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20 maths questions

Q41 Complex Numbers Argand & Loci Circle Equation and Properties via Complex Number Manipulation View
Let complex numbers $\alpha$ and $\frac { 1 } { \bar { \alpha } }$ lie on circles $\left( x - x _ { 0 } \right) ^ { 2 } + \left( y - y _ { 0 } \right) ^ { 2 } = r ^ { 2 }$ and $\left( x - x _ { 0 } \right) ^ { 2 } + \left( y - y _ { 0 } \right) ^ { 2 } = 4 r ^ { 2 }$, respectively. If $z _ { 0 } = x _ { 0 } + i y _ { 0 }$ satisfies the equation $2 \left| z _ { 0 } \right| ^ { 2 } = r ^ { 2 } + 2$, then $| \alpha | =$
(A) $\frac { 1 } { \sqrt { 2 } }$
(B) $\frac { 1 } { 2 }$
(C) $\frac { 1 } { \sqrt { 7 } }$
(D) $\frac { 1 } { 3 }$
Q42 Probability Definitions Probability Using Set/Event Algebra View
Four persons independently solve a certain problem correctly with probabilities $\frac { 1 } { 2 } , \frac { 3 } { 4 } , \frac { 1 } { 4 } , \frac { 1 } { 8 }$. Then the probability that the problem is solved correctly by at least one of them is
(A) $\frac { 235 } { 256 }$
(B) $\frac { 21 } { 256 }$
(C) $\frac { 3 } { 256 }$
(D) $\frac { 253 } { 256 }$
Q43 Indefinite & Definite Integrals Integral Inequalities and Limit of Integral Sequences View
Let $f : \left[ \frac { 1 } { 2 } , 1 \right] \rightarrow \mathbb { R }$ (the set of all real numbers) be a positive, non-constant and differentiable function such that $f ^ { \prime } ( x ) < 2 f ( x )$ and $f \left( \frac { 1 } { 2 } \right) = 1$. Then the value of $\int _ { 1/2 } ^ { 1 } f ( x ) d x$ lies in the interval
(A) $( 2 e - 1,2 e )$
(B) $( e - 1,2 e - 1 )$
(C) $\left( \frac { e - 1 } { 2 } , e - 1 \right)$
(D) $\left( 0 , \frac { e - 1 } { 2 } \right)$
Q44 Curve Sketching Number of Solutions / Roots via Curve Analysis View
The number of points in $( - \infty , \infty )$, for which $x ^ { 2 } - x \sin x - \cos x = 0$, is
(A) 6
(B) 4
(C) 2
(D) 0
Q45 Areas Between Curves Area Involving Piecewise or Composite Functions View
The area enclosed by the curves $y = \sin x + \cos x$ and $y = | \cos x - \sin x |$ over the interval $\left[ 0 , \frac { \pi } { 2 } \right]$ is
(A) $4 ( \sqrt { 2 } - 1 )$
(B) $2 \sqrt { 2 } ( \sqrt { 2 } - 1 )$
(C) $2 ( \sqrt { 2 } + 1 )$
(D) $2 \sqrt { 2 } ( \sqrt { 2 } + 1 )$
Q46 Differential equations Solving Separable DEs with Initial Conditions View
A curve passes through the point $\left( 1 , \frac { \pi } { 6 } \right)$. Let the slope of the curve at each point $( x , y )$ be $\frac { y } { x } + \sec \left( \frac { y } { x } \right) , x > 0$. Then the equation of the curve is
(A) $\quad \sin \left( \frac { y } { x } \right) = \log x + \frac { 1 } { 2 }$
(B) $\quad \operatorname { cosec } \left( \frac { y } { x } \right) = \log x + 2$
(C) $\quad \sec \left( \frac { 2 y } { x } \right) = \log x + 2$
(D) $\quad \cos \left( \frac { 2 y } { x } \right) = \log x + \frac { 1 } { 2 }$
Q47 Sequences and Series Evaluation of a Finite or Infinite Sum View
The value of $\cot \left( \sum _ { n = 1 } ^ { 23 } \cot ^ { - 1 } \left( 1 + \sum _ { k = 1 } ^ { n } 2 k \right) \right)$ is
(A) $\frac { 23 } { 25 }$
(B) $\frac { 25 } { 23 }$
(C) $\frac { 23 } { 24 }$
(D) $\frac { 24 } { 23 }$
Q48 Straight Lines & Coordinate Geometry Point-to-Line Distance Computation View
For $a > b > c > 0$, the distance between $( 1,1 )$ and the point of intersection of the lines $a x + b y + c = 0$ and $b x + a y + c = 0$ is less than $2 \sqrt { 2 }$. Then
(A) $a + b - c > 0$
(B) $a - b + c < 0$
(C) $a - b + c > 0$
(D) $a + b - c < 0$
Q49 Vectors: Lines & Planes Perpendicular/Orthogonal Projection onto a Plane View
Perpendiculars are drawn from points on the line $\frac { x + 2 } { 2 } = \frac { y + 1 } { - 1 } = \frac { z } { 3 }$ to the plane $x + y + z = 3$. The feet of perpendiculars lie on the line
(A) $\frac { x } { 5 } = \frac { y - 1 } { 8 } = \frac { z - 2 } { - 13 }$
(B) $\frac { x } { 2 } = \frac { y - 1 } { 3 } = \frac { z - 2 } { - 5 }$
(C) $\frac { x } { 4 } = \frac { y - 1 } { 3 } = \frac { z - 2 } { - 7 }$
(D) $\frac { x } { 2 } = \frac { y - 1 } { - 7 } = \frac { z - 2 } { 5 }$
Q50 Vectors: Cross Product & Distances View
Let $\overrightarrow { P R } = 3 \hat { i } + \hat { j } - 2 \hat { k }$ and $\overrightarrow { S Q } = \hat { i } - 3 \hat { j } - 4 \hat { k }$ determine diagonals of a parallelogram $P Q R S$ and $\overrightarrow { P T } = \hat { i } + 2 \hat { j } + 3 \hat { k }$ be another vector. Then the volume of the parallelepiped determined by the vectors $\overrightarrow { P T } , \overrightarrow { P Q }$ and $\overrightarrow { P S }$ is
(A) 5
(B) 20
(C) 10
(D) 30
Q51 Sequences and Series Evaluation of a Finite or Infinite Sum View
Let $S _ { n } = \sum _ { k = 1 } ^ { 4 n } ( - 1 ) ^ { \frac { k ( k + 1 ) } { 2 } } k ^ { 2 }$. Then $S _ { n }$ can take value(s)
(A) 1056
(B) 1088
(C) 1120
(D) 1332
Q52 Matrices True/False or Multiple-Select Conceptual Reasoning View
For $3 \times 3$ matrices $M$ and $N$, which of the following statement(s) is (are) NOT correct?
(A) $\quad N ^ { T } M N$ is symmetric or skew symmetric, according as $M$ is symmetric or skew symmetric
(B) $M N - N M$ is skew symmetric for all symmetric matrices $M$ and $N$
(C) $M N$ is symmetric for all symmetric matrices $M$ and $N$
(D) $\quad ( \operatorname { adj } M ) ( \operatorname { adj } N ) = \operatorname { adj } ( M N )$ for all invertible matrices $M$ and $N$
Q53 Applied differentiation Existence and number of solutions via calculus View
Let $f ( x ) = x \sin \pi x , x > 0$. Then for all natural numbers $n , f ^ { \prime } ( x )$ vanishes at
(A) a unique point in the interval $\left( n , n + \frac { 1 } { 2 } \right)$
(B) a unique point in the interval $\left( n + \frac { 1 } { 2 } , n + 1 \right)$
(C) a unique point in the interval $( n , n + 1 )$
(D) two points in the interval $( n , n + 1 )$
Q54 Stationary points and optimisation Geometric or applied optimisation problem View
A rectangular sheet of fixed perimeter with sides having their lengths in the ratio $8 : 15$ is converted into an open rectangular box by folding after removing squares of equal area from all four corners. If the total area of removed squares is 100, the resulting box has maximum volume. Then the lengths of the sides of the rectangular sheet are
(A) 24
(B) 32
(C) 45
(D) 60
Q55 Vectors: Lines & Planes Distance Computation (Point-to-Plane or Line-to-Line) View
A line $l$ passing through the origin is perpendicular to the lines $l _ { 1 } : ( 3 + t ) \hat { i } + ( - 1 + 2 t ) \hat { j } + ( 4 + 2 t ) \hat { k } , - \infty < t < \infty$ $l _ { 2 } : ( 3 + 2 s ) \hat { i } + ( 3 + 2 s ) \hat { j } + ( 2 + s ) \hat { k } , - \infty < s < \infty$ Then, the coordinate(s) of the point(s) on $l _ { 2 }$ at a distance of $\sqrt { 17 }$ from the point of intersection of $l$ and $l _ { 1 }$ is (are)
(A) $\left( \frac { 7 } { 3 } , \frac { 7 } { 3 } , \frac { 5 } { 3 } \right)$
(B) $( - 1 , - 1,0 )$
(C) $( 1,1,1 )$
(D) $\left( \frac { 7 } { 9 } , \frac { 7 } { 9 } , \frac { 8 } { 9 } \right)$
Q56 Binomial Theorem (positive integer n) Determine Parameters from Conditions on Coefficients or Terms View
The coefficients of three consecutive terms of $( 1 + x ) ^ { n + 5 }$ are in the ratio $5 : 10 : 14$. Then $n =$
Q57 Arithmetic Sequences and Series Counting or Combinatorial Problems on APs View
A pack contains $n$ cards numbered from 1 to $n$. Two consecutive numbered cards are removed from the pack and the sum of the numbers on the remaining cards is 1224. If the smaller of the numbers on the removed cards is $k$, then $k - 20 =$
Q58 Independent Events View
Of the three independent events $E _ { 1 } , E _ { 2 }$ and $E _ { 3 }$, the probability that only $E _ { 1 }$ occurs is $\alpha$, only $E _ { 2 }$ occurs is $\beta$ and only $E _ { 3 }$ occurs is $\gamma$. Let the probability $p$ that none of events $E _ { 1 } , E _ { 2 }$ or $E _ { 3 }$ occurs satisfy the equations $( \alpha - 2 \beta ) p = \alpha \beta$ and $( \beta - 3 \gamma ) p = 2 \beta \gamma$. All the given probabilities are assumed to lie in the interval $( 0,1 )$.
$$\text { Then } \frac { \text { Probability of occurrence of } E _ { 1 } } { \text { Probability of occurrence of } E _ { 3 } } =$$
Q59 Conic sections Triangle or Quadrilateral Area and Perimeter with Foci View
A vertical line passing through the point $( h , 0 )$ intersects the ellipse $\frac { x ^ { 2 } } { 4 } + \frac { y ^ { 2 } } { 3 } = 1$ at the points $P$ and $Q$. Let the tangents to the ellipse at $P$ and $Q$ meet at the point $R$. If $\Delta ( h ) =$ area of the triangle $P Q R$, $\Delta _ { 1 } = \max _ { 1/2 \leq h \leq 1 } \Delta ( h )$ and $\Delta _ { 2 } = \min _ { 1/2 \leq h \leq 1 } \Delta ( h )$, then $\frac { 8 } { \sqrt { 5 } } \Delta _ { 1 } - 8 \Delta _ { 2 } =$
Q60 Vectors 3D & Lines Vector Algebra and Triple Product Computation View
Consider the set of eight vectors $V = \{ \mathrm { a } \hat { i } + \mathrm { b } \hat { j } + \mathrm { c } \hat { k } : a , b , c \in \{ - 1,1 \} \}$. Three noncoplanar vectors can be chosen from $V$ in $2 ^ { p }$ ways. Then $p$ is