NEB Class 12 · Past paper
The complete NEB Class 12 2072 exam paper for Mathematics, all 19 questions with solved model answers.
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(a) In how many ways can the letters of "ELEMENT" be arranged so that the vowels are always together? (b) Prove that $\frac{2}{1!}+\frac{4}{3!}+\frac{6}{5!}+\cdots=e$. (c) In a Cayley table for a finite group, why does each element occur exactly once in each row and each column?
(a) ELEMENT has vowels $E,E,E$ (identical) and consonants $L,M,N,T$. Treating the three E's as one block, we arrange the block with $L,M,N,T$: $5!=120$ ways (the block has only one internal arrangement since the E's are identical). (b) T...
(a) Find the equation of the hyperbola with vertex $(8,0)$ passing through $(8\sqrt2,4)$. (b) If $P,Q$ denote $(2,6,2)$ and $(4,5,0)$, find the direction cosines of $PQ$. (c) If $\vec a=(3,-1,-4),\ \vec b=(-2,4,-3)$, find the unit vector along $\vec a-2\vec b$.
(a) Vertex $(8,0)\Rightarrow a=8$, so $\dfrac{x^2}{64}-\dfrac{y^2}{b^2}=1.$ Through $(8\sqrt2,4)$: $\dfrac{128}{64}-\dfrac{16}{b^2}=1\Rightarrow2-\dfrac{16}{b^2}=1\Rightarrow b^2=16.$ Equation $\dfrac{x^2}{64}-\dfrac{y^2}{16}=1.$ (b)
(a) Compute $\int\frac{\coth x,dx}{\sinh x-9,\operatorname{cosech} x}$. (b) Using L'Hospital's rule, evaluate $\lim_{x\to0}\frac{e^x-x-1}{x^2}$. (c) If $(\vec a+\vec b)\cdot(\vec a-\vec b)=0$, prove that $|\vec a|=|\vec b|$.
(a) Write $\coth x=\dfrac{\cosh x}{\sinh x}$ and $\operatorname{cosech}x=\dfrac1{\sinh x}$: $$\int\frac{\cosh x/\sinh x}{\sinh x-9/\sinh x},dx=\int\frac{\cosh x}{\sinh^2x-9},dx.$$ Let $u=\sinh x,\ du=\cosh x,dx$:
(a) Solve $\frac{dy}{dx}+\frac{1+\cos 2y}{1-\cos 2y}=0$. (b) In the data $20,25,30,36,32,43$, find the standard deviation. (c) In a draw of a card from a well-shuffled deck of 52, what is the probability that it is a king or a queen?
(a) Using $1+\cos2y=2\cos^2y,\ 1-\cos2y=2\sin^2y$: $\dfrac{dy}{dx}=-\cot^2y.$ Separate: $\tan^2y,dy=-dx\Rightarrow\int(\sec^2y-1),dy=-x$, so $\tan y-y+x=C.$ (b) $$ \begin{aligned} \bar x &= \dfrac{186}{6} \ &= 31. \end{aligned} $$ Dev...
(a) A person has 12 acquaintances, of whom 8 are relatives. In how many ways can he invite 7 guests so that 5 of them are relatives? (b) Let $(G,)$ be a group, $a,b\in G$; prove (i) $(ab)^{-1}=b^{-1}*a^{-1}$ and (ii) $(a^{-1})^{-1}=a$. OR: Solve for $x$ in a group $G$ (with elements $a,b,c$): $x^2=a^2$ and $x^5=e$.
(a) Choose 5 relatives from 8 and $2$ non-relatives from $4$:
$$ \begin{aligned} \binom85\binom42 &= 56\times6 \ &= 336. \end{aligned} $$
(b) (i)
$$ \begin{aligned} (ab)(b^{-1}a^{-1}) &= a(bb^{-1})a^{-1} \ &= aea^{-1} \ &= e, \end{aligned} $$
so $b^{-1}a^{-1}$ is the inverse of $ab.$ (ii) $a*a^{-1}=e$ shows $a$ is the inverse of $a^{-1}$, i.e. $(a^{-1})^{-1}=a.$
OR. From $x^5=e$ and $x^2=a^2$:
$$ \begin{aligned} x &= x^5\cdot x^{-4} \ &= (x^2)^{-2}\cdot x^{5} \end{aligned} $$
... more directly, since $\gcd(5,2)=1$ there exist integers with $2m+5n=1$ ($m=3,n=-1$):
$$ \begin{aligned} x &= x^{2\cdot3+5\cdot(-1)} \ &= (x^2)^3(x^5)^{-1} \ &= (a^2)^3\cdot e \ &= a^{6}. \end{aligned} $$
Hence $x=a^6.$
(a) If the tangent to the parabola $y^2=12x$ makes an angle $45^\circ$ with the line $x-2y+3=0$, find its equation and point of contact. OR: Find the eccentricity and foci of $\frac{(x+6)^2}{4}+\frac{y^2}{36}=1$. (b) Show that the plane $2x+3y-4z=3$ is parallel to $10x+15y-20z=12$ and perpendicular to $3x+2y+3z=5$.
(a) The line $x-2y+3=0$ has slope $\tfrac12$. If the tangent has slope $m$, $$ \begin{aligned} \tan45^\circ &= \left\dfrac{m-\tfrac12}{1+\tfrac m2}\right \ &= 1 \ \Rightarrow\ m &= 3\text{ or }m \ &= -\tfrac13. \end{aligned} $$ For
(a) Evaluate $\int\frac{dx}{(x-1)^2(x-2)^3}$. (b) Reduce $\frac{dy}{dx}+\frac{y}{x}=y^2$ to linear form and solve it. OR: Solve $\frac{dy}{dx}=\frac{y+1}{x+y+1}$.
(a) Partial fractions: $$\frac{1}{(x-1)^2(x-2)^3}=\frac{-3}{x-1}+\frac{-1}{(x-1)^2}+\frac{3}{x-2}+\frac{-2}{(x-2)^2}+\frac{1}{(x-2)^3}.$$ Integrating, $$\int=-3\ln|x-1|+\frac{1}{x-1}+3\ln|x-2|+\frac{2}{x-2}-\frac{1}{2(x-2)^2}+C.$$
(b) Bernoulli: divide by $y^2$ and put $v=y^{-1}$ ($v'=-y^{-2}y'$): $v'-\dfrac vx=-1.$ I.F. $=\tfrac1x$: $\left(\dfrac vx\right)'=-\dfrac1x\Rightarrow\dfrac vx=-\ln x+C.$ So $\dfrac1y=x(C-\ln|x|)$, i.e. $y=\dfrac{1}{x(C-\ln|x|)}.$
OR. Treat $x$ as function of $y$: $\dfrac{dx}{dy}-\dfrac{x}{y+1}=1.$ I.F. $=\dfrac1{y+1}$: $\dfrac{x}{y+1}=\ln|y+1|+C$, so $x=(y+1)(\ln|y+1|+C).$
(a) Define correlation and find Karl Pearson's coefficient of correlation for $X:20,30,40,50,60$ and $Y:50,46,30,24,8$. (b) The probability of hitting a target is $0.25$. If 8 shots are fired, find the probability that (i) none hits (ii) exactly two hit the target.
(a) Correlation measures the strength and direction of the linear relationship between two variables. With $\bar X=40,\ \bar Y=31.6$, deviations $u=X-40,\ v=Y-31.6$: $$ \begin{aligned} \Sigma uv &= (-20)(18.4)+(-10)(14.4)+0+ (10)(-7.6)+(...
State the binomial theorem. In the expansion of $(1+x)^n$, prove that the sum of the coefficients of the odd terms equals the sum of the coefficients of the even terms, and each equals $2^{n-1}$.
Binomial theorem. $$ \begin{aligned} (1+x)^n &= \displaystyle\sum{r=0}^{n}\binom nr x^r \ &= C0+C1x+C2x^2+\cdots+Cnx^n. \end{aligned} $$ Proof. Put $x=1$: $C0+C1+C2+\cdots+Cn=2^n.$ Put $x=-1$: $C0-C1+C2-\cdots=0$, i.e. $$(C0+C2+C4+\cdot...
Define vector product of two vectors. Prove by vector method that in any triangle $ABC$, $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$.
Vector product. $\vec p\times\vec q$ has magnitude $\vec p\vec q\sin\theta$ and direction perpendicular to both by the right-hand rule. Proof (sine rule). For triangle $ABC$, the sides as vectors satisfy $\vec a+\vec b+\vec c=\vec0$ (tak...
State the mean value theorem and verify it for $f(x)=\sqrt{x^2-4}$ on $[2,4]$. OR: Find from first principles the derivative of $\ln(\cos^{-1}x)$.
Statement. If $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$, then some $c\in(a,b)$ has $f'(c)=\dfrac{f(b)-f(a)}{b-a}.$ Verification. $f(2)=0$, $$ \begin{aligned} f(4) &= \sqrt{12} \ &= 2\sqrt3, \end{aligned} $$ so
(a) Show that the resultant of two equal forces bisects the angle between them. (b) A uniform beam $AB$ is $16$ m long and weighs $50$ kg; weights of $20$ kg and $50$ kg are suspended from $A$ and $B$. At what point must it be supported to rest horizontally? (c) A particle slides down a smooth inclined plane $10$ m long and acquires a velocity $10\sqrt2$ m/s. Find the inclination ($g=10$ m/s$^2$).
(a) Two equal forces $P$ form a rhombus in the parallelogram law; the diagonal (resultant) of a rhombus bisects the vertex angle. Analytically the resultant makes angle $\phi$ with one force where $$ \begin{aligned} \tan\phi &= \dfrac{P...
(a) State and prove Lami's theorem. OR: A body of weight $68$ N is suspended by two strings of length $8$ m and $15$ m whose other ends are attached to two points in a horizontal line $17$ m apart; find the tensions. (b) State the laws of motion and use Newton's law to define an absolute unit of force.
(a) Lami's theorem. For three concurrent forces $P,Q,R$ in equilibrium, $\dfrac{P}{\sin\alpha}=\dfrac{Q}{\sin\beta}=\dfrac{R}{\sin\gamma}$ ($\alpha,\beta,\gamma$ opposite angles). Proof: the forces drawn head-to-tail form a closed triang...
Define coplanar forces. Forces equal to $P,2P,3P,4P$ act along the sides of a square $ABCD$ taken in order; find the magnitude, direction and line of action of the resultant.
Coplanar forces are forces whose lines of action lie in the same plane.
Take $A$ as origin with $AB$ along $+x$, $AD$ along $+y$ (side $a$). Resolving: along $AB$ ($+x$) $P$, along $BC$ ($+y$) $2P$, along $CD$ ($-x$) $3P$, along $DA$ ($-y$) $4P$. $$ \begin{aligned} R_x &= P-3P \ &= -2P, \ R_y &= 2P-4P \ &= -2P. \ |R| &= \sqrt{(-2P)^2+(-2P)^2} \ &= 2\sqrt2,P, \ \tan\theta &= \frac{R_y}{R_x} \ &= 1 \ \Rightarrow\ \theta &= 225^\circ\ (\text{third quadrant}). \end{aligned} $$
Line of action:
$$ \begin{aligned} \text{moment about }A &= 2P\cdot a+3P\cdot a \ &= 5Pa; \end{aligned} $$
the perpendicular distance from $A$ is
$$ \begin{aligned} d &= \dfrac{5Pa}{2\sqrt2,P} \ &= \dfrac{5a}{2\sqrt2} \ &= \dfrac{5\sqrt2,a}{4}. \end{aligned} $$
Define energy and state the principle of conservation of energy. Prove that the sum of the kinetic and potential energy of a moving body remains constant throughout the motion. OR: Describe the motion of a projectile. A stone is thrown horizontally with velocity $\sqrt{2gh}$ from the top of a tower of height $h$. Find where it strikes the ground and the striking velocity.
Energy / conservation. Energy is the capacity to do work; the principle of conservation of energy states that in the absence of dissipative forces the total mechanical energy (K.E. $+$ P.E.) is constant. For a body falling from height $H$, after descending $x$:
$$ \begin{aligned} \text{K.E.} &= \tfrac12m(2gx) \ &= mgx, \end{aligned} $$
$\text{P.E.}=mg(H-x)$,
$$ \begin{aligned} \text{total} &= mgH \ &= \text{constant}. \end{aligned} $$
OR (projectile). Horizontal:
$$ \begin{aligned} x &= ut \ &= \sqrt{2gh},t; \end{aligned} $$
vertical: $h=\tfrac12gt^2\Rightarrow t=\sqrt{2h/g}.$ So
$$ \begin{aligned} \text{horizontal range} &= \sqrt{2gh}\cdot\sqrt{2h/g} \ &= 2h. \end{aligned} $$
Striking velocity components: $u=\sqrt{2gh}$ (horizontal),
$$ \begin{aligned} v_y &= gt \ &= \sqrt{2gh} \end{aligned} $$
(vertical);
$$ \begin{aligned} \text{speed} &= \sqrt{u^2+v_y^2} \ &= \sqrt{2gh+2gh} \ &= 2\sqrt{gh}, \end{aligned} $$
at $45^\circ$ below the horizontal.
(a) Draw the graph of the inequality $3x-3\le5x-y$. (b) Convert the hexadecimal number $70A_{16}$ into binary. (c) Test whether the system $12x+3y-5z=1,\ x+5y+3z=28,\ 3x+7y+13z=1$ is diagonally consistent (dominant).
(a) $3x-3\le5x-y\Rightarrow y\le2x+3.$ Draw the line $y=2x+3$ and shade the region below it (including the line). (b) Convert each hex digit to 4 bits: $7=0111,\ 0=0000,\ A=1010.$ $$ \begin{aligned} 70A{16} &= 0111,0000,10102 \ &= 111...
(a) Using the Gauss-Seidel method, solve $3x+4y+8z=7,\ x+20y+z=-18,\ 25x+y-5z=19$. OR: Use Gauss elimination to solve $4x-y+z=8,\ 2x+5y+2z=3,\ x+2y+4z=11$. (b) Using the bisection method find the root of $x^2+x-4=0$ in $(1,2)$ correct to two decimal places.
(a) Rearrange for dominance: $x=\dfrac{19-y+5z}{25},\ y=\dfrac{-18-x-z}{20},\ z=\dfrac{7-3x-4y}{8}.$ Iterating from $(0,0,0)$ converges to $$ \begin{aligned} x &= 1 \ \ y &= -1 \ \ z &= 1. \end{aligned} $$
OR (Gauss elimination). Eliminating gives $x=1,\ y=-1,\ z=3$ (check eq.1: $4+1+3=8$ ✓).
(b) $f(x)=x^2+x-4$; $f(1)=-2<0$ and $f(2)=2>0$, so a root lies in $[1,2].$ Bisecting: $$ \begin{aligned} f(1.5) &= -0.25\ (<0)\Rightarrow[1.5,2], \ f(1.75) &= 0.8125\ (>0)\Rightarrow[1.5,1.75], \ f(1.625) &= 0.2656\ (>0)\Rightarrow[1.5,1.625], \ f(1.5625) &= 0.0039\ (>0)\Rightarrow[1.5,1.5625], \ f(1.53125) &= -0.1240\ (<0)\Rightarrow[1.53125,1.5625]. \end{aligned} $$ The bracket now narrows about $1.56$, so to two decimal places the root is $\approx\mathbf{1.56}$ (exact $\tfrac{-1+\sqrt{17}}{2}=1.5616$).
math-area-under-curve
Using the Simplex method, find the optimal solution of $z=7x_1+5x_2$ subject to $x_1+2x_2\le6,\ 4x_1+3x_2\le12,\ x_1,x_2\ge0$.
Introduce slacks and evaluate the corner points: $$(0,0)\to0,\quad(3,0)\to21,\quad(0,3)\to15,\quad(1.2,2.4)\to20.4.$$ The binding constraints $x1+2x2=6,\ 4x1+3x2=12$ meet at $(1.2,2.4)$. Comparing values, the simplex terminates at the ve...
Approximate $\int_{-1}^{1}e^x,dx$ using the trapezoidal rule with $n=2$. OR: Evaluate $\int_0^1\sqrt{1+x^3},dx$ using Simpson's $\tfrac13$ rule with $n=4$.
Trapezoidal. $h=1$; nodes $x=-1,0,1$ with $f=e^{-1},1,e=0.3679,1,2.7183.$
$$ \begin{aligned} \int_{-1}^{1}e^x,dx &\approx \frac h2\big[f_0+f_2+2f_1\big] \ &= \frac12\big[0.3679+2.7183+2(1)\big] \ &= 2.5431. \end{aligned} $$
(Exact $e-e^{-1}=2.3504.$)
OR (Simpson). $h=0.25$; $f=\sqrt{1+x^3}$ at $x=0,0.25,0.5,0.75,1$ gives $1,1.00778,1.06066,1.19243,1.41421.$
$$ \begin{aligned} \int_0^1\sqrt{1+x^3},dx &\approx \frac h3\big[f_0+f_4+4(f_1+f_3)+2f_2\big] \ &= \frac{0.25}{3}(13.3364) \ &\approx 1.1114. \end{aligned} $$
(a) In how many ways can the letters of "ELEMENT" be arranged so that the vowels are always together? (b) Prove that . (c) In a Cayley table for a finite group, why does each element occur exactly once in each row and each column?
(a) ELEMENT has vowels (identical) and consonants . Treating the three E's as one block, we arrange the block with : ways (the block has only one internal arrangement since the E's are identical). (b) T...
(a) Find the equation of the hyperbola with vertex passing through . (b) If denote and , find the direction cosines of . (c) If , find the unit vector along .
(a) Vertex , so Through : Equation (b)
(a) Compute . (b) Using L'Hospital's rule, evaluate . (c) If , prove that .
(a) Write and : Let :
(a) Solve . (b) In the data , find the standard deviation. (c) In a draw of a card from a well-shuffled deck of 52, what is the probability that it is a king or a queen?
(a) Using : Separate: , so (b) Dev...
(a) A person has 12 acquaintances, of whom 8 are relatives. In how many ways can he invite 7 guests so that 5 of them are relatives? (b) Let be a group, ; prove (i) and (ii) . OR: Solve for in a group (with elements ): and .
(a) Choose 5 relatives from 8 and non-relatives from :
(b) (i)
so is the inverse of (ii) shows is the inverse of , i.e.
OR. From and :
... more directly, since there exist integers with ():
Hence
(a) If the tangent to the parabola makes an angle with the line , find its equation and point of contact. OR: Find the eccentricity and foci of . (b) Show that the plane is parallel to and perpendicular to .
(a) The line has slope . If the tangent has slope , \begin{aligned} \tan45^\circ &= \left\dfrac{m-\tfrac12}{1+\tfrac m2}\right \\ &= 1 \\ \Rightarrow\ m &= 3\text{ or }m \\ &= -\tfrac13. \end{aligned} For
(a) Evaluate . (b) Reduce to linear form and solve it. OR: Solve .
(a) Partial fractions: Integrating,
(b) Bernoulli: divide by and put (): I.F. : So , i.e.
OR. Treat as function of : I.F. : , so
(a) Define correlation and find Karl Pearson's coefficient of correlation for and . (b) The probability of hitting a target is . If 8 shots are fired, find the probability that (i) none hits (ii) exactly two hit the target.
(a) Correlation measures the strength and direction of the linear relationship between two variables. With , deviations : $$ \begin{aligned} \Sigma uv &= (-20)(18.4)+(-10)(14.4)+0+ (10)(-7.6)+(...
State the binomial theorem. In the expansion of , prove that the sum of the coefficients of the odd terms equals the sum of the coefficients of the even terms, and each equals .
Binomial theorem. Proof. Put : Put : , i.e. $$(C0+C2+C4+\cdot...
Define vector product of two vectors. Prove by vector method that in any triangle , .
Vector product. has magnitude and direction perpendicular to both by the right-hand rule. Proof (sine rule). For triangle , the sides as vectors satisfy (tak...
State the mean value theorem and verify it for on . OR: Find from first principles the derivative of .
Statement. If is continuous on and differentiable on , then some has Verification. , so
(a) Show that the resultant of two equal forces bisects the angle between them. (b) A uniform beam is m long and weighs kg; weights of kg and kg are suspended from and . At what point must it be supported to rest horizontally? (c) A particle slides down a smooth inclined plane m long and acquires a velocity m/s. Find the inclination ( m/s).
(a) Two equal forces form a rhombus in the parallelogram law; the diagonal (resultant) of a rhombus bisects the vertex angle. Analytically the resultant makes angle with one force where $$ \begin{aligned} \tan\phi &= \dfrac{P...
(a) State and prove Lami's theorem. OR: A body of weight N is suspended by two strings of length m and m whose other ends are attached to two points in a horizontal line m apart; find the tensions. (b) State the laws of motion and use Newton's law to define an absolute unit of force.
(a) Lami's theorem. For three concurrent forces in equilibrium, ( opposite angles). Proof: the forces drawn head-to-tail form a closed triang...
Define coplanar forces. Forces equal to act along the sides of a square taken in order; find the magnitude, direction and line of action of the resultant.
Coplanar forces are forces whose lines of action lie in the same plane.
Take as origin with along , along (side ). Resolving: along () , along () , along () , along () .
Line of action:
the perpendicular distance from is
Define energy and state the principle of conservation of energy. Prove that the sum of the kinetic and potential energy of a moving body remains constant throughout the motion. OR: Describe the motion of a projectile. A stone is thrown horizontally with velocity from the top of a tower of height . Find where it strikes the ground and the striking velocity.
Energy / conservation. Energy is the capacity to do work; the principle of conservation of energy states that in the absence of dissipative forces the total mechanical energy (K.E. P.E.) is constant. For a body falling from height , after descending :
,
OR (projectile). Horizontal:
vertical: So
Striking velocity components: (horizontal),
(vertical);
at below the horizontal.
(a) Draw the graph of the inequality . (b) Convert the hexadecimal number into binary. (c) Test whether the system is diagonally consistent (dominant).
(a) Draw the line and shade the region below it (including the line). (b) Convert each hex digit to 4 bits: $$ \begin{aligned} 70A{16} &= 0111,0000,10102 \ &= 111...
(a) Using the Gauss-Seidel method, solve . OR: Use Gauss elimination to solve . (b) Using the bisection method find the root of in correct to two decimal places.
(a) Rearrange for dominance: Iterating from converges to
OR (Gauss elimination). Eliminating gives (check eq.1: ✓).
(b) ; and , so a root lies in Bisecting:
The bracket now narrows about , so to two decimal places the root is (exact ).
Using the Simplex method, find the optimal solution of subject to .
Introduce slacks and evaluate the corner points: The binding constraints meet at . Comparing values, the simplex terminates at the ve...
Approximate using the trapezoidal rule with . OR: Evaluate using Simpson's rule with .
Trapezoidal. ; nodes with
(Exact )
OR (Simpson). ; at gives