NEB Class 11 · Past paper
The complete NEB Class 11 2073 exam paper for Mathematics, all 15 questions with solved model answers.
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(a) Write |x + 2| < 4 without using modulus sign. (b) Find the domain and range of the relation R = {(1, 1), (2, 2), (4, 4)}. What type of relation is this? (c) Examine the symmetry and even or odd nature of the function y = x^3.
(a) $x+2<4\iff -4<x+2<4\iff -6<x<2.$ (b) $R={(1,1),(2,2),(4,4)}$. Domain $={1,2,4}$ (first components); Range $={1,2,4}$ (second components). Each element maps to itself, so $R$ is the identity relation (which is reflexive, symmetr...
(a) Solve: cot^2(x) + cosec^2(x) = 3 (-pi/2 < x < pi/2). (b) Using the principle of Mathematical induction prove that 1 + 3 + 5 + ... to n terms = n^2. (c) If A=(2 3; 4 5) and B=(1 3; 5 7), verify that (A+B)^t = A^t + B^t.
(a) Use $\csc^2x=1+\cot^2x$: $$ \begin{aligned} \cot^2x+1+\cot^2x &= 3\Rightarrow 2\cot^2x \ &= 2\Rightarrow\cot^2x \ &= 1\Rightarrow\cot x \ &= \pm1. \end{aligned} $$ Thus $\tan x=\pm1$. In
(a) Use Cramer's rule to solve 3x + 2y = 8 and 4x + y = 9. (b) Find the conjugate of 1 + i + i^2 + i^3 + i^4. (c) Find the value of P so that equation 5x^2 - Px + 16 = 0 has equal roots.
(a) $D=\begin{vmatrix}3&2\4&1\end{vmatrix}=3-8=-5,\ Dx=\begin{vmatrix}8&2\9&1\end{vmatrix}=8-18=-10,\ Dy=\begin{vmatrix}3&8\4&9\end{vmatrix}=27-32=-5.$ $$ \begin{aligned} x &= \frac{-10}{-5} \ &= 2, \ \qquad y &= \frac{-5}{-5} \ &=...
(a) Determine the value of K so that the line kx + 3y + 10 = 0 will be perpendicular to the line 3x - 2y = 5. (b) Find the equation to the circle which touches the coordinate axes at (a, 0) and (0, a). (c) Find the limit of f(x) = (x^2 - 4)/(x - 2) as x -> 2. Is f(x) continuous? If not, find the point of discontinuity.
(a) Slope of $kx+3y+10=0$ is $-\dfrac{k}{3}$; slope of $3x-2y=5$ is $\dfrac32$. For perpendicularity the product of slopes $=-1$: $$ \begin{aligned} -\frac{k}{3}\cdot\frac32 &= -1\Rightarrow -\frac{k}{2} \ &= -1\Rightarrow \boxed{k=2}. ...
(a) Find dy/dx when y = (x+1)(x+2)(x+3). (b) Evaluate: integral of x dx / (1 - x^2)^(3/2). (c) Find the interval in which the function f(x) = 3x^2 - 6x + 5 is increasing or decreasing.
(a) By the product rule, $$ \begin{aligned} \frac{dy}{dx} &= (x+2)(x+3)+(x+1)(x+3)+(x+1)(x+2) \ &= 3x^2+12x+11. \end{aligned} $$
(b) Let $u=1-x^2\Rightarrow du=-2x,dx$. $$ \begin{aligned} \int\frac{x,dx}{(1-x^2)^{3/2}} &= -\frac12\int u^{-3/2},du \ &= -\frac12\cdot\frac{u^{-1/2}}{-\tfrac12} \ &= u^{-1/2}+C \ &= \frac{1}{\sqrt{1-x^2}}+C. \end{aligned} $$
(c) $f'(x)=6x-6=6(x-1)$.
(a) Define conditional statement. Compute the truth table of the statement (p => q) <=> (~q => ~p). What conclusion can be drawn about the statement from the truth table? OR Define complement of a set. If A and B are the subsets of the universal set U, prove that (A U B)' = A' n B'. (b) Draw the graph of f(x) = x^2 - 6x + 5 indicating its characteristics.
(a) A conditional statement $p\Rightarrow q$ ("if $p$ then $q$") is false only when $p$ is true and $q$ is false; otherwise true. $p$ $q$ $p\Rightarrow q$ $\sim q$ $\sim p$ $\sim q\Rightarrow\sim p$
(a) State and prove Sine law in any triangle. OR Prove that sin(2 sin^-1 x) = 2x sqrt(1 - x^2). (b) Without expanding show that determinant |b c b+c; c a c+a; a b a+b| = 0.
(a) Sine law: In any triangle $ABC$, $\dfrac{a}{\sin A}=\dfrac{b}{\sin B}=\dfrac{c}{\sin C}$. Proof. Drop the perpendicular $AD=h$ from $A$ to $BC$. In right triangle $ABD$, $h=c\sin B$; in $ACD$, $h=b\sin C$. Hence
(a) Using row equivalent or inverse matrix method, solve: x + z = 1, 2y + z = 2, 5x - 9y + 3 = 0. (b) Form the equation whose roots are reciprocal of the roots of x^2 - x + 1 = 0.
(a) From $x+z=1$: $z=1-x$. Substitute in $2y+z=2$: $2y+1-x=2\Rightarrow x=2y-1$. Substitute in $5x-9y+3=0\ (\Rightarrow 5x-9y=-3)$: $5(2y-1)-9y=-3\Rightarrow 10y-5-9y=-3\Rightarrow y=2$. Then $x=2(2)-1=3$ and $z=1-3=-2$. $$\boxed{x=3,\ y...
(a) For what value of 'a', the line x + 3y = a touches the circle x^2 + y^2 - 3x - 3y + 2 = 0. Find also the point of contact. (b) Evaluate: lim(x -> theta) (x cos theta - theta cos x)/(x - theta). OR A function f(x) is defined by f(x) = 3 + x when -3/2 <= x < 0, 3 - x when 0 <= x < 3/2, -3 + x when x >= 3/2. Test the continuity of f(x) at x = 0 and x = 3/2.
(a) Circle: centre $\left(\tfrac32,\tfrac32\right)$, radius $r=\sqrt{\tfrac94+\tfrac94-2}=\sqrt{\tfrac{10}{4}}=\dfrac{\sqrt{10}}{2}$. Tangency: distance from centre to $x+3y-a=0$ equals $r$: $$ \begin{aligned} \frac{\left\tfrac32+\tfrac9...
(a) Find from first principles the derivative of f(x) = sqrt(sin 2x). (b) Using integration, find the area of the circle x^2 + y^2 = 36.
(a) $f(x)=\sqrt{\sin2x}$. $$ \begin{aligned} f'(x) &= \lim{h\to0}\frac{\sqrt{\sin2(x+h)}-\sqrt{\sin2x}}{h} \ &= \lim{h\to0}\frac{\sin2(x+h)-\sin2x}{h\big(\sqrt{\sin2(x+h)}+\sqrt{\sin2x}\big)}. \end{aligned} $$ Now
Let the function f(x) = x^3 and g(x) = Sin x, x in R. Find fog and gof. Is fog = gof? Examine whether f is one to one and onto or not.
$$ \begin{aligned} (f\circ g)(x) &= f\big(\sin x\big) \ &= (\sin x)^3 \ &= \sin^3x. \ (g\circ f)(x) &= g\big(x^3\big) \ &= \sin\big(x^3\big). \end{aligned} $$ In general $\sin^3x\neq\sin(x^3)$ (e.g. at $x=1$: $\sin^3 1\approx0.596$ b...
Find the general term and then find the sum of first n terms of the series n + 2(n-1) + 3(n-2) + ...
The $r$-th term is $tr=r\big(n-(r-1)\big)=r(n-r+1)=r(n+1)-r^2$. $$ \begin{aligned} Sn &= \sum{r=1}^{n}\big[r(n+1)-r^2\big] \ &= (n+1)\sum{r=1}^{n}r-\sum{r=1}^{n}r^2. \ &= (n+1)\cdot\frac{n(n+1)}{2}-\frac{n(n+1)(2n+1)}{6} \ &= \frac{n(...
If P1 and P2 be the lengths of the perpendicular drawn from the points (Cos theta, Sin theta) and (-Sec theta, Cosec theta) on the line x Sec theta + y Cosec theta = 0 respectively, prove that 4/P1^2 - P2^2 = 4. OR If the pair of lines x^2 - 2pxy - y^2 = 0 and x^2 - 2qxy - y^2 = 0 be such that each pair bisects the angles between the other pair, prove that pq = -1.
Main. The line is $x\sec\theta+y\csc\theta=0$, with $\sqrt{\sec^2\theta+\csc^2\theta}=\dfrac{1}{\sin\theta\cos\theta}$ (since $\sec^2\theta+\csc^2\theta=\sec^2\theta\csc^2\theta$). $$ \begin{aligned} P1 &= \frac{\cos\theta\sec\theta+\sin...
State De Moivre's theorem for any positive index n. Using De Moivre's theorem find the square roots of 4 + 4 sqrt(3) i.
De Moivre's theorem. For any positive integer $n$, $(\cos\theta+i\sin\theta)^n=\cos n\theta+i\sin n\theta$. Write $z=4+4\sqrt3,i$ in polar form: $r=\sqrt{4^2+(4\sqrt3)^2}=\sqrt{16+48}=8$, and
Determine the maximum and minimum value of f(x) = 2x^3 - 9x^2 + 12x - 4. Also find the point of inflection, if any. OR If the radius of a circle increases at a uniform rate, prove that its area will increase at a rate which varies as its radius.
Main. $f(x)=2x^3-9x^2+12x-4$. $$ \begin{aligned} f'(x) &= 6x^2-18x+12 \ &= 6(x-1)(x-2) \ &= 0\Rightarrow x \ &= 1 \ \ 2.\qquad f''(x) &= 12x-18. \end{aligned} $$ At $x=1$: $f''(1)=-6<0$ (maximum), $f(1)=2-9+12-4=1$. At $x=2$:
(a) (b) . Domain (first components); Range (second components). Each element maps to itself, so is the identity relation (which is reflexive, symmetr...
(a) Use : Thus . In
(a) $$ \begin{aligned} x &= \frac{-10}{-5} \ &= 2, \ \qquad y &= \frac{-5}{-5} \ &=...
(a) Slope of is ; slope of is . For perpendicularity the product of slopes : $$ \begin{aligned} -\frac{k}{3}\cdot\frac32 &= -1\Rightarrow -\frac{k}{2} \ &= -1\Rightarrow \boxed{k=2}. ...
(a) By the product rule,
(b) Let .
(c) .
(a) A conditional statement ("if then ") is false only when is true and is false; otherwise true.
(a) Sine law: In any triangle , . Proof. Drop the perpendicular from to . In right triangle , ; in , . Hence
(a) From : . Substitute in : . Substitute in : . Then and . $$\boxed{x=3,\ y...
(a) Circle: centre , radius . Tangency: distance from centre to equals : $$ \begin{aligned} \frac{\left\tfrac32+\tfrac9...
(a) . Now
In general (e.g. at : b...
The -th term is . $$ \begin{aligned} Sn &= \sum{r=1}^{n}\big[r(n+1)-r^2\big] \ &= (n+1)\sum{r=1}^{n}r-\sum{r=1}^{n}r^2. \ &= (n+1)\cdot\frac{n(n+1)}{2}-\frac{n(n+1)(2n+1)}{6} \ &= \frac{n(...
Main. The line is , with (since ). $$ \begin{aligned} P1 &= \frac{\cos\theta\sec\theta+\sin...
De Moivre's theorem. For any positive integer , . Write in polar form: , and
Main. . At : (maximum), . At :