1) The document contains an unsolved mathematics paper from 1994 containing 28 multiple choice problems testing concepts in algebra, geometry, trigonometry, and probability. 2) The problems cover topics such as equations of lines and curves, properties of triangles, intersections of circles, inverse trigonometric functions, and probability. 3) For each problem, students must select the single best answer from the four options provided.

1. IITJEE - Past papersMATHEMATICS - UNSOLVED PAPER - 1994

2. SECTION – IFill in the blanksThis question contains ten incomplete statements. In each part, fill in the blanks so that each of the resulting statements is correct.

3. 01ProblemLet n be a positive integer. If the coefficient of 2nd,3rd,and 4th terms in the expansion of are in A>P. Then the value of n is __________ .

4. Problem02Suppose Z1,Z2,Z3 are the vertices of an equilateral triangle inscribed in the circle ________ Z3= _________ .

5. Problem03In a triangle ABC,AD is the altitude from A. Given and __________ .

6. Problem04A circle is inscribed in an equilateral triangle of side a. The area of any square inscribed in the circle is _____________ .

7. Problem05The point of intersection of the tangents at the ends of the latus rectum of the parabola is __________ .

8. Problem06Let P be a variable point on the ellipse With foci .If A is the area of the triangle ,than the maximum value of A is _________ .

9. Problem07Let C be the curve Y3-3XY+2=0 .If H is the set of points on the curve C where the tangent is horizontal and V is the set of points on the curve C where the tangent is vertical. then H=фamd V= ____________ .

10. 08ProblemThe value of is ________ .

11. Problem09A unit vector perpendicular to the plane determined by the points P(1 -1,2),Q(2, 0,-1) and R (0,2,1) is ________ .

12. Problem10If two events A and B are such that __________ .

13. SECTION – ISingle Correct Answer TypeIn the following questions exactly one answer is correct.

14. Problem01The number of points of intersection of the two curves a.0b.1c.2d.

15. 02ProblemLet equalsa.b.c.d.

16. Problem03The equation representsno locus if k > 0an ellipse if k > 0a point if k = 0a hyperbola if k > 0

17. Problem04The area of the region bounded by is211/2none of these

18. Problem05Let p and q be the position vectors of P and Q respectively, with respect to O and .The points R and S divide PQ internally and externally in ratio 2:3 respectively. If OR and OS are perpendicular, thena.9p2=4q2b. 4p2=9q2c. 9p=4qd. 4p=9q

19. Problem06Let αandβ the roots of the equation x2+x+1=0 .The equation whose roots areα19 β 7 isx2-x-1=0x2-x+1=0X2+x-1=0X2+x+1=0

20. 07ProblemLet n be a positive interger such that .Thena.b.c.d.

21. Problem08Let E be the ellipse and C be the circle x2+y2=9.Let P and Q be the points (1,2) and (2,1) respectively .ThenQ lies inside C but outside EQ lies outside both C and EP lies inside both C and EP lies inside C but outside E

22. Problem09If y=4x-5 is tangent to the curve y2=px3+q at (2,3)thenp = 2, q = -7p = -2 , q =7p = -2, q =-7 p = 2, q = 7

23. Problem10Let be distinct real numbers. The points with position vectorsare collinearform an equilateral triangleform a scalene triangle form a right-angled triangle

24. Problem11If w is an imaginary cube root of unity, then the value of isa.b.c.d.

25. Problem12If the lengths of the sides of triangle are 3,5,7 then the largest angle of the triangle is a.b.c.d.

26. Problem13The circles intersect each other in two distinct points if a.b.c.d.

27. Problem14If is equal toa.b.c.d.

28. Problem15A box contains 24 identical balls of which12 are white and 12 are black .The balls are drawn at random from the box one at a time with replacement .The probability that a white ball is drawn for the 4th time on the 7th draw isa.b.c.d.

29. Problem16If we consider only the principal value of the inverse trigonometric functions then the value of isa.b.c.d.

30. Problem17The locus of a variable point whose distance from (-2,0)is 2/3 times its distance from the line isellipse parabolahyperbola none of these

31. Problem18Let .If the ranges of the composition functions respectively, thena.b.c.d.

32. Problem19An unbiased die is tossed until a number greater than 4 appears .The probability that an even number of tosses is needed is 1/22/51/52/3

33. Problem20The equations to a pair of opposite sides of parallelogram are x2-5x+6=0 and y2-6y+5=0 , the equations to its diagonals area.x+4y=13andy=4x-7b.4 x+y=13and4y=x-7c. 4x+y=13andy=4x-7d. y-4x=13andy+4x=7

34. Problem21Let 2 (x is measured in radians).Then x lies in the interval a.b.c.d.

35. Problem22If In (a + c),In(a-c),In(a-2b+c) are in A.P. ,thena. a,b,c are in A.Pb. a2,b2,c2 are in A.Pc. a,b,c are in G.Pd. a,b,c are in H.P

36. Problem23Which one of the following curves cuts the parabola Y2=4ax at right angles?a.x2+y2=a2b. y=e-x/2ac. y=axd. x2=4ay

37. Problem24Let .The number of equations of the form having real roots is15978

38. Problem25The function f defined by isdecreasing for all xdecreasing in and increasing in increasing for all xdecreasing in and increasing in

39. Problem26The vector isa unit vector makes an angle with the vector parallel to the vector perpendicular to the vector

40. Problem27Let A,B,C be three mutually independent events.Consider the two statements . are independent are independentThenBoth are trueOnly is true only is trueNeither . is true .

41. Problem28Let ,where x=0g is differentiable butg’ is not continuous.g is differentiable while f is not.both f and g are differentiableg is differentiable and g’ is continuous.

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