Mathematics NEB 2074 (old course) question paper

a) Prove that $A-\overline{B}=A\cap B$, where A and B are any two sets. [2] b) Let $A=\{a,b\}$, $B=\{b,c\}$ and $C=\{c,d\}$. Find $A\times (B\cup C)$ and $A\times (B\cap C)$. [2] c) Test the periodicity of the function $f(x)=\cos \pi x$ and find its period. [2]

a) Prove that $\sin (2{\sin }^{-1}x)=2x\sqrt{1-x^2}$. [2] b) Using the principle of mathematical induction, prove that: $2+2^2+2^3+\cdots +2^n=2(2^n-1)$. [2] c) If $A=\left(\begin{array}{cc}2& 1\\ 1& -2\end{array}\right)$, find $A{A}^T$. [2]

a) Using Cramer's rule, solve the following equations: $2x+5y=17$, $5x-2y=-1$. [2] b) Find the real numbers x and y if $(x-1)i+(y+1)=(1+i)(4-3i)$. [2] c) Find the value of K so that the equation $3x^2+7x+6-K=0$ has one root equal to zero. [2]

a) Find the equation of a straight line through the mid point of the line segment connecting $(2,-4)$ and $(2,4)$ and parallel to the line $3x-2y=4$. [2] b) Find the equation of a circle having radius 10 units and equations of any two diameters are $x+2y=8$ and $x+y=6$. [2] c) Evaluate: ${\lim }_{x\to p}\frac{x^2-p^2}{\tan (x-p)}$. [2]

a) Find $\frac{dy}{dx}$, when $x=2a\tan \theta$, $y=a\sec \theta$. [2] b) Evaluate: $\int (3\sin x-4{)}^2\cos xdx$ [2] c) Examine whether the function $f(x)=15x^2-14x+1$ is increasing or decreasing at $x=\frac{2}{5}$ and $x=\frac{5}{2}$. [2]

a) Define conjunction of the statements. Prepare a truth table for the compound statement $(p\wedge q)\wedge \sim (p\vee q)$. Draw the conclusion about the statement from the truth table. [4] OR Solve the inequality $|2x-1|\ge 3$ and draw its graph. b) Draw the graph of the function $f(x)=x-x^2$ indicating its characteristics. [4]

a) In any triangle ABC, prove that $\tan \frac{1}{2}(B-C)=\frac{b-c}{b+c}\cot \frac{A}{2}$. [4] OR Solve: $\sec x\tan x=\sqrt{2}$. b) Show that: $\left|\begin{array}{ccc}1& x& x^2\\ 1& y& y^2\\ 1& z& z^2\end{array}\right|=(y-z)(z-x)(x-y)$. [4]

a) Using row-equivalent method or inverse matrix method, solve the following system of equations. $x+y+z=1$, $x+2y+3z=4$, $x+3y+7z=13$. b) If one root of the equation is the square of the other, prove that $b^3+a^{2}c+a{c}^2=3abc$.

a) Find the equations of the tangent and normal to the circle $x^2+y^2-2x-4y+3=0$ at (2, 3). b) Evaluate: ${\lim }_{x\to 2}\frac{\sqrt{x}-\sqrt{6-x}}{x-2}$ OR A function $f(x)$ is defined as follows: $f(x)=\{\begin{array}{ll}2x+3& \text{for }x<1\\ 4& \text{for }x=1\\ 6x-1& \text{for }x>1\end{array}$ Is the function continuous at $x=1$? If not, how can you make it continuous?

a) Find from first principles, the derivative of $\sqrt{2-3x}$. b) Find the area of the region between the curve $y^2=4x$ and the line $x=y$.