Grass-G/jee-main-questions huggingface dataset:
Subject: Mathematics — Hyperbola — Eccentricity
Difficulty: Easy / Moderate/ Tough

Question: If the eccentricity of the hyperbola
$x^{2} - y^{2}\sec^{2}\alpha = 5\ is\ \sqrt{3\ }$times the eccentricity
of the ellipse $x^{2}\sec^{2}\alpha + y^{2} = 25$, then a value of α can
be
nh?kZo`Ùk $x^{2}\sec^{2}\alpha + y^{2} = 25$ dh mRdsUærk
$\ \sqrt{3\ }$xquk gS] rc α


Correct option: 2

Solution:
For the hyperbola
$\frac{x^{2}}{5} - \frac{y^{2}}{5\cos^{2}\alpha} = 1$
vfrijoy; $\frac{x^{2}}{5} - \frac{y^{2}}{5\cos^{2}\alpha} = 1$ ds fy,
We have;gka
$e_{1}^{2} = 1 + \frac{b^{2}}{a^{2}} = 1 + \frac{5\cos^{2}\alpha}{5} = 1 + \cos^{2}\alpha$
For the ellipse $\frac{x^{2}}{25\cos^{2}\alpha} + \frac{y^{2}}{25} = 1$
nh?kZo`r $\frac{x^{2}}{25\cos^{2}\alpha} + \frac{y^{2}}{25} = 1\ $ds
fy,
We have;gka
$e_{2}^{2} = 1 - \frac{25\cos^{2}\alpha}{25} = \sin^{2}\alpha$
Or;k $1 + \cos^{2}\alpha = 3\sin^{2}\alpha$
Or;k $2 = 4\sin^{2}\alpha$
Or;k $sin\alpha = \frac{1}{\sqrt{2}}$
Subject: Mathematics — Properties of triangle — Cosine formula
Difficulty: Tough

Question: The lengths of sides of a triangle are three consecutive natural
numbers and its largest angle is twice the smaller one. Perimeter of the
triangle is

1) 12
2) 14
3) 15
4) 16

Correct option: 3

Solution:
Let the lengths of the sides be n, n + 1, n + 2 where n $\in$
N.
From the question, the largest angle opposite to the side n + 2 is
2$\theta$ while the smallest angle opposite to the side n is $\theta$.
Now,
$cos\theta = \frac{(n + 2)^{2} + (n + 1)^{2} - n^{2}}{2(n + 1)(n + 2)} = \frac{n + 5}{2(n + 2)}$
And $cos2\theta = \frac{n^{2} + (n + 1)^{2} - (n + 2)^{2}}{2n(n + 1)}$
$$= + 1)}$$
 But $cos2\theta = 2\cos^{2}\theta - 1;\ so\ \frac{n - 3}{2n} = 2\left\{ \frac{n + 5}{2(n + 2)} \right\}^{2} - 1$
or $\frac{n - 3}{2n} = \frac{(n + 5)^{2}}{2(n + 2)^{2}} - 1$
or $(n - 3)(n + 2)^{2} = n\left\{ (n + 5)^{2} - 2(n + 2)^{2} \right\}$
or
$(n - 3)\left( n^{2} + 4n + 4 \right) = n\left( - n^{2} + 2n + 17 \right)$
or n^3 + n^2 - 8n - 12 = - n^2 + 2n^2 + 17n
or 2n^3 - n^2 - 25n-12 = ­0
or (n - 4) (2n^2 + 7n + 3) = 0
or n = 4 or 2n^2 + 7n +3 =
Subject: Mathematics — Trigonometry — Trigonometric identities
Difficulty: Easy

Question: If A + B = $\frac{\pi}{4},$  then (tan⁡A+1)(tan⁡B+1) is equal to.

1) 1
2) 2
3) $\sqrt{3}$
4) $\frac{1}{2}$

Correct option: 2

Solution:
$\frac{tanA + tanB}{1 - tanAtanB} = 1$
$\Rightarrow tanA + tanB + tanAtanB + 1 = 1 + =$
$\Rightarrow (1 + tanA)(1 + tanB) = 2$
Subject: Mathematics — Probability — Independent events
Difficulty: Tough

Question: Let |X| denote the number of elements in a set X. Let S = be a sample space, where each element is equally likely
to occur. If A and B are independent events associated with S, then the
number of ordered pairs (A, B) such that [IMAGE] equals.


Correct option: 1

Solution:
Number of ordered pair (A, B)
Subject: Mathematics —  — Equation of sides
Difficulty: Tough

Question: Let the equations of two sides of a triangle be 3x - 2y + 6 = 0
and 4x + 5y - 20 = 0. If the orthocentre of this triangle is at (1,
1), then the equation of its third side is

1) 26x - 122y - 1675 = 0
2) 26x + 61y + 1675 = 0
3) 122y - 26x - 1675 = 0
4) 122y + 26x + 1675 = 0

Correct option: 1

Solution:
4x + 5y = 20..........(i)
3x - 2y + 6 = 0..........(ii)
Orthocenter (1, 1)
Line $$ to 4x + 5y = 20 and passes (1, 1) is 5x - 4y = 1..........(iii)
and line $$to 3x - 2y + 6 = 0 and passes (1, 1) is 2x + 3y = 5..........(iv)
on solving (i) & (iv) pt $\left( \frac{35}{2}, - 10 \right)$
on solving (ii) & (iii) pt $\left( - 13, - \frac{33}{2} \right)$
side BC is y + 10 =
$\frac{- \frac{33}{2} + 10}{- 13 - \frac{35}{2}}\left( x - \frac{35}{2} \right)$
$\Rightarrow y + 10 = \frac{13}{61}\left( x - \frac{35}{2} \right)$
$\Rightarrow 26x - 122y - 1675 = 0$
Subject: Mathematics — Logarithm and its applications — Logarithmic inequality
Difficulty: Moderate

Question: The solution set of the inequality
$\log_{sin\left( \frac{\pi}{3} \right)}\left( x^{2} - 3x + 2 \right) \geq 2$
is

1) $\left( \frac{1}{2},2 \right)$
2) $\left( 1,\frac{5}{2} \right)$
3) $\lbrack\frac{1}{2},1) \cup (2,\frac{5}{2}\rbrack$
4) $(1,2)$

Correct option: 3

Solution:
$\log_{\sin(\pi/3)}\left( x^{2} - 3x + 2 \right) \geq 2$
$\log_{\sin\left( \frac{\pi}{3} \right)}x^{2} - 3x + 2 \geq 2\log_{\sin\left( \frac{\pi}{3} \right)}\sin\left( \frac{\pi}{3} \right)$
$\because\sin\frac{\pi}{3} = \frac{\sqrt{3}}{2} < 1$
$\Rightarrow x^{2} - 3x + 2 \leq \left( \sin\left( \frac{\pi}{3} \right) \right)^{2}$
$\Rightarrow x^{2} - 3x + 2 \leq 3/4$
$\Rightarrow 4x^{2} - 12x + 5 \leq 0$
$\Rightarrow (2x - 5)(2x - 1) \leq 0$
$x \in \left\lbrack \frac{1}{2},\frac{5}{2} \right\rbrack\ \ldots.(A)$
Also, $x^{2} - 3x + 2 > 0$ for log to be defined.
$\Rightarrow (x - 1)(x - 2) > 0$
$\Rightarrow x \in ( - \infty,1) \cup (2,\infty)\ \ \ \ldots.(B)$
Taking intersection of (A) and (B)
$x \in \left\lbrack \frac{1}{2},1 \right) \cup \left( 2,\frac{5}{2} \right\rbrack$
Subject: Mathematics —  — 

Question: The relation R = on the set A = is

1) Reflexive but not symmetric
2) Reflexive but not transitive
3) Symmetric and transitive
4) Neither symmetric nor transitive

Correct option: 1

Solution:
(1, 1), (2, 2) (3, 3) ∈R ⇒R is reflexive
(1, 2) ∈R but (2, 1) ∉R ⇒R is not symmetric
(1, 2) ∈R, (2, 3) ∈R, (1, 3) ∈R ⇒R is transitive.
Subject: Mathematics — Circles — Area
Difficulty: Easy

Question: Area of the region bounded by points (x,y) satisfying x^2 + y^2
≤ |x| + |y| is
x^2 + y^2 ≤ |x| + |y|

1) 2π + 4
2) $\frac{\pi}{2} + 1$
3) π
4) π + 2

Correct option: 4

Solution:
[IMAGE] Required area = 4[A(ΔOPB) + A(semi PCB)]
A(ΔOAB) = $\frac{1}{2}$
A(semi PCB) = $\frac{\pi}{4}$
Hence = 4 $\left( \frac{1}{2} + \frac{\pi}{4} \right)$
= π + 2
Subject: Mathematics — Limits, continuity and derivability — Indeterminate form
Difficulty: Moderate

Question: Suppose that a and b (b $\neq$ a, b > a) are real positive
numbers then the value of [IMAGE] has the value equals to.


Correct option: 4

Solution:
Obviously limit is of the form $1^{\infty}$
Subject: Mathematics — Sets — Power set
Difficulty: Moderate

Question: If a set has 2 elements then the difference between the subsets
of its powers set and number of its subset is

1) 2
2) 8
3) 10
4) 12

Correct option: 4

Solution:
Number of subsets $= 2^{2} = 4$
Number of subsets of power $= 2^{4} = 16$
Subject: Mathematics — Integration — Integration using substitution
Difficulty: Tough

Question: Let n ≥ 2 be a natural number and
$0 < \theta < \pi/2.$ Then $\int\frac{\left( \sin^{n}\theta - sin\theta \right)^{\frac{1}{n}}cos\theta}{\sin^{n + 1}\theta}d\theta$ is
equal to

1) $\frac{n}{n^{2} - 1}\left( 1 - \frac{1}{\sin^{n + 1}\theta} \right)^{\frac{n + 1}{n}} + C$
2) $\frac{n}{n^{2} - 1}\left( 1 - \frac{1}{\sin^{n - 1}\theta} \right)^{\frac{n + 1}{n}} + C$
3) $\frac{n}{n^{2} + 1}\left( 1 - \frac{1}{\sin^{n - 1}\theta} \right)^{\frac{n + 1}{n}} + C$
4) $\frac{n}{n^{2} - 1}\left( 1 + \frac{1}{\sin^{n - 1}\theta} \right)^{\frac{n + 1}{n}} + C$

Correct option: 2

Solution:
$\int\frac{\left( \sin^{n}\theta - sin\theta \right)^{\frac{1}{n}}cos\theta}{\sin^{n + 1}\theta}d\theta$
$= \int\frac{\left( t^{n} - t \right)^{\frac{1}{n}}dt}{t^{n + 1}}$ (Put
$sin\theta = t$)
$= \int\frac{t\left( 1 - \frac{1}{t^{n - 1}} \right)^{\frac{1}{n}}}{t^{n + 1}}dt = \int\frac{\left( 1 - \frac{1}{t^{n - 1}} \right)^{\frac{1}{n}}}{t^{n}}dt$
Put $1 - \frac{1}{t^{n - 1}} = z \Rightarrow \frac{(n - 1)}{t^{n}}$ dt =
dz
$\Rightarrow I = \frac{1}{n - 1}\int z^{\frac{1}{n}}dz = \frac{z^{\frac{1}{n} + 1}}{\left( \frac{1}{n} + 1 \right)(n - 1)} + c = \frac{n\left( 1 - t^{1 - n} \right)^{\frac{1}{n} + 1}}{n^{2} - 1} + c$
$= \frac{n}{n^{2} - 1}\left( 1 - \frac{1}{\sin^{n - 1}\theta} \right)^{\frac{n + 1}{n}} + c$
Subject: Mathematics — Conic Sections — Tangent to parabola
Difficulty: Easy

Question: If two different tangents of y^2 = 4x are normal chords to x^2
= 4by then
 y^2 = 4x

1) $|b| > \frac{1}{2\sqrt{2}}$
2) $|b| < \frac{1}{2\sqrt{2}}$
3) $|b| > \frac{1}{\sqrt{2}}$
4) $|b| < \frac{1}{\sqrt{2}}$

Correct option: 2

Solution:
Tangent to y^2 = 4x → y thok = mx + $\frac{1}{m}$
Normal to x^2 = 4by → y = mx + 2b + $\frac{b}{m^{2}}$
$\Rightarrow$ $\frac{1}{m} = 2b + \frac{b}{m^{2}}$ $\Rightarrow$ (2b)
m^2 - m + b = 0
$\Rightarrow$ D > 0 $\Rightarrow$ $|b| < \frac{1}{2\sqrt{2}}$
Subject: Mathematics — Probability — Definition of probability
Difficulty: Tough

Question: A person throws dice, one the common cube and the other regular
terahedron, the number on the lowest face being taken in the case of a
tetrahedron. The chance that the sum of numbers thrown is not less than
5 is.,d O;fDr nks iklksa dks QSadrk gS ftlesa,d lk/kkj.k ?ku rFkk nwljk

1) $\frac{1}{4}$
2) $\frac{3}{4}$
3) $\frac{4}{5}$
4) $\frac{5}{6}$

Correct option: 2

Solution:
Total number of combinations of numbers on the cube and the
tetrahedon = 6 × 4 = 24
Favourable number of ways of getting a sum not less than 5
= sum of coefficients of x^6, x^7,.........x^10 in the product
= (x + x^2 + x^3 + x^4 + x^5 + x^6) (x + x^2 + x^3 + x^4)
= (x^2 + 2x^3 + 3x^4 + 4x^5 + 4x^6 + 4x^7 + 3x^8 + 2x^9 +
x^10) = 4 + 4 + 4 + 3 + 2 + 1 = 18
$\therefore$Required
probability$= \frac{18}{6 \times 4} = \frac{3}{4}$.
gy VsVªkgsMksu ij dqy = 6
× 4 = 24
5 ls de ugha gksus vuqdwy 
mRikn esa x^6, x^7,.........x^10 ds xq.kkad 
= (x + x^2 + x^3 + x^4 + x^5 + x^6) (x + x^2 + x^3 + x^4)
= (x^2 + 2x^3 + 3x^4 + 4x^5 + 4x^6 + 4x^7 + 3x^8 + 2x^9 +
x^10) = 4 + 4 + 4 + 3 + 2 + 1 = 18
 laHkkouk $= \frac{18}{6 \times 4} = \frac{3}{4}$.
Subject: Mathematics — Properties of triangle — Tangent rule
Difficulty: Easy

Question: In an equilateral $\Delta ABC,r,R,r_{1}$, form (where symbols
have usual meanings)
leckgq f=Hkqt $\Delta ABC,$ $r,R,r_{1}$,

1) an A.P.
2) an H.P.
3) a G.P.
4) an A.G.P

Correct option: 1

Solution:
$a = b = c \Rightarrow s = \frac{3a}{2}$
$r = (s - a)tan\frac{A}{2} = \frac{a}{2\sqrt{3}}$
$R = \frac{abc}{4\Delta} = \frac{a^{3}}{\sqrt{3}a^{2}} = \frac{a}{\sqrt{3}} = \frac{2a}{2\sqrt{3}}$
$r_{1} = Stan\frac{A}{2} = \frac{3a}{2\sqrt{3}}$
$\therefore r,R,r_{1},\ form\ an\ AP$
Subject: Mathematics — Quadratic Equations — Relation between roots and coefficients
Difficulty: Tough

Question: If $\alpha,\beta$ are the roots of $9x^{2} + 6x + 1 = 0$, then
the equation with roots $\frac{1}{\alpha},\frac{1}{\beta}$ is

1) $2x^{2} + 3x + 18 = 0$
2) $x^{2} + 6x - 9 = 0$
3) $x^{2} + 6x + 9 = 0$
4) $x^{2} - 6x + 9 = 0$

Correct option: 3

Solution:
Replace x by $\frac{1}{x}$
Subject: Mathematics — Trigonometry — Complementary angles
Difficulty: Easy

Question: The value of  $\frac{tan20^{\circ}}{cot70^{\circ}}$ is equal
to.

1) 2
2) 3
3) 1
4) (tan20°)^2

Correct option: 3

Solution:
$\cot\left( 90^{\circ} - \theta \right) = \tan\theta$
$\therefore\frac{\tan 20^{\circ}}{\cot 70^{\circ}} = \frac{\tan 20^{\circ}}{\cot\left( 90^{\circ} - 20^{\circ} \right)}$
$= \frac{tan20^{\circ}}{tan20^{\circ}} = 1$
Subject: Mathematics — Limits, continuity and derivability — Differentiability
Difficulty: Tough

Question: The function f(x) = maximum
$\left\{ \sqrt{x(2 - x)},2 - x \right\}$ is non-differentiable at x
equal to

1) 0, 1
2) 1, 3
3) 2, 3
4) 1, 2

Correct option: 4

Solution:
[IMAGE] At x = 2
L.H.D$= h\overset{Lt}{\rightarrow}0^{+}\frac{\sqrt{(2 - h)(2 - (2 - h))} - 0}{- h}$
$= h\overset{Lt}{\rightarrow}0^{+}\frac{\sqrt{2 - h}}{- \sqrt{h}}\left( \frac{\sqrt{2}}{0}\ form\ \right)$
= not finite.
Subject: Mathematics — Vectors — Dot and cross product
Difficulty: Moderate

Question: If [IMAGE] and [IMAGE] then find
the value of[IMAGE].


Answer: 03.0

Solution:
Subject: Mathematics — Binomial Theorem — General Term
Difficulty: Tough

Question: [IMAGE] x^12 ^nC_r ⋅ 2^k, 
[IMAGE] (2r> n)


Answer: 11.5

Solution:
[IMAGE] Coeff. of x^12 = [IMAGE]
Subject: Mathematics — Quadratic Equations — Nature of roots
Difficulty: Tough

Question: The values of for which roots of the equation
$(a - 2)x^{2} - (a - 4)x - 2 = 0$ are real and distinct and their
difference is greater than or equal to 3, are

1) $\ \left\{ \frac{3}{2},2 \right\}$
2) $\left\lbrack \frac{3}{2},3 \right\rbrack - \{ 2\}$
3) $\left( - \infty,\frac{3}{2} \right\rbrack \cup \lbrack 3,\infty)$
4) $\left\lbrack \frac{3}{2},3 \right\rbrack$

Correct option: 2

Solution:
$D > 0 \Rightarrow a^{2} > 0 \Rightarrow a \in R - \{ 0,2\}$
$|Α - \beta| \geq 3$
$$\Rightarrow (\alpha - - 4\alpha\beta \geq 9$$
$\Rightarrow \left( \frac{a - 4}{a - 2} \right)^{2} + \frac{4.2}{a - 2} \geq 9$
$(2a - 3)(a - 3) \leq 0$
$$\Rightarrow a \in \left\lbrack \right\rbrack$$
Subject: Mathematics — Parabola — Area enclosed by parabola
Difficulty: Moderate

Question: A parabola whose axis is parallel to x-axis passes through three
points (2, 0), (0, -1) and (6, 1) then area enclosed by the parabola
and line x - 2y - 8 = 0 is equal to


Correct option: 1

Solution:
Equation of parabola will
Subject: Mathematics — Conic Sections — Hyperbola
Difficulty: Tough

Question: If a variable line has intercepts of e_1 and e_2on the
co-ordinate axes, where $\frac{e_{1}}{2}\&\frac{e_{2}}{2}$are the
eccentricities of a hyperbola and its conjugate, then the line always
touches a fixed circle x^2 + y^2 = r^2, where is
 o ml mRdsUærk,sa
x^2 + y^2 = r^2 djrh

1) 1
2) 2
3) 3
4) 4

Correct option: 2

Solution:
Line is js[kk $xe_{2} + ye_{1} - e_{1}e_{2} = 0$.........(i)
Also rFkk $4\left( e_{1}^{2} + e_{2}^{2} \right) = e_{1}^{2}e_{2}^{2}$.........(ii)
It is tangent to circle if;g o`Ùk
Subject: Mathematics — Vectors — Box product
Difficulty: Tough

Question: [IMAGE] represents the angle between
value of [IMAGE] is [IMAGE] The value of


Correct option: 4

Solution:
Subject: Mathematics — Vectors — Projection formula
Difficulty: Tough

Question: Let [IMAGE] and
$\overrightarrow{b} = \imath + 2j + k$be two vectors. Consider a vector
[IMAGE] R. If the projection of [IMAGE] on
the vector [IMAGE] is [IMAGE], then the
minimum value of [IMAGE] equal to


Correct option: 1

Solution:
[IMAGE] $= 6\left( \alpha^{2} - 2\alpha + 4 \right) =$ min value 18
Subject: Mathematics — Application of derivatives — Monotonocity
Difficulty: Moderate

Question: The interval on which f(x) = 2x^3 + 9x^2 + 12x - 1 is
decreasing in

1) $( - 1,\infty)$
2) $( - 2, - 1)$
3) $( - \infty, - 2)$
4) $( - 1,1)$

Correct option: 2

Solution:
$f(x) = 2x^{3} + 9x^{2} + 12x - 1$
$f'(x) = 6x^{2} + 18x + 12$
For decreasing
$f'(x) < 0$
$6x + 18x + 12 < 0$
$x^{2} + 3x + 2 < 0$
$(x + 1)(x + 2) < 0$
$x \in ( - 2, - 1)$
Subject: Mathematics — Limits — Limits of standard functions
Difficulty: Easy

Question: The value of
$\ \lim_{x \rightarrow 0}\mspace{2mu}\frac{ln(1 + 2sin3x)}{sinx}$ is
equal to


Answer: 6

Solution:
$\lim_{x \rightarrow 0}\mspace{2mu}\frac{\mathcal{l}n(1 + 2sin3x)}{2sin3x} \cdot \frac{2sin3x}{sinx}$
= 1(2×3) = 6
Subject: Mathematics — Probability — Binomial probability distribution
Difficulty: Moderate

Question: In a workshop there are five machines and the probability of any
one of them to be out of service on a day is $\frac{1}{4}$. If the
probability that at most two machines will be out of service on the same
day is $\left( \frac{3}{4} \right)^{3}k$, is equal to



Solution:
Required probability = when no. machine has fault + when only
one machine has fault + when only two machines have fault.
$= \ ^{5}C_{0}\left( \frac{3}{4} \right)^{5} + \ ^{5}C_{1}\left( \frac{1}{4} \right)\left( \frac{3}{4} \right)^{4} + \ ^{5}C_{2}\left( \frac{1}{4} \right)^{2}\left( \frac{3}{4} \right)^{3}$
$= \frac{243}{1024} + \frac{405}{1024} + \frac{270}{1024} = \frac{918}{1024} = \frac{459}{512} = \frac{27 \times 17}{64 \times 8}$
$= \left( \frac{3}{4} \right)^{3} \times k = \left( \frac{3}{4} \right)^{3} \times \frac{17}{8}$
$\therefore k = \frac{17}{8}$
Subject: Mathematics —  — 

Question: A triangle ABC has the feature, [IMAGE] then

1) 2
2) 3
3) 4
4) 5

Correct option: 3

Solution:
= a + b + c
= = sin A + sin B + sin C (using a = 2R sin
A etc.)
⇒ 4 sin A sin B sin C = sin A + sin B + sin C
[IMAGE] Also 2r = R ⇒ triangle is equilateral
Subject: Mathematics — Sets — Operation on sets
Difficulty: Tough

Question: If A and B be two sets such that n(A) = 15, n(B) = 25, then
number of possible values of $n(A\Delta B)$ (symmetric difference of A
and B) is

1) 30
2) 16
3) 26
4) 40

Correct option: 2

Solution:
$n(A\Delta B) = n(A \cup B) - n(A \cap B)$
for maximum $n(A\Delta B),n(A \cup B)$ should be maximum and
$n(A \cap B)$ is minimum
For $n(A \cap B)$ to be minimum, $A \cap B = \phi$
$\Rightarrow n(A \cup B) = 25 + 15 = 40$
$n(A\Delta B) = 40$
For $n(A \cap B)$ to be maximum, $A \subset B$
$n(A \cap B) = 15$
$\Rightarrow n(A \cup B) = 25$
$\Rightarrow n(A\Delta B) = 25 - 15 = 10$
$\Rightarrow \ Range\ of\ (A\Delta B) = \{ 10,\ 12,\ 14,\ 16,\ldots..,\ 40\}$
Subject: Mathematics — Vectors — Volume of Parallelopiped
Difficulty: Tough

Question: The three vectors [IMAGE] taken two at a time
form three planes. The three unit vectors drawn perpendicular to these
three planes form a parallelepiped of volume:
rhu nks&nks ysdj rhu
leryksa dk fuekZ.k A bu rhu leryksa ds yEcor~ rhu bdkbZ
lfn'k,d lekUrj "kV~Qyd

1) $\frac{1}{3}$
2) 4
3) $\frac{3\sqrt{3}}{4}$
4) $\frac{4}{3\sqrt{3}}$

Correct option: 4

Solution:
Subject: Mathematics — Straight Lines — Pair of straight lines
Difficulty: Moderate

Question: If pair of lines [IMAGE] and another pair of lines
given by [IMAGE] have exactly one line common and other
lines represented by them are perpendicular, then

1) a + h = 3
2) a + h = 6
3) angle between the lines represented by first pair is
4) a = 9, h = -4

Correct option: 4

Solution:
[IMAGE] On solving a = 9, h = -4
Subject: Mathematics — Parabola — Parabola and triangle
Difficulty: Tough

Question: An equilateral triangle ABC is inscribed in the parabola
[IMAGE] and one of the sides of the equilateral triangle
has the gradient 2. If the sum of x -coordinates of all the vertices of
the triangle is expressed in the form of [IMAGE] (where
[IMAGE] are co prime natural number), then the value of

1) 6
2) 2
3) 4
4) 8

Correct option: 4

Solution:
[IMAGE] To find [IMAGE] And [IMAGE] Hence, [IMAGE] Now, [IMAGE] Taking the +ve sign, we get
[IMAGE] Taking -ve sign, we get
[IMAGE] We get,
Subject: Mathematics — Integrals — Integration by partial fraction
Difficulty: Moderate

Question: If
$\int\frac{x^{4} + 1}{x\left( x^{2} + 1 \right)^{2}}dx = A\ In|x| + \frac{B}{1 + x^{2}} + C$
then

1) $A = 1,B = - 1$
2) $A = - 1,B = 1$
3) $A = B = 1$
4) $A = B = - 1$

Correct option: 3

Solution:
$\int\frac{x^{4} + 1}{x{(x^{2} + 1)}^{2}}dx = Alog|x| + \frac{B}{1 + x^{2}} + C$
$= \int\frac{(x^{4} + 1)x}{x^{2}{(x^{2} + 1)}^{2}}dx$ Let
$x^{2} + 1 = y,2xdx = dy$
$= \frac{1}{2}\int\frac{(y - 1)^{2} + 1}{(y - 1)y^{2}}dy = \frac{1}{2}\int\frac{y^{2} - 2y + 2}{(y - 1)y^{2}}dy$
$= \frac{y^{2} - 2y + 2}{(y - 1)y^{2}} = \frac{A}{y} + \frac{B}{y^{2}} + \frac{C}{y - 1} = \frac{- 2}{y^{2}} + \frac{1}{y - 1}$
$= \frac{1}{2}\int\frac{y^{2} - 2y + 2}{(y - 1)y^{2}}dy = \frac{1}{2}\int(\frac{- 2}{y^{2}} + \frac{1}{y - 1})dy = \frac{1}{2}(\frac{2}{y} + log(y - 1)) + C$
$= \frac{1}{2}(\frac{2}{x^{2} + 1}) + \frac{1}{2}log|x^{2} + 1 - 1| + C$
$= \frac{1}{x^{2} + 1} + log|x| + C$
$A = 1,B = 1$
Subject: Mathematics — Properties of triangle — Area of triangle
Difficulty: Easy

Question: In $\Delta ABC$, if r=1, R=3 and S=5, then the value of
$a^{2} + b^{2} + c^{2}$ is
$\Delta ABC$ r=1, R=3 rFkk S=5] rc $a^{2} + b^{2} + c^{2}$dk
eku


Answer: 24

Solution:
$r = \frac{\Delta}{S} \Rightarrow \Delta = 5$
$\Delta = \frac{abc}{4R} \Rightarrow abc = 60$
$\Delta^{2} = s(s - a)(s - b)(s - c)$
$5 = (5 - a)(5 - b)(5 - c)$
$= 125 - 25(a + b + c) + 5(ab + bc + ca) - abc$
$a^{2} + b^{2} + c^{2} = 24$
Subject: Mathematics — Binomial Theorem — Sum of Binomial Coefficient
Difficulty: Tough

Question: [IMAGE] x^10 k, k 
 (log_102=0.3010)

1) 6
2) 7
3) 8
4) 5

Correct option: 2

Solution:
Coefficient of x^10 = ^21C_0 + ^21C_1 +..... + ^21C_10
k = 2^21 - 1^ = 220
no. of digits in k = 7
([IMAGE] log_10k = 20 × 0.3010 = 6.020)
Subject: Mathematics — Vectors — Box product
Difficulty: Moderate

Question: [IMAGE] with the plane of [IMAGE] and
[IMAGE] and angle between [IMAGE] is
[IMAGE] then the value of[IMAGE] equals


Correct option: 3

Solution:
Subject: Mathematics — Complex Numbers — Cube root of Unity
Difficulty: Moderate

Question: [IMAGE] ? ω -
[IMAGE] (4)



Solution:
Subject: Mathematics — Complex Numbers — Properties of modulus and argument
Difficulty: Moderate

Question: If z_1, z_2, z_3 are three points lying on the circle |z| =
2 then the minimum value of
$\left| z_{1} + z_{2} \right|^{2} + \left| z_{2} + z_{3} \right|^{2}\left| z_{3} + z_{1} \right|^{2}$
is equal to

1) 6
2) 12
3) 15
4) 24

Correct option: 2

Solution:
$\left| z_{1} + z_{2} \right|^{2} + \left| z_{2} + z_{3} \right|^{2} + \left| z_{3} + z_{1} \right|^{2}$
$\mathbf{=}2\left( \left| z_{1} \right|^{2} + \left| z_{2} \right|^{2} + \left| z_{3} \right|^{2} \right)$+$\left( z_{1}\overrightarrow{z_{2}} + z_{2}\overrightarrow{z_{1}} + z_{2}\overrightarrow{z_{3}} + z_{3}\overrightarrow{z_{2}} + z_{3}\overrightarrow{z_{1}} + z_{1}\overrightarrow{z_{3}} \right)$
Subject: Mathematics — Straight Lines — Locus problem
Difficulty: Tough

Question: Locus of the points which are at equal distance from 3x + 4y -
11 = 0 and 12x + 5y + 2 = 0 and which is nearer to origin is
fcUnqvksa dk fcUnqiFk tks js
2 = 0 ls cjkcj nwjh ij

1) $21x - 77y + 153 = 0$
2) $99x + 77y - 133 = 0$
3) $7x - 11y = 19$
4) None of these buesa

Correct option: 2

Solution:
$\left( \frac{3x + 4y - 11}{5} \right) = - \left( \frac{12x + 5y + 2}{13} \right)$
is required bisector as the angle contains origin
$\left( \frac{3x + 4y - 11}{5} \right) = - \left( \frac{12x + 5y + 2}{13} \right)$
dks.k v}Zd
Subject: Mathematics —  — 

Question: If | Z - i | ≤ 2 and Z1 = 5 + 3i, then the maximum value of |
iZ + Z1 | is

3) 7
4) 9

Correct option: 3

Solution:
| iZ + Z1 | = | i (Z - i) + (Z1 - 1) | ≤ | i (Z - i) | + |
Z1 - 1 |
≤ | i | | Z - i | + | 4 + 3i |
≤ 2 +[IMAGE] = 7
Subject: Mathematics — Limits — Limits of standard functions
Difficulty: Easy

Question: $\lim_{x \rightarrow 0}\mspace{2mu}\frac{\sqrt{1 - cos2x}}{x} =$

1) $\sqrt{2}$
2) $- \sqrt{2}$
3) 0
4) Does not exis

Correct option: 4

Solution:
$\lim_{x \rightarrow 0}\mspace{2mu}\frac{\sqrt{2}|sinx|}{x}$
L.H.L $= - \sqrt{2}$
R.H.L. $= \sqrt{2}$
Both are not equal.
Subject: Mathematics —  — 

Question: The value of [IMAGE] is

4) None of these

Correct option: 3

Solution:
[IMAGE] Let x6 -1 = t
Subject: Mathematics — Probability — Classic definition of Probability
Difficulty: Tough

Question: Let A = set of all 3 × 3 matrices having entries +1 or -1, if a
matrix B from the set A is chosen randomly, then the probability that
the product of the elements of any row or any column of B is -1 is

1) $\frac{1}{32}$
2) $\frac{1}{8}$
3) $\frac{1}{16}$
4) $\frac{9}{32}$

Correct option: 1

Solution:
Total = 2^9
$\ Let\ B = \left| a_{1}\ \ \ \ a_{2}\ \ \ \ a_{3}\ b_{1}\ \ \ \ b_{2}\ \ \ \ b_{3}\ c_{1}\ \ \ \ c_{2}\ \ \ \ c_{3}\ \right|$
The first row has 4 combination
$\{ 1,\ 1, - 1\},\{ - 1,\ 1,\ 1\},\{ 1, - 1,\ 1\},\{ - 1, - 1, - 1\}$.
Now for each combination the rest entries can be filled in 4 ways
Hence, Probability is $\frac{16}{2^{9}} = \frac{1}{32}$
Subject: Mathematics — Pemutation and combination — Permutations and Combinations
Difficulty: Moderate

Question: Number of natural number's less than 7000 using digit 0, 1, 3,
7, 9 (repetition allowed) is

1) 375
2) 275
3) 274
4) 374

Correct option: 4

Solution:
Ist 2st 3st 4st
position position position position
There are 3 way to fill I^st position (0, 1, 3) for remaining digits
there are 5 ways.
(0000 is not included)
hence required ans = 3.5.5.5 - 1 = 375- 1 = 374
Subject: Mathematics — Parabola — Circle and parabola
Difficulty: Moderate

Question: If [IMAGE] then coordinates of point on the
parabola [IMAGE] which is at shortest distance from the


Correct option: 3

Solution:
[IMAGE] Normal to parabola is [IMAGE] Center of circle is (-5, 22)
Subject: Mathematics — Permutations and Combinations — Circular permutation
Difficulty: Moderate

Question: There are n persons sitting around a circular table. They start
singing a 2 minute song in pairs such that no two persons sitting
together will sing together. This process is continued for 28 minutes.


Answer: 22.75

Solution: