CBSE MATHEMATICS • SELECTED CENTRAL & STATE COMPETITIVE EXAMS
AIMSSchool of MathematicsAim through AIMS
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GANITA MANJARI · GRADE 9 · PART II · CHAPTER 9

APS–5M

Think first. Use AI purposefully. Verify always.

Original AIMS written questions. Attempt on paper before comparing reasoning. No automatic marking.

5-mark reasoning

Prove the converse theorem using a constructed right triangle and SSS.

Given triangle ABC: BC equals a, CA equals b, AB equals c. Angle C is not yet proved right. Schematic, not to scale.CBAabcGiven · angle C to be provedConstructed triangle XYZ with right angle Z, YZ equals a, ZX equals b. XY is proved equal to c.ZYXabXY = cConstructed · angle Z = 90°
Original AIMS schematic, not to scale. Matching sides by SSS pairs A with X, B with Y and C with Z.

5-mark reasoning

Prove both directions connecting a positive square with an odd positive factor count.

5-mark reasoning

A designer uses equal-length sticks as diagonals. Only type T → equal diagonals is known. Explain what can and cannot be concluded, then consider the reverse rule.

A quadrilateral whose diagonals join opposite vertices A to C and B to D. No equality or right angle is assumed.ABCD
Original AIMS diagram inspired by the diagonal construction in Figure 9.2. The drawing makes no equality or perpendicularity claim.

5-mark reasoning

Investigate positive integers with exactly four factors. Test the claim that they must be products of two distinct primes.

5-mark reasoning

For real x,y, compare x = y → x2 = y2 and x = y → x3 = y3, including their converses.