝑥𝑖 by 𝑥𝑖- 𝑖2. If 𝑥 is the mean of new data, then the greatest integer less than or equal to 𝑥 is

Q84.A rectangle R with end points of the one of its sides as (1, 2) and (3, 6) is inscribed in a circle. If the equation of a diameter of the circle is 2x −y + 4 = 0, then the area of R is _____.

Q84.If the coefficients of x and x2 in the expansion of (1 + x)p(1 −x)q, p, q ≤15 , are −3 and −5 respectively, then the coefficient of x3 is equal to ______.

Q84.If the mean deviation about the mean of the numbers 1, 2, 3, … … , 𝑛, where 𝑛 is odd, is , then 𝑛 is equal 𝑛 to ______.

Q84.If a1(> 0), a2, a3, a4, a5 are in a G.P. , a2 + a4 = 2a3 + 1 and 3a2 + a3 = 2a4 , then a2 + a4 + 2a5 is equal to _____. JEE Main 2022 (26 Jun Shift 2) JEE Main Previous Year Paper

Q84.The number of one-one functions f : {a, b, c, d} →{0, 1, 2, … , 10} such that 2f(a) −f(b) + 3f(c) + f(d) = 0 is _____ −3x −7 if x ⩽−1Q85. ⎧ 2x2 The number of points where the function f(x) = [4x2 −1] if −1 < x < 1 , where [t] denotes the ⎨ ⎩|x + 1| + |x −2| if x ⩾1 greatest integer ⩽t, is discontinuous is ______ π π

Q84.If sin2(10°) sin(20°) sin(40°) sin(50°) sin(70°) = α− 161 sin(10°), then 16 + α−1 is equal to _____.

Q84.Let the coefficients of the middle terms in the expansion of 4 6 + , β > 0 , βx) , (1 −3βx)2 and (1 −β2 x) ( √61 respectively form the first three terms of an A.P. If d is the common difference of this A.P., then 50 −2d is β2 equal to _____ .

Q84.Let 𝐴𝐵 be a chord of length 12 of the circle 169 𝑥- 22 + 𝑦+ 12 = 4 JEE Main 2022 (29 Jul Shift 2) JEE Main Previous Year Paper If tangents drawn to the circle at points 𝐴 and 𝐵 intersect at the point 𝑃, then five times the distance of point 𝑃 from chord 𝐴𝐵 is equal to _____.

Q84.If x→1(lim sin(3x2−4x+1)−x2+12x3−7x2+ax+b ) = −2, then the value of (a −b) is equal to

Q84.Let a circle C : (x −h)2 + (y −k)2 = r2, k > 0, touch the x-axis at (1, 0). If the line x + y = 0 intersects the circle C at P and Q such that the length of the chord PQ is 2, then the value of h + k + r is equal to _____.

Q84.The number of solutions of the equation sin x = cos2 x in the interval (0, 10) is ______. 2k . If (I −M 2)N = −2I , then the

Q84.A ray of light passing through the point P(2, 3) reflects on the X -axis at point A and the reflected ray passes through the point Q(5, 4). Let R be the point that divides the line segment AQ internally into the ratio 2 : 1 . Let the co-ordinates of the foot of the perpendicular M from R on the bisector of the angle PAQ be (α, β). Then, the value of 7α + 3β is equal to _____.

Q85.A circle of radius 2 unit passes through the vertex and the focus of the parabola y2 = 2x and touches the 2 parabola y = (x −14 ) + α, where α > 0 . Then (4α −8)2 is equal to ______. Q86. ⎡ 14 28 −14 ⎤ The positive value of the determinant of the matrix A , whose Adj(Adj(A)) = −14 14 28 , is ______. ⎣ 28 −14 14 ⎦

Q85.If 40C0 + 41C1 + 42C2 + ⋯+ 60C20 = mn × 60C20 where m & n are co-prime, then m + n is equal to and let L2 be the line passing through the origin and

Q85.Let S = {θ ∈(0, 2π) : 7 cos2 θ −3 sin2 θ −2 cos2 2θ = 2}. Then, the sum of roots of all the equations x2 −2(tan2 θ + cot2 θ)x + 6 sin2 θ = 0 θ ∈S, is _______.

Q85.Let the common tangents to the curves 4(x2 + y2) = 9 and y2 = 4x intersect at the point Q. Let an ellipse, centered at the origin O, has lengths of semi-minor and semi-major axes equal to OQ and 6, respectively. If e and l respectively denote the eccentricity and the length of the latus rectum of this ellipse, then l is equal to e2 ______.

Q85.Let P1 be a parabola with vertex (3, 2) and focus (4, 4) and P2 be its mirror image with respect to the line x + 2y = 6. Then the directrix of P2 is x + 2y = _____.

Q85.Let M = 0 −α , where α is a non-zero real number and N = ∑49k=1 M [α 0 ] positive integral value of α is ______.

Q85.Let the lines y + 2x = √11 + 7√7 and 2y + x = 2√11 + 6√7 be normal to a circle C , then the value of C : (x −h)2 + (y −k)2 = r2 . If the line √11y −3x = 5√773 + 11 is tangent to the circle (5h −8k)2 + 5r2 is equal to ______.

Q85.The sum of diameters of the circles that touch (i) the parabola 75𝑥2 = 645𝑦- 3 at the point 5, 5 and (ii) the 𝑦- axis, is equal to _____ .

Q85.The equations of the sides AB, BC and CA of a triangle ABC are 2x + y = 0, x + py = 15a and x −y = 3 respectively. If its orthocentre is (2, a), −12 < a < 2 , then p is equal to

Q85.If 1 + (2 + 49C1 + 49C2 + … . +49C49)(50C2 + 50C4 + … . . +50C50) is equal to 2n. m, where m is odd, then n + m is equal to _____ .

Q85.Let x = sin(2 tan−1 α) and y = sin( 12 tan−1 43 ). If S = {α ∈R : y2 = 1 −x}, then ∑α∈S 16α3 is equal to _______.