Mathematics 12 past paper 2022

$x = a \cos \theta, y = b \sin \theta$ are parametric equations of:

MCQ1 marksannual

Let the real valued functions $f$ and $g$ be defined by $f(x) = 4x + 1$ and $g(x) = 2x^2 + 5x$. Obtain the expression for: a. $f(g(x))$ b. $g(f(x))$ c. $f(f(x))$ d. $g(g(x))$

Part B4 marksannual

If $\theta$ is measured in radian then prove that $\lim_{\theta \to 0} \frac{\sin \theta}{\theta} = 1$. a. Draw the figure and give explanation. b. Find area of triangles in figure…

Part C8 marksannual

A function $f: X \rightarrow Y$ defined by $f(x) = a, \forall x \in X, a \in Y$ is called:

MCQ1 marksannual

For the real valued function $f(x) = \sqrt{x^3 + 4}$, find $f^{-1}(x)$. Also verify $f(f^{-1}(x)) = x$.

Part B4 marksannual

Let $f(x) = \begin{cases} mx+3 & \text{if } x 3 \end{cases}$ **a.** Find $\lim_{x \rightarrow 3^-} f(x)$ and $\lim_{x \rightarrow 3^+} f(x)$ **b.** Find the $\lim_{x \rightarrow 3}…

Part C8 marksannual

Which of the following represents $f^{-1}(5)$ if $f(x) = x^{\frac{1}{3}} + 2$?

MCQ1 marksannual

Evaluate $\lim_{x \to 0} \frac{\sqrt{x+5} - \sqrt{5}}{x}$

Part B4 marksannual

Consider the function $f(x) = \sin x + \frac{1}{\sqrt{2}} \cos 2x$ where $x \in (0, 2\pi)$. Find the extreme values of the function in the interval $x \in (0, 2\pi)$: a. Find funct…

Part C8 marksannual

If $f(x) = \sqrt{x^2 - 1}$, then the **Domain** of $f$ is:

MCQ1 marksannual

Evaluate $\lim_{x \rightarrow 0} \frac{\csc x - \cot x}{x}$.

Part B4 marksannual

The perimeter of a triangle is 18 centimetres. If one side is of length 8 cm, what are the lengths of the other sides for maximum area of the triangle? **a.** Find function $f(x)$…

Part C8 marksannual

In which of the following intervals, $f(x) = 4x - 2x^2$ is increasing?

MCQ1 marksannual

Find $\frac{dy}{dx}$ if $x = \frac{3at}{1+t^3}$ and $y = \frac{3at^2}{1+t^3}$

Part B4 marksannual

Integrate $\int \frac{2x+5}{(x-3)^2(x^2-x+5)} dx$: a. Resolve $\frac{2x+5}{(x-3)^2(x^2-x+5)}$ into Partial fractions. b. After Partial Fraction, integrate the result.

Part C8 marksannual

What result occurs in evaluating $\lim_{x \rightarrow 3} \frac{x - 3}{\sqrt{3} - \sqrt{x}}$?

MCQ1 marksannual

If $y = \sqrt{\sin x + \sqrt{\sin x + \sqrt{\sin x + \dots \infty}}}$, prove that $(2y - 1) \frac{dy}{dx} = \cos x$.

Part B4 marksannual

Evaluate the integral $\int \frac{2x^2 + 5x + 3}{(x - 2)^2(x^2 + x + 1)} dx$ **a.** Resolve $\frac{2x^2 + 5x + 3}{(x - 2)^2(x^2 + x + 1)}$ into Partial fractions **b.** After Parti…

Part C8 marksannual

What result will occur, in evaluating $\lim_{n \to \infty} \left(1 + \frac{2}{n}\right)^{3n}$?

MCQ1 marksannual

If $y = \tan(4 \tan^{-1} \frac{x}{4})$, show that $\frac{dy}{dx} = \frac{16(1+y^2)}{16+x^2}$

Part B4 marksannual

If $f(x) = \cos x$, then what is the value of $f'(\sin^{-1} 3x)$?

MCQ1 marksannual

Show that $\sin(x + h) = \sin x + h \cos x - \frac{h^2}{2!} \sin x - \frac{h^3}{3!} \cos x + \dots$ by using **Taylor's Series**.

Part B4 marksannual

The diagram shows a triangle ABC where $A(-2, 3)$, $B(4, 5)$, and $C(6, 2)$ are vertices of $\triangle ABC$. **a.** Find the slopes of sides $\overline{AB}$, $\overline{BC}$ and $\…

Part C8 marksannual

For a function $f(x) = a \sin 3x$ and $f'(\frac{\pi}{3}) = 6$, then what is the value of $a$?

MCQ1 marksannual

Use implicit rule to find the second derivative of the function $y = x + \tan^{-1} y$

Part B4 marksannual

Find the maximum and minimum values of $f$ and $g$ defined as $f(x,y) = 3x + 5y$ and $g(x,y) = 6x + 8y$ under the constraints: $2x - 3y \leq 6$, $2x + y \geq 2$, $2x + 3y \leq 12$,…

Part C8 marksannual

If $f(x) = \ln x^2$, then what is the value of $f''(\sqrt{5})$?

MCQ1 marksannual

An agent wishes to purchase a number of chairs and tables. He has only Rs. 12000 to invest and has space at most for 28 items. A chair costs him Rs. 480 and a table costs Rs. 300.…

Part C8 marksannual

$\frac{d}{dx} (\sec^{-1} x + \csc^{-1} x) =$

MCQ1 marksannual

If $x = \cos \theta$ and $y = \cos n\theta$, show that $(1-x^2)y_2 - xy_1 + n^2y = 0$

Part B4 marksannual

Find the equations of tangent and normal lines at a point $(3, \frac{12}{5})$ to ellipse $\frac{x^2}{25} + \frac{y^2}{9} = 1$. For what value of $c$ the line $x + y + c = 0$ will t…

Part C8 marksannual

$(1 + x^2) \frac{d}{dx} (\tan^{-1} x + \cot^{-1} x) =$

MCQ1 marksannual

Evaluate $\int \frac{dx}{3x(\ln 3x)^4}$.

Part B4 marksannual

Find the Centre, Foci, Eccentricity, Vertices and Equation of directrices of the conic $25x^2 + 4y^2 - 250x - 16y + 541 = 0$.

Part C8 marksannual

Find the area between the $x$-axis and the curve $f(x) = x^2 - 2x$ from $x = 0$ to $x = 3$

Part B4 marksannual

The integral $\int \frac{dx}{x \ln x}$ is equal to:

MCQ1 marksannual

Evaluate $\int_{0}^{3} \frac{x^3 + 9x + 3}{x^2 + 9} dx$.

Part B4 marksannual

Which one of the following results occurs from the integral $\int_{0}^{2} \frac{dx}{x^2 + 4}$?

MCQ1 marksannual

Evaluate $\int x^3 \sqrt{1+x^2} dx$

Part B4 marksannual

What is the value of $k$ if $\int_{0}^{1} (3x + k) dx = 2$?

MCQ1 marksannual

Solve the differential equation $\frac{dy}{dx} + \frac{4xy}{4y + 2} = x$.

Part B4 marksannual

If $\int_{0}^{2} f(x) dx = 3$, then what is the value of $k$ if $\int_{0}^{2} (3f(x) + 4) dx = k$?

MCQ1 marksannual

Find the point two-fifth of the way along the line segment $A(-3,5)$ to $B(5,3)$.

Part B4 marksannual

What is the area between the $x$-axis and the curve $y = \sin x$ from $x=0$ to $x=\pi$?

MCQ1 marksannual

Find an equation of the perpendicular bisector of a line joining the points $A(5, 6)$ and $B(8, 4)$.

Part B4 marksannual

The points $A(2,5)$ and $B(3,-2)$ are the ends of a diameter of a circle, what is the radius of the circle?

MCQ1 marksannual

Find the angle $\theta$ from the lines $L_1$ to $L_2$ where: $L_1: 7x + 3y - 9 = 0$ $L_2: 5x - 2y + 2 = 0$

Part B4 marksannual

The equation of a line $\frac{x}{P \sec \alpha} + \frac{y}{P \csc \alpha} = 1$ is called:

MCQ1 marksannual

Find the value of $k$ such that the lines $2x - 2y + 2 = 0$, $3x - 5y - 1 = 0$ and $2x + ky + 8 = 0$ meet at a point.

Part B4 marksannual

A line cuts the $x$-axis at $(2,0)$ and $y$-axis at $(0,-4)$, then equation of the line is:

MCQ1 marksannual

Graph the feasible solution region of the system of linear inequalities by shading, also find the corner points: $3x + 7y \leq 21, x - y \leq 3, x \geq 0, y \geq 0$

Part B4 marksannual

For what value of $k$ are the lines $kx - 2y + 5 = 0$ and $x - 2ky + 3 = 0$ parallel?

MCQ1 marksannual

Graph the feasible region of the system of linear inequalities by shading: $$5x + 7y \leq 35, \quad -x + 3y \leq 3, \quad x \geq 0, \quad y \geq 0$$

Part B4 marksannual

Pair of lines represented by Homogeneous equation $ax^2 + 2hxy + by^2 = 0$ through origin will be real and coincident if:

MCQ1 marksannual

Find the equation of parabola with focus $(1,3)$ and vertex $(4,3)$.

Part B4 marksannual

The equation of the vertical line through $(-6, 5)$ is:

MCQ1 marksannual

Find the equation of a circle passing through the points $A(2, 3)$ and $B(0, 2)$ having its centre on the line $3x + 2y - 3 = 0$.

Part B4 marksannual

The solution set of $2y + 5 > 4y - 3$ is:

MCQ1 marksannual

Find the equation of parabola, with Directrix $y = 3$ and vertex $(2,2)$.

Part B4 marksannual

Which point satisfies the inequality $x + 2y < 6$?

MCQ1 marksannual

Find the equation of the Parabola with focus $(3, 2)$ and directrix $2x - y + 5 = 0$.

Part B4 marksannual

The line $y = mx + c$ will be tangent to a circle $x^2 + y^2 = a^2$ if:

MCQ1 marksannual

Write the equation of ellipse with vertices at $(-1,2)$ and $(7,2)$ and 2 is the length of semi minor axis whereas major axis is horizontal.

Part B4 marksannual

Find the equation of the tangent to the hyperbola $9x^2 - 4y^2 = 36$ parallel to the line $3x + 2y + 7 = 0$.

Part B4 marksannual

What is the length of the latus rectum of the parabola $x^2 = 5y$?

MCQ1 marksannual

Prove that $\sin(\alpha - \beta) = \sin \alpha \cos \beta - \cos \alpha \sin \beta$ using vectors.

Part B4 marksannual

What is the eccentricity of an ellipse $\frac{x^2}{16} + \frac{y^2}{4} = 1$?

MCQ1 marksannual

Find the scalar $\alpha$ so that vectors $3\hat{i} + \alpha\hat{j} + 4\hat{k}$ and $4\hat{i} + 5\hat{j} + \alpha\hat{k}$ are perpendicular to each other.

Part B4 marksannual

Which one of the following represents the graph of $9x^2 - 18x + 4y^2 + 8y - 23 = 0$?

MCQ1 marksannual

Find constant $\alpha$ so that vectors are coplanar: $\mathbf{i} - \alpha \mathbf{j} - \mathbf{k}$, $\mathbf{i} + \mathbf{j} + 2\mathbf{k}$ and $\alpha \mathbf{i} - \mathbf{j} + \m…

Part B4 marksannual

What is the length of the latus rectum of the hyperbola $\frac{x^2}{16} - \frac{y^2}{9} = 1$?

MCQ1 marksannual

Find the volume of the tetrahedron whose vertices are $A(-2, 1, 4)$, $B(3, 2, 5)$, $C(-3, -5, 0)$, and $D(5, 8, 9)$.

Part B4 marksannual

What are the foci of the hyperbola $x^2 - y^2 = 4$?

MCQ1 marksannual

What is the projection of vector $-2\hat{i} + 3\hat{j} + 7\hat{k}$ on $2\hat{j} + \hat{k}$?

MCQ1 marksannual

If $\mathbf{u} = \mathbf{i} + 3\mathbf{j} - 4\mathbf{k}$ and $\mathbf{w} = \lambda \mathbf{i} + 9\mathbf{j} - 12\mathbf{k}$ are parallel, then what is the value of $\lambda$?

MCQ1 marksannual

What is the angle between the vectors $\hat{i} + \hat{j}$ and $\hat{j} + \hat{k}$?

MCQ1 marksannual

What is the volume of a parallelepiped whose edges are $2\mathbf{i} - 4\mathbf{j} + 5\mathbf{k}$, $2\mathbf{i} - 3\mathbf{j} + 6\mathbf{k}$, and $-\mathbf{j} - \mathbf{k}$?

MCQ1 marksannual

For what value of $\alpha$ are the vectors $2\hat{i}$, $\hat{j} + \hat{k}$ and $\hat{i} + \alpha\hat{j} + 2\hat{k}$ coplanar?

MCQ1 marksannual