Practise questions | Salam Academy

SA-FP-MAT-0033 marksL3 ExamNon-calculatorDeterminants

Find

\det\begin{pmatrix}2&1&0\\1&3&1\\0&1&2\end{pmatrix}

.

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SA-FP-MAT-0043 marksL3 ExamNon-calculatorDeterminants

Find the value of

k

for which

\begin{pmatrix}k&2\\3&k-1\end{pmatrix}

is singular, given

k>0

.

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SA-FP-MTR-0033 marksL3 ExamNon-calculatorInvariance

Find the gradients of the invariant lines through

O

of

\begin{pmatrix}3&1\\2&2\end{pmatrix}

.

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SA-FP-MTR-0043 marksL3 ExamNon-calculatorInvariance

\mathbf M=\begin{pmatrix}2&1\\1&2\end{pmatrix}

has a line of invariant points

y=kx

. Find

k

.

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SA-FP-NT-0033 marksL3 ExamNon-calculatorFermat

Find

2^{100}\bmod7

.

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SA-FP-POL-0023 marksL3 ExamCalculatorAreas

Find the area enclosed by the cardioid

r=2(1+\cos\theta)

(to 3 s.f.).

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SA-FP-POL-0043 marksL3 ExamCalculatorAreas

Find the area swept by

r=\theta

from

\theta=0

to

\theta=\pi

(to 3 s.f.).

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SA-FP-RA-0023 marksL3 ExamCalculatorReduction formulae

I_n=\int_0^{\pi/2}\sin^nx\,dx

satisfies

I_n=\frac{n-1}nI_{n-2}

,

I_0=\frac\pi2

. Find

I_4

(to 4 s.f.).

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SA-FP-RA-0033 marksL3 ExamNon-calculatorArc length

Find the length of

y=\frac23x^{3/2}

from

x=0

to

x=3

.

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SA-FP-RAT-0033 marksL3 ExamNon-calculatorRange

Find the greatest value of

y=\dfrac{x}{x^2+1}

.

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SA-FP-RR-0044 marksL3 ExamNon-calculatorSecond order

u_{n+2}=5u_{n+1}-6u_n

with

u_0=1

,

u_1=4

. Find

u_3

using the closed form.

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SA-FP-TF-0034 marksL3 ExamCalculatorEquations

Solve

3\sin x+4\cos x=4

for

0<x<\pi

(radians, 4 s.f.).

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SA-FP-TF-0044 marksL3 ExamNon-calculatorIntegrals

Evaluate

\displaystyle\int_0^{\pi/2}\frac{dx}{1+\sin x}

.

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SA-FP-TL-0043 marksL3 ExamNon-calculatorLeibniz

Use Leibniz's theorem to find

\frac{d^3}{dx^3}(x^2e^x)

at

x=0

.

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SA-FP-VD-0034 marksL3 ExamCalculatorSkew lines

Find the shortest distance between

\mathbf r=s\begin{pmatrix}1\\0\\0\end{pmatrix}

and

\mathbf r=\begin{pmatrix}0\\1\\2\end{pmatrix}+t\begin{pmatrix}0\\1\\1\end{pmatrix}

(to 3 s.f.).

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SA-FP-VP-0033 marksL3 ExamCalculatorIntersections

Find the acute angle between the line with direction

\begin{pmatrix}1\\1\\1\end{pmatrix}

and the plane

x+2y+z=5

, in degrees to 1 d.p.

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SA-FP-VP-0043 marksL3 ExamCalculatorCross product

Find the area of the triangle formed by vectors

\mathbf a=\begin{pmatrix}2\\1\\0\end{pmatrix}

and

\mathbf b=\begin{pmatrix}1\\3\\2\end{pmatrix}

from a common vertex (to 3 s.f.).

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SA-IAL-M1-DY-0085 marksL3 ExamCalculatorTowing

A car of mass

900

kg tows a trailer of mass

300

kg along a straight horizontal road using a light horizontal tow bar. The driving force is

2000

N. The resistances are

200

N on the car and

100

N on the trailer. Find (i) the acceleration, to 3 significant figures, and (ii) the tension in the tow bar.

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SA-IAL-P1-IN-0055 marksL3 ExamCalculatorFinding the curve

\dfrac{dy}{dx}=3x^{\frac12}-2

, and the curve passes through

(4,10)

. Find

y

in terms of

x

.

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SA-IAL-S2-CD-0064 marksL3 ExamCalculatorNormal approximations

X\sim B(80,0.3)

. Use a suitable approximation to find

P(X\le20)

, correct to 3 significant figures.

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