Class 12 Mathematics Exam Prep
Full explanation, worked examples & memory tricks in the app
Class 12 Mathematics Exam Prep
English 27 leveled MCQs Free
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What you'll study, topic by topic
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Sample questions with model answers
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5 marks Long answer
Show that $R = \{(a, b) : a \le b^2\}$ on R is neither reflexive, nor symmetric, nor transitive.
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R is neither reflexive, symmetric, nor transitive.
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5 marks Long answer
Prove that $\tan^{-1} \frac{1}{11} + \tan^{-1} \frac{2}{24} = \tan^{-1} \frac{1}{2}$.
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Use the tangent addition formula.
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5 marks Long answer
Simplify $\tan^{-1} \left(\frac{\cos x - \sin x}{\cos x + \sin x}\right)$, $0 < x < \frac{\pi}{2}$.
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The result is $\frac{\pi}{4} - x$.
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5 marks Long answer
If $A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$ and $B = \begin{pmatrix} 2 & 0 \\ 1 & 3 \end{pmatrix}$, prove that $(AB)' = B'A'$.
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Use properties of transpose.
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5 marks Long answer
Using properties of determinants, prove $\begin{vmatrix} 1 & a & a^2 \\ 1 & b & b^2 \\ 1 & c & c^2 \end{vmatrix} = (a - b)(b - c)(c - a)$.
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Apply determinant properties.
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5 marks Long answer
Evaluate $\int \frac{x}{(x+1)(x+2)} dx$.
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The result is $\frac{1}{2} \ln |x+1| - \frac{1}{2} \ln |x+2| + C$.
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5 marks Long answer
Evaluate $\int_0^{\pi/4} \tan^2 x \, dx$.
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The result is $1$.
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5 marks Long answer
Find the area in the first quadrant bounded by $y^2 = 4x$ and $x = 3$.
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Area is $6$.
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5 marks Long answer
Find the general solution of $\frac{dy}{dx} = \frac{1+y^2}{1+x^2}$.
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The solution is $y = \tan^{-1}(x) + C$.
The full exam-length answer is in the app.
5 marks Long answer
If $\mathbf{a} = 2\mathbf{i} + 2\mathbf{j} - 5\mathbf{k}$ and $\mathbf{b} = 2\mathbf{i} + \mathbf{j} + 3\mathbf{k}$, find the unit vector in the direction of $\mathbf{a} + \mathbf{b}$.
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The unit vector is $\frac{1}{\sqrt{30}}(4\mathbf{i} + 3\mathbf{j} - 2\mathbf{k})$.
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5 marks Long answer
Find the Cartesian equation of the line through $(2, -1, 4)$ and $(1, 1, -2)$.
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The equation is $\frac{x-2}{-1} = \frac{y+1}{2} = \frac{z-4}{-6}$.
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5 marks Long answer
A family has two children. Given that at least one child is a boy, find the conditional probability that both children are boys.
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The probability is $\frac{1}{3}$.
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5 marks Long answer
Using the matrix method $X = A^{-1}B$, solve $x - y + 2z = 7$, $3x + 4y - 5z = -5$, $2x - y + 3z = 12$.
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Use $X = A^{-1}B$ to find $(x, y, z)$.
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5 marks Long answer
Using partial fractions, evaluate $\int \frac{2x - 3}{(x^2 - 1)(2x + 3)} dx$.
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Apply partial fraction decomposition.
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5 marks Long answer
Find the particular solution of $\frac{dy}{dx} + y \cot x = 2x + x^2 \cot x \quad (x \neq 0)$, given that $y = 0$ when $x = \frac{\pi}{2}$.
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Use integrating factor method.
The full exam-length answer is in the app.
5 marks Long answer
Find the shortest distance between the skew lines $r = (\hat{i} + 2\hat{j} + \hat{k}) + \lambda(\hat{i} - \hat{j} + \hat{k})$, $r = (2\hat{i} - \hat{j} - \hat{k}) + \mu(2\hat{i} + \hat{j} + 2\hat{k})$.
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Use the formula for distance between skew lines.
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5 marks Long answer
Machines A, B and C produce 25%, 35% and 40% of the bolts; of these, 5%, 4% and 2% are defective. A bolt drawn at random is found defective. Using Bayes' theorem, find the probability that it was produced by machine B.
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Apply Bayes' theorem to find $P(B|D)$.
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5 marks Long answer
If $A = \begin{pmatrix} 1 & -1 & 2 \\ 2 & 1 & -3 \\ 1 & 1 & 1 \end{pmatrix}$, find $A^{-1}$. Hence solve $x - y + z = 4$, $2x + y - 3z = 0$, $x + y + z = 2$.
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Use matrix inversion to solve for $(x, y, z)$.
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3 marks Medium answer
Let $f : \mathbb{R} \setminus \{-4/3\} \to \mathbb{R} \setminus \{4/3\}$ be defined by $f(x) = \frac{4x}{3x+4}$. Show that $f$ is one-one and onto. Also find $f^{-1}$.
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Show $f$ is one-one by proving $f(x_1) = f(x_2) \Rightarrow x_1 = x_2$.
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3 marks Medium answer
Express $A = \begin{pmatrix} 3 & 5 \\ 1 & -1 \end{pmatrix}$ as the sum of a symmetric matrix and a skew-symmetric matrix.
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The symmetric part is $\begin{pmatrix} 3 & 2 \\ 2 & -1 \end{pmatrix}$ and skew-symmetric part is $\begin{pmatrix} 0 & 3 \\ -3 & 0 \end{pmatrix}$.
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3 marks Medium answer
If $A = \begin{pmatrix} 2 & 3 \\ 1 & 2 \end{pmatrix}$, prove that $A^2 - 4A + I = O$. Hence find $A^{-1}$.
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Use $A^{-1} = \frac{1}{-1}(A - I)$.
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3 marks Medium answer
Evaluate $\int x \sin^{-1} x \, dx$.
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Use integration by parts.
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3 marks Medium answer
Using properties of definite integrals, prove $\int_0^{\pi} \frac{x \sin x}{1 + \cos^2 x} \, dx = \frac{\pi^2}{4}$.
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Apply symmetry and substitution.
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3 marks Medium answer
Using integration, find the total area enclosed by the ellipse $\frac{x^2}{16} + \frac{y^2}{9} = 1$.
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Area is $12\pi$.
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3 marks Medium answer
Show that the differential equation $(x - y) \, dy - (x + y) \, dx = 0$ is homogeneous and find its solution.
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The solution is $y = Cx$.
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3 marks Medium answer
Using the vector method, find the area of the triangle with vertices $A(1, 2, 3)$, $B(2, -1, 1)$ and $C(-1, 2, 3)$.
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Area is $\frac{3\sqrt{3}}{2}$.
The full exam-length answer is in the app.
3 marks Medium answer
A die is thrown twice and the sum of the numbers is 6. Find the conditional probability that the number 4 appears at least once.
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The probability is $\frac{1}{5}$.
The full exam-length answer is in the app.
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