NEB Class 11 · Past paper
The complete NEB Class 11 2070 exam paper for Mathematics, all 15 questions with solved model answers.
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(a) If p and q are any two statements, prove that p v q = q v p. (b) Let A={a, b}, B={b, c} and C={c, d}. Find Ax(B U C) and Ax(B n C). (c) Test the even or odd nature and the symmetricity of the function f(x) = x^4 + 3x^2 + 1.
(a) Commutativity of disjunction $p \lor q \equiv q \lor p$. Build the truth table. $p$ $q$ $p\lor q$ $q\lor p$ ------------ T T T T T F T T F T T T F F F F The columns for $p\lor q$ and $q\lor p$ are identical for every combination, hen...
(a) Prove that cos(sin^-1 u + cos^-1 v) = vsqrt(1-u^2) - usqrt(1-v^2). (b) Using principle of mathematical induction, prove that 1/2 + 1/4 + 1/8 + ... + 1/2^n = 1 - 1/2^n. (c) If A=(4 -5; 3 6) and B=(2 3; -1 -2), find B^T A^T.
(a) Let $\sin^{-1}u=A$ and $\cos^{-1}v=B$, so $\sin A=u,\ \cos A=\sqrt{1-u^2}$ and $\cos B=v,\ \sin B=\sqrt{1-v^2}$. Then $$ \begin{aligned} \cos(A+B) &= \cos A\cos B-\sin A\sin B \ &= \sqrt{1-u^2},v-u\sqrt{1-v^2} \ &= v\sqrt{1-u^2}-u\sqrt{1-v^2}. \end{aligned} $$ Hence $\cos(\sin^{-1}u+\cos^{-1}v)=v\sqrt{1-u^2}-u\sqrt{1-v^2}$.
(b) Let $P(n):\ \dfrac12+\dfrac14+\cdots+\dfrac{1}{2^n}=1-\dfrac{1}{2^n}$. For $n=1$: LHS $=\tfrac12$, RHS $=1-\tfrac12=\tfrac12$. True. Assume $P(k)$ true: $\displaystyle\sum_{r=1}^{k}\frac{1}{2^r}=1-\frac{1}{2^k}$. Then $$ \begin{aligned} \sum_{r=1}^{k+1}\frac{1}{2^r} &= 1-\frac{1}{2^k}+\frac{1}{2^{k+1}} \ &= 1-\frac{2}{2^{k+1}}+\frac{1}{2^{k+1}} \ &= 1-\frac{1}{2^{k+1}}, \end{aligned} $$ which is $P(k+1)$. By induction $P(n)$ holds for all $n\in\mathbb{N}$.
(c) $A^{T}=\begin{pmatrix}4&3\-5&6\end{pmatrix},\ B^{T}=\begin{pmatrix}2&-1\3&-2\end{pmatrix}$. $$ \begin{aligned} B^{T}A^{T} &= \begin{pmatrix}2&-1\3&-2\end{pmatrix}\begin{pmatrix}4&3\-5&6\end{pmatrix} \ &= \begin{pmatrix}8+5&6-6\12+10&9-12\end{pmatrix} \ &= \begin{pmatrix}13&0\22&-3\end{pmatrix}. \end{aligned} $$
(a) Applying Cramer's rule, solve: 3x + 4/y = 10, -2x + 3/y = -1. (b) Find the values of x and y if (x+2) + yi = (3+i)(1-2i). (c) Find the quadratic equation whose one root is 2 + sqrt(3).
(a) Put $u=\dfrac1y$: $\ 3x+4u=10,\ -2x+3u=-1$. $$ \begin{aligned} D &= \begin{vmatrix}3&4\-2&3\end{vmatrix} \ &= 9+8 \ &= 17, \ \quad Dx &= \begin{vmatrix}10&4\-1&3\end{vmatrix} \ &= 30+4 \ &= 34, \ \quad Du &= \begin{vmatrix}3&...
(a) Find the equation of the line passing through the middle point of the line segment connecting (2, -4) and (2, 4) and parallel to the line 3x - 2y = 4. (b) Find the centre and radius of the circle x^2 + y^2 + 4x - 6y + 4 = 0. (c) Evaluate: lim(x -> pi/4) (sec^2 x - 2)/(tan x - 1).
(a) Midpoint of $(2,-4)$ and $(2,4)$ is $\left(\dfrac{2+2}{2},\dfrac{-4+4}{2}\right)=(2,0)$. Line $3x-2y=4$ has slope $\dfrac32$; a parallel line has the same slope. $$ \begin{aligned} y-0 &= \tfrac32(x-2)\Rightarrow 2y \ &= 3x-6\Righta...
(a) Find dy/dx when x - y = tan(xy). (b) Evaluate: integral of (1/x) sin(log x) dx. (c) A stone thrown into a pond produces circular ripples which expand from the point of impact. If the radius of the ripple increases at the rate of 3.5 cm/sec, find how fast the area is growing when the radius is 15 cm. (pi = 22/7).
(a) Differentiate $x-y=\tan(xy)$ with respect to $x$: $$1-\frac{dy}{dx}=\sec^2(xy)\left(y+x\frac{dy}{dx}\right).$$ Collecting $\dfrac{dy}{dx}$: $$ \begin{aligned} 1-y\sec^2(xy) &= \frac{dy}{dx}\big(1+x\sec^2(xy)\big)\ \Rightarrow\ \frac{...
(a) Define union and intersection of two sets. If A, B and C are any three non-empty sets, prove that A U (B n C) = (A U B) n (A U C). OR Solve the inequality: 6 + 5x - x^2 >= 0. (b) Using different characteristics, sketch the graph of y = (x-1)(x-2)(x-3).
(a) Union $A\cup B={x:x\in A \text{ or } x\in B}$; Intersection $A\cap B={x:x\in A \text{ and } x\in B}$. Proof of $A\cup(B\cap C)=(A\cup B)\cap(A\cup C)$ (element method). Let $x\in A\cup(B\cap C)$. Then $x\in A$ or $x\in B\cap C$. ...
(a) Solve: sin^2(theta) - 2cos(theta) + 1/4 = 0. OR If a^4 + b^4 + c^4 = 2c^2(a^2 + b^2), prove that C = 45 deg or 135 deg. (b) Prove that determinant |a+x b c; a b+y c; a b c+z| = xyz(1 + a/x + b/y + c/z).
(a) Use $\sin^2\theta=1-\cos^2\theta$: $$ \begin{aligned} 1-\cos^2\theta-2\cos\theta+\tfrac14 &= 0\Rightarrow \cos^2\theta+2\cos\theta-\tfrac54 \ &= 0. \ \cos\theta &= \frac{-2\pm\sqrt{4+5}}{2} \ &= \frac{-2\pm3}{2} \ &= \tfrac12\ \t...
(a) Using row equivalent matrix method or inverse matrix method, solve: 9y - 5x = 3, x + z = 1, z + 2y = 2. (b) If one root of the equation ax^2 + bx + c = 0 be the square of the other, prove that b^3 + a^2 c + a c^2 = 3abc.
(a) Write the system as $-5x+9y=3,\ x+z=1,\ 2y+z=2$. From $x+z=1$: $z=1-x$. Substitute in $2y+z=2$: $2y+1-x=2\Rightarrow x=2y-1$. Substitute in $-5x+9y=3$: $-5(2y-1)+9y=3\Rightarrow -10y+5+9y=3\Rightarrow y=2$. Then $x=2(2)-1=3$ and $z=1-3=-2$. $$\boxed{x=3,\ y=2,\ z=-2}$$ Check: $z+2y=-2+4=2$. Correct.
(b) Let the roots be $\alpha$ and $\alpha^2$. Then $$ \begin{aligned} \alpha+\alpha^2 &= -\frac{b}{a}, \ \qquad \alpha\cdot\alpha^2 &= \alpha^3 \ &= \frac{c}{a}. \end{aligned} $$ Cube the sum: $(\alpha+\alpha^2)^3=\alpha^3+\alpha^6+3\alpha^3(\alpha+\alpha^2)$, i.e. $$ \begin{aligned} \left(-\frac{b}{a}\right)^3 &= \frac{c}{a}+\left(\frac{c}{a}\right)^2+3\cdot\frac{c}{a}\left(-\frac{b}{a}\right). \ -\frac{b^3}{a^3} &= \frac{c}{a}+\frac{c^2}{a^2}-\frac{3bc}{a^2}. \end{aligned} $$ Multiply by $a^3$: $-b^3=a^2c+ac^2-3abc$, hence $\boxed{b^3+a^2c+ac^2=3abc}$.
(a) Find the value of k so that the line 4x + 3y + k = 0 may touch the circle x^2 + y^2 - 4x + 10y + 4 = 0. (b) Evaluate: lim(x -> infinity) sqrt(x) (sqrt(x) - sqrt(x-a)). OR A function f(x) is defined by f(x) = kx+3 for x >= 2, and 3x-1 for x < 2. Find the value of k so that f(x) is continuous at x = 2.
(a) Circle: centre $(2,-5)$, radius $r=\sqrt{4+25-4}=\sqrt{25}=5$. For tangency, distance from centre to the line $=r$: $$ \begin{aligned} \frac{4(2)+3(-5)+k}{\sqrt{16+9}} &= 5\Rightarrow\frac{k-7}{5} \ &= 5\Rightarrowk-7 \ &= 25. \ ...
(a) Find from first principles, the derivative of sqrt(1+x). (b) Find the area enclosed by the axis of x and the curve y = 3x - 5x^2.
(a) With $f(x)=\sqrt{1+x}$, $$ \begin{aligned} f'(x) &= \lim{h\to0}\frac{\sqrt{1+x+h}-\sqrt{1+x}}{h} \ &= \lim{h\to0}\frac{(1+x+h)-(1+x)}{h\big(\sqrt{1+x+h}+\sqrt{1+x}\big)} \ &= \lim{h\to0}\frac{1}{\sqrt{1+x+h}+\sqrt{1+x}} \ &= \frac...
Define composite function of two functions f and g. Let f: R -> R and g: R -> R be two functions defined by f(x) = 2x^2 - 3 and g(x) = 3x + 2. Determine (fog)(x), (gof)(x) and (gog)(x). Is (fog)(x) = (gof)(x)? Are the functions (fog)(x) and (gog)(x) one to one?
Composite function: For $f:A\to B$ and $g:B\to C$, the composite $g\circ f:A\to C$ is defined by $(g\circ f)(x)=g\big(f(x)\big)$. With $f(x)=2x^2-3,\ g(x)=3x+2$: $$ \begin{aligned} (f\circ g)(x) &= f(3x+2) \ &= 2(3x+2)^2-3 \ &= 2(9x^2+...
Sum to infinity the following series: 1 - 5a + 9a^2 - 13a^3 + ... to infinity (-1 < a < 1).
This is an arithmetico-geometric series: the coefficients $1,5,9,13,\dots$ form an A.P. with first term $1$ and common difference $4$, multiplied by powers of the common ratio $r=-a$ (since the signs alternate). For $r<1$,
Find the equations of the bisectors of the angles between the lines 4x - 3y + 1 = 0 and 12x - 5y + 7 = 0 and prove that the bisectors are at right angles to each other. OR Find the condition under which the equation ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0 may represent a pair of lines.
Bisectors. They satisfy $\dfrac{4x-3y+1}{\sqrt{4^2+3^2}}=\pm\dfrac{12x-5y+7}{\sqrt{12^2+5^2}}$, i.e. $\dfrac{4x-3y+1}{5}=\pm\dfrac{12x-5y+7}{13}$. $$13(4x-3y+1)=\pm5(12x-5y+7).$$ ($+$):
Define absolute value of a complex number. If Z and W are two complex numbers, prove that |z + w| <= |z| + |w|. OR Find the cube roots of unity. Also, establish the properties of cube roots of unity.
Modulus. For $z=x+iy$, the absolute value (modulus) is $z=\sqrt{x^2+y^2}$, the distance of $z$ from the origin. Proof of $z+w\lez+w$. For any complex numbers, $$ \begin{aligned} z+w^2 &= (z+w)(\overline{z+w}) \ &= z^2+w^2+z\bar w+\bar z...
Write the criteria for the function y = f(x) to have local maxima and local minima at a point. Find the local maxima and local minima of the function f(x) = 2x^3 - 9x^2 - 24x + 3. Also find the point of inflection.
Criteria. At an interior point $x=c$ where $f'(c)=0$: - local maximum if $f''(c)<0$, - local minimum if $f''(c)0$. For $f(x)=2x^3-9x^2-24x+3$: $$ \begin{aligned} f'(x) &= 6x^2-18x-24 = 6(x^2-3x-4) = 6(x-4)(x+1) = 0\Rightarrow x = -1,\ 4....
(a) Commutativity of disjunction . Build the truth table. ------------ T T T T T F T T F T T T F F F F The columns for and are identical for every combination, hen...
(a) Let and , so and . Then
Hence .
(b) Let . For : LHS , RHS . True. Assume true: . Then
which is . By induction holds for all .
(c) .
(a) Put : . $$ \begin{aligned} D &= \begin{vmatrix}3&4\-2&3\end{vmatrix} \ &= 9+8 \ &= 17, \ \quad Dx &= \begin{vmatrix}10&4\-1&3\end{vmatrix} \ &= 30+4 \ &= 34, \ \quad Du &= \begin{vmatrix}3&...
(a) Midpoint of and is . Line has slope ; a parallel line has the same slope. $$ \begin{aligned} y-0 &= \tfrac32(x-2)\Rightarrow 2y \ &= 3x-6\Righta...
(a) Differentiate with respect to : Collecting : $$ \begin{aligned} 1-y\sec^2(xy) &= \frac{dy}{dx}\big(1+x\sec^2(xy)\big)\ \Rightarrow\ \frac{...
(a) Union ; Intersection . Proof of (element method). Let . Then or . ...
(a) Use : $$ \begin{aligned} 1-\cos^2\theta-2\cos\theta+\tfrac14 &= 0\Rightarrow \cos^2\theta+2\cos\theta-\tfrac54 \ &= 0. \ \cos\theta &= \frac{-2\pm\sqrt{4+5}}{2} \ &= \frac{-2\pm3}{2} \ &= \tfrac12\ \t...
(a) Write the system as . From : . Substitute in : . Substitute in : . Then and . Check: . Correct.
(b) Let the roots be and . Then
Cube the sum: , i.e.
Multiply by : , hence .
(a) Circle: centre , radius . For tangency, distance from centre to the line : $$ \begin{aligned} \frac{4(2)+3(-5)+k}{\sqrt{16+9}} &= 5\Rightarrow\frac{k-7}{5} \ &= 5\Rightarrowk-7 \ &= 25. \ ...
(a) With , $$ \begin{aligned} f'(x) &= \lim{h\to0}\frac{\sqrt{1+x+h}-\sqrt{1+x}}{h} \ &= \lim{h\to0}\frac{(1+x+h)-(1+x)}{h\big(\sqrt{1+x+h}+\sqrt{1+x}\big)} \ &= \lim{h\to0}\frac{1}{\sqrt{1+x+h}+\sqrt{1+x}} \ &= \frac...
Composite function: For and , the composite is defined by . With : $$ \begin{aligned} (f\circ g)(x) &= f(3x+2) \ &= 2(3x+2)^2-3 \ &= 2(9x^2+...
This is an arithmetico-geometric series: the coefficients form an A.P. with first term and common difference , multiplied by powers of the common ratio (since the signs alternate). For ,
Bisectors. They satisfy , i.e. . ():
Modulus. For , the absolute value (modulus) is , the distance of from the origin. Proof of . For any complex numbers, $$ \begin{aligned} z+w^2 &= (z+w)(\overline{z+w}) \ &= z^2+w^2+z\bar w+\bar z...
Criteria. At an interior point where : - local maximum if , - local minimum if . For : $$ \begin{aligned} f'(x) &= 6x^2-18x-24 = 6(x^2-3x-4) = 6(x-4)(x+1) = 0\Rightarrow x = -1,\ 4....