МЭСК ашық сұрақтары

Іздеу

Түрі	Сұрақ	Пункт суреттері	Тақырып	Сынып	Жыл	Ұпай	МЭСК компоненті	Емтихан түрі	Оқу мақсаты	Олимпиада	Автор	Тегтер	Суреттер
МЭСК	Complete the following tasks involving integrals. a) Find each of the following integrals, simplifying your answer wherever possible. i) 2 ұпай $\int 8e^{2-5x}\,dx$. ii) 2 ұпай $\int\frac{6}{\sqrt[3]{4x-7}}\,dx$. iii) 2 ұпай $\int\cos\left(\frac{7x-9}{3}\right)\,dx$. b) Use the derivative of a logarithmic function to evaluate the definite integral. i) 2 ұпай Differentiate $\ln(3x^2-8)$ with respect to $x$. ii) 4 ұпай 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.	-	Анықталмаған және анықталған интегралдар	12	-	12	2	Емтихан	N/A	N/A	N/A	N/A	-
МЭСК	Complete the following differentiation tasks. a) Differentiate each of the following with respect to $x$, simplifying your answer wherever possible. i) 2 ұпай $\sqrt{3x^2+5x}$. ii) 2 ұпай $\arcsin(3x)$. b) 4 ұпай 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 $x$.	-	Құрама және кері тригонометриялық функциялардың туындысы	12	-	8	2	Емтихан	N/A	N/A	N/A	N/A	-
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 ұпай Write down the value of $a$. ii) 3 ұпай Show that $b=32$. iii) 6 ұпай Find the area of the shaded region.	-	Параболаға жүргізілген жанама және аудан	12	-	10	2	Емтихан	N/A	N/A	N/A	N/A
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 $a$.	-	Биномдық жіктеудегі коэффициенттер	12	-	4	2	Емтихан	N/A	N/A	N/A	N/A	-
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 ұпай Find the coordinates of $A$. b) 2 ұпай Write down a vector equation for the line $l_2$. c) 2 ұпай 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 ұпай The acute angle between $AP$ and $l_2$ is $\theta$. Find the value of $\cos\theta$. e) 2 ұпай A point $E$ lies on the line $l_2$. Given that $AP=PE$, find the area of triangle $APE$. f) 5 ұпай Find the coordinates of the two possible positions of $E$.	-	Кеңістіктегі параллель түзулер және векторлар	12	-	15	2	Емтихан	N/A	N/A	N/A	N/A	-
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 ұпай Find $\int(2x-1)^{\frac32}\,dx$, giving your answer in its simplest form. b) 2 ұпай Find the value of $k$. c) 4 ұпай Find the exact value of the volume of the solid generated.	-	Дәрежелік функцияны интегралдау және айналу денесінің көлемі	12	-	8	2	Емтихан	N/A	N/A	N/A	N/A
МЭСК	Complete the following tasks involving integrals. a) Find each of the following integrals, simplifying your answer wherever possible. i) 3 ұпай $\int\frac{7x^2-2}{x}\,dx$. ii) 2 ұпай $\int\sin\left(\frac{2x}{3}-\pi\right)\,dx$. b) 4 ұпай Evaluate $\int_3^6\frac{1}{\sqrt[4]{5x-14}}\,dx$.	-	Анықталмаған және анықталған интегралдар	12	-	9	2	Емтихан	N/A	N/A	N/A	N/A	-
МЭСК	Complete the following differentiation tasks. a) Differentiate each of the following with respect to $x$, simplifying your answer wherever possible. i) 2 ұпай $\ln(4x^2-3x-5)$. ii) 2 ұпай $e^{\sqrt{x}}$. iii) 3 ұпай $\frac{a+b\sin x}{a-b\sin x}$, where $a$ and $b$ are constants. b) 3 ұпай 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.	-	Логарифмдік, экспоненциалдық және бөлшек функциялардың туындысы	12	-	10	2	Емтихан	N/A	N/A	N/A	N/A	-
МЭСК	Given that $x=\arctg t$, $y=\ln t$, where $t>0$. a) 4 ұпай Find an expression for $\frac{dy}{dx}$ in terms of $t$. b) 5 ұпай Find the value of $x$ for which $\frac{d^2y}{dx^2}=0$.	-	Параметрлік қисықтың бірінші және екінші туындылары	12	-	9	2	Емтихан	N/A	N/A	N/A	N/A	-
МЭСК	Complete the tasks involving trigonometric equations. a) 6 ұпай 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 ұпай Without carrying out any calculations, explain why there are no values of $\phi$ in the range $0^\circ\leq\phi\leq90^\circ$ which satisfy $4\sec\phi+3\cosec\phi=6$.	-	Косеканс арқылы тригонометриялық теңдеу	12	-	7	2	Емтихан	N/A	N/A	N/A	N/A	-
МЭСК	Find each of the following, simplifying your answer wherever possible. i) 2 ұпай $\int\cos(2-5x)\,dx$. ii) 2 ұпай $\int\frac4{e^{3x-2}}\,dx$. iii) 2 ұпай $\int\frac5{\frac16x-3}\,dx$.	-	Анықталмаған интегралдар	12	-	6	2	Емтихан	N/A	N/A	N/A	N/A	-
МЭСК	Differentiate each of the following with respect to $x$, simplifying your answer wherever possible. i) 2 ұпай $\frac{1}{\sqrt[4]{9-4x^5}}$. ii) 3 ұпай $\frac{3+2x^3}{7-x^3}$.	-	Құрама және бөлшек функциялардың туындысы	12	-	5	2	Емтихан	N/A	N/A	N/A	N/A	-
МЭСК	Find all values of $\theta$ in the range $0^\circ\leq\theta\leq360^\circ$ satisfying $8\tg^2\theta-5\sec^2\theta=7+4\sec\theta$.	-	Секанс арқылы тригонометриялық теңдеу	12	-	6	2	Емтихан	N/A	N/A	N/A	N/A	-
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 ұпай Given that vector $\overrightarrow{PA}$ is perpendicular to $l$, find the value of $p$. b) 5 ұпай Given also that $B$ is a point on $l$ such that $\angle BPA=45^\circ$, find the coordinates of the two possible positions of $B$.	-	Кеңістіктегі түзу және перпендикуляр вектор	12	-	9	2	Емтихан	N/A	N/A	N/A	N/A	-
МЭСК	Use the binomial expansion to complete the following tasks. a) 6 ұпай Show that $\sqrt{\frac{1+x}{1-x}}\approx1+x+\frac12x^2$, where $\lvert x\rvert<1$. b) 3 ұпай 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.	-	Биномдық жіктеу және жуықтау	12	-	9	2	Емтихан	N/A	N/A	N/A	N/A	-
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.	-	Қисық астындағы аудан	12	-	4	2	Емтихан	N/A	N/A	N/A	N/A
МЭСК	The curve $C$ has equation $y=-2x^3+12x^2-18x+5$. a) 6 ұпай Find the coordinates and the nature of each of the stationary points of $C$. b) 2 ұпай Sketch $C$, indicating the coordinates of each of the stationary points. c) 2 ұпай Given that the equation $-2x^3+12x^2-18x+5=k$ has three distinct real roots, find the range of possible values for $k$.	-	Кубтық функцияның стационар нүктелері және графигі	12	-	10	2	Емтихан	N/A	N/A	N/A	N/A	-
МЭСК	Solve the equation $6x^3+13x^2-10x-24=0$.	-	Үшінші дәрежелі теңдеуді шешу	12	-	6	2	Емтихан	N/A	N/A	N/A	N/A	-
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 $a=6$.	-	Қалдық туралы теорема	12	-	2	2	Емтихан	N/A	N/A	N/A	N/A	-
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 $a$.	-	Туынды арқылы тұрақтыны анықтау	12	-	4	2	Емтихан	N/A	N/A	N/A	N/A	-

1-20 / 1162