IOE Model Test 1

A comprehensive practice test with 140 questions across all four subjects to thoroughly test your IOE entrance preparation.

Note: The actual IOE entrance exam is a Computer-Based Test (CBT) with 100 questions worth 140 marks in 2 hours. This practice set contains extra questions for thorough preparation.

Marking Scheme: +1 or +2 for correct (depending on question), -10% for wrong, 0 for unanswered

Mathematics (45 Questions)

Q1. If $x>1$, then $2\tan^{-1}(x)+\sin^{-1}\left(\dfrac{2x}{1+ x^2}\right)$ is equal to:

a) $0$
b) $\pi$
c) $\dfrac{\pi}{2}$
d) $4\tan^{-1}(x)$

Show Answer

Answer: d) $4\tan^{-1}(x)$

$2\tan^{-1}(x)+\sin^{-1}\left(\dfrac{2x}{1+ x^2}\right)$

$x= \tan A \to A = \tan^{-1} x$

$= 2 A+\sin^{-1}\left(\dfrac{2 \tan A}{1+ \tan^2A}\right)$

$= 2 A+\sin^{-1}\left(\sin 2A\right)$

$= 2A + 2A$

$= 4A$

$= 4 \tan^{-1} x$

Q2. If two matrices A and B are of order $p \times q$ and $r \times s$ respectively, can be subtracted only, if

a) $p=q,r=s$
b) $p=q$
c) None of these
d) $p=r,q=s$

Show Answer

Answer: d) $p=r,q=s$

If two matrices A and B can be subtracted if

row of A = row of b $\implies$ $p=r$

column of A = column of b $\implies$ $q=s$

Q3. If the circle $\text{x}^2 + \text{y}^2 + \text{2a'x} + \text{2b'y} + \text{c'} = 0 $
and $\text{2x}^2 + \text{2y}^2 + 2ax + 2by +c = 0$ cut orthogonally then :

a) $\text{aa'} + \text{bb'} = \dfrac{\text{c}}{2} + \dfrac{\text{c'}}{2}$
b) $\text{aa'} + \text{bb'} = \text{c} + \text{c'}$
c) $\text{aa'} +\text{bb'} = \text{c}+\dfrac{\text{c'}}{2}$
d) $\text{aa'} +\text{bb'} =\dfrac{\text{c}}{2}+\text{c'}$

Show Answer

Answer: d) $\text{aa'} +\text{bb'} =\dfrac{\text{c}}{2}+\text{c'}$

$\text{x}^2 + \text{y}^2 + \text{2a'x} + \text{2b'y} + \text{c'} = 0 $ ... (i)

$\text{2x}^2 + \text{2y}^2 + 2ax + 2by +c = 0$

$\text{x}^2 + \text{y}^2 + ax + by +c/2 = 0$ ... (ii)

$2(ff' + gg') = c+c'$

$2(a' \dfrac{a}{2} + b' \dfrac{b}{2}) = c' + \dfrac{c}{2}$

$aa' + bb' =c' + \dfrac{c}{2} $

Q4. If $y = mx$ is one of the bisectors of angle between the lines represented by $ax^2 - 2hxy + by^2 = 0$

a) $h( 1+ m^2 ) + m ( a + b ) = 0$
b) $h ( 1 + m^2 ) + m ( a- b ) = 0$
c) $h ( 1- m^2 ) + m( a + b ) = 0$
d) $h( 1 - m^2 ) + m ( a- b ) = 0$

Show Answer

Answer: d) $h( 1 - m^2 ) + m ( a- b ) = 0$

The equation of bisector is

$h(x^2 -y^2) = (a-b) xy$

It should satisfy the$ y= mx$

$-h(x^2 -(mx)^2) = (a-b) mx^2$

$h(1-m^2) + (a-b) m =0$

Q5. $A \cap (B-A) =$

a) $\phi$
b) $A \cap B$
c) B
d) A

Show Answer

Answer: a) $\phi$

$A \cap (B-A) = \phi$

Q6. $\arg(5 - \sqrt 3 i)$

a) $\tan^{-1} \left(\dfrac{\sqrt 3}{5}\right)$
b) $\tan^{-1} \left(-\dfrac{\sqrt 3}{5}\right)$
c) $\tan^{-1} \left(\dfrac{5}{\sqrt 3}\right)$
d) $\tan^{-1} \left(-\dfrac{5}{\sqrt 3}\right)$

Show Answer

Answer: b) $\tan^{-1} \left(-\dfrac{\sqrt 3}{5}\right)$

$5- \sqrt{3} i$

$\tan \theta= \dfrac{|y|}{|x|}= \dfrac{ \sqrt{3}}{5}$

$ \theta = \tan^{-1} \dfrac{\sqrt 3}{5}$

(Since, $(5,- \sqrt{3})$ lies on $4^{ \text {th}}$ quadrant)

$ \therefore$ principal argument $= - \theta= - \tan^{-1} \dfrac{\sqrt 3}{5} =\tan^{-1} \left(-\dfrac{\sqrt 3}{5}\right)$

Q7. $\displaystyle{\lim_{x \to 3}} \dfrac{x^2-9}{x-3}$ equals:

a) 9
b) 6
c) 3
d) Doesnt exist

Show Answer

Answer: b) 6

$\displaystyle{\lim_{x \to 3}} \dfrac{x^2-9}{x-3}$

$\displaystyle{\lim_{x \to 3}} \dfrac{(x+3(x-3)}{x-3}$

$3 +3 =6$

Q8. $\displaystyle{\int \dfrac{e^x -1}{e^x +1}} dx =$

a) $\log ((e^{x/2})+(e^{-x/2}) + c)$
b) $\log ((e^{x/2})- (e^{-x/2}) + c)$
c) $2 \log ((e^{x/2})+ (e^{-x/2}) + c)$
d) $2 \log ((e^{x/2}) - (e^{-x/2}) + c)$

Show Answer

Answer: c) $2 \log ((e^{x/2})+ (e^{-x/2}) + c)$

$\displaystyle{\int \dfrac{e^x -1}{e^x +1}} dx $

$\displaystyle{\int \dfrac{e^x}{e^x +1}} dx- \displaystyle{\int \dfrac{1}{e^x +1}} dx $

$\displaystyle{\int \dfrac{e^x}{e^x +1}} dx- \displaystyle{\int \dfrac{1+ e^x -e^x}{e^x +1}} dx $

$\log (e^x + 1) -x + \ln (e^x +1) + c $

$\log (e^x + 1) - \log e^x + \log (e^x +1) + c $

$\log \dfrac{(e^x +1)^2}{(e^x)} + c$

$\log \dfrac{(e^{2x} +2e^x + 1)}{(e^x)} + c$

$\log (e^{x} +2 + e^{-x}) + c$

$\log (e^{x/2} + e^{-x/2})^2 + c$

$2 \log (e^{x/2}+ (e^{-x/2}) + c)$

Q9. If $\sin^{-1}\dfrac{x}{5}+\cos^{-1}\dfrac{4}{5}=\dfrac{\pi}{2}$, then x is equal to:

a) $1$
b) $4$
c) $3$
d) $5$

Show Answer

Answer: b) $4$

$\sin^{-1}\dfrac{x}{5}+\cos^{-1}\dfrac{4}{5}=\dfrac{\pi}{2}$

$\sin^{-1}\dfrac{x}{5}=\dfrac{\pi}{2}- \cos^{-1}\dfrac{4}{5}$

$\sin^{-1}\dfrac{x}{5}=\sin^{-1}\dfrac{4}{5}$

$x=4$

Q10. $\vec{A} $ and $\vec{B}$ are two unit vectors having angle $\theta$ between them then, $\sin \theta$ is equal to

a) $|\vec{A} - \vec{B}|$
b) $|\vec{A} \times \vec{B}|$
c) $|\vec{A} + \vec{B}|$
d) $|\vec{A} \cdot \vec{B}|$

Show Answer

Answer: b) $|\vec{A} \times \vec{B}|$

Q11. If $^{42}C_{r+1} = ^{42}C_{3r+1}$ then value of ‘$r$’ is

a) 18
b) 0
c) 21
d) 10

Show Answer

Answer: d) 10

We know, when $^nC_r = ^nC_s$, then$ r + s = n$ Hence, $(r + 1) + (3r + 1) = 42$ i.e., $r = 10$

Q12. $ \displaystyle{\int \dfrac{1}{\sqrt{1+x^2}} \log( x+ \sqrt{1+x^2}) dx } =$

a) $[\log( x+ \sqrt{1+x^2})]^2$ + c
b) $\log( x+ \sqrt{1+x^2})$ + c
c) $\dfrac{1}{2} [\log( x+ \sqrt{1+x^2})]^2$ + c
d) $\dfrac{1}{2} \log( x+ \sqrt{1+x^2})$ + c

Show Answer

Answer: c) $\dfrac{1}{2} [\log( x+ \sqrt{1+x^2})]^2$ + c

$y= \log( x+ \sqrt{1+x^2})$

$\dfrac{dy}{dx} = \dfrac{1}{\log( x+ \sqrt{1+x^2})} (1 + \dfrac{2x}{\sqrt{1+x^2}}) $

$\dfrac{dy}{dx} = \dfrac{1}{\sqrt{1+x^2}}$

$ \displaystyle{\int \dfrac{1}{\sqrt{1+x^2}} \log( x+ \sqrt{1+x^2}) dx } = \displaystyle{\int ydy } = \dfrac{y^2}{2} + c$

Q13. If AB is valid , which of the following is true ?

a) $(AB)^T = B^TA^T$
b) $(AB)^T = (BA)^T$
c) $A^TB^T = B^TA^T$
d) $A^TB^T = (AB)^T$

Show Answer

Answer: a) $(AB)^T = B^TA^T$

$(AB)^T = B^TA^T$

Q14. The area bounded by curve $y = 2x – 4 , y = 1$ and y – axis is

a) 25/4 sq. units
b) 4 sq. unit
c) 25/2 sq . units
d) 10 sq. unit

Show Answer

Answer: a) 25/4 sq. units

Area of right angle triangle = $\dfrac{1}{2} \times bh = \dfrac{1}{2} \times \dfrac{5}{2} \times 5 = \dfrac{25}{4}$ sq. units

Q15. The pair of straight lines joining the origin to the common points of
$x^2 + y^2 = 4$ and $y = 3 x + c$ are perpendicular then $c^2$ equals :

a) $\dfrac{1}{5}$
b) 13
c) 20
d) 5

Show Answer

Answer: c) 20

$x^2 + y^2 = 4$ and $y = 3 x + c \to 1 = \dfrac{y-3x}{c}$

$x^2 + y^2 = 4(\dfrac{y-3x}{c})^2$

Coefficient of $x^2 =1 - \dfrac{36}{c^2} $

Coefficient of $y^2 =1 - \dfrac{4}{c^2} $

For perpendicular:

Coefficient of $x^2$ + Coefficient of $y^2 =0$

$1 - \dfrac{36}{c^2} + 1 - \dfrac{4}{c^2}=0 $

$2= \dfrac{40}{c^2}$

$c^2 =20$

Q16. If $A = {3^{2n}-1: n\in \mathbb N}, B = {8n: n \in \mathbb N },$ then

a) $A=B$
b) $B \subset A$
c) $A \subset B$
d) $A \cap B = \phi$

Show Answer

Answer: c) $A \subset B$

$A = {8,80,728,6560, ...}$

$B = {8,16,24,...80,...,728,...6560, ...}$

So, $A \subset B$

Q17. $\int\dfrac{5^xdx}{\sqrt{1-{25}^x}} =$

a) $\dfrac{1}{\log5}\sin ^{-1} (25x) + c$
b) $\dfrac{1}{\log5} \sin ^{-1} (5x) + c $
c) $\log5 \sin ^{-1} (25x) +c$
d) $\log5 \sin ^{-1} (5x) + c$

Show Answer

Answer: b) $\dfrac{1}{\log5} \sin ^{-1} (5x) + c $

$\int\dfrac{5^xdx}{\sqrt{1-\ {25}^x}}$

Put $y = 5x$

$dy = 5x \log5 dx \to \dfrac{1}{\log5} dy = 5xdx$

$= \dfrac{1}{\log5} \int\dfrac{dy}{\sqrt{1-\ y^2}}$

$= \dfrac{1}{\log5} \sin ^{-1} (y) + c$

$= \dfrac{1}{\log5} \sin ^{-1} (5x) + c$

Q18. If $z=3−4i$, then $z^4−3z^3+3z^2+99z−95$ is equal to

a) 6
b) -4
c) -5
d) 5

Show Answer

Answer: d) 5

Q19. The maximum value of $7 \cos Œ∏ + 24 \sin Œ∏$ is

a) $\infty$
b) 23
c) 25
d) 31

Show Answer

Answer: c) 25

Maximum of $a \cos x + b \sin x$ is $ \sqrt{a^2 + b^2} $

So, Maximum = $\sqrt{7^2+24^2} = 25$

Q20. If $\begin{bmatrix} x+y & 2x+z \ x-y & 2z+w \end{bmatrix} =\begin{bmatrix} 4 & 7 \ 0 & 10 \end{bmatrix}$ , then

a) $x=4,y=2,z=2,w=3$
b) $x=3,y=4,z=2,w=2$
c) $x=2,y=3,z=4,w=2$
d) $x=2,y=2,z=3,w=4$

Show Answer

Answer: d) $x=2,y=2,z=3,w=4$

$\begin{bmatrix} x+y & 2x+z \ x-y & 2z+w \end{bmatrix} =\begin{bmatrix} 4 & 7 \ 0 & 10 \end{bmatrix}$

From equality of matrix,

$x+y = 4$

$2x+z = 7$

$x-y = 0$

$2z+w = 10$

solving we get, $x=2,y=2,z=3,w=4$

Q21. The sum of $2$ numbers is $100$ and ratio of their AM and GM is $5:4$ then numbers are

a) $40, 60$
b) $10, 90$
c) $50, 50$
d) $20, 80$

Show Answer

Answer: d) $20, 80$

$a+b =100$ ... (i)

$\dfrac{AM}{GM} = \dfrac{5}{4}$

$\dfrac{\dfrac{a+b}{2}}{\sqrt{ab}} = \dfrac{5}{4}$

$\sqrt{ab} = 40$

$ab = 1600$

$a-b = \sqrt{(a+b)^2 - 4ab} = \sqrt{10000-4 \times 1600} = \sqrt{3600} = 60$ ... (ii)

Solving (i) and (ii), we get

$a=80, b =20$

Q22. if $|\overrightarrow{a} + \overrightarrow{b}| = |\overrightarrow{a} - \overrightarrow{b}|$ then the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ such that $a,b \neq 0$ is

a) $\dfrac{\pi}{2}$
b) $\dfrac{\pi}{6}$
c) $\dfrac{\pi}{3}$
d) $\dfrac{\pi}{4}$

Show Answer

Answer: a) $\dfrac{\pi}{2}$

$|\overrightarrow{a} + \overrightarrow{b}| = |\overrightarrow{a} - \overrightarrow{b}|$

$|\overrightarrow{a} + \overrightarrow{b}|^2 = |\overrightarrow{a} - \overrightarrow{b}|^2$

$a^2 + 2\overrightarrow{a}. \overrightarrow{b} + b^2 = a^2 - 2\overrightarrow{a} . \overrightarrow{b} + b^2$

$4\overrightarrow{a} . \overrightarrow{b}=0$

$4ab \cos \theta =0$

$a,b \neq 0$

$\cos \theta =0$

$\theta = \dfrac{\pi}{2}$

Q23. If the arithmetic mean of 20 values is 10, then sum of these 20 values is:

a) 20
b) 200
c) 30
d) 10

Show Answer

Answer: b) 200

$AM = \dfrac{sum}{n}$

sum $ = AM \times n = 10 \times 20 =200$

Q24. $\int \dfrac{ \ln x}{x^3} dx$

a) $\dfrac{2 \ln x + 1}{4x^2} + c$
b) $\dfrac{2 \ln x - 1}{4x^2} + c$
c) $-\dfrac{2 \ln x + 1}{4x^2} + c$
d) $\dfrac{\ln x - 1}{4x^2} + c$

Show Answer

Answer: c) $-\dfrac{2 \ln x + 1}{4x^2} + c$

Q25. The eccentricity of ellipse $9x^2+5y^2-18x-20y-16=0$ is:

a) 3/5
b) 2/5
c) 2/3
d) 4/9

Show Answer

Answer: c) 2/3

Q26. If $(a_{11}=9, a_{12}=4, a_{21}=2, a_{22}=1)$ then $A + A^{-1}=$

a) $-5I$
b) $-10I$
c) $10I$
d) $5I$

Show Answer

Answer: c) $10I$

$A= \begin{bmatrix} 9 & 4 \ 2 & 1 \end{bmatrix}$

$A^{-1}=\dfrac{1}{1} \begin{bmatrix} 1 & -4 \ -2 & 9 \end{bmatrix}=\begin{bmatrix} 1 & -4 \ -2 & 9 \end{bmatrix}$

$A+ A^{-1} = \begin{bmatrix} 10 & 0 \ 0 & 10 \end{bmatrix} = 10 I$

Q27. If $\theta$ is the angle between $\overrightarrow{a}, \overrightarrow{b}
$ and $\overrightarrow{a}.\overrightarrow{b}
= \sqrt{3} \overrightarrow{a} \times \overrightarrow{b}$ then $\theta$ equal to

a) $\dfrac{\pi}{3}$
b) $3\pi$
c) $\dfrac{\pi}{6}$
d) $\pi$

Show Answer

Answer: c) $\dfrac{\pi}{6}$

$\overrightarrow{a}.\overrightarrow{b}
= \sqrt{3} \overrightarrow{a} \times \overrightarrow{b}$

$ab \cos \theta= \sqrt{3} ab \sin \theta$

$\cot \theta = \sqrt{3}$

$\theta= \dfrac{\pi}{6}$

Q28. If $3a=b+c,$ then $\tan \dfrac{C}{2} \tan \dfrac{B}{2}$ is

a) -12
b) 12
c) -1
d) 1

Show Answer

Answer: b) 12

Q29. A line making angles $Œ±, Œ≤ , Œ≥$ with Z- axis , x ‚Äìaxis and y- axis respectively . Then direction cosines of lines are

a) $\cos^2Œ≤ , \cos^2Œ≥ , \cos^2 Œ±$
b) $\cos Œ≤ , \cos Œ≥ , \cos Œ±$
c) $2 \cosŒ≤ , 2 \cosŒ≥ , 2 \cos Œ±$
d) $\cos 2Œ≤ , cos 2Œ≥ , cos 2Œ±$

Show Answer

Answer: b) $\cos Œ≤ , \cos Œ≥ , \cos Œ±$

A line making angles $Œ±, Œ≤ , Œ≥$ with Z- axis , x ‚Äìaxis and y- axis respectively . Then direction cosines of lines are $\cos Œ≤ , \cos Œ≥ , \cos Œ±$

Q30. If $y = a \cos (\log x) + b \sin (\log x)$, then

a) $x^2 \dfrac{d^2y}{dx^2} - x \dfrac{dy}{dx} + y =0$
b) $x^2 \dfrac{d^2y}{dx^2} + x \dfrac{dy}{dx} - y =0$
c) $x^2 \dfrac{d^2y}{dx^2} - x \dfrac{dy}{dx} - y =0$
d) $x^2 \dfrac{d^2y}{dx^2} + x \dfrac{dy}{dx} + y =0$

Show Answer

Answer: a) $x^2 \dfrac{d^2y}{dx^2} - x \dfrac{dy}{dx} + y =0$

$y = a \cos (\log x) + b \sin (\log x)$

$\dfrac{dy}{dx} = \dfrac{a}{x} - \sin (\log x)+ \dfrac{b}{x} \cos (\log x)$

$x \dfrac{dy}{dx} = - a \sin (\log x) + b \cos (\log x)$

$x \dfrac{d^2y}{dx^2} + \dfrac{dy}{dx} = -[ \dfrac{a}{x} (\cos \log x) + \dfrac{b}{x} (\sin \log x)]$

$x^2 \dfrac{d^2y}{dx^2} - x \dfrac{dy}{dx} + y =0$

Q31. The sum of squares of first 5 natural numbers is ,

a) $55$
b) $25 $
c) $165$
d) $625 $

Show Answer

Answer: a) $55$

Sum of squares of n natural number is given by:

$S_n = \dfrac{n(n+1)(2n+1)}{6}$

$S_5 = \dfrac{5(5+1)(2 \times 5 +1)}{6} = 55$

Q32. The locus of the midpoints of those chords which are drawn from the vertex of
the parabola $\text{y}^2 = 4ax$ is :

a) $\text{y}^2$ = 16ax
b) $\text{y}^2$ = 8ax
c) $\text{y}^2$ = 4ax
d) $\text{y}^2$ = 2ax

Show Answer

Answer: d) $\text{y}^2$ = 2ax

The point in parabola is given by $(at^2, 2at)$

Other end of chord is $(0,0)$

Hence,

the midpoint is $(x,y) = (\dfrac{at^2}{2}, at)$

$2x= at^2$

$2x= \dfrac{a^2t^2}{a}$

$2x= \dfrac{y^2}{a}$

$y^2 =2ax$

Q33. $\sin^{-1}‚Å°x-\cos^{-1}‚Å°x=\dfrac{œÄ}{6}$ then $x=$

a) $\dfrac{\sqrt 3}{2}$
b) $1$
c) $\dfrac 1 2$
d) $0$

Show Answer

Answer: a) $\dfrac{\sqrt 3}{2}$

$\sin^{-1}‚Å°x-\cos^{-1}‚Å°x=\dfrac{œÄ}{6}$

$\sin^{-1}‚Å°x+\cos^{-1}‚Å°x=\dfrac{œÄ}{2}$

Adding:

$2 \sin^{-1} x = \dfrac{2\pi}{3}$

$\sin^{-1} x = \dfrac{\pi}{3}$

$x =\sin \dfrac{\pi}{3} = \dfrac{\sqrt 3}{2}$

Q34. If two tangents drawn from a point P to the parabola $y^2 = 4x$ are at right angles then the locus of P is :

a) x + 1 = 0
b) 2x$ - $1 = 0
c) 2x + 1 = 0
d) x $-$ 1 = 0

Show Answer

Answer: a) x + 1 = 0

The locus of the point of intersection of perpendicular tangents to a parabola is its directrix. Hence,

the locus of P is x = -1.

Q35. $\begin{bmatrix}1 & 2 & 3 \ 1 & 2 & 3 \ -1 & -2 & -3 \end{bmatrix}$

a) idempotent
b) nilpotent
c) orthogonal
d) involuntary

Show Answer

Answer: b) nilpotent

Let $A= \begin{bmatrix}1 & 2 & 3 \ 1 & 2 & 3 \ -1 & -2 & -3 \end{bmatrix}$

$\begin{bmatrix}1 & 2 & 3 \ 1 & 2 & 3 \ -1 & -2 & -3 \end{bmatrix}$$\begin{bmatrix}1 & 2 & 3 \ 1 & 2 & 3 \ -1 & -2 & -3 \end{bmatrix}= \begin{bmatrix}0&0&0\0&0&0\0&0&0 \end{bmatrix}$

$A^2 =O$

Hence it is nilpotent matrix.

Q36. The angle between $ \overrightarrow{a} \times \overrightarrow{b}$ and $ \overrightarrow{b} \times \overrightarrow{a}$ is

a) $90^{\circ}$
b) $45^{\circ}$
c) $180^{\circ}$
d) $0^{\circ}$

Show Answer

Answer: c) $180^{\circ}$

If $ \overrightarrow{a} \times \overrightarrow{b} = \overrightarrow{c}$

$ \overrightarrow{b} \times \overrightarrow{a} = -\overrightarrow{c}$

$\overrightarrow{c}$ and $- \overrightarrow{c}$ are at $180^{\circ}$ angle.

Q37. What is the area bounded by $y^2=4x$ and double ordinate at point (1,2) is:

a) 8/3
b) 2/3
c) 4/3
d) 16/3

Show Answer

Answer: a) 8/3

The area bounded by the parabola $y^2 = 4ax$ and its double ordinate x =b is

$A = \dfrac{8}{3} b \sqrt{ab} = \dfrac{8}{3}$

$a=b=1$

Q38. $20i + j , 5i + pj$ and $10i - j $ are collinear. The value of ‚Äòp‚Äô is

a) 2
b) -2
c) 1
d) -1

Show Answer

Answer: b) -2

$a= 20i + j , b= 5i + pj$ and $c =10i - j $

Slope of (ab) =slope of (ac)

Or, $\dfrac{(p-1)}{(5-20)} = \dfrac{(-1-1)}{(10-20)} $

$p = -2$

Q39. If $5, x, y, z$ and $405$ are in G.P., find the value of $z$.

a) $162$
b) $81$
c) $135$
d) $202.5$

Show Answer

Answer: c) $135$

$a=5$

$ar^4 = 405$

$r^4 = 81$

$r= 3$

$\therefore z= ar^3 = 5 \times 3^3 = 135$

Q40. If the points ($a, 0$) ,($0, b$) and ($x, y$) are collinear then

a) $ax + by + ab = 0$
b) $ax + by -ab = 0$
c) $bx + ay - ab = 0$
d) $bx + ay + ab = 0$

Show Answer

Answer: c) $bx + ay - ab = 0$

($a, 0$) ,($0, b$) and ($x, y$) are collinear

$\text{Slope from first two points = Slope from second two points}$

$\dfrac{b-0}{0-a} = \dfrac{y-b}{x-0}$

$bx = -ay +ab$

$bx + ay - ab = 0$

Q41. Sum to infinity of the series $1+ \dfrac{2}{3}+ \dfrac{3}{3^{2}}+ \dfrac{4}{3^{3}}+\ldots$ is

a) $\dfrac 2 3$
b) $\dfrac{15} 7$
c) $\dfrac 4 9$
d) $\dfrac{9} 4$

Show Answer

Answer: d) $\dfrac{9} 4$

$a=1, \quad d=1, \quad r= \dfrac{1}{3}$

$$
S _{ \infty}= \dfrac{ a }{1- r }+ \dfrac{ dr }{(1- r )^{2}}= \dfrac{9}{4}
$$

Q42. The area bounded by the curve $y^2 = 8x$ and $x^2=8y$ is:

a) $\cfrac{64}{6}$
b) $\cfrac{64}{5}$
c) $\cfrac{64}{3}$
d) $\cfrac{64}{7}$

Show Answer

Answer: c) $\cfrac{64}{3}$

$\text{Area }(A) = \displaystyle \int_0^8 (y_1 - y_2) dx \ = \displaystyle \int_0^8 \left( \sqrt{8x} - \cfrac{x^2}{8} \right) dx \ = \left[ \sqrt{8} \cfrac{x^{1/2+1}}{3/2} - \cfrac{x^3}{24} \right]_0^8 \ = \cfrac{64}{3} \text{ sq units}$

Q43. The equation of the parallel to x-axis at a distance of 3 units on the up side from the origin

a) y = 3
b) x = -3
c) y = -3
d) x = 3

Show Answer

Answer: a) y = 3

At top from origin, parallel to x ‚Äì axis is y = a. so, it is y = 3 as a = 3

Q44. If $\overrightarrow{a} + \overrightarrow{b} = \overrightarrow{c}$ and $| \overrightarrow{a}| =5, |\overrightarrow{b}|=6, |\overrightarrow{c}| =9$, then the angle between $ \overrightarrow{a}$ and $ \overrightarrow{b}$ is

a) $\dfrac{\pi}{3}$
b) $\cos^{-1} \dfrac{1}{3}$
c) $\cos^{-1} \dfrac{1}{4}$
d) $\dfrac{\pi}{4}$

Show Answer

Answer: b) $\cos^{-1} \dfrac{1}{3}$

$\overrightarrow{a} + \overrightarrow{b} = \overrightarrow{c}$

$(\overrightarrow{a} + \overrightarrow{b})^2 = (\overrightarrow{c})^2$

$a^2 + 2\overrightarrow{a}.\overrightarrow{b} + b^2 = c^2$

$a^2 + 2ab \cos \theta + b^2 = c^2$

$\cos \theta =\dfrac{c^2 -a^2-b^2}{2ab}$

$\cos \theta =\dfrac{9^2 -5^2-6^2}{2 \times 5 \times 6} = \dfrac{1}{3}$

Q45. $\dfrac{dy}{dx} = e^{3x-2y} + x^2 e^{-2y}$ is

a) $\dfrac{e^{3y} (e^{2x}+x^3)}{6} + c$
b) $\dfrac{e^{2y}}{3} = \dfrac{e^{3x}}{3} + \dfrac{x^2}{2} + c$
c) $\dfrac{e^{2y}}{2} = \dfrac{e^{3x}}{3} + \dfrac{x^3}{3} + c$
d) $\dfrac{e^{2y} (e^{3x}+x^3)}{6} + c$

Show Answer

Answer: c) $\dfrac{e^{2y}}{2} = \dfrac{e^{3x}}{3} + \dfrac{x^3}{3} + c$

$ \dfrac{dy}{dx} = e^{3x-2y} + x^2 e^{-2y}$

$\dfrac{dy}{dx} = e^{-2y} (e^{3x} + x^2)$

separating the variable

$e^{2y} dy = (e^{3x}+x^2)dx$

Integrating

$\int e^{2y} dy =\int e^{3x}+x^2 dx$

$\dfrac{e^{2y}}{2} = \dfrac{e^{3x}}{3} + \dfrac{x^3}{3} + c$

Physics (45 Questions)

Q1. $2x^2 + kxy +2y^2 =0$ represents a pair of coincident straight lines if

a) $k=4$
b) $k=0$
c) $k=3$
d) $k=2$

Show Answer

Answer: a) $k=4$

Here $a=2, h = k/2, b =2$

$abc + 2fgh -af^2-bg^2 - ch^2=0 $

$2 \times 2 \times 0 + 2 \times 0 \times 0 \times k/2 - 2 \times 0^2 - 2 \times 0^2 - 0 \times (k/2)^2 =0$

$0=0$

So, it is satisfied by any value of k.

For coincident:

$h^2 = ab$

$k=4$

Q2. The energy released per fission of $U^{235}$ in 200 Mev. A reactor using $U^{235}$ as fuel is producing 1000 Kw power. The number of $U^{235}$ nuclei undergoing fission per sec is approximately

a) $931$
b) $2 \times 10^8$
c) $3 \times 10^{16}$
d) $10^6$

Show Answer

Answer: c) $3 \times 10^{16}$

Q3. The most appropriate material for making a cooking pot is the one having [ BPKIHS - 07 ]

a) High specific heat and low conductivity
b) Low specific heat and low conductivity
c) High specific heat and low conductivity
d) Low specific heat and high conductivity

Show Answer

Answer: d) Low specific heat and high conductivity

Knowledge based question

Q4. A stone is thrown at an angle of 45 degree horizontal with kinetic energy ‘K’ . The kinetic energy at highest point is

a) $K $
b) $ 4K $
c) $ 2K $
d) $\dfrac{K}{2}$

Show Answer

Answer: d) $\dfrac{K}{2}$

$KE_i = \dfrac{1}{2} m u^2 = K$

$KE_h = \dfrac{1}{2} m u^2 \cos^2 \theta = K_i \cos^2 \theta = K \cos^2 45^{\circ} = \dfrac{K}{2}$

Q5. A stone dropped from a height covers $\frac{5}{9}$ part of total distance in last second. Then initial height will be

a) 60 m
b) 30 m
c) 45 m
d) 90 m

Show Answer

Answer: c) 45 m

$h_{1} = \dfrac{g}{2} ( 2 t - 1) $ $\$or, $\dfrac{5}{9} H = \dfrac{g}{2} ( 2t -1)$ $\$or, $\dfrac{5}{9} \left( \dfrac{1}{2} gt^{2} \right) = \dfrac{g}{2}( 2t -1)$ $\$or, $5t^{2} = 9 (2t -1) $ $\$ On solving, t = 3second $\$$H = \dfrac{1}{2} gt^{2}$ $\$$= \dfrac{1}{2} \times 10 \times (3)^{2} = 45$m

Q6. The net capacitance of system of capacitors in fig. between points A and B is:

a) 1 $\mu$F
b) 3 $\mu$F
c) 2 $\mu$F
d) 4 $\mu$F

Show Answer

Answer: c) 2 $\mu$F

Upper 2 $\mu$F and 2 $\mu$F are in series as:$\$ $\text{C}'=2 \times \dfrac{2}{2} +2=2\times\dfrac{2}{4}=1$ $\$ Now, 1$\mu$F and 1 $\mu$F are in parallel $\$So,$\$ $\text{C}'$=1+1 $\$ $\text{C}'=2\mu$F $\$ $\text{C}\text{AB}=1+1$ $\$ $\text{C}\text{AB}=2\mu$F

Q7. Light of wavelength $\lambda$ is incident on a slit of which d . The resulting diffraction pattern is observed on a screen at a distance D . The linear width of the principal maximum is equal to the width of the slit if D equals :

a) $\dfrac{d}{\lambda}$
b) $\dfrac{2 \lambda}{d}$
c) $\dfrac{2\lambda^2}{d}$
d) $\dfrac{d^2}{2 \lambda}$

Show Answer

Answer: d) $\dfrac{d^2}{2 \lambda}$

The linear width of central principal maximum $= \dfrac{2λD}{d} \$ If $d = \dfrac{nλ}{d}$

Q8. A cubical room is formed with 6 plane mirrors. An object started to move along the diagonal of floor. The velocity of image in two adjacent walls is $ 20 \sqrt 2 \ cm/s $ then the velocity of image along the diagonal of roof is:

a) $ 20 \sqrt 2 \ cm/s $
b) $ 40 \ cm/s $
c) $ 40 \sqrt 2 \ cm/s $
d) $ 20 \ cm/s $

Show Answer

Answer: b) $ 40 \ cm/s $

$ \text {Velocity of the object along the diagonal of the floor is given by} $ $\$ $ \sin 45^ {\circ}=\dfrac {20\sqrt 2}{v_0} $ $\$ $ or, \dfrac {1}{\sqrt 2}= \dfrac {20\sqrt 2}{v_0} $ $\$ $ v_0 = 40 \ cm/s $ $\$ $ \text {Diagonal of floor and diagonal of roof are parallel, hence the velocity of the object on the roof = 40 cm/s} $

Q9. An electric current is flowing in a circular coil of radius $a$ at what distance from the centre on the axis of the coil will the magnetic field be 1/8th off its value at is

a) $\sqrt{3} a$
b) $2a$
c) $a$
d) $\sqrt{2} a$

Show Answer

Answer: a) $\sqrt{3} a$

Field at a distance $X$ on the axis from the centre of the coil $\$ $B_1 =\dfrac{ \mu_0Ia^2}{2(x^2+ a^2)^{3/2}}$ $\$Field at the centre $\$ $B_2 = \dfrac{\mu_0I}{2a}$ $\$Or, $B_1 = \dfrac{1}{8} B_2$ $\$ Or, $\dfrac{\mu_0Ia^2}{2(x^2+ a^2)^{3/2}} = \dfrac{1}{8}\dfrac{\mu_0I}{2a}$ $\$ Or, $x = \sqrt{3}$ a

Q10. The dimensional formula of surface tension is:

a) $[MLT^{-1}]$
b) $[ML^{-1}T^{-1}]$
c) $[MLT^{2}]$
d) $[ML^{0}T^{-2}]$

Show Answer

Answer: d) $[ML^{0}T^{-2}]$

Surface tension = $\dfrac{\text{force }}{\text{length}}$

$= \dfrac{[MLT^{-2} ]}{[L]} = [ML^{0}T^{-2}]$

Q11. An inductor of inductance 1.8 H is supplied by source of $V = 282.8 \sin(314 t)$, volts. The rms current flowing in the circuit is

a) 0.45 A
b) 0.7 A
c) 0.5 A
d) 0.35 A

Show Answer

Answer: d) 0.35 A

$ v=282.8 \sin (314 t)$

Comparing with

$V=V_0\sin (2\pi ft) \newline V_0 = 282.8 V \implies v_{rms} = \dfrac{V_0}{\sqrt{2}}=\dfrac{282.8}{\sqrt{2}}=200V $

And $ 2\pi f=314 \newline Now, I_{rms} = \dfrac{V_{rms}}{X_L} = \dfrac{200}{314 \times 1.8 } = 0.35 A$

Q12. A fielder can throw a cricket ball to maximum horizontal distance of 100 m . How high the fielder can throw the same ball ?

a) 50 m
b) 25 m
c) 40 m
d) 100 m

Show Answer

Answer: a) 50 m

$R = \dfrac{u^{2} \sin2\theta }{g}$ $\$ For $R_{max}, \theta = 45^{\circ} $ $\$So , $R_{max} = \dfrac{u^{2}}{g} = 100m$ $\$and $H = \dfrac{u^{2} \sin 2 \theta }{2g} $ $\$$\$For$ H_{max} , \theta = 90^{\circ} $So, $H_{max} = \dfrac{u^{2}}{2g} = \dfrac{1}{2} \times 100 = 50$ m

Q13. The resistance of wire at $20^o C$ is 20ohm and at $500^oC$ is 60 ohm. At what temperature, resistance will be 25ohm?

a) $60^oC$
b) $80^oC$
c) $50^oC$
d) $70^oC$

Show Answer

Answer: b) $80^oC$

$R-R_0 = R_0 \alpha \Delta \theta$

$20-R_0 = R_0 \alpha \times 20$ ...(i)

$60- R_0 = R_0 \alpha \times 500$... (ii)

Dividing (i) by (ii)

$\dfrac{20-R_0}{60-R_0} = \dfrac{2}{50}$

$1000 - 50 R_0 = 120 - 2R_0$

$880 = 48R_0$

$R_0 = 18.333$

Subtracting (i) from (ii)

$40 = R_0 \alpha \times 480$

$\alpha = \dfrac{40}{18.333 \times 480}$

At 25 degree celsius

$25 - 18.33 = 18.33 \times \dfrac{40}{18.333 \times 480} \times T$

$6.666 = \dfrac{T}{12}$

$T = 80^\circ$

Q14. Velocity of a particle is given by $v = a + \dfrac{b}{t} + ct^{2}$ . The unit of b will be:

a) $ ms^{2}$
b) $ ms^{-1} $
c) $ms^{-2} $
d) $ m$

Show Answer

Answer: d) $ m$

unit of $v$ = unit of $\dfrac{b}{t}$

unit of $b$ = unit of $vt =$ unit of length $= m$

Q15. The magnetic field at a distance d from a short bar magnet in longitudinal and transverse position are in the ratio :

a) $1 : 1$
b) $2 : 1$
c) $1 : 2$
d) $0 : 1$

Show Answer

Answer: b) $2 : 1$

$B_{longitudinal}=\dfrac{\mu_o}{4\pi}\times\dfrac{2M}{d^3}$ $\$ and, $B_{transverse}=\dfrac{\mu_o}{4\pi}\times\dfrac{M}{d^3}$ $\$ $B_{longitudinal}\ :\ B_{transverse}=2\ :\ 1$

Q16. An automobile engine of mass $M$ accelerates and a constant power $P$ is applied by the engine . The instantaneous speed of the engine will be

a) $\left(\dfrac{Pt}{M}\right)^{1/2}$
b) $\left(\dfrac{Pt}{2M}\right)^{1/2}$
c) $\left(\dfrac{2Pt}{M}\right)^{1/2}$
d) $\left(\dfrac{Pt}{4M}\right)^{1/2}$

Show Answer

Answer: c) $\left(\dfrac{2Pt}{M}\right)^{1/2}$

$P=Fv=Ma v=M a\dfrac{dv}{dt}v$

$\displaystyle \int v dv= \displaystyle \int \dfrac{P}{M} dt$

$\dfrac{v^2}{2}=\dfrac{Pt}{M}$

$v= \left(\dfrac{2Pt}{M}\right)^{1/2}$

Q17. In the network of resistances, the effective resistances between points A and B is

a) 8 R
b) $\dfrac{8}{3}$ R
c) $\dfrac{5}{3}$ R
d) 5 R

Show Answer

Answer: c) $\dfrac{5}{3}$ R

The upper portion of the circuit is balanced Wheatstone's bridge. Hence has equivalent resistance R. $\$ $R_{AB} = R + \dfrac{R \times 2R}{R + 2R}$ $\$ $R_{AB} = R + \dfrac{2R \times R}{3R}$ $\$ $R_{AB} = \dfrac{5}{3} R$

Q18. A body of mass 0.4 kg is whirled in a vertical circle making 2 rev/sec. If the radius of the circle is 2 m, then tension in the string when the body is at the top of the circle, is

a) 41.56 N
b) 109.86 N
c) 89.86 N
d) 115.86 N

Show Answer

Answer: d) 115.86 N

Tension at the top of the circle,$T=mω^2r−mg$

$T=0.4×4π^2n^2×2−0.4×9.8=115.86 N$

Q19. If pressure amplitude of a sound wave is tripled, the intensity of sound increases by a factor of:

a) 27
b) 3
c) 6
d) 9

Show Answer

Answer: d) 9

$\dfrac{I_2}{ I_1} = ( \dfrac{3P_0}{ P_0})^2 = 9$

Q20. The M.I of two spheres of equal masses about their respective diameters are same . If one of them is solid and other is hollow , then the ratio of their radii ( solid to hollow ) , will be

a) $ 3 : 5 $
b) $\sqrt 5 : \sqrt 3 $
c) $\sqrt 3 : \sqrt 5 $
d) $ 5: 3 $

Show Answer

Answer: b) $\sqrt 5 : \sqrt 3 $

$ \text {M.I of solid sphere about diameter,} I_s = \dfrac {2}{5}M{R_s}^2 $ $\$ $ \text {M.I of hollow sphere about diameter,} I_h = \dfrac {2}{3}M{R_h}^2 $ $\$ $ I_s=I_h $ $\$ $ or, \dfrac {2}{5}M{R_s}^2 = I_h = \dfrac {2}{3}M{R_h}^2 $ $\$ $ \therefore \dfrac {R_s}{R_h} =\sqrt 5 : \sqrt 3 $

Q21. The square of resultant of two equal vectors is equal to three times their dot product. The angle between them is:

a) $60^\circ$
b) $90^\circ$
c) $30^\circ$
d) $45^\circ$

Show Answer

Answer: a) $60^\circ$

$\vec{R} =\vec{A} + \vec{A}$

$R^2 = A^2 + 2A^2 \cos \theta + A^2= 3 \times \vec{A}\cdot\vec{A}$

$2A^2 + 2A^2 \cos \theta = 3 A^2$

$\cos \theta = \dfrac{1}{2} \to \theta = 60^\circ$

Q22. Activity of $1.85 \times 10^{12}$ dps in nearly equivalent to

a) 40 curie
b) 60 curie
c) 30 curie
d) 50 curie

Show Answer

Answer: b) 60 curie

Q23. A parallel plate capacitor is charged and then isolated . When the effect of increasing the plate separation on charge , potential capacitance , respectively ?

a) constant , increases , decreases
b) constant , decreases , decreases
c) constant , decreases , increases
d) increases , decreases , decreases

Show Answer

Answer: a) constant , increases , decreases

When the capacitor is disconnected , its charge remains constant i.e. Q = constant $ \ C = \dfrac{\epsilon_o A}{d}\ \Rightarrow C \alpha \dfrac{1}{d} \text{As d increases , C decreases .} \ \text{Now , } Q= CV \ \Rightarrow V \alpha \dfrac{1}{C}\ \text{when Q is constant} \ \text{As C decreases , V increases .}$

Q24. ${12} Al^{27}$ is a stable isotope, ${13}Al^{29}$ is expected to be disintegrated by

a) Proton emission
b) Neutron emission
c) $\beta$ - emmission
d) $\alpha$ - emmission

Show Answer

Answer: c) $\beta$ - emmission

Q25. Which of the following depends on the resistance of galvanometer?

a) None of these
b) Current sensitivity
c) Voltage sensitivity
d) Both (a) and (b)

Show Answer

Answer: c) Voltage sensitivity

Q26. The Rydberg constant for an electron revolving around hydrogen atom is R. The Rydberg constant for electron revolving about 10 times ionized sodium atom will be

a) R
b) less than R
c) arbitrary
d) greater than R

Show Answer

Answer: d) greater than R

Q27. A metallic sphere with an internal cavity weighs 40g in air and 20g in water. The density of material is 8g/cc then volume of cavity is

a) 20 cc
b) 5 cc
c) 10 cc
d) 15 cc

Show Answer

Answer: d) 15 cc

Weight of sphere in air = 40 gwt

weight of sphere in water = 20 gwt

Loss in weight = (40-20) gwt = 20 gwt

Mass of water displaced = 20 gwt

Volume of water displaced =$ 20 cm^3$

Actual volume of sphere = Volume of water displaced =20 cm^3$

Volume of material in sphere = $\dfrac{40}{8} = 5 cm^3$

Volume of cavity = $20 -5 = 15 cc$

Q28. Two convex lens having focal length 20 cm are separated by distance 10 cm. The final image of object placed at 10 cm from first lens is:

a) ∞
b) 45 cm
c) 60cm
d) 20cm

Show Answer

Answer: b) 45 cm

From first lens

$\dfrac{1}{20} = \dfrac{1}{10} + \dfrac{1}{v}$

$v= -10 cm$

This is a virtual image.

For second lens the object distance is $10 +20 = 30 cm$

$\dfrac{1}{v_2} = \dfrac{1}{10} - \dfrac{1}{30} = \dfrac{1}{15}$

$v_2 = 15 cm$

Total distance = 30 + 15 = 45 cm

Q29. A large open tank has two holes in the wall. One is a square hole of side L at a depth h from the top and the other is a circular hole of radius R at a depth 4h from the top. When the tank is completely filled with water, the quantities of water flowing out per second from both holes are the same. Then, R is equal to

a) $ \dfrac{L}{\sqrt{4\pi}}$
b) $ \dfrac{L}{\sqrt{2\pi}}$
c) $ \dfrac{L}{\sqrt{8\pi}}$
d) $ \dfrac{L}{\sqrt{\pi}}$

Show Answer

Answer: b) $ \dfrac{L}{\sqrt{2\pi}}$

$A_1v_1 = A_2 v_2$

$\sqrt{2gh} L^2 = \pi R^2 \sqrt{2g \times 4h} $

$L^2 = 2 \pi R^2$

$R = \dfrac{L}{\sqrt{2\pi}}$

Q30. If n drops, each of capacitance C, coalesce to form a single big drop, then the ratio of the energy stored in the big drop to that in each small drop will be

a) $n^{2/3}:1$
b) $n^{5/3}:1$
c) $n^{2}:1$
d) $n:1$

Show Answer

Answer: b) $n^{5/3}:1$

Q31. The deflection in moving coil galvanometer is reduced to half , when it is shunted with a 40 $\Omega$, coil .The resistance of the galvanometer is

a) 60 $\Omega$,
b) 20 $\Omega$,
c) 40 $\Omega$,
d) 10 $\Omega$,

Show Answer

Answer: c) 40 $\Omega$,

Here , $I_g = \dfrac{1}{2} I$ ; $S = 40 \Omega$ G = ? $\ G = \dfrac{I – I_g}{I_g}\times S = \dfrac{I-\dfrac{I}{2}\times40}{\dfrac{I}{2}} = 40 \Omega $.

Q32. A nucleus with a excess of neutrons may decay radio activity with the emission of:

a) neutron
b) positron
c) electron
d) proton

Show Answer

Answer: c) electron

Q33. The ratio of charge to potential of a body is known as

a) Resistance
b) Conductance
c) Capacitance
d) Inductance

Show Answer

Answer: c) Capacitance

Q34. The angle of prism is A and the minimum deviation is (180$^{\circ}$-2A$ ).Then the total refractive index of the material of the prism is :

a) $\cot(\dfrac{\text{A}}{2})$
b) $\sin(\dfrac{\text{A}}{2})$
c) $\cos(\dfrac{\text{A}}{2})$
d) $\tan(\dfrac{\text{A}}{2})$

Show Answer

Answer: a) $\cot(\dfrac{\text{A}}{2})$

$\mu= \dfrac{\sin \sin[\dfrac{\text{A}+\delta_\text{m}}{2}]}{\sin \sin \dfrac{\text{A}}{2}}=\dfrac{\sin \sin[\dfrac{\text{A}+180^{\circ}-2\text{A}}{2}]}{\sin \sin \dfrac{\text{A}}{2}}$ $\$ $\dfrac{\sin \sin[90-\dfrac{\text{A}}{2}]}{\sin \sin\dfrac{\text{A}}{2}}=\dfrac{\cos \cos\dfrac{\text{A}}{2}}{\sin \sin \dfrac{\text{A}}{2}}=\cot \cot \dfrac{\text{A}}{2}$

Q35. The air bubble in a glass cube is seen 10cm when viewed from one face and 6cm from the opposite face. The length of cube is (µ = 1.5)

a) 32 cm
b) 24 cm
c) 16 cm
d) 18 cm

Show Answer

Answer: b) 24 cm

$\mu\ = x/ x_1 + x_2\to 1.5 = x/10 + 6 \to x = 24 cm$

Q36. The dimensional formula of relative permeability is:

a) $[MLT^{‚àí2}A^{‚àí1}]$
b) $[M^0L^0T^0A^0]$
c) $[MLT^{‚àí2}A^{‚àí2}]$
d) $[MLT^{‚àí1}A^{‚àí2}]$

Show Answer

Answer: b) $[M^0L^0T^0A^0]$

The relative permeability is given by $\mu_r = \dfrac{\mu}{\mu_0}$

It is a dimensionless quantity. Other dimensionless quantities are relative permittivity, relative density or specific gravity, strain, pure numbers, angle, refractive index, poisson's ratio, power factor, magnification etc.

But the relative velocity has the dimension of velocity.

Q37. Two capacitors of capacitances C and C/2 are connected in parallel with V volt battery. Work done in charging fully both the capacitors is

a) $\dfrac{1}{2} CV^2$
b) $ CV^2$
c) $\dfrac{3}{4} CV^2$
d) $\dfrac{1}{4} CV^2$

Show Answer

Answer: c) $\dfrac{3}{4} CV^2$

Q38. The projection of vector $ 2 \hat{i} + 3 \hat{j} $ on $ \hat{i} + \hat{j} $ is

a) $\dfrac{5}{\sqrt{2}}$
b) $5\sqrt{2}$
c) $\dfrac{10}{\sqrt{2}}$
d) $10\sqrt{2}$

Show Answer

Answer: a) $\dfrac{5}{\sqrt{2}}$

Projection of $\overrightarrow{A}$ on $\overrightarrow{B}$ = $\overrightarrow{A}.\hat{B} =(2 \hat{i} + 3 \hat{j} ).\dfrac{\hat{i} + \hat{j}}{\sqrt{2}} = \dfrac{5}{\sqrt{2}}$

Q39. Which of the following is deflected by electric field

a) X-ray
b) $\alpha$ ray
c) $\beta$ ray
d) Neutrons

Show Answer

Answer: b) $\alpha$ ray

Q40. A closed organ pipe and open organ pipe have their first overtones in unison . Their lengths are in the ratio :

a) 2 : 3
b) 3 : 4
c) 4 : 5
d) 1 : 2

Show Answer

Answer: b) 3 : 4

Q41. Two particle $X$ and $Y$ having equal charges after being accelerated through the same p.d. enter a region of uniform magnetic field and describe circular paths of radii $R_1$ and $R_2$ respectively. The ratio of mass of $X$ to that of $Y$ is

a) $\sqrt{\dfrac{R_1}{R_2}}$
b) $\dfrac{R_2}{R_1}$
c) $\dfrac{R_1}{R_2}$
d) $\left(\dfrac{R_1}{R_2}\right)^2$

Show Answer

Answer: d) $\left(\dfrac{R_1}{R_2}\right)^2$

$R = \dfrac{mv}{Bq} =\dfrac{p}{Bq} = \dfrac{\sqrt{2mK}}{Bq} = \dfrac{\sqrt{2mqV}}{Bq}$ $\$ $m \alpha R^2$ $\$ $\dfrac{m_x}{m_y} = (\dfrac{R_1}{R_2})^2$

Q42. If g be the acceleration due to gravity on earth’s surface and R be the radius of earth then escape velocity of a particle at height R will be:

a) $ \sqrt {2gR} $
b) $ \dfrac {\sqrt {gR}} {2} $
c) $ \sqrt {gR} $
d) $\dfrac {1}{2} \sqrt {gR} $

Show Answer

Answer: c) $ \sqrt {gR} $

$ \text {Escape velocity from a height h is} $ $\$ $ v_e = \sqrt {\dfrac {2GM}{R+h}} $ $\$ $ If \ h = R \ then,$ $\$ $ v_e = \sqrt {\dfrac {2GM}{2R}}=\sqrt {\dfrac {GM}{R}}=\sqrt {\dfrac {GM}{R^2}\cdot R}=\sqrt{gR}$ $\$ $ (Note : \ v_e \ from \ earth's \ surface \ is \ \sqrt {2gr}=11.2 \ km/s) $

Q43. The range and time of flight of a projectile are related by $R = 5T^2$. The angle of projection is:

a) $60^{\circ} $
b) $45^{\circ}$
c) $90^{\circ}$
d) $30^{\circ}$

Show Answer

Answer: b) $45^{\circ}$

$R = 5T^{2}$

or, $\dfrac{u^{2} \times 2 \sin\theta \times \cos\theta}{g} = 5 \times 4 \times \dfrac{u^{2}\sin^{2}\theta}{g^{2}}$

or, $\tan\theta = 1 = \tan 45^{\circ}$ $\$$\theta = 45^{\circ}$

Q44. The thickness of layer of ice on surface of lake is 20 m. A hole is made in ice layer. Then minimum length of rope required to take a bucket full of water is (Density of ice= 900 kg/m^3)

a) 8 m
b) 1 m
c) 2m
d) 4 m

Show Answer

Answer: c) 2m

$(1- \dfrac{\sigma}{\rho}) = \dfrac{1}{10}^{th}$ part will be outside water

i.e $\dfrac{1}{10} \times 20 = 2 m$

Q45. The effective power in diopter if lenses of focal length +10 cm and – 20 cm are combined is [ MOE 2056 ]

a) -5
b) +3
c) +5
d) -2

Show Answer

Answer: c) +5

$ F_{eq} = \dfrac {f_1 \cdot f_2}{f_2-f_1} $ $\$ $ \text {Note: Don’t put sign convention in this equation put the value; see text} $

Chemistry (25 Questions)

Q1. The bond angle in geometry of haloalkane involving triple bond is :

a) $45^\circ$
b) $90^\circ$
c) $180^\circ$
d) $120^\circ $

Show Answer

Answer: c) $180^\circ$

If a double bond is formed, the angle between the bonds is $120°$, and if a triple bond is involved, the angle is $180°$.

Q2. Solubility of salt $\ce{A2B3}$ is $\ce{1 \times 10^{-4} mol l^{-1}}$, its solubility product is

a) $1.08 \times 10^{-18}$
b) $1.08 \times 10^{-20}$
c) $1.08 \times 10^{-16}$
d) $1.08 \times 10^{-20}$

Show Answer

Answer: a) $1.08 \times 10^{-18}$

Solubility of salt $\ce{A2B3}$ is $1×10^{−4}$

$SP = (2s)^2 (3s)^3 = 108 s^5 = 108 \times (1 \times 10^{-4})^5= 1.08 \times 10^{-18}$

Q3. The eq wt of $\ce{FeC2O4}$ in the following reaction is

$\ce{FeC2O4 + H2SO4 + KMnO4 -> Fe2(SO_4)3 + CO2 + MnSO4 + K2SO4 + H2O}$

a) $M$
b) $\dfrac M 2$
c) $\dfrac{M}{10}$
d) $\dfrac M 3$

Show Answer

Answer: d) $\dfrac M 3$

$\ce{\overset{+2}{Fe} \overset{+3}{C_2}O4→\overset{+3}{Fe_2}(SO_4)_3 + \overset{+4}{C}O2 }$

Both Fe and C undergo oxidation

Total increase in O.N.$\ce{=1+1×2=3}$

Eq wt=$\dfrac M 3$

Q4. The general formulas $\ce{C_nH_{2n}O_2}$ could be for open chain

a) Diketones
b) Diols
c) Dialdehydes
d) Carboxylic acids

Show Answer

Answer: d) Carboxylic acids

Q5. Bonding of $\ce{CO3^{- -}}$ is

a) $\ce{sp^3d}$
b) $\ce{sp^3}$
c) $\ce{sp^2}$
d) $\ce{sp}$

Show Answer

Answer: c) $\ce{sp^2}$

In carbonate ion, the carbon atom is $\ce{sp^2}$ hybridized and forms pi bond.

Q6. Oxidation number of phosphorus in $\ce{H4P2O5 , H4P2O6 and H4P2O7}$ are respectively

a) 3,4,5
b) 5,4,3
c) 5,3,4
d) 3,4,4

Show Answer

Answer: a) 3,4,5

In $\ce{H4P2O5, 4 \times 1 + x \times 2 + 5 \times (- 2) = 0 }$ i.e x=3

in $\ce{H4P2O6, 4 \times 1 + x \times 2 + 6 \times (-2) = 0}$ i.e., x = 4

In $\ce{H4P2O7, 4 \times 1 + x \times 2 + 7 \times (- 2) = 0}$ i.e., x = 5

Q7. Which of these is the strongest reducing agent?

a) $\ce{Rb}$
b) $\ce{Li}$
c) $\ce{Na}$
d) $\ce{K}$

Show Answer

Answer: b) $\ce{Li}$

Lithium is the strongest reducing agent because of lower reduction potential (i.e it has lower tendency to acquire electrons.) Just because it loses electrons very easily,so when it combines with any other element in order to form a compound it gives its electron to that element and reduce it. That's why it is a strong reducing agent.

Q8. What is the IUPAC name of the following compound?

$\ce{CH3-CH(CH3)-CH2-CH(NH2)-CH2-CH3}$

a) 5-methyl-3-aminohexane
b) 5-methyl-3-hexanamine
c) 2-methyl-4-hexanamine
d) 2-methyl-4-aminohexane

Show Answer

Answer: b) 5-methyl-3-hexanamine

The compound contains longest chain of 6C atoms
and amino group. Hence it is an alkanamine.

Q9. The vapour densities of two gases are in the ratio of $1:3$. Their molecular masses are in the ratio of:

a) $1:3 $
b) $ 2:3 $
c) $ 1:2 $
d) $3:1$

Show Answer

Answer: a) $1:3 $

$\ce{Molecular Mass = 2 \times Vapour Density}$

$\ce{Molecular Mass \propto Vapour Density}$

Q10. Elimination of $\ce{HBr}$ from 2-bromobutane results in the formation of:

a) Equimolar mixture of 1- and 2-butene
b) Predominantly 2-butene
c) Predominantly 2-butyne
d) Predominantly 1-butene

Show Answer

Answer: b) Predominantly 2-butene

In this elimination reaction of alkyl halide major product is produced according to Saytzeff’s rule. This states that when two alkenes may be formed, then alkene which is most substituted one predominates. Therefore, predominantly (80%) 2-butene will be produced.

$\ce{CH3CH2CH(Br)CH3 ->[alc. KOH] \underset{2-butene, 80%}{CH3-CH=CH-CH3} + \underset{1-butene, 20%}{CH3-CH2-CH=CH2}}$

Q11. If 4 g of oxygen diffuse through a very narrow hole, how much hydrogen would have diffused under identical conditions

a) 1/4 g
b) 1 g
c) 16 g
d) 64 g

Show Answer

Answer: b) 1 g

Q12. Benzene cannot be prepared by:

a) Polymerization of Ethyne
b) Reduction of cyclohexane
c) Reduction of phenol in presence of Zn dust
d) Decarboxylation of soda lime

Show Answer

Answer: b) Reduction of cyclohexane

Polymerization of Ethyne results in Benzene.

Reduction of phenol in presence of Zn dust results in Benzene.

Decarboxylation of soda lime results in Benzene.

Q13. Which of the following is the methylating agent ?

a) all of these
b) $\ce{ CH3I}$
c) $\ce{C2H5Br }$
d) $\ce{ C2H5CI }$

Show Answer

Answer: b) $\ce{ CH3I}$

$\ce{CH3-I}$ add $\ce{-CH3}$ group i.e., methyl group and is known as ethylating agent.

Q14. Which bond angle would result in the maximum dipole moment for the diatomic molecule $\ce{XY_2}$?

a) $ 150^{\circ} $
b) $ 180^{\circ} $
c) $ 90^{\circ} $
d) $ 120^{\circ} $

Show Answer

Answer: c) $ 90^{\circ} $

The dipole moment of two dipoles inclined at an angle θ
is given by the equation

$\mu = \sqrt{X^2 + Y^2 + 2XY \cos \theta}$

The angle increases from $90^\circ -180^\circ$, the value of cos θ becomes more and more -ve and hence resultant decreases. Thus, dipole moment is maximum when $θ=90^\circ$.

Q15. In which compound, the oxidation state of phosphorus is +4

a) PCl5
b) P4O11
c) P4O8
d) NCl5

Show Answer

Answer: c) P4O8

Q16. Law of multiple proportions can be used to determine

a) none
b) Equivalent masses
c) Molecular masses of gases
d) Atomic mass of a gas

Show Answer

Answer: b) Equivalent masses

Q17. An electrolytic cell contains a solution of $\ce{Ag2SO4}$ and have platinum electrodes. A current is passed until 1.6 g of $\ce{O2}$ has been liberated at anode. The amount of silver deposited at cathode would be

a) 0.8 g
b) 107.88 g
c) 21.60 g
d) 1.6 g

Show Answer

Answer: c) 21.60 g

no of equiv of oxygen $=\dfrac{1.6}{8}= 0.2 $

no of equiv of Ag deposited=0.2

mass of Ag deposited$= 0.2×108$ g $=21.6$ g

Q18. $\ce{CH3COOH}$ behaves as a strong acid in

a) $\ce{HNO3}$
b) $\ce{HF}$
c) $\ce{H2O}$
d) $\ce{NH3}$

Show Answer

Answer: d) $\ce{NH3}$

The ammonium ion ($\ce{NH4+}$) is a weak acid in water ($\ce{K_a = 6 \times 10^{-10}}$), but it is a strong acid in ammonia

Q19. Benzene hexachloride is

a) 1, 1, 1, 6, 6, 6-hexachlorocyclohexane
b) 1, 1-phenyl-6, 6-chlorohexane
c) 1, 6-phenyl-1, 6-chlorohexane
d) 1, 2, 3, 4, 5, 6-hexachlorocyclohexane

Show Answer

Answer: d) 1, 2, 3, 4, 5, 6-hexachlorocyclohexane

Q20. When $\ce{SO_2}$ is passed through a solution of $\ce{H_2S}$ is water

a) Sulphuric acid is formed
b) Sulphur is precipitated
c) No change
d) A clear solution is formed

Show Answer

Answer: b) Sulphur is precipitated

$\ce{SO2 + H2S -> S + H2O }$

Q21. Chlorine reacts with ethanol to give

a) Chloral
b) Chloroform
c) Ethyl chloride
d) Acetaldehyde

Show Answer

Answer: a) Chloral

$C{{H}{3}}C{{H}{2}}OH+C{{l}{2}}\to C{{H}{3}}CHO+2HCl$

$C{{H}{3}}CHO+3C{{l}{2}}\to \underset{\text{Chloral}}{\mathop{CC{{l}_{3}},CHO}},+3HCl$

Q22. Number of neutrons present in Protium is

a) 3
b) 2
c) Zero
d) 1

Show Answer

Answer: c) Zero

Protium has one proton and no neutrons.

Q23. The basic character of amines is due to:

a) high electronegativity of nitrogen
b) lone pair of electrons on nitrogen atom
c) tetrahedral structure
d) presence of nitrogen atom

Show Answer

Answer: b) lone pair of electrons on nitrogen atom

Basic nature of amines arises due to presence of lone
pair of $\ce{e–}$ s on the N-atom, which can be shared with an electron deficient species.

Q24. Sodium thiosulphate is prepared by

a) Boiling $\ce{Na2SO3}$ solution with S in acidic medium
b) Reducing $\ce{Na2SO4}$ solution with $\ce{H2S}$
c) Neutralising $\ce{H2S2O3}$ solution with $\ce{NaOH}$
d) Boiling $\ce{Na2SO3}$ solution with S in alkaline medium

Show Answer

Answer: d) Boiling $\ce{Na2SO3}$ solution with S in alkaline medium

$\ce{Na2SO3 +S ->[NaOH] Na2S2O3}$

Q25. Consider the gaseous state of Cr. The no. of electron with azimuthal q.n. $l=1$ and $l=2$ are respectively

a) 12 and 5
b) 16 and 5
c) 12 and 4
d) 16 and 4

Show Answer

Answer: a) 12 and 5

The electronic configuration of Cr along with the values of the quantum numbers n and l are shown.

	n	l
$1s^2$	1	0
$2s^2$	2	0
$2p^6$	2	1
$3s^2$	3	0
$3p^6$	3	1
$3d^5$	3	2
$4s^1$	4	0

Thus the number of electrons with l=1 is 12

and the number of electrons with l=2 is 5

English (25 Questions)

Q1. Antonym of 'sapient' is:

a) hunched
b) simian
c) simple
d) strong

Show Answer

Answer: c) simple

sapient means possessing great wisdom, or sage; one meaning of simple is deficient in intelligence

Q2. The antonym of ‚Äòlaconic‚Äô is .............

a) flagrant
b) wicked
c) wimple
d) verbose

Show Answer

Answer: d) verbose

Q3. If you‚Ä¶‚Ä¶..believe what he says ask your mother.

a) will not
b) couldn‚Äôt
c) doesn‚Äôt
d) don‚Äôt

Show Answer

Answer: d) don‚Äôt

Q4. He said," I bought a micro-bus yesterday,"

a) He said that he had bought a micro-bus the previous day.
b) He said that he has bought a micro-bus the previous day
c) He told me that he had bought a micro-bus die previous day.
d) He said that she had bought a micro-bus yesterday

Show Answer

Answer: c) He told me that he had bought a micro-bus die previous day.

Q5. I said to him, "Who are you?"

a) I asked him who was he.
b) I said to him who he was.
c) I asked him who he was.
d) I asked him who are you.

Show Answer

Answer: c) I asked him who he was.

Q6. The active of ‚ÄòHe got himself promoted to manager.‚Äô is ‚Ä¶‚Ä¶‚Ä¶.

a) He promoted to manager, himself.
b) He promoted himself to manager.
c) He himself has promoted to manager.
d) He promotes to manager himself.

Show Answer

Answer: b) He promoted himself to manager.

Q7. The effects of the sun's rays .............. devastating.

a) is proving
b) was proving
c) prove
d) a proves

Show Answer

Answer: c) prove

Q8. If only they labored hard they‚Ä¶‚Ä¶..high marks.

a) would be secured
b) secured
c) secure
d) would secure

Show Answer

Answer: d) would secure

Q9. An enthusiastic person like you should work‚Ä¶‚Ä¶‚Ä¶‚Ä¶‚Ä¶.

a) energetic
b) energy
c) energetically
d) energize

Show Answer

Answer: c) energetically

Q10. Which of these sentences has BOTH commas in the right places?

a) The school which, had only just opened, burnt down.
b) The school, which had only just opened, burnt down.
c) The school which, had only just opened burnt, down.
d) The school, which had only just, opened burnt down.

Show Answer

Answer: b) The school, which had only just opened, burnt down.

Q11. The word ‚Äògarish‚Äô is synonymous to .............

a) gaudy
b) lusty
c) prominent
d) thoughtful

Show Answer

Answer: a) gaudy

Q12. An introvert person is

a) notcomfortable around people
b) both ((a) and ((c)
c) who solves problems through discussions
d) comfortable around people

Show Answer

Answer: a) notcomfortable around people

Q13. A series of programmes....................on television.

a) are being aired
b) have been aired
c) was aired
d) has aired

Show Answer

Answer: d) has aired

Q14. ............ has offered me this opportunity.

a) either
b) neither
c) any
d) no one

Show Answer

Answer: d) no one

Q15. Which of the words is opposite in meaning to ‚Äòsensible‚Äô .............

a) responsible
b) practical
c) realistic
d) foolish

Show Answer

Answer: d) foolish

Q16. They_____ fresh apples here.

a) sold
b) sells
c) selling
d) sell

Show Answer

Answer: d) sell

Q17. Which of the following words is not five-syllabic word?

a) Visibility
b) Capacity
c) Possibility
d) Capability

Show Answer

Answer: b) Capacity

Q18. Find out the correct one.My hair needs‚Ä¶‚Ä¶‚Ä¶‚Ä¶

a) cutting
b) to have cut
c) to cut
d) having cut

Show Answer

Answer: a) cutting

Q19. The dictionary spelling of $/ \int I \rho /$ is

a) ship
b) sip
c) sheep
d) slip

Show Answer

Answer: a) ship

Q20. Which of the following is a short vowel?

a) /i:/
b) /ɒ/
c) / ɔ:/
d) /a:/

Show Answer

Answer: b) /ɒ/

Q21. ........... is a beautiful country.

a) My
b) Your
c) Ours
d) Her

Show Answer

Answer: c) Ours

Q22. Neither of the photos............... good.

a) looks
b) have looked
c) look
d) arc looking

Show Answer

Answer: a) looks

Q23. Select the right pattern for the sentence, What you said sounds good.

a) Interrogative Clause + V + O
b) S + V + Adverb
c) S + V + Od
d) Nominal clause + V + Adjective.

Show Answer

Answer: d) Nominal clause + V + Adjective.

Q24. The word ‘subscription’ gets its primary stress on its ………. Syllable .

a) 1st
b) 4th
c) 3rd
d) 2nd

Show Answer

Answer: d) 2nd

Q25. Select the right sentence for the pattern, S + V + Subject complement.

a) Birds fly in the sky.
b) The milk has turned sour.
c) The sun rose.
d) I lent her my pen.

Show Answer

Answer: b) The milk has turned sour.

How Did You Score?

Count your correct and wrong answers, then calculate:

Score = (Correct x 1) - (Wrong x 0.1)

Score Range	Assessment
120+	Excellent — Pulchowk level
100-119	Very Good — competitive for top colleges
80-99	Good — keep practicing
Below 80	Needs more preparation

Want more practice? Try IOE Model Test 2 or take a timed mock test for a real exam experience with automatic scoring.

Related Guides

IOE Entrance Exam Guide — eligibility, exam pattern, important dates
IOE Entrance Preparation Guide — 3-month study plan
IOE Entrance Tips — How to Score 100+ — time management and exam strategy
IOE Entrance Syllabus — chapter-wise syllabus organized by section

IOE Model Test 1 — Full Practice Exam

IOE Model Test 1

Mathematics (45 Questions)

Physics (45 Questions)

Chemistry (25 Questions)

English (25 Questions)

How Did You Score?

Related Guides