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Original exam-style questions, mapped to your syllabus. Answers are checked for mathematical equivalence and every attempt updates your mastery.

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2730 questions · page 114 of 137 · choose a topic above to narrow them down

SA-FP-IC-0032 marksL2 StandardCalculatorIntegrating
Evaluate ∫01dx1+x2\displaystyle\int_0^1\frac{dx}{1+x^2} (to 4 s.f.).

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Salam Academy Original Practice · Proposed mark scheme
SA-FP-IND-0012 marksL2 StandardNon-calculatorDivisibility
Let f(n)=7n−1f(n)=7^n-1. Find the constant cc such that f(k+1)−7f(k)=cf(k+1)-7f(k)=c.

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SA-FP-IND-0032 marksL2 StandardNon-calculatorMatrices and sequences
M=(1201)\mathbf M=\begin{pmatrix}1&2\\0&1\end{pmatrix} and it can be proved that Mn=(12n01)\mathbf M^n=\begin{pmatrix}1&2n\\0&1\end{pmatrix}. Find the top-right entry of M5\mathbf M^5.

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SA-FP-IND-0042 marksL2 StandardNon-calculatorMatrices and sequences
A sequence has u1=2u_1=2 and un+1=3un−2u_{n+1}=3u_n-2. It can be proved that un=3n−1+1u_n=3^{n-1}+1. Find u4u_4.

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SA-FP-IND-G012 marksL2 StandardNon-calculatorSeries
In proving ∑r=1nr(r+1)=13n(n+1)(n+2)\displaystyle\sum_{r=1}^n r(r+1)=\tfrac13n(n+1)(n+2), the sum to k+1k+1 terms is 13k(k+1)(k+2)+(k+1)(k+2)\tfrac13k(k+1)(k+2)+(k+1)(k+2). This factorises to 13(k+1)(k+2)(k+a)\tfrac13(k+1)(k+2)(k+a). Find aa.

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Salam Academy Original Practice · Proposed mark scheme
SA-FP-INEQ-0013 marksL2 StandardNon-calculatorRational inequalities
1x−1>1\dfrac1{x-1}>1 has solution a<x<ba<x<b. Find bb.

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Salam Academy Original Practice · Proposed mark scheme
SA-FP-INEQ-0032 marksL2 StandardNon-calculatorRational inequalities
Find the smallest positive integer satisfying 3x+2≤1\dfrac3{x+2}\le1.

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Salam Academy Original Practice · Proposed mark scheme
SA-FP-INEQ-0042 marksL2 StandardNon-calculatorRational inequalities
How many integers xx with −5≤x≤5-5\le x\le5 satisfy x−3x+1>0\dfrac{x-3}{x+1}>0?

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Salam Academy Original Practice · Proposed mark scheme
SA-FP-INEQ-G012 marksL2 StandardNon-calculatorRational inequalities
Find the critical values when solving x+1x−2<2\dfrac{x+1}{x-2}<2.

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SA-FP-MAC-0022 marksL2 StandardNon-calculatorDeriving
Find the coefficient of x3x^3 in the series for ln⁡(1+x)\ln(1+x) by differentiation (f′′′(0)/3!f'''(0)/3!).

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SA-FP-MAC-0042 marksL2 StandardNon-calculatorApplications
Use cos⁡x≈1−x22\cos x\approx1-\frac{x^2}2 to estimate cos⁡0.2\cos0.2.

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SA-FP-MAT-0012 marksL2 StandardNon-calculatorOperations
A=(1234)\mathbf A=\begin{pmatrix}1&2\\3&4\end{pmatrix}, B=(0152)\mathbf B=\begin{pmatrix}0&1\\5&2\end{pmatrix}. Find the bottom-left entry of AB\mathbf{AB}.

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SA-FP-MAT-0022 marksL2 StandardNon-calculatorInverses
A=(5273)\mathbf A=\begin{pmatrix}5&2\\7&3\end{pmatrix}. Find the top-right entry of A−1\mathbf A^{-1}.

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SA-FP-MTR-0012 marksL2 StandardNon-calculatorStandard matrices
The matrix (12−323212)\begin{pmatrix}\frac12&-\frac{\sqrt3}2\\\frac{\sqrt3}2&\frac12\end{pmatrix} is a rotation about OO. Find the anticlockwise angle in degrees (0≤θ<3600\le\theta<360).

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Salam Academy Original Practice · Proposed mark scheme
SA-FP-MTR-0022 marksL2 StandardNon-calculatorCombining
P=(2001)\mathbf P=\begin{pmatrix}2&0\\0&1\end{pmatrix} (stretch) and Q=(0−110)\mathbf Q=\begin{pmatrix}0&-1\\1&0\end{pmatrix} (rotation). The point (1,3)(1,3) undergoes P\mathbf P then Q\mathbf Q. Find the xx-coordinate of its image.

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Salam Academy Original Practice · Proposed mark scheme
SA-FP-NT-0012 marksL2 StandardNon-calculatorEuclid
Use the Euclidean algorithm to find gcd⁡(252,198)\gcd(252,198).

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Salam Academy Original Practice · Proposed mark scheme
SA-FP-NT-0022 marksL2 StandardNon-calculatorCongruences
Solve 3x≡4(mod7)3x\equiv4\pmod7, giving 0≤x<70\le x<7.

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SA-FP-POL-0012 marksL2 StandardCalculatorAreas
Use 12∫r2 dθ\frac12\int r^2\,d\theta to find the area enclosed by r=3r=3 (to 3 s.f.).

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SA-FP-POL-0032 marksL2 StandardNon-calculatorConverting
The curve r=2cos⁡θr=2\cos\theta is a circle. Find its radius.

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Salam Academy Original Practice · Proposed mark scheme
SA-FP-RA-0012 marksL2 StandardCalculatorReduction formulae
Using the same formula, find I2I_2 (to 4 s.f.).

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Salam Academy Original Practice · Proposed mark scheme