# November 2023

Q) If sin θ = , prove that . Ans:  Let’s start LHS: =  =  ∵ 1- cos2 θ= sin2 θ ∴ LHS =  =  =  =  = cot θ ∵ sin θ =     ∴ cot θ = ∴ LHS = = RHS Hence Proved !

Q) In the given figure, tangents PQ and PR are drawn to a circle such that ∠RPQ = 300. A chord RS is drawn parallel to tangent PQ. Find the ∠RQS. Ans: In △PRQ, PQ and PR are tangents from an external point P to circle. ∴ PR = PQ Since the angles opposites to equal sides

Q) Mrs. Gupta arranged some snacks for her child’s birthday party. After the guest left she had some food left over. She did not want to waste food and so she contacted a local NGO. She gave 60 pieces of pastries,168 pieces of cookies, and 330 chocolate bars to the team. Now the NGO workers want

Q) Find the area of the shaded region in Figure, where arcs drawn with centres A, B, C and D intersect in pairs at mid-points P, Q, R and S of the sides AB, BC, CD and DA respectively of a square ABCD. (Use π = 22/7) Ans:  We are given that an arc is drawn

Q) If a, b and c are the sides of a right angled triangle, where c is hypotenuse, then prove that the radius of the circle whichtouches the sides of the triangle is given by r = Ans: Let’s consider a right angled triangle ABC with sides a, b & c.Its ∠ A is right angle

Q) A number consists of two digits. Where the number is divided by the sum of its digits, the quotient is 7. If 27 is subtracted from the number, the digits interchange their places, find the number Ans: Let’s consider X and Y are the digits of the given number. Hence the given number is 10 X

Q) Solve for x and y: = -1; = 3 Ans: Let’s take the equations one by one: = -1 3X + 4 Y = – 6 ………. (i) Similarly, = 3 We can write this as: = 3 3X – Y = 9 ….. (ii) By solving equations (i) and (ii), we get: X =

Q)  If 𝛼, β are zeroes of quadratic polynomial x2 – 2x + 3, find the polynomial whose roots are:1. 𝛼 + 2, 𝛽 + 22. Ans: Given polynomial equation x2 – 2x + 3 = 0 Comparing with standard polynomial, ax2 + b x + c = 0, we get, a =  1, b =

Q)  In figure, PQ is a tangent from an external point P to a circle with centre O and OP cuts the circle at T and QOR is adiameter. If ∠POR = 130° and S is a point on the circle, find ∠1 +∠2 Ans: In the given diagram, it is given that: ∠ POR

Q)  In Fig. AD bisects ∠A, AB = 12 cm, AC = 20 cm and BD = 5 cm, determine CD. Ans: ∵  AD bisects ∠ A We know that, according to the angle bisector theorem, the angle bisector of a triangle divides the opposite side into two parts that are proportional to the other

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