OCR A-Level Maths (PURE) EXAM TEST BANK WITH ALL VERSIONS OF THE EXAM WITH ALLMODULES COVERED | ACCURATE AND VERIFIED QUESTIONS AND ANSWERS FOR GUARANTEED PASS| LATEST UPDATE A tangent to a circle is ______ to the radius at the point of contact. A. P

A tangent to a circle is ______ to the radius at the point of contact. A. Parallel B. Equal C. Perpendicular D. Tangential Correct answer: C Rationale: A tangent meets the circle at exactly one point and is perpendicular to the radius at that point. The perpendicular bisector of a chord will go through the… A. Midpoint of the arc B. Circumference C. Centre of the circle D. Tangent Correct answer: C Rationale: The perpendicular bisector of any chord passes through the centre. The angle in a semicircle is always… A. Acute B. Obtuse C. A right angle D. Straight Rationale: By circle theorems, the angle subtended by a diameter is 90°. If PRQ = 90°, then R lies… A. Inside the circle B. Outside the circle C. On the circle with diameter PQ D. At the midpoint of PQ Correct answer: C Rationale: A right angle subtended by PQ means PQ is the diameter. To find the centre of a circle given any three points: A. Join the three points B. Find midpoint of one side C. Find perpendicular bisectors of two chords and their intersection D. Average coordinates Correct answer: C Rationale: The centre is at the intersection of perpendicular bisectors. If f(p)=0 then (x−p) is a factor of f(x). This is called: A. Remainder Theorem B. Factor Theorem C. Binomial Theorem D. Product Rule Correct answer: B Rationale: A root makes the polynomial zero, so (x−p) is a factor. Starting from known facts and using logical steps to reach a conclusion is: A. Proof by exhaustion B. Proof by contradiction C. Proof by deduction D. Counter-example Rationale: Deduction builds logically from accepted truths. Breaking a statement into smaller cases and proving each separately is: A. Deduction B. Exhaustion C. Contradiction D. Induction Correct answer: B Rationale: All possible cases are checked. An example that disproves a statement is: A. Deduction B. Example proof C. Counter-example D. Assumption Correct answer: C Rationale: One counter-example disproves a universal claim. Which row of Pascal’s Triangle gives coefficients of (a+b)ⁿ? A. nth row B. (n−1)th row C. (n+1)th row D. 2n row Correct answer: C Rationale: Counting begins at row 1 for n=0. n! equals: A. n² B. n + (n−1) C. n × (n−1) × … × 1 D. n(n−1) Rationale: Factorial multiplies all positive integers down to 1. nCr equals: A. n!/r B. n!/(r!(n−r)!) C. r!/n! D. n−r Correct answer: B Rationale: Standard combination formula. The binomial expansion of (a+b)ⁿ is: A. aⁿ + bⁿ B. aⁿ + nC₁aⁿ⁻¹b + … + bⁿ C. (a+b)n D. na + nb Correct answer: B Rationale: Uses binomial coefficients nCr. If x is small, the first few terms of a binomial expansion are used to: A. Find exact values B. Factorise expressions C. Approximate complicated expressions D. Solve equations Correct answer: C Rationale: Higher powers become negligible. a² = b² + c² − 2bc cosA is the: A. Sine Rule B. Cosine Rule C. Pythagoras D. Double angle formula B Rationale: Generalised Pythagoras theorem. a/sinA = b/sinB = c/sinC is the: A. Cosine Rule B. Sine Rule C. Tangent Rule D. Ratio Rule Correct answer: B Rationale: Relates sides to opposite angles. sinθ = sin(180−θ) explains the: A. Double angle B. Ambiguous case of sine rule C. Cosine rule D. Complement rule Correct answer: B Rationale: Two angles share same sine value. In the third quadrant (CAST diagram): A. All positive B. Only sin positive C. Only tan positive D. Only cos positive Correct answer: C Rationale: Tangent is positive in QIII. sin²θ + cos²θ ≡ ? A. 0 B. 1 C. tanθ D. −1 B Rationale: Fundamental trig identity. sinθ / cosθ = ? A. cotθ B. secθ C. tanθ D. 1 Correct answer: C Rationale: Definition of tanθ. λa represents: A. Unit vector B. Any vector parallel to a C. Perpendicular vector D. Zero vector Correct answer: B Rationale: Scalar multiple gives parallel vector. The gradient of the tangent equals: A. Average gradient B. Gradient at a given point C. Zero D. Constant Correct answer: B Rationale: Derivative gives instantaneous rate of change. If f(x)=axⁿ, then f′(x)= A. anxⁿ⁻¹ B. nx C. ax D. n A Rationale: Power rule. If f″(a)0 at a stationary point, it is a: A. Maximum B. Minimum C. Inflection D. Root Correct answer: B Rationale: Positive second derivative means concave up. If y = e^(kx), then dy/dx = A. e^(kx) B. k e^(kx) C. kx D. x e^(kx) Correct answer: B Rationale: Chain rule. loga(x) + loga(y) = A. loga(x+y) B. loga(xy) C. loga(x−y) D. xy Correct answer: B Rationale: Product rule of logs.