IOE Model Test 2

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. In triangle ABC, angles A,B and C are on AP.
If b:c = 3:2 , Find A.

a) 30°
b) 60°
c) 75°
d) 90°

Show Answer

Answer: c) 75°

Q2. The value of $\displaystyle \lim _{x\to 1}\left(1-x\right)\tan\left(\dfrac{\pi :x}{2}\right)$ is equal to

a) $\dfrac{\pi}{2}$
b) $\pi$
c) $\dfrac{2}{\pi}$
d) $-2\pi$

Show Answer

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

$\displaystyle \lim _{x\to 1}\left(1-x\right)\tan\left(\dfrac{\pi :x}{2}\right)$

$\displaystyle \lim _{x\to 1} \dfrac{1-x}{\cot (\dfrac{\pi x}{2})} (0/0)$

Using L Hospital Rule:

$\displaystyle \lim _{x\to 1} \dfrac{-1}{- \cosec^2 (\dfrac{\pi x}{2})\dfrac{\pi}{2}} (0/0)$

$\dfrac{2}{\pi}$

Q3. The equation of the chord of the parabola $y^2 = 16x$ with midpoint $( 1, 2)$ is

a) $4x + y - 2 = 0 $
b) $4x + y + 2 = 0$
c) $4x- y + 2 = 0 $
d) $4x - y - 2 = 0 $

Show Answer

Answer: d) $4x - y - 2 = 0 $

equation of chord having $(1,2)$ as the midpoint of the parabola is

$T= S_1$

$yy_1 -2a(x+x_1) = y_1^2 -16x_1$

$2y - 8(x+1) = 4 - 16$

$2y - 8x-8 = -12$

$2y - 8x + 4 =0$

$y - 4x + 2 =0$

$4x -y-2 =0$

Q4. The domain of the function $y = \sqrt{a^2-x^2}$ is:

a) $ [-a, a] $
b) $\mathbb R$
c) $ {-a, a }$
d) $\mathbb R-{\pm a}$

Show Answer

Answer: a) $ [-a, a] $

$a^2-x^2 \geq 0$

$a^2\geq x^2$

$a \geq |x|$

$|x| \leq a$

$x \in [-a,a]$

Q5. The Cartesian equation of a line AB is
$\dfrac{(2x-1)}{\sqrt{3}}= \dfrac{(y+2)}{2}= \dfrac{(z-3)}{3}$
Find the direction cosines of a line parallel to $AB$.

a) $\dfrac{\sqrt{3}}{\sqrt{55}} ,\dfrac{4}{\sqrt{55}} ,\dfrac{6}{\sqrt{55}}$
b) $\sqrt{3},2,3$
c) $\dfrac{\sqrt{3}}{\sqrt{17}} ,-\dfrac{2}{\sqrt{17}} ,-\dfrac{3}{\sqrt{17}}$
d) $\sqrt{3},-4,-6$

Show Answer

Answer: a) $\dfrac{\sqrt{3}}{\sqrt{55}} ,\dfrac{4}{\sqrt{55}} ,\dfrac{6}{\sqrt{55}}$

$\dfrac{(2x-1)}{\sqrt{3}}= \dfrac{(y+2)}{2}= \dfrac{(z-3)}{3}$

$\dfrac{(x-1/2)}{\sqrt{3}}= \dfrac{(y+2)}{4}= \dfrac{(z-3)}{6}$

$a = \sqrt{3}, b=4, c=6$

$\sqrt{\sqrt{3}^2 + 4^2 + 6^2} = \sqrt{55}$

$\dfrac{\sqrt{3}}{\sqrt{55}} ,\dfrac{4}{\sqrt{55}} ,\dfrac{6}{\sqrt{55}}$

Q6. If $\overrightarrow{a}= \overrightarrow{i} - 2\overrightarrow{j} + \overrightarrow{k}$, $\overrightarrow{b} = p \overrightarrow{i} - 5\overrightarrow{j} + 3\overrightarrow{k}$, $\overrightarrow{c} = 5 \overrightarrow{i} - 9\overrightarrow{j} + 4\overrightarrow{k}$ are coplanar then $p =$

a) $3$
b) $-3$
c) $-2$
d) $2$

Show Answer

Answer: d) $2$

$\begin{vmatrix} 1 & -2 & 1 \ p & -5 & 3 \ 5 & -9 & 4 \end{vmatrix}=0$

$1(-20 +27) +2(4p-15) -9p+25 =0$

$7 +8p-30 -9p+25 =0$

$p=2$

Q7. Differential coefficient of $x^4 + 4^x$ with respect to $x$ is

a) $4x^3 + \log 4$
b) $4x^3 + 4x \log x $
c) $3x^3 + 4x \log 4 $
d) $4x^3 + 4^x \log 4 $

Show Answer

Answer: d) $4x^3 + 4^x \log 4 $

$\dfrac{d}{dx} (x^4 + 4^x)$

= $4x^3 + 4^x \log 4$

Q8. The value of $(1, 0)^{81} =$

a) $-i$
b) $-1$
c) $i$
d) $1 $

Show Answer

Answer: d) $1 $

$(1, 0)^{81} = 1^{81}=1$

Q9. If A is a square matrix such that $A^2 = I$ then $(A - I)^3 + (A + I)^3 - 7A$ is equal to

a) A
b) I-A
c) I+A
d) 3A

Show Answer

Answer: a) A

$ A^2=I \newline A.A^2=I.A \implies A^3=A \newline$

Now $ (A-I)^3 + (A+I)^3-7A = A^3-I^3 - 3AI(A-I)+A^3+I^3+3AI.(A+I)-7A \newline =2A + 3A^3 \times 2I -7A \newline =A $

Q10. If in the determinant $\Delta =\left| \begin{matrix} {{a}{1}} & {{b}{1}} & {{c}{1}} \ {{a}{2}} & {{b}{2}} & {{c}{2}} \ {{a}{3}} & {{b}{3}} & {{c}_{3}} \ \end{matrix} \right|$ etc. be the co-factors of $a_1,b_1,c_1$ etc., then which of the following relations is incorrect

a) $\ce{a3A3 +b3B3 +c3C3=Δ}$
b) $\ce{a1A2 +b1B2 +c1C2=Δ}$
c) $\ce{ a1A1 +b1B1 +c1C1=Δ}$
d) $\ce{a2A2 +b2B2 +c2C2=Δ}$

Show Answer

Answer: b) $\ce{a1A2 +b1B2 +c1C2=Δ}$

It is a fundamental concept.

Q11. The maximum value of $7 \cos θ + 24 \sin θ$ is

a) 23
b) $\infty$
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 = $\sqrt{7^2+24^2} = 25$

Q12. The equations of the line which passes through the point ($1,-2$) and cuts off equal
intercepts from the axes is

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

Show Answer

Answer: d) $x + y + 1 = 0$

The intercept is equal so, $a=b$

$\dfrac{x}{a} + \dfrac{y}{a} =1$

$x+ y =a$ is the equation.

This lines passes through $(1,-2)$

$1-2 =a$

$a=-1$

Hence $x+y =-1 $ or $x+y+1 =0$ is the equation of required line.

Q13. If $5, x, y, z$ and $405$ are in G.P., find the value of $z$.

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

Show Answer

Answer: b) $135$

$a=5$

$ar^4 = 405$

$r^4 = 81$

$r= 3$

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

Q14. The value of $(1+i)^{8}+(1-i)^{8}=$

a) $2^{8}$
b) $2^{5}$
c) $2^{4}$
d) $2^{6}$

Show Answer

Answer: b) $2^{5}$

We know:

$(1+i)^n = (-4)^Q (1+i)^R$

$(1+i)^{8}+(1-i)^{8}$

$(-4)^2 + (-4)^2$

$16 + 16$

$32$

$2^5$

Q15. $\displaystyle \int\left[\cfrac{1}{\ln x}-\cfrac{1}{(\ln x)^{2}}\right] d x=$

a) $\cfrac{\ln x}{x^2}+c$
b) $\cfrac{x}{\ln x}+c$
c) $\cfrac{\ln x}{x}+c$
d) $\cfrac{2\ln x}{x}+c$

Show Answer

Answer: b) $\cfrac{x}{\ln x}+c$

Solution:

Put $\ln x=t \quad \Rightarrow x=e^{t} \rightarrow d x=e^{t} d t$

$\displaystyle \therefore ,I=\int\left(\cfrac{1}{t}-\cfrac{1}{t^{2}}\right) e^{t} d t=e^{t} \cdot \cfrac{1}{t}+c=\cfrac{x}{\ln x}+c$

Q16. If the circles $\text{x}^2 + \text{y}^2 - 2x + 4y + 5 = 0$ and $\text{3x}^2 + \text{3y}^2 - 6x +ky - 7 = 0$ are concentric then the values of k is :

a) 12
b) 8
c) 4
d) $- 12$

Show Answer

Answer: d) $- 12$

$\text{x}^2 + \text{y}^2 - 2x + 4y + 5 = 0$

center (1,2)

$\text{3x}^2 + \text{3y}^2 - 6x +ky - 7 = 0$

$\text{x}^2 + \text{y}^2 - 2x + \dfrac{k}{3} y - \dfrac{7}{3} = 0$

center $(1, -\dfrac{k}{6})$

$-\dfrac{k}{6} =2 \to k =-12$

Q17. The roots of $z^ { 3} + 8 i=0$ are:

a) $-2 i,-2 i \omega,-2 i \omega^{2}$
b) $1,2 \omega, 2 \omega^{2}$
c) $2, \omega, \omega^{2}$
d) 2i, $2i \omega, 2 \ {i} \omega^{2}$

Show Answer

Answer: d) 2i, $2i \omega, 2 \ {i} \omega^{2}$

$z^{3}+8 i=0$

$z^{3}=-8 i$

$z^{3}=(2 i)^{3}$

$z=2 i, 2 i \omega, 2 i \omega^{2}$

Q18. If $OP= <3, 1, -3 > OQ= <4, -2, 1>$ the direction cosines of PQ are:

a) $\dfrac{1}{\sqrt{26}}, \dfrac{3}{\sqrt{26}}, -\dfrac{4}{\sqrt{26}}$
b) $-\dfrac{1}{\sqrt{26}}, \dfrac{3}{\sqrt{26}}, \dfrac{4}{\sqrt{26}}$
c) $\dfrac{1}{\sqrt{26}}, \dfrac{3}{\sqrt{26}}, \dfrac{4}{\sqrt{26}}$
d) $\dfrac{1}{\sqrt{26}}, -\dfrac{3}{\sqrt{26}}, \dfrac{4}{\sqrt{26}}$

Show Answer

Answer: d) $\dfrac{1}{\sqrt{26}}, -\dfrac{3}{\sqrt{26}}, \dfrac{4}{\sqrt{26}}$

Q19. The vertices of conic described by $3x^2 - 4y^2 = 36$ is

a) $(0, \pm 3 ) $
b) $ ( 0, \pm 2 \sqrt{3} ) $
c) $( \pm 3, 0)$
d) $ ( \pm 2 \sqrt{3}, 0 ) $

Show Answer

Answer: d) $ ( \pm 2 \sqrt{3}, 0 ) $

The conic is $3x^2 - 4y^2 = 36$

i.e., $\dfrac{x^2}{12} - \dfrac{y^2}{9} = 1$ (hyperbola with a = $\sqrt{12}$ and b = 3) Hence for a > b, Vertices = ($\sqrt{12}, 0$) = $ ( \pm 2 \sqrt{3}, 0 ) $

Q20. If the difference between two numbers is $48$ and the difference between their AM and GM is $18$. Find the bigger number

a) $96$
b) $49$
c) $54$
d) $60$

Show Answer

Answer: b) $49$

$a-b = 48$

$(\sqrt a - \sqrt b)(\sqrt a + \sqrt b)=48$ ... (i)

$\dfrac{a+b}{2}- \sqrt{ab} = 18$

$a+b-2 \sqrt{ab} = 36$

$(\sqrt a - \sqrt b)^2 = 36$

$(\sqrt a - \sqrt b) = 6$ ... (ii)

Dividing (i) by (ii)

$(\sqrt a + \sqrt b)= 8$ ... (iii)

Adding equation (ii) and (iii)

$2\sqrt a = 14$

$\sqrt a =7$

$a=49$

Q21. The scalar product of two vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ inclined at angle $\theta$ is 0 only if

a) $\overrightarrow{a} = 0$ or $\overrightarrow{b} = 0$ or $\overrightarrow{a},\overrightarrow{b}$ are perpendicular
b) None
c) $\overrightarrow{b} =0$
d) $\overrightarrow{a} = 0$

Show Answer

Answer: a) $\overrightarrow{a} = 0$ or $\overrightarrow{b} = 0$ or $\overrightarrow{a},\overrightarrow{b}$ are perpendicular

$\overrightarrow{a} .\overrightarrow{b} = ab \cos \theta = 0 \iff a=0, b=0 or \theta =90$

Q22. Which one of the following measures of mark is the most suitable central location for computing intelligence of students:

a) Mode
b) Median
c) G. M
d) A. M

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Answer: b) Median

Median will be the best measure for calculating the average intelligence of students in a class. It is the value that divides the series into two equal parts. So, number of students below and above the average intelligence can easily be estimated by median.

Q23. If roots of equation $x^2-px + q = 0$ are in the ratio $2 : 3$ then

a) $25p = 6q^2$
b) $6p = 25q^2$
c) $6q = 25p^2 $
d) $25q = 6p^2$

Show Answer

Answer: d) $25q = 6p^2$

$(m+n)^2 ac = mn b^2$

$5^2 \times 1 \times q = 2 \times 3 \times p^2$

$25q = 6p^2$

Q24. If sum of first 2n terms of 2,5,8,11 …………sequence is equal to sum of n terms of 57,59,61 ………… sequence , the value of ‘n’ is equal to :

a) 11
b) 9
c) 10
d) 12

Show Answer

Answer: a) 11

Sequence 2, 5, 8, 11, …….. has $a_1 = 2$ and $d_1 = 3$

Sequence 57, 59, 61, …… has $a_2 = 57$ and $d_2 = 2$

So, $S_{2n} = S_n$

or, $\dfrac{2n}{2} [2a_1 + (2n – 1) d_1] = \dfrac{n}{2} [2a_2 + (n – 1) d_2] $

or, $n [2 \times 2+ (2n – 1) 3] = \dfrac{n}{2} [114 + (n-1) 2]$

or, $8 + 12n - 6 = 114 + 2n -2$

or, $10 n = 110$

or, $n=11$

Q25. If $\overrightarrow{u} = \overrightarrow{a}- \overrightarrow{b}$ and $\overrightarrow{v} = \overrightarrow{a} + \overrightarrow{b}$ and $a=b=2$ then $|\overrightarrow{u} \times \overrightarrow{v}|$ is

a) $2\sqrt{16 -(\overrightarrow{a}.\overrightarrow{b})^2}$
b) $3\sqrt{16 -(\overrightarrow{a}.\overrightarrow{b})^2}$
c) $\sqrt{16 -(\overrightarrow{a}.\overrightarrow{b})^2}$
d) None

Show Answer

Answer: a) $2\sqrt{16 -(\overrightarrow{a}.\overrightarrow{b})^2}$

$|\overrightarrow{u} \times \overrightarrow{v}|= |(\overrightarrow{a}- \overrightarrow{b}) \times (\overrightarrow{a}+\overrightarrow{b})| =(\overrightarrow{a}\times \overrightarrow{a} + \overrightarrow{a} \times \overrightarrow{b} - \overrightarrow{b} \times \overrightarrow{a} + \overrightarrow{b} \times b) = 2 \overrightarrow{a} \times \overrightarrow{b} = 2 \sqrt{a^2b^2 - (\overrightarrow{a}.\overrightarrow{b})^2} = 2 \sqrt{16 - (\overrightarrow{a}.\overrightarrow{b})^2}$

Q26. If $\displaystyle \int \dfrac{1}{f(x)} dx = 2\log(f(x)) + c$, then find $f(x).$

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

Show Answer

Answer: b) $\dfrac{x}{2} + c$

Q27. If $f:R\to R$ is defined by $f(x)=|x|$, then

a) The function $f^{-1}(x)$ doesn't exist
b) $f^{-1}(x) = \dfrac{1}{|x|}$
c) $f^{-1}(x) = \dfrac{1}{x}$
d) $f^{-1}(x) = -x$

Show Answer

Answer: a) The function $f^{-1}(x)$ doesn't exist

Q28. If $\overrightarrow{a}= \overrightarrow{i} - 2\overrightarrow{j} + \overrightarrow{k}$, $\overrightarrow{b} = p \overrightarrow{i} - 5\overrightarrow{j} + 3\overrightarrow{k}$, $\overrightarrow{c} = 5 \overrightarrow{i} - 9\overrightarrow{j} + 4\overrightarrow{k}$ are coplanar then $p =$

a) $-2$
b) $3$
c) $2$
d) $-3$

Show Answer

Answer: c) $2$

$\begin{vmatrix} 1 & -2 & 1 \ p & -5 & 3 \ 5 & -9 & 4 \end{vmatrix}=0$

$1(-20 +27) +2(4p-15) -9p+25 =0$

$7 +8p-30 -9p+25 =0$

$p=2$

Q29. A card is drawn from a well shuffled pack of cards . The probability of getting a queen of club or king heart is

a) $\dfrac{1}{52}$
b) None
c) $\dfrac{1}{26}$
d) $\dfrac{1}{18}$

Show Answer

Answer: c) $\dfrac{1}{26}$

Probability of getting a queen of club $= \dfrac{1}{52}$

Probability of getting a king heart$= \dfrac{1}{52}$

The probability of getting a queen of club or king heart is $= \dfrac{1}{52} + \dfrac{1}{52} = \dfrac{1}{26}$

Q30. The number of real solutions of $\cos ^{-1}x + \cos ^{-1}2x + \pi =0$ is

a) 2
b) 1
c) 0
d) $\infty$

Show Answer

Answer: c) 0

$\cos ^{-1}x + \cos ^{-1}2x + \pi =0$

$\cos ^{-1}x + \cos ^{-1}2x =- \pi $

$\cos^{-1} [x \times 2x + \sqrt{1-x^2} \sqrt{1- 4x^2}] = - \pi$

$x \times 2x + \sqrt{1-x^2} \sqrt{1- 4x^2} = -1 $

$ \sqrt{1-x^2} \sqrt{1- 4x^2} = -(1 +2x^2)$

The square roots values doesn't result in negative values.

Q31. The collection of intelligence students in a class is

a) a singleton set
b) not a set
c) a finite set
d) well defined non collection

Show Answer

Answer: b) not a set

The collection of intelligence students in a class is not well defined so it's not set.

Q32. If $A ⊆ B$ , then $A \cap B$ is equal to

a) $A^c$
b) $B^c$
c) $B$
d) $A$

Show Answer

Answer: d) $A$

$A ⊆ B$ is same is to say that $A$ is part of $B$. i.e. whole $A$ is common part of $A \cap B$

$\therefore A \cap B = A$

Q33. in a right angled triangle ABC $\sin^2A+ \sin^2B+\sin^2C =$

a) 2
b) 0
c) 1
d) -1

Show Answer

Answer: a) 2

Q34. A relation $f:N \rightarrow N$ is defined by $y=x^2 \forall x \in N$, then f is :

a) Not a function
b) Many to one and into
c) One-to-one and onto
d) One-to-one and into

Show Answer

Answer: d) One-to-one and into

Let $x_1, x_2 \in N$ and $f(x_1) = f(x_2)$. Then

$x_1^2 = x_2^2$

Both $x_1$ and $x_2$ are positive.

Hence, $x_1 = x_2$

Hence, f is one to one.

And,

$f^{-1}(x) = \sqrt{x}$

For x= 3 in the range of f(x), the pre-image doesn't exist in N.

Hence it is into function.

Q35. The value of $i+ i^2+ i^3+ i^4 =$

a) 1
b) 0
c) $i$
d) $-i$

Show Answer

Answer: b) 0

$i+ i^2+ i^3+ i^4 =0$

The sum of consecutive powers of i is 0.

Q36. The equation of parabola whose vertex and focus are (0, 4) and (0, 2) is given by :

a) $\text{x}^2 + 8\text{y} - $32 = 0
b) $\text{y}^2 - 8\text{x} - $32 = 0
c) $\text{y}^2 + 8\text{x} - $32 = 0
d) $\text{y}^2 + 8\text{x} - $32 = 0

Show Answer

Answer: a) $\text{x}^2 + 8\text{y} - $32 = 0

The vertex and focus lie in y -axis and focus is below vertex. so, it opens downwards.

$(x-h)^2 = -4a(y-k)$ is the general equation of such parabola.

$(x-0)^2 = -4 \times 2(y-4)$

$x^2 = -8y +32$

$x^2 + 8y -32=0$

Q37. The area of curve $x^2+y^2=2ax$ is:

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

Show Answer

Answer: a) $\pi a^2$

The equation of the circle is: $x^{2}-2 a x+y^{2}=0$ $(x-a)^{2}+y^{2}=a^{2}$ radius $=a$ area of circle $=\pi a^{2}$

Q38. The $7^{th}$ term in the expansion of$\left(x+\dfrac{1}{2x}\right)^{10}$ is:

a) $\dfrac{105}{32x^2}$
b) $\dfrac{105}{8x^2}$
c) $\dfrac{105}{16}x^2$
d) $\dfrac{105}{32}x^2$

Show Answer

Answer: a) $\dfrac{105}{32x^2}$

$t_{6+1} = ^{10}C_6 x^{4}\left(\dfrac{1}{2x}\right)^6 = 210 \times \dfrac{1}{64x^2} = \dfrac{105}{32x^2}$

Q39. If $\overrightarrow D$ is the midpoint of line $ \overrightarrow{BC}$ in ∆ABC, then which of the following is true?

a) $2\overrightarrow{AB} = \overrightarrow{AD} - \overrightarrow{AC} $
b) $2\overrightarrow{AC} = \overrightarrow{AB} + \overrightarrow{AD} $
c) $2\overrightarrow{AD} = \overrightarrow{AB} - \overrightarrow{AC} $
d) $2\overrightarrow{AD} = \overrightarrow{AB} + \overrightarrow{AC} $

Show Answer

Answer: d) $2\overrightarrow{AD} = \overrightarrow{AB} + \overrightarrow{AC} $

Q40. $\dfrac{dy}{dx} = e^{3x-2y} + x^2 e^{-2y}$ is

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

Show Answer

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

$ \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

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

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

Q41. The coefficient $x^6$ in$ (1 + x^2 + x^3 + x^5)^7$ is:

a) 56
b) 91
c) 42
d) 84

Show Answer

Answer: a) 56

$(1 + x^2 + x^3 + x^5)^7$

$(1(1 + x^2) + x^3(1 + x^2))^7$

$[(1+x^2)(1+x^3)]^7$

$[(^7C_0 + ^7C_1 x^2 + ^7C_2 x^4 + ^7C_3 x^6 +...)(^7C_0 + ^7C_1 x^3 + ^7C_2 x^6 + ...)]$\

The coefficient $x^6$

$x^6[^7C_0 \times ^7C_2 + ^7C_3 \times ^7C_0]$

$(21+35) x^6$

Q42. In ∆ABC, $\tan A + \tan B + \tan C – \tan A \tan B \tan C =?$

a) $1$
b) $\cot A$
c) $0$
d) $\tan A$

Show Answer

Answer: c) $0$

Q43. If semi perimeter is equal to ex – radius of a triangle , the triangle is

a) right angle
b) isosceles
c) scalene
d) equilateral

Show Answer

Answer: a) right angle

Semi perimeter = ex – radius

Or, $s = r_1$

Or, $s =\dfrac{\Delta}{s-a} $

Or, $s(s-a) = \Delta$

Or, $s(s-a) = \sqrt{s(s-a)(s-b)(s-c)}$

Or, $\sqrt{\dfrac{(s-b)(s-c)}{s(s-a)}} =1$

Or, $\tan (\dfrac{A}{2}) =\tan (\dfrac{\pi}{4})$ i.e., $A = \dfrac \pi 2$

Q44. The equation of the plane passing through the intersection of the planes $x + 2y + 3z = 4$ and $4x + 3y + 2z = -1$ and the origin is

a) $-17x + 14y + 11z = 0$
b) $17x + 14y -11z = 0$
c) $17x - 14 y + 11z = 0$
d) $17x + 14y + 11z = 0$

Show Answer

Answer: d) $17x + 14y + 11z = 0$

The required plane is
$(4x + 3y + 2z + 1) + \lambda (x + 2y + 3z - 4) = 0$

Since it passes through the origin $1 - 4\lambda = 0$

$\lambda = \dfrac{1}{4}$

$\therefore 4x + 3y + 2z + 1 + \left( \dfrac{x + 2y + 3z -4}{4} \right) = 0$

i.e., $ 16x + 12y + 8z +4 + x + 2y + 3z -4 = 0$

$i.e, 17x + 14 y + 11z = 0$.

Q45. Relation between length of side and circum-radius of an equilateral triangle is:

a) $a=3 R$
b) $a = \sqrt 3 R$
c) $R = 3a$
d) $a= R$

Show Answer

Answer: b) $a = \sqrt 3 R$

Physics (45 Questions)

Q1. A given ray of light suffers minimum deviation in an equilateral prism P. Additional prism Q and R of identical shape and of the same material as P are now added as shown in the figure. The ray will now suffer

a) greater deviation
b) no deviation
c) same deviation as before
d) total internal reflection

Show Answer

Answer: c) same deviation as before

Q2. A 15 g bullet with the speed of 1000 m/ s strikes a wooden target and penetrates a distance of 10 cm. The force by which bullet hit the target is:

a) $7.5 \times 10^{4}$ N
b) $ 1.5 \times 10^{4} $N
c) $15 \times 10^{4} $N
d) $2.5 \times 10^{4} $N

Show Answer

Answer: a) $7.5 \times 10^{4}$ N

u = 1000 m/ s , v = 0, s = 10 cm = 0.1 m$\$$v^{2} = u^{2} + 2as$$\$or, $a = \dfrac{u^{2}}{2s} = - \dfrac{1000^{2}}{2} \times 0.1 = -5 \times 10^{6} m/s^{2}$$\$or, $F = ma = 15 \times 10^{-3} \times 5 \times 10^{6}$$\$or, $F = ma = 7.5 \times 10^{4}$ N

Q3. A car starts from rest and moves on a surface in which coefficient of friction between road and tyres increases linearly with distance ( $x$ ) . The car moves with maximum possible acceleration . The K.E. of the car ( $E$ ) will depend on $x$ as :

a) $E \propto x $
b) $E \propto \dfrac{1}{x}$
c) $E \propto x^2$
d) $E \propto \dfrac{1}{x^2} $

Show Answer

Answer: c) $E \propto x^2$

$\mu = kx$

$a = \mu g = kxg$

$v = u + at = 0 + kx \times gt$

$K . E . = E = \dfrac{1}{2} mv^2 = \dfrac{1}{2} m k^2 x^2 g^2 t^2$

$E \propto x^2$

Q4. Elasticity does not depend upon

a) impurities mixed
b) nature of material
c) temperature
d) mass

Show Answer

Answer: d) mass

Elasticity depends upon temperature, nature of material and impurities mixed.

Q5. If $y= \log x$ find $\dfrac{dx}{dy}$

a) $\dfrac{1}{x}$
b) $e^x$
c) $x$
d) $\dfrac{1}{e^x}$

Show Answer

Answer: c) $x$

$y= \log x$

$\dfrac{dy}{dx} = \dfrac{1}{x}$

$\dfrac{dx}{dy} = x$

Q6. The corresponding wavelength of photon having an energy 2 eV is

a) $5.2 \times 10^{-7}$ m
b) $ 6.2 \times 10^{-7}$ m
c) $ 8.2 \times 10^{-7}$ m
d) $7.2 \times 10^{-7}$ m

Show Answer

Answer: b) $ 6.2 \times 10^{-7}$ m

E = hc / λ

λ = hc / E

Q7. A current carrying coil is placed with its axis parallel to N – S direction. Let horizontal component of earth’s magnetic field be $H_0$ and magnetic field inside the loop is H. If a magnet is suspended inside the loop it makes an angle $\theta$ with H then $\theta$ is equal to:

a) $\operatorname{cosec}^{-1}\left(\frac{H}{H_o}\right)$
b) $\tan^{-1}\left(\dfrac{H}{H_o}\right)$
c) $\tan^{-1}\left(\dfrac{H_o}{H}\right)$
d) $\cot^{-1}\left(\dfrac{H_o}{H}\right)$

Show Answer

Answer: b) $\tan^{-1}\left(\dfrac{H}{H_o}\right)$

$\tan\theta=\left(\dfrac{H}{H_o}\right)$ $\$ $\theta=\tan^{-1}\left(\dfrac{H}{H_o}\right)$

Q8. If the refractive index of water is $\dfrac{4}{3}$ and that of given slab of glass immersed in it is $\dfrac{5}{3}$ . What is the critical angle for a ray of light tending to go from glass to water ?

a) none .
b) $sin^{-1}(\dfrac{4}{5})$
c) $sin^{-1}(\dfrac{3}{5})$
d) $sin^{-1}(\dfrac{3}{4})$

Show Answer

Answer: b) $sin^{-1}(\dfrac{4}{5})$

$\sin\text{C}=\dfrac{\mu_{rarer}}{\mu_{denser}}=\dfrac{\mu_w}{\mu_g}$ $\$ or,$\sin\text{C}=\dfrac{\dfrac{2}{3}}{\dfrac{5}{3}}=\dfrac{4}{5} \rightarrow C=\sin^{-1}(\dfrac{4}{5})$

Q9. The angle between geographical meridian and magnetic meridian is

a) angle of declination
b) longitude
c) angle of dip
d) latitude

Show Answer

Answer: a) angle of declination

Definition of angle of declination

Q10. A glass prism of refractive index is 1.3 is immersed in water ($_a\mu_g = \dfrac{4}{3}$ ) . A light beam incident normally on the face AB is totally reflected to reach the face BC if

a) $\sin\theta<\dfrac{8}{9}$
b) $\sin<\dfrac{2}{3}$
c) $\sin\theta>\dfrac{8}{9}$
d) $\dfrac{2}{3}>\sin\theta<\dfrac{8}{9}$

Show Answer

Answer: c) $\sin\theta>\dfrac{8}{9}$

$\$ $\sin\text{C}=\dfrac{1}{_w\mu_g}=\dfrac{1}{\dfrac{\mu_g}{\mu_w}}$ $\$ $\dfrac{\mu_w}{\mu_g}=\dfrac{4}{3\times 1.5}=\dfrac{4}{4.5}=\dfrac{8}{9}$ $\$For total internal reflection $\$ $\theta>\text{C}$[ On solving we get angle of incidence on AC = $\theta$] $\$ i.e $\sin \theta>\dfrac{8}{9}$

Q11. Time (T), velocity (C) and angular momentum (h) are chosen as fundamental quantities instead of mass, length and time. In terms of these, the dimensions of mass would be :

a) $[T^{-1}C^{-2}h^{-1}]$
b) $[TC^{-2}h]$
c) $[T^{-1}C^{2}h]$
d) $[T^{-1}C^{-2}h]$

Show Answer

Answer: d) $[T^{-1}C^{-2}h]$

Q12. A convex lens of focal length 0.5 m and concave lens of focal length 1 m are combined . The power of the resulting lens will be

a) -0.5 D
b) 0.5 D
c) 1 D
d) -1 D

Show Answer

Answer: c) 1 D

$ \text {The combined focal length is given by} $ $\$ $ \dfrac {1}{f} = \dfrac {1}{f_1}+\dfrac {1}{f_2}=\dfrac {1}{0.5}-\dfrac {1}{1}= 1 \ m $

Q13. The density of a cubical body is measured by measuring mass and side. The maximum error in measuring density if the error in measuring sides and mass are 3% and 4% respectively.

a) $5%$
b) $15% $
c) $ 7% $
d) $ 13% $

Show Answer

Answer: d) $ 13% $

$\rho = \dfrac{m}{l^{3}}$

$ \dfrac{\Delta \rho }{\rho } = \dfrac{\Delta m}{m} + \dfrac{3 \Delta l}{l}%$

$= 4% + 3\times 3%$

$= 13%$

Q14. 1000 identical drops of water each charged to the potential V at the surface combine to form a single drop. The potential on the surface will be:

a) 1000 V
b) 100 V
c) 10 V
d) V

Show Answer

Answer: b) 100 V

$\dfrac{\text{V}_2}{\text{V}_1}=(\text{n})^{\dfrac{2}{3}}$ $\$ or,$\text{V}_2=(1000)^{\dfrac{2}{3}}\text{V}$=100V

Q15. A 5kg stone falls from a height of 100m and penetrates 2m is a layer of sand . The time of penetration is :

a) 0.089s
b) 7.14s
c) 14.28s
d) 0.89s

Show Answer

Answer: a) 0.089s

Initial acceleration $a = g$ $\$ Velocity ($v$) of stone as it hits the land is given by $\$ $V' = u' + 2as$ $\$ $V' = 0 + 2g$ $\times$ 100 $\$ $v = 200g$ $\$ When stone traverses in initial velocity $u' = v = 200g$, $\$ final velocity $v' = 0$ $\$ $v'^2 = u'^2 + 2as$ $\$ $0 = 200 g + 2a$ $\times$ 2 $\$ or, $a = - 50 g = 50 g$ (retardation) $\$ Now, $v' = u' + at$ $\$ $0 = \sqrt{200 g} - 50 gt$ $\$ $t = \dfrac{\sqrt{200 g}}{50 g} = 0 . 089s$

Q16. A radioactive source has a half - life of 3 hrs. A freshly prepared sample of the same emits radiation 16 times to permissible safe value. The minimum time after which it would be possible work safely with source is

a) 6 hr
b) 12 hr
c) 24 hr
d) 18 hr

Show Answer

Answer: b) 12 hr

Q17. Coulomb force between two charges kept at certain distance is $F$ , if half of separation between is filled with dielectric of constant $K = 4$ , then the new Coulomb's force is

a) $\dfrac{9F}{4}$
b) $36 F$
c) $\dfrac{F}{36}$
d) $\dfrac{4F}{9}$

Show Answer

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

Q18. A uniform metallic disc has its M.I $I_0$ about its diameter . Then its M.I about an axis through its rim perpendicular to the plane will be

a) $\dfrac {5}{2} I_0$
b) $\dfrac {3}{2} I_0$
c) $6 I_0$
d) $4 I_0$

Show Answer

Answer: c) $6 I_0$

$ \text {According to perp. axis theorem} $ $\$ $ I_d+I_d=I $ $\$ $ or, 2I_0= \dfrac {MR^2}{2} $ $\$ $ I_0 = \dfrac {MR^2}{4} $ $\$ $ \text {i.e. M.I of disc about its diameter} $ $\$ $ = \dfrac {MR^2}{4}=I_0 $ $\$ $ I’=I+MR^2 $ $\$ $ = \dfrac {MR^2}{2}+MR^2 = \dfrac {3MR^2}{2} = 6I_0 $

Q19. Speed of sound is maximum in

a) dry air
b) moist air
c) Equal in all
d) normal air

Show Answer

Answer: b) moist air

Q20. If the maximum height of a projectile is increased by $10%$ keeping $\theta$ same, then the time of flight increases by

a) $ 20 %$
b) $ 10 % $
c) $5 % $
d) $ 15 % $

Show Answer

Answer: c) $5 % $

$H_{max} = \dfrac{gT^{2}}{8} \Rightarrow H_{max} \propto T^{2}$ $\$$\dfrac{\Delta H_{max}}{H_{max}} \times 100 % = 2 \times \dfrac{\Delta T} \times 100 % $ $\$or, $10 = 2 \times ( \dfrac{\Delta T}{T} \times 100 % ) $ $\$$\dfrac{\Delta T}{T} \times 100% = 5 %$

Q21. If an object is placed 10 cm in front of a convex mirror of focal length 20 cm, then distance of the image from the mirror is:

a) $\dfrac{10}{3}$ cm
b) $\dfrac{40}{3}$ cm
c) $10$ cm
d) $-\dfrac{20}{3}$ cm

Show Answer

Answer: d) $-\dfrac{20}{3}$ cm

Q22. An ammeter shows a current flowing through it . Now if the equal resistance to ammeter is joined parallel then

a) The reading in ammeter will be exactly halved .
b) The reading in ammeter will be exactly one eight .
c) The reading in ammeter will be exactly one fourth .
d) The reading in ammeter will be exactly doubled .

Show Answer

Answer: a) The reading in ammeter will be exactly halved .

We know that, $\$ Ammeter and resistor are connected in parallel $\ \text{so, } V_1=V_2 \ \text{or, } I_1 r = I_2 r \ \text{i.e. } I_1=I_2 \ \text{and} \ I=I_1+I_2 \ \text{so, }I_1=\dfrac{I}{2} \ \text{Therefore, } I_1=I_2= \dfrac{I}{2}$

Q23. In a LRC series circuit , the voltage along L, C and R is 50 V . The voltage across LC combination is

a) $50$ V
b) $0$ V
c) $50\sqrt{2}$ V
d) $100$ V

Show Answer

Answer: b) $0$ V

Q24. If $\vec{A} , \vec{B} , \vec{C}$ have magnitude 6, 8 &
10 respectively, $\vec{A} + \vec{B} = \vec{C}$ then angle between $\vec{A}$ and $\vec{B}$ is:

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

Show Answer

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

Here, $\vec{A} + \vec{B} = \vec{C} $

So, $A^{2} + 2AB \cos \theta + B^{2} = C^{2} $

or, $36 + 2 AB \cos \theta + 64 = 100$

or, $\cos \theta = 0$

$\theta = 90^{\circ}$

Addition of vectors follow commutative law and associative law.

Q25. A convex lens

a) Always form virtual image
b) Diverges light rays
c) converges light rays
d) always form real image

Show Answer

Answer: c) converges light rays

Q26. The specific heat at constant volume of air is $1 KJ / Kg K$. The specific heat of air at constant pressure is if density of air at NTP is $1.293 kg / m ^{3}$ :

a) $1.286 KJ / kg ^{0} k$
b) $0.114 KJ / kg ^{0} k$
c) $1.83 KJ / kg ^{0} k$
d) $1.48 KJ / kg ^{0} k$

Show Answer

Answer: a) $1.286 KJ / kg ^{0} k$

$r = P { d } / \rho{0} T {0}=1.01 \times 10^{5} / 1.293 \times 273$ $\$ $C{p}=C_{v}+r$

Q27. A person is standing on a railway, when a train is approaching him, the frequency of whistle heard by him is 220 Hz,but when train has crossed him, the frequency heard by him is 184 Hz. The actual frequency of the whistle is:

a) 202 Hz
b) 200 Hz
c) 207 Hz
d) $\sqrt{184}\times 220$ Hz

Show Answer

Answer: b) 200 Hz

$ f_1 = (\frac{v}{v}-v_s)\times f $\$ $$\frac{f}{f_1}$$ = (v-\frac{v_s}{v})$

Q28. Two satellites A and B, ratio of masses 3:1 are in circular orbites of radius r and 4r. Then ratio of total mechanical energy of A and B is:

a) 3:1
b) 3:4
c) 1:3
d) 12:1

Show Answer

Answer: d) 12:1

$TE = - \dfrac{GMm}{2r}$

$TE \propto \dfrac{m}{r}$

Q29. Angle of deviation when passing through a prism is greatest for light :

a) red
b) violet
c) yellow
d) blue

Show Answer

Answer: b) violet

Deviation produced by a prism ($\delta$)=A ($\mu$-1 ). Since μ is higher for lower wavelength . $\delta$ will be higher for light of lower wavelength i. e $\delta_{violet}>\delta_{red}$

Q30. An object weighs 30 g in air and 25 g when totally immersed in water density of object is:

a) 5 g/ cc
b) 6 g/ cc
c) 4 g/ cc
d) 8 g/ cc

Show Answer

Answer: b) 6 g/ cc

$\text{Loss in wt } = \text{upthrust} \ \text{or, } 30 – 25 = V \sigma_w \ \text{or, } V = 5 cc \text{or, } \rho = \dfrac{30}{ 5} = 6 g/ cc$

Q31. Newton’s corpuscular theory couldn’t explain[ BP 2010 ]

a) Diffraction
b) Refraction
c) Rectilinear propagation
d) Reflection

Show Answer

Answer: a) Diffraction

$\bullet$ Newton’s corpuscular theory is based on the basis of rectilinear propagation of light. $\$ $\bullet$This theory explains :- Reflection of light , refraction of light and rectilinear propagation of light. $\$ $\bullet$This theory can’t explain : Interference of light , diffraction of light and polarization of light .

Q32. What will be the maximum height attained by a body projected with a velocity equal to one third of the escape velocity from the surface of earth?

a) $\dfrac{R}{8}$
b) $\dfrac{8R}{9}$
c) $\dfrac{R}{9}$
d) $\dfrac{9R}{8}$

Show Answer

Answer: a) $\dfrac{R}{8}$

$\dfrac{1}{2} m(\dfrac{v_e}{3})^2 = \dfrac{GMm}{R} - \dfrac{GMm}{r}$

$\dfrac{1}{2} m \dfrac{2gR}{9} = {GMm} (\dfrac{1}{R} - \dfrac{1}{r})$

$ \dfrac{gR}{9} = {GM} (\dfrac{1}{R} - \dfrac{1}{r})$

$\dfrac{gR}{9} = {gR^2} (\dfrac{1}{R} - \dfrac{1}{r})$

$\dfrac{1}{9} = {R} (\dfrac{1}{R} - \dfrac{1}{r})$

$r = \dfrac{9}{8} R$

Distance from surface = $r- R = \dfrac{R}{8}$

Q33. A rectangular vessel when full of water takes 10 minutes to be emptied through an orifice in its bottom . How many time will it take to be emptied when half filled with water ?

a) 5 minutes
b) 9 minutes
c) 3 minutes
d) 7 minutes

Show Answer

Answer: d) 7 minutes

$t \propto \sqrt{H}$

$t' = \dfrac{10}{\sqrt{2}} = 7.07 s$

Q34. Which of the following has maximum energy?

a) cosmic ray
b) $\gamma$ ray
c) X-ray
d) UV rays

Show Answer

Answer: a) cosmic ray

Q35. The half-life of the radioactive isotope in a source if its mass decreases from $24$ g to $6$g over a period of 60 days?

a) $12$ days
b) $6$ days
c) $30$ days
d) $60$ days

Show Answer

Answer: c) $30$ days

Q36. A tyre at $200^\circ C$ and $4$ atm pressure bursts suddenly then what is the final temperature of the tyre? $(γ=1.4)$

a) $212.24^\circ C$
b) $112.24^\circ C$
c) $72.57^\circ C$
d) $345^\circ C$

Show Answer

Answer: c) $72.57^\circ C$

It is adiabatic process:

$T_1^\gamma P_1^{1- \gamma} = T_2^\gamma P_2^{1- \gamma}$

$(\dfrac{T_1}{T_2})^\gamma = (\dfrac{P_2}{P_1})^{1- \gamma}$

$(\dfrac{200 + 273}{T_2})^\gamma = (\dfrac{1 }{3})^{1- \gamma}$

$(\dfrac{200 + 273}{T_2}) = (\dfrac{1 }{3})^{\frac{1- \gamma}{\gamma}}$

$T_2 = \dfrac{473}{1.368} = 345.76 K= 72.57^\circ C$

Q37. The pair NOT having identical dimensions is:

a) Moment of inertia and moment of force
b) Impulse and linear momentum
c) Planck‚Äôs constant and angular momentum
d) Young‚Äôs modulus and pressure

Show Answer

Answer: a) Moment of inertia and moment of force

impulse $= F \times t = MLT^{-2} \times T =[MLT^{-1}]$

linear momentum $= m \times v = M \times LT^{-1}$= $= [MLT^{-1}]$

Planck‚Äôs constant $= \dfrac{E}{\nu} = \dfrac{ML^{2}T^{-2} }{ T^{-1}} = [ML^{2}T^{-1}]$

Angular momentum = mvr $ = M \times LT^{-1}\times L = [ML^{2}T^{-1}]$

Moment of inertia $= mr^{2} = [ML^{2}] $

Moment of force / torque = $F \times r= MLT^{-2} \times L= [ML^{2}T^{-2}]$

Q38. A block rests on a rough inclined plane making an angle of 30° with the horizontal. The coefficient of static friction between the block and plane is 0.80 and frictional force on block is 10N, then mass of block is

a) 2.5 kg
b) 4 kg
c) 1.6 kg
d) 2 kg

Show Answer

Answer: d) 2 kg

$f - mg \sin \theta =0$

$10 - m \times 10 \times \dfrac{1}{2}$

$m=2 kg$

Q39. A quantity x is given by $\dfrac{IFv^2}{WL^4}$ in terms of moment of inertia I, force F, velocity v, work W and Length L. The dimensional formula for x is same as that of

a) coefficient of viscosity
b) force constant
c) planck's constant
d) energy density

Show Answer

Answer: d) energy density

Q40. If the pressure amplitude in a sound wave is tripled, then by what factor the intensity of the sound wave is increase(d)

a) 3
b) $\sqrt{3}$
c) 9
d) 6

Show Answer

Answer: c) 9

$I = \dfrac{{P_o}^2}{2\rho v} \propto {P_o}^2$

Q41. The station is broadcasting the waves of wavelength 600 m. If the radiating power of transmitter is 10 KW, then the number of photons radiated per second is

a) $3.0 \times 10^{31}$
b) $ 3.0 \times 10^{35}$
c) $3.0\times 10^{20 }$
d) $3.0 \times 10^{12}$

Show Answer

Answer: a) $3.0 \times 10^{31}$

P = w / t = E / t = λ / t

n = P t λ / hc

Q42. A short bar magnet of magnetic moment $0.4 Am^2$ is placed in a uniform magnetic field of $0.16 T$. The magnet is in stable equilibrium when the potential energy is

a) $0.64 J $
b) $ -0.064 J $
c) $-0.64 J $
d) $ 0.064 J$

Show Answer

Answer: b) $ -0.064 J $

$PE = - MB \cos \theta$ For stable equilibrium, potential energy is minimum i.e., $\theta= 0^o$

$PE = - 0.4 \times 0.16 = - 0.064 J$

Q43. In Rutherford scattering experiment. What will be the current angle to alpha scattering for an impact parameter b = 0?

a) $270^o$
b) $90^o$
c) $0^o$
d) $180^o$

Show Answer

Answer: d) $180^o$

Q44. Two short magnets have equal pole strengths but one is twice as long as the other . The shorter magnet is placed $20 cm$ in $\tan A$ position from the compass needle . The longer magnet must be placed on the other side of the magnetometer for the deflection at a distance equal to

a) $20 \left(2\right)^{\dfrac{3}{3}} cm$
b) $20 \left(2\right)^{\dfrac{1}{2}} cm$
c) $20 cm$
d) $20 \left(2\right)^{\dfrac{2}{3}} cm$

Show Answer

Answer: b) $20 \left(2\right)^{\dfrac{1}{2}} cm$

$Here,\ d=20cm,\ M_2=2M_1,d_2=?$ $\$ $\dfrac{M_2}{M_1}=\dfrac{d_2^3}{d_1^3}=2$ $\$ $d_2=2^{\dfrac{1}{3}}d_1$ $\$ $=20\left(2\right)^{-\dfrac{1}{3}}cm$

Q45. Two waves are represented as : $\ y_1=20 sin \pi \theta \text{ and } y_2=40sin100 \pi \theta .$ The ratio $\dfrac{l_2}{l_1}$ is given by: [ MOE 2063 ]

a) none of the above
b) 4 : 1
c) 3 : 1
d) 1 : 4

Show Answer

Answer: a) none of the above

$I =2 \pi^2 n^2a^2 \rho v , y = a sin wt \$

Chemistry (25 Questions)

Q1. The end product B in the sequence of reactions $R-X\xrightarrow{C{{N}^{-}}}A\xrightarrow{NaOH}B$ is

a) Sodium salt of carboxylic acid
b) An alkane
c) A carboxylic acid
d) A ketone

Show Answer

Answer: a) Sodium salt of carboxylic acid

$R-X\xrightarrow{KCN}R-CN\underset{{{H}{2}}O}{\mathop{\xrightarrow{NaOH}}}, R-COONa+N{{H}{3}}$

Q2. The dissociation constant of weak acid is $4.9 \times 10^{-9}$, its percentage of ionization at 1 M is

a) 0.007%
b) 0.0007%
c) 0.7%
d) 0.07 %

Show Answer

Answer: a) 0.007%

$k_a = c\alpha^2$

$\ce{\alpha = \sqrt{\dfrac{k_a}{c}} = \sqrt{\dfrac{4.9 \times 10^{-9}}{1}} =0.00007 i.e.e 0.007 %}$

Q3. Which of the following isomers of the pentane gives four monochloropentane on chlorination?

a) neo-pentane
b) n-pentane
c) 2,2-dimethylpropane
d) isopentane

Show Answer

Answer: d) isopentane

A) On photochlorination of n-pentane, 3 monochlorinated products are obtained.

B) On photochlorination of iso-pentane, 4 monochlorinated products are obtained.

C) On photochlorination of neo-pentane, 1 monochlorinated product is obtained.

D) On photochlorination of 2,2-dimethylpropane, 1 monochlorinated product is obtained.

Q4. $\ce{CuSO4.5H2O}$ is blue in colour because

a) $\ce{Cu^{++}}$ ions absorb all colour except red from the white light
b) It contains water of crystallization
c) $\ce{Cu^{++}}$ ions absorb red light
d) $\ce{SO4^{- -}}$ ions absorb red light

Show Answer

Answer: b) It contains water of crystallization

The presence of water of crystallisation facilitates the transition which transmits color to the compound.

Q5. Which of the following transitions involves maximum amount of energy?

a) $\ce{M^{2+}(g) -> M^{3+}(g)}$
b) $\ce{M(g) -> M^+(g)}$
c) $\ce{M-(g) -> M(g)}$
d) $\ce{M^+(g) -> M^{2+}(g)}$

Show Answer

Answer: a) $\ce{M^{2+}(g) -> M^{3+}(g)}$

Greater the effective nuclear charge, more tightly the electrons will be held by the nucleus and hence larger energy will be required to remove further electron. Hence among given options ,

$\ce{M^{2+} →M^{3+}}$

transition will require maximum energy.

Q6. Charring of sugar by conc. $\ce{H2SO4}$ is its

a) Dehydrogenation action
b) Oxidizing action
c) Reducing action
d) Dehydrating action

Show Answer

Answer: d) Dehydrating action

Charring of sugar, when it is treated with sulphuric acid($\ce{H2SO4}$) is due to dehydration. In this reaction water is removed from the sugar.

Q7. Internal energy is an example of

a) State function
b) None of these
c) Path function
d) Both A and B

Show Answer

Answer: a) State function

The functions whose value depends only on the state of a system are known as state functions.

Q8. Sodium reacts with water more vigorously than lithium because it

a) Is more electronegative
b) Is a metal
c) Is more electropositive
d) Has higher atomic weight

Show Answer

Answer: c) Is more electropositive

As one moves down the group, the electropositive nature rises. Therefore, sodium is more electropositive than lithium and its size is larger as well. As a result, Sodium reacts with water more violently than Lithium.

Q9. In rn1 g of a metal A displaces m2 g of another metal B from its salt solution and if their equivalent masses are $E_1$ and $E_2$ respectively, then the equivalent mass of A can be expressed as:

a) $E_1=\sqrt{\dfrac{m_1 \times E_2}{m_2}}$
b) $E_1=\dfrac{m_2 \times E_2}{m_1}$
c) $E_1=\dfrac{m_1 \times E_2}{m_2}$
d) $E_1=\dfrac{m_2 \times m_1}{E_2}$

Show Answer

Answer: c) $E_1=\dfrac{m_1 \times E_2}{m_2}$

Q10. Atomic number of next inert gas to be discovered will be

a) 118
b) 87
c) 104
d) 132

Show Answer

Answer: a) 118

Electronic configuration of next inert gas

$\ce{=[Rn] 7s^2 5f^{14} 6d^{10} 7p^6}$

So Atomic number = 118

Q11. Ionization energy of hydrogen is

a) Lesser than that of chlorine
b) Equal to that of chlorine
c) Much higher than that of chlorine
d) Slightly higher than that of chlorine

Show Answer

Answer: d) Slightly higher than that of chlorine

IE of H is 1312 kJ/mole. IE of Cl is 1255 kJ/mole.

Q12. C.N.G. is called

a) complicated natural gas
b) catalyzed natural gas
c) condensed natural gas
d) compressed natural gas

Show Answer

Answer: d) compressed natural gas

CNG, also known as compressed natural gas, is an eco-friendly alternative to gasoline.

Q13. Which one of the following orders of acid strength is correct

a) $\ce{RCOOH >HOH >HC≡CH >ROH}$
b) $\ce{RCOOH >ROH >HOH >HC≡CH}$
c) $\ce{RCOOH>HC≡CH>HOH>ROH}$
d) $\ce{RCOOH >HOH >ROH >HC≡CH}$

Show Answer

Answer: d) $\ce{RCOOH >HOH >ROH >HC≡CH}$

Q14. Which of the following is not a reducing agent?

a) $\ce{NaNO_2}$
b) $\ce{SnCl_2}$
c) $\ce{NaNO_3}$
d) $\ce{HI}$

Show Answer

Answer: c) $\ce{NaNO_3}$

Sodium nitrate $\ce{NaNO3}$ is a strong oxidizing agent, it is not a reducing agent.

Q15. $\ce{C2H5CHO and (CH3)2CO}$ can be distinguished by testing with

a) Hydroxylamine
b) Phenyl hydrazine
c) Sodium bisulphite
d) Fehling solution

Show Answer

Answer: d) Fehling solution

${{C}{2}}{{H}{5}}CHO+2C{{u}^{+2}}+5O{{H}^{-}}\to \underset{\operatorname{Re}d,,ppt}{\mathop{C{{u}{2}}O}},+3{{H}{2}}O +{{C}{2}}{{H}{5}}CO{{O}^{-}}$

$C{{H}{3}}COC{{H}{3}}+2C{{u}^{+2}}+5O{{H}^{-}}\to \text{No},,\text{reaction}$

Q16. The trace metal present in insulin is

a) Manganese
b) Cobalt
c) Iron
d) Zinc

Show Answer

Answer: d) Zinc

Insulin is secreted in the form of zinc crystals. Zinc is required to maintain the structural integrity of the insulin. It works as a cofactor for the enzyme which is related to glucose metabolism. So, the element present in the trace in insulin is zinc.

Q17. The reactivity of the alkali metal sodium with water, is made use of

a) In drying of ammonia solution
b) In drying of alcohols
c) In drying of benzene
d) As a general drying agent

Show Answer

Answer: c) In drying of benzene

Q18. Law of multiple proportions can be used to determine

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

Show Answer

Answer: d) Equivalent masses

Q19. Name the organic compound

a) 4,4-Diethyl-1-chloro butanol
b) 1-chloro-4-methyl pentan-2-ol
c) 2-methyl-1-chloro-4-butanol
d) 1-chloro-4-hydro-2-butanol

Show Answer

Answer: b) 1-chloro-4-methyl pentan-2-ol

Q20. Compounds formed by $\ce{sp^3d}$ hybridization have structure.

a) Planar
b) Trigonal bipyramidal
c) pyramidal
d) angular

Show Answer

Answer: b) Trigonal bipyramidal

$\ce{sp^3d}$ hybridization is trigonal bipyramidal structures no lone pair of electrons on central atom.

Q21. N–ethyl benzene sulphonyl amide is strongly acidic and soluble in alkali due to presence of:

a) strong electron withdrawing sulphonyl group.
b) weak electron withdrawing sulphonyl group.
c) weak electron donating sulphonyl group.
d) strong electron donating sulphonyl group.

Show Answer

Answer: a) strong electron withdrawing sulphonyl group.

The hydrogen attached to nitrogen in sulphonamide
is strongly acidic due to the presence of strong
electron withdrawing sulphonyl group. Hence, it is
soluble in alkali.

Q22. Which of the following amines can be prepared by Gabriel method?

(i) $\ce{CH_3CH_2NH_2}$

(ii) $\ce{(CH_3)_2CHNH_2}$

(iii) $\ce{(CH_3)_3CNH_2}$

(iv) $\ce{C_6H5NH_2}$

a) (i), (ii) and (iii)
b) (ii) and (iv)
c) (i) and (iii)
d) (i) and (ii)

Show Answer

Answer: d) (i) and (ii)

For the preparation of $\ce{Me3CNH2}$, the required alkyl
halide is $\ce{Me3CX}$ which will react with potassium
phthalimide, a strong base, to form alkene rather than
substituted product. For preparing $\ce{C6H5NH2, C6H5Cl}$ will be the starting halide in which Cl is non-reactive.

Q23. Hydrolysis of calcium carbide (magnesium carbide) yields.

a) Methane
b) Ethyne
c) Ethane
d) Benzene

Show Answer

Answer: b) Ethyne

Q24. Which of the following is not correct?

a) Ethyl amine and aniline dissolve in HCl
b) Ethyl amine and aniline both react with HNO₂ in cold to give hydroxy compounds
c) Ethyl amine and aniline both react with CHCl₃ and KOH to form unpleasant smelling compound
d) Ethyl amine and aniline have –NH₂ group

Show Answer

Answer: b) Ethyl amine and aniline both react with HNO₂ in cold to give hydroxy compounds

Nitrous acid reacts differently with aliphatic and
aromatic amines in cold.

Q25. Acetone and acetaldehyde are

a) Functional isomers
b) Position isomers
c) Chain isomers
d) Not isomers

Show Answer

Answer: d) Not isomers

English (25 Questions)

Q1. Synonym of appall

a) dismay
b) covered
c) confuse
d) delirious

Show Answer

Answer: a) dismay

to appall is to overcome with shock, or to dismay

Q2. Among the following sentences, Which one is correct?

a) The aim & motive of Hari is to succeed
b) The aim & motive of Hari has to succeed
c) The aim & motive of Hari were to succeed
d) The aim & motive of Hari are to succeed

Show Answer

Answer: a) The aim & motive of Hari is to succeed

Q3. The word ‚Äòinterdict‚Äô is synonymous to .............

a) prohibit
b) dispute
c) decide
d) fret

Show Answer

Answer: a) prohibit

Q4. The word ‚Äòtypical‚Äô is the antonym of

a) uncharacteristic
b) habitual
c) prototypical
d) stereotypical

Show Answer

Answer: a) uncharacteristic

Q5. The word ‘Patriot’ gets its primary stress on its ……………….. syllable.

a) second
b) none
c) first
d) third

Show Answer

Answer: c) first

Q6. ‚ÄòTanks and policemen surrounded the house.‚Äô is the active voice of:

a) The house was surrounded with tanks and policemen.
b) The house is being surrounded with tanks and policemen.
c) The house is surrounded with tanks and policemen.
d) The house had been surrounded with tanks and policemen.

Show Answer

Answer: a) The house was surrounded with tanks and policemen.

Q7. Select the appropriate one. What about ‚Ä¶‚Ä¶‚Ä¶‚Ä¶ for a drive instead?

a) to be going
b) going
c) to go
d) to have gone

Show Answer

Answer: b) going

Q8. Dr. Sharma is fairly‚Ä¶‚Ä¶‚Ä¶‚Ä¶‚Ä¶. . This quality has helped him invent great theories.

a) imaginative
b) imagination
c) imaginatory
d) imagining

Show Answer

Answer: a) imaginative

Q9. Which of the following word has five syllables?

a) university
b) unitary
c) universal
d) universe

Show Answer

Answer: a) university

Q10. The word ‚Äòminimalism‚Äô receives the primary stress on the ... ... syllable.

a) second
b) first
c) fourth
d) third

Show Answer

Answer: b) first

Q11. He said to me," Open the door."

a) He requested me to open the door.
b) He told me to open the door.
c) He told me open the door.
d) He asked me for opening the door.

Show Answer

Answer: b) He told me to open the door.

Q12. Which of the following words doesn"t have the /æ/ sound?

a) gang
b) grace
c) grab
d) gap

Show Answer

Answer: b) grace

Q13. Which of the following nouns can have the suffix ‚Äòship‚Äô removed and ‚Äòdom‚Äô replaced to form another noun?

a) fellowship
b) friendship
c) freeship
d) partnership

Show Answer

Answer: c) freeship

Q14. One who cannot be changed or reformed

a) Hardened
b) Incurable
c) Invulnerable
d) Incorrigible

Show Answer

Answer: d) Incorrigible

Q15. Choose the opposite of ‚Äòtransparent‚Äô.............

a) opaque
b) clear
c) see-through
d) glossy

Show Answer

Answer: a) opaque

Q16. As long as they saved some of the money we‚Ä¶‚Ä¶..to send them to England every summer.

a) will try
b) had tried
c) would try
d) will have tried

Show Answer

Answer: c) would try

Q17. He appeared‚Ä¶‚Ä¶‚Ä¶ a disciplinary committee.

a) upon
b) after
c) before
d) between

Show Answer

Answer: c) before

Q18. The right option for ‚Äòat the spur of the moment‚Äô is . . . .

a) Without Delay
b) Great Moment
c) Impulsive
d) Difficult Moment

Show Answer

Answer: c) Impulsive

Q19. Bag and baggage

a) Leave
b) With all one's belongings
c) Without any belonging
d) All the clothing

Show Answer

Answer: b) With all one's belongings

Q20. The primary stress of the word ‚Äòthesaurus‚Äô is on the ... ... syllable.

a) fourth
b) second
c) third
d) first

Show Answer

Answer: b) second

Q21. Synonym of apportion

a) squabble
b) cut
c) divide
d) decide

Show Answer

Answer: c) divide

to apportion is to divide and share out

Q22. The ‚Ä¶‚Ä¶‚Ä¶ chart that you have prepared is not up to our satisfaction.

a) different
b) differences
c) differentiation
d) differing

Show Answer

Answer: c) differentiation

Q23. The word ‚Äòromance‚Äô has its primary stress on the ... ... syllable.

a) third
b) fourth
c) first
d) second

Show Answer

Answer: d) second

Q24. ‚Äòput one‚Äôs foot down‚Äô signifies to . . .

a) To demand
b) To do something stupid
c) To take rest
d) To be firm about something

Show Answer

Answer: d) To be firm about something

Q25. / tʌŋ/ is the phonetic transcription of_____

a) Tongue
b) Tone
c) Toy
d) Ton

Show Answer

Answer: a) Tongue

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

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IOE Model Test 2

Mathematics (45 Questions)

Physics (45 Questions)

Chemistry (25 Questions)

English (25 Questions)

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