Permutation-Combination & Coordinate-3

Q1.The focal distance of a point on a parabola y₂ = 8x is 4, its coordinates are

(a)

(2, 4)

(b)

(– 2, 4)

(c)

(– 2, – 4)

(d)

none of these

Q2.The equation of the parabola whose focus is (3, – 4) and directrix is the line x + y – 2 = 0, is

(a)	x² + 2xy + y² – 8x + 20y + 46 = 0	(b)	x² – 2xy + y² – 8x + 20y + 46 = 0
(c)	x² – 2xy + y² + 8x – 20y + 46 = 0	(d)	None of these

Q3.The equation of the tangent to the parabola y² = 6x at the point whose ordinate is 6, is

(a)

x + 2y + 6 = 0

(b)

2x – y + 6 = 0

(c)

x – 2y + 6 = 0

(d)

None of these

Q4.The normal to the parabola y² = 8x at (2, 4) meets the parabola again at

(a)

(18, 12)

(b)

(18, – 12)

(c)

(– 18, 12)

(d)

None of these

Q5.The equation of the tangent to the parabola y² = 16x inclined at an angle of 60° to x – axis, is

(a)

(b)

(c)

(d)

None of these

Q6.The point of intersection of two perpendicular tangents to a parabola lies on the

(a)

axis

(b)

tangent at the vertex

(c)

directrix

(d)

None of these

Q7.The pole of the line 2x = y with respect to the parabola y² = 2x is

(a)

(b)

(c)

(d)

None of these

Q8.The locus of the poles of tangents to the parabola y² = 4ax w.r.t. the parabola y² = 4bx is the parabola

(a)

(b)

(c)

(d)

Q9.The length of latus rectum of the parabola 4y² + 2x – 20y + 17 = 0 is

(a)

(b)

(c)

1/2

(d)

Q10.If the focal distance of a point on the parabola y² = 4x is 6, then the coordinates of the point are

(a)

(b)

(c)

(d)

None of these

Q11.The equation of the ellipse referred to its axes as the axes of coordinates with length of major axis 8 and eccentricity

, is

(a)

4x² + 3y² = 48

(b)

3x² + 4y² = 48

(c)

4x² + 3y² = 24

(d)

None of these

Q12.The equation of the ellipse with focus (– 1, 1), directrix x – y + 3 = 0 and eccentricity

, is

(a)	7x² + 2xy + 7y² + 10x + 10y + 7 = 0	(b)	7x² + 2xy + 7y² + 10x – 10y + 7 = 0
(c)	7x² + 2xy + 7y² + 10x – 10y – 7 = 0	(d)	None of these

Q13.The equation of the normal to the ellipse

at the end of the latus rectum in the first quadrant, is

(a)

x + ey – ae³ = 0

(b)

x – ey + ae³ = 0

(c)

x – ey – ae³ = 0

(d)

None of these

Q14.The condition that the line x cos a + y sin a = p may be a tangent to the ellipse

(a)	a² cos² a + b² sin² a = p²	(b)	a² cos² a + b² sin² a = 2p²
(c)	a² sin² a + b² cos² a = p²	(d)	None of these

Q15.If the normal at the end of a latus rectum of an ellipse passes through one extremity of the minor axis, then

(a)

e⁴ + e² – 1 = 0

(b)

e⁴ – e² + 1 = 0

(c)

e⁴ – e² – 1 = 0

(d)

None of these

Q16.In how many ways can 5 beads out 7 different beads be strung into a string?

(a)

504

(b)

2520

(c)

252

(d)

none of these

Q17.A person has 12 friends, out of them 8 are his relatives. In how many ways can he invite his 7 friends so as to include his 5 relatives?

(a)

⁸C₃ x ⁴C₂

(b)

¹²C₇

(c)

¹²C₅ x ⁴C₃

(d)

none of these

Q18.It is essential for a student to pass in 5 different subjects of an examination then the no. of method so that he may failure

(a)

(b)

(c)

(d)

Q19.The number of ways of dividing 20 persons into 10 couples is

(a)

(b)

²⁰C₁₀

(c)

(d)

none of these

Q20.The number of words by taking 4 letters out of the letters of the word ‘COURTESY’, when T and S are always included are

(a)

120

(b)

720

(c)

360

(d)

none of these

Q21.The number of ways to put five letters in five envelopes when one letter is kept in right envelope and four letters in wrong envelopes are–

(a)

(b)

(c)

(d)

Q22.

is equal to

(a)

⁵¹C₄

(b)

⁵²C₄

(c)

⁵³C₄

(d)

none of these

Q23.A candidate is required to answer 6 out of 10 questions which are divided into two groups each containing 5 questions and he is not permitted to attempt more than 4 from each group. The number of ways in which he can make up his choice is

(a)

100

(b)

200

(c)

300

(d)

400

Q24.Out of 10 white, 9 black and 7 red balls, the number of ways in which selection of one or more balls can be made, is

(a)

881

(b)

891

(c)

879

(d)

892

Q25.The number of diagonals in an octagon are

(a)

(b)

(c)

(d)

none of these

Q26.Out of 10 given points 6 are in a straight line. The number of the triangles formed by joining any three of them is

(a)

100

(b)

150

(c)

120

(d)

none of these

Q27.In how many ways the letters AAAAA, BBB, CCC, D, EE, F can be arranged in a row when the letter C occur at different places?

(a)

(b)

(c)

(d)

none of these

Q28.A is a set containing n elements. A subset P of A is chosen. The set A is reconstructed by replacing the elements of P. A subset Q of A is again chosen. The number of ways of chosen P and Q so that P Ç Q = f is

(a)

2²ⁿ – ²ⁿC_n

(b)

2ⁿ

(c)

2ⁿ – 1

(d)

3ⁿ

Q29.A parallelogram is cut by two sets of m lines parallel to the sides, the number of parallelograms thus formed is

(a)

(b)

(c)

(d)

Q30.Along a railway line there are 20 stations. The number of different tickets required in order so that it may be possible to travel from every station to every station is

(a)

380

(b)

225

(c)

196

(d)

105

Q31.The number of ordered triplets of positive integers which are solutions of the equation x + y + z = 100 is

(a)

5081

(b)

6005

(c)

4851

(d)

none of these

Q32.The number of numbers less than 1000 that can be formed out of the digits 0, 1, 2, 3, 4 and 5, no digit being repeated, is

(a)

130

(b)

131

(c)

156

(d)

none of these

Q33.A variable name in certain computer language must be either a alphabet or alphabet followed by a decimal digit. Total number of different variable names that can exist in that language is equal to

(a)

280

(b)

290

(c)

286

(d)

296

Q34.The total number of ways of selecting 10 balls out of an unlimited number of identical white, red and blue balls is equal to

(a)

¹²C₂

(b)

¹²C₃

(c)

¹⁰C₂

(d)

¹⁰C₃

Q35.Total number of ways in which 15 identical blankets can be distributed among 4 persons so that each of them get atleast two blankets equal to

(a)

¹⁰C₃

(b)

⁹C₃

(c)

¹¹C₃

(d)

none of these

Q36.The number of ways in which three distinct numbers in AP can be selected from the set {1, 2, 3, …, 24}, is equal to

(a)

(b)

132

(c)

198

(d)

none of these

Q37.The number of ways of distributing 8 identical balls in 3 distinct boxes so that none of the boxes is empty is:

(a)

(b)

(c)

3⁸

(d)

⁸C₃

Q38.The number of ways in which 6 men and 5 women can dine at a round table if no two women are to sit together is given by:

(a)

6! x 5!

(b)

(c)

5! x 4!

(d)

7! x 5!

Q39.If ⁿC_r denotes the number of combinations of n things taken r at a time, then the expression ⁿC_{r + 1} + ⁿC_{r – 1} + 2 x ⁿC_r equals:

(a)

^{n + 2}C_r

(b)

^{n + 2}C_{r + 1}

(c)

^{n + 1}C_r

(d)

^{n + 1}C_{r + 1}

Q40.If the letters of the word SACHIN are arranged in all possible ways and these are written out as in dictionary, then the word SACHIN appears at serial number

(a)

600

(b)

601

(c)

602

(d)

603

AIEEE Minor Test #3