Practice Questions

Q77.For 𝑥∈𝑅, Let [𝑥] denotes the greatest integer ≤𝑥, then the sum of the series -1 + -1 - 1 + -1 - 2 + . . . . . + -1 - 99 is 3 3 100 3 100 3 100 (1) -131 (2) -153 (3) -135 (4) -133

Q77.Let f(x) = 15–|x –10|; x ∈R. Then the set of all values of x, at which the function g(x) = f(f(x)) is not differentiable, is: (1) {5, 10, 15} (2) {10} (3) {10, 15} (4) {5, 10, 15, 20} √2cosx−1 π cotx−1 , x ≠π 4 is continuous, then k is equal to

Q77. x sinθ cosθ x sin2θ cos2θ If Δ1 = −sinθ −x 1 and Δ2 = −sin2θ −x 1 , x ≠0; then for all θ ∈(0, π2 ) : cosθ 1 x cos2θ 1 x (1) Δ1 + Δ2 = −2(x3 + x −1) (2) Δ1 −Δ2 = x(cos2θ −cos4θ) (3) Δ1 + Δ2 = −2x3 (4) Δ1 −Δ2 = −2x3

Q77.In a class of 140 students numbered 1 to 140 , all even numbered students opted Mathematics course, those whose number is divisible by 3 opted Physics course and those whose number is divisible by 5 opted Chemistry course. Then the number of students who did not opt for any of the three courses is: (1) 42 (2) 1 (3) 38 (4) 102

Q77.The sum of the real roots of the equation 𝑥 -6 -1 2 -3𝑥 𝑥- 3 = 0, is equal to: -3 2𝑥 𝑥+ 2 (1) 0 (2) -4 (3) 6 (4) 1 JEE Main 2019 (10 Apr Shift 2) JEE Main Previous Year Paper

Q78.If the system of equations 2x + 3y −z = 0, x + ky −2z = 0 and 2x −y + z = 0 has a non-trivial solution (x, y, z), then xy + yz + xz + k is equal to (1) −14 (2) 21 (3) −4 (4) 34

Q78.Let N be the set of natural numbers and two functions f and g be defined as f, g : N →N such that n+1 2 , if n is odd f(n) = n if n is even { 2 , and g(n) = n −(−1)n. Then fog is: (1) onto but not one-one (2) Both one-one and onto (3) One-one but not onto (4) Neither one-one nor onto K be the set of all points

Q78.If the function f defined on ( 6 , π3 ) by f(x) = π { k, x = 4 (1) 1 (2) 1 2 (3) 2 (4) 1 √2

Q78.If cos-1 2 cos-1 3 𝜋 𝑥> 3 then 𝑥 is equal to : 3𝑥+ 4𝑥= 2 4, (1) √145 (2) √145 10 11 (3) √146 (4) √145 12 12 1 1

Q78.Let f(x) = x2, x ∈R . For any A ⊆R, define g(A) = {x ∈R : f(x) ∈A} . If S = [0, 4] , then which one of the following statements is not true? (1) g(f(S)) ≠S (2) f(g(S)) ≠f(S) (3) f(g(S)) = S (4) g(f(S)) = g(S)

Q78.If the system of linear equations 2x + 2y + 3z = a 3x −y + 5z = b x −3y + 2z = c where, a, b, care non- zero real numbers, has more than onc solution, then (1) b −c + a = 0 (2) b −c −a = 0 (3) a + b + c = 0 (4) b + c −a = 0

Q78.For 𝑥 𝜖 (0, 3 ), let 𝑓𝑥= 𝑔𝑥= tan𝑥 and ℎ𝑥= 1 - 𝑥2 . If ϕ𝑥= ( ho𝑓) og ) ( 𝑥) , then ϕ 𝜋 is equal to: 2 √𝑥, 1 + 𝑥2 3 𝜋 5𝜋 (1) tan⁡ (2) tan⁡ 12 12 7𝜋 11𝜋 (3) tan⁡ (4) tan⁡ 12 12

Q78.The set of all values of λ for which the system of linear equations x −2y −2z = λx x + 2y + z = λy −x −y = λz has a non-trivial solution : (1) is an empty set (2) contains more than two elements (3) is a singleton (4) contains exactly two elements

Q78.If the system of linear equations x −4y + 7z = g; 3y −5z = h ; −2x + 5y −9z = k is consistent, then: (1) g + h + 2k = 0 (2) g + 2h + k = 0 (3) 2g + h + k = 0 (4) g + h + k = 0

Q78.If the system of linear equations 𝑥- 2𝑦+ 𝑘𝑧= 1 2𝑥+ 𝑦+ 𝑧= 2 3𝑥- 𝑦- 𝑘𝑧= 3 has a solution 𝑥, 𝑦, 𝑧, 𝑧≠0, then 𝑥, 𝑦 lies on the straight line whose equation is: (1) 4𝑥- 3𝑦- 4 = 0 (2) 3𝑥- 4𝑦- 4 = 0 (3) 3𝑥- 4𝑦- 1 = 0 (4) 4𝑥- 3𝑦- 1 = 0

Q78.An ordered pair (α, β) for which the system of linear equations (1 + α)x + βy + z = 2 αx + (1 + β)y + z = 3 αx + βy + 2z = 2 has a unique solution, is : (1) (−3, 1) (2) (1, −3) (3) (2, 4) (4) (−4, 2) JEE Main 2019 (12 Jan Shift 1) JEE Main Previous Year Paper

Q78.The number of functions f from {1, 2, 3, … , 20} onto {1, 2, 3, … , 20} such that f(k) is a multiple of 3, whenever k is a multiple of 4 is: (1) 65 × (15)! (2) 5! × 6! (3) (15)! × 6! (4) 56 × 15

Q78.Let 𝜆 be a real number for which the system of linear equations 𝑥+ 𝑦+ 𝑧= 6, 4𝑥+ 𝜆𝑦- 𝜆𝑧= 𝜆- 2 and 3𝑥+ 2𝑦- 4𝑧= - 5 has infinitely many solutions. Then 𝜆 is a root of the quadratic equation: (1) 𝜆2 + 3𝜆- 4 = 0 (2) 𝜆2 - 𝜆- 6 = 0 (3) 𝜆2 - 3𝜆- 4 = 0 (4) 𝜆2 + 𝜆- 6 = 0 – 1 𝑦 𝑦

Q78.If the system of equations x + y + z = 5, x + 2y + 3z = 9, x + 3y + αz = β has inifinitely many solutions, then β −α equals (1) 8 (2) 21 (3) 5 (4) 18 Q79. ⎡−2 4 + d (sin θ) −2 ⎤ Let d ∈R, and A = 1 (sin θ) + 2 d , θ ∈[0, 2π]. If the minimum value of det(A) is ⎣ 5 (2 sin θ) −d (−sin θ) + 2 + 2d ⎦ 8, then a value of d is: + + (1) 2(√2 2) (2) 2(√2 1) (3) −5 (4) −7 . Let S be the set of points in the interval (−4, 4) at which f is not

Q78.If 𝑓𝑥= log𝑒 11 +- 𝑥𝑥, 1 + 𝑥2 (1) 𝑓𝑥2 (2) 2𝑓𝑥2 (3) – 2𝑓𝑥 (4) 2𝑓𝑥 sin𝑥 𝜋 then 𝑑𝑦 is equal to

Q79.If x = sin−1(sin 10) and y = cos−1 (cos 10), then y −x is equal to: (1) 10 (2) π (3) 0 (4) 7π

Q79.If 2𝑦= cot-1√3cos𝑥+ 2 ∀𝑥∈0, cos𝑥- √3sin𝑥 2, 𝑑𝑥 (1) 𝜋 - 𝑥 (2) 2𝑥- 𝜋 (3) 𝑥- 𝜋 (4) None of these 6 3 6

Q79.Let f : R →R be defined by f(x) = x , x ∈R. Then the range of f is 1+x2 (1) [−12 , 12 ] (2) R −[−1, 1] (3) R −[−12 , 12 ] (4) (−1, 1) −{0} and g(x) = |η(x)| + f(x ∣). Then, in the interval (−2, 2), g is:

Q79.For 𝑥∈𝑅- 0, 1, let 𝑓1𝑥= 𝑥, 𝑓2𝑥= 1 - 𝑥 and 𝑓3𝑥= 1 - 𝑥 be three given functions. If a function, 𝐽𝑥 satisfies 𝑓2𝑜𝐽𝑜𝑓1𝑥= 𝑓3𝑥 then 𝐽𝑥 is equal to: (1) 𝑓3𝑥 (2) 1 𝑥𝑓3𝑥 (3) 𝑓1𝑥 (4) 𝑓2𝑥

Q79.Let ∑10k=1 f(a + k) = 16(210 −1), where the function f satisfies f(x + y) = f(x)f(y) for all natural numbers x, y and f(1) = 2. Then the natural number 'a' is: (1) 3 (2) 16 (3) 4 (4) 2