IIT-JEE Mathematics Paper 2

Mathematics Online : It’s all about Mathematics pankaj.baluja@mathematicsonline.co.in IIT JEE Screening Exam Questions -2 -Visit us at: www.mathematicsonline.co.in MATHEMATICS ONLINE IIT JEE SCREENING QUESTIONS CHAPTER WISE 1. TRIGONOMETRIC RATIOS 1. For 0 < φ < < π/2, if x = , y = ∑φ, z = φ sin2n φ , then 2nn0∞=∑cos n 0sin ∞= 2n n 0cos ∞= ∑2n (a) xyz = xz + y (b*) xyz = xy + z (c) xyz = yz + x (d) None of these 2. If K = sin (π/18) sin (5π /18) sin (7π/18), then the numerical value of K is (a*) 1/8 (b) 1/16 (c) 1/2 (d) None of these 3. If A > 0, B > 0 and A = B = π/3, then the maximum value of tan A tan B is 3 (d) 1/3 (a) 1 (b*) 1/3 (c) 4. The expression 4433sinsin(3)⎤−α⎢⎥⎣⎦ 2 ⎡ π ⎛ ⎞ − α + π ⎜ ⎟ ⎝ ⎠ –2 6 6 sin sin (5 ) 2 ⎡ π ⎤ ⎛ ⎞ + α + π−α ⎜ ⎟ ⎢ ⎥ ⎝ ⎠ ⎣ ⎦ is equal to (a) 0 (b*) 1 (c) 3 (d) sin 4α + cos 6α 5. 3(sin x – cos x)4 + 6 (sin x + cos x)2 + 4 (sin6x + cos6x) = (a) 11 (b) 12 (c*) 13 (d) 14 6. sec2θ = 2 4xyy) + (x is true, if and only if-(a) x + y ≠ 0 (b*) x = y, x ≠ 0 (c) x = y (d) x ≠ 0, y ≠ 0 7. The number of values of x where the function f(x) = cos x + cos ( 2x) n r f 0 attains its maximum is-(a) 0 (b*) 1 (c) 2 (d) Infinite 8. Which of the following number(s) is rational -(a) sin 15º (b) cos 15º (c*) sin15º cos 15º (d) sin 15º cos 75º b = ∑ sinr θ every value of θ, then 9. Let n be an odd integer. If sin n θ = (a) b0 = 1, b1 = 3 (b*) b0 = 0, b1 = n (c) b0 = – 1, b1 = n (d) b0 = 0, b1 = n2 + 3n = 3 10. The function f(x) = sin4x + cos4 x increases if-38π (c) 38π < x < 58π (d) 58π < x < 34π 8π (b*) 4π < x < (a) 0 < x < 11. In a triangle PQR, ∠R = 2π. If tan P2 ⎛ ⎞ ⎝ ⎠ Q2 ⎛ ⎞⎟ ⎝ ⎠ ⎜ and tan ⎟⎜ are the roots of the equation Ax2 = bx + c = 0 (a ≠ 0), then-(a*) a + b = 0 (b) b + c = a (c) a + c = b (d) b = c 12. For a positive integer n, tan 2θ ⎛ ⎞⎟ ⎝ ⎠ ⎜ (1 + sec θ) (1 + sec 2θ) (1 + se 4 θ)… (1 + sec 2nθ). Then-Lot fn(θ) = Mathematics Online : It’s all about Mathematics pankaj.baluja@mathematicsonline.co.in IIT JEE Screening Exam Questions -3 -Visit us at: www.mathematicsonline.co.in MATHEMATICS ONLINE IIT JEE SCREENING QUESTIONS CHAPTER WISE (a) f2 16 π ⎛ ⎞⎟ ⎝ ⎠ ⎜ = 2 (b*)3f 1 32 π ⎛ ⎞ = ⎜ ⎟ ⎝ ⎠ (c) 464π f 0 ⎛ ⎞ = ⎜ ⎟ ⎝ ⎠ (d) None of these 13. Let f (θ) = sin θ (sin θ + sin 3θ). Then f(θ) (a) ≥ 0 only when θ ≥ 0 (b) ≤ 0 for all real θ (c*) ≥ 0 for all real θ (d) ≤ 0 only when θ ≤ 0 2π 14. If α + β = and β + γ = a, then tanα equals-(a) 2(tanβ + tanγ) (b) tan β + tan γ (c*) tan β + 2 tan γ (d) 2 tan β + tan γ 15. The maximum value of (cos α1). (cos α2)………. (cos αn), 2π Under the restrictions 0 ≤ α1, α2… αn ≤ and (cot α1). (cot α2). (cot α3)……(cot αn) = 1 is (a*) n /2 1 2 n 1 2 1 2n (b) (c) (d) 1 12 13 16. If θ & φ are acute angles such that sin θ = and cos φ = then θ + φ lies in-(a) ,⎛⎤⎜ (b*) 3 2 π π⎥ ⎝ ⎦ 2 , 2 3 π π ⎥ ⎝ ⎦ ⎛⎤⎜ (c) 2 5 , 2 3 π π ⎛ ⎞ ⎜ ⎟ ⎝ ⎠ 6π ⎛⎞⎜⎟ π ⎝ ⎠ (d) 1e 17. cos (α + β) = , cos (α – β) = 1 find no. of ordered pair of (α, β), – π ≥ α, β ≤ π (a) 0 (b) 1 (c) 2 (d) 4 ANSWER KEY Q.No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 Ans. a a b b c b b c b b a c c c a b d 2. TRIGONOMETRIC EQUATION 1. The number of solutions of the equation tan x + sec x = 2 cos x lying in the interval [0, 2π] is (a) 0 (b) 1 (c*) 2 (d) 3 2. Let 2 sin2 x + 3 sin x – 2 > 0 and x2 – x – 2 < 0 (x is measured in radians). Then x lies in the interval (a) 5 , 6 6 π π ⎝ ⎠ ⎛⎞⎜⎟ (b) 5 1, 6π ⎝ ⎠ ⎛⎞⎜⎟ (c) (– 1, 2) (d*) − , 2 6π ⎝ ⎠ ⎛⎞⎜⎟ 3. The number of all possible triplets (a1, a2, a3) such that a1 + a2 cos 2x + a3 sin2x = 0 for all x is (a) 0 (b) 1 (c) 2 (d*) infinite 4. The smallest positive root of the equation tan x – x = 0 lies on (a) 0, 2π ⎛ ⎞ ⎜ ⎟ ⎝ ⎠ (b) , 2π ⎞ π⎟ ⎝ ⎠ ⎛⎜ (c*) 3, 2π ⎛⎜ (d) ⎞ π ⎟ ⎝ ⎠ 32π, 2π ⎝ ⎠ ⎛⎞⎜⎟ Mathematics Online : It’s all about Mathematics pankaj.baluja@mathematicsonline.co.in IIT JEE Screening Exam Questions -4 -Visit us at: www.mathematicsonline.co.in MATHEMATICS ONLINE IIT JEE SCREENING QUESTIONS CHAPTER WISE 5. General value of θ satisfying equation tan2θ + sec 2 θ = 1 is (a) nπ (b) nπ + 3π (c) nπ + 3π (d*) all of these 6. The parameter, on which the value of the determinant 2 a d)x d)x + − + 1acos(pd)xcospxcos(psin(pd)xsinpxsin(p− does not depend upon is (a)a (b) d (c*) p (d) x 23π 32 7. The solution set of the system of equations : x + y = , cos x + cos y = , where x and y are real in: (a) a finite non-empty set (b*) null set (c) ∞ (d) none of these 8. The number of value of x in the interval [0, 5π] satisfying the equation 3 sin3x – 7 sin x + 2 = 0 is (a) 0 (b) 5 (c*) 6 (d) 10 9. The number of distinct real roots of sinxcosxcosxsinxcosxcosx cos x cos x sin x = 0 in the interval [–π/4, π/4] is-(a) 0 (b) 2 (c*) 1 (d) 3 10. The number of integral values of k for which the equation 7 cos x + 5 sin x = 2 k + 1 has a solution is-(a) 4 (b*) 8 (c) 10 (d) 12 ANSWER KEY Q.No. 1 2 3 4 5 6 7 8 9 10 Ans. c d d c d c b c c b 3. INVERSE TRIGONOMETRIC FUNCTIONS 5π 1. If sin–1x = , x ∈ (–1, 1), then cos–1x = (a) 310π 510π (b) (c) 310π − (d) 910π 2. tan(cos–1 x) is equals to-(a) 2 1 x x− (b) 2 x 1 x + (c) 2 1 x x+ (d) 21x − 3. If we consider only the principal values of the inverse trigonometric functions, then the value of tan 1cossin52−− 1 1 417 − ⎛ ⎞ ⎝ ⎠ ⎜⎟ is 29 3 (a) 29 3 (b) (c) 3 29 (d) 329 Mathematics Online : It’s all about Mathematics pankaj.baluja@mathematicsonline.co.in IIT JEE Screening Exam Questions -5 -Visit us at: www.mathematicsonline.co.in MATHEMATICS ONLINE IIT JEE SCREENING QUESTIONS CHAPTER WISE x x 1 2 2 π + + = 4. The number of real solution of tan–1 x(x 1 + ) + sin–1 is-(a) Zero (b) One (c) Two (d) Infinite 5. If sin–1 462xxx24−+−⎜ .............. ⎛ ⎞ ⎝ ⎠⎟ = 2π 2 for 0 < |x| < , then x equals (a) 12 12 (d) –1 (b) 1 (c) – 6. For which value of x, sin (cos–1 (x + 1)) = cos (tan–1x) (a) 1/2 (b) 0 (c) 1 (d) –1/2 ANSWER KEY Q.No. 1 2 3 4 5 6 Ans. a a d c b d 4. PROPERTIES OF TRIANGLE 1. If in a triangle ABC 2cosAcosB2cosCabc++ a b bc ca = + then the value of the angle A. (a) π/3 (b) π (c*) π /2 (d) π /6 2. In a ∆ABC , if cosAcosBab==cosCc and the side a = 2, then are a of the triangle is 3 2 (d*) 3 (a) 1 (b) 2 (c) 3. In a ∆ ABC, AD is the altitude from A. Given b > c, ∠ C = 23º and AD = 2 2 abc b, then ∠ B. c − (a) 67º (b*) 113º (c) 157º (d) None of these 4. The sides of a triangle inscribed in a given circle subtend angles α, β, γ at the centre. The minimum value of the A.M. of cos 2π⎛⎞⎜⎟⎝⎠ α + , cos 2π ⎛β + ⎜⎝ ⎠ 2π⎞ γ + ⎟ ⎝ ⎠ ⎞⎟ and cos ⎛⎜ is equal to (a) 3 2 3 2 (b*) – (c) 23 − (d) 2 , Let D divide BC internally in the ratio 1 : 3 Then sinsin BAD CAD ∠∠ equal to 3π 4π 5. In a triangle ABC, ∠ B = and ∠ C = 13 (a*) 16 (b) (c) 13 (d) 23 6. There exists at triangle ABC satisfying the conditions: Mathematics Online : It’s all about Mathematics pankaj.baluja@mathematicsonline.co.in IIT JEE Screening Exam Questions -6 -Visit us at: www.mathematicsonline.co.in MATHEMATICS ONLINE IIT JEE SCREENING QUESTIONS CHAPTER WISE 2π 2π 2π (a*) b sin A = a, A < or b sin A < a, A < , b > a (b) b sin A > a, A < 2π (d) None of these (c) b sin A > a, A < 7. If in a triangle PQR, sin P, sin Q and sin R are in A.P., then (a) the altitudes are in A.P. (b*) the altitudes are in H.P. (c) the medians are in G.P. (d) the medians are in A.P. 8. If the radius of circumcircle of an isosceles triangle PQR is equal to PQ (= PR), then the angle P is (a) 6π (b) 2π (c) 3π (d*) 23π 9. If the vertices P, Q, R of a triangle PQR and rational points, then which of the following points of the triangle PQR is (are) always rational point(s) ? (a*) Centroid (b) Incentre (c) Circumecentre (d) orthocentre 10. In a triangle PQR, ∠R = 2π. If tan P2 ⎟ ⎝ ⎠ ⎛⎞⎜and tan Q2 ⎛ ⎞⎟ ⎝ ⎠ ⎜ are the roots of the equation ax2 + bx + c = 0 (a ≠ 0), then (a*) a + b = c (b) b + c = a (c) a + c = b (d) b = c 11. In a ∆ABC, 2ac sin AB C 2 − + ⎛ ⎞ ⎜ ⎟ ⎝ ⎠ = (a) a2 + b2 – c2 (b*) c2 + a2 – b2 (c) b2 – c2 – a2 (d) c2 – a2 0– b2 12. If the angles of a triangle are in ratio 4 : 1 : 1 then the ratio of the longest side and perimeter of triangle is : (a) 1 2 3 + (b) 2 3 2 − (c*) 32 3 + (d) none of these 3 13. Of the sides a, b, c of a triangle are such that a : b : c : : 1 : : 2, them A : B : C is-(a) 3 : 2 : 1 (b) 3 : 1 : 2 (c) 1 : 3 : 2 (d*) 1 : 2 : 3 14. In any ∆ABC having sides a, b, c opposite to angles A, B, C respectively, then-A2 A2 B C 2− (a*) a sin B C 2− ⎛ ⎞⎟ ⎝ ⎠ ⎜ = (b – c) cos (b) a cos = (b – c) sin A2 B C 2+ B C 2+ A2 = (b + c) sin (d) a sin = (b + c) cos (c) a cos ANSWER KEY Q.No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Ans. c d b b a a b d a A b c d a 5. RADII OF CIRCLE 1. A regular polygon of nine sides, each of length 2 is inscribed in a circle. The radius of the circle is: (a*) cosec 9 π ⎛ ⎞⎟ ⎝ ⎠ 3π ⎛ ⎞⎟ ⎝ ⎠ ⎜ (b) cosec ⎜ (c) cot 9π ⎛ ⎞ ⎝ ⎠ 9π ⎛ ⎞ ⎝ ⎠ ⎜⎟ ⎜⎟ (d) tanMathematics Online : It’s all about Mathematics pankaj.baluja@mathematicsonline.co.in IIT JEE Screening Exam Questions -7 -Visit us at: www.mathematicsonline.co.in MATHEMATICS ONLINE IIT JEE SCREENING QUESTIONS CHAPTER WISE 2. In a triangle ABC, a : b : c = 4 : 5 : 6. The ratio of the radius of the circumcircle to that of the incircle is (a*) 16/7 (b) 7/16 (c) 16/3 (d) none of these 3. Let A0A1A2A3A4A5 be a regular hexagon inscribed in a circle of unit radius. Then the product of the lengths of the line segments A0A1,A0A2, and A0A4 is-(a) 3/4 (b) 3 (c*) 3 (d) 3 33 /2 2π 4. In a triangle ABC, let ∠C = . If r is the in radius and R is the circumradius of the triangle, then 2(r + R) is equal to-(a*) a + b (b) b + c (c) c + a (d) a + b + c 5. Which of the following pieces of data does ΝΟΤ uniquely determine an acute angled triangle ABC (R being the radius of the circumcircle)-(a) a, sin A, sin B (b) a, b, c (c) a, sin B, R (d*) a, sin A, R 6. In any equilateral ∆, three circles of radii one are touching to the sides given as in the figure then area of the ∆ (a*) 64 (b) 3 + 12 8 3 + (c) 74 (d) 3 + 72 +43 ANSWER KEY Q.No. 1 2 3 4 5 6 Ans. a a c a d a 6. COMPLEX NUMBER 1. The equation not representing a circle is given by-(a) Re 1 z 1 z +− ⎝ ⎠ ⎛⎞⎜⎟ = 0 (b) zziziz 1 0 + − + = (c) are z1z1⎜+2 − π ⎛ ⎞=⎟⎝ ⎠ (d*) z 1 z 1 −+ = 1 2. If z is a complex number such that z ≠ 0 and Re(z) = 0, then-(a) Re(z2) = 0 (b*) Im (z2) = 0 (c) Re(z2) = Im(z2) (d) none of these 3. If α and β are different complex numbers with |β| = 1, then 1β − α − αβ is equal to-(a) 0 (b) 1/2 (c*) 1 (d) 2 4. The smallest positive integer n for which (1 + i)2n = (1 – i)2n is -(a) 4 (b) 9 (c*) 2 (d) 12 5. If β and β are two fixed non-zero complex numbers and ‘z’ a variable complex number. If the lines zzα+α 1 0 + = and zzβ+β 1 0 − = are mutually perpendicular, then-(a) 0αβ+ (b) αβ = 0 − αβ = αβ (c) 0 αβ−αβ = (d*) 0 αβ+ αβ = Mathematics Online : It’s all about Mathematics pankaj.baluja@mathematicsonline.co.in IIT JEE Screening Exam Questions -8 -Visit us at: www.mathematicsonline.co.in MATHEMATICS ONLINE IIT JEE SCREENING QUESTIONS CHAPTER WISE 6. If z1 = 8 + 4i, z2 = 6 + 4i and arg 12zzzz− 4 ⎛ ⎞ π =⎟⎝ ⎠ ⎜−, then z satisfies-2 (a) |z – 7 – 4i| = 1 (b*) |z – 7 – 5i| = 3 (c) |z – 4i| = 8 (d) |z – 7i| = 7. If ω is an imaginary cube root of unity, then the value of sin 10( 3)4 π ⎡ ⎤ π− ω+ω2⎢ ⎥ ⎣ ⎦ is-(a) 3 2 12 − − (b) (c*) 12 (d) 32 3 8. If z1, z2, z3 are vertices of an equilateral triangle inscribed in the circle |z| = 2 and If z1 = 1 + I , then-3 3 (a*) z2 = – 2, z3 = 1 – i (b) z2 = 2, z3 = 1 – i 3 3 3 + (c) z2 = – 2, z3 = – i (d) z2 = – 1 – i , z3 = – 1 – I 9. If ω (≠ 1) is a cube root of unity and (1 + ω)7 = A + B ω, then A & b are respectively the numbers (a) 0 , 1 (b*) 1, 1 (c) 1. 0 (d) –1, 1 10. Let z & ω be two non zero compelx numbers such that |z| = |ω| and Arg z + Arg ω = π, then z equal: (a) ω (b) – ω (c) ω (d*) – ω 11. Let z & ω be two complex numbers such that |z| ≤ 1, |ω| ≤ 1 and |z + iω| = |z i | − ω = 2, then z equals: (a) 1 or i (b) i or – i (c*) 1 or –1 (d) i or – 1 12. If (ω ≠ 1) is a cube root of unity then 2 2 2 1 1 ω ω − − 11i1i1ii1++ω−−−−+ω− = (a*) 0 (b) 1 (c) (d) ω 13. The value of the expression 1.(2 – ω). (2 – ω2) + 2. (3 – ω) (3 – ω2) +………+ (n – 1) (n – ω) (n – ω2), where ω is an imaginary cube root of unity is-(a) 2 1) 2+ ⎛ ⎞⎟ ⎝ ⎠ n(n⎜ (b*) 2 n ⎛ ⎞ n(n1)2+ − ⎜ ⎟ ⎝ ⎠ (c) 2 n ⎛ ⎞ +⎟⎝ ⎠ n(n1)2+⎜ (d) none of the above 14. 6i43203−3i 1 i 1i − (a) 128 ω (b) – 128 ω (c) 128 ω = x + iy, then (a) x = 3, y = 1 (b) x = 1, y = 3 (c) x = 0, y = 3 (d*) x = 0, y = 0 15. If ω is an imaginary cube root of unity, then (1 + ω – ω 2)7 equals 2 (d*) – 128 ω2 Mathematics Online : It’s all about Mathematics pankaj.baluja@mathematicsonline.co.in IIT JEE Screening Exam Questions -9 -Visit us at: www.mathematicsonline.co.in MATHEMATICS ONLINE IIT JEE SCREENING QUESTIONS CHAPTER WISE 16. The value n n1 i ) + + = of the sum 13n1(i=∑, where 1 − equals (a) i (b*) i – 1 (c) – i (d) = 0 17. = 1 − If I , then 4 + 5 334⎛⎞⎜⎟⎜⎟⎝⎠ + 3 1 3 2 2 − − + 365 1 3 2 2 ⎛ ⎞ ⎜ ⎟ ⎜ ⎟⎠ is equal to i −+⎝ 3 + I 3 (c*) i 3 – i (b) – 1 (d)(a) 1 – 3 18. If z1, z2, z3 are complex numbers such that |z1| = |z 2| = |z3| = 1211zz 31 1 z + + = , then |z1 + z2 + z3| is-(a*) equal to 1 (b) less than 1 (c) greater than 3 (d) equal to 3 9. If arg (z) < 0, then arg (–z)–arg(z) = 1 (a*) π (b) – π (c) 2π − (d) 2 π 20. The complex numbers z1,z2 and z3 satisfying 1323zzzz−−=− 1 i 3 2 are the vertices of a triangles which is (a) of area zero (b) right angled isosceles 1. If z1 and z2 be the nth roots of unity which subtend right angle at the origin. Then n must be of the form 2. For all complex numbers z1,z2 satisfying |z1| = 12 and |z3 – 3 – 4i| = 5, then minimum value of (b*) 2 (c) 7 (d) 17 23. Let ω + i (c*) equilateral (d) obtuse angled isosceles 2 (a) 4 k + 1 (b) 4k + 2 (c) 4k + 3 (d*) 4k 2 |z1 – z2| is -(a) 0 3 /2. Then the value of the = – 1/2 determinant 2 2 2 41 11111−ω ω ω ω is -(a) 3ω (b*) 3 ω (ω – 1) (c) 3 ω (d) 3ω (1 – ω) . If |z| = 1, z ≠ – = 2 z 1 z 1 −+ then real part of w = ? 241 and w (a) 2 11| − |z+(b) 2 1 | z 1| + (c) 2|z1|+2 (d*) 0 5. If ω is cube root of unity (ω ≠ 1) then the least value of n, where n is positive integer such that *) 3 (c) 2 (d) 3 6. a, b, c are variable integers not all equal and w ≠ 1, w is cube root of unity, then minimum value of x = |z + bw + cw2| is 2 (1 ω2)n = (1 + ω4)n is (a) 2 (b 2 (a) 0 (b*) 1 (c) 2 (d) 3 Mathematics Online : It’s all about Mathematics pankaj.baluja@mathematicsonline.co.in IIT JEE Screening Exam Questions -10 -Visit us at: www.mathematicsonline.co.in MATHEMATICS ONLINE IIT JEE SCREENING QUESTIONS CHAPTER WISE 27. Four points P(–1, 0) Q (1, 0), R (2 – 1, 2), S ( 2 – 1, – 2 ) are given on a complex plane then equation of the locus of the shaded region excluding the boundaries 4π 2π (b) |z + 1| > 2 &| arg (2 + 1) | < (a*) |z + 1| > 2 & arg (z + 1) < 4π 2π (d) |z – 1| > 2 &| arg (2 – 1)| < (c) |z – 1| > 2 & | arg (z – 1)| < ANSWER KEY Q.No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Ans. d b c c d b c a b d c a b d d Q.No. 16 17 18 19 20 21 22 23 24 25 26 27 Ans. b c a a c d b b d b b a 7. PROGRESSIONS 1. Let an be nth term of the G.P. of positive numbers. Let 100n1a= 2n =α ∑ 100 2n 1 n 1 a − = and = β ∑ α ≠ β , such that then the common ratio is-(a*) α/β (b) β/α (c*) (/) α β (d) (/) β α 120 = cos 2. If the sum of first n natural numbers is 1/5 times the sum of their squares, then the value of n is-(a) 5 (b) 6 (c*) 7 (d) 8 3. If ratio of H.M. and G.M. between two positive numbers a and b (a > b) is 4 : 5, then a : 5, then a : b is -(a) 1 : 1 (b) 2 : 1 (c*) 4 : 1 (d) 3 : 1 4. If f(x) is a function satisfying f(x + y) = f(x) f(y) for all x, y ∈ N such that f(1) = 3 and ,Then the value of n is-nx1f(x)=∑ (a*) 4 (b) 5 (c) 6 (d) None of these 5. log3 2, log6 2 and log12 2 are in-(a) A.P. (b) G.P. (c*) H.P. (d) None of these 6. For 0 < φ < π/2 if x = 2nn0∞= φ ∑, 2nin n 0 y s ∞= = φ ∑ 2n 2n n 0 z cos sin ∞= ; = φ φ ∑ , the-(a) xyz = xz + y (b*) zyz = xy + z (c) xyz = yz + x (d) None of these 7. If ln (a + c), ln(c – a), ln( a – 2b + c) are in A.P., then-(a) a, b, c are in A.P. (b) a2, b2, c2 are in A.P. (c) a, b, c are in G.P. (d*) a, b, c are in H.P. 8. For a real number x,[x], denotes the integral part of x. The value of 11112....221002100⎡⎤⎡⎤⎡⎤++++++⎢⎥⎢⎥⎢⎥⎣⎦⎣⎦⎣⎦ 1 99 2 100 ⎡ ⎤ + ⎢ ⎥ ⎣ ⎦is (a) 49 (b*) 50 (c) 48 (d) 51 Mathematics Online : It’s all about Mathematics pankaj.baluja@mathematicsonline.co.in IIT JEE Screening Exam Questions -11 -Visit us at: www.mathematicsonline.co.in MATHEMATICS ONLINE IIT JEE SCREENING QUESTIONS CHAPTER WISE 12 9. If the sum of n terms of an AlP. Is nP + n (n – 1) Q then its common difference is-(a) P + Q (b) 2P + 3Q (c) 2Q (d*) Q 10. If p,q, r in A.P. and are positive, the roots of the quadratic equation px2 + qx + r = 0 are all real for-(a*) r7 3 p −≥ (b) p 7 4 3 r − < (c) all p and r (d) No. p and r 11. If cos (x – y), cos x and cos (x + y) are in H.P., then cos x sec (y/2) equals-2 (d) None of these (a) 1 (b) 2 (c*) 12. If x be the AM and y,z be two GM’s between two positive numbers, then 3 3 y z xyz + is equal to-(a) 1 (b*) 2 (c) 3 (d) 4 13. Let Tr be the rth term of an A.P., for r = 1, 2, 3,……..if for some positive integers m, n we have Tm = 1n 1m and Tn = , then Tmn equals-(a) 1 mn 1 1 m n + (c*) 1 (d) 0 (b) 14. If x > 1, y > 1, z > 1 are in G.P., then 11,,1nx1ny++ll 1 1 n z +l are in-(a) A.P. (b) H.P. (c*) G.P. (d) None of these 15. Let a1, a2,…..a10 be in A.P. and h1, h2,…….h10 be in H.P. If a1 = h1 = 2 and a10 = h10 = 3, then a4 h7 is-(a) 2 (b) 3 (c) 5 (d*) 6 16. The harmonic mean of the root of the equation ()()5 2 5 2 x 4 + − + x + 8 + 5 = 0 is-(a) 2 (b*) 4 (c) 6 (d) 8 17. If x1, x2, x3 as well as y1, y2, y3 and in G.P. with the same common ratio, then the points (x1, y1), (x2, y2) and (x3, y3-) (a*) lie on a straight line (b) lie on an ellipse (c) lie on a circle (d) are vertices of a triangle 18. The sum of the integers from 1 to 100 which are not divisible by 3 or 5 is-(a) 2489 (b) 4735 (c) 2317 (d*) 2632 34 19. Consider an infinite geometric series with first term a and common ratio r. If its sum is 4 and the second term is , then-(a) a = 74, r = 37 38 (b) a = 2, r = (c) a = 32, r = 12 14 (d*) a = 3, r = 20. Let α, β be the roots of x2 – x + p = 0 and γ, δ be the roots of x2 – 4x + q = 0. If α, β, γ, δ are in G.P., then the integral values of p and q respectively, are-(a*) –2, – 32 (b) – 2, 3 (c) – 6, 3 (d) – 6,–32 Mathematics Online : It’s all about Mathematics pankaj.baluja@mathematicsonline.co.in IIT JEE Screening Exam Questions -12 -Visit us at: www.mathematicsonline.co.in MATHEMATICS ONLINE IIT JEE SCREENING QUESTIONS CHAPTER WISE 21. Let the positive numbers a, b, c, d be in A.P. Then abc, abd, acd, bcd are-(a) Not in A.P./G.P./H.P. (b) in A.P. (c) in G.P. (d*) in H.P. 22. If the sum of the first 2n terms of the A.P. 2, 5, 8,….. is equal to the sum of the first n terms of the A.P. 57, 59, 61,….. then n equals-(a) 10 (b) 12 (c*) 11 (d) 13 23. If a1, a2,……,an are positive real numbers whose product is a fixed number c, then the minimum value of a1 + a2 +………+ an –1 + an is-(a*) n (c)1/n (b) (n + 1)c1/n (c) 2nc1/n (d) (n + 1) (2n)1/n 24. An infinite G.P., with first term x & sum of the series is 5 then-(a) x ≥ 10 (b*) 0 < x < 10 (c) x < – 10 (d) – 10 < c < 0 ANSWER KEY Q.No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Ans. a c c a c b d b d a c b c b d Q.No. 16 17 18 19 20 21 22 23 24 Ans. b a d d a d c a b 8. PERMUTATION & COMBINATION 1. If n is an integer between 0 an d21; then the minimum value of n ! (21 – n)! is-(a) 9! 12! (b) 10! 11! (c) 20! (d) 2! 2. The number of divisors of 9600 including 1 and 9600 are-(a) 60 (b) 58 (c) 48 (d) 46 3. A polygon has 44 diagonals, then the number of its sides are-(a) 11 (b) 7 (c) 8 (d) none of these 4. An n-digit number is a positive number with exactly n digits. Nine hundred distinct n-digit numbers are to be formed using only the three digits 2, 5, and 7. The smallest value of n for which this is possible is-(a) 6 (b) 7 (c) 8 (d) 9 5. How many different nine digit numbers can be formed from the number 223355888 by rearranging its digits so that odd digits occupy even position ? (a) 60 (b) 36 (c) 160 (d) 180 6. nC1 + 2nCr + 1 + nCr + 2 is equal to (2 ≤ r ≤ n) (a) 2. nCr+2 (b) n+1Cr+1 (c) n + 2Cr + 2 (d) none of these 7. The number of arrangement of the letters of the word BANANA in which the two N’s do nto appear adjacently is-(a) 40 (b) 60 (c) 80 (d) 100 8. No. of points with integer coordinates lie inside the triangle whose vertices are (0, 0), (0, 21), (21, 0) is : (a) 190 (b) 185 (c) 210 (d) 230 9. A rectangle has sides of (2m – 1) & (2n – 1) units as shown in the figure composed of squares having edge length one unit then no. of rectangles which have odd unit length (a) m2 – n2 (b) m (m + 1) n (n + 1) (c) 4m + n – 2 (d) m2n2 Mathematics Online : It’s all about Mathematics pankaj.baluja@mathematicsonline.co.in IIT JEE Screening Exam Questions -13 -Visit us at: www.mathematicsonline.co.in MATHEMATICS ONLINE IIT JEE SCREENING QUESTIONS CHAPTER WISE ANSWER KEY Q.No. 1 2 3 4 5 6 7 8 9 Ans. b c a b a c a a d 9. BINOMIAL THEOREM 1. If the sum of the coefficients in the expansion of (α2x2 – 2αx + 1)51 vanishes, then the value of α is-(a) 0 (b) –1 (c*) 1 (d) –2 2. The expansion [x + (x3 – 1)1/2]5 = [x – (x3 – 1)1/2]5 is a polynomial of degree-(a) 5 (b) 6 (c*) 7 (d) 8 3. If the rth term in the expansion of (x/3 – 2/x2)10 contains x4, then r is equal to-(a) 2 (b*) 3 (c) 4 (d) 5 4. The coefficient of x53 in the expansion is -1001001mm0C(x3)=−∑ 00 m m 2 − 2n 1)C , − (a) 100C47 (b) 100C53 (c*) – 100C53 (d) – 100C100 5. The value of C0 + 3C1 + 5C2 + 7C3 +……..+ (2n + 1) cn is equal to-(a) 2n (b) 2n + n.2n – 1 (c*) 2n. (n + 1) (d) None of these 6. The largest term in the expansion of (3 + 2x)50 where x = 1/5 is-(a) 5th (b) 51th (c*) 6th and 7th (d) 8th 7. where n is an even integer is 222012CCC..........(−+ (a) 2nCn (b) (–1)n 2nCn (c) (–1)n 2nCn –1 (d*) None of these 8. The co-efficient of the term independent of x in the expansion of 10 2 x 3 3 2x ⎛ ⎞ ⎝ ⎠ +⎜⎟⎜⎟ is (a) 9/4 (b) 3/4 (c*) 5/4 (d) 7/4 9. The sum of the rational terms in the expansion of ( )10 1/5 2 3 + is-(a*) 41 (b) 42 (c) 40 (d) 43 10. If an = n n r 0 = ∑11C then n n r 0 1 rC = ∑ equals-(a) (n – 1) an (b) nan (c*) 1/2 nan (d) None of these 11. If n is an odd natural number, then r n r 0 r ( 1) C − ( ) n 1 r 2 − − + n 1 1 1 + − n=∑ equal (a*) 0 (b) 1/n (c) n/2n (d) none of these 12. If in the expansion of (1 + x)m (1 – x)n, the coefficients of x and x2 are 3 and -6 respectively, then m is-(a) 6 (b) 9 (c*) 12 (d) 24 13. For 2 ≤ r ≤ n, = ……… ()()nnrr2+ (a) () (b*) 2( ) n 1 r 1 + + (c) 2( ) n2r (d) + ( ) n2r +Mathematics Online : It’s all about Mathematics pankaj.baluja@mathematicsonline.co.in IIT JEE Screening Exam Questions -14 -Visit us at: www.mathematicsonline.co.in MATHEMATICS ONLINE IIT JEE SCREENING QUESTIONS CHAPTER WISE ab 14. In the binomial expansion of (a – b)n, n ≥ 5, the sum of the 5th and 6th terms is zero. Then equals-(a) n 5 6− n 4 5− 5 n 4 − (b*) (c) (d) 6n 5 − 15. Find coefficient of t24 in the expansion of (1 + t2)12 (1 + t12) (1 + t24) is (a*) 12C6 + 2 (b) 12C6 + 1 (c) 12C6 + 3 (d) 12C6 16. If n-1Cr = (k2 – 3) nCr + 1, then k lies between (a) (– ∞, – 2) (b) (2, ∞) (c) 3,3⎡ ⎤ − ⎣ ⎦ (d*) (3, 2⎤⎦ ( ) 30 1 ) 30 11 17. ( – () ( +…………..+300)3010() ( ) 30 30 ) 30 10 ) 60 20 3020 = (a*) ( (b) ( (c)( ) 31 10 (d) ( ) 31 11 ANSWER KEY Q.No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 Ans. c c b c c c d c a c a c b b a d a 10. QUADRATIC EQUATION 1. If satisfies the equation x2 – 9x + 8 = 0, find the value of 246{(sinxsinxsinx..e++ ..... ) n 2} ∞ l cocosx s xsin x + , 0 < x < 2π (a*) 11 3 + (b) 1 1 3 − (c) 2 1 2 − (d) None of these 2. If the roots of the equation (x – a) (x – b) – k = 0 be c & d then find the equation whose roots are a & b. (a*) (x – c) (x – d) + k = 0 (b) (x + c) (x – a) + k = 0 (c) (x – c) + (x – a) = 0 (d) None of these 3. The set of values of p for which the roots of the equation 3x2 = 2x + p (p – 1) = 0 are of opposite sign is-(a) (– ∞, 0) (b*) (0, 1) (c) (1, ∞) (d) (0, ∞) 4. Let p,q ∈ {1, 2, 3, 4}. The number of equations of the form px2 + qx + 1 = 0 having real roots is-(a) 15 (b) 9 (c*) 7 (d) 8 5. Let α and β be the roots of the equation x2 + x + 1 = 0. The equation whose roots are α19, β7 is (a) x2 – x – 1 (b) x2 – x + 1 = 0 (c) x2 + x – 1 = 0 (d*) x2 + x + 1 = 0 6. If p,q are roots of the equation x2 + px + q = 0, then-(a) p = 1 (b) p = – 2 (c*) p = 1 or 0 (d) p = – 2 or 0 7. Let p and q are roots of the equation x2 – 2x + A = 0 and r, s are roots of x2 – 18x + B = 0 if p < q < r < s are in A.P. then the value of A and B are-(a) –7, –33 (b) –7, –37 (c*) –3, 77 (d) None of these 8. The equation (x1)(x1)(+−−= 4x 1) − has-(a*) No solution (b) One solution (c) Two solutions (d) More than 2 soluions Mathematics Online : It’s all about Mathematics pankaj.baluja@mathematicsonline.co.in IIT JEE Screening Exam Questions -15 -Visit us at: www.mathematicsonline.co.in MATHEMATICS ONLINE IIT JEE SCREENING QUESTIONS CHAPTER WISE 9. The sum of all real roots of the equation |x – 2|2 + |x – 2| – 2 = 0 is (a) 2 (b*) 4 (c) 1 (d) None of these 10 The number of values of x in the interval [0, 5π] satisfying the equation 3sin2x – 7 sinx + 2 = 0 is-(a) 0 (b) 5 (c*) 6 (d) 10 11. If the roots of the equation x2 – 2ax + a2 + a – 3 = 0 are real and less than 3, then-(a*) a < 2 (b) 2 ≤ a ≤ 3 (c) 3 < a ≤ 4 (d) a > 4 5 12. The harmonic mean of the roots of the equation (5 x2 – 2) + (4 5) + x + 8 + 2 = 0 is-(a) 2 (b*) 4 (c) 6 (d) 8 13. In a ∆PQR, ∠R =2π. If tan P2 and tan Q2 are the roots of the equation ax2 + bx + c = 0 (a ≠ 0), then-(a*) a + b = c (b) b + c = a (c) c + a = b (d) b = c 14. For the equation 3x2 + px + 3 = 0, p > 0, if one of the roots is square of the other, then p is equal to-(a) 1/3 (b) 1 (c*) 3 (d) 2/3 15. If α and β (α < β), are the roots of the equation x2 + bx + c = 0, where c < 0 < b, then (a) 0 < α < β (b*) α < 0 < β < |α| (c) α < β < 0 (d) α < 0 < |α| < β 16. If b > a, then the equation (x – a) (x – b) – 1 = 0, has-(a) both roots in [a, b] (b) both roots in (– ∞, a) (c) both roots in (b, + ∞) (d*) one root in (– ∞, a) and other in (b, + ∞) 17. Let α, β be the roots of x2 – x + p = 0 and γ,δ be the roots of x2 – 4x + q = 0. If α,β,γ,δ are in G.P., then the integral values of p and q respectively, are-(a*) –2, – 32 (b) –2, 3 (c) –6, 3 (d) –6, –32 18. The set of all real numbers x for which x2 – |x + 2| + x > 0, is-(a) (– ∞, – 2) ∪ (2, ∞) (b*) (,2) ( 2, ) −∞−∪ ∞ (c) (– ∞, – 1) ∪ (1, ∞) (d) (2, ) ∞ 19. If one root of the equation x2 + px + q = 0 is square of the other then for any p & q, it will satisfy the relation-(a*) p3 – q (3p – 1) + q2 = 0 (b) p3 – q (3p + 1) + q2 = 0 (c) p3 + q (3p – 1) + q2 = 0 (d) p3 + q (3p + 1) + q2 = 0 20. Let x2 + 2ax + 10 – 3a > 0 for every real value of x, then-(a) a > 5 (b) a < – 5 (c*) – 5 < a < 2 (d) 2 < a < 5 21. α, β are roots of equation ax2 + bx + c = 0 and α + β, α2 + β2 , α2 + β3 are in G.P., ∆ = b2 – 4ac, then (a) ∆b = 0 (b) bc ≠ 0 (c) ∆ ≠ 0 (d*) ∆ = 0 ANSWER KEY Q.No. 1 2 3 4 5 6 7 8 9 10111213141516171819 20 21 Ans. a a b c d c c a b c a b a c b d a b a c d 11. LOGARITHMS & MODULUS FUNCTION 1. The domain of the function 0.5 g x) (lo is-Mathematics Online : It’s all about Mathematics pankaj.baluja@mathematicsonline.co.in IIT JEE Screening Exam Questions -16 -Visit us at: www.mathematicsonline.co.in MATHEMATICS ONLINE IIT JEE SCREENING QUESTIONS CHAPTER WISE (a*) (1, ∞) (b) (0, ∞) (c) (0, 1) (d) (0.5, 1) 2. The number log2 7 is-(a) an integer (b) a rational number (c*) an irrational number (d) a prime number 3. Find the no. of solution log4 (x – 1) = log2 (x – 3) (a) 3 (b*) 1 (c) 2 (d) 0 4. For all x ∈ (0, 1) (a) ex < 1 + x (b*) loge (1 + x) < x (c) sin x > x (d) loge x > x 5. The set of all real numbers x for which x2 – |x + 2| + x > 0, is-(a) (– ∞, – 1) ∪ (2, ∞) (b*) (– ∞, – 2) ∪ ( 2 , ∞) (c) (– ∞, – 1) ∪ (1, ∞) (d) (2 , ) ∞ ANSWER KEY Q.No.1 2 3 4 5 Ans.c c b b b 12. POINT 1. If the sum of the distances of a point from two perpendicular lines in a plane is 1, then its locus is (a*) square (b) circle (c) straight line (d) two intersecting lines 2. If P (1, 0), Q (–1, 0) and R (2, 0) are three given points, then the locus of S satisfying the relation SQ2 + SR2 = 2SP2 is (a) a st. line || to x-axis (b*) a circle thro’ the origin (c) a circle with centre at the origin (d) a st. line || to y-axis 3. The orthocenter of the triangle with vertices (31)2⎡⎤−⎢⎥⎣⎦ 2, , 1122⎛⎞ , −⎜⎟⎝⎠ and 1 2 2 ⎛ ⎞ − ⎜ ⎟ ⎝ ⎠ is-(a) 3 3 , 2 6−3⎡⎤⎢⎥⎣⎦ (b*) 1 2, 2 ⎡ ⎤ −⎢⎥ ⎣ ⎦ (c) 5 3 2 4 4 ,⎡ ⎤ − −⎢ ⎥ ⎣ ⎦ (d) 1 1 , 2 2 ⎡ ⎤ − ⎢ ⎥ ⎣ ⎦ 4. The orthocenter of the triangle formed by the lines xy = 0 and x + y = 1 is (a) 1 1 , 2 2 ⎛ ⎞ ⎜ ⎟ ⎝ ⎠ 1 1 , 3 3 ⎛ ⎞ ⎜ ⎟ ⎝ ⎠ 1 1 , 4 4 ⎛ ⎞ ⎝ ⎠ ⎜⎟ (b) (c*) (0, 0) (d) 5. The diagonals of parallelogram PQRS are along the lines x + 3y = 4 and 6x – 2y = 7. Then PQRS must be a (a) rectangle (b) square (c) cyclic quadrilateral (d*) rhombus 6. If the vertices P, Q, R of a triangle PQR are rational points, which of the following points of the triangle PQR is (are) not always rational points (s) ? (a) Centroid (b*) Incentre (c) Circumcentre (d) Orthocentre 7. If P (1, 2), Q (4, 6) R (5, 7) and S(a, b) are the vertices of a parallelogram PQRS, then (a) a = b, b = 4 (b) a = 3, b = 4 (c*) a = 2, b = 3 (d) a = 3 , b = 5 8. If x1, x2, x3 as well as y1, y2, y3 are in G.P. wih the same common ratio, then the points (x1, y1), (x2, y2) and (x3, y3) Mathematics Online : It’s all about Mathematics pankaj.baluja@mathematicsonline.co.in IIT JEE Screening Exam Questions -17 -Visit us at: www.mathematicsonline.co.in MATHEMATICS ONLINE IIT JEE SCREENING QUESTIONS CHAPTER WISE (a*) lie on straight line (b) lie on an ellipse (c) lie on a circle (d) are vertices of a triangle (1, (0, 0) and (2, 0) is , 3) 9. The incentre of the triangle with vertices (a) 3 1, 2 ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ 2 3 , 3 2 ⎛ ⎞⎟⎟⎝ ⎠ (b) 2 1 , 3 ⎛ ⎞ ⎜ ⎟ ⎝ ⎠ ⎜⎜ (d*) 1 1, 3 ⎛ ⎞ ⎝ ⎠ ⎜⎟ 3 (c) 10. Orthocentre of the triangle whose vertices are A (0, 0), B (3, 4) &C (4, 0) is: (a*) 33, 4 ⎛ ⎞ ⎝ ⎠ 5 3, 4 ⎛ ⎞ ⎝ ⎠ ⎜⎟ (b) ⎜⎟ (c) (3, 12) (d) (2, 0) ANSWER KEY Q.No 1 2 3 4 5 6 7 8 9 10Ans. a d b c d b c a d a 13. STRAIGHT LINE 1. The equation of the lines through the points (2, 3) and making an intercept of length 2 units between the lines y + 2x = 3 and y + 2x + 5 are (a) x + 3 = 0 (b) y – 2 = 0 (c*) x – 2 = 0 (d) None of these 3x + 4y = 12 4x – 3y = 6 3x + 4y = 18 2. let the algebraic sum of the perpendicular distances from the points A (2, 0) (0, 2) C(1, 1) to a variable line be zero. Then all such lines: (a) passes through the point (–1, 1) (b*) passes through the fixed point (1, 1) (c) touches some fixed circle (d) None of these 3. If one of the diagonals of a square is along the line x = 2y and one of its vertices is (3, 0) then its sides through this vertex are given by the equations (a*) y – 3x + 9 = 0, 3y + x – 3 = 0 (b) y + 3x + 9 = 0, 3y + x – 3 = 0 (c) 3x2 – 3y2 + 8xy + 10x + 15y + 20 = 0 (d) 3x2 – 3y2 – 8xy – 10x – 15y – 20 = 0 4. All points lying inside the triangle formed by the points (1, 3), (5, 0), (–1, 2) satisfy: (a*) 3x + 2y ≥ 0 (b) 2x + y – 13 ≥ 0 (c) –2x + y ≥ 0 (d) None of these 5. Let PQR be a right angled isosceles triangle, right angled at P(2, 1). If the equation of the line QR is 2x + y = 3, then the equation representing the pair of lines PQ and PR is-(a) 3x2 – 3y2 + 8xy + 20x + 10y + 25 = 0 (b*) 3x2 – 3y2 + 8xy – 20x – 10y + 25 = 0 (c) 3x2 – 3y2 + 8xy + 10x + 15y + 29 = 0 (d) 3x2 – 3y2 – 8xy – 10x – 15y – 20 = 0 6. Let PS be the median of the triangle with vertices P(2, 2), Q(6, –1) and R(7, 3). The equation of the line passing through (1, –1) and parallel to PS is-(a) 2x – 9y – 7 = 0 (b) 2x – 9y – 11 = 0 (c)2x + 9y – 11 = 0 (d*) 2x + 9y + 7 = 0 7. Find the number of integer value of m which makes the x coordinates of point of intersection of lines. 3x + 4y = 9 and y = mx + 1 integer. (a*) 2 (b) 0 (c) 4 (d) 1 8. Area of the parallelogram formed by the linen y = mx, y = mx + 1, y = nx, y = nx + 1 is (a) |m + n|/(m – n)2 (b) 2/|m + n| (c) 1/|m + n| (d*) 1/|m – n| Mathematics Online : It’s all about Mathematics pankaj.baluja@mathematicsonline.co.in IIT JEE Screening Exam Questions -18 -Visit us at: www.mathematicsonline.co.in MATHEMATICS ONLINE IIT JEE SCREENING QUESTIONS CHAPTER WISE 9. A straight line through the origin O meets the parallel lines 4x + 2y = 9 and 2x + y + 6 = 0 at the points P and Q respectively. Then the point O divides the segment PQ in the ratio-(a) 1 : 2 (b*) 3 : 4 (c) 2 : 1 (d) 4 : 3 10. Let P = (–1, 0), Q = (0, 0) and R =(3, 3 3) be three points. Then the equation of the bisector of the angle PQR is-3 (a) (3x + y = 0 (b) x + /2) x = 0 (c*) 3x + y = 0 (d) x + ( 3/2) y = 0 11. Let 0 α < π/2 be a fixed angle. If P = (cos θ, sin θ) and Q = (cos (α – θ)), sin (α – θ)) then Q is obtained from P by-(a) clockwise rotation around origin through an angle α (b) anticlockwise rotation around origin through an angle α (c) reflection in the line through origin with slope tan α (d*) reflection in the line through origin with slope tan α/2 12. A pair of st. linen x2 – 8x + 12 = 0 & y2 – 14y + 45 = 0 are forming a square. What is the centre of circle inscribed in the square: (a) (3, 2) (b) (7, 4) (c*) (4, 7) (d) (0, 1) 13. Area of the triangle formed by the line x + y = 3 and the angle bisector of the pair of lines x2 – y2 + 2y = 1, is-(a) 1 (b) 3 (c*) 2 (d) 4 ANSWER KEY Q.No.1 2 3 4 5 6 7 8 9 10111213 Ans.c b a a b d a d b c d c c 14. CIRCLE 1. The centre of the circle passing through points (0, 0), (1,0) and touching the circle x2 + y2 = 9 is (a) (3/2, 1/2) (b) (1/2, 3/2) (c) (1/2, 1/2) (d) (1/2, – 21/2) 2. The equation of the circle which touches both the axes and the straight line 4x + 3y = 6 in the first quadrant and lies below it is (a) 4x2 + 4y2 – 4x – 4y + 1 = 0 (b) x2 + y2 – 6x – 6y + 9 = 0 (c) x2 + y2 – 6x – y + 9 = 0 (d) 4(x2 + y2 – x – 6y) + 1 = 0 3. The slope of the tangent at the point (h, h) of the circle x2 + y2 = a2 is-(a) 0 (b) 1 (c) –1 (d) depends on h 4. The co-ordinates of the point at which the circles x2 + y2 – 4x – 2y – 4 = 0 and x2 + y2 – 12x – 8y – 36 = 0 touch each other are-(a) (3, – 2) (b) (–2, 3) (c) (3, 2) (d) None of these 5. Given that two circles x2 + y2 = r2 and x2 + y2 – 10x + 16 = 0, the value of r such that they intersect in real and distinct points is given by-(a) 2 < r < 8 (b) r = 2 ro r = 8 (c) (3, 2) (d) None of these 6. The distance from the centre of the circle x2 + y2 = 2x to the straight line passing through the points of intersection of the two circles. x2 + y2 + 5x – 8y + 1 = 0 and x2 + y2 – 3x + 7y – 25 = 0 is-(a) 1 (b) 3 (c) 2 (d ) 1/3 7. The intercept on the line y = x by the circle x2 + y2 – 2x = 0 is AB. Equation of the circle with AB as a diameter is-(a) x2 + y2 + x + y = 0 (b) – x2 + y2 = x – y = 0 (c) x2 + y2 – x – y = 0` (d) None of these Mathematics Online : It’s all about Mathematics pankaj.baluja@mathematicsonline.co.in IIT JEE Screening Exam Questions -19 -Visit us at: www.mathematicsonline.co.in MATHEMATICS ONLINE IIT JEE SCREENING QUESTIONS CHAPTER WISE 8. The angle between a pair of tangents from a point P to the circle x2 + y2 + 4x – 6y + 9 sin2 α + 13 cos2 α = 0 is 2α. The equation of the locus of P is-(a) x2 + y2 + 4x – 6y + 4 = 0 (b) x2 + y2 + 4x – 6y – 9 = 0 (c) x2 + y2 + 4x – 6y – 4 = 0 (d) x2 + y2 + 4x – 6y + 9 = 0 9. Two vertices of an equilateral triangle are (–1, 0) and (1, 0) and its circumcircle is-(a) x2 + 2 1 43 ⎛ ⎞ − = ⎟ ⎝ ⎠ y3⎜ (b) x2 – 1 43 3 ⎛ ⎞ y+ = ⎝ ⎠ ⎜⎟ (c) x2 + 2 1 43 ⎛ ⎞ − = ⎝ ⎠ y3⎜⎟ (d) None of these 10. If a circle passes thro’ the points of intersection of the co-ordinate axes with the lines λx – y + 1 = 0 and x – 2y + 3 = 0, then the value of λ is-(a) 2 (b) 4 (c) 6 (d) 3 11. The number of common tangents to the circles x2 + y2 = 4 and x2 + y2 – 6x – 8y = 24 is (a) 0 (b) 1 (c) 3 (d) 4 12. If two distinct chords drawn from the point (p,q) on the circle x2 + y2 = px + qy (where pq ≠ 0) are bisected by the x-axis then (a) p2 = q2 (b) p2 = 8q2 (c) p2 < 8q2 (d) p2 > 8q2 13. Let L1 be a straight line passing through the origin and L2 be the straight line x + y = 1. If the intercepts made by the circle x2 + y2 – x + 3y = 0 on L1 and L2 are equal, then which of the following equations can represent L1 (a) x + y = 0 (b) x – y = 0 (c) x + 7y = 0 (d) None of these 14. If the circles x2 + y2 + 2x + 2ky + 6 = 0 and x2 + y2 + 2ky + k = 0 intersect orthogonally, then k is-(a) 2 or –3/2 (b) –2 or –3/2 (c) 2 or 3/2 (d) –2 or 3/2 15. The triangle PQR is inscribed in the circle x2 + y2 = 25. If Q and R have co-ordinates (3, 4) and (–4, 3) respectively, then angle QPR is equal to-(a) π/2 (b) π/3 (c) π/4 (d) π/6 16. Let PQ and RS be tangents at the extremities of the diameter PR of a circle of radius r. if PS and RQ intersect at a point X on the circumference of the circle, then 2r equals (c) 2PPQQ.RS RS + (d) 2 2 RS 2+ PQ (a) PQ (b) .RS PQ RS 2+ 17. If the tangent at the point P on the circle x2 + y2 + 6x + 6y = 2 meets the straight line 5x – 2y + 6 = 0 at a point Q on the y-axis, then the length of PQ is-2 5 (c) 5 (d) 3 (a) 4 (b) 18. If a > 2b > 0 then the positive value of m for which y = mx – b 2 1 m + is a common tangent to x2 + y2 = b2 and (x – a) + y2 = b2 is-2 2 a 4b 2b − (a) 22b 2b a 4 − (b) (c) 2b a 2b − (d) b a 2b − 19. Diameter of the given circle x2 + y2 – 2x – 6y + 6 = 0 is the chord of another circle C having centre (2, 1), the radius of the circle C is-Mathematics Online : It’s all about Mathematics pankaj.baluja@mathematicsonline.co.in IIT JEE Screening Exam Questions -20 -Visit us at: www.mathematicsonline.co.in MATHEMATICS ONLINE IIT JEE SCREENING QUESTIONS CHAPTER WISE 3 (b) 2 (c) 3 (d) 1 (a) 20. Locus of the centre of circle touching to the x-axis & the circle x2 + (y – 1)2 = 1 externally is (a) {(0, y); y ≤ 0} ∪ (x2 = 4y) (b) {(0, y); ≤ 0} ∪ (x2 = y) (c) {(x, y); y ≤ y} ∪ (x2 = 4y) (d) {(0, y); y ≥ 0} ∪ (x2 + (x2 + (y – 1)2 = 4 ANSWER KEY Q.No. 1 2 3 4 5 6 7 8 9 10111213141516 17 18 19 20Ans. d a c d a c c d a a b d c a c a c a c a 15. PARABOLA 1. The point of intersection of the tangents at the ends of the latus retum of the parabola y2 = 4x is… (a) (–1, 0) (b) (1, 0) (c) (0,1) (d) None of these 2. Consider a circle with centre lying on the focus of the parabola y2 = 2px such that it touches the directrix of the parabola. Then a point of intersection of the circle and the parabola is (a) (p/2, p) (b) (–p/2, p) (c) (–p/2, –p) (d) None of these 3. The curve described parametrically by x = t2 + t + 1, y = t2 – t + 1 represents-(a) a pair of st. lines (b) an ellipse (c) a parabola (d) a hyperbola 4. If x + y = k is normal to y2 = 12x, then k is-(a) 3 (b) 9 (c) –9 (d) –3 5. If the line x – 1 = 0 is the directrix of the parabola y2 – kx + 8 = 0, then one of the values of k is-(a) 1/8 (b) 8 (c) 4 (d) 1/4 6. Above x-axis, the equation of the common tangents to the circle (x – 3)2 + y2 = 9 and parabola y2 = 4x is-(a)3y 3x 1=+ (b) 3y (x 3) = − + (c)3y (d) 3y(=− 3x 1) + x 3 = + 7. The equation of the directrix of the parabola y2 + 4y + 4x + 2 = 0 is-32 32 (a) x = – 1 (b) x = 1 (c) x = – (d) x = 8. The locus of the mid-point of the line segment joining the focus to a moving point on the parabola y2 = 4ax is another parabola with directrix-(a) x = – a (b) x = – a/2 (c) x = 0 (d) x = a/2 9. If focal chord of y2 = 16x touches (x – 6)2 + y2 = 2 then slope of such chord is-12 (a) 1, –1 (b) 2, – (c) 12 , – 2 (d) 2, – 2 10. Angle between the tangents drawn from (1, 4) to the parabola y2 = 4x is-(a) 2π 3π (b) (c) 6 π 4π (d) 11. A tangent at any point P (1, 7) the parabola y = x2 + 6, which is touching to the circle x2 + y2 + 16x + 12 y + c = 0 at point Q, then Q = is (a) (–6, –7) (b) (–10, –15) (c) (–9, –7) (d) (–6, –3) Mathematics Online : It’s all about Mathematics pankaj.baluja@mathematicsonline.co.in IIT JEE Screening Exam Questions -21 -Visit us at: www.mathematicsonline.co.in MATHEMATICS ONLINE IIT JEE SCREENING QUESTIONS CHAPTER WISE ANSWER KEY Q.No. 1 2 3 4 5 6 7 8 9 1011Ans. a a c b c c d c a b a 16. FUNCTION 1. If function f(x) = 12– tan x 2 π ⎛ ⎞ ⎜ ⎟ ⎝ ⎠; (–1 < x < 1) and g (x) = 2 4x 34x+−, then the domain of gof is-1 1 , 2 2 ⎡ ⎤ ⎢ ⎥ ⎣ ⎦ (a) (–1, 1) (b*) − (c) 1 1, 2 (d) 12,1⎡ ⎤ − ⎢⎥ ⎡ ⎤ − − ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ 2. If f(x) = cos [π2]x + cos [–π2]x, where [x] stands for the greatest integer function, then (a*) f 2π ⎛ ⎞⎟ ⎝ ⎠ 4π ⎛ ⎞⎟ ⎝ ⎠ ⎜ = 2 (d) None of these ⎜ = – 1 (b) f(π) = 1 (c) f 3. The value of b and c for which the identity f(x + 1) – f(x) = 8x + 3 is satisfied, where f(x) = bx2 + cx + d, are (a) b = 2, c = 1 (b*) b = 4, c = – 1 (c) b = – 1, c = 4 (d) None 4. Let f(x) = sin x and g(x) = ln|x|. If the ranges of the composities functions fog and gof are R1 and R2 respectively, then-(a) R1 = {u: –1 < u < 1}, R2 = {v : – ∞ < v < 0} (b) R1 = {u : – ∞ < u ≤ 0}, R2 = {v: –1 ≤ v ≤ 1} (c) R1 = {u: –1 < u < 1}, R2 = {v : – ∞ < v < 0} (d*) R1 = {u: – 1 ≤ u ≤ 1}, R2 = {v : – ∞ < v ≤ 0} 5. Let 2 sin2 x + 3 sin x – 2 > 0 and x2 – x – 2 < 0 (x is measured in radians). Then x lies in the interval 5 1, 6π ⎛ ⎞ − ⎜ ⎟ ⎝ ⎠ (a) , 6 6 π 5π ⎛ ⎞ ⎝ ⎠ ⎜ (b)⎟ , 2 6π ⎛ ⎞ ⎝ ⎠ ⎜⎟ (c) (–1, 2) (d*) 6. Let f(x) = (x + 1)2 – 1, (x ≥ – 1). Then the set S = {x : f (x) = f–1(x)} is-(a) Empty (b*) {0, –1} (c) {0, 1, –1} (d) 3i330,1,, i 3 2 2 ⎧ ⎫ −+−−⎪ ⎪ −⎨ ⎬ ⎪ ⎪ ⎩ ⎭ 7. If f(1) = 1 and f(n + 1) = 2f(n) + 1 if n ≥ 1, then f(n) is-(a) 2n + 1 (b) 2n (c*) 2n – 1 (d) 2n – 1 – 1 8. Let f : R → R be given by f(x) = (x + 1)2 – 1. Then f–1(x) = x 1 + (a*) – 1 + x 1 + (b) – 1 – (c) does not exist because if not one-one (d) does not exist because f is not onto 9. If f is an even function defined on the interval (–5, 5), then the real values of x satisfying the equation f(x) = f x 1 x 2 + ⎛ ⎞ ⎜ ⎟ + ⎝ ⎠ Mathematics Online : It’s all about Mathematics pankaj.baluja@mathematicsonline.co.in IIT JEE Screening Exam Questions -22 -Visit us at: www.mathematicsonline.co.in MATHEMATICS ONLINE IIT JEE SCREENING QUESTIONS CHAPTER WISE (a*) 15,−±−3 5 2 2± (b) 13,−±−3 3 2 2± (c) 2 5 2 − ± (d) None of these 10. Let f(x) = [x] sin [x 1] ⎛ ⎞ π⎜⎟ + ⎝ ⎠ [1,0)} , where [.] denotes the greatest integer function. The domain of f is…… (a) {x ∈ R| x ∈ [–1, 0) } (b) {xR|x∈∉ (c*) {x (d) None of these R|x[–∈∉ 1, 0)} 54 ⎛ ⎞ ⎜ ⎟ ⎝ ⎠ 11. If f(x) = sin2x + sin2 cos x cos x 3π +⎜ ⎛⎞⎟⎝⎠ and g = 1, then (gof) (x) = (a) – 2 (b) – 1 (c) 2 (d*) 1 2 n x ) (si, then 12. If g (f(x)) = |sin x| and f(g(x)) = x (a*) f(x) = sin2x, g(x) = (b) f(x) = sin x, g(x) = |x| x (c) f(x) = x2, g(x) = sin (d) f and g cannot be determined 13. If f(x) = 3x – 5, then f–1(x) (a) is given by 1 3x 5 − x 5 3+ (b*) is given by (c) does not exist because f is not one-one (d) does not exist because f is not onto 14. If the function f:[1, ∞] → [1, ∞) is defined by f(x) = 2x(x – 1), then f–1 (x) is (a) x(x 1) 12 − ⎛ ⎞ ⎝ ⎠ ⎜⎟ (b*) 2 og 21(114lx)++ (c) 2 1 (1 1 4log x ) 2 − + (d) not defined 15. The domain of definition of the function y(x) given by the equation 2x + 2y = 2 is-(a) 0 < x ≤ 1 (b) 0 ≤ x ≤ 1 (c) – ∞ < x ≤ 0 (d*) – ∞ < x < 1 16. Let f(θ) = sinθ (sinθ + sin3θ), then f(θ) (a) ≥ 0 only when θ ≥ 0 (b) ≤ 0 for all θ (c*) ≥ 0 for all real θ (d) ≤ 0 only when θ ≤ 0 17. The number of solutions of log4 (x – 1) = log2 (x – 3) is-(a) 3 (b*) 1 (c) 2 (d) 0 x x 1 α+ 18. Let f(x) = , x ≠ – 1, then for what value of α, f{f(x)} = x. (a) 2 (b) – 2 (c) 1 (d*) – 1 Mathematics Online : It’s all about Mathematics pankaj.baluja@mathematicsonline.co.in IIT JEE Screening Exam Questions -23 -Visit us at: www.mathematicsonline.co.in MATHEMATICS ONLINE IIT JEE SCREENING QUESTIONS CHAPTER WISE 19. The domain of definition of f(x) = 22log (x 3) x 3x 2 + + + is-(a) R/{–2, –2} (b) (–2, ∞) (c) R/{–1, –2, –3} (d*) (–3, ∞)/{–1, –2} 1x then f–1(x) equals-20. If f : [1, ∞) → [2, ∞) is given by f(x) = x + (a*) 2 x x2 + −4 (b) 2 x 1 x − (c) 2 x x 4 2 − − (d) 1 + 2 x 4 − 21. Suppose f(x) = (x + 1)2 for x ≥ – 1. If g(x) is the function whose graph is the reflection of the graph of f(x) with respect to the line y = x, then g(x) equals-(a) – x – 1, x ≥ 0 (b) 2 1 (x 1) + , x > – 1 (c) x, x ≥ – 1 (d*) 1 + x – 1, x ≥ 0 22. Let function f : R → R be defined by f(x) = 2x + sin x for x ∈ R. Then f is-(a*) one to one and onto (b) one to one but NOT onto (c) onto but NOT one to one (d) neither one to one onto x 1 x + 23. Let f(x) = defined as [0, ∞) → [0, ∞), f(x) is-(a)one one & onto (b*) one-one but not onto (c) not one-one but onto (d) neither one-one nor onto 24. Find the range of f(x) = 22 x x 2 x x 1 + + + + is-(a) (1, ∞) (b) 11 1, 7 ⎛ ⎞⎟ ⎝ ⎠ ⎜ (c*) 7 1, 3 ⎛ ⎞⎟ ⎝ ⎠ ⎜ (d) 7 1, 5 ⎛ ⎞ ⎝ ⎠ ⎜⎟ 25. Domain of f(x) = 1sin(2x)− /6 + π is-(a*) 1 1 , 4 2 ⎡ ⎤ ⎣ ⎦ 1 1 , 2 2 ⎡ ⎤ −⎢⎥ ⎣ ⎦ −⎢ (b)⎥ (c) 1 1 , 4 4 ⎡ ⎤ −⎢ ⎥ ⎣ ⎦ (d) 1 1 , 2 4 ⎡ ⎤ −⎢ ⎥ ⎣ ⎦ 26. Let f(x) = sin x + cos x & g (x) = x2 – 1, then g(f(x)) will be invertible for the domain-(a) x ∈ [0, π] (b*) x ∈ , 4 4 π π ⎡ ⎤ ⎣ ⎦ 0, 2π ⎡ ⎤ ⎢ ⎥ ⎣ ⎦ −⎢⎥ (c) x ∈ (d) x ∈ ,0 2π ⎡ ⎞ − ⎟ ⎣ ⎠ x Q x Q ∈ ∈ ∉ ∉ ⎢ 27. f(x) = xxQ0;g(x)0xQx⎧⎧=⎨⎨⎩⎩ then (f – g) is (a*) one – one, onto (b) neither one-one, nor onto (c) one-one but not onto (d) onto but not one-one 1n ⎛ ⎞⎟ ⎝ ⎠ 28. f : R → R, f ⎜ = 0 n ∈ I, n ≥ 1 then (a) f(x) = 0 for x ∈ [0, 1] (b) f(x) = 0 for x ∈ R pankaj.baluja@mathematicsonline.co.in IIT JEE Screening Exam Questions -24 -Visit us at: www.mathematicsonline.co.in Mathematics Online : It’s all about Mathematics MATHEMATICS ONLINE IIT JEE SCREENING QUESTIONS CHAPTER WISE (c*) f(0) = 0 = f ’ (0) (d) f(x) = 0 = f ’ (x) can not be ANSWER KEY Q.No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Ans. b a b d d b c a a c d a b b D Q.No. 16 17 18 19 20 21 22 23 24 25 26 27 28 Ans. c b d d a d a b c a b a c 17. LIMITS 1. x/42cLimcot→π osx 1 x 1− − = 12 (a) 12 (b*) (c) 12 2 (d) 1 2. 40x(2x1)(Lim(2x3→∞+ 5 45 4x 1) ) + − = (a) 16 (b) 24 (c*) 32 (d) 8 x 0 1 (1 cos 2x) 2 Lim x → − 3. = (a) 1 (b) –1 (c) 0 (d*) None nx x Lim e →∞ 4. x = 0 for (a) no value of n (b*) n is any whole number (c) n = 0 only (d) n = 2 only 5. 1 x 2x − ⎡ ⎤ ⎢ ⎥ ⎣ ⎦ x0Limtan→ = 12 (c) 2 (d) ∞ (a) 0 (b*) 6. 1/x x ⎫ ⎛ ⎞ ⎨ ⎬ ⎟ ⎝ ⎠ ⎩ ⎭ x0Limtan4→⎧π+⎜ = (a) 1 (b) – 1 (c*) e2 (d) e Mathematics Online : It’s all about Mathematics pankaj.baluja@mathematicsonline.co.in IIT JEE Screening Exam Questions -25 -Visit us at: www.mathematicsonline.co.in MATHEMATICS ONLINE IIT JEE SCREENING QUESTIONS CHAPTER WISE 7. 2 1/x 22 xx ⎛ ⎞ ⎜ ⎟ ⎝ ⎠ x015Lim13→++ = (a) e2 (b) e (c) e–2 (d*) e–1 8. The value of 2h0log(12h)2log(limh→+− 1 h) − is-(a*) 1 (b) –1 (c) 0 (d) None of these 9. x11cos2(Limx1→− x 1) − − = 2 (a*) Does not exist because LHL ≠ RHL (b) Exists and it equals – 2 (c) Does not exist because x – 1 → 0 (d) Exists and it equals 10. 2 tan x x) x0xtan2x2xLim(1cos2→−− is-(a*) 12 12 − (b) – 2 (c) 2 (d) 11. For x ∈ R, x x 3 x 2 − ⎛ ⎞ ⎜ ⎟ + ⎝ ⎠ xLim→∞ = (a) e (b) e–1 (c*) e–5 (d) e5 12. 2 2 cos x) x0sin(Limx→π equals-(a) – π (b*) π (c) 2π (d) 1 13. The value of integer n; for which x x e ) − − nx0(cosx1)(cosLimx→ is a finite non zero number-(a) 1 (b) 2 (c*) 3 (d) 4 14. Let f : R → R such that f(1) = 3 and f ’(1) = 6. then 1/x x) ) ⎛ ⎞⎟ ⎝ ⎠ x0f(1Limf(1→+⎜ equals-(a) 1 (b) e1/2 (c*) e2 (d) e3 15. If 2x0(sinnx)[(an)nxLimx→−−tan x] = 0 then the value of a is-(a) 1n 1 + (b) n n 1 + (c*) 1n n+ (d) n 16. If f(x) is a differentiable function and f ’(2) = 6, f ’(1) = 4, f ’(c) represents the differentiation of f(x) at x = c, then 22x0f(22hh)Limf(1hh)→++++ f (2) f (1) − − (a) may exist (b) will not exist (c*) is equal to 3 (d) is equal to – 3 Mathematics Online : It’s all about Mathematics pankaj.baluja@mathematicsonline.co.in IIT JEE Screening Exam Questions -26 -Visit us at: www.mathematicsonline.co.in MATHEMATICS ONLINE IIT JEE SCREENING QUESTIONS CHAPTER WISE 17. Let f(x) be strictly increasing and differentiable, then 2x0f(xLimf(x→ ) f (x) ) f (0) − − is-(a) 1 (b*) – 1 (c) 0 (d) 2 ANSWER KEY Q.No. 1 2 3 4 5 6 7 8 9 101112131415 16 17 Ans. b c d b b c a b a a c b c c c c b 18. CONTINUITY 1. If f(x) = 21cos4x,whenxxa,whex,when16x)4−⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪+−⎪⎩ 0 nx0x 0 < = > is continuous at x = 0, then the value of ‘a’ will be-(a) 8 (b) – 8 (c) 4 (d) None 2. The following functions are continuous on (0, π) (a) tan x (b) xsinx;0sin(x);22x /2 x < ≤ π π ⎧⎪⎨ππ+<⎪⎩ <π (c) 3 x 4 x 1,0232sinx,94 π ⎧⎪⎪⎨π⎪⎪⎩ < ≤ < < π (d) None of these 3. If f(x) = xsinx,when0sin(x),when22x 2 x π ⎧⎪⎪⎨ππ⎪π+⎪⎩ < ≤ < < π , then-2π 2π (b) f(x) is continuous at x = (a) f(x) is discontinuous at x = (c) f(x) is continuous at x = 0 (d) None of these 4. The function f(x) = [x] cos {(2x – 1)/2}π, [] denotes the greatest integer function, is discontinuous at (a) all x (b) all integer points (c) no x (d) x which is not an integer Mathematics Online : It’s all about Mathematics pankaj.baluja@mathematicsonline.co.in IIT JEE Screening Exam Questions -27 -Visit us at: www.mathematicsonline.co.in MATHEMATICS ONLINE IIT JEE SCREENING QUESTIONS CHAPTER WISE xy ⎛ ⎞⎟ ⎝ ⎠ 5. Let f(x) be defined for all x > 0 and be continuous. Let f(x) satisfy f ⎜ = f(x) – f(y) for all x, y & f(e) = 1 1x ⎛ ⎞ ⎜ ⎟ ⎝ ⎠ → 0 as x → 0 (a) f(x) is bounded (b) f (c) x f(x) → 1 as x → 0 (d) f(x) = log x 6. The function f(x) = [x]2 = [x2] (where [y] is the greatest integer less than or equal to y), is discontinuous at-(a) All integers (b) All integers except 0 and 1 (c) All integers except 0 (d) All integers except 1 ANSWER KEY Q.No.1 2 3 4 5 6 Ans.a c a c d d 19. DIFFERENTIATION 6π 1. The derivative of function cot–1 [(cos 2x)1/2] at x = is (a*) (2/3)1/2 (b) (1/3)1/2 (c) 31/2 (d) 61/2 2. Indicate the correct alternative: Let [x] denote the greater integer ≤ x and f(x) = [tan2x], then (a) Li f(x) does not exist (b*) f(x) is continous at x = 0 x 0 m → (c) f(x) is not differentiable at x = 0 (d) f ’(0) = 1 dy dx 3. If y = sec tan–1 x then = (a) x/(1 + x2) (b) x 2 ) (1x+ (c) 21 ) + 1/(x (d*) x/2 ) (1x+ 4. If f(x) = tan–1 1 sinx 1 sinx +− 0 ≤ x ≤ π/2, the f’ (π/6) is (a) – 14 (b) – 12 (c) 14 (d*) 12 x 0 0 x0 5. g(x) = x f(x), where f(x) = xsin(1/x), ≠ ⎧⎨= ⎩ at x = 0 (a*) g is differentiable but g’ is not continuous (b) both f and g are differentiable (c) g is diffentiable but g’ is continuous (d) None of these 6. Let fxyf(x)⎛⎞=⎜⎟⎝⎠ f (y) 2 2 + + for all real x and y and f ’ (1) = – 1, then f ’(2) = (a) 1/2 (b) 1 (c*) –1 (d) –1/2 Mathematics Online : It’s all about Mathematics pankaj.baluja@mathematicsonline.co.in IIT JEE Screening Exam Questions -28 -Visit us at: www.mathematicsonline.co.in MATHEMATICS ONLINE IIT JEE SCREENING QUESTIONS CHAPTER WISE 7. Let f (x) = 3xsinx61pp−2 3 cosx 0p where p is a constant. Then 33 d dx [f(x)] at x = 0 is (a) p (b) p + p3 (c) p + p2 (d*) Independent of p 8. Let F(x) = f(x) g(x) h(x) for all real x, where f(x), g(x) and h(x) are differentiable functions at some point x0. F’ (x0) = 21 F(x0), f ’(x0) = 4f(x0), g’(x0) = – 7g(x0) and h’(x0) = Kh(x0), then K = (a) 12 (b*) 24 (c) 6 (d) 18 9. Let h(x) = min {x, x2}, for every real number of x. Then-(a*) h is not differentiable at two values of x (b) h is differentiable for all x (c) h’ (x) = 0, for all x > 1 (d) None of these 10. The function f(x) = (x2 – 1) |x2 – 3x + 2| + cos (|x|) is not differentiable at. (a) – 1 (b) 0 (c) 1 (d*) 2 11. If x2 + y2 = 1, then (a) yy” – 2(y’)2 + 1 = 0 (b*) yy” + (y’)2 + 1 = 0 (c) yy” – (y’)2 – 1 = 0 (d) yy” + 2(y’)2 + 1 = 0 12. Let f : R → R is a function which is defined by f (x) = max {x x3} set of points on which f(x) is not differentiable is (a) {–1, 1} (b*) {–1, 0} (c) {0, 1} (d) {–1, 0, 1} 13. Find left and hand derivative at x = k, k ∈ I.f(x) = [x] sin (πx) (a*) (–1)k (k – 1)π (b) (– 1)k – 1 (k – 1) π (c) (–1)k (k – 1) π (d) (– 1)k – 1 (k – 1) π 14. Which of the following functions is differentiable at x = 0 ? (a) cos (|x|) + |x| (b) cos (|x|) – |x| (c) sin (|x|) + |x| (d*) sin (|x|) – |x| 15. Let f : R → R be such that f(1) = 3 and f ’(1) = 6. Then 1/x x) ) ⎛ ⎞⎟ ⎝ ⎠ x0f(1limf(1→+⎜ equals-(a) 1 (b) e1/2 (c*) e2 (d) e3 16. The domain of the derivative of the function f(x) = 1tanxif|x1(|x|1)if|2−⎧⎪⎨ | 1 x | 1 ≤ − > ⎪⎩ is-(a*) R – {0} (b) R – {1} (c) R – {– 1} (d) R – {–1, 1} 17. Let y be a function of x, such that log (x + y) – 2xy = 0, then y’(0) is-(a)0 (b*) 1 (c) 1/2 (d) 3/2 18. If x cos y + y cos x = π, then y’ (0) = (a*) π (b) – π (c) 0 (d) 1 19. S is a set of polynomial of degree less then or equal to 2 f(0) = 0 f(1) = 1 f ’ (x) > 0; ∈ [0, 1] then set S = (a) φ (b) ax + (1 – a) x2 ; a ∈ R (c) ax + (1 – a) x2 ; 0 < a < ∞ (d*) ax + (1 – a) x2 ; 0 < a < 2 Mathematics Online : It’s all about Mathematics pankaj.baluja@mathematicsonline.co.in IIT JEE Screening Exam Questions -29 -Visit us at: www.mathematicsonline.co.in MATHEMATICS ONLINE IIT JEE SCREENING QUESTIONS CHAPTER WISE 20. If f (1) = 1; f(2) = 4, f(3) = 9 & f is twice differentiable then (a*) f ” (x) = for all x ∈ [1, 3] (b) f ” (x) = f ’ (x) = 5 ; x ∈ [1, 3] (c) f ” (x) = 2 for only x ∈ [1, 3] (d) ax + (1 – a) x2 ; for x ∈ (1, 3) 21. f(x) = | |x| – 1| is not differentiable at x = (a*) 0, ± 1 (b) ± 1 (c) 0 (d) 1 ANSWER KEY Q.No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Ans. a b d d a c d b a d b d a d c Q.No. 16 17 18 19 20 21 Ans. d b a d a a 20. TANGENT & NORMAL 1. The co-ordinates of the point on the curve y = x2 + 3x + 4 the tangent at which passes through the origin is equal to (a*) (2, 14) (–2, 2) (b) (2, 14), (–2, –2) (c) (2, 14) (2, 2) (d) None of these 2. If the parametric equation of a curve is given by x = et cos t, y = et sin t then the tangent to the curve at the point t = π/4 makes with the axis of x the angle (a) 0 (b) π/4 (c) π/3 (d*) π/2 3. The curve y – exy + x = 0 has a vertical tangent at the point-(a) (1, 1) (b) at no point (c) (0, 1) (d*) (1, 0) 4. If y = 4x – 5 is tangent to the curve y2 = px3 + q at (2, 3), then (a*) p = 2, q = – 7 (b) p = –2, q = 7 (c) p = – 2, q = – 7 (d) p = 2, q = 7 5. The curve y = ax3 + bx2 + cx + 5 touches the x-axis at P(–2, 0) and cuts the y-axis at a point Q where its gradient is 3. The a, b, c are respectively (a*) –1/2, –3/4, 3 (b) 3, –1/2, –4 (c) –1/2, –7/4, 2 (d) None of these 6. Let C be the curve y3 – 3xy + 2 = 0. If H be the set of points on the curve C, where tangent is horizontal and V is the set of points on the curve C where the tangent is vertical, then H = … V =… (a*) φ, (1, 1) (b) φ, (2, 1) (c) φ, (0, 1) (d) None of these 7. On the ellipse 4x2 + 9y2 = 1, the points at which the tangents are parallel to the line 8x = 9y are-(a) 21,or55⎛⎞⎜⎟⎝⎠ 1 2,5 5 ⎛ ⎞ ⎜ ⎟ ⎝ ⎠ (b*) 21,or55⎛⎞ 2 1,5 5 ⎛ ⎞ − −⎜⎟ ⎝ ⎠ ⎜⎟⎝⎠ (c) 2 1 , 5 5 ⎛ ⎞ − − ⎟ ⎝ ⎠ ⎜ (d) 1 2 , 5 5 ⎛ ⎞ − − ⎝ ⎠ ⎜⎟ 8. If x + y = K is normal to y2 = 12, then K is-(a) 3 (b*) 9 (c) – 9 (d) – 3 9. If the normal to the curve y = f(x) at the point (3, 4) makes an angle 3π/4 with the positive x-axis, then f ’(3) = Mathematics Online : It’s all about Mathematics pankaj.baluja@mathematicsonline.co.in IIT JEE Screening Exam Questions -30 -Visit us at: www.mathematicsonline.co.in MATHEMATICS ONLINE IIT JEE SCREENING QUESTIONS CHAPTER WISE (a) –1 (b) 34 − (c) 43 (d*) 1 10. The triangle formed by the tangent to the curve f(x) = x2 + bx – b at the point (1, 1) and the co-ordinate axis, lies in the first quadrant. If its area is 2, then the value of b is (a) –1 (b) 3 (c*) –3 (d) 1 11. The point(s) on the curve y3 + 3x2 = 12y where the tangent is vertical, is (are) (a) 4 , 2 3 ± ⎛⎞−⎜⎟⎝⎠ (b) 11, 1 3 ⎛ ⎞ ± ⎝ ⎠ ⎜⎟⎜⎟ (c) (0, 0) (d*)4± , 2 3 ⎛ ⎞ ⎝ ⎠ ⎜⎟ 12. The equation of the common tangent to the curves y2 = 8x and xy = – 1 is-(a) 3y = 9x + 2 (b) y = 2x + 1 (c) 2y = x + 8 (d*) y = x + 2 13. According to mean value theorem in the interval x ∈ [0, 1] which of the following does not follow-(a*) 2f(x)xx;=−⎛⎞=−⎜⎟⎝⎠ 1 1 ;x 2 2 1 1 x 2 2 <≥ (b) sinxf(x);xx1;= 0x0 ≠ = = (c) f(x) = x|x| (d) f(x)= |x| 14. If focal chord of y2 = 16 x touches (x – 6)2 + y2 = 2 then slope of such chord is 12 (a*) 1, –1 (b) 2, – (c) 12 , –2 (d) 2, –2 15. Let f(x) = xα log x for x > 0 & f(0) = 0 follows Rolle’s theorem for [0, 1] then α is-(a) –2 (b) –1 (c) 0 (d*) 1/2 ANSWER KEY Q.No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Ans. a d d a a a b b d c d d a a d 21. MONOTONICITY 1. If f(x) = then f(x) is-23x12x1,137x,2x3⎧+−−⎨−<≤⎩ x 2 ≤ ≤ (a) Increasing in [–1, 2] (b) Continuous in [–1, 3] (c) Greatest at x = 2 (d*) All above correct 2. The function f defined by f(x) = (x + 2) e–x is-(a) Decreasing for all x (b) Decreasing in (–∞, –1) and increasing (–1, ∞) (c) Increasing for all x (d*) Decreasing in (–1, ∞) and increasing in (–∞, –1) 3. Function f(x) = log(log( x) e x) π ++ is decreasing in the interval-(a) (–∞,∞) (b*) (0, ∞) (c) (–∞, 0) (d) No where Mathematics Online : It’s all about Mathematics pankaj.baluja@mathematicsonline.co.in IIT JEE Screening Exam Questions -31 -Visit us at: www.mathematicsonline.co.in MATHEMATICS ONLINE IIT JEE SCREENING QUESTIONS CHAPTER WISE x sin x x tan x 4. If f(x) = and g(x) = , where 0 < x ≤ 1, then in this interval-(a) Both f(x) and g(x) are increasing functions (b) Both f(x) and g(x) are decreasing function (c*) f(x) is an increasing function (d) g(x) is an increasing function 5. Let h(x) = f(x) – (f(x))2 + (f(x))3 for every real number x. Then-(a*) h is increasing whenever f is increasing (b) h is increasing whenever f is decreasing (c) h is decreasing whenever f is increasing (d) nothing can be said in general 6. The function f(x) is defined by f(x) = (x + 2) e–x is-38π (c) 38π < x < 58π (d*) 58π < x < 34π 8π (b) 4π < x < (a) 0 < x < 7. The function f(x) = sin4 x + cos4 x increases if-38π (c) 38π < x < 58π (d) 58π < x < (a) 0 < x < 8π (b*) 4π < x < 34π 2)dx − 8. Let f(x) = . Then f decreases in the interval-xe(x1)(x−∫ (a) (–∞, –2) (b) (–2, –1) (c*) (1, 2) (d) (2, + ∞) 9. Consider the following statement S and R-, 2π ⎛ ⎞ π S : Both sin x and cos x are decreasing function in the interval ⎜⎟ ⎝ ⎠ R : If a differentiable function decreases in an interval (a, b), then its derivative also decreases in (a, b) Which of the following is true ? (a) Both S and R are wrong (b) Both S and R are correct, but R is not the correct explanation for S (c) S is correct and R is the correct explanation for S (d*) S is correct and R is wrong 10. Let f(x) = x ex(1 – x), then f(x) is-(a*) Increasing on [–1/2, 1] (b) Decreasing on R (c) Increasing on R (d) Decreasing on [–1/2, 1] 11. The length of a longest interval in which the function 3 sin x – 4 sin3 x is increasing, is-(a*) π/3 (b) π /2 (c) 3π /2 (d) π 12. f(x) = x2 – 2bx + 3c2 & g(x) = – x2 – 2cx + b2 if the minimum value of f(x) is always greater than maximum value of g(x) then. 2 | b | (b) c (c) 2b > c (d) 2b < − |c|< 2 | b | 2 2 2x 1 t e + − (a*) |c| > 13. Let f(x) = , x ∈ (–∞, ∞) then the interval for which f(x) is increasing is x∫ (a*) (–∞, 0] (b) [0, ∞) (c) [–2, 2] (d) no where 14. Let f(x) = x3 + bx2 + cx + d; 0 < b2 < c then f(x)-(a*) is strictly increasing (b) has local maxima Mathematics Online : It’s all about Mathematics pankaj.baluja@mathematicsonline.co.in IIT JEE Screening Exam Questions -32 -Visit us at: www.mathematicsonline.co.in MATHEMATICS ONLINE IIT JEE SCREENING QUESTIONS CHAPTER WISE (c) has local minima (d) is bounded curve ANSWER KEY Q.No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Ans. d d b c a d b c d a a A a a 22. MAXIMA & MINIMA 1. If A > 0, B > 0 and A + B = π/3, then the maximum value of tanA tan B is…….. (a*)1/3 (b) 2/3 (c) 1/2 (d) None of these 2. The function f(x) = |px – q| + r|x|, x ∈ (–∞, ∞) where p > 0, q > 0, r > 0, assumes its minimum value only at one point if (a) p ≠ q (b) r ≠ q (c*) r ≠ p (d) p = q = r 3. On the interval [0, 1], the function x25 (1 – x 75) takes its maximum value of the point-14 (a) 0 (b*) (c) 12 (d) 13 4. The number of values of x where the function f(x) = cosx + cos (2 attains its maximu is x) 3 5 (t 3) − x | 2 < ≤ (a) 0 (b*) 1 (c) 2 (d) infinte 5. The function f(x) = dt has a local minimum at x = xt1t(e1)(t1)(t2)−−−−∫ (a) 0, 4 (b*) 1, 3 (c) 0, 2 (d) 2, 4 6. Let f(x) = , then at x = 0, f has-|x|for0|1forx0⎧⎨=⎩ (a*) a local maximum (b) no local maximum (c) a local minimum (d) no extremum 7. Let f(x) = (1 + b2) x2 + 2bx + 1 and m (b) is minimum value of f(x). As b varies, the range of m(b) is-(a) [0, 1] (b) [0, 1/2] (c) [1/2, 1] (d*) (0, 1] 8. The value of ‘θ’; θ ∈ [0, π] for which the sum of intercepts on coordinate axes cut by tangent at point (3 cos θ, sin θ) to ellipse 3 2 x27 + y2 = 1 is minimum is: (a*) 6 π (b) 3π (c) 4π (d) 8π 9. If f(x) = 2xx+ + 2 2 tanx xα + , α ∈ (0, π/2) , x > 0 then value of f(x) is greater than or equal to: (a) 2 (b*) 2 tan α (c) 52 (d) sec α ANSWER KEY Mathematics Online : It’s all about Mathematics pankaj.baluja@mathematicsonline.co.in IIT JEE Screening Exam Questions -33 -Visit us at: www.mathematicsonline.co.in MATHEMATICS ONLINE IIT JEE SCREENING QUESTIONS CHAPTER WISE Q.No. 1 2 3 4 5 6 7 8 9 Ans. a c b b b a d a b 23. INDEFINITE INTEGRATION 1. 3(3x(x1)+∫ 1) (x 1) − + dx equal to-I (a*) 14 14 log |x – 1| – 112(x1)( c x 1) − + −− log |x + 1| – (b) 14 14 log |x + 1| – 2 c x 1) 112(x1)(− + −− log |x – 1| – 14 14 log |x – 1| – 1 1 (x 1) − − 2(x1)− + c log |x + 1| + (c) (d) None of these 2. The value of the integral 3 54 cos x sin x 2cosxsinx++∫dx is-(a) sin x – 6 tan–1 (sin x) + c (b) sin x – 2 (sin x)–1 + c (c*) sin x – 2 (sin x)–1 – 6 tan–1 (sin x) + c (d) sin x – 2 (sin x)–1 + 5 tan–1 (sin x) + c 3. dx(xp)(xp)−−∫ (x q) − is equal to-(a) 2xpqx−−pcq − + (b*)2xpqx qcp − − + −− (c) 1(xp) (x q) − − + c (d) None of these 4. x 2 1) xe ) + (xx(1+∫dx is equal-(a*) log xxxe11xe1x⎛⎞+⎜⎟++⎝⎠ x c e + (b) log x x c e x11xe1x⎛⎞+ + ⎜⎟++⎝⎠ (c) log x1xe1xe1x⎛⎞+⎜⎟+⎝⎠ x x c e + + (d) None of these 5. dx(sinx4)(sinx1)ta=+−∫ Axn 1 2 − + B tan–1 (f(x)) + c, then-(a) A = 15, B = – 2 5 15, f(x) = 4 tan x 3 15+ pankaj.baluja@mathematicsonline.co.in IIT JEE Screening Exam Questions -34 -Visit us at: www.mathematicsonline.co.in Mathematics Online : It’s all about Mathematics MATHEMATICS ONLINE IIT JEE SCREENING QUESTIONS CHAPTER WISE x 4 tan 1 2 15 ⎛ ⎞ + ⎜ ⎟ ⎝ ⎠ (b) A = – 15, B = 115 , f(x) = (c) A = 25, B = – 2 5 5 , f(x) = 4 tan x 1 5 + x 4 tan 1 25 ⎛ ⎞+ ⎜ ⎟ ⎝ ⎠ (d*) A = 25, B = – 215 , f(x) = 6. cosxcosx∫ sin x sin x −+ (2 + 2 sin 2x) dx is equal to (a*) sin 2x + c (b) cos 2x + c (c) tan 2x + c (d) None of these 7. 2dx(2x7)x7−−∫ x 12 + is equal to-(a) 2 sec–1 (2x – 7) + c (b*) sec–1 (2x – 7) + c (c) 12 sec–1 (2x – 7) + 2 (d) None of these 1/2 1 x dxx 1 x ⎛ ⎞ −⎜⎟ ⎜ ⎟ + ⎝ ⎠ ∫ 8. is equal to-(a) 1 x − ⎡ ⎤ ⎢ ⎥ ⎣ ⎦ 11xlogcosx+−+⎢⎥ + c (b*) – 2 1 x − 11xlogcosx⎡ ⎤ +−−⎢ ⎥ ⎢ ⎥ ⎣ ⎦ + c (c) – 2 1 x − ⎡ ⎤⎥ ⎢ ⎥ ⎣ ⎦ xlogcos11x−⎢+− + c (d) None of thse x dx 2 ⎛ ⎞⎟ ⎝ ⎠ cosxlogtan⎜∫is equal to-9. x2 x tan 2 ⎛ ⎞ ⎝ ⎠ (a) sin x log ⎜⎟ + c (b*) sin x log tan – x + c x tan 2 ⎛ ⎞⎟ ⎝ ⎠ ⎜ + x + c (d) None of these (c) sin x log 10. 322x3x(x1)(x++++∫ 2 dx 1) is equal to-(a) 21221x13logtanx x 4 (x 1) 2 x 1 − + + − + + c+ (b) 2 1 2 2 1 (x 1) 3 x log tan x c 4 x 1 2 x 1 − + + + + + + (c*) 2121x13logtanx4(x1)2−+++ 2x c x 1 ++ + (d) None of these ANSWER KEY pankaj.baluja@mathematicsonline.co.in IIT JEE Screening Exam Questions -35 -Visit us at: www.mathematicsonline.co.in Mathematics Online : It’s all about Mathematics MATHEMATICS ONLINE IIT JEE SCREENING QUESTIONS CHAPTER WISE Q.No. 1 2 3 4 5 6 7 8 9 10 Ans. a c b a d a b b b c 24. DEFINITE INTEGRATION 1. dx where a and b are integers is equal to-2 n bx) 2 (1 x ) − (cosaxsiπ−π−∫ (a) –π (b) 0 (c) π (d*) 2 π 2. The value of sin x cos2 x dx is-π−π∫ (a*) 0 (b) π – π3/3 (c) 2π – π 3 (d) 2272− π x | dx π 3. Integral is equal to-10|sin2∫ (a) 0 (b) 1π − (c) 1π (d*) 2π /3 0 cos x 3 4sinx π + ∫ dx = k log 3 2 3 3 ⎛ ⎞⎟⎟⎝ ⎠ +⎜⎜ then k is-4. If (a) 12 13 (b) (c*) 14 (d) 18 5. The value of 3/20π∫ 3 dx 1 tan + is (a) 0 (b) 1 (c) π/2 (d*) π/4 6. The value of 3/4/41sππφ∫ d in φ + φ is……. (a*) ( 2−1)π (b) ( 2 1) π + (c) ( 2 ) 2π − (d) None 7. 32x(5x)−+∫ dx x = (a*) 1/2 (b) 1/3 (c) 1/5 (d) None 8. If f(x) = A sin (πx/2) + B, f ’ 12⎛⎞⎜⎝⎠⎟ = 2 and 1 0 2A f (x)dx = π ∫ n x]dx , then the constants A and B are-(a) π/2 and π/2 (b) 2/π and 3π (c) 0 and –4/π (d*) 4/π and 0 9. The value of , where [] represents the greatest integer function is: 2[2siππ∫ (a*) 53π − (b) – π (c) 53π (d) – 2π Mathematics Online : It’s all about Mathematics pankaj.baluja@mathematicsonline.co.in IIT JEE Screening Exam Questions -36 -Visit us at: www.mathematicsonline.co.in MATHEMATICS ONLINE IIT JEE SCREENING QUESTIONS CHAPTER WISE x1 dt t ∫ 10. The function L(x) = satisfied the equation xy ⎛ ⎞ ⎜ ⎟ ⎝ ⎠ (a) L (x + y) = L (x) + L(y) (b) L = L(x) = L(y) (c*) L(xy) = L(x) + L(y) (d) None of these 11. If for a non-zero x, a f(x) + b f 1 1 5 x x = − f (x) ⎛⎞⎜⎟⎝⎠, where a ≠ b, then 21∫dx = (a) 22alog25aab⎜+1 7b2 ⎛ ⎞ + + ⎟ ⎝ ⎠ (b*)22alog25aab−1 7b2 ⎛ ⎞ − + ⎜ ⎟ ⎝ ⎠ (c) – 22alog25aab⎜+1 7b2 ⎛ ⎞ + − ⎟ ⎝ ⎠ (d) None of these 12. The value of 2 x cos x 1 a + π−π∫ dx, a > 0 is-(a) π (b) a π (c*) 2π (d) 2π 13. Let ddxF(x) = sin x e x , x > 0. If 2 sin x 2ex 4 t dt, k 1 k − ∫ k 1 k − 41∫dx = F(K) – F(1), then one of the possible values of K is-(a) 2 (b) 4 (c) 8 (d*) 16 14. If g (x) = then g(x + π) equals-xacos∫ (a*) g(x) + g(π) (b) g(x) – g(π) (c) g(x) g(π) (d) g(x)/g(π) 15. Let f be a positive function, let I1 = x. f[x (1 – x)] dx & I2 = ∫ f[x (1 – x)] dx, where (2k – 1) > 0, then 12 II x tf (t)dt, ∫ 1 2)dx = is (a) 2 (b) k (c*) 1/2 (d) 1 16. If then the value of f(1) is-x10f(t)dtx=+∫ (a*) 1/2 (b) 0 (c) 1 (d) –1/2 17. 10tan(1xx−−+∫ 12 (a*) log 2 (b) log (c) π log 2 (d) 1 log 2 2 π 18. For n > 0 22n2n2n0xsinxsinxcosπ+∫ dx x = (a*) π (b) π (c) 2π (d) 3π Mathematics Online : It’s all about Mathematics pankaj.baluja@mathematicsonline.co.in IIT JEE Screening Exam Questions -37 -Visit us at: www.mathematicsonline.co.in MATHEMATICS ONLINE IIT JEE SCREENING QUESTIONS CHAPTER WISE 19. Let f(x) = x – [x], for every real number x, where [x] is the integral part of x. Then is-11−∫ f (x)dx 12 (a*) 1 (b) 2 (c) 0 (d) 20. 3/4/4ππ∫ dx 1 cosx + is equal to-12 (d) – (a*) 2 (b) – 2 (c) 12 x] dx 21. If for a real number y, [y] is the greatest integer less than or equal to y, then the value of the integral is 3/2/2[2sinππ∫ (a) – π (b) 0 (c*) – π/2 (d) π/2 22. 2 xx dx 1 a cosπ−π+∫, a > 0 (a) π (b) πa (c*) π/2 (d) 2π 23. The value of the integral 21eelo−∫e g x dx x is (a) 32 (b*) 52 |x| 2; otherw ⎧ < ⎨⎩ 32 f (x)dx − (c) 3 (d) 5 24. If f(x) = Then cosxesin;2ise = ∫ f (t) ∫ (a) 0 (b) 1 (c*) 2 (d) 3 25. Let g(x) = where x0dt 12 12 ≤ f(t) ≤ 1, t ∈ [0, 1] and 0 ≤ f(t) ≤ for t ∈ [1, 2]. Then (a) – 3 1 2) 2 2< g(≤ (b*) 0 ≤ g(2) < 2 (c)3 5 2) 2 2 g(< ≤ f (t)dt (d) 2 < g(2) < 4 26. Let f: (0, ∞) → R and F(x2) = If F(x2) = x2 (1 + x), then f(4) equals-2x0∫ (a) 5 (b) 7 (c*) 4 (d) 2 4 27. The integral 1212[x]n−+∫l 1 x 1 x ⎛ + ⎞ ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ − ⎝ ⎠ ⎝ ⎠ dx equals-pankaj.baluja@mathematicsonline.co.in IIT JEE Screening Exam Questions -38 -Visit us at: www.mathematicsonline.co.in Mathematics Online : It’s all about Mathematics MATHEMATICS ONLINE IIT JEE SCREENING QUESTIONS CHAPTER WISE (a*) –1/2 (b) 0 (c) 1 (d) 2ln(1/2) 28. Let T > 0 be a fixed real number. Suppose f is a continuous function such that for al x ∈ R, f(x + T) = f(x). If I = then the value of is-T0∫f (x)dx 2x)dx 33T3f(+∫ (a) –3/2 I (b) ± 1/2 (c*) ± 12 (d) 0 and 1 29. Let f(x) = x1∫2 2 t dt. − Then the real roots of the equation x2 – f ’(x) = 0 are-(a*) ± 1 (b) ±1/(c) ± 2 12 n t) dt, (d) 0 and 1 30. I(m, n) = then Im, n = ? 1m0t(1+∫ (a) I(m, n) = (m 1, n 1) m 1 + − + + In.m1 (b) (m(m,n)I1I.m1 1, n 1) m 1 + − =+ + (c*) n(m,n)n.I2I1m (m 1, n 1) m 1 + − + + =− (d)n(m(m,n)n.I2I1m 1, n 1) m 1 + − + + =+ 31. If 2t0xf(x)dx∫ 5 2 t 5 = for t > 0, then f(4/25) is-(a) – 25 (b) 0 (c*) 25 (d) 1 10 1 x dx 1 x −+ ∫ 32. equals to-(a) 1 2π 1 2π − + (b*) (c) 1 (d) π 33. 0322[x3x3x3(x1)cos(x1−+++++∫ )]dx + = sinx = − ∫ (a*) 4 (b) 0 (c) –1 (d) 1 34. 0 ≤ x ≤ 12sinxtf(t)dt1;2π, then f 13 ⎛ ⎞⎟ ⎝ ⎠ ⎜ is-13 3 (c) 1 (d) (a*) 3 (b)ANSWER KEY Q.No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 Ans. d a d c d a a d a c b c d a c a a Q.No. 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 Mathematics Online : It’s all about Mathematics pankaj.baluja@mathematicsonline.co.in IIT JEE Screening Exam Questions -39 -Visit us at: www.mathematicsonline.co.in MATHEMATICS ONLINE IIT JEE SCREENING QUESTIONS CHAPTER WISE Ans. a a a c c b c b c a c a c c b a a 25. AREA UNDER THE CURVE 2 2 1 x + 1. The area between the curves y = x2 and y = is-(a) π – 13 23 (b) π – 2 (c*) π – (d) π + 23 2. The area of the region bounded by y = |x – 1| and y = 1 is (a*) 1 (b) 2 (c) 1/2 (d) None of these 3. The slope of the tangent to the curve y = f(x) at a point (x, y) is 2x + 1 and the curve passes through (1, 2). The area of the region bounded by the curve, the x-axis and the line x = 1 is-(a) 5/3 units (b*) 5/6 units (c) 6/5 units (d) 6 units 4. Let An be the area bounded by the curve y = (tan x)n and the lines x = 0, y = 0, x = π/4. If n ≥ 2, then An + An + 2 is equal to-(a*) 1n 1 + 1n (b) (c) 1 n 1 − (d) None of these 1b 5. If the area bounded by the curves y = x – bx2 and y = x2, where b > 0 is maximum, then b = (a) 0 (b*) 1 (c) 2 (d) None of these 6. Let f(x) = Maximum [x2, (1 – x)2, 2x (1 – x)] where 0 ≤ x ≤ 1. The area of the region bounded by the curves y = f(x), x-axis x = 0 and x = 1 is-17 27 14 27 (a*) (b) (c) 19 27 (d) None of these 7. For which of the following values of m, is the area of the region bounded by the curve y = x – x2 and the line y = mx equals 9/2 (a) –4 (b) –2 (c) 2, –4 (d*) 4, – 2 8. The triangle formed by the tangent to the curve f(x) = x2 + bx – b at the point (1, 1) and the coordinates axes, lies in the first quadrant. If its area is 2, then the value of b is-(a) –1 (b) 3 (c) – 3 (d) 1 9. The area bounded by the curves y = |x| – 1 and y = – |x| + 1 is-(a) 1 (b) 2 (c*) 2 (d) 4 2 x , x = 2y + 3 & x-axis lying in 1st quadrant is-10. Area of the region bounded by y = (a)23 (b) 18 (c*) 9 (d) 34/3 11. The are a of quadrilateral formed by tangents at the ends of latus rectum of ellipse 2 2 x y 1 9 5 + = is-27 4 (d) 27 2 (a) 9 (b*) 27 (c) pankaj.baluja@mathematicsonline.co.in IIT JEE Screening Exam Questions -40 -Visit us at: www.mathematicsonline.co.in Mathematics Online : It’s all about Mathematics MATHEMATICS ONLINE IIT JEE SCREENING QUESTIONS CHAPTER WISE 12. If area bounded by the curves x = ay2 and y = ax2 is 1, then a equals-13 (a*) 13 (b) (c) 12 16 (d) 14 13. Find the area between the curves y = (x – 1)2, y = (x + 1)2 and y = (a*) 13 (b) 23 (c) 43 (d) 16 ANSWER KEY Q.No. 1 2 3 4 5 6 7 8 9 10 11 12 13 Ans. c a b a b a d c c c b a a 26. DIFFERENTIAL EQUATION 1. The differential equation whose solution is (x – h)2 + (y – k)2 = a2 is (where a is a constant)-(a) 32dx1dx⎡⎤⎛⎞+⎢⎥⎜⎟⎝⎠⎢⎥⎣⎦ 2 2 2 d y a dx = (b*)322dy1adx⎡⎤⎛⎞+=⎢⎥⎜⎟⎝⎠⎢⎥⎣⎦ 2 2 2 d y dx ⎛ ⎞ ⎜ ⎟ ⎝ ⎠ 2 3 2 2 2 dy d y 1 a dx dx ⎛ ⎞ ⎡ ⎤ ⎛ ⎞ + = ⎜ ⎟ ⎜ ⎟ ⎢ ⎥ ⎝ ⎠ ⎣ ⎦ ⎝ ⎠ (d) None of these (c) dy dx 2. The solution of the differential equation (2x – 10y3) + y = 0 is-(a) x + y = ce2x (b) y2 = 2x3 + c (c*) xy2 = 2y5 + c (d) x(y2 + xy) = 0 3. A curve y = f(x) passes thro’ the point P(1, 1). The normal to the curve at P is a (y – 1) = 0. If the slope of the tangent at any point on the curve is proportional to the ordinate of the point, then the equation of the curve is-(a*) y = eK(x – 1) (b) y = eKx (c) y = eK(x – 2) (d) None of these dy dx 4. The equation of the curve passing through origin and satisfying the differential equation = sin (10x + 6y) is-(a*)115tanytan343tan−⎛⎜− 4x 5x 4x 3 ⎞ = −⎟ ⎝ ⎠ (b)115tanytan343tan−+ 4x 5x 4x 3 ⎛ ⎞ = − ⎜ ⎟ ⎝ ⎠ (c) 113tanytan343tan−+=− 4x 5x 4x 3 ⎛ ⎞− ⎜ ⎟ ⎝ ⎠ 5 x c e + (d) None of these 5. A curve C has the property that if the tangent drawn at any point P on C meets the coordinate axis at A and B, then P is the midpoint of AB. If the curve passes through the point (1, 1) then the equation of the curve is-(a) xy = 2 (b) xy = 3 (c*) xy = 1 (d) None of these 6. The order of the differential equation whose general solution is given by y = (c1 + c2) cos (x + c3) – c4 , where c1, c2, c3, c4, c5 are arbitrary constant is-(a) 5 (b) 4 (c*) 3 (d) 2 7. The differential equation representing the family of curve y2 = 2x (x, where c is a positive parameter, is of-c) + (a*) Order 1, degree 3 (b) Order 2, degree 2 (c) Degree 3, order 3 (d) Degree 4, order 4 Mathematics Online : It’s all about Mathematics pankaj.baluja@mathematicsonline.co.in IIT JEE Screening Exam Questions -41 -Visit us at: www.mathematicsonline.co.in MATHEMATICS ONLINE IIT JEE SCREENING QUESTIONS CHAPTER WISE 8. The solution of the differential equation 2dydyxdxdx⎛⎞ y 0 − + = ⎜⎟⎝⎠ is-(a) y = 2 (b) y = 2x (c*) y = 2x – 4 (d) y = 2x2 – 4 dy dt 9. Let (1 + t) – ty = 1, y (0) = – 1. find y(t) t = 1 ? (a*) 12 − (b) 12 1 e 2 − (c) (d) 12 e+ 10. If y = y(x) satisties 2sin1y++ x dy dx ⎛ ⎞ ⎜ ⎟ ⎝ ⎠ = – cos x such that y(0) = 1 then y (π/2) is equal to-(a) 3/2 (b) 5/2 (c*) 1/3 (d) 1 11. (x2 + y2) dy = xy dx (initial value problem), y > 0, x > 0, y (1) = 1, y(x0) then find x0 = ? (a) 2 e 1 2− (b) 2 2e 1 − (c) 2 2 e − (d*) 3e 12. xdy – ydx = y2dy, y > 0 & y(1) = 1 then find y (–3) = ? (a*) 3 (b) 2 (c) 4 (d) 5 ANSWER KEY Q.No. 1 2 3 4 5 6 7 8 9 10 11 12 Ans. b c a a c c a c a c d a 27. VECTOR 1. A unit vector coplanar with i + j + 2k and i + 2j + k and perpendicular to i + j + k is (a*) j k2 − j k2 − ± (b) (c) j k2 + − (d) None of these 2. A unit vector in xy-plane that makes an angle of 45º with the vector i + j and an angle of 60º with the vector 3i – 4j is (i j) 2 + (c) (i j) 2 − (d*) None of these (a) i (b) 12 3. If x and y are two unit vectors and φ is the angle between them, then |x – y| is equal to (a) 0 (b) π/2 (c) 12 sinφ (d*) 1 s 2 coφ 4. Let a, b, c be distinct non-negative numbers. If the vectors ai + aj + ck, I + k and ci + cj + bk lie in a plane, then c is-(a) The Arithmetic Mean of a and b (b*) The Geometric mean of an and b (c) The Harmonic mean of and b (d) Equal to zero+ pankaj.baluja@mathematicsonline.co.in IIT JEE Screening Exam Questions -42 -Visit us at: www.mathematicsonline.co.in Mathematics Online : It’s all about Mathematics MATHEMATICS ONLINE IIT JEE SCREENING QUESTIONS CHAPTER WISE b→r a b → → → × = 5. If the non-zero vectors and a→ are perpendicular to each other, then the solution of the equation is -(a*) 1rxaa.a→→→→=+ (a b) → → × (b) 1rxb (a b) b.b→→→→ → → = + × a b → → → = × b a → → → (c) rx (d) rx= × 6. Let α, β, γ be distinct real numbers. The points with position vectors αi + βj + γk, βi + γj + αk, γi + αj + βk-(a) Are collinear (b*) Form an equilateral triangle (c) Form an isosceles triangle (d) Form a right angled triangle 13 7. The vector (2i – 2j + k) is (a*) A unit vector (b) Makes an angle π/3 with the vector 2i –4j + 3k (c) Parallel to the vector 3i + 2j – 2k (d) None of these 8. Let a = i – j, b = j – k, c = k – i. If d is a unit vector such that a. d = 0 = [b, c, d], then d equals (a*) ± i j 2 6 + − k (b)± i j k 3 + − (c)i j k 3 + + a, b, c → → → (d) ± k 9. Let u, v, w be vectors such that u + v + w = 0. If |u| = 3, |v| = 4, |w| = 5. Then the value of the u.v. + v. w + w. u is-(a*) 47 (b) – 25 (c) 0 (d) 25 10. A, B and C are three non coplanar vectors, then (A + B + C). ((A + B) × (A + C)) equals (a) 0 (b) [A, B, C] (c) 2[A, B, C] (d*) – [A, B, C] b c 2 → → a(bc)→→→ + ×× = then the angle between 11. If are non-coplanar unit vectors such that and a→b→ is-(a*) 34π (b) 4π (c) 2 π (d) π 12. A vector has components 2p and 1 with respect to a rectangular Cartesian system. The sytem is rotated thro’a certain angle about the origin in the counterclockwise sense. If, with respect to new system, has components p + 1 and 1, then a→ a→ 13 13 (d) p = 1 or p = – 1 (c) p = – 1 or p = (a) p = 0 (b*) p = 1 or p = – 13. If b→ and are any two perpendicular unit vectors and is any vector, then c→a→ 2a.(bc)(a.b)c(a.c)b|bc|→→→→→→→→→→→×++× (b c) → →× is equal to-(a) b→→ → (b*) a (c) c (d) None of these pankaj.baluja@mathematicsonline.co.in IIT JEE Screening Exam Questions -43 -Visit us at: www.mathematicsonline.co.in Mathematics Online : It’s all about Mathematics MATHEMATICS ONLINE IIT JEE SCREENING QUESTIONS CHAPTER WISE 14. Let where O, A, c are non-collinear. Let p denote the are of the quadrilateral OABC and q denote the area of the parallelogram with OA and OC as adjacent sides. The OAa,OB10aand→→==uuuruuur OC b→= uuur pq is equal to-(a) 4 (b*) 6 (c) 1|2 a b | | a | → → → − (d) None of these 15. Let be three mutually perpendicular vectors of the same magnitude. If a vector satisfies the equation. , then is given by p, q, r → → → x→ r ] → → → → → → → → → → → → × − × + × − × + × − × x→p[(xq)p]q[(xr)qr[(xp)0→= (a)1(p q 2 2 → → + − r)→ (b*) 1 (p q r ) 2 → → → + + (c)1(pq r ) 3 → → → + + (d)1(2p3 q r ) → → → + − b and c→| | = a b) (a c)] (b c).(b c) → → → → → → → → + × + × × + a, b and c → → → b 0 → → 16. If are vectors such that |b, then [( = a,→→c|→→ (a) 1 (b) –1 (c*) 0 (d) None of these 17. If be three vectors having magnitudes 1, 1 and 2 respectively. If a(ac)→→→××− = a and c → → , then the acute angle between is-(a) 4π 6π (b*) (c) 3π (d) None of these 3 18. If a = i + j + k, b = 4i + 3j + 4k and c = I + αj + βk are linearly dependent vectors and |c| =, then (a) α = 1, β = – 1 (b) α = 1, β = ± 1 (c) α = – 1, β = ± 1 (d*) α = ± 1, β = 1 19. For three vectors u, v, w which of the following expressions is not equal to any of remaining three ? (a) u. (v × w) (b) (v × w). u (c*) v. (u × w) (d) (u × v). w 20. Which of the following expression of meaningful ? (a*) u. (v × w) (b) (u. v). w (c) (u . v)w (d) None of these 2 21. Let a = 2i + j – 2k and b = I + j. if c is vector such that a . c = |c|, |c – a| = 2 and the angle between (a × b) and c is 30º. Then |(a × b) × c| = (a) 23 32 (c) 2 (d) 3 (b*) 22. Let a = 2i + j + k, b = I + 2j – k and a unit vector c be coplanar. If c perpendicular to a, the c = (a) 12 13 15 (–j + k) (b) (–i – j –k) (c*) (i – 2j) (d) 13 u and v → → w→u) → → (i – j – k) 23. Let a and b be two non-collinear unit vectors. If u = a – (a . b) b and v = a × b, then |v| is (a) |u| + |u + a| (b*) |u| + |u . a| (c) |u| + |u . b| (d) |u| + u . (a + b) 24. Let be unit vectors. If is a vector such that + (ww→v→+ = u v).w| → → → × , then |( (a) ≤ 1/3 (b*) ≤ 1/2 (c) >1/3 (d) ≥ 1/2 25. If the vector a,b and c form the sider BC, CA and AB respectively of a triangle ABC, the-(a*) a . b + b . c + c . a = 0 (b) a × b = b × c = c × a pankaj.baluja@mathematicsonline.co.in IIT JEE Screening Exam Questions -44 -Visit us at: www.mathematicsonline.co.in Mathematics Online : It’s all about Mathematics MATHEMATICS ONLINE IIT JEE SCREENING QUESTIONS CHAPTER WISE (c) a . b = b . c = c . a (d) a × b + c × c + c × a = 0 26. Let the vector a, b, c and d be such that (a × b) × (c × d) = 0. Let P1 and P2 be planes determined by the pairs of vectors a, b, c and d respectively. Then the angle between P1 and P2 is-(a*) 0 (b) π/4 (c) π/3 (d) π /2 27. Let be the position vectors of three vertices. A, B, C of a triangle respectively. Then the area of this triangle is given by-a, b, c → → → c a → → × (a) ab (b)bc→→→→×+×+1(a2× b). c → → → (c*)1|abbcc→→→×+×+ a a | → → → × a, b, c → → → 2 2 c a | → → + − a and b → → 5a 4b → → − a and b → → (d) None of these 28. Let a = i – k, b = xi + j + (1 – x)k and c = yi + xj + (1 + x – y)k. Then [a b c] depends on-(a) only x (b) only y (c*) neither x nor y (d) both x and y 29. If are unit vectors , then does not exceed-2|ab||bc||→→→→−+− (a) 4 (b*) 9 (c) 8 (d) 6 30. If are two unit vectors such that are perpendicular to each other then the angle between is-a2band→→+ 1 1 s 3 − ⎛ ⎞ ⎜ ⎟ ⎝ ⎠ co (d) (a) 45º (b*) 60º (c) 1 27 − ⎛ ⎞ ⎝ ⎠ k → → → → → → = + − = U → VW] → → → co⎜⎟ s 31. Let . If is a unit vector; then the maximum value of the scalar triple product is-V2ijkandWi3→+[U (a) –1 (b)10 6 + (c*)59 (d)60 32. If → = I + aj + k; a b→c→ = j + ak; = ai + k, then find the value of ‘a’ for which volume of parallelepiped formed by these three vectors as coterminous edges, is minimum. (a) 3 (b) 3 (c*) 13 (d) 13 = + + → → × = 33. If → and then aijka.b1→→=abjk− b→ is equal to-(a) 2i (b) I – j + k (c*) i (d) 2j – k 34. A unit vector is orthogonal to 5i – 2j + 6k and is coplanar to 2i – 5j + 3k and I – j + k then the vector, is-(a*) 3j k 10 − (b) 2j 5k 29 + (c) 6j (d)2i2j k 3 + − 5k 61 − ANSWER KEY Q.No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 Ans. a d c b a b a a b d a b b b b c B Q.No. 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 Mathematics Online : It’s all about Mathematics pankaj.baluja@mathematicsonline.co.in IIT JEE Screening Exam Questions -45 -Visit us at: www.mathematicsonline.co.in MATHEMATICS ONLINE IIT JEE SCREENING QUESTIONS CHAPTER WISE Ans. d c a b a c b b a c c b b c c c a 28. PROBABILITY 1. India plays two matches each with West-indies and Australia. In any match the probability of India getting points 0, 1 and 2 are 0.45, 0.05 and 0.50 respectively. Assuming that the out comes are independent, the probability of Indies getting at least 7 points is-(a) 0.8750 (b*) 0.0875 (c) 0.0626 (d) 0.0250 2. An unbiased die with faces marked 1,2,3,4,5 and 6 is rolled four times. Out of four face values obtained, the probability that the minimum face value is not less than 2 and the maximum face value is not greater than 5 is then-(a*) 16/81 (b) 1/81 (c) 80/81 (d) 15/81 3. Let E and F be two independent events. The probability that both E and F happen is 1/12 and the probability that neither E nor F happens is 1/2. Then-(a*) p(E) = 1/3, p(F) = 1/4 (b) p(E) = 1/2, p(E) = 1/6 (c) p(E) = 1/6, p(F) = 1/2 (d) None of these 4. You are given a box with 20 cards in it. 10 of these cards have letter I printed on them. The other ten have the letter T printed on the. If you pick up 3 cards at random and keep them in same order, the probability of making the word I.I.T. is-(a) 9 80 (b*) 18 (c) 427 (d) 538 5. Three identical dice are rolled. The probability that the same number will appear on each of them is-16 1 36 (a) (b*) (c) 1 18 (d) 3 28 6. The probability of India winning a test match against West Indies is ½. Assuming independence from match to match the probability that in a 5 match series. India’s second with occurs at the third test is-(a*) 1/8 (b) 1/4 (c) 1/2 (d) 1/3 7. Three of the six vertices of a regular hexagon are chosen at random. The probability that the triangle with these vertices is equilateral, equals-(a) 1/2 (b) 1/5 (c*) 1/10 (d) 1/20 8. Three numbers are chosen at random without replacement from {1, 2, 3,…10}. The probability that the minimum of the chosen numbering is 3 or their maximum is 5, (a*) 7/40 (b) 5/40 (c) 11/40 (d) None of these 9. Seven white balls and three black balls are randomly placed in a row. The probability that no two black balls are placed adjacently equals-(a) 1/2 (b*) 7/15 (c) 2/15 (d) 1/3 10. If from each of the three boxes containing 3 2hite and 1 black, 2 2hite and 2 black, white and 3 black balls, one ball is drawn at random, then the probability that 2 white and 1 black ball will be drawn is-(a*) 13/32 (b) 1/4 (c) 1/32 (d) 3/16 11. There are four machines and it is known that exactly two of them are faulty. They are tested, one by one, in a random order till both the faulty machines are identified. Then the probability that only two tests are needed is-(a) 1/3 (b*) 1/6 (c) 1/2 (d) 1/4 12. A fair coin is tossed repeatedly. If tail appears on first four tosses, then the probability of head appearing on fifth toss equals-(a*)1/2 (b) 1/32 (c) 31/32 (d) 1/5 Mathematics Online : It’s all about Mathematics pankaj.baluja@mathematicsonline.co.in IIT JEE Screening Exam Questions -46 -Visit us at: www.mathematicsonline.co.in MATHEMATICS ONLINE IIT JEE SCREENING QUESTIONS CHAPTER WISE 13. If the integers m and n are chosen at random between 1 and 100, then the probability that a number of the form 7m + 7n is divisible by 5 equals-(a) 1/4 (b) 1/7 (c*) 1/8 (d) 1/49 14. The probabilities that a student passes in Mathematics, Physics and Chemistry are m, p and c, respectively. Of these subjects the student have a 75% chance of passing in atleast one, a 50% change of passing in atleast two, and a 40% chance of passing in exactly two. Which of the following relations are true ? (a) p + m + c = 19/20 (b*) p + m + c = 27/20 (c) pmc = 1/4 (d) None of these 15. A coin has probability p of showing head when tossed. It is tossed in times. Let pn denote the probability that no two (or more) consecutive heads occurs, then (a) p1 = 1 (b) p2 = 1 – p2 (c) pn = (1 – p)pn –1 + p(1 –p)pn–2 for all n ≥ 3 (d*) All of these P(A B C) ∩ ∩ 16. Given that P(B) = ¾, P(A ∩ B ∩ C) = 1/3, = 1/3 then find probability of B ∩ C, when A,B,C are negotiations of A,B,C respectively, is (a) 2/3 (b*) 1/12 (c) 1/15 (d) 1/4 17. Two numbers are chosen, one by one (with out replacement) from the set of numbers A = {1, 2, 3, 4, 5, 6} The probability that minimum value of chosen number is less than 4 is (a) 1/15 (b) 14/15 (c) 1/5 (d*) 4/4 18. Three distinct numbers are chosen randomly from first 100 natural number, then probability that all are divisible by 2 and 3 both is (a) 4/33 (b) 4/35 (c) 4/25 (d*) 4/115 19. While throwing a dice getting one an even no. of throws has probability P, then P is equal to (a) 1/6 (b) 5/36 (c) 6/11 (d*) 5/11 ANSWER KEY Q.No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Ans. b a a d b a c a b a b a c b D Q.No.16 17 18 19 Ans. b d d d 29. MATRICES & DETERMINANTS 1. If Dr = r1r1nn22.32131−−αβ−− r 1 n4.5 5 1− γ− n r r 1 D = , then the value of ∑ (a*) 0 (b) α β γ (c) α + β + γ (d) α.2n + β.3n + γ.4n 2. If a, b,c are in G.P., then the value of determinant ∆ =ab ax b b c bx c 0 axbbxc ++ ++ is-(a) 1 (b*) 0 (c) –1 (d) None of thees Mathematics Online : It’s all about Mathematics pankaj.baluja@mathematicsonline.co.in IIT JEE Screening Exam Questions -47 -Visit us at: www.mathematicsonline.co.in MATHEMATICS ONLINE IIT JEE SCREENING QUESTIONS CHAPTER WISE 3. If a1, a2……..form a G.P. and ai > 0 for all i ≥ 1, then ∆ = mm1m3m4m6m7logalogalogalogalogalogalogalogalogam 2 m 5 m 8 + + ++ + + + + is equal to-(a) log (am + 8) – log (am) (b) log (am + 8) + log am (c*) zero (d) log2 am + 4 4. If the system of equations x + ay + az = 0; bx + y + bz = 0; cx + cy + z = 0 where a, b and c are non-zero and non-unity, has a non-trivial solution, then the value of ab1a1 c b 1 c + + − − − is (a) zero (b) 1 (c*) –1 (d) 2 2 2 abc c ab+ + 5. If ω(≠ 1) is a cube root of unity, then 2 2 2 1 1 11i1i1ii1++ω−−−−+ω− ω ω − − equals (a) 0 (b) 1 (c*) i (d) ω 6. The determinant xpyxypzy0xpy++ yz yp z + + = 0 if-(a) x, y, z are in A.P. (b*) x, y, z are in G.P. (c) x, y, z are in H.P. (d) xy, yz, zx are in A.P. 7. Let f(x) = 3xsinx61pp−2 3 cosx 0p where p is a constant. Then 33 d dx [f(x)] at x = 0 is-(a) p (b) p + p2 (c) p + p3 (d*) independent of p 8. The parameter on which the value of the determinant 2 ap d)x p d)x + − + 1acos(pd)xcospxcos(sin(pd)xsinpxsin(− does not depend upon is-(a) a (b*) p (c) d (d) x 9. If 6i43203−3i 1 i 1i − = x + iy, then-(a) x = 3, y = 1 (b) x = 1, y = 3 (c) x = 0, y = 3 (d*) x = 0, y = 0 10. If f(x) = 1x2xx(x1)(x3x(x1)x(x1)(x2)(x1)x(x+−−−− x 1 1)x 1) + + − then f(100) is equal to-(a*) –1, 2 (b) 1, 2 (c) 0, 1 (d) –1, 1 pankaj.baluja@mathematicsonline.co.in IIT JEE Screening Exam Questions -48 -Visit us at: www.mathematicsonline.co.in Mathematics Online : It’s all about Mathematics MATHEMATICS ONLINE IIT JEE SCREENING QUESTIONS CHAPTER WISE 11. If the system of equations x – Ky – z = 0, Kx – y – z = 0, x + y – z = 0 has a non-zero solution, then the possible values of K are-(a) –1, 2 (b) 1, 2 (c) 0, 1 (d*) –1, 1 12. The number of distinct real roots of sinxcosxcosxsinxcosxcosx cos x cos x sin x = 0 in the interval – 4π ≤ x ≤ 4π is-(a) 0 (b) 2 (c*) 1 (d) 3 13. The number of values of K for which the system of equations, (K + 1)x + 8y = 4K and Kx + (K + 3) y = 3K – 1 has infinitely many solutions, is (a) 0 (b*) 1 (c) 2 (d) Infinite 14. Let ω = –1 3 i 2 2 + . Then the value of the determinant ∆ = 2 2 2 41 11111−−ω ω ω ω α⎢ 1 0 5 1 ⎡ ⎤ ⎢ ⎥ ⎣ ⎦ 2 2α ⎡ ⎤ ⎢ ⎥ α ⎣ ⎦1 0 0 0 2 − 1 0 0 0 1 0 0 0 1 ⎡ ⎤ ⎢⎢⎢ ⎥ ⎣ ⎦ is (a) 3ω (b*) 3ω (ω – 1) (c)3 ω2 (d) 3ω(1 – ω) 15. If A = , B = and A2 = B, then 011⎡⎤⎥⎣⎦ (a*) Statement is not true for any real value of α (b) α = 1 (c) α = – 1 (d) α = 4 16. If x + ay = 0; y + az = 0; z + ax = 0, then value of ‘a’ for which system of equations will have infinite number of solution is (a) a = 1 (b) a = 0 (c*) a = – 1 (d) no value of a 17. = A &|A3| = 125, then α is-(a) 0 (b) ±2 (c*) ± 3 (d) ± 5 18. If the system of equations 2x – y – 2z = 2; x – 2y + z = – 4; x + y + λz = 4 has no solutions then λ is equal to (a) –2 (b) 3 (c) 0 (d*) –3 19. Let A = & I = and A–1 = 0112⎡⎤⎢⎥⎢⎥⎢⎥⎣⎦⎥⎥ 16 1 1 0 1 ⎢ [A2 + cA + dI], find ordered pair (c, d) ? (a) (6, 11) (b) (–6, –11) (c*) (–6, 11) (d) (6, –11) 3 1 2 2 1 3 2 2 ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ − ⎢ ⎥ ⎣ ⎦ 20. Let a matrix A = & P = ⎡⎤⎥⎣⎦Q = PAPT where PT is transpose of matrix P. Find PT Q2005 P is Mathematics Online : It’s all about Mathematics pankaj.baluja@mathematicsonline.co.in IIT JEE Screening Exam Questions -49 -Visit us at: www.mathematicsonline.co.in MATHEMATICS ONLINE IIT JEE SCREENING QUESTIONS CHAPTER WISE (a*) (b) 1 2005 0 1 ⎡ ⎤ ⎢ ⎥ ⎣ ⎦ 120053601514200512005 3 ⎡ ⎤ +⎢ ⎥ −⎢ ⎥ ⎣ ⎦ (c)120053214200512005+⎢ 005 3 ⎡ ⎤ − ⎢ ⎥ ⎣ ⎦ 2005 2005 0 1 ⎥ (d) ⎡ ⎤ ⎢ ⎥ ⎣ ⎦ ANSWER KEY Q.No. 1 2 3 4 5 6 7 8 9 10111213141516 17 18 19 20Ans. a b c c c b d b d a d c b b a c c d c a 30. ELLIPSE 1. Let P be a variable point on the ellipse 2 2 2 2 x y 1 a b + = with foci F1 and F2. If A is the area of the triangle PF1F2, then the maximum value of A is-12 abc (d) None of these (a) 2abe (b*) abe (c) 2. Let E be the ellipse 2 2 x y 1 9 4 + = and C be the circle x2 + y2 = 9. Let P and Q be the points (1, 2) and (2, 1) respectively. Then (a) Q lies inside C but outside E (b) Q lies outside both C and E (c) P lies inside both C and E (d*) P lies inside C but outside E 3. The radius of the circle passing thro’ the foci of the ellipse 22xy169 1 + = and having its centre (0, 3) is-(a*) 4 (b) 3 (c) 12 (d) 72 4. If P (x, y), F1 = (3, 0), F2 (–3, 0) and 16x2 + 25y2 = 400, then PF1 + PF2 equals-(a) 8 (b) 6 (c*) 10 (d) 12 5. On the ellipse 4x2 + 9y2 = 1, then points at which the tangents are parallel to 8x = 9y are-(a) 2 1 , 5 5 ⎛ ⎞⎟ ⎝ ⎠ ⎜ (b*) 21,or55⎛⎞ 2 1,5 5 ⎛ ⎞ − −⎜⎟ ⎝ ⎠ ⎜⎟⎝⎠ (c) 2 1 , 5 5 ⎛ ⎞ − − ⎟ ⎝ ⎠ ⎜ (d) 3 2 , 5 5 ⎛ ⎞ − − ⎜⎟ ⎝ ⎠ 6. An ellipse has OB as semi-minor axis. F and F’ are its foci and the angle FBF’ is a right angle. Then the eccentricity of the ellipse is-12 (a) (b*) 12 (c) 23 (d) 13 7. A tangent to the ellipse x2 + 4y2 = 4 meets the ellipse x2 + 2y2 = 6 at P and Q. The angle between the tangents at P and Q of the ellipse x2 + 2y2 = 6 is-Mathematics Online : It’s all about Mathematics pankaj.baluja@mathematicsonline.co.in IIT JEE Screening Exam Questions -50 -Visit us at: www.mathematicsonline.co.in MATHEMATICS ONLINE IIT JEE SCREENING QUESTIONS CHAPTER WISE 2π (a*) (b) 3π (c) 4 π (d) 6 π 8. The number of values of c such that the straight line y = 4x + c touches the curve 2x4 2 y 1 + = is (a) 0 (b) 1 (c*) 2 (d) infinite 9. Locus of middle point of segment of tangent to ellipse x2 + 2y2 = 2 which is intercepted between the coordinate axes, is-(a*) 221= 1 1 2x 4y + (b) 2 2 1 1 1 4x 2y += (c) 2 2 x y 1 2 4 + = (d) 2 2 x y 1 4 2 + = 10. A tangent is drawn at some point P of the ellipse 2 22 x y 1 a b 2+ = is intersecting to the coordinate axes at points A & B then minimum area of the ∆PAB is-(a*) ab (b) 2 2 a b 2+ (c) 2 2 a b 4+ (d) 2 2 ab 3 ab+ − ANSWER KEY Q.No. 1 2 3 4 5 6 7 8 9 10Ans. b d a c b b a c a a 31. HYPERBOLA 1. A variable straight line of slope 4 intersects the hyperbola xy = 1 at two point. The locus of the point which divides the line segment between these two points in the ratio 1 : 2 is (a*) 16x2 + 1= xy + y2 = 2 (b) 16x2 – 10 xy + y2 = 2 (c) 16x2 + 10 xy + y2 = 4 (d) None of these 2. If the circle x2 + y2 = a2 intersects the hyperbola xy = c2 in four points P(x1, y1), Q (x2, y2), R(x3, y3), S(x4, y4), then (a*) x1 + x2 + x3 + x4 = 0 (b) y1 + y2 + y3 + y4 = 2 (c) x1x2x3x4 = 2c4 (d) y1y2y3y4 = 2c4 3. If a circle cuts the rectangular hyperbola xy = 1 in the points (x1, yr) whre r = 1, 2, 3, 4, then (a) x1x2x3x4 = 2 (b*) x1x2x3x4 = 1 (c) x1+x2+x3+x4 = 0 (d) y1 + y2 + y3 + y4 = 0 4. If x = 9 is the chord of contact of the hyperbola x2 – y2 = 9, then the equation of the corresponding pair of tangents is-(a) 9x2 – 8y2 + 18 x –9 = 0 (b*) 9x2 – 8y2 – 18x + 9 = 0 (c) 9x2 – 8y2 – 18x – 9 = 0 (d) 9x2 – 8y2 + 18x + 9 = 0 2π 5. Let P (a sec θ, b tan) and Q (a sec φ, b tan φ) where θ + φ = , be two points on the hyperbola 2 2 2 2 x y 1 a b − = . If (h, k) is the point of intersection of the normals at P and Q, then K is equal to-(a) 2 2 a b a+ (b) – 2 2 a b a+ (c) 2 2 a b b+ (d*) – 2 2 a b b+ pankaj.baluja@mathematicsonline.co.in IIT JEE Screening Exam Questions -51 -Visit us at: www.mathematicsonline.co.in Mathematics Online : It’s all about Mathematics MATHEMATICS ONLINE IIT JEE SCREENING QUESTIONS CHAPTER WISE 6. 2 22 1 = α α 2xycossin− represents family of hyperbolas, where α varies then (a) e remains constant (b*) Abscissas of foci remain constant (c) equation of directrices remain constant (d) Abscissas of vertices remains constant 6 y = 2 touches the curve x2 – 2y2 = 4, is-7. The point at which the line 2x + (a*)(4 (b) , 6) − (6 (c) ,1) 1 1 , 6 ⎛ ⎞⎟ ⎝ ⎠ 2⎜ (d) , 6π ⎛ ⎞ π ⎝ ⎠ ⎜⎟ ANSWER KEY Q.No. 1 2 3 4 5 6 7 Ans. a a b b d b a 32. 3-DIMENSIONAL GEOMETRY 1. If line x4y211−−==2 k2− lies in the plane 2x – 4y + z = 7 then the value of k = ? (a) k = – 7 (b*) k = 7 (c) k = – 7 (d) no value of k 2. Two lines x1y1z1x3and2341−+−−===y k z 2 1 − = intersect at a point then k is-(a) 3/2 (b*) 9/2 (c) 2/9 (d) 2 3. A plane at a unit distance from origins cuts at three axes at P, Q, R points. ∆PQR has centroid at (x, y, z) point and satisfies to 2 2 2 1 1 1 k x y z + + = , then k = (a*) 9 (b) 1 (c) 3 (d) 4 ANSWER KEY Q.No.1 2 3 Ans.b b a