All questions
- Show that the line joining the points A(4, 8) and B(5, 5) is parallel to the line joining the points C(2, 4) and D(1, 7). Answer: Both lines have slope -3, so line AB ∥ line CD. Slope of AB =5-8/5-4=-3 . Slope of CD =7-4/1-2=-3 . Lines with equal slopes are para... 1
- Show that points P(1, -2), Q(5, 2), R(3, -1), S(-1, -5) are the vertices of a parallelogram. Answer: PQ ∥ RS and QR ∥ SP, so □PQRS is a parallelogram. Slope =y₂-y₁/x₂-x₁ . Slope of PQ =2+2/5-1=1 ; slope of RS =-5+1/-1-3=1 . Slope of... 1
- Show that the □PQRS formed by P(2, 1), Q(-1, 3), R(-5, -3) and S(-2, -5) is a rectangle. Answer: Opposite sides are equal and the diagonals are equal, so □PQRS is a rectangle. Distance formula: d=√((x₂-x₁)²+(y₂-y₁)²) . PQ=√(9+4)=... 1
- Find the lengths of the medians of a triangle whose vertices are A(-1, 1), B(5, -3) and C(3, 5). Answer: AD = 5, BE = 2√13, CF = √37 Midpoints: D of BC =(4,1) , E of CA =(1,3) , F of AB =(2,-1) . AD=√((4+1)²+(1-1)²)=5 BE=√((1-5)²+(3+3)²)... 1
- Find the coordinates of centroid of the triangles if points D(-7, 6), E(8, 5) and F(2, -2) are the mid points of the sides of that triangle. Answer: (1, 3) Let the vertices be A(x₁,y₁), B(x₂,y₂), C(x₃,y₃) with D, E, F the midpoints of the sides. Adding the three midpoint equations... 1
- Show that A(4, -1), B(6, 0), C(7, -2) and D(5, -3) are vertices of a square. Answer: All sides equal √5 and both diagonals equal √10, so ABCD is a square. Distance formula: d=√((x₂-x₁)²+(y₂-y₁)²) . AB=√(4+1)=sqrt5 , B... 1
- Find the coordinates of circumcentre and radius of circumcircle of ΔABC if A(7, 1), B(3, 5) and C(2, 0) are given. Answer: Circumcentre (25/6, 13/6), radius 13sqrt2/6 Let P(x, y) be the circumcentre: PA = PB = PC. PA²=PB² : (x-7)²+(y-1)²=(x-3)²+(y-5)² -14... 1
- Given A(4, -3), B(8, 5). Find the coordinates of the point that divides segment AB in the ratio 3 : 1. Answer: (7, 3) Section formula: if P divides AB in the ratio m:n , then P=(mx₂+nx₁/m+n,\ my₂+ny₁/m+n) . x=3(8)+1(4)/4=7, y=3(5)+1(-3)/4=3 1
- Find the type of the quadrilateral if points A(-4, -2), B(-3, -7), C(3, -2) and D(2, 3) are joined serially. Answer: □ABCD is a parallelogram. Distance formula: d=√((x₂-x₁)²+(y₂-y₁)²) . AB=√(1+25)=√(26) , BC=√(36+25)=√(61) , CD=√(1+25)=√(26) , DA=√(... 1
- The line segment AB is divided into five congruent parts at P, Q, R and S such that A-P-Q-R-S-B. If point Q(12, 14) and S(4, 18) are given find the coordinates of A, P, R, B. Answer: A(20, 10), P(16, 12), R(8, 16), B(0, 20) R is the midpoint of QS: (12+4/2,14+18/2)=(8,16) . Q is the midpoint of PR, so P=(2·12-8,\... 1
- Find the coordinates of the centre of the circle passing through the points P(6, -6), Q(3, -7) and R(3, 3). Answer: (3, -2) Let the centre be C(x, y); CP = CQ = CR. CQ²=CR² : (x-3)²+(y+7)²=(x-3)²+(y-3)² gives 14y+49=-6y+9 , so y=-2 . CP²=CQ² : (x-6... 1
- Find the possible pairs of coordinates of the fourth vertex D of the parallelogram, if three of its vertices are A(5, 6), B(1, -2) and C(3, -2). Answer: D(7, 6) if the parallelogram is ABCD; the other possible positions are D(3, 6) and D(-1, -10). Diagonals of a parallelogram bisect e... 1
- Find the slope of the diagonals of a quadrilateral with vertices A(1, 7), B(6, 3), C(0, -3) and D(-3, 3). Answer: Slope of AC = 10, slope of BD = 0 Slope =y₂-y₁/x₂-x₁ . Diagonal AC: -3-7/0-1=-10/-1=10 . Diagonal BD: 3-3/-3-6=0/-9=0 (BD is paralle... 1
- sinθ cosecθ = ? (A) 1 (B) 0 (C) 1/2 (D) √2 Answer: (A) 1 cscθ=1/sinθ , so sinθ·cscθ=sinθ·dfrac1sinθ=1 . 1
- cosec45° = ? (A) 1/√2 (B) √2 (C) √3/2 (D) 2/√3 Answer: (B) √2 sin45^°=dfrac1sqrt2 , so csc45^°=dfrac1sin45^°=sqrt2 . Option (A) is sin45^° itself and (D) is csc60^° . 1
- 1 + tan²θ = ? (A) cot²θ (B) cosec²θ (C) sec²θ (D) tan²θ Answer: (C) sec²θ Dividing sin²θ+cos²θ=1 by cos²θ gives tan²θ+1=sec²θ . (The identity with csc²θ is 1+cot²θ=csc²θ .) 1
- When we see at a higher level, from the horizontal line, angle formed is ........ (A) angle of elevation. (B) angle of depression. (C) 0 (D) straight angle. Answer: (A) angle of elevation. When the object is above the observer's horizontal line of sight, the line of vision is raised and the angle... 1
- If sinθ = 11/61, find the values of cosθ using trigonometric identity. Answer: cosθ = 60/61 sin²θ+cos²θ=1 cos²θ=1-(11/61)²=3721-121/3721=3600/3721 cosθ=60/61 (taking θ as an acute angle, so cosθ is positive). 1
- Find the distance between the points W(-7/2, 4), X(11, 4). Answer: WX = 29/2 Distance formula: d=√((x₂-x₁)²+(y₂-y₁)²) . Both points have y-coordinate 4, so WX is parallel to the X-axis. WX=√((11+7/2)... 1
- Determine whether the points are collinear.A(1, –3), B(2, –5), C(–4, 7) Answer: Yes, the points A, B and C are collinear. Slope of AB = -5 - (-3)/2 - 1 = -2/1 = -2 . Slope of BC = 7 - (-5)/-4 - 2 = 12/-6 = -2 . T... 1
- Determine whether the points are collinear: L(-2, 3), M(1, -3), N(5, 4). Answer: L, M, N are not collinear. Three points are collinear if the sum of two of the distances equals the third. LM=√(3²+(-6)²)=√(45)=3sqr... 1
- Determine whether the points are collinear: R(0, 3), D(2, 1), S(3, -1). Answer: R, D, S are not collinear. Three points are collinear if the sum of two of the distances equals the third. RD=√(2²+(-2)²)=sqrt8=2sqr... 1
- Determine whether the points are collinear: P(-2, 3), Q(1, 2), R(4, 1). Answer: P, Q, R are collinear. Three points are collinear if the sum of two of the distances equals the third. PQ=√(3²+(-1)²)=√(10) QR=√(3²+... 1
- Find the point on the X-axis which is equidistant from A(-3, 4) and B(1, -4). Answer: (-1, 0) A point on the X-axis is P(x, 0). PA = PB, so PA²=PB² : (x+3)²+(0-4)²=(x-1)²+(0+4)² x²+6x+9+16=x²-2x+1+16 8x=-8\ ⇒\ x=-1 The... 1