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Sample Test ( AIEEE )
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Minor Test #2

### AIEEE Minor Test #2

Circle, Pairs of Straight Lines, Permutation and Combination

Q1. The equation of the circle, whose centre is the point of intersection of the lines 2x – 3y + 4 = 0 and 3x + 4x – 5 = 0 and passes through the origin, is

 (a) 17(x2 + y2) + 2x – 44y = 0 (b) 17(x2 + y2) + 2x + 44y = 0 (c) 17(x2 + y2) + 2x – 44y = 0 (d) None of these

Q2. The equation of the circle which touches the axis of y at a distance + 4 from the origin and cuts off an intercept 6 from the axis of x is

 (a) x2 + y2 – 10x – 8y + 16 = 0 (b) x2 + y2 + 10x – 8y + 16 = 0 (c) x2 + y2 – 10x + 8y + 16 = 0 (d) None of these

Q3. A circle of radius 2 lies in the first quadrant and touches both the axes of coordinates. The equation of the circle with centre at (6, 5) and touching the above circle externally is

 (a) x2 + y2 + 12x – 10y + 52 = 0 (b) x2 + y2 – 12x + 10y + 52 = 0 (c) x2 + y2 – 12x – 10y + 52 = 0 (d) None of these

Q4. The equation of the circle which has two normals (x – 1) (y – 2) = 0 and a tangent 3x + 4y = 6 is

 (a) x2 + y2 – 2x – 4y + 4 = 0 (b) x2 + y2 + 2x – 4y + 5 = 0 (c) x2 + y2 = 5 (d) (x + 3)2 + (y – 4)2 = 5

Q5. The coordinates of the middle point of the chord which the circle x2 + y2 + 4x – 2y – 3 = 0 cuts off on the line y = x + 2, are

 (a) (b) (c) (d)

Q6. The equation of tangent to the circle x2 + y2 = 25, which is inclined at an angle of 30° to the axis of x, is

 (a) (b) (c) (d)

Q7. The limiting points of the coaxal system determined by the circle x2 + y2 – 2x – 6y + 9 = 0 and x2 + y2 + 6x – 2y + 1 = 0 are

 (a) (b) (c) (d) None of these

Q8.The number of common tangents to the circle x2 + y2 = 4 and x2 + y2 – 8x + 12 = 0 is

 (a) 1 (b) 2 (c) 3 (d) 4

Q9.The equation of the circle passing through (1, 0) and (0, 1) and having smallest possible radius is

 (a) x2 + y2 – x – y = 0 (b) x2 + y2 + x + y = 0 (c) x2 + y2 – 2x – y = 0 (d) x2 + y2 – x – 2y = 0

Q10.The radical axis of the circles, belonging to the coaxal system of circles whose limiting points are (1, 3) and (2, 6), is

 (a) x – 3y – 15 =  0 (b) x + 3y – 15 =  0 (c) x – 3y + 15 =  0 (d) 2x + 3y – 15 =  0

Q11. The equation of the circle, which touches the circle x2 + y2 – 6x + 6y + 17 = 0 externally and to which the lines x2 – 3xy – 3x + 9y = 0 are normal, is

 (a) x2 + y2 – 6x – 2y – 1 = 0 (b) x2 + y2 + 6x – 2y + 1 = 0 (c) x2 + y2 – 6x – 2y + 1 = 0 (d) None of these

Q12.A line meets the coordinate axes in A and B, and a circle is circumscribing triangle AOB where O is the origin. If m, n are the distances of the tangents to this circle at the origin from the points A and B respectively, then the diameter of the circle is

 (a) m(m + n) (b) n(m + n) (c) m – n (d) m + n

Q13. Four distinct points (2K, 3K) , (1, 0), (0, 1) and (0, 0) lie on a circle when

 (a) all values of K are integral (b) 0 < K < 1 (c) K < 0 (d) For two values of K

Q14.The length of the tangent from any point on the circle 15x2 + 15y2 – 48x + 64y = 0 to the two circles

5x2 + 5y2 – 24x + 32y + 75 = 0 and 5x2 + 5y2 – 48x + 64y + 300 = 0 are in the ratio of

 (a) 1 :  2 (b) 2 : 3 (c) 3 : 4 (d) none of these

Q15.The circles x2 + y2 + x + y = 0 and x2 + y2 + x – y = 0 intersect at an angle of

 (a) p/6 (b) p/4 (c) p/3 (d) p/2

Q16. The difference of the tangents of the angles which the lines

x2(sec 2q - sin 2q) – 2xy tan q + y2 sin 2q = 0 with the x – axis is

 (a) 2 tan q (b) 2 (c) 2 cot q (d) sin 2 q

Q17. If the line y = cuts the curve x3 + y3 + 3xy + 5x2 + 3y3 + 4x + 5y -1 = 0 at the points A, B, C then OA . OB. OC is

 (a) (b) (c) (d) None of these

Q18.A variable line drawn through the point (1, 3) meets the x-axis at A and y-axis at B. If the rectangle OAPB is completed, where ‘O’ is the origin, then locus of ‘P’ is

 (a) (b) x + 3y = 1 (c) (d) 3x + y = 1

Q19.ABC is a right angled isosceles triangle, right angled at A(2, 1). If the equation of side BC is 2x + y = 3, then the combined equation of lines AB and AC is

 (a) 3x2 – 3y2 + 8xy + 20x + 10y + 25 = 0 (b) 3x2 – 3y2 + 8xy – 20x – 10y + 25 = 0 (c) 3x2 – 3y2 + 8xy + 10x + 15y + 20 = 0 (d) None of these

Q20. If the line y = x is one of the angle bisector of the pair of lines ax2 + 2hxy + by2 = 0, then

 (a) a + b = 0 (b) h = 0 (c) a – b = 0 (d) None of these

Q21. If the sum of the slopes of the lines given by x2 – 2cxy – 7y2 = 0 is four times their product, then c has the value:

 (a) 1 (b) – 1 (c) 2 (d) – 2

Q22. If one of the lines given by 6x2 – xy + 4cy2 = 0 is 3x + 4y = 0, then c equals:

 (a) 1 (b) – 1 (c) 3 (d) – 3

Q23. If the pair of straight lines x2 – 2pxy – y2 = 0 and x2 – 2qxy – y2 = 0 be such that each pair bisects the angle between the other pair, then

 (a) p = q (b) p = – q (c) pq = 1 (d) pq = – 1

Q24. The orthocenter of the triangle formed by the lines xy = 0 and x + y = 1 is

 (a) (b) (c) (0, 0) (d)

Q25. 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 represents L1?

 (a) x + y = 0 (b) x – y = 0 (c) x + 7y = 0 (d) x – 7y = 0

Q26.The number of numbers are there between 100 and 1000 in which all the digits are distinct is

 (a) 648 (b) 548 (c) 448 (d) none of these

Q27.The number of arrangements of the letters of the word ‘CALCUTTA’ is

 (a) 5040 (b) 2550 (c) 40320 (d) 10080

Q28. How many different words can be formed with the letters of the word “PATLIPUTRA” without changing the position of the vowels and consonants?

 (a) 2160 (b) 180 (c) 720 (d) none of these

Q29. How many different words ending and beginning with a consonant can be formed with the letters of the word ‘EQUATION’?

 (a) 720 (b) 4320 (c) 1440 (d) none of these

Q30.The number of 4 digit numbers divisible by 5 which can be formed by using the digits 0, 2, 3, 4, 5 is

 (a) 36 (b) 42 (c) 48 (d) none of these

Q31.The number of ways in which 5 biscuits can be distributed among two children is

 (a) 32 (b) 31 (c) 30 (d) none of these

Q32.How many five-letter words containing 3 vowels and 2 consonants can be formed using the letters of the word “EQUATION” so that the two consonants occur together?

 (a) 1380 (b) 1420 (c) 1440 (d) none

Q33.If the letters of the word ‘RACHIT’ are arranged in all possible ways and these words are written out as in a dictionary, then the rank of this word is

 (a) 365 (b) 702 (c) 481 (d) none of these

Q34.On the occasion of Dipawali festival each student of a class sends greeting cards to the others. If there are 20 students in the class, then the total number of greeting cards exchanged by the students is

 (a) 20C2 (b) 2 . 20C2 (c) 2 . 20P2 (d) none of these

Q35.The sum of the digits in the unit place of all the numbers formed with the help of 3, 4, 5, 6 taken all at a time is

 (a) 18 (b) 108 (c) 432 (d) 144

Q36.How many six digits numbers can be formed in decimal system in which every succeeding digit is greater than its preceding digit

 (a) 9P6 (b) 10P6 (c) 9P3 (d) none of these

Q37.How many ways are there to arrange the letters in the work GARDEN with the vowels in alphabetical order?

 (a) 120 (b) 240 (c) 360 (d) 480

Q38.A five-digit numbers divisible by 3 is to be formed using the numerals 0, 1, 2, 3, 4 and 5, without repetition. The total number of ways this can be done is

 (a) 216 (b) 240 (c) 600 (d) 3125

Q39.How many different nine digit numbers can be formed from the number 223355888 by rearranging its digits so that the odd digits occupy even positions?

 (a) 16 (b) 36 (c) 60 (d) 180

Q40.The number of arrangements of the letters of the word BANANA in which the two N’s do not appear adjacently is

 (a) 40 (b) 60 (c) 80 (d) 100

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