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Math

Relations and Functions

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Question: Write the identity relation on set A = {a, b, c}.

SETS

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Question: If A = {1, 2, 3, 4} define relations on A which have properties of being reflexive, symmetric, and transitive.

INVERSE TRIGONOMETRIC FUNCTIONS

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Question: \( 4 \tan^{-1}(1/5) = \tan^{-1}(1/70) + \tan^{-1}(1/99) + \frac{\pi}{4} \)

MATRICES

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Question: The number of solutions of the system of equations: 2x + y − z = 7 x − 3y + 2z = 1 x + 4y − 3z = 5 (a) 3 (b) 2 (c) 1 (d) 0

DETERMINANTS

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Question: Without expanding, show that the value of each of the determinants is zero: ∣∣ ∣ ∣∣ 1 a a^2 - bc 1 b b^2 - ac 1 c c^2 - ab ∣∣ ∣ ∣∣

CONTINUITY AND DIFFERENTIABILITY

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Ashutosh
Let $f: R \rightarrow R$ be a continuous function such that $f(3 x)-f(x)=x$. If $f(8)=7$, then $f(14)$ is equal to:

APPLICATION OF DERIVATIVES

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Question: If the tangent line at a point (x, y) on the curve y = f(x) is parallel to the x-axis, then write the value of dy/dx.

INTEGRALS

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Question: Evaluate: (i) ∫ 1 / √(1 − cos^2(x)) dx (ii) ∫ 1 / √(1 + cos(x)) dx

APPLICATION OF INTEGRALS

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Question: Draw a rough sketch of the region \(\{(x,y) : y^2 \leq 3x, 3x^2 + 3y^2 \leq 16\}\) and find the area enclosed by the region using the method

DIFFERENTIAL EQUATIONS

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Determine the order and degree of each of the following differential equation. State also whether they are linear or non-linear: \[ (xy^2 + x) \, dx +

VECTOR ALGEBRA

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Question: Let \(\vec{a} = \hat{i} + \hat{j} + \hat{k}\), \(\vec{b} = \hat{i} - \hat{j} + 2\hat{k}\) and \(\vec{c} = x\hat{i} + (x-2)\hat{j} - \hat{k}\

THREE DIMENSIONAL GEOMETRY

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If (2, 3, 5) is one end of a diameter of the sphere \(x^2 + y^2 + z^2 - 6x - 12y - 2z + 20 = 0\), then the coordinates of the other end of the diamete

LINEAR PROGRAMMING

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PROBABILITY

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Question: Two aeroplanes I and II bomb a target in succession. The probabilities of I and II scoring a hit correctly are 0.3 and 0.2, respectively. Th
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