Practice Questions

Q83.Let the area enclosed by the lines x + y = 2, y = 0 , x = 0 and the curve f(x) = min{x2 + 43 , 1 + [x]} where [x] denotes the greatest integer ≤x, be A . Then the value of 12A is

Q83.Some couples participated in a mixed doubles badminton tournament. If the number of matches played, so that no couple played in a match, is 840, then the total numbers of persons, who participated in the tournament, is ________.

Q83.Consider the triangles with vertices A(2, 1), B(0, 0) and C(t, 4), t = [0, 4]. If the maximum and the minimum perimeters of such triangles are obtained at t = α and t = β respectively, then 6α + 21β is equal to ___________.

Q83.Let Δ be the area of the region {(x, y) ∈R2 : x2 + y2 ≤21, y2 ≤4x, x ≥1}. Then 21 (Δ √7 equal to (1) 2√3 −13 (2) √3 −23 (3) 2√3 −23 (4) √3 −43

Q83.Let A be the area of the region {(x, y) : y ≥x2, y ≥(1 −x)2, y ≤2x(1 −x)}. Then 540A is equal to y(1) = 0 is

Q83.Let y = y(x), y > 0, be a solution curve of the differential equation (1 + x2)dy = y(x −y)dx. If y(0) = 1 = β, then and y(2√2) = + + 2√2) (2) e3β−1 e(5 √2) (1) e3β−1 = e(3 = + + 2√2) (4) eβ−1 e−2(5 √2) (3) eβ−1 = e−2(3

Q84.Let 𝑆 be the set of values of λ, for which the system of equations 6𝜆𝑥- 3𝑦+ 3𝑧= 4𝜆2, 2𝑥+ 6𝜆𝑦+ 4𝑧= 1 and 3𝑥+ 2𝑦+ 3𝜆𝑧= 𝜆 has no solution. Then,12 ∑𝜆∈𝑆𝜆 is equal to _______. 2𝑥

Q84.Let αx = exp(xβyγ) be the solution of the differential equation 2x2ydy −(1 −xy2)dx = 0 , x > 0, y(2) = √loge 2 . Then α + β −γ equals : (1) 1 (2) −1 (3) 0 (4) 3 →

Q84.Let the point 𝑝, 𝑝+ 1 lie inside the region 𝐸= 𝑥, 𝑦: 3 - 𝑥≤𝑦≤√9 - 𝑥2 , 0 ≤𝑥≤3 . If the set of all values of 𝑝 is the interval 𝑎, 𝑏, then 𝑏2 + 𝑏- 𝑎2 is equal to ________ .

Q84.The remainder, when 7103 is divided by 17, is

Q85.Let →a = −ˆi −ˆj + ˆk,→a⋅ b = 1 and →a× b = ˆi −ˆj. Then →a−6 b is equal to (1) 3(ˆi −ˆj −ˆk) (2) 3(ˆi + ˆj + ˆk) + (3) 3(ˆi −ˆj ˆk) (4) 3(ˆi + ˆj −ˆk)

Q85.Let 𝑆= {1, 2, 3, 4, 5, 6}. Then the number of oneone functions 𝑓: 𝑆→𝑃( 𝑆) , where 𝑃( 𝑆) denote the power set of 𝑆, such that 𝑓( 𝑛) ⊂𝑓( 𝑚) where 𝑛< 𝑚 is

Q85.The foci of a hyperbola are ( ± 2, 0 ) and its eccentricity is 32. A tangent, perpendicular to the line 2𝑥+ 3𝑦= 6, is drawn at a point in the first quadrant on the hyperbola. If the intercepts made by the tangent on the 𝑥- and 𝑦-axes are 𝑎 and 𝑏 respectively, then |6𝑎| + | 5𝑏| is equal to

Q85.Let →a,→b and→cbe three non zero vectors such that →b ⋅→c= 0 and →a× (→b ×→c) →b−→c → → → → → is equal to × × b ⋅ d =→a⋅ b, then (→a b) ⋅(→c d) (1) 3 (2) 1 4 2 (3) −14 (4) 41

Q85.Suppose ∑𝑟=20230 𝑟2 · 2023𝐶𝑟= 2023 × 𝛼× 22022, then the value of 𝛼 is

Q86.Let →a = ˆi + 2ˆj + 3ˆk and b = ˆi + ˆj −ˆk. If →cis a vector such that →a⋅→c= 11, b ⋅(→a×→c) 2 is equal to −√3→b , then →a×→c

Q86.Let 𝑎∈ℤ and 𝑡 be the greatest integer ≤𝑡, then the number of points, where the function 𝑓𝑥= 𝑎+ 13 sin𝑥, 𝑥∈0, 𝜋 is not differentiable, is ____________

Q86.In the figure, θ1 + θ2 = π2 and √3BE θ1 then the perimeter (in unit) of ∆CED is equal to

Q86.If 1 1 / 1 𝑛 𝑥21 + 𝑥14 + 𝑥72𝑥14 + 3𝑥7 + 6 7𝑑𝑥= where 𝑙, 𝑚, 𝑛∈𝑁, 𝑚 and 𝑛 are co-prime then 𝑙+ 𝑚+ 𝑛 ∫0 𝑙11𝑚/ is equal to _____ .

Q86.Let →a = 2ˆi −7ˆj + 5ˆk , b = ˆi + ˆk and→c= ˆi + 2ˆj −3ˆk be three given vectors. If→ris a vector such that →r×→a =→c×→a and→r⋅→b = 0 , then →r is equal to: (1) 11 7 √2 (2) 117 (3) 11 5 √2 (4) √9147

Q86.Let →a = ˆi + 2ˆj + λˆk, b = 3ˆi −5ˆj −λˆk, →a⋅→c= 7 , 2( ⋅→c)

Q86.The vector →a = −ˆi + 2ˆj + ˆk is rotated through a right angle, passing through the y-axis in its way and the → → resulting vector is b. Then the projection of 3→a+ √2 b on →c= 5ˆi + 4ˆj + 3ˆk is (1) 3√2 (2) 1 (3) √6 (4) 2√3

Q86.Let a common tangent to the curves 𝑦2 = 4𝑥 and 𝑥- 42 + 𝑦2 = 16 touch the curves at the points 𝑃 and 𝑄. Then 𝑃𝑄2 is equal to ________.

Q87.Let the line L pass through the point (0, 1, 2), intersect the line x−12 = y−23 = z−34 and be parallel to the plane 2x + y −3z = 4. Then the distance of the point P(1, –9, 2) from the line L is (1) √74 (2) √69 (3) √54 (4) 9 = −5. If P

Q87.Let for 𝑥∈𝑅, 𝑓𝑥= 𝑥+ 𝑥 and 𝑔𝑥= 𝑥, 𝑥< 0 . Then area bounded by the curve 𝑦= 𝑓𝑜𝑔𝑥 and the lines 2 𝑥2, 𝑥≥0 𝑦= 0, 2𝑦- 𝑥= 15 is equal to _____ . 2