Maths Exam Prep 2026-27
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Maths Exam Prep 2026-27
English 30 leveled MCQs Free
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8 marks Long answer
Obtain the simple logical expression of the following. Draw the corresponding switching circuit: $\neg (p \wedge q) \vee (\neg p \wedge q) \vee (p \wedge \neg q)$.
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Expression simplifies to $\neg p \vee \neg q$.
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8 marks Long answer
The volume of a sphere increase at the rate of $20 \text{ cm}^3/\text{sec}$. Find the rate of change of its surface area when its radius is 5 cm.
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Rate of change of surface area is $40\pi$ cm$^2$/sec.
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8 marks Long answer
Prove that, The acute angle $\theta$ between the lines represented by $ax^2 + 2hxy + by^2 = 0$ is given by $\tan \theta = |\frac{2\sqrt{h^2 - ab}}{a + b}|$, and Find the acute angle between lines represented by: $x^2 + xy = 0$.
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Angle $\theta = 45^\circ$.
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8 marks Long answer
Find the value of $x$ for which the function $f(x) = x^3 - 12x^2 - 144x + 13$ (a) increasing (b) decreasing.
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Increasing for $x < 4$ and decreasing for $x > 4$.
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5 marks Long answer
Examine whether each of the following statement patterns is a tautology or a contradiction or a contingency: $(p \Rightarrow q) \leftrightarrow (\neg p \vee q)$.
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It is a tautology.
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5 marks Long answer
Write the negation of the statements: (a) The number is neither odd nor perfect square. (b) The number is an even number if and only if it is divisible by 2.
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Negation: (a) The number is odd or perfect square. (b) The number is not an even number or not divisible by 2.
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5 marks Long answer
In $\triangle ABC$, prove that if $a = 18, b = 24, c = 30$ then find the values of $\cos A$.
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$\cos A = \frac{b^2 + c^2 - a^2}{2bc} = \frac{24^2 + 30^2 - 18^2}{2 \cdot 24 \cdot 30} = \frac{1}{2}$
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5 marks Long answer
Find the value of $k$ if the following equation represent a pair of lines: $3x^2 + 10xy + 3y^2 + 16y + k = 0$.
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Value of $k = -64$.
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5 marks Long answer
Prove the Cosine Rule: In $\triangle ABC$, $a^2 = b^2 + c^2 - 2bc \cos A$.
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Using the law of cosines derived from the coordinates of points A, B, and C.
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5 marks Long answer
Find the condition that the line $4x + 5y = 0$ coincides with one of the lines given by $ax^2 + 2hxy + by^2 = 0$.
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Condition: $\frac{a}{4} = \frac{b}{-5} = \frac{h}{0}$.
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5 marks Long answer
Find the approximate value of $\tan^{-1}(0.999)$.
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$\tan^{-1}(0.999) \approx 0.7854$.
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5 marks Long answer
If $y = \sqrt{\cos x + \sqrt{\cos x + \sqrt{\cos x + \dots}}}$, then show that $\frac{dy}{dx} = \frac{\sin x}{1 - 2y}$.
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Using implicit differentiation and the chain rule.
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5 marks Long answer
Evaluate: $\int \frac{\log(\log x)}{x \log x} dx$.
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Result is $\frac{1}{2} \log^2(\log x) + C$.
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5 marks Long answer
Evaluate: $\int \frac{x^2}{x+2} dx$.
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$\frac{x^3}{3} - 2x^2 + 4x + C$.
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5 marks Long answer
Differentiate the given function w.r.t. $x$: $[((\tan x)^{\tan x})^{\tan x}]$.
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Using the chain rule, $\frac{d}{dx} = \tan^x \sec^2 x + \tan^{\tan x} \cdots$.
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5 marks Long answer
Prove it: $\sin^{-1}(\frac{3}{5}) + \cos^{-1}(\frac{12}{13}) = \sin^{-1}(\frac{56}{65})$.
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Using the identity $\sin^{-1} x + \cos^{-1} x = \frac{\pi}{2}$.
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5 marks Long answer
Express the following equation in matrix form and solve them by the method of reduction: $x - y + z = 1$, $2x - y = 1$, $3x + 3y - 4z = 2$.
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Matrix form: $\begin{bmatrix} 1 & -1 & 1 \\ 2 & -1 & 0 \\ 3 & 3 & -4 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 1 \\ 1 \\ 2 \end{bmatrix}$.
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5 marks Long answer
If $A = \begin{bmatrix} 1 & 2 \\ -1 & 3 \end{bmatrix}$, verify that $A (\text{adj} A) = (\text{adj} A) A = |A| I$.
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Verification shows $A (\text{adj} A) = 5I$.
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5 marks Long answer
Differentiate $\sin^{-1}(\frac{1 - x^2}{1 + x^2})$ w.r.t. $\cos^{-1}(\frac{1 - x^2}{1 + x^2})$.
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Using implicit differentiation, the result is $1$.
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5 marks Long answer
A wire of length 36 meter is bent in the form of a rectangle. Find its dimensions if the area of the rectangle is maximum.
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Dimensions are $9m \times 9m$.
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5 marks Long answer
If $y = e^{m \tan^{-1} x}$, show that $(1 + x^2) \frac{d^2y}{dx^2} + (2x - m) \frac{dy}{dx} = 0$.
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Using differentiation and substitution.
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4 marks Medium answer
If $A = \begin{bmatrix} 1 & 2 \\ -1 & 1 \end{bmatrix}$ and $A (\text{adj} A) = kI$ then find the value of $k$.
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k = |A| = 3
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4 marks Medium answer
Find the combined equation of lines $x + y - 2 = 0$ and $2x - y + 2 = 0$.
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Combined equation: $x^2 + y^2 - 2x - 2y + 2 = 0$
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4 marks Medium answer
Evaluate: $\int \frac{1}{25 - 9x^2} dx$.
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$\frac{1}{15} \tan^{-1}\left(\frac{3x}{5}\right) + C$
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4 marks Medium answer
Find the derivative of inverse function of $y = x^2 e^x$.
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$\frac{dy^{-1}}{dx} = \frac{1}{(2xe^x + x^2e^x)}$
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1 marks Short answer
If $p \wedge q$ is false and $p \vee q$ is true, the _______ is not true.
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$p \Rightarrow q$
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1 marks Short answer
The principal solution of equation $\cot \theta = 3$ _______.
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$\frac{5\pi}{6}$
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1 marks Short answer
If the lines represented by $kx^2 - 3xy + 6y^2 = 0$ are perpendicular to each other then _______.
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$k=-6$
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1 marks Short answer
Rolle's theorem is satisfied for the function $f(x) = \sin x + \cos x + 7, x \in [0, 2\pi]$. Find $c = $ _______.
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$\frac{5\pi}{4}$
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1 marks Short answer
If $f'(x) = x \cdot f(1) = 11/2$ then $f(x) = $ _______.
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$\frac{x^2}{2} + \frac{3}{2x^2}$
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