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Maths 12 Past Paper 2022 | Notely

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All Maths 12 papers

Class 12 · Past paper

Maths 12 past paper 2022

78 published questions in this index. The full practice view lets you work through them and track your own attempts.

36 MCQs42 written questionsPractise this paper

Question index

1
$x = a \cos \theta, y = b \sin \theta$ are parametric equations of:
MCQ1 marksannual
1
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
1
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
1
A function $f: X \rightarrow Y$ defined by $f(x) = a, \forall x \in X, a \in Y$ is called:

This is a skimmable index, not a replacement for the study surface. Open the practice view to answer questions and use the linked study material.

MCQ1 marksannual

1

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

1

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

2

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

MCQ1 marksannual

2

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

Part B4 marksannual

2

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

2

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

MCQ1 marksannual

2

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

Part B4 marksannual

2

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

3

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

MCQ1 marksannual

3

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

Part B4 marksannual

3

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

3

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

MCQ1 marksannual

3

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

3

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

4

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

MCQ1 marksannual

4

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

4

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

MCQ1 marksannual

4

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

4

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

5

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

MCQ1 marksannual

5

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

Part B4 marksannual

5

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

5

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

MCQ1 marksannual

5

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

6

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

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6

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

Part B4 marksannual

6

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

6

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

MCQ1 marksannual

6

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

Part B4 marksannual

6

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

Part C8 marksannual

7

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

Part B4 marksannual

7

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

MCQ1 marksannual

7

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

Part B4 marksannual

8

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

MCQ1 marksannual

8

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

Part B4 marksannual

8

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

MCQ1 marksannual

8

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

Part B4 marksannual

9

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

9

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

Part B4 marksannual

9

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

MCQ1 marksannual

9

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

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10

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

10

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

10

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

MCQ1 marksannual

10

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

11

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

MCQ1 marksannual

11

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

11

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

MCQ1 marksannual

11

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

12

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

MCQ1 marksannual

12

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

Part B4 marksannual

12

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

MCQ1 marksannual

12

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

13

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

MCQ1 marksannual

13

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

Part B4 marksannual

13

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

MCQ1 marksannual

13

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

Part B4 marksannual

14

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

MCQ1 marksannual

14

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

14

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

15

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

MCQ1 marksannual

15

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

Part B4 marksannual

15

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

MCQ1 marksannual

15

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

16

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

MCQ1 marksannual

16

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

16

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

MCQ1 marksannual

16

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

17

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

MCQ1 marksannual

18

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

MCQ1 marksannual

19

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

19

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

MCQ1 marksannual

20

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

20

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