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Q 1. The number of integral values of {tex} m , {/tex} for which the {tex} x {/tex} -coordinate of the point of intersection of the lines {tex} 3 x + 4 y = 9 {/tex} and {tex} y = m x + 1 {/tex} is also an integer is

2

B

0

C

4

D

1

Explanation

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Q 2. If the coordinates of the middle point of the portion of a line intercepted between coordinate axes {tex} ( 3,2 ) , {/tex} then the equation of the line will be

{tex} 2 x + 3 y = 12 {/tex}

B

{tex} 3 x + 2 y = 12 {/tex}

C

{tex} 4 x - 3 y = 6 {/tex}

D

{tex} 5 x - 2 y = 10 {/tex}

Explanation




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Q 3. A line through point {tex} A ( - 5 , - 4 ) {/tex} meets the lines {tex} x + 3 y + 2 = 0, {/tex} {tex} 2 x + y + 4 = 0 {/tex} and {tex} x - y - 5 = 0 {/tex} at points {tex} B , C {/tex} and {tex} D , {/tex} respectively. If {tex} \left( \frac { 15 } { \mathrm { AB } } \right) ^ { 2 } + \left( \frac { 10 } { \mathrm { AC } } \right) ^ { 2 } = \left( \frac { 6 } { \mathrm { AD } } \right) ^ { 2 } , {/tex} then the equation of the line is

{tex} 2 x + 3 y + 22 = 0 {/tex}

B

{tex} 5 x - 4 y + 7 = 0 {/tex}

C

{tex} 3 x - 2 y + 3 = 0 {/tex}

D

None of these

Explanation





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Q 4. The equation of perpendicular bisectors of sides {tex} A B {/tex} and {tex} A C {/tex} of a triangle {tex} A B C {/tex} are {tex} x - y + 5 = 0 {/tex} and {tex} x + 2 y = 0 {/tex} , respectively. If point {tex} A {/tex} is {tex} ( 1 , - 2 ) , {/tex} then the equation of line {tex} B C {/tex} is

A

{tex} 23 x + 14 y - 40 = 0 {/tex}

B

{tex} 14 x - 23 y + 40 = 0 {/tex}

C

{tex} 23 x - 14 y + 40 = 0 {/tex}

{tex} 14 x + 23 y - 40 = 0 {/tex}

Explanation





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Q 5. The medians {tex} A D {/tex} and {tex} B E {/tex} of a triangle with vertices {tex} A ( 0 , b ) , B ( 0,0 ) {/tex} and {tex} C ( a , 0 ) {/tex} are perpendicular to each other if

A

{tex} a = \sqrt { 2 b } {/tex}

B

{tex} a = - \sqrt { 2 } b {/tex}

Both {tex} ( \mathrm { A } ) {/tex} and {tex} ( \mathrm { B } ) {/tex}

D

None of these

Explanation


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Q 6. Let {tex} P S {/tex} be the median of the triangle with vertices {tex} P ( 2,2 ) , Q ( 6 , - 1 ) {/tex} and {tex} R ( 7,3 ) . {/tex} Then the equation of the line passing through {tex} ( 1 , - 1 ) {/tex} and parallel to {tex} P S {/tex} is

A

{tex} 2 x - 9 y - 7 = 0 {/tex}

B

{tex} 2 x - 9 y - 11 = 0 {/tex}

C

{tex} 2 x + 9 y - 11 = 0 {/tex}

{tex} 2 x + 9 y + 7 = 0 {/tex}

Explanation




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Q 7. The equation of straight line passing through {tex} ( - a , 0 ) {/tex} and making the triangle with axes of area {tex} T {/tex} is

A

{tex} 2 T x + a ^ { 2 } y + 2 a T = 0 {/tex}

{tex} 2 T x - a ^ { 2 } y + 2 a T = 0 {/tex}

C

{tex} 2 T x - a ^ { 2 } y - 2 a T = 0 {/tex}

D

None of these

Explanation



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Q 8. The equations of two equal sides of an isosceles triangle are {tex} 7 x - y + 3 = 0 {/tex} and {tex} x + y - 3 = 0 {/tex} and the third side passes through the point {tex} ( 1 , - 10 ) {/tex} . The equation of the third side is

A

{tex} y = \sqrt { 3 } x + 9 {/tex} but not {tex} x ^ { 2 } - 9 y ^ { 2 } = 0 {/tex}

B

{tex} 3 x + y + 7 = 0 {/tex} but not {tex} x - 3 y - 31 = 0 {/tex}

{tex} 3 x + y + 7 = 0 {/tex} or {tex} x - 3 y - 31 = 0 {/tex}

D

Neither {tex} 3 x + y + 7 {/tex} nor {tex} x - 3 y - 31 = 0 {/tex}

Explanation



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Q 9. If the equation of base of an equilateral triangle is {tex} 2 x - y = 1 {/tex} and the vertex is {tex} ( - 1,2 ) , {/tex} then the length of the side of the triangle is

{tex} \sqrt { \frac { 20 } { 3 } } {/tex}

B

{tex} \frac { 2 } { \sqrt { 15 } } {/tex}

C

{tex} \sqrt { \frac { 8 } { 15 } } {/tex}

D

{tex} \sqrt { \frac { 15 } { 2 } } {/tex}

Explanation




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Q 10. If {tex} x _ { 1 } , x _ { 2 } , x _ { 3 } {/tex} and {tex} y _ { 1 } , y _ { 2 } , y _ { 3 } {/tex} are both in GP, with the same common ratio, then the points {tex} \left( x _ { 1 } , y _ { 1 } \right) , \left( x _ { 2 } , y _ { 2 } \right) {/tex} and {tex} \left( x _ { 3 } , y _ { 3 } \right) {/tex}

Lie on a straight line

B

Lie on an ellipse

C

Lie on a circle

D

Are vertices of a triangle

Explanation

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Q 11. In what direction a line be drawn through the point {tex} ( 1,2 ) {/tex} so that its points of intersection with the line {tex} x + y = 4 {/tex} is at a distance {tex} \sqrt { 6 } / 3 {/tex} from the given point

A

{tex} 30 ^ { \circ } {/tex}

B

{tex} 45 ^ { \circ } {/tex}

C

{tex} 60 ^ { \circ } {/tex}

{tex} 75 ^ { \circ } {/tex}

Explanation





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Q 12. If straight lines {tex} a x + b y + p = 0 {/tex} and {tex} x \cos \alpha + y \sin \alpha - p = 0 {/tex} include an angle {tex} \pi / 4 {/tex} between them and meet the straight line {tex} x \sin \alpha - y \cos \alpha = 0 {/tex} in the same point, then the value of {tex} a ^ { 2 } + b ^ { 2 } {/tex} is equal to

A

1

2

C

3

D

4

Explanation





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Q 13. Given vertices {tex} A ( 1,1 ) , B ( 4 , - 2 ) {/tex} and {tex} C ( 5,5 ) {/tex} of a triangle, then the equation of the perpendicular dropped from {tex} C {/tex} to the the the interior bisector of the angle {tex} A {/tex} is

A

{tex} y - 5 = 0 {/tex}

{tex} x - 5 = 0 {/tex}

C

{tex} y + 5 = 0 {/tex}

D

{tex} x + 5 = 0 {/tex}

Explanation



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Q 14. The equation of the line joining the point {tex} ( 3,5 ) {/tex} to the point of intersection of the lines {tex} 4 x + y - 1 = 0 {/tex} and {tex} 7 x - 3 y - 35 = 0 {/tex} is equidistant from the points {tex} ( 0,0 ) {/tex} and {tex} ( 8,34 ) {/tex}

TRUE

B

FALSE

C

Nothing can be said

D

None of these

Explanation

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Q 15. The line {tex} 3 x + 2 y = 24 {/tex} meets {tex} y {/tex} -axis at point {tex} A {/tex} and {tex} x {/tex} -axis at point {tex} B {/tex} . The perpendicular bisector of {tex} A B {/tex} meets the line through {tex} ( 0 , - 1 ) {/tex} parallel to {tex} x {/tex} -axis at point {tex} C . {/tex} The area of the triangle {tex} A B C {/tex} is

A

182 sq. units

91 sq. units

C

48 sq. units

D

None of these

Explanation




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Q 16. The equation of the locus of foot of perpendiculars drawn from the origin to the line passing through a fixed point {tex} ( a , b ) {/tex} is

{tex} x ^ { 2 } + y ^ { 2 } - a x - b y = 0 {/tex}

B

{tex} x ^ { 2 } + y ^ { 2 } + a x + b y = 0 {/tex}

C

{tex} x ^ { 2 } + y ^ { 2 } - 2 a x - 2 b y = 0 {/tex}

D

None of these

Explanation




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Q 17. The orthocentre of the triangle formed by the lines {tex} x y = 0 {/tex} and {tex} x + y = 1 {/tex} is

{tex}\mathrm{(0,0)}{/tex}

B

{tex} \left( \frac { 1 } { 2 } , \frac { 1 } { 2 } \right) {/tex}

C

{tex} \left( \frac { 1 } { 3 } , \frac { 1 } { 3 } \right) {/tex}

D

{tex} \left( \frac { 1 } { 4 } , \frac { 1 } { 4 } \right) {/tex}

Explanation

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Q 18. The lines joining the origin to the points of intersection of the line {tex} y = m x + c {/tex} and the circle {tex} x ^ { 2 } + y ^ { 2 } = a ^ { 2 } {/tex} will be mutually perpendicular if

A

{tex} a ^ { 2 } \left( m ^ { 2 } + 1 \right) = c ^ { 2 } {/tex}

B

{tex} a ^ { 2 } \left( m ^ { 2 } - 1 \right) = c ^ { 2 } {/tex}

{tex} a ^ { 2 } \left( m ^ { 2 } + 1 \right) = 2 c ^ { 2 } {/tex}

D

{tex} a ^ { 2 } \left( m ^ { 2 } - 1 \right) = 2 c ^ { 2 } {/tex}

Explanation



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Q 19. Two of the lines represented by the equation {tex} a y ^ { 4 } + b x y ^ { 3 } + c x ^ { 2 } {/tex} {tex} y ^ { 2 } + d x ^ { 3 } y + e x ^ { 4 } = 0 {/tex} will be perpendicular when

{tex} ( b + d ) ( a d + b e ) + ( e - a ) ^ { 2 } {/tex}
{tex}( a + c + e ) = 0 {/tex}

B

{tex} ( b + d ) ( a d + b e ) + ( e + a ) ^ { 2 } {/tex}
{tex}( a + c + e ) = 0 {/tex}

C

{tex} ( b - d ) ( a d - b e ) + ( e - a ) ^ { 2 }{/tex}
{tex}( a + c + e ) = 0 {/tex}

D

{tex} ( b - d ) ( a d - b e ) + ( e + a ) ^ { 2 }{/tex}
{tex} ( a + c + e ) = 0 {/tex}

Explanation




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Q 20. The area of the triangle bounded by the straight line ax + by + c = 0, (a, b, c ≠ 0) and the coordinate axes is

A

$\frac{1}{2}\frac{a^{2}}{|\text{bc}|}$

$\frac{1}{2}\frac{c^{2}}{|\text{ab}|}$

C

$\frac{1}{2}\frac{b^{2}}{\left| \text{ac} \right|}$

D

0

Explanation

The vertices of the triangle are $A\left( 0,\ 0 \right),\ B\left( - \frac{c}{a},\ 0 \right)$ and $(C\left( 0,\ - \frac{c}{b} \right)$

$\therefore Area\ of\ \mathrm{\Delta}\ = \frac{1}{2}\left| \begin{matrix} 0 & 0 & 1 \\ - \frac{c}{a} & 0 & 1 \\ 0 & - \frac{c}{b} & 1 \\ \end{matrix} \right|$

$= \frac{1}{2}1\left| \left\{ \left( - \frac{c}{a} \right)\left( - \frac{c}{b} \right) - 0 \right\} \right|$

$= \frac{c^{2}}{2\left| \text{ab} \right|}$

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Q 21. The points (1, 1), ( − 5, 5) and (13, λ) lie on the same straight line, if λ is equal to

A

7

− 7

C

± 7

D

0

Explanation

The equation of any line passing through (1, 1) and ( − 5, 5) is

$y - 1 = \frac{5 - 1}{- 5 - 1}\left( x - 1 \right)$

⇒ − 6(y−1) = 4(x−1)

Since, the point (13, λ) lies on this line.

∴ − 6(λ−1) = 4(13−1) ⇒ λ = − 7

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Q 22. The equations to the straight lines passing through the origin and making an angle α with the straight line y + x = 0 are given by

x2 + 2xysec 2 α + y2 = 0

B

x2 − 2 xysec 2 α + y2 = 0

C

x2 + 2 xycos 2 α + y2 = 0

D

None of these

Explanation

The equations of the lines passing through the origin and making angle α with y + x = 0 are

$y - 0 = \frac{- 1 \pm \tan\alpha}{1 \pm \tan\alpha}\left( x - 0 \right)\text{\ \ \ }\left\lbrack Using\ :y - y_{1} = \frac{m \pm \tan\alpha}{1 \mp \tan\alpha}\left( x - x_{1} \right) \right\rbrack$

$\Rightarrow y + \frac{1 - \tan\alpha}{1 + \tan\alpha}x = 0\ \text{and\ }y + \frac{1 + \tan\alpha}{1 - \tan\alpha}x = 0$

The combined equations of these two lines is

$\left( y + \frac{1 - \tan\alpha}{1 + \tan\alpha} \right)\left( y + \frac{1 + \tan\alpha}{1 - \tan\alpha} \right) = 0$

$\Rightarrow y^{2} + x^{2} + 2\text{xy}\left( \frac{1 + \tan^{2}\alpha}{1 - \tan^{2}\alpha} \right) = 0$

⇒ x2 + 2xysec 2 α + y2 = 0

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Q 23. Consider the points A ≡ (3, 4), B ≡ (7, 13). If ‘P’ be a point on the line y = x, such that PA + PB is minimum, then coordinates of P is

A

$\left( \frac{13}{7},\frac{13}{7} \right)$

B

$\left( \frac{23}{7},\frac{23}{7} \right)$

$\left( \frac{31}{7},\frac{31}{7} \right)$

D

$\left( \frac{33}{7},\frac{33}{7} \right)$

Explanation

Points (3, 4) and (7, 13) are on the same side of straight line y = x. Take image of A about y = x ie, A ≡ (4, 3)

Now, P is a intersection point of line y = x and AB

Equation of line AB is $y - 3 = \frac{10}{3}\left( x - 4 \right)$

⇒ 3y − 9 = 10x − 40

⇒ 10x − 3y = 31

$\Rightarrow \left( \frac{31}{7},\frac{31}{7} \right)\ \text{satisfy\ the\ line}\ A^{''}B\ \text{such\ that\ }\text{PA} + \text{PB}\ \text{is\ minimum}$

$\therefore\text{Coordinates\ of\ }P\text{\ are\ }\left( \frac{31}{7},\frac{31}{7} \right)$

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Q 24.

The length of the perpendicular from the origin of the line

$\frac{x\sin\alpha}{b} - \frac{y\cos\alpha}{a} - 1 = 0\ \text{is}$

A

$\frac{|\text{ab}|}{\sqrt{a^{2}\cos^{2}\alpha - b^{2}\sin^{2}\alpha}}$

B

$\frac{|\text{ab}|}{\sqrt{a^{2}\cos^{2}\alpha + b^{2}\sin^{2}\alpha}}$

C

$\frac{|\text{ab}|}{\sqrt{a^{2}\sin^{2}\alpha - b^{2}\cos^{2}\alpha}}$

$\frac{|\text{ab}|}{\sqrt{a^{2}\sin^{2}\alpha + b^{2}\cos^{2}\alpha}}$

Explanation

The length of perpendicular from the origin to the line

$\frac{x\sin\alpha}{b} - \frac{y\cos\alpha}{a} - 1 = 0\ \text{is}$

$d = \frac{\left| 0 - 0 - 1 \right|}{\sqrt{\frac{\sin^{2}\alpha}{b^{2}} + \frac{\cos^{2}\alpha}{a^{2}}}}$

$= \frac{|\text{ab}|}{\sqrt{a^{2}\sin^{2}\alpha + b^{2}\cos^{2}\alpha}}$

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Q 25. The angle between the straight line $x - y\sqrt{3} = 5$ and $\sqrt{3}x + y = 7$ is

90

B

60

C

75

D

30

Explanation

Given equation is compared with a1x + b1y = 0 and a2x + b2y = 0

Now, $a_{1}a_{2} + b_{1}b_{2} = \left( 1 \right)\left( \sqrt{3} \right) + \left( - \sqrt{3} \right)\left( 1 \right) = 0$

∴ Lines are perpendicular

Hence, θ = 90