Q) If the sum of first m terms of an A.P. is same as sum of its first n terms (m ≠ n), then show that the sum of its first (m + n) terms is zero. Ans: Step 1: We know that the sum of first m terms of an AP is given by: Sm […]
July 2024
Q) In an A.P., the sum of three consecutive terms is 24 and the sum of their squares is 194. Find the numbers. Ans: Step 1: Let’s consider that the middle term is A, first term is A – D and third term is A + D, here D is the common difference. By given 1st
Q) In the given figure, PQ is tangent to a circle centred at O and ∠BAQ = 30°; show that BP = BQ. Ans: Here, We need to prove that BP = BQ, it means we need to get ∠ BQP = ∠ BPQ Step 1: Let’s start with given diagram. Since AB is a straight
Q) In the given figure, AB, BC, CD and DA are tangents to the circle with centre O forming a quadrilateral ABCD. Show that angle AOB+ angle COD = 1800 Ans: Let’s draw a diagram and connect O with all vertices of Quadrilateral ABCD and al touch points on its circumference: Let’s start with Δ
Q) Prove that : Ans: Let’s start from simplifying the LHS: LHS = = = Since, we need to get demnominator in simplified form, hence let’s multiply nominator and denominator by cos θ + sin θ – 1, we get: LHS = = = = = = We know that sin2 θ + cos2
Prove that : 1 + sec θ – tan θ / 1 + sec θ + tan θ = 1 – sin θ / cos θ Read More »
Q) The perimeter of a certain sector of a circle of radius 5.6 m is 20.0 m. Find the area of the sector. Ans: We know that the perimeter of a circle’s sector making θ angle, is given by: P = 2 R + 2 π R ∴ π R = ……..(i) And the area
Q) If √2 is given as an irrational number, then prove that (5 – 2√2) is an irrational number. Ans: STEP BY STEP SOLUTION Let’s start by considering 5 – 2 √2 is a rational number. ∴ 5 – 2 √2 = (here p and q are integers and q ≠ 0) ∴ –
Q) Check whether 6n can end with the digit 0 for any natural number n. Ans: For any number to end with the digit 0, it will be divisible by 10 Since 10 = 2 x 5, therefore, a number ending with digit 0 will be divisible by 2 and 5 both. Next, since factors of
Check whether 6n can end with the digit 0 for any natural number n . Read More »
Q) In the figure, E is a point on side CB produced of an isosceles triangle ABC with AB = AC. If AD ⊥ BC and EF ⊥ AC, prove that Δ ABD ~ Δ ECF. Ans: Since Δ ABC is an isosceles triangle, hence AB = AC ∴ ∠ B = ∠ C ………..
In the figure, E is a point on side CB produced of an isosceles triangle ABC Read More »
Q) Prove that A(4, 3), B(6, 4), C(5, 6), D(3, 5) are the vertices of a square ABCD. Ans: Let’s plot the points on the graph: Step 1: Now for a quadrilateral ABCD to be a square, required conditions are: i) its all four sides should be equal i.e. AB = BC = CD = AD
Prove that A(4, 3), B(6, 4), C(5, 6), D(3, 5) are the vertices of a square ABCD. Read More »