Practice Questions

Q70.A hyperbola passes through the point 𝑃√2, √3 and has foci at ± 2, 0. Then the tangent to this hyperbola at 𝑃 also passes through the point (1) 3√2, 2√3 (2) 2√2, 3√3 (3) √3, √2 (4) -√2, - √3 JEE Main 2017 (02 Apr) JEE Main Previous Year Paper cot𝑥- cos𝑥

Q70.If y = mx + c is the normal at a point on the parabola y2 = 8x whose focal distance is 8 units, then |c| is equal to: (1) 8√3 (2) 10√3 (3) 2√3 (4) 16√3

Q70.If a point P(0, −2) and Q is any point on the circle, x2 + y2 −5x −y + 5 = 0 , then the maximum value of (PQ)2 is (1) 8 + 5√3 (2) 47+10√6 2 (3) 14 + 5√3 (4) 25+ √6 2

Q71. lim equals 𝑥→𝜋 𝜋- 2𝑥3 2 1 1 (1) (2) 24 16 (3) 1 (4) 1 8 4

Q71.Consider an ellipse, whose center is at the origin and its major axis is along the x-axis. If its eccentricity is 3 5 and the distance between its foci is 6, then the area (in sq. units) of the quadrilateral inscribed in the ellipse, with the vertices as the vertices of the ellipse, is: (1) 32 (2) 80 (3) 40 (4) 8

Q71. The eccentricity of an ellipse having centre at the origin, axes along the co-ordinate axes and passing through the points (4, −1) and (−2, 2) is (1) √3 (2) √3 2 4 (3) 2 (4) 1 √5 2

Q72.The statement 𝑝→𝑞→~𝑝→𝑞→𝑞 is (1) A tautology (2) Equivalent to ~𝑝→𝑞 (3) Equivalent to 𝑝→~𝑞 (4) A fallacy

Q72. lim √3x−3 is equal to x→3 √2x−4− √2 (1) 1 (2) 1 √2 2√2 (3) √3 (4) √3 2

Q72.The contrapositive of the statement 'If two numbers are not equal, then their squares are not equal', is (1) If the squares of two numbers are equal, then the (2) If the squares of two numbers are not equal, then numbers are not equal the numbers are equal (3) If the squares of two numbers are not equal, then (4) If the squares of two numbers are equal, then the the numbers are not equal numbers are equal

Q73.The sum of 100 observations and the sum of their squares are 400 & 2475, respectively. Later on, three observations 3, 4 & 5 were found to be incorrect. If the incorrect observations are omitted, then the variance of the remaining observations is (1) 8. 25 (2) 8. 50 (3) 9. 00 (4) 8. 00

Q73.A box contains 15 green and 10 yellow balls. If 10 balls are randomly drawn, one-by-one, with replacement, then the variance of the number of green balls drawn is: 12 (1) (2) 6 5 (3) 4 (4) 6 25

Q73.The proposition (~p) ∨(p ∧~q) is equivalent to (1) p →∼q (2) p∧∼q (3) q →p (4) none

Q74.The mean age of 25 teachers in a school is 40 years. A teacher retires at the age of 60 years and a new teacher is appointed in his place. If the mean age of the teachers in this school now is 39 years, then the age (in years) of the newly appointed teacher is (1) 35 (2) 40 (3) 25 (4) 30

Q74.Let a vertical tower 𝐴𝐵 have its end 𝐴 on the level ground. Let 𝐶 be the mid-point of 𝐴𝐵 and 𝑃 be a point on the ground such that 𝐴𝑃= 2𝐴𝐵. If ∠𝐵𝑃𝐶= 𝛽, then tan𝛽 is equal to: (1) 6 (2) 1 7 4 2 4 (3) (4) 9 9

Q74.For two 3 × 3 matrices A and B , let A + B = 2B′ and 3A + 2B = I3, where B′ is the transpose of B and I3 is 3 × 3 identity matrix. Then : (1) 10A + 5B = 3I3 (2) 3A + 6B = 2I3 (3) 5A + 10B=2I3 (4) B + 2A = I3

Q75.If x = a, y = b, z = c is a solution of the system of linear equations x + 8y + 7z = 0 9x + 2y + 3z = 0 x + y + z = 0 Such that the point (a, b, c) lies on the plane x + 2y + z = 6 , then 2a + b + c equals: (1) 2 (2) −1 (3) 1 (4) 0

Q75.If 𝐴= 2 -3 , then Adj3𝐴2 + 12𝐴 is equal to: -4 1 (1) 72 -84 (2) 51 63 -63 51 84 72 (3) 51 84 (4) 72 -63 63 72 -84 51

Q75.Let A be any 3 × 3 invertible matrix. Then which one of the following is not always true? (1) adj (adj (A)) = |A|2. (adj (A))−1 (2) adj (adj (A)) = |A|. (adj (A))−1 (3) adj (adj (A)) = |A| . A (4) adj (A) = |A|. A−1

Q76.The number of real values of λ for which the system of linear equations, 2x + 4y −λz = 0 , 4x + λy + 2z = 0 and λx + 2y + 2z = 0 , has infinitely many solutions, is: (1) 3 (2) 1 (3) 2 (4) 0 Q77. ⎧ 0 cos x −sin x ⎫ π If S = x ∈[0, 2π] : sin x 0 cos x = 0 , then ∑x ∈S tan( 3 + x) is equal to: ⎨ ⎬ ⎩ cos x sin x 0 ⎭ (1) 4 + 2√3 (2) −4 -2 √3 (3) −2 + √3 (4) -2 −√3 |x| < 12 , x ≠0, is equal to:

Q76.A value of x satisfying the equation sin[cot−1(1 + x)] = cos[tan−1x], is: JEE Main 2017 (09 Apr Online) JEE Main Previous Year Paper (1) −12 (2) 0 (3) −1 (4) 21

Q76.If 𝑆 is the set of distinct values of 𝑏 for which the following system of linear equations 𝑥+ 𝑦+ 𝑧= 1 𝑥+ 𝑎𝑦+ 𝑧= 1 𝑎𝑥+ 𝑏𝑦+ 𝑧= 0 has no solution, then 𝑆 is: (1) An empty set (2) An infinite set (3) A finite set containing two or more elements (4) A singleton

Q77.The function 𝑓 : 𝑅→-1 1 defined as 𝑓𝑥= 𝑥 is: 2, 2 1 + 𝑥2, (1) Invertible (2) Injective but not surjective (3) Surjective but not injective (4) Neither injective nor surjective

Q77.The function f : N →I defined by f(x) = x −5[ x5 ] , where N is the set of natural numbers and [x] denotes the greatest integer less than or equal to x, is: (1) one-one but not onto (2) one-one and onto (3) neither one-one nor onto (4) onto but not one-one Q78. 4 tantan 5x4x π 5 ) , 0 < x < 2 π The value of k which the function f(x) = is continuous at x = 2 , is 2 π {( k + 5 , x = 2 (1) 2 5 (2) −25 (3) 17 (4) 3 20 5 , then λ + k is equal to

Q78.Let 𝑎, 𝑏, 𝑐∈𝑅 . If 𝑓𝑥= 𝑎𝑥2 + 𝑏𝑥+ 𝑐 is such that 𝑎+ 𝑏+ 𝑐= 3 and 𝑓𝑥+ 𝑦= 𝑓𝑥+ 𝑓𝑦+ 𝑥𝑦, ∀ 𝑥, 𝑦∈𝑅 , 10 then ∑ 𝑓(𝑛) is equal to: 𝑛= 1 (1) 330 (2) 165 (3) 190 (4) 255 1 6𝑥√𝑥

Q78.The value of tan−1[ √1+x2−√1+x2+ √1−x2√1−x2 ], (1) π 4 + 21 cos−1x2 (2) π4 −cos−1x2 (3) π 4 −12 cos−1x2 (4) π4 + cos−1x2