1995 (AIEEE / IIT-JEE, in the JEE Main bank) questions

Mathematics

1. The value of ∫ (cos^3 x + cos^5 x)/(sin^2 x + sin^4 x) dx is

   Indefinite Integration

2. Let z and w be two complex numbers such that |z| ≤ 1, |w| ≤ 1 and |z + iw| = |z - iw| = 2, then z equals

   Complex Numbers and Quadratic Equations

3. If 0 < P(A) < 1, 0 < P(B) < 1 and P(A ∪ B) = P(A) + P(B) - P(A)P(B), then

   Probability · with solution

4. Let z and w be two non-zero complex numbers such that |z| = |w| and arg(z) + arg(w) = π, then z equals

   Complex Numbers and Quadratic Equations

5. If a, b, c are non-coplanar unit vectors such that a × (b × c) = dfrac(b + c)√(2), then the angle between a and b is

   Vector Algebra · with solution

6. If A and B are square matrices of equal degree, then which one is correct among the following?

   Matrices and Determinants · with solution

7. If ω (≠ 1) is a cube root of unity and (1 + ω)^7 = A + Bω, then A and B are respectively

   Complex Numbers and Quadratic Equations · with solution

8. In a triangle ABC, ∠ B = (π)/3 and ∠ C = (π)/4. Let D divides BC internally in the ratio 1:3, then (sin ∠ BAD)/(sin ∠ CAD) is equal to

   Trigonometry · with solution

9. If a, b and c are three non-coplanar vectors, then (a + b + c) · [(a + b) × (a + c)] equals

   Vector Algebra

10. 3(sin x - cos x)^4 + 6(sin x + cos x)^2 + 4(sin^6 x + cos^6 x) equals

    Trigonometry · with solution

11. 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

    Vector Algebra · with solution

12. The general value of θ satisfying the equation 2sin^2θ - 3sinθ - 2 = 0, is

    Trigonometry · with solution

13. Let u, v and w be vectors such that u + v + w = 0. If |u| = 3, |v| = 4 and |w| = 5, then u · v + v · w + w · u is

    Vector Algebra · with solution

14. Three of the six vertices of a regular hexagon are chosen at random. The probability that the triangle with three vertices is equilateral, equals

    Probability · with solution

15. The probability of India winning a test match against West Indies is 1/2. Assuming independence from match to match the probability that in a 5 match series India's second win occurs at third test, is

    Probability · with solution

16. On the interval [0,1], the function x^25(1-x)^75 takes its maximum value at the point

    Application of Derivatives · with solution

17. The value of int_0^2π [2sin x] dx, where [·] represents the greatest integral functions, is

    Definite Integrals and Areas

18. The orthocentre of the triangle formed by the lines xy = 0 and x + y = 1, is

    Straight Lines and Pair of Lines · with solution

19. If f(x) = Asin ((π x)/2 ) + B, f' (1/2 ) = √(2) and int_0^1 f(x) dx = 2A/(π), then constants A and B are

    Definite Integrals and Areas · with solution