NEB Class 12 · Past paper
The complete NEB Class 12 2071 exam paper for Mathematics, all 19 questions with solved model answers.
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(a) A man has 5 friends. In how many ways can he invite one or more of them to dinner? (b) Find the coefficient of $x$ in the expansion of $\left(x^2+\frac{a^2}{x}\right)^5$. (c) Show that multiplication is a binary operation on $S={-1,0,1}$.
(a) Each friend is either invited or not; excluding the case of none, the number of ways is $$2^5-1=31.$$ (b) General term $$ \begin{aligned} T{r+1} &= \binom5r(x^2)^{5-r}\left(\tfrac{a^2}{x}\right)^r \ &= \binom5r a^{2r}x^{10-3r}. \end...
(a) Find the eccentricity and foci of the hyperbola $\frac{x^2}{9}-\frac{y^2}{16}=1$. (b) Find the direction cosines of the line through $M(-2,4,3)$ and $N(-1,2,5)$. (c) Show that the points with position vectors $7\vec j+10\vec k,\ -\vec i+6\vec j+6\vec k$ and $-4\vec i+9\vec j+6\vec k$ form an isosceles triangle.
(a) $a^2=9,\ b^2=16$. $$ \begin{aligned} e &= \sqrt{1+\tfrac{b^2}{a^2}} \ &= \sqrt{1+\tfrac{16}{9}} \ &= \tfrac53. \ \text{Foci } (\pm ae,0) &= \left(\pm3\cdot\tfrac53,0\right) \ &= (\pm5,0). \end{aligned} $$ (b) $\vec{MN}=(1,-2,2)$,...
(a) Using L'Hospital's rule, evaluate $\lim_{x\to0}\frac{e^x+e^{-x}-2\cos x}{\sin^2 x}$. (b) Evaluate $\int\frac{dx}{\sqrt{2ax+x^2}}$. (c) Find a unit vector perpendicular to both $3\vec i+\vec j+2\vec k$ and $2\vec i-2\vec j+4\vec k$.
(a) This is a $0/0$ form. Differentiating numerator and denominator twice (each intermediate form is again $0/0$): $$ \begin{aligned} \lim{x\to0}\frac{e^x+e^{-x}-2\cos x}{\sin^2x} &= \lim{x\to0}\frac{e^x+e^{-x}+2\cos x}{2\cos2x} \ &= \f...
(a) Solve $\frac{dy}{dx}+4x=2e^{2x}$. (b) If $n=10,\ \Sigma x=120,\ \Sigma x^2=1530$, find the standard deviation and coefficient of variation. (c) Two coins are tossed simultaneously. Find the sample space and the probability that both are heads.
(a) $\dfrac{dy}{dx}=2e^{2x}-4x$; integrate: $$ \begin{aligned} y &= \int(2e^{2x}-4x),dx \ &= e^{2x}-2x^2+C. \end{aligned} $$ (b) $$ \begin{aligned} \bar x &= \dfrac{120}{10} \ &= 12. \ \sigma &= \sqrt{\dfrac{\Sigma x^2}{n}-\bar x^2} ...
(a) In how many ways can the letters of "COMPUTER" be arranged so that (i) all vowels are always together (ii) the relative positions of vowels and consonants are unchanged? (b) A binary operation $$ on $S={a,b,c}$ is given by a Cayley table (row/col order $a,b,c$: $a!:!a,b,c;\ b!:!b,c,a;\ c!:!c,a,b$). Show that $(S,)$ is a group. OR: Let $a,b,c$ be elements of a group $(G,)$: (i) if $ab=b$ prove $a=e$; (ii) if $a*b=e$ prove $b=a^{-1}$.
(a) COMPUTER has 8 distinct letters; vowels $O,U,E$ (3), consonants $C,M,P,T,R$ (5). (i) Treat the vowels as one block: $6!$ arrangements of block+consonants, times $3!$ within the block: $$ \begin{aligned} 6!\times3! &= 720\times6 \ &=...
(a) Find the equation of the parabola in the standard form $y^2=4ax$. OR: Find the equation of the ellipse whose distance between the two foci is 8 and semi-latus rectum is 6. (b) Find the equation of the plane through $(-1,1,1)$ and $(1,-1,1)$ and perpendicular to $x+2y+2z=5$.
(a) Let the focus be $S(a,0)$ and directrix $x=-a$. For a point $P(x,y)$ on the parabola, $SP=$ distance from directrix: $$ \begin{aligned} \sqrt{(x-a)^2+y^2} &= x+a\ \Rightarrow\ (x-a)^2+y^2 \ &= (x+a)^2\ \Rightarrow\ y^2 \ &= 4ax. \e...
(a) Evaluate $\int\frac{dx}{1-3\sin x}$. (b) Solve $xy\frac{dy}{dx}=x^2+y^2$. OR: Solve $\frac{dy}{dx}+\frac{2x}{1+x^2},y=\frac{1}{(1+x^2)^2}$.
(a) Put $t=\tan\tfrac x2$: $$ \begin{aligned} \int\frac{2,dt}{t^2-6t+1} &= \int\frac{2,dt}{(t-3)^2-8} \ &= \frac{1}{2\sqrt2}\ln\left\frac{t-3-2\sqrt2}{t-3+2\sqrt2}\right+C,\quad t=\tan\tfrac x2. \end{aligned} $$ (b) Homogeneous:
(a) Calculate Karl Pearson's correlation coefficient between height (cm) and weight (kg): Height $160,162,165,161,163$; Weight $63,62,64,60,61$. (b) In a city 60% of recorded births are male; 5 birth records are selected. Find the probability that (i) three are male (ii) more than 4 are male.
(a) With $\bar H=162.2$ and $\bar W=62$, form the deviations $h=H-\bar H$ and $k=W-\bar W$: $h=-2.2,-0.2,2.8,-1.2,0.8$ and $k=1,0,2,-2,-1.$ $$ \begin{aligned} \Sigma hk &= (-2.2)(1)+(-0.2)(0)+(2.8)(2)+(-1.2)(-2)+(0.8)(-1) \ &= 5.0, \ ...
Show that $1+\frac{1+2}{2!}+\frac{1+2+3}{3!}+\frac{1+2+3+4}{4!}+\cdots=\frac{3e}{2}$.
The $n$th term is $$ \begin{aligned} \dfrac{1+2+\cdots+n}{n!} &= \dfrac{n(n+1)/2}{n!} \ &= \dfrac{n^2+n}{2,n!}. \end{aligned} $$ So the sum is $$ \begin{aligned} S &= \frac12\sum{n=1}^{\infty}\frac{n^2+n}{n!} \ &= \frac12\left(\sum{n=...
Define scalar product of two vectors. Prove by vector method that $\cos(A+B)=\cos A\cos B-\sin A\sin B$.
Scalar product.
$$ \begin{aligned} \vec p\cdot\vec q &= |\vec p||\vec q|\cos\theta \ &= p_1q_1+p_2q_2. \end{aligned} $$
Proof. Take unit vectors $\hat u=(\cos A,\sin A)$ (angle $A$ above the $x$-axis) and $\hat v=(\cos B,-\sin B)$ (angle $B$ below the $x$-axis). The angle between them is $A+B$, so $$\hat u\cdot\hat v=\cos(A+B).$$ In components $\hat u\cdot\hat v=\cos A\cos B-\sin A\sin B.$ Hence $\cos(A+B)=\cos A\cos B-\sin A\sin B.\ \blacksquare$
State Rolle's theorem, interpret it geometrically, and verify it for $f(x)=x(x-3)^2$ on $[0,3]$. OR: Find from first principles the derivative of $\ln!\left(\sin\frac{x}{a}\right)$.
Statement. If $f$ is continuous on $[a,b]$, differentiable on $(a,b)$ and $f(a)=f(b)$, then there is $c\in(a,b)$ with $f'(c)=0.$ Geometrically, the tangent at $x=c$ is horizontal (parallel to the $x$-axis). Verification. $f(0)=0=f(3)$.
(a) Forces $7p,5p,8p$ acting on a particle are in equilibrium. Find the angle between the latter pair. (b) Two unlike parallel forces, the greater $75$ N, have a resultant $25$ N. Find the ratio of the distances of the resultant from the components. (c) A ball is thrown vertically upwards with velocity $30$ m/s. Find the time to reach the ground again ($g=10$ m/s$^2$).
(a) In equilibrium the resultant of $5p$ and $8p$ balances $7p$. If $\theta$ is the angle between $5p$ and $8p$: $$ \begin{aligned} (7p)^2 &= (5p)^2+(8p)^2+2(5p)(8p)\cos\theta\Rightarrow49 \ &= 89+80\cos\theta\Rightarrow\cos\theta \ &=...
(a) The resultant of two forces $P$ and $Q$ equals $\sqrt3,Q$ and makes $30^\circ$ with $P$. Show that $P=Q$ or $P=2Q$. OR: State and prove Lami's theorem. (b) A gun of mass $400$ kg fires a shot of mass $3$ kg with velocity $200$ m/s. Find the constant force which, acting on the gun, would stop it after a recoil of $2.5$ m.
(a) Let $\alpha$ be the angle between $P,Q$ and $R=\sqrt3Q$. Resolving perpendicular to $P$: $$ \begin{aligned} Q\sin\alpha &= R\sin30^\circ \ &= \tfrac{\sqrt3}{2}Q \ \Rightarrow\ \sin\alpha &= \tfrac{\sqrt3}{2}\Rightarrow\alpha \ &= ...
Define potential energy and kinetic energy of a body. Prove that the sum of the K.E. and P.E. of a freely falling body at any instant is constant. OR: A cannonball has the same range $R$ on a horizontal plane for two different angles of projection. If $H$ and $H'$ are the greatest heights in the two paths, prove $R^2=16HH'$.
Energy. Kinetic energy is the energy of motion, $\tfrac12mv^2$; potential energy (gravitational) is the energy of position, $mgh$.
Conservation. Drop a body from height $H$; after falling $x$ its speed is $v^2=2gx$, so
$$ \begin{aligned} \text{K.E.} &= \tfrac12m(2gx) \ &= mgx \end{aligned} $$
and P.E. $=mg(H-x)$.
$$ \begin{aligned} \text{Total} &= mgx+mg(H-x) \ &= mgH, \end{aligned} $$
a constant independent of $x$.
OR (range). Equal range occurs for complementary angles $\theta,,90^\circ-\theta$. $H=\dfrac{u^2\sin^2\theta}{2g},\ H'=\dfrac{u^2\cos^2\theta}{2g}$, so $HH'=\dfrac{u^4\sin^2\theta\cos^2\theta}{4g^2}.$ Also
$$ \begin{aligned} R &= \dfrac{u^2\sin2\theta}{g} \ &= \dfrac{2u^2\sin\theta\cos\theta}{g}, \end{aligned} $$
giving
$$ \begin{aligned} R^2 &= \dfrac{4u^4\sin^2\theta\cos^2\theta}{g^2} \ &= 16HH'.\ \blacksquare \end{aligned} $$
Define moment of a force about a point. Prove that the algebraic sum of the moments of two intersecting forces about any point in their plane equals the moment of their resultant about the same point (Varignon's theorem).
Moment. The moment of a force about a point is force $\times$ perpendicular distance from the point to the line of action. Varignon's theorem. Let two forces meet at $O'$ and let $\vec R$ be their resultant (also through $O'$). Take mome...
(a) Determine graphically the feasible region for $3x+4y\le24,\ x\ge2,\ y\ge1$. (b) Convert the hexadecimal number $AB5_{16}$ to decimal. (c) Using the trapezoidal rule, evaluate $\int_0^2(2x^2-1),dx$ with $n=4$.
(a) The lines $3x+4y=24,\ x=2,\ y=1$ bound the region. Corner points: $(2,1),\ (2,4.5)$ (from $x=2$ on $3x+4y=24$), and $\left(\tfrac{20}{3},1\right)$ (from $y=1$). Shade the triangle enclosed by these vertices. (b) $$ \begin{aligned} AB...
(a) Solve by Gauss elimination: $x-2y+3z=2,\ 2x-3y+z=1,\ 3x-y+2z=9$. OR: Solve by the inverse-matrix method: $x_1-2x_2-x_3=1,\ x_1-x_2+2x_3=9,\ 2x_1-3x_2-x_3=4$. (b) Estimate $\int_0^{\pi}\sin x,dx$ using Simpson's $\tfrac13$ rule with $n=6$.
(a) Eliminating $x$: R2$-2$R1 gives $y-5z=-3$; R3$-3$R1 gives $5y-7z=3$. Then $5(5z-3)-7z=3\Rightarrow18z=18\Rightarrow z=1,\ y=2$, $$ \begin{aligned} x &= 2+2(2)-3(1) \ &= 3. \ \boxed{x=3,\ y=2,\ z=1.} \end{aligned} $$ OR (inverse mat...
Using the Simplex method, maximize $U=25x+45y$ subject to $x+3y\le21,\ 2x+3y\le24,\ x,y\ge0$.
Introduce slacks and evaluate the corner points of the feasible region: $$(0,0)\to0,\quad(12,0)\to300,\quad(0,7)\to315,\quad(3,6)\to345.$$ The binding constraints $x+3y=21$ and $2x+3y=24$ meet at $x=3,\ y=6$ (subtract to get $x=3$, then ...
Using the bisection method, find a root of $2x^3-5x+2=0$ between $1$ and $2$ with error less than $10^{-2}$. OR: Derive the Newton-Raphson formula and use it to find a positive root of $x^3+3x-5=0$ in $(1,2)$ correct to $3$ decimal places.
Bisection. $f(x)=2x^3-5x+2$; $f(1)=-1<0,\ f(2)=80.$ Midpoints: $$ \begin{aligned} f(1.5) &= 1.25\ (\to[1,1.5]) \ \ f(1.25) &= -0.344 \ \ f(1.375) &= +0.320 \ \ f(1.3125) &= -0.041 \ \ f(1.34375) &= +0.135. \end{aligned} $$ The interv...
(a) A man has 5 friends. In how many ways can he invite one or more of them to dinner? (b) Find the coefficient of in the expansion of . (c) Show that multiplication is a binary operation on .
(a) Each friend is either invited or not; excluding the case of none, the number of ways is (b) General term $$ \begin{aligned} T{r+1} &= \binom5r(x^2)^{5-r}\left(\tfrac{a^2}{x}\right)^r \ &= \binom5r a^{2r}x^{10-3r}. \end...
(a) Find the eccentricity and foci of the hyperbola . (b) Find the direction cosines of the line through and . (c) Show that the points with position vectors and form an isosceles triangle.
(a) . (b) ,...
(a) Using L'Hospital's rule, evaluate . (b) Evaluate . (c) Find a unit vector perpendicular to both and .
(a) This is a form. Differentiating numerator and denominator twice (each intermediate form is again ): $$ \begin{aligned} \lim{x\to0}\frac{e^x+e^{-x}-2\cos x}{\sin^2x} &= \lim{x\to0}\frac{e^x+e^{-x}+2\cos x}{2\cos2x} \ &= \f...
(a) Solve . (b) If , find the standard deviation and coefficient of variation. (c) Two coins are tossed simultaneously. Find the sample space and the probability that both are heads.
(a) ; integrate: (b) $$ \begin{aligned} \bar x &= \dfrac{120}{10} \ &= 12. \ \sigma &= \sqrt{\dfrac{\Sigma x^2}{n}-\bar x^2} ...
(a) In how many ways can the letters of "COMPUTER" be arranged so that (i) all vowels are always together (ii) the relative positions of vowels and consonants are unchanged? (b) A binary operation on is given by a Cayley table (row/col order : ). Show that is a group. OR: Let be elements of a group : (i) if prove ; (ii) if prove .
(a) COMPUTER has 8 distinct letters; vowels (3), consonants (5). (i) Treat the vowels as one block: arrangements of block+consonants, times within the block: $$ \begin{aligned} 6!\times3! &= 720\times6 \ &=...
(a) Find the equation of the parabola in the standard form . OR: Find the equation of the ellipse whose distance between the two foci is 8 and semi-latus rectum is 6. (b) Find the equation of the plane through and and perpendicular to .
(a) Let the focus be and directrix . For a point on the parabola, distance from directrix: $$ \begin{aligned} \sqrt{(x-a)^2+y^2} &= x+a\ \Rightarrow\ (x-a)^2+y^2 \ &= (x+a)^2\ \Rightarrow\ y^2 \ &= 4ax. \e...
(a) Evaluate . (b) Solve . OR: Solve .
(a) Put : \begin{aligned} \int\frac{2\,dt}{t^2-6t+1} &= \int\frac{2\,dt}{(t-3)^2-8} \\ &= \frac{1}{2\sqrt2}\ln\left\frac{t-3-2\sqrt2}{t-3+2\sqrt2}\right+C,\quad t=\tan\tfrac x2. \end{aligned} (b) Homogeneous:
(a) Calculate Karl Pearson's correlation coefficient between height (cm) and weight (kg): Height ; Weight . (b) In a city 60% of recorded births are male; 5 birth records are selected. Find the probability that (i) three are male (ii) more than 4 are male.
(a) With and , form the deviations and : and $$ \begin{aligned} \Sigma hk &= (-2.2)(1)+(-0.2)(0)+(2.8)(2)+(-1.2)(-2)+(0.8)(-1) \ &= 5.0, \ ...
Show that .
The th term is So the sum is $$ \begin{aligned} S &= \frac12\sum{n=1}^{\infty}\frac{n^2+n}{n!} \ &= \frac12\left(\sum{n=...
Define scalar product of two vectors. Prove by vector method that .
Scalar product.
Proof. Take unit vectors (angle above the -axis) and (angle below the -axis). The angle between them is , so In components Hence
State Rolle's theorem, interpret it geometrically, and verify it for on . OR: Find from first principles the derivative of .
Statement. If is continuous on , differentiable on and , then there is with Geometrically, the tangent at is horizontal (parallel to the -axis). Verification. .
(a) Forces acting on a particle are in equilibrium. Find the angle between the latter pair. (b) Two unlike parallel forces, the greater N, have a resultant N. Find the ratio of the distances of the resultant from the components. (c) A ball is thrown vertically upwards with velocity m/s. Find the time to reach the ground again ( m/s).
(a) In equilibrium the resultant of and balances . If is the angle between and : $$ \begin{aligned} (7p)^2 &= (5p)^2+(8p)^2+2(5p)(8p)\cos\theta\Rightarrow49 \ &= 89+80\cos\theta\Rightarrow\cos\theta \ &=...
(a) The resultant of two forces and equals and makes with . Show that or . OR: State and prove Lami's theorem. (b) A gun of mass kg fires a shot of mass kg with velocity m/s. Find the constant force which, acting on the gun, would stop it after a recoil of m.
(a) Let be the angle between and . Resolving perpendicular to : $$ \begin{aligned} Q\sin\alpha &= R\sin30^\circ \ &= \tfrac{\sqrt3}{2}Q \ \Rightarrow\ \sin\alpha &= \tfrac{\sqrt3}{2}\Rightarrow\alpha \ &= ...
Define potential energy and kinetic energy of a body. Prove that the sum of the K.E. and P.E. of a freely falling body at any instant is constant. OR: A cannonball has the same range on a horizontal plane for two different angles of projection. If and are the greatest heights in the two paths, prove .
Energy. Kinetic energy is the energy of motion, ; potential energy (gravitational) is the energy of position, .
Conservation. Drop a body from height ; after falling its speed is , so
and P.E. .
a constant independent of .
OR (range). Equal range occurs for complementary angles . , so Also
giving
Moment. The moment of a force about a point is force perpendicular distance from the point to the line of action. Varignon's theorem. Let two forces meet at and let be their resultant (also through ). Take mome...
(a) Determine graphically the feasible region for . (b) Convert the hexadecimal number to decimal. (c) Using the trapezoidal rule, evaluate with .
(a) The lines bound the region. Corner points: (from on ), and (from ). Shade the triangle enclosed by these vertices. (b) $$ \begin{aligned} AB...
(a) Solve by Gauss elimination: . OR: Solve by the inverse-matrix method: . (b) Estimate using Simpson's rule with .
(a) Eliminating : R2R1 gives ; R3R1 gives . Then , OR (inverse mat...
Using the Simplex method, maximize subject to .
Introduce slacks and evaluate the corner points of the feasible region: The binding constraints and meet at (subtract to get , then ...
Using the bisection method, find a root of between and with error less than . OR: Derive the Newton-Raphson formula and use it to find a positive root of in correct to decimal places.
Bisection. ; Midpoints: The interv...