Mathematics 12 past paper 2023

For what value of $a$ the vectors $2\underline{i}+4\underline{j}-5\underline{k}$ and $-4\underline{i}-8\underline{j}+a\underline{k}$ are parallel?

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Let $f(x)=x^{2}+2$ and $g(x)=x+1$ then: (a) Find $g \circ f(x)$ (b) Find $x$ for which $f \circ g(x)=11$

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Let $f(x)= \begin{cases}2-x, & 0 \leq x<1 \\ k, & x=1 \\ 2x-1, & 1<x \leq 2\end{cases}$ then: a. What are domain and range of $f(x)$? b. Find the value of $k$ for which $f(x)$ is c…

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What is the value of $f(-1)$ if $f^{-1}(x)=\frac{8-x}{2}$?

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For the function $f(x)= \begin{cases}3x+4, & 0 \leq x<3 \\ 16-x, & 3 \leq x<12 \\ x, & 12 \leq x<14\end{cases}$ (a) Explain whether $\lim_{x \rightarrow 3} f(x)$ exists (b) Discuss…

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A box has square base and the sum of one side of the base and height of the box is 12. If the length of one side of box is $x$ cm: a. Express volume $V$ of the box in terms of $x$.…

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What is the value of $\lim_{x \rightarrow 0} \frac{1-\cos x}{x^{2}}$?

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If $x=a \cos^{3} \theta, y=b \sin^{3} \theta$ show that $a \frac{dy}{dx}+b \tan \theta=0$ (Chain Rule)

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For a curve $f(x)=x^{2}-2x, x \in [-1,3]$: a. Find $\int_{-1}^{3} f(x) dx$. b. Sketch the graph of $f(x)$ and shade the area bounded by $x$-axis and the curve $f(x)$. c. Find the a…

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Which of the following is equal to $y_{2}$ if $xy=2$?

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If $e^{x}+e^{y}=e^{x+y}$ then find the value of $\frac{dy}{dx}$ at $(1,1)$

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If the sides of a triangle $\triangle ABC$ are $7x-y-10=0$, $10x+y-41=0$, and $3x+2y+3=0$ then: a. Find the vertices $A, B, C$ of the triangle. b. Find the equations of altitudes o…

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What is the value of $\frac{d}{dx}[f \circ g(x)]$ if $f(x)=e^{x}$ and $g(x)=\sin x$?

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Show that $\cos(x+h)=\cos x-h \sin x-\frac{h^{2}}{2!} \cos x+\frac{h^{3}}{3!} \sin x+\dots$

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A car detailing company performs two types of detailing: deluxe and ordinary. The deluxe detailing requires 1 hour inspection and 3 hours maintenance time. While an ordinary detail…

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Which of the following is the derivative of $x^{x}$ w.r.t. $x$?

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Find two positive integers whose sum is 9 and product of one with square of the other is maximum.

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Find the centre, foci, eccentricity, vertices and directrices of the given conic $9x^{2}-y^{2}-36x-6y+18=0$.

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The derivative of an odd function is always:

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The side of a cube is measured to be 20 cm with a maximum error of 0.12 cm in its measurement. Find the maximum error in the calculated volume of the cube.

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Which of the following is equal to $\int(\csc^{2} x-1) dx$?

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Evaluate $\int \frac{\cos x}{\sin x \cdot \ln \sin x} dx$

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For what value of $x$ the function $f(x)=x^{2}-3x+7$ has a **critical point**?

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Find the equation of the curve for which $\frac{dy}{dx}+\frac{2xy}{2y+1}=x$. The curve passes through the point $(2,1)$

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If $\int_{a}^{\frac{\pi}{4}} \cos 2x \, dx = \frac{1}{2}$ then what should be the value of $a$?

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Which of the following is the evaluation of $\int \frac{e^{x}}{e^{x}-3} dx$?

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If $\int_{-3}^{3}(x^{3}+kx^{2}) dx=54$ find the value of $k$.

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The slope and $y$-intercept of the line $2x+y-3=0$ are:

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Find the area of a triangle with one vertex as point of intersection of the lines $x+y-5=0$ and $2x-y+2=0$, and points $(2,-3)$ and $(3,4)$ are other two vertices.

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Perpendicular distance of the line $x+2y+4=0$ from the point $(1,0)$ is:

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Find the equation of a circle concentric with the circle $x^{2}+y^{2}-8x+4=0$ and is tangent to the line $x+2y+6=0$.

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If line $l_1$ is making angle $45°$ with the $x$-axis, what is the slope?

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Which one is the solution of inequality $-20 > -2x$?

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Verify that $\underline{a}$ and $\underline{b}$ are perpendicular to $\underline{b} \times \underline{a}$ where $\underline{a}=3\underline{i}-\underline{j}+5\underline{k}$ and $\un…

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If $y^{2}-y=\sin x$ then which one is equal to $\frac{dy}{dx}$?

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Prove that in any triangle $\triangle ABC$, $b=c \cos A+a \cos C$.

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What is the length of latus rectum of parabola $(x+1)^{2}=8y+8$?

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What is the centre of ellipse $\frac{(2x+1)^{2}}{9}+\frac{(y-1)^{2}}{5}=1$?

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If $\theta$ is the angle between two vectors $\underline{a}$ and $\underline{b}$, and $\sin \theta=\frac{1}{2}$, $|\underline{a}|=|\underline{b}|=\sqrt{2}$ then what is the value o…

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When a force $\underline{F}=3\underline{i}+2\underline{j}-4\underline{k}$ is applied to an object a displacement $\underline{d}=4\underline{i}+\alpha\underline{j}+4\underline{k}$ o…

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