NEB Class 12 · Past paper
The complete NEB Class 12 2070 exam paper for Mathematics, all 19 questions with solved model answers.
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(a) In how many ways can the letters of the word "ELEMENT" be arranged? (b) Show that $\log_e 2 = \frac{1}{1\cdot2}+\frac{1}{3\cdot4}+\frac{1}{5\cdot6}+\cdots$ (c) Show that multiplication is a binary operation on the set $S={-1,0,1}$.
(a) ELEMENT has 7 letters with E repeated 3 times. $$ \begin{aligned} N &= \frac{7!}{3!} \ &= \frac{5040}{6} \ &= 840. \end{aligned} $$ (b) Using $\loge(1+x)=x-\tfrac{x^2}{2}+\tfrac{x^3}{3}-\cdots$ at $x=1$:
(a) Find the eccentricity and foci of the ellipse $3x^2+4y^2=36$. (b) Find the equation of the plane which makes equal intercepts on the axes and passes through $(2,3,4)$. (c) The vertices $A,B,C$ of a triangle are $(2,-1,-3),(4,2,3)$ and $(6,3,4)$. Show that $\vec{AB}=(2,3,6)$ and $AC=9$.
(a) Divide by 36: $\dfrac{x^2}{12}+\dfrac{y^2}{9}=1$, so $a^2=12,\ b^2=9$. $$ \begin{aligned} e &= \sqrt{1-\frac{b^2}{a^2}} \ &= \sqrt{1-\frac{9}{12}} \ &= \sqrt{\tfrac14} \ &= \frac12. \ \text{Foci } (\pm ae,0) &= \left(\pm 2\sqrt3...
(a) Find the equation of the tangent to the curve $y=2x^3-5x^2+8$ at $(2,4)$. (b) Evaluate $\int\frac{dx}{\sqrt{(x-\alpha)(x-\beta)}},\ (\beta>\alpha)$. (c) Find the sine of the angle between the vectors $2\vec i-\vec j+\vec k$ and $3\vec i+4\vec j-\vec k$.
(a) $\dfrac{dy}{dx}=6x^2-10x$; at $x=2$, $$ \begin{aligned} \text{slope} &= 24-20 \ &= 4. \end{aligned} $$ Tangent: $$ \begin{aligned} y-4 &= 4(x-2)\ \Rightarrow\ y \ &= 4x-4. \end{aligned} $$ (b) Complete the square:
(a) Solve $x^2,dy-y^2,dx=0$. (b) If $n=10,\ \Sigma X=60,\ \Sigma Y=60,\ \Sigma X^2=400,\ \Sigma Y^2=580,\ \Sigma XY=415$, find the correlation coefficient. (c) A card is drawn from a well-shuffled deck of 52. Find the probability that it is a King or a Diamond.
(a) Separate: $\dfrac{dy}{y^2}=\dfrac{dx}{x^2}\Rightarrow -\dfrac1y=-\dfrac1x+c$, i.e. $$\frac1x-\frac1y=C.$$ (b) $$ \begin{aligned} r &= \frac{n\Sigma XY-\Sigma X,\Sigma Y}{\sqrt{(n\Sigma X^2-(\Sigma X)^2)(n\Sigma Y^2-(\Sigma Y)^2)}} ...
From 6 gentlemen and 4 ladies a committee of 5 is to be formed. In how many ways can this be done so as to include at least two gentlemen? OR: If $a,b$ are elements of a group $(G,\circ)$ prove that the equation $a\circ x=b$ has a unique solution in $(G,\circ)$.
Committee. $$ \begin{aligned} \text{Total ways} &= \binom{10}{5} \ &= 252. \end{aligned} $$ Exclude committees with fewer than 2 gentlemen: 0 gentlemen needs 5 ladies from 4 (impossible, $\binom{4}{5}=0$); 1 gentleman gives
(a) Prove that the line $3x+4y+6=0$ is tangent to the parabola $2y^2=9x$ and find its point of contact. OR: Deduce the equation of a hyperbola with a focus at $(6,0)$ and a vertex at $(4,0)$. (b) Find the angle between two straight lines whose direction cosines are $l_1,m_1,n_1$ and $l_2,m_2,n_2$.
(a) Parabola $y^2=\tfrac92x$, so $4a=\tfrac92,\ a=\tfrac98$. Line: $y=-\tfrac34x-\tfrac32$, giving $m=-\tfrac34,\ c=-\tfrac32$. Tangency condition $c=\dfrac{a}{m}$: $$ \begin{aligned} \dfrac{9/8}{-3/4} &= -\dfrac32 \ &= c \end{aligned} ...
(a) Evaluate $\int\frac{dx}{1+2\sin x}$. (b) Solve $(1+x^2)\frac{dy}{dx}+2xy=4x^2$. OR: Solve $(x^2+y^2),dy=xy,dx$.
(a) Put $t=\tan\tfrac x2$, $\sin x=\dfrac{2t}{1+t^2}$, $dx=\dfrac{2,dt}{1+t^2}$: $$ \begin{aligned} \int\frac{2,dt}{t^2+4t+1} &= \int\frac{2,dt}{(t+2)^2-3} \ &= \frac{1}{\sqrt3}\ln\left\frac{t+2-\sqrt3}{t+2+\sqrt3}\right+C,\quad t=\t...
(a) Distribution A: mean 100, median 90, S.D. 10. Distribution B: mean 90, median 80, S.D. 10. Is A the same as B regarding degree of variation and skewness? (b) A coin is tossed 5 times. Find the probability of getting (i) two heads (ii) at least two heads.
(a) Coefficient of variation $=\dfrac{\sigma}{\bar x}\times100$: $$ \begin{aligned} CVA &= \frac{10}{100}\times100 \ &= 10%, \ CVB &= \frac{10}{90}\times100 \ &= 11.11%. \end{aligned} $$ So B is more variable (less consistent). Pear...
If $(1+x)^n=C_0+C_1x+C_2x^2+\cdots+C_nx^n$, prove that $C_0C_n+C_1C_{n-1}+\cdots+C_nC_0=\dfrac{2n!}{(n!)^2}$.
Consider the identity $(1+x)^n(1+x)^n=(1+x)^{2n}.$ Left side: $\left(\sum{r=0}^n Cr x^r\right)\left(\sum{s=0}^n Cs x^s\right)$. The coefficient of $x^n$ collects all $r+s=n$: $$C0Cn+C1C{n-1}+C2C{n-2}+\cdots+CnC0.$$ Right side: coefficien...
Define scalar product of two vectors. Prove by vector method that $\cos(A-B)=\cos A\cos B+\sin A\sin B$.
Scalar product. For vectors $\vec p,\vec q$ with angle $\theta$ between them, $\vec p\cdot\vec q=|\vec p||\vec q|\cos\theta$; in components $\vec p\cdot\vec q=p_1q_1+p_2q_2.$
Proof. Take unit vectors in the plane making angles $A$ and $B$ with the $x$-axis: $$ \begin{aligned} \hat u &= (\cos A,\sin A) \ \qquad \hat v &= (\cos B,\sin B). \end{aligned} $$ The angle between them is $A-B$, so
$$ \begin{aligned} \hat u\cdot\hat v &= |\hat u||\hat v|\cos(A-B) \ &= \cos(A-B). \end{aligned} $$
But in components $\hat u\cdot\hat v=\cos A\cos B+\sin A\sin B.$ Therefore $$\cos(A-B)=\cos A\cos B+\sin A\sin B.\qquad\blacksquare$$
State the mean value theorem, interpret it geometrically, and verify it for $f(x)=x(x-1)^2$ on $[0,2]$. OR: Find from first principles the derivative of $x^x$.
Statement. If $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$, there exists $c\in(a,b)$ with $f'(c)=\dfrac{f(b)-f(a)}{b-a}.$ Geometrically, the tangent at $x=c$ is parallel to the chord joining $(a,f(a))$ and $(b,f(b)).$ Verif...
(a) Two forces acting at $45^\circ$ have a resultant $\sqrt{10}$ N; if one force is $\sqrt2$ N, find the other. (b) Find two like parallel forces $2.5$ m apart equivalent to a force of $30$ N, one line of action being $50$ cm from the given force. (c) A ball is projected vertically upwards with velocity $40$ m/s. Find its velocity and position after $3$ s ($g=10$ m/s$^2$).
(a) $R^2=P^2+Q^2+2PQ\cos45^\circ$ with $R=\sqrt{10},P=\sqrt2$: $$ \begin{aligned} 10 &= 2+Q^2+2\sqrt2,Q\cdot\tfrac{1}{\sqrt2} \ &= 2+Q^2+2Q \ \Rightarrow\ Q^2+2Q-8 &= 0\ \Rightarrow\ Q \ &= 2\text{ N}. \end{aligned} $$ (b) Let forces...
(a) A body of weight $65$ N is suspended by two strings of lengths $5$ and $12$ m attached to two points in the same horizontal line $13$ m apart; find the tensions. OR: State and prove Lami's theorem. (b) If $a,b,c$ are the spaces described by a particle during the $p$th, $q$th, $r$th seconds, prove $a(q-r)+b(r-p)+c(p-q)=0$.
(a) Since $5^2+12^2=13^2$, the strings are perpendicular at the body. Resolving, the tension in a string equals $W\times\dfrac{\text{other side}}{\text{hypotenuse}}$: $$ \begin{aligned} T5 &= 65\cdot\frac{12}{13} \ &= 60\text{ N}, \ T{...
A projectile thrown from a point on a horizontal plane returns to the plane in $4$ s at a distance $60$ m in front of the point of projection. Find the velocity of projection ($g=10$ m/s$^2$). OR: A bullet of mass $20$ g is fired horizontally into a suspended stationary block of mass $380$ g with velocity $200$ m/s. Find the common velocity if the bullet embeds, and the loss of K.E. ($g=10$ m/s$^2$).
Projectile. Time of flight $$ \begin{aligned} T &= \dfrac{2u\sin\theta}{g} \ &= 4 \ \Rightarrow\ u\sin\theta &= 20. \end{aligned} $$ Range $$ \begin{aligned} R &= u\cos\theta\cdot T \ &= 60 \ \Rightarrow\ u\cos\theta &= 15. \ u &= ...
Define moment of a force about a point. Prove that the algebraic sum of the moments of two like parallel forces about any point in their plane equals the moment of their resultant about the same point.
Moment. The moment of a force about a point is the product of the force and the perpendicular distance from the point to the line of action; it measures the turning effect. Theorem. Let like parallel forces $P,Q$ act at $A,B$; their resu...
(a) Draw the graph of $x+y\le6,\ 2x+y\ge8,\ y\ge0$ and shade the feasible region. (b) Convert the decimal number $3058$ to hexadecimal. (c) Using the trapezoidal rule, evaluate $\int_0^3(3x^2-4x),dx$ with $n=3$.
(a) Boundary lines $x+y=6$ and $2x+y=8$ meet at $(2,4)$; with $y\ge0$ the feasible region is bounded by $y=0$, lies below $x+y=6$ and above $2x+y=8$. Corner points $(4,0),(6,0),(2,4)$; shade the triangle they enclose. (b) Repeated divisi...
(a) Solve by Gaussian elimination: $x+3y-2z=5,\ 3x+5y+6z=7,\ 2x+4y+3z=8$. OR: Solve by Gauss-Seidel: $3x+y-z=2,\ 2x-5y+z=20,\ x-3y-8z=3$. (b) Using Simpson's $\tfrac13$ rule, evaluate $\int_0^1\frac{dx}{1+x}$ with $n=4$.
(a) Eliminate $x$: R2$-3$R1 and R3$-2$R1 give $-4y+12z=-8\Rightarrow -y+3z=-2$ and $-2y+7z=-2$. Solving these, $z=2,\ y=8$; back-substitute $$ \begin{aligned} x &= 5-3(8)+2(2) \ &= -15. \ \boxed{x=-15,\ y=8,\ z=2.} \end{aligned} $$ OR ...
Using the Simplex method, maximize $P=50x_1+80x_2$ subject to $x_1+2x_2\le32,\ 3x_1+4x_2\le84,\ x_1,x_2\ge0$.
Introduce slacks: $x1+2x2+s1=32,\ 3x1+4x2+s2=84.$ Starting from the origin, $x2$ enters (most negative reduced cost $-80$); the ratio test ${32/2,84/4}={16,21}$ makes $s1$ leave, so $x2=16$. Then $x1$ enters and the pivot moves to wh...
Applying the bisection method, find the root of $x^3-4x+1=0$ lying between $1$ and $2$ correct to $2$ decimal places. OR: Using the Newton-Raphson method, find the positive root of $x^3-18=0$ in $(2,3)$.
Bisection. $f(x)=x^3-4x+1$; $f(1)=-2<0,\ f(2)=10.$ Successive midpoints: $$ \begin{aligned} &f(1.5)=-1.625\ (\to[1.5,2]),\ f(1.75)=-0.641\ (\to[1.75,2]),\ &f(1.875)=+0.092\ (\to[1.75,1.875]),\ f(1.8125)=-0.297,\ f(1.8438)=-0.108,\ f(1.8...
(a) In how many ways can the letters of the word "ELEMENT" be arranged? (b) Show that (c) Show that multiplication is a binary operation on the set .
(a) ELEMENT has 7 letters with E repeated 3 times. (b) Using at :
(a) Find the eccentricity and foci of the ellipse . (b) Find the equation of the plane which makes equal intercepts on the axes and passes through . (c) The vertices of a triangle are and . Show that and .
(a) Divide by 36: , so . $$ \begin{aligned} e &= \sqrt{1-\frac{b^2}{a^2}} \ &= \sqrt{1-\frac{9}{12}} \ &= \sqrt{\tfrac14} \ &= \frac12. \ \text{Foci } (\pm ae,0) &= \left(\pm 2\sqrt3...
(a) Find the equation of the tangent to the curve at . (b) Evaluate . (c) Find the sine of the angle between the vectors and .
(a) ; at , Tangent: (b) Complete the square:
(a) Solve . (b) If , find the correlation coefficient. (c) A card is drawn from a well-shuffled deck of 52. Find the probability that it is a King or a Diamond.
(a) Separate: , i.e. (b) $$ \begin{aligned} r &= \frac{n\Sigma XY-\Sigma X,\Sigma Y}{\sqrt{(n\Sigma X^2-(\Sigma X)^2)(n\Sigma Y^2-(\Sigma Y)^2)}} ...
From 6 gentlemen and 4 ladies a committee of 5 is to be formed. In how many ways can this be done so as to include at least two gentlemen? OR: If are elements of a group prove that the equation has a unique solution in .
Committee. Exclude committees with fewer than 2 gentlemen: 0 gentlemen needs 5 ladies from 4 (impossible, ); 1 gentleman gives
(a) Prove that the line is tangent to the parabola and find its point of contact. OR: Deduce the equation of a hyperbola with a focus at and a vertex at . (b) Find the angle between two straight lines whose direction cosines are and .
(a) Parabola , so . Line: , giving . Tangency condition : $$ \begin{aligned} \dfrac{9/8}{-3/4} &= -\dfrac32 \ &= c \end{aligned} ...
(a) Evaluate . (b) Solve . OR: Solve .
(a) Put , , : $$ \begin{aligned} \int\frac{2,dt}{t^2+4t+1} &= \int\frac{2,dt}{(t+2)^2-3} \ &= \frac{1}{\sqrt3}\ln\left\frac{t+2-\sqrt3}{t+2+\sqrt3}\right+C,\quad t=\t...
(a) Coefficient of variation : So B is more variable (less consistent). Pear...
If , prove that .
Consider the identity Left side: . The coefficient of collects all : Right side: coefficien...
Define scalar product of two vectors. Prove by vector method that .
Scalar product. For vectors with angle between them, ; in components
Proof. Take unit vectors in the plane making angles and with the -axis:
The angle between them is , so
But in components Therefore
State the mean value theorem, interpret it geometrically, and verify it for on . OR: Find from first principles the derivative of .
Statement. If is continuous on and differentiable on , there exists with Geometrically, the tangent at is parallel to the chord joining and Verif...
(a) Two forces acting at have a resultant N; if one force is N, find the other. (b) Find two like parallel forces m apart equivalent to a force of N, one line of action being cm from the given force. (c) A ball is projected vertically upwards with velocity m/s. Find its velocity and position after s ( m/s).
(a) with : (b) Let forces...
(a) A body of weight N is suspended by two strings of lengths and m attached to two points in the same horizontal line m apart; find the tensions. OR: State and prove Lami's theorem. (b) If are the spaces described by a particle during the th, th, th seconds, prove .
(a) Since , the strings are perpendicular at the body. Resolving, the tension in a string equals : $$ \begin{aligned} T5 &= 65\cdot\frac{12}{13} \ &= 60\text{ N}, \ T{...
A projectile thrown from a point on a horizontal plane returns to the plane in s at a distance m in front of the point of projection. Find the velocity of projection ( m/s). OR: A bullet of mass g is fired horizontally into a suspended stationary block of mass g with velocity m/s. Find the common velocity if the bullet embeds, and the loss of K.E. ( m/s).
Projectile. Time of flight Range $$ \begin{aligned} R &= u\cos\theta\cdot T \ &= 60 \ \Rightarrow\ u\cos\theta &= 15. \ u &= ...
Moment. The moment of a force about a point is the product of the force and the perpendicular distance from the point to the line of action; it measures the turning effect. Theorem. Let like parallel forces act at ; their resu...
(a) Draw the graph of and shade the feasible region. (b) Convert the decimal number to hexadecimal. (c) Using the trapezoidal rule, evaluate with .
(a) Boundary lines and meet at ; with the feasible region is bounded by , lies below and above . Corner points ; shade the triangle they enclose. (b) Repeated divisi...
(a) Solve by Gaussian elimination: . OR: Solve by Gauss-Seidel: . (b) Using Simpson's rule, evaluate with .
(a) Eliminate : R2R1 and R3R1 give and . Solving these, ; back-substitute OR ...
Using the Simplex method, maximize subject to .
Introduce slacks: Starting from the origin, enters (most negative reduced cost ); the ratio test makes leave, so . Then enters and the pivot moves to wh...
Applying the bisection method, find the root of lying between and correct to decimal places. OR: Using the Newton-Raphson method, find the positive root of in .
Bisection. ; Successive midpoints: $$ \begin{aligned} &f(1.5)=-1.625\ (\to[1.5,2]),\ f(1.75)=-0.641\ (\to[1.75,2]),\ &f(1.875)=+0.092\ (\to[1.75,1.875]),\ f(1.8125)=-0.297,\ f(1.8438)=-0.108,\ f(1.8...