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20 maths questions

Q41 Complex numbers 2 Roots of Unity and Cyclotomic Properties View
Let $w = \frac { \sqrt { 3 } + \mathrm { i } } { 2 }$ and $P = \left\{ w ^ { n } : n = 1,2,3 , \ldots \right\}$. Further $H _ { 1 } = \left\{ \mathrm { z } \in \mathbb { C } : \operatorname { Re } z > \frac { 1 } { 2 } \right\}$ and $H _ { 2 } = \left\{ \mathrm { z } \in \mathbb { C } : \operatorname { Re } z < \frac { - 1 } { 2 } \right\}$, where $\mathbb { C }$ is the set of all complex numbers. If $z _ { 1 } \in P \cap H _ { 1 }$, $z _ { 2 } \in P \cap H _ { 2 }$ and $O$ represents the origin, then $\angle z _ { 1 } O z _ { 2 } =$
(A) $\frac { \pi } { 2 }$
(B) $\frac { \pi } { 6 }$
(C) $\frac { 2 \pi } { 3 }$
(D) $\frac { 5 \pi } { 6 }$
Q42 Exponential Equations & Modelling Solve Exponential Equation for Unknown Variable View
If $3 ^ { x } = 4 ^ { x - 1 }$, then $x =$
(A) $\frac { 2 \log _ { 3 } 2 } { 2 \log _ { 3 } 2 - 1 }$
(B) $\frac { 2 } { 2 - \log _ { 2 } 3 }$
(C) $\frac { 1 } { 1 - \log _ { 4 } 3 }$
(D) $\frac { 2 \log _ { 2 } 3 } { 2 \log _ { 2 } 3 - 1 }$
Q43 Matrices Matrix Power Computation and Application View
Let $\omega$ be a complex cube root of unity with $\omega \neq 1$ and $P = \left[ p _ { i j } \right]$ be a $n \times n$ matrix with $p _ { i j } = \omega ^ { i + j }$. Then $P ^ { 2 } \neq 0$, when $n =$
(A) 57
(B) 55
(C) 58
(D) 56
Q44 Modulus function Graphing functions involving modulus View
The function $f ( x ) = 2 | x | + | x + 2 | - | | x + 2 | - 2 | x | |$ has a local minimum or a local maximum at $x =$
(A) $- 2$
(B) $\frac { - 2 } { 3 }$
(C) $2$
(D) $\frac { 2 } { 3 }$
Q45 Sequences and Series Limit Evaluation Involving Sequences View
For $a \in \mathbb { R }$ (the set of all real numbers), $a \neq - 1$, $$\lim _ { \mathrm { n } \rightarrow \infty } \frac { \left( 1 ^ { a } + 2 ^ { a } + \ldots + \mathrm { n } ^ { a } \right) } { ( n + 1 ) ^ { a - 1 } [ ( n a + 1 ) + ( n a + 2 ) + \ldots + ( n a + n ) ] } = \frac { 1 } { 60 }$$ Then $a =$
(A) 5
(B) 7
(C) $\frac { - 15 } { 2 }$
(D) $\frac { - 17 } { 2 }$
Q46 Circles Circle Equation Derivation View
Circle(s) touching $x$-axis at a distance 3 from the origin and having an intercept of length $2 \sqrt { 7 }$ on $y$-axis is (are)
(A) $x ^ { 2 } + y ^ { 2 } - 6 x + 8 y + 9 = 0$
(B) $x ^ { 2 } + y ^ { 2 } - 6 x + 7 y + 9 = 0$
(C) $x ^ { 2 } + y ^ { 2 } - 6 x - 8 y + 9 = 0$
(D) $x ^ { 2 } + y ^ { 2 } - 6 x - 7 y + 9 = 0$
Q47 Vectors 3D & Lines MCQ: Relationship Between Two Lines View
Two lines $L _ { 1 } : x = 5 , \frac { y } { 3 - \alpha } = \frac { z } { - 2 }$ and $L _ { 2 } : x = \alpha , \frac { y } { - 1 } = \frac { z } { 2 - \alpha }$ are coplanar. Then $\alpha$ can take value(s)
(A) 1
(B) 2
(C) 3
(D) 4
Q48 Sine and Cosine Rules Circumradius or incircle radius computation View
In a triangle $P Q R$, $P$ is the largest angle and $\cos P = \frac { 1 } { 3 }$. Further the incircle of the triangle touches the sides $P Q , Q R$ and $R P$ at $N , L$ and $M$ respectively, such that the lengths of $P N , Q L$ and $R M$ are consecutive even integers. Then possible length(s) of the side(s) of the triangle is (are)
(A) 16
(B) 18
(C) 24
(D) 22
Q49 Complex Numbers Argand & Loci Intersection of Loci and Simultaneous Geometric Conditions View
Let $S = S _ { 1 } \cap S _ { 2 } \cap S _ { 3 }$, where $$S _ { 1 } = \{ z \in \mathbb { C } : | \mathrm { z } | < 4 \} , \quad S _ { 2 } = \left\{ z \in \mathbb { C } : \operatorname { Im } \left[ \frac { z - 1 + \sqrt { 3 } i } { 1 - \sqrt { 3 } i } \right] > 0 \right\}$$ and $S _ { 3 } = \{ z \in \mathbb { C } : \operatorname { Re } z > 0 \}$.
Area of $S =$
(A) $\frac { 10 \pi } { 3 }$
(B) $\frac { 20 \pi } { 3 }$
(C) $\frac { 16 \pi } { 3 }$
(D) $\frac { 32 \pi } { 3 }$
Q50 Complex Numbers Argand & Loci Distance and Region Optimization on Loci View
Let $S = S _ { 1 } \cap S _ { 2 } \cap S _ { 3 }$, where $$S _ { 1 } = \{ z \in \mathbb { C } : | \mathrm { z } | < 4 \} , \quad S _ { 2 } = \left\{ z \in \mathbb { C } : \operatorname { Im } \left[ \frac { z - 1 + \sqrt { 3 } i } { 1 - \sqrt { 3 } i } \right] > 0 \right\}$$ and $S _ { 3 } = \{ z \in \mathbb { C } : \operatorname { Re } z > 0 \}$.
$\min _ { z \in S } | 1 - 3 i - z | =$
(A) $\frac { 2 - \sqrt { 3 } } { 2 }$
(B) $\frac { 2 + \sqrt { 3 } } { 2 }$
(C) $\frac { 3 - \sqrt { 3 } } { 2 }$
(D) $\frac { 3 + \sqrt { 3 } } { 2 }$
Q51 Probability Definitions Finite Equally-Likely Probability Computation View
A box $B _ { 1 }$ contains 1 white ball, 3 red balls and 2 black balls. Another box $B _ { 2 }$ contains 2 white balls, 3 red balls and 4 black balls. A third box $B _ { 3 }$ contains 3 white balls, 4 red balls and 5 black balls.
If 1 ball is drawn from each of the boxes $B _ { 1 } , B _ { 2 }$ and $B _ { 3 }$, the probability that all 3 drawn balls are of the same colour is
(A) $\frac { 82 } { 648 }$
(B) $\frac { 90 } { 648 }$
(C) $\frac { 558 } { 648 }$
(D) $\frac { 566 } { 648 }$
Q52 Conditional Probability Bayes' Theorem with Production/Source Identification View
A box $B _ { 1 }$ contains 1 white ball, 3 red balls and 2 black balls. Another box $B _ { 2 }$ contains 2 white balls, 3 red balls and 4 black balls. A third box $B _ { 3 }$ contains 3 white balls, 4 red balls and 5 black balls.
If 2 balls are drawn (without replacement) from a randomly selected box and one of the balls is white and the other ball is red, the probability that these 2 balls are drawn from box $B _ { 2 }$ is
(A) $\frac { 116 } { 181 }$
(B) $\frac { 126 } { 181 }$
(C) $\frac { 65 } { 181 }$
(D) $\frac { 55 } { 181 }$
Q53 Differential equations Qualitative Analysis of DE Solutions View
Let $f : [ 0,1 ] \rightarrow \mathbb { R }$ (the set of all real numbers) be a function. Suppose the function $f$ is twice differentiable, $f ( 0 ) = f ( 1 ) = 0$ and satisfies $f ^ { \prime \prime } ( x ) - 2 f ^ { \prime } ( x ) + f ( x ) \geq e ^ { x } , x \in [ 0,1 ]$.
Which of the following is true for $0 < x < 1$?
(A) $0 < f ( x ) < \infty$
(B) $- \frac { 1 } { 2 } < f ( x ) < \frac { 1 } { 2 }$
(C) $- \frac { 1 } { 4 } < f ( x ) < 1$
(D) $- \infty < f ( x ) < 0$
Q54 Applied differentiation MCQ on derivative and graph interpretation View
Let $f : [ 0,1 ] \rightarrow \mathbb { R }$ (the set of all real numbers) be a function. Suppose the function $f$ is twice differentiable, $f ( 0 ) = f ( 1 ) = 0$ and satisfies $f ^ { \prime \prime } ( x ) - 2 f ^ { \prime } ( x ) + f ( x ) \geq e ^ { x } , x \in [ 0,1 ]$.
If the function $\mathrm { e } ^ { - x } f ( x )$ assumes its minimum in the interval $[ 0,1 ]$ at $x = \frac { 1 } { 4 }$, which of the following is true?
(A) $f ^ { \prime } ( x ) < f ( x ) , \frac { 1 } { 4 } < x < \frac { 3 } { 4 }$
(B) $f ^ { \prime } ( x ) > f ( x ) , \quad 0 < x < \frac { 1 } { 4 }$
(C) $f ^ { \prime } ( x ) < f ( x ) , \quad 0 < x < \frac { 1 } { 4 }$
(D) $f ^ { \prime } ( x ) < f ( x ) , \frac { 3 } { 4 } < x < 1$
Q55 Conic sections Focal Chord and Parabola Segment Relations View
Let $P Q$ be a focal chord of the parabola $y ^ { 2 } = 4 a x$. The tangents to the parabola at $P$ and $Q$ meet at a point lying on the line $y = 2 x + a , a > 0$.
Length of chord $P Q$ is
(A) $7 a$
(B) $5 a$
(C) $2 a$
(D) $3 a$
Q56 Conic sections Focal Chord and Parabola Segment Relations View
Let $P Q$ be a focal chord of the parabola $y ^ { 2 } = 4 a x$. The tangents to the parabola at $P$ and $Q$ meet at a point lying on the line $y = 2 x + a , a > 0$.
If chord $P Q$ subtends an angle $\theta$ at the vertex of $y ^ { 2 } = 4 a x$, then $\tan \theta =$
(A) $\frac { 2 } { 3 } \sqrt { 7 }$
(B) $\frac { - 2 } { 3 } \sqrt { 7 }$
(C) $\frac { 2 } { 3 } \sqrt { 5 }$
(D) $\frac { - 2 } { 3 } \sqrt { 5 }$
Q57 Stationary points and optimisation Geometric or applied optimisation problem View
A line $L : y = m x + 3$ meets $y$-axis at $E ( 0,3 )$ and the arc of the parabola $y ^ { 2 } = 16 x$, $0 \leq y \leq 6$ at the point $F \left( x _ { 0 } , y _ { 0 } \right)$. The tangent to the parabola at $F \left( x _ { 0 } , y _ { 0 } \right)$ intersects the $y$-axis at $G \left( 0 , y _ { 1 } \right)$. The slope $m$ of the line $L$ is chosen such that the area of the triangle $E F G$ has a local maximum.
Match List I with List II and select the correct answer using the code given below the lists:
List I
  • [P.] $m =$
  • [Q.] Maximum area of $\triangle EFG$ is
  • [R.] $y_0 =$
  • [S.] $y_1 =$

List II
  1. $\frac{1}{2}$
  2. $4$
  3. $2$
  4. $1$

Codes:
PQRS
(A)4123
(B)3412
(C)1324
(D)1342
Q58 Reciprocal Trig & Identities View
Match List I with List II and select the correct answer using the code given below the lists:
List I
  • [P.] $\left( \frac { 1 } { y ^ { 2 } } \left( \frac { \cos \left( \tan ^ { - 1 } y \right) + y \sin \left( \tan ^ { - 1 } y \right) } { \cot \left( \sin ^ { - 1 } y \right) + \tan \left( \sin ^ { - 1 } y \right) } \right) ^ { 2 } + y ^ { 4 } \right) ^ { 1 / 2 }$ takes value
  • [Q.] If $\cos x + \cos y + \cos z = 0 = \sin x + \sin y + \sin z$ then possible value of $\cos \frac { x - y } { 2 }$ is
  • [R.] If $\cos \left( \frac { \pi } { 4 } - x \right) \cos 2 x + \sin x \sin 2 x \sec x = \cos x \sin 2 x \sec x + \cos \left( \frac { \pi } { 4 } + x \right) \cos 2 x$ then possible value of $\sec x$ is
  • [S.] If $\cot \left( \sin ^ { - 1 } \sqrt { 1 - x ^ { 2 } } \right) = \sin \left( \tan ^ { - 1 } ( x \sqrt { 6 } ) \right) , x \neq 0$, then possible value of $x$ is

List II
  1. $\frac { 1 } { 2 } \sqrt { \frac { 5 } { 3 } }$
  2. $\sqrt { 2 }$
  3. $\frac { 1 } { 2 }$
  4. $1$

Codes:
PQRS
(A)4312
(B)4321
(C)3421
(D)3412
Q59 Vectors: Lines & Planes Find Cartesian Equation of a Plane View
Consider the lines $L _ { 1 } : \frac { x - 1 } { 2 } = \frac { y } { - 1 } = \frac { z + 3 } { 1 } , L _ { 2 } : \frac { x - 4 } { 1 } = \frac { y + 3 } { 1 } = \frac { z + 3 } { 2 }$ and the planes $P _ { 1 } : 7 x + y + 2 z = 3 , P _ { 2 } : 3 x + 5 y - 6 z = 4$. Let $a x + b y + c z = d$ be the equation of the plane passing through the point of intersection of lines $L _ { 1 }$ and $L _ { 2 }$, and perpendicular to planes $P _ { 1 }$ and $P _ { 2 }$.
Match List I with List II and select the correct answer using the code given below the lists:
List I
  • [P.] $a =$
  • [Q.] $b =$
  • [R.] $c =$
  • [S.] $d =$

List II
  1. $13$
  2. $-3$
  3. $1$
  4. $-2$

Codes:
PQRS
(A)3241
(B)1342
(C)3214
(D)2413
Q60 Vectors: Cross Product & Distances View
Match List I with List II and select the correct answer using the code given below the lists:
List I
  • [P.] Volume of parallelepiped determined by vectors $\vec { a } , \vec { b }$ and $\vec { c }$ is 2. Then the volume of the parallelepiped determined by vectors $2 ( \vec { a } \times \vec { b } ) , 3 ( \vec { b } \times \vec { c } )$ and $( \vec { c } \times \vec { a } )$ is
  • [Q.] Volume of parallelepiped determined by vectors $\vec { a } , \vec { b }$ and $\vec { c }$ is 5. Then the volume of the parallelepiped determined by vectors $3 ( \vec { a } + \vec { b } ) , ( \vec { b } + \vec { c } )$ and $2 ( \vec { c } + \vec { a } )$ is
  • [R.] Area of a triangle with adjacent sides determined by vectors $\vec { a }$ and $\vec { b }$ is 20. Then the area of the triangle with adjacent sides determined by vectors $( 2 \vec { a } + 3 \vec { b } )$ and $( \vec { a } - \vec { b } )$ is
  • [S.] Area of a parallelogram with adjacent sides determined by vectors $\vec { a }$ and $\vec { b }$ is 30. Then the area of the parallelogram with adjacent sides determined by vectors $( \vec { a } + \vec { b } )$ and $\vec { a }$ is

List II
  1. $100$
  2. $30$
  3. (values as given in the list)

Codes:
PQRS
(A)4231
(B)2314
(C)3412
(D)1432