Practice Questions

Q83.Let a circle C of radius 5 lie below the x-axis. The line L1 = 4x + 3y + 2 passes through the centre P of the circle C and intersects the line L2 : 3x −4y −11 = 0 at Q . The line L2 touches C at the point Q . Then the distance of P from the line 5x −12y + 51 = 0 is

Q83.Different A.P.'s are constructed with the first term 100, the last term 199, And integral common differences. The sum of the common differences of all such, A.P's having at least 3 terms and at most 33 terms is.

Q83.If two tangents drawn from a point (α, β) lying on the ellipse 25x2 + 4y2 = 1 to the parabola y2 = 4x are such that the slope of one tangent is four times the other, then the value of (10α + 5)2 + (16β2 + 50)2 equals ______

Q83.Let A = {1, 2, 3, 4, 5, 6, 7} . Define B ={ T ⊆A : either 1 ∉T or 2 ∈T } and C ={ T ⊆A : T the sum of all the elements of T is a prime number.} Then the number of elements in the set B ∪C is _______. Q84. 1 a a 1 48 2160 Let A = ⎡0 1 b ⎤ , a, b ∈R. If for some n ∈N, An = ⎡0 1 96 ⎤ then n + a + b is equal to _______. 0 0 1 0 0 1 ⎣ ⎦ ⎣ ⎦

Q84.Let 𝐶𝑟 denote the binomial coefficient of 𝑥𝑟 in the expansion of 1 + 𝑥10. If for 𝛼, 𝛽∈𝑅, 𝛼× 211 𝐶1 𝐶2 𝐶1 + 3 · 2𝐶2 + 5 · 3𝐶3 + … upto 10 terms = (𝐶0 + 2 + 3 + … upto 10 terms) then the value of 2𝛽- 1 𝛼+ 𝛽 is equal to _____. 𝜋 7𝜋

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.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.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 _____.

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 _____.

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.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 𝑆= 𝑥, 𝑦∈ℕ× ℕ: 9𝑥- 32 + 16𝑦- 42 ≤144 and 𝑇= 𝑥, 𝑦∈ℝ× ℝ: 𝑥- 72 + y - 42 ≤36 The 𝑛𝑆∩𝑇 is equal to ______. Q86. 1 -1 2 3 Let 𝑥= 1 and 𝐴= 0 1 6 . For 𝑘∈ℕ, if 𝑋'𝐴𝑘𝑋= 33, then 𝑘 is equal to 1 0 0 -1

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 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.For the hyperbola 𝐻: 𝑥2 - 𝑦2 = 1 and the ellipse 𝐸: 𝑥2 + 𝑦2 = 1, 𝑎> 𝑏> 0, let the 𝑎2 𝑏2 (1) eccentricity of 𝐸 be reciprocal of the eccentricity of 𝐻, and 𝐾 be a common tangent of 𝐸 and 𝐻. (2) the line 𝑦= √ 52𝑥+ Then 4𝑎2 + 𝑏2 is equal to 100 Q86. 𝑥 𝑥+ 2cos𝑥3 + 2𝑥+ 2cos𝑥2 + 3sin𝑥+ 2cos𝑥 lim is equal to 𝑥→0 𝑥+ 23 + 2𝑥+ 22 + 3sin𝑥+ 2 JEE Main 2022 (28 Jul Shift 1) JEE Main Previous Year Paper

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 _____ .

Q86.The sum of the maximum and minimum values of the function f(x) = |5x −7| + [x2 + 2x] in the interval [ 54 , 2], where [t] is the greatest integer ≤t, is ______.

Q86.Let S = [−π, π2 ) −{−π2 , −π4 , −3π4 , π4 }. Then the number of elements in the set A = ∈S : tan + √5 = √5 {θ θ(1 tan(2θ)) −tan(2θ)} is _____ .

Q86.Suppose a class has 7 students. The average marks of these students in the mathematics examination is 62 , and their variance is 20 . A student fails in the examination if he/she gets less than 50 marks, then in worst case, the number of students can fail is where i = √−1. Then, the number of elements in the set

Q86.Let 𝑓𝑥= 2𝑥2 + 1 and 𝑔𝑥= 2𝑥- 3, 𝑥< 0 , where 𝑡 is the greatest integer ≤𝑡. Then, in the open interval 2𝑥+ 3, 𝑥≥0 -1, 1, the number of points where fog is discontinuous is equal to ______.

Q86.Let the mirror image of a circle c1 : x2 + y2 −2x −6y + α = 0 in line y = x + 1 be c2 : 5x2 + 5y2 + 10gx +10fy + 38 = 0. If r is the radius of circle c2 , then α + 6r2 is equal to ______

Q86.Let S be the set containing all 3 × 3 matrices with entries from {−1, 0, 1} . The total number of matrices A ∈S such that the sum of all the diagonal elements of ATA is 6 is ______.

Q86.Let a line L1 be tangent to the hyperbola x216 −y24 = 1 perpendicular to L1 . If the locus of the point of intersection of L1 and L2 is (x2 + y2)2 = αx2 + βy2 , then α + β is equal to ______. Q87. ⎡ 0 1 0 ⎤ 2 Let X = 0 0 1 , Y = αl + βX + γX and Z = α2I −αβX + (β2 −αγ)X 2, α, β, γ ∈R. ⎣ 0 0 0 ⎦ 1 −2 1 5 5 5 ⎡ ⎤ If Y−1 = 0 51 −25 , then (α −β + γ)2 is equal to ______. 1 ⎣ 0 0 5 ⎦ is equal to _____.

Q87.If 𝑡 denotes the greatest integer ≤𝑡, then number of points, at which the function 𝑓𝑥= 42𝑥+ 3 + 1 9𝑥+ - 12𝑥+ 20 is not differentiable in the open interval -20, 20, is ______. 2

Q87.Let f and g be twice differentiable even functions on (−2, 2) such that f( 41 ) = 0, f( 21 ) = 0, f(1) = 1 and g( 34 ) = 0, g(1) = 2 Then, the minimum number of solutions of f(x)g′′(x) + f ′(x)g′′(x) = 0 in (−2, 2) is equal to _____.