Mathematics NEB Class 11 2072 question paper

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1. 120726 marksHideAnswer

   (a) Construct a truth table of ~(p v q) ^ q. (b) Let f: R -> R and g: R -> R be defined by f(x) = 2x + 1 and g(x) = 3x - 1, find (gof)(x) and (fog)(x). (c) Examine the even or odd nature and the symmetricity of the function y = 10^x - 10^-x.

   (a) Truth table of $\sim(p\lor q)\wedge q$: $p$ $q$ $p\lor q$ $\sim(p\lor q)$ $\sim(p\lor q)\wedge q$ --------------- T T T F F T F T F F F T T F F F F F T F The final column is always F, so the statement is a contradiction. (b)

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2. 220726 marksHideAnswer

   (a) Prove that sin(2 sin^-1 x) = 2x sqrt(1 - x^2). (b) Prove by the principle of mathematical induction that 1/(1.2) + 1/(2.3) + 1/(3.4) + ... + 1/(n(n+1)) = n/(n+1). (c) If A=(3 -5; 4 6) and B=(0 3; 1 -2) find (AB)^T.

   (a) Let $\sin^{-1}x=\theta$, so $\sin\theta=x$ and $\cos\theta=\sqrt{1-x^2}$. Then $$ \begin{aligned} \sin(2\sin^{-1}x) &= \sin2\theta \ &= 2\sin\theta\cos\theta \ &= 2x\sqrt{1-x^2}. \end{aligned} $$

   (b) $P(n):\ \displaystyle\sum_{r=1}^{n}\frac{1}{r(r+1)}=\frac{n}{n+1}$. $n=1$: LHS $=\tfrac12$, RHS $=\tfrac12$. True. Assume $P(k)$: $\displaystyle\sum_{r=1}^{k}\frac{1}{r(r+1)}=\frac{k}{k+1}$. Then $$ \begin{aligned} \sum_{r=1}^{k+1}\frac{1}{r(r+1)} &= \frac{k}{k+1}+\frac{1}{(k+1)(k+2)} \ &= \frac{k(k+2)+1}{(k+1)(k+2)} \ &= \frac{(k+1)^2}{(k+1)(k+2)} \ &= \frac{k+1}{k+2}. \end{aligned} $$ This is $P(k+1)$, so by induction the result holds for all $n\in\mathbb{N}$.

   (c) $AB=\begin{pmatrix}3&-5\4&6\end{pmatrix}\begin{pmatrix}0&3\1&-2\end{pmatrix}=\begin{pmatrix}0-5&9+10\0+6&12-12\end{pmatrix}=\begin{pmatrix}-5&19\6&0\end{pmatrix}.$ $$(AB)^{T}=\begin{pmatrix}-5&6\19&0\end{pmatrix}.$$

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3. 320726 marksNumericalHideAnswer

   (a) Solve by Cramer's rule: 3x + 4y = -2, 5x - 7y = 24. (b) Express the complex number i - sqrt(3) into polar form. (c) Find the value of k so that the equation 3x^2 + 7x + 6 - k = 0 has one root equal to zero.

   (a) $D=\begin{vmatrix}3&4\5&-7\end{vmatrix}=-21-20=-41,\ Dx=\begin{vmatrix}-2&4\24&-7\end{vmatrix}=14-96=-82,\ Dy=\begin{vmatrix}3&-2\5&24\end{vmatrix}=72+10=82.$ $$ \begin{aligned} x &= \frac{-82}{-41} \ &= 2, \ \qquad y &= \frac{8...

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4. 420726 marksNumericalHideAnswer

   (a) Find the equation of a straight line through the centroid of the triangle with vertices at (3, -4), (-2, 1) and (5, 0) and perpendicular to the line x - 3y = 4. (b) Find the equation of the circle with centre at (p, q) and radius sqrt(p^2 + q^2). (c) Evaluate: lim(x -> infinity) (sqrt(x) - sqrt(x-3)).

   (a) Centroid $G=\left(\dfrac{3-2+5}{3},\dfrac{-4+1+0}{3}\right)=(2,-1)$. Line $x-3y=4$ has slope $\tfrac13$; the perpendicular line has slope $-3$. $$ \begin{aligned} y-(-1) &= -3(x-2)\Rightarrow y+1 \ &= -3x+6\Rightarrow \boxed{3x+y-5=...

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5. 520726 marksNumericalHideAnswer

   (a) If y = e^(sin(log x)), find dy/dx. (b) Evaluate: integral of (1 - 1/x^2) e^(x + 1/x) dx. (c) If f(x) = 2x^3 - 6x^2 + 5, determine where the graph of the function is concave upward.

   (a) Using the chain rule, $$ \begin{aligned} \frac{dy}{dx} &= e^{\sin(\log x)}\cdot\cos(\log x)\cdot\frac{1}{x} \ &= \frac{e^{\sin(\log x)}\cos(\log x)}{x}. \end{aligned} $$ (b) Note that

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6. 620728 marksHideAnswer

   (a) If A and B are non-empty subsets of the universal set U, prove that (i) (A U B)^c = A^c n B^c (ii) (A n B)^c = A^c U B^c. OR Define absolute value of a real number. If x in R and a is any positive real number, prove that |x| < a implies -a < x < a and conversely. (b) Sketch the graph of y = x - x^2 indicating its characteristics.

   (a) De Morgan's laws. (i) $x\in(A\cup B)^c\iff x\notin A\cup B\iff x\notin A\text{ and }x\notin B\iff x\in A^c\text{ and }x\in B^c\iff x\in A^c\cap B^c$. Hence $(A\cup B)^c=A^c\cap B^c$. (ii)

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7. 720728 marksHideAnswer

   (a) Solve: sec x . tan x = sqrt(2). OR In any triangle ABC, prove that tan((B-C)/2) = ((b-c)/(b+c)) cot(A/2). (b) Show that determinant |x y z; x^2 y^2 z^2; yz zx xy| = (y-z)(z-x)(x-y)(yz+zx+xy).

   (a) $\sec x\tan x=\dfrac{\sin x}{\cos^2x}=\sqrt2\Rightarrow \sin x=\sqrt2\cos^2x=\sqrt2(1-\sin^2x)$. Let $s=\sin x$: $\sqrt2,s^2+s-\sqrt2=0\Rightarrow s=\dfrac{-1\pm3}{2\sqrt2}$, so $s=\dfrac{1}{\sqrt2}$ or $s=-\sqrt2$ (rejected).

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8. 820728 marksNumericalHideAnswer

   (a) Using row equivalent matrix or inverse matrix method, solve: x + 4y + z = 18, 3x + 3y - 2z = 2, -4y + z = -7. (b) If the roots of the equation (a^2 + b^2)x^2 - 2(ac + bd)x + (c^2 + d^2) = 0 are equal, prove that a/b = c/d.

   (a) From the third equation $z=4y-7$. Substitute in the first: $$ \begin{aligned} x+4y+(4y-7) &= 18\Rightarrow x+8y \ &= 25\Rightarrow x \ &= 25-8y. \end{aligned} $$ Substitute both in the second: $3(25-8y)+3y-2(4y-7)=2$ $$ \begin{alig...

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9. 920728 marksNumericalHideAnswer

   (a) Find the equation of the tangent to the circle x^2 + y^2 = a^2 at the point (x1, y1). (b) Evaluate: lim(x -> theta) (x cos theta - theta cos x)/(x - theta). OR A function f(x) is defined as f(x) = (x^2 - x - 6)/(x^2 - 2x - 3) for x != 3, and 5/3 for x = 3. Prove that f(x) is discontinuous at x = 3. Can the definition of f(x) for x = 3 be modified so as to make it continuous there?

   (a) For the circle $x^2+y^2=a^2$, differentiate: $2x+2y\dfrac{dy}{dx}=0\Rightarrow\dfrac{dy}{dx}=-\dfrac{x}{y}$. Slope at $(x1,y1)$ is $-\dfrac{x1}{y1}$. Tangent: $$ \begin{aligned} y-y1 &= -\frac{x1}{y1}(x-x1)\Rightarrow yy1-y1^2 \ &= ...

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10. 1020728 marksNumericalHideAnswer

    (a) Find from first principles, the derivative of tan(3x - 4). (b) Find the area of the region between the curve y^2 = 4x and the line y = x.

    (a) Let $f(x)=\tan(3x-4)$. $$ \begin{aligned} f'(x) &= \lim_{h\to0}\frac{\tan(3x+3h-4)-\tan(3x-4)}{h} \ &= \lim_{h\to0}\frac{\sin\big((3x+3h-4)-(3x-4)\big)}{h,\cos(3x+3h-4)\cos(3x-4)} \ &= \lim_{h\to0}\frac{\sin 3h}{3h}\cdot\frac{3}{\cos(3x+3h-4)\cos(3x-4)} \ &= \frac{3}{\cos^2(3x-4)} \ &= 3\sec^2(3x-4). \end{aligned} $$

    (b) Curves $y^2=4x$ and $y=x$ meet where $x^2=4x\Rightarrow x=0,4$ (points $(0,0),(4,4)$). For $0\le x\le4$ the parabola $y=2\sqrt x$ lies above the line $y=x$: $$ \begin{aligned} A &= \int_0^4\big(2\sqrt x-x\big),dx \ &= \left[\frac{4}{3}x^{3/2}-\frac{x^2}{2}\right]_0^4 \ &= \frac{4}{3}(8)-8 \ &= \frac{32}{3}-8 \ &= \frac{8}{3}. \ \boxed{A=\frac{8}{3}\ \text{square units}}. \end{aligned} $$

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11. 1120726 marksNumericalHideAnswer

    Let a function f: A -> B be defined by f(x) = (x-1)/(x+2) with A = {-1, 0, 1, 2, 3, 4} and B = {-2, 1, -1/2, 0, 1/2, 1/4, 2/5}. Find the range of f. Is the function f one to one and onto both? If not, how can you make it one to one and onto both?

    Compute $f(x)=\dfrac{x-1}{x+2}$ for each element of $A$: $$ \begin{aligned} f(-1) &= \frac{-2}{1} \ &= -2, \ \quad f(0) &= \frac{-1}{2} \ &= -\tfrac12, \ \quad f(1) &= \frac{0}{3} \ &= 0, \ f(2) &= \frac{1}{4} \ &= \tfrac14, \ \q...

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12. 1220726 marksNumericalHideAnswer

    The sum of three numbers in A.P. is 36. When the numbers are increased by 1, 4, 43 respectively, the resulting numbers are in G.P. Find the numbers.

    Let the three numbers in A.P. be $a-d,\ a,\ a+d$. Their sum is $3a=36\Rightarrow a=12$. So the numbers are $12-d,\ 12,\ 12+d$. After increasing by $1,4,43$: $$13-d,\quad 16,\quad 55+d\quad\text{are in G.P.}$$ G.P. condition: $16^2=(13-d)(55+d)$. $$ \begin{aligned} 256 &= 715+13d-55d-d^2\Rightarrow d^2+42d-459 \ &= 0. \ d &= \frac{-42\pm\sqrt{1764+1836}}{2} \ &= \frac{-42\pm60}{2} \ &= 9\ \text{or}\ -51. \end{aligned} $$

    • $d=9$: numbers $\boxed{3,\ 12,\ 21}$ (increased: $4,16,64$, ratio $4$).
    • $d=-51$: numbers $63,\ 12,\ -39$ (increased: $64,16,4$, ratio $\tfrac14$).

    Both satisfy the conditions.

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13. 1320726 marksNumericalHideAnswer

    Find the equations of the bisectors of the angles between the lines 4x - 3y + 1 = 0 and 12x - 5y + 7 = 0, and prove that the bisectors are at right angles to each other. Also identify the bisector of the angle between the lines containing the origin.

    The angle bisectors satisfy $\dfrac{4x-3y+1}{5}=\pm\dfrac{12x-5y+7}{13}$, i.e. $13(4x-3y+1)=\pm5(12x-5y+7)$. ($+$): $52x-39y+13=60x-25y+35\Rightarrow 4x+7y+11=0$. ($-$): $52x-39y+13=-60x+25y-35\Rightarrow 7x-4y+3=0$. Bisectors:

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14. 1420726 marksHideAnswer

    If z and w are two complex numbers, prove that: |z + w|^2 = |z|^2 + |w|^2 + 2 Re(z w-bar).

    For complex numbers, $u^2=u\bar u$ and $\overline{z+w}=\bar z+\bar w$. Hence $$ \begin{aligned} z+w^2 &= (z+w)\overline{(z+w)} \ &= (z+w)(\bar z+\bar w) \ &= z\bar z+z\bar w+w\bar z+w\bar w. \end{aligned} $$ Now $z\bar z=z^2$,

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15. 1520726 marksNumericalHideAnswer

    If y = f(x) represents a certain curve, what do f'(x) > 0, f'(x) < 0 and f'(x) = 0 at a point represent? Find the interval in which the function f(x) = 2x^3 - 15x^2 + 36x + 1 is increasing or decreasing. Also find the point of inflection. OR A spherical ball of salt dissolving in water decreases its volume at the rate of 0.75 cm^3/min. Find the rate at which the radius of the salt is decreasing when its radius is 6 cm.

    Geometrical meaning: at a point, $f'(x)0$ means the curve is increasing (rising), $f'(x)<0$ means decreasing (falling), and $f'(x)=0$ means the tangent is horizontal (a possible turning point / stationary point). For

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