= 0 1 (3 4 ) 1 (4 4) + 1 (4 3) = = + 1 = 0 = 1 = ± 1 ]

Size: px
Start display at page:

Download "= 0 1 (3 4 ) 1 (4 4) + 1 (4 3) = = + 1 = 0 = 1 = ± 1 ]"

Transcription

1 STRAIGHT LINE [STRAIGHT OBJECTIVE TYPE] Q. If the lines x + y + = 0 ; x + y + = 0 and x + y + = 0, where + =, are concurrent then (A) =, = (B) =, = ± (C) =, = ± (D*) = ±, = [Sol. Lines are x + y + = 0 ; x + y + = 0 and x + y + = 0, where + = = 0 ( ) ( ) + ( ) = + + = + = 0 = = ± ] Q. Given the family of lines, a (x + y + 6) + b (x + y + ) = 0. The line of the family situated at the greatest distance from the point P (, ) has equation : (A*) x + y + 8 = 0 (B) 5x + y + 0 = 0 (C) 5x + 8y + 0 = 0 (D) none [Hint : point of intersection is A (, 0). The required line will be one which passes through (, 0) and is perpendicular to the line joining (, 0) and (, ) or taking (, ) as centre and radius equal to PA draw a circle, the required line will be a tangent to the circle at (, 0) ] Q. A variable rectangle PQRS has its sides parallel to fixed directions. Q and S lie respectively on the lines x = a, x = a and P lies on the x axis. Then the locus of R is (A*) a straight line (B) a circle (C) a parabola (D) pair of straight lines [Hint: Mid point of QS = Mid point of PS 0 = h + x x = h hence p ( h, 0) equation of PQ y = mx + c passes through ( h, 0) c = mh y = mx + mh since Q lies on it = ma + mh y now m QR = m = y K a h m = m ( a h ) K a h simplifying h + mk = a + (a + h)m a st. line] Q. A rectangular billiard table has vertices at P(0, 0), Q(0, 7), R(0, 7) and S (0, 0). A small billiard ball starts at M(, ) and moves in a straight line to the top of the table, bounces to the right side of the table, then comes to rest at N(7, ). The y-coordinate of the point where it hits the right side, is (A*).7 (B).8 (C).9 (D) none Use Image method Q.5 B and C are points in the xy plane such that A(, ) ; B (5, 6) and AC = BC. Then (A) ABC is a unique triangle (B) There can be only two such triangles. (C) No such triangle is possible (D*) There can be infinite number of such triangles. [Hint: Locus of C is a circle 8(x + y ) 88x 0y + 5 = 0 (D)] Q.B on Straight line and circle []

2 Q.6 A ray of light passing through the point A (, ) is reflected at a point B on the x axis and then passes through (5, ). Then the equation of AB is : (A*) 5x + y = (B) 5x y = (C) x + 5y = (D) x 5y = 6 [Hint : a = 5 a a = 5 Q.7 m, n are integer with 0 < n < m. A is the point (m, n) on the cartesian plane. B is the reflection of A in the line y = x. C is the reflection of B in the y-axis, D is the reflection of C in the x-axis and E is the reflection of D in the y-axis. The area of the pentagon ABCDE is (A) m(m + n) (B*) m(m + n) (C) m(m + n) (D) m(m + n) [Sol. Area of rectangle BCDE = mn m(m n) Area of ABC = = m mn area of pentagon = mn + m mn = m + mn Ans. ] Q.8 The area enclosed by the graphs of x + y = and x = is (A) (B) (C) 6 (D*) 8 [Hint: figure is a parallelogram ] Area = = 8 Ans. ] Q.9 If P = (, 0) ; Q = (, 0) and R = (, 0) are three given points, then the locus of the points S satisfying the relation, SQ + SR = SP is : (A) a straight line parallel to xaxis (B) a circle passing through the origin (C) a circle with the centre at the origin (D*) a straight line parallel to yaxis. Q.0 The equation of the pair of bisectors of the angles between two straight lines is, x 7xy y = 0. If the equation of one line is y x = 0 then the equation of the other line is : (A*) x 8y = 0 (B) x + y = 0 (C) 8x + y = 0 (D) x y = 0 [ Hint : (y x) (y mx) = mx xy (m + ) + y = 0 the equation to the pair of bisectors are : x m y = xy m x 7xy y m (m ) = 7 or 8m = m = ] 8 Q. Two points A(x, y ) and B(x, y ) are chosen on the graph of f (x) = ln x with 0 < x < x. The points C and D trisect line segment AB with AC < CB. Through C a horizontal line is drawn to cut the curve at E(x, y ). If x = and x = 000 then the value of x equals [Sol. (A*) 0 (B) 0 (C) (0) / (D) (0) / Using section formula Q.B on Straight line and circle []

3 000 a = = 0 ln000 ln000 b = b = now line y = b intersects the curve y = ln x b = ln x...() from () and ()...() ln000 = ln x ln (000) / = ln x x = (000) / = 0 Ans. ] Q. If L is the line whose equation is ax + by = c. Let M be the reflection of L through the y-axis, and let N be the reflection of L through the x-axis. Which of the following must be true about M and N for all choices of a, b and c? (A) The x-intercepts of M and N are equal. (B) The y-intercepts of M and N are equal. (C*) The slopes of M and N are equal. (D) The slopes of M and N are reciprocal. [Sol. Reflecting a graph over the x-axis results in the line M whose equation is ax by = c, while a reflection through the y-axis results in the line N whose equation is ax + by = c. Both clearly have slope equal to a/b (from, say, the slope-intercept form of the equation.) ] Q. The distance between the two parallel lines is unit. A point 'A' is chosen to lie between the lines at a distance 'd' from one of them. Triangle ABC is equilateral with B on one line and C on the other parallel line. The length of the side of the equilateral triangle is d d (A) d d (B*) (C) d d (D) d d [ Hint : x cos ( + 0º) = d () and x sin = d () dividing cot = d, squaring equation () and putting d the value of cot, x = (d d + ) x = d d Q. Given A(0, 0) and B(x, y) with x (0, ) and y > 0. Let the slope of the line AB equals m. Point C lies on the line x = such that the slope of BC equals m where 0 < m < m. If the area of the triangle ABC can be expressed as (m m ) f (x), then the largest possible value of f (x) is (A) (B) / (C) / (D*) /8 [Sol. Let the coordinates of C be (, c) c y c mx m = ; m x = x m m x = c m x (m m )x = c m c = (m m )x + m...() ] Q.B on Straight line and circle []

4 now area of ABC = 0 x 0 m x = [cx m x] = c [(( m m)x m)x m x] = [( m m)x mx mx] = (m m )(x x ) (x > x in (0, ) Hence, f (x) = (x x ); f (x)] max = 8 when x = ] Q.5 Consider a parallelogram whose sides are represented by the lines x + y = 0; x + y 5 = 0; x y = 0 and x y =. The equation of the diagonal not passing through the origin, is (A) x y + 5 = 0 (B) 9x y + 5 = 0 (C) x 9y 5 = 0 (D*) x y 5 = 0 [Sol. Equation of AC (x + y)(x y) = (x y )(x + y 5) or 5(x y) + (x + y) 5 = 0 x y 5 = 0 Ans. equation of BD is (x + y)(x y ) = (x y)(x + y 5) 5(x y) (x + y) = 0 9x 9y = 0 ] Q.6 What is the y-intercept of the line that is parallel to y = x, and which bisects the area of a rectangle with corners at (0, 0), (, 0), (, ) and (0, )? (A) (0, 7) (B) (0, 6) (C*) (0, 5) (D) (0, ) [Sol. Area of OADE = Area of ABCD since Area of ADD' = Area of A'DA so area of rectangle AA'OE = Area of rectangle D'DCB BD' = OA h = h h + h =...() m AD = 0 h h = h h =...() 0 from () and () h = = h = 5 5 Equation of line AD is y 0 = x put x = 0, get y = 5 (0, 5) Ans. ] Q.7 Given A (, ) and AB is any line through it cutting the x-axis in B. If AC is perpendicular to AB and meets the y-axis in C, then the equation of locus of mid- point P of BC is (A*) x + y = (B) x + y = (C) x + y = xy (D) x + y = [ Hint: y = m (x ) Q.B on Straight line and circle []

5 y = (x ) m h = m k = + m locus is x + y = ] Q.7 Given x y = and ax + by = are two variable lines, 'a' and 'b' being the parameters connected by a b the relation a + b = ab. The locus of the point of intersection has the equation (A*) x + y + xy = 0 (B) x + y xy + = 0 (C) x + y + xy + = 0 (D) x + y xy = 0 [Sol. Let (h, k) be point of intersection then h k a b ah kb b a multiply h k hk( ) a b h k hk x y xy 0 given a b a b ab b a Note that the locus is not physically viable ] Q.8 Triangle formed by the lines x + y = 0, x y = 0 and lx + my =. If l and m vary subject to the condition l + m = then the locus of its circumcentre is (A*) (x y ) = x + y (B) (x + y ) = (x y ) (C) (x + y ) = x y (D) (x y ) = (x + y ) [Sol. Coordinates of circumcentre l m, l m m l h = l l m...() and m k = l m...() also l + m = square and add () and () and then subtract () and () l m h + k = ( l m ) l m and h k = ( l m ) hence h + k = (h k ) locus is x + y = (x y ) ] = ( l m = l m ) Q.B on Straight line and circle [5]

6 Q.9 Area of the triangle formed by the line x + y = and the angle bisectors of the line pair x y + y = 0 is (A*) / (B) (C) / (D) [Sol. x (y y + ) = 0 x (y ) = 0 (x + y )(x y + ) = 0 Area = = Ans. ] Q.0 If the straight lines, ax + amy + = 0, b x + (m + ) b y + = 0 and cx + (m + )cy + = 0, m 0 are concurrent then a, b, c are in : (A) A.P. only for m = (B) A.P. for all m (C) G.P. for all m (D*) H.P. for all m. [Sol. D = a b c am b(m ) = 0 ; c(m ) a b c 0 b c a(b c) +(bc bc) = 0 ab ac + bc = 0 b(a + c) = ac b = ac a c = 0 a, b, c are in H.P. ] Q. A is a point on either of two lines y + x= at a distance of units from their point of intersection. The co-ordinates of the foot of perpendicular from A on the bisector of the angle between them are [Sol. (A), (B*) (0, 0) (C), (D) (0, ) Draw figure y x for x > 0 y x for x < 0 ] Q. P is a point inside the triangle ABC. Lines are drawn through P, parallel to the sides of the triangle. The three resulting triangles with the vertex at P have areas, 9 and 9 sq. units. The area of the triangle ABC is [Sol. (A) (B) (C) (D*) 9 q = a ly or p a 7 q a...() and...() r a...() () + () + () gives p q r = a = = ] Q.B on Straight line and circle [6]

7 Q. Let PQR be a right angled isosceles triangle, right angled at P (, ). If the equation of the line QR is x + y =, then the equation representing the pair of lines PQ and PR is (A) x y + 8xy + 0x + 0y + 5 = 0 (B*) x y + 8xy 0x 0y + 5 = 0 (C) x y + 8xy + 0x + 5y + 0 = 0 (D) x y 8xy 0x 5y 0 = 0 Q. An equilateral triangle has each of its sides of length 6 cm. If (x, y ) ; (x, y ) and (x, y ) are its vertices then the value of the determinant, x x x y y y is equal to : (A) 9 (B) (C) 86 (D*) 97 [Hint: x x x y y y = A D = A D = 6 = 97 Ans. ] Q.5 A graph is defined in polar co-ordinates as r() = cos +. The smallest x-coordinates of any point on the graph is (A*) 6 (B) 8 (C) (D) [Sol. x-coordinate = r cos = (cos + ) cos = cos + Hence smallest is 6 Ans. ] cos = (cos + ) 6 Paragraph for Question Nos. 6 to 8 Consider a general equation of degree, as x 0xy + y + 5x 6y = 0 Q.6 The value of '' for which the line pair represents a pair of straight lines is (A) (B*) (C) / (D) Q.7 For the value of obtained in above question, if L = 0 and L = 0 are the lines denoted by the given line pair then the product of the abscissa and ordinate of their point of intersection is (A) 8 (B) 8 (C*) 5 (D) 5 Q.8 If is the acute angle between L = 0 and L = 0 then lies in the interval (A) (5, 60 ) (B) (0, 5 ) (C) (5, 0 ) (D*) (0, 5 ) [Sol. (i) a = ; h = 5; b = ; g = 5 ; f = 8, c = 5 5 ()( ) + ( 8) ( 5) (6) + 5 = = 0 00 = 00 = Ans. Q.B on Straight line and circle [7]

8 (ii) x 0xy + y + 5x 6y = 0 consider the homogeneous part x 0xy + y x 6xy xy + y or x(x y) y(x y) or (x y)(x y) x 0xy + y + 5x 6y (x 6y + A)(x y + B) solving A = ; B = hence lines are x 6y = 0 and x y + = 0 7 solving intersection point 0, product = 5 Ans. (iii) tan = h ab a b = 5 = 7 (0, 5 ) Ans. ] Paragraph for Question Nos. 9 to Consider a family of lines (a + )x (a + )y (a + ) = 0 where a R Q.9 The locus of the foot of the perpendicular from the origin on each member of this family, is (A) (x ) + (y + ) = 5 (B) (x ) + (y + ) = 5 (C) (x + ) + (y ) = 5 (D*) (x ) + (y ) = 5 Q.0 A member of this family with positive gradient making an angle of with the line x y =, is (A*) 7x y 5 = 0 (B) x y + = 0 (C) x + 7y = 5 (D) 5x y = 0 Q. Minimum area of the triangle which a member of this family with negative gradient can make with the positive semi axes, is (A) 8 (B) 6 (C*) (D) [Sol. Given (a + )x (a + )y (a + ) = 0 (x y ) + a(x y ) = 0 family of lines passes through the fixed point P which is the intersection of x y = and x y = solving P(, ), now (i) k k = h h locus is x(x ) + y(y ) = 0 x + y y x = 0 5 x (y ) (x ) + (y ) = 5 Ans. (ii) We have y = m(x )...() this makes an angle of / with x y = with slope / m ( ) m = ; m = ± m = + m (with + ve sign) m = 7 m with ve sign m = m 7m = m = 7 (rejected) hence the line is y = 7(x ) 7x y 5 = 0 Ans. Q.B on Straight line and circle [8]

9 (iii) Again y = m(x ) x = 0; y = m; y = 0, x = m A = ( m) (m < 0) m A = m + = + m m let m = M (M > 0) m A = + M + = M = M M area is minimum if M = m = A min = 8 A min = Ans. ] 8 M M [MULTIPLE OBJECTIVE TYPE] Q. If x y = is a line through the intersection of x y = and x y = and the lengths of the c d a b b a perpendiculars drawn from the origin to these lines are equal in lengths then : (A*) = a b c d (B) (C*) a b = (D) none c d [Hint : Use the condition of concurrency for three lines ] = a b c d Q. A and B are two fixed points whose co-ordinates are (, ) and (5, ) respectively. The co-ordinates of a point P if ABP is an equilateral triangle, is/are : (A*), (B*), (C), (D), [ Hint : use parametric ] Q. Straight lines x + y = 5 and x y = intersect at the point A. Points B and C are chosen on these two lines such that AB = AC. Then the equation of a line BC passing through the point (, ) is (A*) x y = 0 (B*) x + y = 0 (C) x + y 9 = 0 (D) x y + 7 = 0 [Hint: Note that the lines are perpendicular. Find the equation of the lines through (, ) and parallel to the bisectors of the given lines, the slopes of the bisectors being / and ] Q.5 The straight lines x + y = 0, x + y = 0 and x + y = 0 form a triangle which is (A*) isosceles (B) right angled (C*) obtuse angled (D) equilateral Q.B on Straight line and circle [9]

10 CIRCLE [STRAIGHT OBJECTIVE TYPE] Q. The line x y + = 0 is tangent to the circle at the point (, 5) and the centre of the circles lies on x y =. The radius of the circle is [Sol. (A*) 5 (B) 5 (C) 5 (D) 5 x y + = 0 is tangent slope of line OA = equation of OA, (y 5) = (x ) y 0 = x + x + y = intersection with x y = will give coordinates of centre solving we get (8, ) distance OA = ( 8 ) ( 5) = 6 9 = 5 = 5 ] Q. If the circles x + y + ax + cy + a = 0 and x + y ax + dy = 0 intersect in two distinct points P and Q then the line 5x + by a = 0 passes through P and Q for (A) exactly one value of a (B*) no value of a (C) infinitely many values of a (D) exactly two values of a [Sol. S S = 5ax + (c d)y + a + = 0 and 5x + by a = 0 must represent the same line a c d a b a ab = c d and + a + a + = 0 a is imaginary only so no value of a (B) ] Q. Four unit circles pass through the origin and have their centres on the coordinate axes. The area of the quadrilateral whose vertices are the points of intersection (in pairs) of the circles, is [Sol. (A) sq. unit (B) sq. units (C*) sq. units (D) can not be uniquely determined, insufficient data Ar. (square of sides ) = sq. units ] Q.B on Straight line and circle [0]

11 Q. To which of the following circles, the line y x + = 0 is normal at the point,? (A) x y 9 (B) x y 9 (C) x + (y ) = 9 (D*) (x ) + y = 9 [Hint: Line must pass through the centre of the circle ] Q.5 Three concentric circles of which the biggest is x + y =, have their radii in A.P. If the line y = x + cuts all the circles in real and distinct points. The interval in which the common difference of the A.P. will lie is [Sol. (A) 0, (B) 0, (C*) 0, (D) none Equation of circle are x + y = ; x + y = ( d) ; x + y = ( d) solve any of circle with line y = x + e.g. x + y = ( d) x + x + d d = 0 cuts the circle in real and distinct point hence > 0 d d + > 0 d = ] Q.6 The circle with equation x + y = intersects the line y = 7x + 5 at two distinct points A and B. Let C be the point at which the positive x-axis intersects the circle. The angle ACB is (A) tan (B) tan (C*) tan () (D) tan [Sol. Solving y = 7x + 5 and the circle x + y = [th, ] A, 5 5 and B m OA =, 5 ; m OB = 5 Hence AOD = 90 ACB = = tan () Ans. ] Q.7 The number of common tangents of the circles (x + )² + (y )² = 9 and (x )² + (y + )² = is (A) 0 (B*) (C) (D) Q.8 In the xy-plane, the length of the shortest path from (0, 0) to (, 6) that does not go inside the circle (x 6) + (y 8) = 5 is 5 (A) 0 (B) 0 5 (C*) 0 + [Sol. Let O = (0, 0), P = (6, 8) and Q = (, 6). As shown in the figure the shortest route consists of tangent OT, minor arc TR and tangent RQ. Since OP = 0, PT = 5, and OTP = 90, (D) Q.B on Straight line and circle []

12 it follows that OPT = 60 and OT = 5. By similar reasoning, QPR = 60 and QR = 5. Because O, P and Q are collinear (why?), 5 RPT = 60, so arc TR is of length. 5 Hence the length of the shortest route is ( 5 ) + Ans. ] Q.9 Triangle ABC is right angled at A. The circle with centre A and radius AB cuts BC and AC internally at D and E respectively. If BD = 0 and DC = 6 then the length AC equals (A) 6 (B*) 6 6 (C) 0 (D) [Sol. b + r = (6)...() Also CD CB = CE CX (using power of the point C) 6 6 = (b r)(b + r) b r = 6 6 from () and () b = 6(6 + 6) = 6 5 b = 6 6 b = 6 6 Ans. ] Q.0 The equation of a line inclined at an angle to the axis X, such that the two circles x + y =, x + y 0x y + 65 = 0 intercept equal lengths on it, is (A*) x y = 0 (B) x y + = 0 (C) x y + 6 = 0 (D) x y 6 = 0 [Sol. Let equation of line be y = x + c y x = c...() perpendicular from (0, 0) on () is c = c In AON, c = AN and in CPM, c = CM perpendicular from (5, 7) on line y x = c = c Given AN = CM = c = 9 ( c) c = equation of line y = x of x y = 0 ] Q. If x = is the chord of contact of the circle x y = 8, then the equation of the corresponding pair of tangents, is (A) x 8y + 5x + 79 = 0 (B*) x 8y 5x + 79 = 0 (C) x 8y 5x 79 = 0 (D) x 8y = 79 Q.B on Straight line and circle []

13 [Sol. Equation of tangent to the circle at A,6 x + and at B, 6 6 y = 8...() x 6 y = 8...() hence equation of the line pair (x + y 7)(x y 7) = 0 [th, --007] x 8y 7(x + y ) 7(x y ) + 79 = 0 x 8y 5x + 79 = 0 Ans.] Q. Let C and C are circles defined by x + y 0x + 6 = 0 and x + y + 0x + = 0. The length of the shortest line segment PQ that is tangent to C at P and to C at Q is (A) 5 (B) 8 (C*) 0 (D) [Sol. Centres are (0, 0) and ( 5, 0) r = 6; r = 9 d = 5 r + r < d circles are separated PQ = l = d (r r ) = 65 5 = 0 Ans. ] Q. If the straight line y = mx is outside the circle x + y 0y + 90 = 0, then (A) m > (B) m < (C) m > (D*) m < [Sol. Centre (0,0), radius 0. 0 Distance of (0,0) from y = mx is greater than 0 i.e. 0 < m < ] m Q. The centre of the smallest circle touching the circles x + y y = 0 and x + y 8x 8y + 9 = 0 is (A) (, ) (B) (, ) (C) (, 7) (D*) (, 5) Q.5 Suppose that two circles C and C in a plane have no points in common. Then (A) there is no line tangent to both C and C. (B) there are exactly four lines tangent to both C and C. (C) there are no lines tangent to both C and C or there are exactly two lines tangent to both C and C. (D*) there are no lines tangent to both C and C or there are exactly four lines tangent to both C and C. [Hint: or ] Q.6 A variable line moves in such way that the product of the perpendiculars from (a, 0) and (0, 0) is equal to k. The locus of the feet of the perpendicular from (0, 0) upon the variable line is a circle, the square of whose radius is (Given: a < k ) (A*) a a k k (B) (C) a k + (D) k Q.B on Straight line and circle [] a

14 Sol. Let the equation of the variable line is [th, , P-] xx + yy (x + y ) = 0 x y p p = x y ax (x x y i.e. x + y ax = k locus is x + y ax ± k = 0 y ) = k (x, y ) Variable line x m= y r a = (± k ) +ve sign r = p p (0,0) (a,0) a k (not possible as r becomes ve) ve sign r a = + k Ans. ] Q.7 The shortest distance from the line x + y = 5 to the circle x + y = 6x 8y is equal to (A*) 7/5 (B) 9/5 (C) /5 (D) /5 [Sol. Centre: (, ) and r = 5 perpendicular distance from (, ) on x + y 5 = 0 is p = = 5 7 d = 5 = Ans. ] 5 5 Q.8 If the circle C : x + y = 6 intersects another circle C of radius 5 in such a manner that the common chord is of maximum length and has a slope equal to /, then the co-ordinates of the centre of C are 9 (A), (B*) , 5 (C) 9, (D) 5 5 [Hint : Equation of common chord is y = x x y = 0 family of circle x + y + (x y) = 0 equate radius of this circle as 5. Alternately...] Q.9 The points (x, y ), (x, y ), (x, y ) and (x, y ) are always : (A) collinear (B*) concyclic (C) vertices of a square (D) vertices of a rhombus [Hint: All the points lie on the circle (x x ) (x x ) + (y y ) (y y ) = 0 Note that points form a rectangle] 5 9, 5 Q.0 Let C be the circle of radius unity centred at the origin. If two positive numbers x and x are such that the line passing through (x, ) and (x, ) is tangent to C then (A*) x x = (B) x x = (C) x + x = (D) x x = Q.B on Straight line and circle []

15 [Sol. Let the equation tangent in parametric form is x cos + y sin = put (x, ) x cos sin = x cos = + sin...() now put (x, ) x cos = sin...() () () x x cos = sin = cos x x = Ans. Q. Tangents are drawn from any point on the circle x + y = R to the circle x + y = r. If the line joining the points of intersection of these tangents with the first circle also touch the second, then R equals (A) r (B*) r (C) r (D) r 5 [HInt: ] Q. The locus of the middle points of the system of chords of the circle x² + y² = 6 which are parallel to the line y = x + 5 is (A) x = y (B*) x + y = 0 (C) y + x = 0 () y = x Q. The distance between the chords of contact of tangents to the circle x + y + gx + fy + c = 0 from the origin and the point (g, f) is [Sol. (A) g f (B) g f c (C*) g f c g f Equation of chord or contact of the pair of tangent from (0, 0) and (g, f) are gx + fy + c = 0...() and gx + fy + g f (D) c = 0...() g f c g f These lines are parallel hence distance = g c g f f c ] Q. The locus of the center of the circles such that the point (, ) is the mid point of the chord 5x + y = 6 is (A*) x 5y + = 0 (B) x + 5y = 0 (C) x + 5y + = 0 (D) none [ Hint : Slope of the given line = 5/ 5 f. g = 5 + 5f = + g locus is x 5y + = 0 ] Q.B on Straight line and circle [5]

16 Q.5 The points A (a, 0), B (0, b), C (c, 0) and D (0, d) are such that ac = bd and a, b, c, d are all non-zero. Then the points (A) form a parallelogram (B) do not lie on a circle (C) form a trapezium (D*) are concyclic [Sol. If OA OC = OB OD (Power of point) then points are concylic a c = bd (true) ] Q.6 The locus of the centers of the circles which cut the circles x + y + x 6y + 9 = 0 and x + y 5x + y = 0 orthogonally is : (A) 9x + 0y 7 = 0 (B) x y + = 0 (C*) 9x 0y + = 0 (D) 9x + 0y + 7 = 0 [Hint: Let out circle be x + y + gx + fy + c = 0 conditions ( g) ( ) + ( f ) () = c + 9 and ( g) (5/) + ( f ) ( ) = c ag 0 f = locus of centre 9x 0y + = 0 ] Q.7 If the angle between the tangents drawn from P to the circle x + y + x 6y + 9 sin + cos =0 is, then the locus of P is (A) x + y + x 6y + = 0 (B) x + y + x 6y 9 = 0 (C) x + y + x 6y = 0 (D*) x + y + x 6y + 9 = 0 [Sol. C(, ); R = sin cos = sin R = sin (,)C sin Now use sin = CP = CP Hence locus is (x + ) + (y ) = ] P(x,y) y sin Q.8 The locus of the mid points of the chords of the circle x² + y² + x 6y = 0 which subtend an angle of radians at its circumference is : (A) (x )² + (y + )² = 6.5 (B*) (x + )² + (y )² = 6.5 (C) (x + )² + (y )² = 8.75 (D) (x + )² + (y + )² = 8.75 [ Hint : radius = 5 p = 5 cos 60º =.5 locus is (h + ) + (k ) = 6.5 ] Q.9 The angle at which the circles (x ) + y = 0 and x + (y ) = 5 intersect is (A) (B*) 6 (C) (D) [Hint: cos = = = ] Q.B on Straight line and circle [6]

17 Q.0 A circle of radius unity is centred at origin. Two particles start moving at the same time from the point (, 0) and move around the circle in opposite direction. One of the particle moves counterclockwise with constant speed v and the other moves clockwise with constant speed v. After leaving (, 0), the two particles meet first at a point P, and continue until they meet next at point Q. The coordinates of the point Q are (A) (, 0) (B) (0, ) (C) (0, ) (D*) (, 0) [Sol. The particle which moves clockwise is moving three times as fast as the particle moving anticlockwise This tells us that in the time that the clockwise particle travels (/) th of the way around the circle the anticlockwise particle will travel (/) th of the way around the circle and so the nd particle will meet at p (0, ). Using the same logic they will meet at Q (, 0) when they meet the nd time] Q. The value of 'c' for which the set, { (x, y)x + y + x } {(x, y)x y + c 0} contains only one point in common is : (A) (, ] [, ) (B) {, } (C) { } (D*) { } [Hint: x + y + x = 0; centre (, 0) and rad. = line x y + c = 0 c ; c = ; c = ± c = or ] Q. Locus of the middle points of a system of parallel chords with slope, of the circle x + y x y = 0, has the equation (A*) x + y = 0 (B) x y = 0 (C) x y = 0 (D) x + y 5 = 0 [Hint: Locus will be a line with slope / and passing through the centre (, ) of the circle y = (x ) y = x + x + y = 0 Ans. ] Q. Two circles are drawn through the points (, 0) and (, ) to touch the axis of y. They intersect at an angle (A*) cot (B) cos 5 (C) (D) tan [Hint : The two circles are x + y x + y + = 0 and x + y 0x 6y + 9 = 0 (use family of circles) ] Q. A foot of the normal from the point (, ) to a circle is (, ) and a diameter of the circle has the equation x y = 0. Then the equation of the circle is (A) x + y y + = 0 (B) x + y y + = 0 (C*) x + y x = 0 (D) x + y x + = 0 [Sol. Diameter of the circle is given as x y = 0...() slope of PN: = Q.B on Straight line and circle [7]

18 equation of normal through PN is y = (x ) x y = 0...() solving () and () centre is (, 0) hence equation of the circle is (x ) + y = ( ) + x + y x = 0 Ans. ] Q.5 A rhombus is inscribed in the region common to the two circles x + y x = 0 and x + y + x = 0 with two of its vertices on the line joining the centres of the circles. The area of the rhombous is : (A*) 8 sq.units (B) sq.units (C) 6 sq.units (D) none [Hint : circles with centre (, 0) and (, 0) each with radius y axis is their common chord. The inscribed rhombus has its diagonals equal to and A = d d = 8 ] Q.6 A circle is drawn touching the xaxis and centre at the point which is the reflection of (a, b) in the line y x = 0. The equation of the circle is (A) x + y bx ay + a = 0 (B*) x + y bx ay + b = 0 (C) x + y ax by + b = 0 (D) x + y ax by + a = 0 k b [Sol. h a also a h and b k a b = k h and touches the x-axis k b = h a or a b = h k lies on y-axis g c = 0 a h = b k centre (b, a) g = c = b equation x + y hx ay + b = 0 ] Q.7 A(, 0) and B(0, ) and two fixed points on the circle x + y =. C is a variable point on this circle. As C moves, the locus of the orthocentre of the triangle ABC is (A*) x + y x y + = 0 (B) x + y x y = 0 (C) x + y = (D) x + y + x y + = 0 [Sol. Let C (cos, sin ); H(h, k) is the orthocentre of the ABC B(0, ) h = + cos k = + sin (x ) + (y ) = x + y x y + = 0 ] C(cos, sin ) O A(, 0) Q.B on Straight line and circle [8]

19 Q.8 A line meets the co-ordinate axes in A and B. A circle is circumscribed about the triangle OAB. If d and d are the distances of the tangent to the circle at the origin O from the points A and B respectively, the diameter of the circle is : (A) d d (B) [Sol. Let the circle be x + y + gx + fy = 0 Tangent at the origin is gx + fy = 0 d = d = g g g f f f d d (C*) d + d (D) d d g f diameter d d d d Q.9 In a right triangle ABC, right angled at A, on the leg AC as diameter, a semicircle is described. The chord joining A with the point of intersection D of the hypotenuse and the semicircle, then the length AC equals to (A) AB AD AB AD (B) [Sol. l x = y l x l x = y (l + x ) l (x y ) = x y l = x xy y = AB AD AB AD (C) where l = AC; x = AB, y = AD AB AD AB AD Ans. ] AB AD (D*) AB AD AB AD Q.0 A straight line l with equation x y + 0 = 0 meets the circle with equation x + y = 00 at B in the first quadrant. A line through B, perpendicular to l cuts the y-axis at P (0, t). The value of 't' is (A) (B) 5 (C*) 0 (D) 5 [Sol. slope of l = slope of l = equation of l y = (x 0) y + x = 0 Hence t = 0 Ans.] Q. B and C are fixed points having coordinates (, 0) and (, 0) respectively. If the vertical angle BAC is 90º, then the locus of the centroid of the ABC has the equation : (A*) x + y = (B) x + y = (C) 9 (x + y ) = (D) 9 (x + y ) = [Hint : Let A (a, b) and G (h. k) Now A, G, O are collinear h = 0. a a = h and similarly b = k. Now (a, b) lies on the circle x + y = 9 A ] ] Q.B on Straight line and circle [9]

20 Paragraph for Question Nos. to Let C be a circle of radius r with centre at O. Let P be a point outside C and D be a point on C. A line through P intersects C at Q and R, S is the midpoint of QR. Q. For different choices of line through P, the curve on which S lies, is (A) a straight line (B) an arc of circle with P as centre (C) an arc of circle with PS as diameter (D*) an arc of circle with OP as diameter Q. Let P is situated at a distance 'd' from centre O, then which of the following does not equal the product (PQ) (PR)? (A) d r (B) PT, where T is a point on C and PT is tangent to C (C) (PS) (QS)(RS) (D*) (PS) Q. Let XYZ be an equilateral triangle inscribed in C. If,, denote the distances of D from vertices X, Y, Z respectively, the value of product ( + ) ( + ) ( + ), is [Sol. (i) (A*) 0 (B) 8 (C) 6 Locus of S is a part of circle with OP as diameter passing inside the circle 'C' C (D) None of these N Q O M R S(h,k) (ii) (PR) (PQ) = PT = (PN)(PM) = (d r) (d + r) = d r = (PS SR) (PS + SQ) = PS SQ ( SQ = SR) = PS (SQ)(SR) (PQ) (PR) (PS) (iii) Using Ptolemy's theorem, (YD) (XZ) = (XY)(ZD) + (YZ) (XD) = XZ (ZD + XD) { (XY = YZ = ZX) } = + (A) Alternatively: P Y X D Z (YD) (XY) = (YD) from XYD (YD) = + (YD) (XY)...() (YD) = + (YD) (YZ)...() ()-() ( ) = = + ] (YD) (YZ) = (YD) from YZD Q.B on Straight line and circle [0]

21 Paragraph for question nos. 5 to 7 Consider circles S : x + y + x = 0 S : x + y = 0 S : x + y + y = 0 Q.5 The radius of the circle which bisect the circumferences of the circles S = 0 ; S = 0 ; S = 0 is (A) (B) (C*) (D) 0 Q.6 If the circle S = 0 is orthogonal to S = 0 ; S = 0 and S = 0 and has its centre at (a, b) and radius equals to 'r' then the value of (a + b + r) equals (A) 0 (B) (C) (D*) Q.7 The radius of the circle touching S = 0 and S = 0 at (, 0) and passing through (, ) is (A) (B) (C*) (D) S S [Sol. (,0) O (,0) (i) S S = 0 (,0) r (a,b) r (0,0) (0, ) S r = a + b + = (a + ) + b + and (a + ) + b + = a + (b + ) + a + = 0 a = b a = b = r = 9 r = Ans. (ii) S S = 0 x = S S y = Radical centre = (, ) radius L T = S = equation of circle is (x ) + (y ) = radius = and a = ; b = a + b + r = Ans. Q.B on Straight line and circle []

22 x = (,) (iii) (,0) family of circles touches the line x = 0 at (, 0) is (x ) + (y 0) + (x ) = 0 passing through (, ) + + = 0 = x + y 6x + 5 = 0 radius 9 5 = Ans. ] Paragraph for Question Nos. 8 to 50 Consider a line pair ax + xy y 5x + 5y + c = 0 representing perpendicular lines intersecting each other at C and forming a triangle ABC with the x-axis. Q.8 If x and x are intercepts on the x-axis and y and y are the intercepts on the y-axis then the sum (x + x + y + y ) is equal to (A) 6 (B*) 5 (C) (D) Q.9 Distance between the orthocentre and circumcentre of the triangle ABC is (A) (B) (C*) 7/ (D) 9/ Q.50 If the circle x + y y + k = 0 is orthogonal with the circumcircle of the triangle ABC then 'k' equals (A) / (B) (C) (D*) / [Sol. (i) As lines are perpendicular a = 0 a = (coefficient of x + coefficient of y = 0) using = 0 c = (D abc + fgh af bg ch ) hence the two lines are x + y = 0 and x y + = 0 x intercepts y intercepts (ii) (CM) = (iii) = x ; x y ; y = = = = Circumcircle of ABC 9 5 x + x + y + y = 5 Ans. CM = 7 Ans. x (x ) y 0 (x + )(x ) + y = 0 (x + y ) 5x = 0 x + y 5 x = 0...() given x + y y + k = 0 which is orthogonal to () using the condition of orthogonality we get, = k k = Ans. ] Q.B on Straight line and circle []

23 Paragraph for question nos. 5 to 5 Consider a circle x + y = and a point P(, ). denotes the angle enclosed by the tangents from P on the circle and A, B are the points of contact of the tangents from P on the circle. Q.5 The value of lies in the interval (A) (0, 5 ) (B) (5, 0 ) (C) 0, 5 ) (D*) (5, 60 ) Q.5 The intercept made by a tangent on the x-axis is (A) 9/ (B*) 0/ (C) / (D) / Q.5 Locus of the middle points of the portion of the tangent to the circle terminated by the coordinate axes is (A*) x + y = (B) x + y = (C) x + y = (D) x y = [Sol. Tangent y = m(x ) mx y + ( m) = 0 m p = m = ( m) = + m m m = 0 m = 0 or m = tan = (5, 60 ) Hence equation of tangent is y = and (with infinite intercept on x-axis) or y = (x ) y 6 = x 6 x y 0 = 0 0 x-intercept = Ans.(ii) (B) Variable line with mid point (h, k) x h y, it touches the circle x k + y = h k = h locus is x k + y = Ans.(iii) (A)] [MULTIPLE OBJECTIVE TYPE] Q.5 Let L be a line passing through the origin and L be the line x + y =. If the intercepts made by the circle x + y x + y = 0 on L and L are equal then the equation of L can be (A) x + y = 0 (B*) x y = 0 (C*) x + 7y = 0 (D) x 7y = 0 [Sol. The chords are of equal length, then the distances of the centre from the lines are equal. Let L be y mx = 0. Centre is, m m 7m 6m = 0 m =, ] 7 Q.B on Straight line and circle []

24 Q.55 Consider the circles S : x + y = and S : x + y x y + = 0 which of the following statements are correct? (A*) Number of common tangents to these circles is. (B*) If the power of a variable point P w.r.t. these two circles is same then P moves on the line x + y = 0. (C) Sum of the y-intercepts of both the circles is 6. (D*) The circles S and S are orthogonal. [Sol. S : x + y = and S = x + y x y + = 0 centre: (0, 0); radius = centre : (, ); radius = (A) d = distance between centres = 5 r + r = r r = r r < d < r + r these circles are intersecting. number of common tangents is. (A) is correct (B) P(h, k) power of point P is same w.r.t. these two circles. h k = h k h k = h k + h + k 8 = 0 x + y = 0 (B) is correct (C) y intercept of S is = y intercept of S is = 0 sum of y-intercept = (C) is incorrect (D) (0 + 0) = + circle is orthogonal. (D) is correct] Q.56 If al bm + dl + = 0, where a, b, d are fixed real numbers such that a + b = d then the line lx + my + = 0 touches a fixed circle : (A*) which cuts the xaxis orthogonally (B) with radius equal to b (C*) on which the length of the tangent from the origin is d b (D) none of these. [Hint : (d b) l + dl + = bm d l + dl + = b (l + m ) d m = b centre (d, 0) and radius b (x d) + y = b ] Q.57 Three distinct lines are drawn in a plane. Suppose there exist exactly n circles in the plane tangent to all the three lines, then the possible values of n is/are (A*) 0 (B) (C*) (D*) [Sol. Case-: If lines form a triangle then n = i.e. excircles and incircle Case-: If lines are concurrent or all parallel then n = 0 Case-: If two are parallel and third cuts then n = hence (A), (C), (D) ] Q.B on Straight line and circle []

25 Q.58 Consider the circles S : x + y + x + y + = 0 S : x + y x + = 0 S : x + y + 6y + 5 = 0 Which of this following statements are correct? (A) Radical centre of S, S and S lies in st quadrant. (B*) Radical centre of S, S and S lies in th quadrants. (C*) Radius of the circle intersecting S, S and S orthogonally is. (D*) Circle orthogonal to S, S and S has its x and y intercept equal to zero. [Hint: radical centre is (, ) radius of the circle orthogonally to S, S, S is and has the equation x + y x + y + = 0 ] Q.59 Locus of the intersection of the two straight lines passing through (, 0) and (, 0) respectively and including an angle of 5 can be a circle with [Sol. m = (A) centre (, 0) and radius. (B) centre (, 0) and radius. (C*) centre (0, ) and radius. (D*) centre (0, ) and radius. k k k h ; m = k h ; tan 5 = h h k h h + k = ± k locus is x + y ± y = 0 k(h h ) = h k x + (y ± ) = (0, ) or (0, ) and radius = (C) and (D) Ans. ] [MATCH THE COLUMN] Q.60 Column-I Column-II (A) If the straight line y = kx K I touches or passes outside (P) the circle x + y 0y + 90 = 0 then k can have the value (B) Two circles x + y + px + py 7 = 0 (Q) and x + y 0x + py + = 0 intersect each other orthogonally then the value of p is (C) If the equation x + y + x + = 0 and x + y y + 8 = 0 (R) represent real circles then the value of can be (D) Each side of a square is of length. The centre of the square is (, 7). (S) 5 One diagonal of the square is parallel to y = x. The possible abscissae of the vertices of the square can be [Ans. (A) P, Q, R; (B) Q, R; (C) Q, R, S; (D) P, S] [Sol. (A) x + k x 0kx + 90 = 0 x ( + k ) 0kx + 90 = 0 Q.B on Straight line and circle [5]

26 D 0 00k 90( + k ) 0 0k 9 9k 0 k 9 0 k [, ] p p (B) 5 p = 6 5p + p + 6 = 0 p 5p + 6 = 0 p = or Ans. (C) r = 0 (, ] [, )...() r = (, ] [, )...() () () is (, ] [, ) Ans. (D) Ans. {, 5}] Q.6 Column-I Column-II (A) Two intersecting circles (P) have a common tangent (B) Two circles touching each other (Q) have a common normal (C) Two non concentric circles, one strictly inside (R) do not have a common normal the other (D) Two concentric circles of different radii (S) do not have a radical axis. [Ans. (A) P, Q; (B) P, Q; (C) Q; (D) Q, S] Q.6 Column-I Column-II (A) Let 'P' be a point inside the triangle ABC and is equidistant (P) centroid from its sides. DEF is a triangle obtained by the intersection of the external angle bisectors of the angles of the ABC. With respect to the triangle DEF point P is its (B) Let 'Q' be a point inside the triangle ABC (Q) orthocentre If (AQ)sin A = (BQ)sin B = (CQ)sin C then with respect to the triangle ABC, Q is its (C) Let 'S' be a point in the plane of the triangle ABC. If the point is (R) incentre such that infinite normals can be drawn from it on the circle passing through A, B and C then with respect to the triangle ABC, S is its (D) Let ABC be a triangle. D is some point on the side BC such that (S) circumcentre the line segments parallel to BC with their extremities on AB and AC get bisected by AD. Point E and F are similarly obtained on CA and AB. If segments AD, BE and CF are concurrent at a point R then with respect to the triangle ABC, R is its [Ans. (A) Q; (B) R; (C) S; (D) P] Q.B on Straight line and circle [6]

(D) (A) Q.3 To which of the following circles, the line y x + 3 = 0 is normal at the point ? 2 (A) 2

(D) (A) Q.3 To which of the following circles, the line y x + 3 = 0 is normal at the point ? 2 (A) 2 CIRCLE [STRAIGHT OBJECTIVE TYPE] Q. The line x y + = 0 is tangent to the circle at the point (, 5) and the centre of the circles lies on x y = 4. The radius of the circle is (A) 3 5 (B) 5 3 (C) 5 (D) 5

More information

QUESTION BANK ON STRAIGHT LINE AND CIRCLE

QUESTION BANK ON STRAIGHT LINE AND CIRCLE QUESTION BANK ON STRAIGHT LINE AND CIRCLE Select the correct alternative : (Only one is correct) Q. If the lines x + y + = 0 ; 4x + y + 4 = 0 and x + αy + β = 0, where α + β =, are concurrent then α =,

More information

Q.2 A, B and C are points in the xy plane such that A(1, 2) ; B (5, 6) and AC = 3BC. Then. (C) 1 1 or

Q.2 A, B and C are points in the xy plane such that A(1, 2) ; B (5, 6) and AC = 3BC. Then. (C) 1 1 or STRAIGHT LINE [STRAIGHT OBJECTIVE TYPE] Q. A variable rectangle PQRS has its sides parallel to fied directions. Q and S lie respectivel on the lines = a, = a and P lies on the ais. Then the locus of R

More information

DEEPAWALI ASSIGNMENT CLASS 11 FOR TARGET IIT JEE 2012 SOLUTION

DEEPAWALI ASSIGNMENT CLASS 11 FOR TARGET IIT JEE 2012 SOLUTION DEEPAWALI ASSIGNMENT CLASS FOR TARGET IIT JEE 0 SOLUTION IMAGE OF SHRI GANESH LAXMI SARASWATI Director & H.O.D. IITJEE Mathematics SUHAG R. KARIYA (S.R.K. Sir) DOWNLOAD FREE STUDY PACKAGE, TEST SERIES

More information

QUESTION BANK ON. CONIC SECTION (Parabola, Ellipse & Hyperbola)

QUESTION BANK ON. CONIC SECTION (Parabola, Ellipse & Hyperbola) QUESTION BANK ON CONIC SECTION (Parabola, Ellipse & Hyperbola) Question bank on Parabola, Ellipse & Hyperbola Select the correct alternative : (Only one is correct) Q. Two mutually perpendicular tangents

More information

TARGET : JEE 2013 SCORE. JEE (Advanced) Home Assignment # 03. Kota Chandigarh Ahmedabad

TARGET : JEE 2013 SCORE. JEE (Advanced) Home Assignment # 03. Kota Chandigarh Ahmedabad TARGT : J 01 SCOR J (Advanced) Home Assignment # 0 Kota Chandigarh Ahmedabad J-Mathematics HOM ASSIGNMNT # 0 STRAIGHT OBJCTIV TYP 1. If x + y = 0 is a tangent at the vertex of a parabola and x + y 7 =

More information

PREPARED BY: ER. VINEET LOOMBA (B.TECH. IIT ROORKEE) 60 Best JEE Main and Advanced Level Problems (IIT-JEE). Prepared by IITians.

PREPARED BY: ER. VINEET LOOMBA (B.TECH. IIT ROORKEE) 60 Best JEE Main and Advanced Level Problems (IIT-JEE). Prepared by IITians. www. Class XI TARGET : JEE Main/Adv PREPARED BY: ER. VINEET LOOMBA (B.TECH. IIT ROORKEE) ALP ADVANCED LEVEL PROBLEMS Straight Lines 60 Best JEE Main and Advanced Level Problems (IIT-JEE). Prepared b IITians.

More information

Trans Web Educational Services Pvt. Ltd B 147,1st Floor, Sec-6, NOIDA, UP

Trans Web Educational Services Pvt. Ltd B 147,1st Floor, Sec-6, NOIDA, UP Solved Examples Example 1: Find the equation of the circle circumscribing the triangle formed by the lines x + y = 6, 2x + y = 4, x + 2y = 5. Method 1. Consider the equation (x + y 6) (2x + y 4) + λ 1

More information

So, eqn. to the bisector containing (-1, 4) is = x + 27y = 0

So, eqn. to the bisector containing (-1, 4) is = x + 27y = 0 Q.No. The bisector of the acute angle between the lines x - 4y + 7 = 0 and x + 5y - = 0, is: Option x + y - 9 = 0 Option x + 77y - 0 = 0 Option x - y + 9 = 0 Correct Answer L : x - 4y + 7 = 0 L :-x- 5y

More information

VKR Classes TIME BOUND TESTS 1-7 Target JEE ADVANCED For Class XI VKR Classes, C , Indra Vihar, Kota. Mob. No

VKR Classes TIME BOUND TESTS 1-7 Target JEE ADVANCED For Class XI VKR Classes, C , Indra Vihar, Kota. Mob. No VKR Classes TIME BOUND TESTS -7 Target JEE ADVANCED For Class XI VKR Classes, C-9-0, Indra Vihar, Kota. Mob. No. 9890605 Single Choice Question : PRACTICE TEST-. The smallest integer greater than log +

More information

Q1. If (1, 2) lies on the circle. x 2 + y 2 + 2gx + 2fy + c = 0. which is concentric with the circle x 2 + y 2 +4x + 2y 5 = 0 then c =

Q1. If (1, 2) lies on the circle. x 2 + y 2 + 2gx + 2fy + c = 0. which is concentric with the circle x 2 + y 2 +4x + 2y 5 = 0 then c = Q1. If (1, 2) lies on the circle x 2 + y 2 + 2gx + 2fy + c = 0 which is concentric with the circle x 2 + y 2 +4x + 2y 5 = 0 then c = a) 11 b) -13 c) 24 d) 100 Solution: Any circle concentric with x 2 +

More information

CONCURRENT LINES- PROPERTIES RELATED TO A TRIANGLE THEOREM The medians of a triangle are concurrent. Proof: Let A(x 1, y 1 ), B(x, y ), C(x 3, y 3 ) be the vertices of the triangle A(x 1, y 1 ) F E B(x,

More information

VAISHALI EDUCATION POINT (QUALITY EDUCATION PROVIDER)

VAISHALI EDUCATION POINT (QUALITY EDUCATION PROVIDER) BY:Prof. RAHUL MISHRA Class :- X QNo. VAISHALI EDUCATION POINT (QUALITY EDUCATION PROVIDER) CIRCLES Subject :- Maths General Instructions Questions M:9999907099,9818932244 1 In the adjoining figures, PQ

More information

Topic 2 [312 marks] The rectangle ABCD is inscribed in a circle. Sides [AD] and [AB] have lengths

Topic 2 [312 marks] The rectangle ABCD is inscribed in a circle. Sides [AD] and [AB] have lengths Topic 2 [312 marks] 1 The rectangle ABCD is inscribed in a circle Sides [AD] and [AB] have lengths [12 marks] 3 cm and (\9\) cm respectively E is a point on side [AB] such that AE is 3 cm Side [DE] is

More information

Q.1 If a, b, c are distinct positive real in H.P., then the value of the expression, (A) 1 (B) 2 (C) 3 (D) 4. (A) 2 (B) 5/2 (C) 3 (D) none of these

Q.1 If a, b, c are distinct positive real in H.P., then the value of the expression, (A) 1 (B) 2 (C) 3 (D) 4. (A) 2 (B) 5/2 (C) 3 (D) none of these Q. If a, b, c are distinct positive real in H.P., then the value of the expression, b a b c + is equal to b a b c () (C) (D) 4 Q. In a triangle BC, (b + c) = a bc where is the circumradius of the triangle.

More information

l (D) 36 (C) 9 x + a sin at which the tangent is parallel to x-axis lie on

l (D) 36 (C) 9 x + a sin at which the tangent is parallel to x-axis lie on Dpp- to MATHEMATICS Dail Practice Problems Target IIT JEE 00 CLASS : XIII (VXYZ) DPP. NO.- to DPP- Q. If on a given base, a triangle be described such that the sum of the tangents of the base angles is

More information

Chapter (Circle) * Circle - circle is locus of such points which are at equidistant from a fixed point in

Chapter (Circle) * Circle - circle is locus of such points which are at equidistant from a fixed point in Chapter - 10 (Circle) Key Concept * Circle - circle is locus of such points which are at equidistant from a fixed point in a plane. * Concentric circle - Circle having same centre called concentric circle.

More information

Part (1) Second : Trigonometry. Tan

Part (1) Second : Trigonometry. Tan Part (1) Second : Trigonometry (1) Complete the following table : The angle Ratio 42 12 \ Sin 0.3214 Cas 0.5321 Tan 2.0625 (2) Complete the following : 1) 46 36 \ 24 \\ =. In degrees. 2) 44.125 = in degrees,

More information

STRAIGHT LINES EXERCISE - 3

STRAIGHT LINES EXERCISE - 3 STRAIGHT LINES EXERCISE - 3 Q. D C (3,4) E A(, ) Mid point of A, C is B 3 E, Point D rotation of point C(3, 4) by angle 90 o about E. 3 o 3 3 i4 cis90 i 5i 3 i i 5 i 5 D, point E mid point of B & D. So

More information

PRACTICE QUESTIONS CLASS IX: CHAPTER 4 LINEAR EQUATION IN TWO VARIABLES

PRACTICE QUESTIONS CLASS IX: CHAPTER 4 LINEAR EQUATION IN TWO VARIABLES PRACTICE QUESTIONS CLASS IX: CHAPTER 4 LINEAR EQUATION IN TWO VARIABLES 1. Find the value of k, if x =, y = 1 is a solution of the equation x + 3y = k.. Find the points where the graph of the equation

More information

( 1 ) Show that P ( a, b + c ), Q ( b, c + a ) and R ( c, a + b ) are collinear.

( 1 ) Show that P ( a, b + c ), Q ( b, c + a ) and R ( c, a + b ) are collinear. Problems 01 - POINT Page 1 ( 1 ) Show that P ( a, b + c ), Q ( b, c + a ) and R ( c, a + b ) are collinear. ( ) Prove that the two lines joining the mid-points of the pairs of opposite sides and the line

More information

POINT. Preface. The concept of Point is very important for the study of coordinate

POINT. Preface. The concept of Point is very important for the study of coordinate POINT Preface The concept of Point is ver important for the stud of coordinate geometr. This chapter deals with various forms of representing a Point and several associated properties. The concept of coordinates

More information

Objective Mathematics

Objective Mathematics . A tangent to the ellipse is intersected by a b the tangents at the etremities of the major ais at 'P' and 'Q' circle on PQ as diameter always passes through : (a) one fied point two fied points (c) four

More information

chapter 1 vector geometry solutions V Consider the parallelogram shown alongside. Which of the following statements are true?

chapter 1 vector geometry solutions V Consider the parallelogram shown alongside. Which of the following statements are true? chapter vector geometry solutions V. Exercise A. For the shape shown, find a single vector which is equal to a)!!! " AB + BC AC b)! AD!!! " + DB AB c)! AC + CD AD d)! BC + CD!!! " + DA BA e) CD!!! " "

More information

1 is equal to. 1 (B) a. (C) a (B) (D) 4. (C) P lies inside both C & E (D) P lies inside C but outside E. (B) 1 (D) 1

1 is equal to. 1 (B) a. (C) a (B) (D) 4. (C) P lies inside both C & E (D) P lies inside C but outside E. (B) 1 (D) 1 Single Correct Q. Two mutuall perpendicular tangents of the parabola = a meet the ais in P and P. If S is the focus of the parabola then l a (SP ) is equal to (SP ) l (B) a (C) a Q. ABCD and EFGC are squares

More information

COORDINATE GEOMETRY BASIC CONCEPTS AND FORMULAE. To find the length of a line segment joining two points A(x 1, y 1 ) and B(x 2, y 2 ), use

COORDINATE GEOMETRY BASIC CONCEPTS AND FORMULAE. To find the length of a line segment joining two points A(x 1, y 1 ) and B(x 2, y 2 ), use COORDINATE GEOMETRY BASIC CONCEPTS AND FORMULAE I. Length of a Line Segment: The distance between two points A ( x1, 1 ) B ( x, ) is given b A B = ( x x1) ( 1) To find the length of a line segment joining

More information

SOLVED SUBJECTIVE EXAMPLES

SOLVED SUBJECTIVE EXAMPLES Example 1 : SOLVED SUBJECTIVE EXAMPLES Find the locus of the points of intersection of the tangents to the circle x = r cos, y = r sin at points whose parametric angles differ by /3. All such points P

More information

( 1 ) Find the co-ordinates of the focus, length of the latus-rectum and equation of the directrix of the parabola x 2 = - 8y.

( 1 ) Find the co-ordinates of the focus, length of the latus-rectum and equation of the directrix of the parabola x 2 = - 8y. PROBLEMS 04 - PARABOLA Page 1 ( 1 ) Find the co-ordinates of the focus, length of the latus-rectum and equation of the directrix of the parabola x - 8. [ Ans: ( 0, - ), 8, ] ( ) If the line 3x 4 k 0 is

More information

LLT Education Services

LLT Education Services 8. The length of a tangent from a point A at distance 5 cm from the centre of the circle is 4 cm. Find the radius of the circle. (a) 4 cm (b) 3 cm (c) 6 cm (d) 5 cm 9. From a point P, 10 cm away from the

More information

SYSTEM OF CIRCLES OBJECTIVES (a) Touch each other internally (b) Touch each other externally

SYSTEM OF CIRCLES OBJECTIVES (a) Touch each other internally (b) Touch each other externally SYSTEM OF CIRCLES OBJECTIVES. A circle passes through (0, 0) and (, 0) and touches the circle x + y = 9, then the centre of circle is (a) (c) 3,, (b) (d) 3,, ±. The equation of the circle having its centre

More information

21. Prove that If one side of the cyclic quadrilateral is produced then the exterior angle is equal to the interior opposite angle.

21. Prove that If one side of the cyclic quadrilateral is produced then the exterior angle is equal to the interior opposite angle. 21. Prove that If one side of the cyclic quadrilateral is produced then the exterior angle is equal to the interior opposite angle. 22. Prove that If two sides of a cyclic quadrilateral are parallel, then

More information

CIRCLES. ii) P lies in the circle S = 0 s 11 = 0

CIRCLES. ii) P lies in the circle S = 0 s 11 = 0 CIRCLES 1 The set of points in a plane which are at a constant distance r ( 0) from a given point C is called a circle The fixed point C is called the centre and the constant distance r is called the radius

More information

Objective Mathematics

Objective Mathematics . In BC, if angles, B, C are in geometric seq- uence with common ratio, then is : b c a (a) (c) 0 (d) 6. If the angles of a triangle are in the ratio 4 : :, then the ratio of the longest side to the perimeter

More information

+ 2gx + 2fy + c = 0 if S

+ 2gx + 2fy + c = 0 if S CIRCLE DEFINITIONS A circle is the locus of a point which moves in such a way that its distance from a fixed point, called the centre, is always a constant. The distance r from the centre is called the

More information

6 CHAPTER. Triangles. A plane figure bounded by three line segments is called a triangle.

6 CHAPTER. Triangles. A plane figure bounded by three line segments is called a triangle. 6 CHAPTER We are Starting from a Point but want to Make it a Circle of Infinite Radius A plane figure bounded by three line segments is called a triangle We denote a triangle by the symbol In fig ABC has

More information

PAIR OF LINES-SECOND DEGREE GENERAL EQUATION THEOREM If the equation then i) S ax + hxy + by + gx + fy + c represents a pair of straight lines abc + fgh af bg ch and (ii) h ab, g ac, f bc Proof: Let the

More information

1. The unit vector perpendicular to both the lines. Ans:, (2)

1. The unit vector perpendicular to both the lines. Ans:, (2) 1. The unit vector perpendicular to both the lines x 1 y 2 z 1 x 2 y 2 z 3 and 3 1 2 1 2 3 i 7j 7k i 7j 5k 99 5 3 1) 2) i 7j 5k 7i 7j k 3) 4) 5 3 99 i 7j 5k Ans:, (2) 5 3 is Solution: Consider i j k a

More information

Time : 2 Hours (Pages 3) Max. Marks : 40. Q.1. Solve the following : (Any 5) 5 In PQR, m Q = 90º, m P = 30º, m R = 60º. If PR = 8 cm, find QR.

Time : 2 Hours (Pages 3) Max. Marks : 40. Q.1. Solve the following : (Any 5) 5 In PQR, m Q = 90º, m P = 30º, m R = 60º. If PR = 8 cm, find QR. Q.P. SET CODE Q.1. Solve the following : (ny 5) 5 (i) (ii) In PQR, m Q 90º, m P 0º, m R 60º. If PR 8 cm, find QR. O is the centre of the circle. If m C 80º, the find m (arc C) and m (arc C). Seat No. 01

More information

PRACTICE TEST 1 Math Level IC

PRACTICE TEST 1 Math Level IC SOLID VOLUME OTHER REFERENCE DATA Right circular cone L = cl V = volume L = lateral area r = radius c = circumference of base h = height l = slant height Sphere S = 4 r 2 V = volume r = radius S = surface

More information

Einstein Classes, Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road New Delhi , Ph. : ,

Einstein Classes, Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road New Delhi , Ph. : , MCN COMPLEX NUMBER C The complex number Complex number is denoted by ie = a + ib, where a is called as real part of (denoted by Re and b is called as imaginary part of (denoted by Im Here i =, also i =,

More information

CO-ORDINATE GEOMETRY. 1. Find the points on the y axis whose distances from the points (6, 7) and (4,-3) are in the. ratio 1:2.

CO-ORDINATE GEOMETRY. 1. Find the points on the y axis whose distances from the points (6, 7) and (4,-3) are in the. ratio 1:2. UNIT- CO-ORDINATE GEOMETRY Mathematics is the tool specially suited for dealing with abstract concepts of any ind and there is no limit to its power in this field.. Find the points on the y axis whose

More information

0112ge. Geometry Regents Exam Line n intersects lines l and m, forming the angles shown in the diagram below.

0112ge. Geometry Regents Exam Line n intersects lines l and m, forming the angles shown in the diagram below. Geometry Regents Exam 011 011ge 1 Line n intersects lines l and m, forming the angles shown in the diagram below. 4 In the diagram below, MATH is a rhombus with diagonals AH and MT. Which value of x would

More information

It is known that the length of the tangents drawn from an external point to a circle is equal.

It is known that the length of the tangents drawn from an external point to a circle is equal. CBSE -MATHS-SET 1-2014 Q1. The first three terms of an AP are 3y-1, 3y+5 and 5y+1, respectively. We need to find the value of y. We know that if a, b and c are in AP, then: b a = c b 2b = a + c 2 (3y+5)

More information

Mathematics. A basic Course for beginers in G.C.E. (Advanced Level) Mathematics

Mathematics. A basic Course for beginers in G.C.E. (Advanced Level) Mathematics Mathematics A basic Course for beginers in G.C.E. (Advanced Level) Mathematics Department of Mathematics Faculty of Science and Technology National Institute of Education Maharagama Sri Lanka 2009 Director

More information

2. (i) Find the equation of the circle which passes through ( 7, 1) and has centre ( 4, 3).

2. (i) Find the equation of the circle which passes through ( 7, 1) and has centre ( 4, 3). Circle 1. (i) Find the equation of the circle with centre ( 7, 3) and of radius 10. (ii) Find the centre of the circle 2x 2 + 2y 2 + 6x + 8y 1 = 0 (iii) What is the radius of the circle 3x 2 + 3y 2 + 5x

More information

BOARD QUESTION PAPER : MARCH 2016 GEOMETRY

BOARD QUESTION PAPER : MARCH 2016 GEOMETRY BOARD QUESTION PAPER : MARCH 016 GEOMETRY Time : Hours Total Marks : 40 Note: (i) Solve All questions. Draw diagram wherever necessary. (ii) Use of calculator is not allowed. (iii) Diagram is essential

More information

1 / 23

1 / 23 CBSE-XII-07 EXAMINATION CBSE-X-009 EXAMINATION MATHEMATICS Series: HRL Paper & Solution Code: 0/ Time: Hrs. Max. Marks: 80 General Instuctions : (i) All questions are compulsory. (ii) The question paper

More information

IMPORTANT QUESTIONS FOR INTERMEDIATE PUBLIC EXAMINATIONS IN MATHS-IB

IMPORTANT QUESTIONS FOR INTERMEDIATE PUBLIC EXAMINATIONS IN MATHS-IB ` KUKATPALLY CENTRE IMPORTANT QUESTIONS FOR INTERMEDIATE PUBLIC EXAMINATIONS IN MATHS-IB 017-18 FIITJEE KUKATPALLY CENTRE: # -97, Plot No1, Opp Patel Kunta Huda Park, Vijaynagar Colony, Hyderabad - 500

More information

Question 1 ( 1.0 marks) places of decimals? Solution: Now, on dividing by 2, we obtain =

Question 1 ( 1.0 marks) places of decimals? Solution: Now, on dividing by 2, we obtain = Question 1 ( 1.0 marks) The decimal expansion of the rational number places of decimals? will terminate after how many The given expression i.e., can be rewritten as Now, on dividing 0.043 by 2, we obtain

More information

(b) the equation of the perpendicular bisector of AB. [3]

(b) the equation of the perpendicular bisector of AB. [3] HORIZON EDUCATION SINGAPORE Additional Mathematics Practice Questions: Coordinate Geometr 1 Set 1 1 In the figure, ABCD is a rhombus with coordinates A(2, 9) and C(8, 1). The diagonals AC and BD cut at

More information

SECTION A(1) k k 1= = or (rejected) k 1. Suggested Solutions Marks Remarks. 1. x + 1 is the longest side of the triangle. 1M + 1A

SECTION A(1) k k 1= = or (rejected) k 1. Suggested Solutions Marks Remarks. 1. x + 1 is the longest side of the triangle. 1M + 1A SECTION A(). x + is the longest side of the triangle. ( x + ) = x + ( x 7) (Pyth. theroem) x x + x + = x 6x + 8 ( x )( x ) + x x + 9 x = (rejected) or x = +. AP and PB are in the golden ratio and AP >

More information

Berkeley Math Circle, May

Berkeley Math Circle, May Berkeley Math Circle, May 1-7 2000 COMPLEX NUMBERS IN GEOMETRY ZVEZDELINA STANKOVA FRENKEL, MILLS COLLEGE 1. Let O be a point in the plane of ABC. Points A 1, B 1, C 1 are the images of A, B, C under symmetry

More information

Downloaded from

Downloaded from Triangles 1.In ABC right angled at C, AD is median. Then AB 2 = AC 2 - AD 2 AD 2 - AC 2 3AC 2-4AD 2 (D) 4AD 2-3AC 2 2.Which of the following statement is true? Any two right triangles are similar

More information

1. Matrices and Determinants

1. Matrices and Determinants Important Questions 1. Matrices and Determinants Ex.1.1 (2) x 3x y Find the values of x, y, z if 2x + z 3y w = 0 7 3 2a Ex 1.1 (3) 2x 3x y If 2x + z 3y w = 3 2 find x, y, z, w 4 7 Ex 1.1 (13) 3 7 3 2 Find

More information

AREAS OF PARALLELOGRAMS AND TRIANGLES

AREAS OF PARALLELOGRAMS AND TRIANGLES AREAS OF PARALLELOGRAMS AND TRIANGLES Main Concepts and Results: The area of a closed plane figure is the measure of the region inside the figure: Fig.1 The shaded parts (Fig.1) represent the regions whose

More information

Unit 8. ANALYTIC GEOMETRY.

Unit 8. ANALYTIC GEOMETRY. Unit 8. ANALYTIC GEOMETRY. 1. VECTORS IN THE PLANE A vector is a line segment running from point A (tail) to point B (head). 1.1 DIRECTION OF A VECTOR The direction of a vector is the direction of the

More information

1 / 23

1 / 23 CBSE-XII-017 EXAMINATION CBSE-X-008 EXAMINATION MATHEMATICS Series: RLH/ Paper & Solution Code: 30//1 Time: 3 Hrs. Max. Marks: 80 General Instuctions : (i) All questions are compulsory. (ii) The question

More information

RMT 2013 Geometry Test Solutions February 2, = 51.

RMT 2013 Geometry Test Solutions February 2, = 51. RMT 0 Geometry Test Solutions February, 0. Answer: 5 Solution: Let m A = x and m B = y. Note that we have two pairs of isosceles triangles, so m A = m ACD and m B = m BCD. Since m ACD + m BCD = m ACB,

More information

Properties of the Circle

Properties of the Circle 9 Properties of the Circle TERMINOLOGY Arc: Part of a curve, most commonly a portion of the distance around the circumference of a circle Chord: A straight line joining two points on the circumference

More information

Udaan School Of Mathematics Class X Chapter 10 Circles Maths

Udaan School Of Mathematics Class X Chapter 10 Circles Maths Exercise 10.1 1. Fill in the blanks (i) The common point of tangent and the circle is called point of contact. (ii) A circle may have two parallel tangents. (iii) A tangent to a circle intersects it in

More information

Maharashtra Board Class X Mathematics - Geometry Board Paper 2014 Solution. Time: 2 hours Total Marks: 40

Maharashtra Board Class X Mathematics - Geometry Board Paper 2014 Solution. Time: 2 hours Total Marks: 40 Maharashtra Board Class X Mathematics - Geometry Board Paper 04 Solution Time: hours Total Marks: 40 Note: - () All questions are compulsory. () Use of calculator is not allowed.. i. Ratio of the areas

More information

Maharashtra State Board Class X Mathematics - Geometry Board Paper 2016 Solution

Maharashtra State Board Class X Mathematics - Geometry Board Paper 2016 Solution Maharashtra State Board Class X Mathematics - Geometry Board Paper 016 Solution 1. i. ΔDEF ΔMNK (given) A( DEF) DE A( MNK) MN A( DEF) 5 5 A( MNK) 6 6...(Areas of similar triangles) ii. ΔABC is 0-60 -90

More information

SSC CGL Tier 1 and Tier 2 Program

SSC CGL Tier 1 and Tier 2 Program Gurudwara Road Model Town, Hisar 9729327755 www.ssccglpinnacle.com SSC CGL Tier 1 and Tier 2 Program -------------------------------------------------------------------------------------------------------------------

More information

INVERSION IN THE PLANE BERKELEY MATH CIRCLE

INVERSION IN THE PLANE BERKELEY MATH CIRCLE INVERSION IN THE PLANE BERKELEY MATH CIRCLE ZVEZDELINA STANKOVA MILLS COLLEGE/UC BERKELEY SEPTEMBER 26TH 2004 Contents 1. Definition of Inversion in the Plane 1 Properties of Inversion 2 Problems 2 2.

More information

2007 Fermat Contest (Grade 11)

2007 Fermat Contest (Grade 11) Canadian Mathematics Competition An activity of the Centre for Education in Mathematics and Computing, University of Waterloo, Waterloo, Ontario 007 Fermat Contest (Grade 11) Tuesday, February 0, 007 Solutions

More information

IMPORTANT QUESTIONS FOR INTERMEDIATE PUBLIC EXAMINATIONS

IMPORTANT QUESTIONS FOR INTERMEDIATE PUBLIC EXAMINATIONS ` KUKATPALLY CENTRE IMPORTANT QUESTIONS FOR INTERMEDIATE PUBLIC EXAMINATIONS IN MATHS-IB FIITJEE KUKATPALLY CENTRE: # -97, Plot No1, Opp Patel Kunta Huda Park, Vijaynagar Colony, Hyderabad - 57 Ph: 4-646113

More information

MATHEMATICS. IMPORTANT FORMULAE AND CONCEPTS for. Summative Assessment -II. Revision CLASS X Prepared by

MATHEMATICS. IMPORTANT FORMULAE AND CONCEPTS for. Summative Assessment -II. Revision CLASS X Prepared by MATHEMATICS IMPORTANT FORMULAE AND CONCEPTS for Summative Assessment -II Revision CLASS X 06 7 Prepared by M. S. KUMARSWAMY, TGT(MATHS) M. Sc. Gold Medallist (Elect.), B. Ed. Kendriya Vidyalaya GaCHiBOWli

More information

Plane geometry Circles: Problems with some Solutions

Plane geometry Circles: Problems with some Solutions The University of Western ustralia SHL F MTHMTIS & STTISTIS UW MY FR YUNG MTHMTIINS Plane geometry ircles: Problems with some Solutions 1. Prove that for any triangle, the perpendicular bisectors of the

More information

Calgary Math Circles: Triangles, Concurrency and Quadrilaterals 1

Calgary Math Circles: Triangles, Concurrency and Quadrilaterals 1 Calgary Math Circles: Triangles, Concurrency and Quadrilaterals 1 1 Triangles: Basics This section will cover all the basic properties you need to know about triangles and the important points of a triangle.

More information

0811ge. Geometry Regents Exam

0811ge. Geometry Regents Exam 0811ge 1 The statement "x is a multiple of 3, and x is an even integer" is true when x is equal to 1) 9 ) 8 3) 3 4) 6 In the diagram below, ABC XYZ. 4 Pentagon PQRST has PQ parallel to TS. After a translation

More information

2013 Sharygin Geometry Olympiad

2013 Sharygin Geometry Olympiad Sharygin Geometry Olympiad 2013 First Round 1 Let ABC be an isosceles triangle with AB = BC. Point E lies on the side AB, and ED is the perpendicular from E to BC. It is known that AE = DE. Find DAC. 2

More information

CLASS IX : CHAPTER - 1 NUMBER SYSTEM

CLASS IX : CHAPTER - 1 NUMBER SYSTEM MCQ WORKSHEET-I CLASS IX : CHAPTER - 1 NUMBER SYSTEM 1. Rational number 3 40 is equal to: (a) 0.75 (b) 0.1 (c) 0.01 (d) 0.075. A rational number between 3 and 4 is: (a) 3 (b) 4 3 (c) 7 (d) 7 4 3. A rational

More information

Sample Question Paper Mathematics First Term (SA - I) Class IX. Time: 3 to 3 ½ hours

Sample Question Paper Mathematics First Term (SA - I) Class IX. Time: 3 to 3 ½ hours Sample Question Paper Mathematics First Term (SA - I) Class IX Time: 3 to 3 ½ hours M.M.:90 General Instructions (i) All questions are compulsory. (ii) The question paper consists of 34 questions divided

More information

0114ge. Geometry Regents Exam 0114

0114ge. Geometry Regents Exam 0114 0114ge 1 The midpoint of AB is M(4, 2). If the coordinates of A are (6, 4), what are the coordinates of B? 1) (1, 3) 2) (2, 8) 3) (5, 1) 4) (14, 0) 2 Which diagram shows the construction of a 45 angle?

More information

Nozha Directorate of Education Form : 2 nd Prep. Nozha Language Schools Ismailia Road Branch

Nozha Directorate of Education Form : 2 nd Prep. Nozha Language Schools Ismailia Road Branch Cairo Governorate Department : Maths Nozha Directorate of Education Form : 2 nd Prep. Nozha Language Schools Sheet Ismailia Road Branch Sheet ( 1) 1-Complete 1. in the parallelogram, each two opposite

More information

Geometry 3 SIMILARITY & CONGRUENCY Congruency: When two figures have same shape and size, then they are said to be congruent figure. The phenomena between these two figures is said to be congruency. CONDITIONS

More information

0610ge. Geometry Regents Exam The diagram below shows a right pentagonal prism.

0610ge. Geometry Regents Exam The diagram below shows a right pentagonal prism. 0610ge 1 In the diagram below of circle O, chord AB chord CD, and chord CD chord EF. 3 The diagram below shows a right pentagonal prism. Which statement must be true? 1) CE DF 2) AC DF 3) AC CE 4) EF CD

More information

Mathematics Revision Guides Vectors Page 1 of 19 Author: Mark Kudlowski M.K. HOME TUITION. Mathematics Revision Guides Level: GCSE Higher Tier VECTORS

Mathematics Revision Guides Vectors Page 1 of 19 Author: Mark Kudlowski M.K. HOME TUITION. Mathematics Revision Guides Level: GCSE Higher Tier VECTORS Mathematics Revision Guides Vectors Page of 9 M.K. HOME TUITION Mathematics Revision Guides Level: GCSE Higher Tier VECTORS Version:.4 Date: 05-0-05 Mathematics Revision Guides Vectors Page of 9 VECTORS

More information

Fill in the blanks Chapter 10 Circles Exercise 10.1 Question 1: (i) The centre of a circle lies in of the circle. (exterior/ interior) (ii) A point, whose distance from the centre of a circle is greater

More information

Pre RMO Exam Paper Solution:

Pre RMO Exam Paper Solution: Paper Solution:. How many positive integers less than 000 have the property that the sum of the digits of each such number is divisible by 7 and the number itself is divisible by 3? Sum of Digits Drivable

More information

10. Circles. Q 5 O is the centre of a circle of radius 5 cm. OP AB and OQ CD, AB CD, AB = 6 cm and CD = 8 cm. Determine PQ. Marks (2) Marks (2)

10. Circles. Q 5 O is the centre of a circle of radius 5 cm. OP AB and OQ CD, AB CD, AB = 6 cm and CD = 8 cm. Determine PQ. Marks (2) Marks (2) 10. Circles Q 1 True or False: It is possible to draw two circles passing through three given non-collinear points. Mark (1) Q 2 State the following statement as true or false. Give reasons also.the perpendicular

More information

Secondary School Certificate Examination Syllabus MATHEMATICS. Class X examination in 2011 and onwards. SSC Part-II (Class X)

Secondary School Certificate Examination Syllabus MATHEMATICS. Class X examination in 2011 and onwards. SSC Part-II (Class X) Secondary School Certificate Examination Syllabus MATHEMATICS Class X examination in 2011 and onwards SSC Part-II (Class X) 15. Algebraic Manipulation: 15.1.1 Find highest common factor (H.C.F) and least

More information

Practice Test Geometry 1. Which of the following points is the greatest distance from the y-axis? A. (1,10) B. (2,7) C. (3,5) D. (4,3) E.

Practice Test Geometry 1. Which of the following points is the greatest distance from the y-axis? A. (1,10) B. (2,7) C. (3,5) D. (4,3) E. April 9, 01 Standards: MM1Ga, MM1G1b Practice Test Geometry 1. Which of the following points is the greatest distance from the y-axis? (1,10) B. (,7) C. (,) (,) (,1). Points P, Q, R, and S lie on a line

More information

y mx 25m 25 4 circle. Then the perpendicular distance of tangent from the centre (0, 0) is the radius. Since tangent

y mx 25m 25 4 circle. Then the perpendicular distance of tangent from the centre (0, 0) is the radius. Since tangent Mathematics. The sides AB, BC and CA of ABC have, 4 and 5 interior points respectively on them as shown in the figure. The number of triangles that can be formed using these interior points is () 80 ()

More information

0811ge. Geometry Regents Exam BC, AT = 5, TB = 7, and AV = 10.

0811ge. Geometry Regents Exam BC, AT = 5, TB = 7, and AV = 10. 0811ge 1 The statement "x is a multiple of 3, and x is an even integer" is true when x is equal to 1) 9 2) 8 3) 3 4) 6 2 In the diagram below, ABC XYZ. 4 Pentagon PQRST has PQ parallel to TS. After a translation

More information

Created by T. Madas 2D VECTORS. Created by T. Madas

Created by T. Madas 2D VECTORS. Created by T. Madas 2D VECTORS Question 1 (**) Relative to a fixed origin O, the point A has coordinates ( 2, 3). The point B is such so that AB = 3i 7j, where i and j are mutually perpendicular unit vectors lying on the

More information

1 / 24

1 / 24 CBSE-XII-017 EXAMINATION CBSE-X-01 EXAMINATION MATHEMATICS Paper & Solution Time: 3 Hrs. Max. Marks: 90 General Instuctions : 1. All questions are compulsory.. The question paper consists of 34 questions

More information

ieducation.com Tangents given as follows. the circle. contact. There are c) Secant:

ieducation.com  Tangents given as follows. the circle. contact. There are c) Secant: h Tangents and Secants to the Circle A Line and a circle: let us consider a circle and line say AB. There can be three possibilities given as follows. a) Non-intersecting line: The line AB and the circle

More information

1 Line n intersects lines l and m, forming the angles shown in the diagram below. 4 In the diagram below, MATH is a rhombus with diagonals AH and MT.

1 Line n intersects lines l and m, forming the angles shown in the diagram below. 4 In the diagram below, MATH is a rhombus with diagonals AH and MT. 1 Line n intersects lines l and m, forming the angles shown in the diagram below. 4 In the diagram below, MATH is a rhombus with diagonals AH and MT. Which value of x would prove l m? 1) 2.5 2) 4.5 3)

More information

FILL THE ANSWER HERE

FILL THE ANSWER HERE HOM ASSIGNMNT # 0 STRAIGHT OBJCTIV TYP. If A, B & C are matrices of order such that A =, B = 9, C =, then (AC) is equal to - (A) 8 6. The length of the sub-tangent to the curve y = (A) 8 0 0 8 ( ) 5 5

More information

1 What is the solution of the system of equations graphed below? y = 2x + 1

1 What is the solution of the system of equations graphed below? y = 2x + 1 1 What is the solution of the system of equations graphed below? y = 2x + 1 3 As shown in the diagram below, when hexagon ABCDEF is reflected over line m, the image is hexagon A'B'C'D'E'F'. y = x 2 + 2x

More information

CBSE CLASS X MATH -SOLUTION Therefore, 0.6, 0.25 and 0.3 are greater than or equal to 0 and less than or equal to 1.

CBSE CLASS X MATH -SOLUTION Therefore, 0.6, 0.25 and 0.3 are greater than or equal to 0 and less than or equal to 1. CBSE CLASS X MATH -SOLUTION 011 Q1 The probability of an event is always greater than or equal to zero and less than or equal to one. Here, 3 5 = 0.6 5% = 5 100 = 0.5 Therefore, 0.6, 0.5 and 0.3 are greater

More information

Mathematics 2260H Geometry I: Euclidean geometry Trent University, Fall 2016 Solutions to the Quizzes

Mathematics 2260H Geometry I: Euclidean geometry Trent University, Fall 2016 Solutions to the Quizzes Mathematics 2260H Geometry I: Euclidean geometry Trent University, Fall 2016 Solutions to the Quizzes Quiz #1. Wednesday, 13 September. [10 minutes] 1. Suppose you are given a line (segment) AB. Using

More information

HANOI OPEN MATHEMATICAL COMPETITON PROBLEMS

HANOI OPEN MATHEMATICAL COMPETITON PROBLEMS HANOI MATHEMATICAL SOCIETY NGUYEN VAN MAU HANOI OPEN MATHEMATICAL COMPETITON PROBLEMS HANOI - 2013 Contents 1 Hanoi Open Mathematical Competition 3 1.1 Hanoi Open Mathematical Competition 2006... 3 1.1.1

More information

MAT1035 Analytic Geometry

MAT1035 Analytic Geometry MAT1035 Analytic Geometry Lecture Notes R.A. Sabri Kaan Gürbüzer Dokuz Eylül University 2016 2 Contents 1 Review of Trigonometry 5 2 Polar Coordinates 7 3 Vectors in R n 9 3.1 Located Vectors..............................................

More information

NATIONAL SENIOR CERTIFICATE GRADE 11

NATIONAL SENIOR CERTIFICATE GRADE 11 NATIONAL SENIOR CERTIFICATE GRADE MATHEMATICS P NOVEMBER 05 MARKS: 50 TIME: 3 hours This question paper consists of 5 pages and a 4-page answer book. Mathematics/P DBE/November 05 CAPS Grade INSTRUCTIONS

More information

Name: GEOMETRY: EXAM (A) A B C D E F G H D E. 1. How many non collinear points determine a plane?

Name: GEOMETRY: EXAM (A) A B C D E F G H D E. 1. How many non collinear points determine a plane? GMTRY: XM () Name: 1. How many non collinear points determine a plane? ) none ) one ) two ) three 2. How many edges does a heagonal prism have? ) 6 ) 12 ) 18 ) 2. Name the intersection of planes Q and

More information

SURA's Guides for 3rd to 12th Std for all Subjects in TM & EM Available. MARCH Public Exam Question Paper with Answers MATHEMATICS

SURA's Guides for 3rd to 12th Std for all Subjects in TM & EM Available. MARCH Public Exam Question Paper with Answers MATHEMATICS SURA's Guides for rd to 1th Std for all Subjects in TM & EM Available 10 th STD. MARCH - 017 Public Exam Question Paper with Answers MATHEMATICS [Time Allowed : ½ Hrs.] [Maximum Marks : 100] SECTION -

More information

WBJEE Answer Keys by Aakash Institute, Kolkata Centre

WBJEE Answer Keys by Aakash Institute, Kolkata Centre WBJEE - 08 Answer Keys by, Kolkata Centre MATHEMATICS Q.No. 0 A B C D 0 C D A B 0 B D A C 04 C B A B 05 C A C C 06 A C D C 07 B A C C 08 B *C,D C A 09 C D D B 0 D A C D B A B C C D A B B A A C 4 C C B

More information

[STRAIGHT OBJECTIVE TYPE] Q.4 The expression cot 9 + cot 27 + cot 63 + cot 81 is equal to (A) 16 (B) 64 (C) 80 (D) none of these

[STRAIGHT OBJECTIVE TYPE] Q.4 The expression cot 9 + cot 27 + cot 63 + cot 81 is equal to (A) 16 (B) 64 (C) 80 (D) none of these Q. Given a + a + cosec [STRAIGHT OBJECTIVE TYPE] F HG ( a x) I K J = 0 then, which of the following holds good? (A) a = ; x I a = ; x I a R ; x a, x are finite but not possible to find Q. The minimum value

More information