IIT JEE & NEET Question Bank with Solution: Co-ordinate Geometry : Mathematics MCQ

Q.1. If a line lies in the octant OXYZ and It makes equal angles with the axes, then [ MP PET 1988 ]

A) $I equals m equals n equals fraction numerator 1 over denominator square root of 3 end fraction$

B) $I equals m equals n equals plus-or-minus fraction numerator 1 over denominator square root of 3 end fraction$

C) $I equals m equals n equals negative fraction numerator 1 over denominator square root of 3 end fraction$

D) $I equals m equals n equals plus-or-minus fraction numerator 1 over denominator square root of 2 end fraction$

Q.2. If the co-ordinates of the points P, Q, R, S be (1, 2, 3), (4, 5, 7), (-4, 3, -6) and (2, 0, 2) respectively, then

A) $P Q space parallel to space R S$

B) $P Q space perpendicular R S$

C) $P Q equals R S$

D) None of these

Explanation-

Find angle between the lines PQ and RS, we get that neither $P Q space parallel to space R S$ nor $P Q space perpendicular R S$ . Also, $P Q not equal to R S.$

Q.3. The co-ordinates of a point in the xy plane which is equidistant from the three points A, B and C whose position vectors are i, j and k, is

A) (1, 1, 0)

B) (0, 0, 0)

C) (0, 1, 0)

D) (1, -1, 0)

Explanation-

A point has the same distance from points A, B and Cin xy plane because point is on the origin whose coordinate is (0, 0, 0).

Q.4. The equation of a line passing through the point (- 3, 2, - 4) and equally to the axes, are

A) $x minus 3 equals y plus 2 equals z minus 4$

B) $x plus 3 equals y minus 2 equals z plus 4$

C) $fraction numerator x plus 3 over denominator 1 end fraction equals fraction numerator y minus 2 over denominator 2 end fraction equals fraction numerator z plus 4 over denominator 3 end fraction$

D) None of these

Explanation-

Required equation of line is

$fraction numerator open parentheses x plus 3 close parentheses over denominator 1 end fraction equals fraction numerator y minus 2 over denominator 1 end fraction equals fraction numerator z plus 4 over denominator 1 end fraction.$

Q.5. A plane making intercepts of lengths 1, 2, 3 on the rectangular coordinate axes is represented by the equation [ MP PET 2007 ]

A) $x plus y divided by 2 plus z divided by 3 equals 1$

B) $x plus 2 y plus 3 z equals 1$

C) $x minus 1 equals y minus 2 equals z minus 3$

D) None of these

Explanation-

Formula $x over a plus y over b plus z over c equals 1 rightwards double arrow x over 1 plus y over 2 plus z over 3 equals 1.$

Q.6. The direction cosines of three lines passing through the origin are $l subscript 1 comma space m subscript 1 space comma n subscript 1 space semicolon space l subscript 2 comma space m subscript 2 comma space n subscript 2 space a n d space l subscript 3 comma space m subscript 3 comma space n subscript 3$ . The lines will be coplanar, if

A) $negative open vertical bar table row cell l subscript 1 end cell cell n subscript 1 end cell cell m subscript 1 end cell row cell l subscript 2 end cell cell n subscript 2 end cell cell m subscript 2 end cell row cell l subscript 3 end cell cell n subscript 2 end cell cell m subscript 2 end cell end table close vertical bar equals 0$

B) $space open vertical bar table row cell l subscript 1 end cell cell m subscript 2 end cell cell n subscript 3 end cell row cell l subscript 2 end cell cell m subscript 3 end cell cell n subscript 1 end cell row cell l subscript 3 end cell cell m subscript 1 end cell cell n subscript 2 end cell end table close vertical bar equals 0 space$

C) $l subscript 1 space l subscript 2 space l subscript 3 space end subscript plus m subscript 1 space m subscript 2 space m subscript 3 space plus n subscript 1 space n subscript 2 space n subscript 3 space equals 0$

D) None of these

Explanation-

(a) Here, three given lines are coplanar if they have common perpendicular

Let d. c. 's of common perpendicular be l, m, n

$rightwards double arrow l l subscript 1 plus m m subscript 1 plus n n subscript 1 equals 0 space space space space space l l subscript 2 plus m m subscript 2 plus n n subscript 2 equals 0 a n d space space l l subscript 3 plus m m subscript 3 plus n n subscript 3 equals 0$ $............ left parenthesis i right parenthesis ............. left parenthesis i i right parenthesis ................. left parenthesis i i i right parenthesis$

Solving (ii) and (iii), we get

$fraction numerator l over denominator m subscript 2 n subscript 3 minus n subscript 2 m subscript 3 end fraction equals fraction numerator m over denominator n subscript 2 l subscript 3 minus n subscript 3 l subscript 2 end fraction equals fraction numerator n over denominator l subscript 2 m subscript 3 minus l subscript 3 m subscript 2 end fraction equals k rightwards double arrow l equals k open parentheses m subscript 2 n subscript 3 minus n subscript 2 m subscript 3 close parentheses comma space m equals k open parentheses n subscript 2 l subscript 3 minus n subscript 3 l subscript 2 close parentheses comma space n equals k open parentheses l subscript 2 m subscript 2 minus l subscript 3 m subscript 2 close parentheses$

Substituting in (i), we get

$l subscript 1 open parentheses m subscript 2 n subscript 3 minus n subscript 2 m subscript 3 close parentheses plus m subscript 1 open parentheses n subscript 2 l subscript 2 minus n subscript 3 l subscript 2 close parentheses plus n subscript 1 open parentheses l subscript 2 m subscript 3 minus l subscript 3 m subscript 2 close parentheses equals 0 space rightwards double arrow open vertical bar table row cell l subscript 1 end cell cell m subscript 1 end cell cell n subscript 1 end cell row cell l subscript 2 end cell cell m subscript 2 end cell cell n subscript 2 end cell row cell l subscript 3 end cell cell m 3 end cell cell n subscript 3 end cell end table close vertical bar equals 0 rightwards double arrow negative open vertical bar table row cell l subscript 1 end cell cell n subscript 1 end cell cell m subscript 1 end cell row cell l subscript 2 end cell cell n subscript 2 end cell cell m subscript 2 end cell row cell l subscript 3 end cell cell n subscript 3 end cell cell m subscript 3 end cell end table close vertical bar equals 0.$

Q.7. A point moves in such a way that the sum of its distance from xy - plane and yz - plane remains equal to its distance from zx - plane. The locus of the point is

A) $x minus y plus z equals 2$

B) $x plus y minus z equals 0$

C) $x minus y plus z equals 0$

D) $x minus y minus z equals 2$

Explanation-

According to question, z + x = y or x - y+ z = 0.

Q.8. The line drawn from (4, -1,2) to the point (-3,2,3) meets a plane at right angles at the point (-10,5,4) then the equation of plane is [DSSE 1985]

A) 7x-3y-z+89=0

B) 7x+3y+z+89=0

C) 7x-3y+z+89=0

D) None of these

Explanation-

Required plane is $7 vertical line left parenthesis x plus 10 right parenthesis minus 3 left parenthesis y minus 5 right parenthesis minus left parenthesis z minus 4 right parenthesis equals 0 o r space 7 x minus 3 y minus z plus 89 equals 0$

Q.9. If a plane cuts off intercepts $O A equals a comma space O B equals b comma space O C equals c space$ from the co-ordinate axes, then the area of the triangle ABC =

A) $1 half square root of b squared c squared plus c squared a squared plus a squared b squared end root$

B) $1 half open parentheses b c plus c a plus a b close parentheses$ .

C) $1 half a b c$

D) $1 half square root of open parentheses b minus c close parentheses squared plus open parentheses c minus a close parentheses squared plus open parentheses a minus b close parentheses squared end root$

Explanation-

(a) Length of sides are $square root of a squared plus b squared end root space comma square root of b squared plus c squared end root space comma square root of c squared plus a squared end root$ respectively

Now use $increment equals 1 divided by 2 square root of s open parentheses s minus a close parentheses open parentheses s minus b close parentheses open parentheses s minus c close parentheses end root space.$

Trick : Put $a equals 2 comma space b equals 2 comma space c equals 2 comma$ then sides will be $2 square root of 2 comma space 2 square root of 2 space a n d space 2 square root of 2$ i.e.

equilateral triangle. So area of this triangle will be $increment equals fraction numerator square root of 3 over denominator 4 end fraction cross times open parentheses 2 square root of 2 close parentheses squared equals square root of 3 space s q. space u n i t s$

Now, option (a) $rightwards double arrow increment equals 1 divided by square root of 16 plus 16 plus 16 end root$

$equals 1 divided by 2 cross times 4 square root of 3 equals 2 square root of 3.$

Q.10. Distance of the point (2, 3, 4) from the plane $3 x minus 6 y plus 2 z plus 11 equals 0 space i s$ [ MP PET 1990, 96, 2001, 03, 13; Odisha JEE 2010 ]

A) 1

B) 2

C) 3

D) 0

Explanation-

Required distance $equals open vertical bar fraction numerator 6 minus 18 plus 8 plus 11 over denominator 7 end fraction close vertical bar equals 1.$

IIT JEE & NEET Question Bank with Solution

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Tuesday, August 24, 2021

Co-ordinate Geometry : Mathematics MCQ

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