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Открытые вопросы МЭСК

Открытые вопросы МЭСК на трёх языках. Используйте фильтры для поиска.

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МЭСК

Complete the following tasks involving integrals.

a)

Find each of the following integrals, simplifying your answer wherever possible.

i)
2 баллов

$\int 8e^{2-5x}\,dx$.

ii)
2 баллов

$\int\frac{6}{\sqrt[3]{4x-7}}\,dx$.

iii)
2 баллов

$\int\cos\left(\frac{7x-9}{3}\right)\,dx$.

b)

Use the derivative of a logarithmic function to evaluate the definite integral.

i)
2 баллов

Differentiate $\ln(3x^2-8)$ with respect to $x$.

ii)
4 баллов

Use your answer to b(i) to evaluate $\int_2^6\frac{3x}{3x^2-8}\,dx$. Give your answer in the form $\ln k$, where $k$ is an integer whose value is to be found.

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Анықталмаған және анықталған интегралдар12-122Экзамен
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МЭСК

Complete the following differentiation tasks.

a)

Differentiate each of the following with respect to $x$, simplifying your answer wherever possible.

i)
2 баллов

$\sqrt{3x^2+5x}$.

ii)
2 баллов

$\arcsin(3x)$.

b)
4 баллов

By first writing $y=\arcctg x$ as $x=\ctg y$ and then assuming the derivative of $\ctg y$, find $\frac{dy}{dx}$ in terms of $x$.

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Құрама және кері тригонометриялық функциялардың туындысы12-82Экзамен
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МЭСК

The diagram shows a sketch of the curve $y=16-x^2$, which intersects the negative $x$-axis at the point $P(a,0)$. The tangent to the curve at $P$ intersects the $y$-axis at the point $Q(0,b)$.

i)
1 баллов

Write down the value of $a$.

ii)
3 баллов

Show that $b=32$.

iii)
6 баллов

Find the area of the shaded region.

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Параболаға жүргізілген жанама және аудан12-102Экзамен
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МЭСК

In the binomial expansion of $(a+2x)^6$, where $a\neq0$, the coefficient of the term in $x^2$ is equal to the coefficient of the term in $x$. Find the value of $a$.

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Биномдық жіктеудегі коэффициенттер12-42Экзамен
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МЭСК

With respect to a fixed origin $O$, the line $l_1$ is given by $\mathbf r=\begin{pmatrix}8\\1\\-3\end{pmatrix}+\mu\begin{pmatrix}-5\\4\\3\end{pmatrix}$, where $\mu$ is a scalar parameter. The point $A$ lies on $l_1$ where $\mu=1$. The point $P$ has position vector $\begin{pmatrix}1\\5\\2\end{pmatrix}$. The line $l_2$ passes through $P$ and is parallel to $l_1$.

a)
1 баллов

Find the coordinates of $A$.

b)
2 баллов

Write down a vector equation for the line $l_2$.

c)
2 баллов

Find the exact value of the distance $AP$. Give your answer in the form $k\sqrt2$, where $k$ is a constant to be determined.

d)
3 баллов

The acute angle between $AP$ and $l_2$ is $\theta$. Find the value of $\cos\theta$.

e)
2 баллов

A point $E$ lies on the line $l_2$. Given that $AP=PE$, find the area of triangle $APE$.

f)
5 баллов

Find the coordinates of the two possible positions of $E$.

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Кеңістіктегі параллель түзулер және векторлар12-152Экзамен
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МЭСК

The curve $C$ has equation $y=(2x-1)^{\frac34}$, $x\geq\frac12$. The curve cuts the line $y=8$ at the point $P(k,8)$. The finite region $S$ is bounded by the curve $C$, the $x$-axis, the $y$-axis and the line $y=8$. This region is rotated through $2\pi$ radians about the $x$-axis to form a solid of revolution.

a)
2 баллов

Find $\int(2x-1)^{\frac32}\,dx$, giving your answer in its simplest form.

b)
2 баллов

Find the value of $k$.

c)
4 баллов

Find the exact value of the volume of the solid generated.

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Дәрежелік функцияны интегралдау және айналу денесінің көлемі12-82Экзамен
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МЭСК

Complete the following tasks involving integrals.

a)

Find each of the following integrals, simplifying your answer wherever possible.

i)
3 баллов

$\int\frac{7x^2-2}{x}\,dx$.

ii)
2 баллов

$\int\sin\left(\frac{2x}{3}-\pi\right)\,dx$.

b)
4 баллов

Evaluate $\int_3^6\frac{1}{\sqrt[4]{5x-14}}\,dx$.

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Анықталмаған және анықталған интегралдар12-92Экзамен
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МЭСК

Complete the following differentiation tasks.

a)

Differentiate each of the following with respect to $x$, simplifying your answer wherever possible.

i)
2 баллов

$\ln(4x^2-3x-5)$.

ii)
2 баллов

$e^{\sqrt{x}}$.

iii)
3 баллов

$\frac{a+b\sin x}{a-b\sin x}$, where $a$ and $b$ are constants.

b)
3 баллов

By first writing $\ctg x=(\tg x)^{-1}$ and assuming the derivative of $\tg x$, find an expression for $\frac{d}{dx}(\ctg x)$. Simplify your answer.

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Логарифмдік, экспоненциалдық және бөлшек функциялардың туындысы12-102Экзамен
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МЭСК

Given that $x=\arctg t$, $y=\ln t$, where $t>0$.

a)
4 баллов

Find an expression for $\frac{dy}{dx}$ in terms of $t$.

b)
5 баллов

Find the value of $x$ for which $\frac{d^2y}{dx^2}=0$.

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Параметрлік қисықтың бірінші және екінші туындылары12-92Экзамен
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МЭСК

Complete the tasks involving trigonometric equations.

a)
6 баллов

Find all values of $\theta$ in the range $0^\circ\leq\theta\leq360^\circ$ satisfying $7\cosec^2\theta-4\ctg^2\theta=16+5\cosec\theta$.

b)
1 баллов

Without carrying out any calculations, explain why there are no values of $\phi$ in the range $0^\circ\leq\phi\leq90^\circ$ which satisfy $4\sec\phi+3\cosec\phi=6$.

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Косеканс арқылы тригонометриялық теңдеу12-72Экзамен
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МЭСК

Find each of the following, simplifying your answer wherever possible.

i)
2 баллов

$\int\cos(2-5x)\,dx$.

ii)
2 баллов

$\int\frac4{e^{3x-2}}\,dx$.

iii)
2 баллов

$\int\frac5{\frac16x-3}\,dx$.

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Анықталмаған интегралдар12-62Экзамен
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МЭСК

Differentiate each of the following with respect to $x$, simplifying your answer wherever possible.

i)
2 баллов

$\frac{1}{\sqrt[4]{9-4x^5}}$.

ii)
3 баллов

$\frac{3+2x^3}{7-x^3}$.

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Құрама және бөлшек функциялардың туындысы12-52Экзамен
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МЭСК

Find all values of $\theta$ in the range $0^\circ\leq\theta\leq360^\circ$ satisfying $8\tg^2\theta-5\sec^2\theta=7+4\sec\theta$.

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Секанс арқылы тригонометриялық теңдеу12-62Экзамен
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МЭСК

With respect to a fixed origin $O$, the line $l$ has equation $\mathbf r=\begin{pmatrix}13\\8\\1\end{pmatrix}+\lambda\begin{pmatrix}2\\2\\-1\end{pmatrix}$, where $\lambda$ is a scalar parameter. The point $A$ lies on $l$ and has coordinates $(3,-2,6)$. The point $P$ has position vector $-p\mathbf i+2p\mathbf k$ relative to $O$, where $p$ is a constant.

a)
4 баллов

Given that vector $\overrightarrow{PA}$ is perpendicular to $l$, find the value of $p$.

b)
5 баллов

Given also that $B$ is a point on $l$ such that $\angle BPA=45^\circ$, find the coordinates of the two possible positions of $B$.

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Кеңістіктегі түзу және перпендикуляр вектор12-92Экзамен
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МЭСК

Use the binomial expansion to complete the following tasks.

a)
6 баллов

Show that $\sqrt{\frac{1+x}{1-x}}\approx1+x+\frac12x^2$, where $\lvert x\rvert<1$.

b)
3 баллов

Substitute $x=\frac1{26}$ into $\sqrt{\frac{1+x}{1-x}}=1+x+\frac12x^2$ to obtain an approximation to $\sqrt3$. Give your answer in the form $\frac ab$, where $a$ and $b$ are integers.

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Биномдық жіктеу және жуықтау12-92Экзамен
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МЭСК

The diagram shows a sketch of the curve $y=3x^2-\frac14x^3$. The shaded region is bounded by the curve, the $x$-axis and the lines $x=2$, $x=6$. Find the area of this shaded region.

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Қисық астындағы аудан12-42Экзамен
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МЭСК

The curve $C$ has equation $y=-2x^3+12x^2-18x+5$.

a)
6 баллов

Find the coordinates and the nature of each of the stationary points of $C$.

b)
2 баллов

Sketch $C$, indicating the coordinates of each of the stationary points.

c)
2 баллов

Given that the equation $-2x^3+12x^2-18x+5=k$ has three distinct real roots, find the range of possible values for $k$.

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Кубтық функцияның стационар нүктелері және графигі12-102Экзамен
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МЭСК

Solve the equation $6x^3+13x^2-10x-24=0$.

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Үшінші дәрежелі теңдеуді шешу12-62Экзамен
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МЭСК

When $ax^3+13x^2-10x-24$ is divided by $x+3$, the remainder is $-39$. Write down an equation satisfied by $a$ and hence show that $a=6$.

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Қалдық туралы теорема12-22Экзамен
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МЭСК

Given that $y=ax^{\frac43}+24x^{\frac12}$ and that $\frac{dy}{dx}=\frac{11}{2}$ when $x=64$, find the value of the constant $a$.

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Туынды арқылы тұрақтыны анықтау12-42Экзамен
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