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19 maths questions

Q21 Vectors: Lines & Planes Parallelism Between Line and Plane or Constraint on Parameters View
Let $P ( 3,2,6 )$ be a point in space and $Q$ be a point on the line
$$\vec { r } = ( \hat { i } - \hat { j } + 2 \hat { k } ) + \mu ( - 3 \hat { i } + \hat { j } + 5 \hat { k } )$$
Then the value of $\mu$ for which the vector $\overrightarrow { P Q }$ is parallel to the plane $x - 4 y + 3 z = 1$ is
(A) $\frac { 1 } { 4 }$
(B) $- \frac { 1 } { 4 }$
(C) $\frac { 1 } { 8 }$
(D) $- \frac { 1 } { 8 }$
Q22 Circles Circle Equation Derivation View
Tangents drawn from the point $P ( 1,8 )$ to the circle
$$x ^ { 2 } + y ^ { 2 } - 6 x - 4 y - 11 = 0$$
touch the circle at the points $A$ and $B$. The equation of the circumcircle of the triangle $P A B$ is
(A) $x ^ { 2 } + y ^ { 2 } + 4 x - 6 y + 19 = 0$
(B) $x ^ { 2 } + y ^ { 2 } - 4 x - 10 y + 19 = 0$
(C) $x ^ { 2 } + y ^ { 2 } - 2 x + 6 y - 29 = 0$
(D) $x ^ { 2 } + y ^ { 2 } - 6 x - 4 y + 19 = 0$
Q23 Differential equations Integral Equations Reducible to DEs View
Let $f$ be a non-negative function defined on the interval $[ 0,1 ]$. If
$$\int _ { 0 } ^ { x } \sqrt { 1 - \left( f ^ { \prime } ( t ) \right) ^ { 2 } } d t = \int _ { 0 } ^ { x } f ( t ) d t , \quad 0 \leq x \leq 1 ,$$
and $f ( 0 ) = 0$, then
(A) $f \left( \frac { 1 } { 2 } \right) < \frac { 1 } { 2 }$ and $f \left( \frac { 1 } { 3 } \right) > \frac { 1 } { 3 }$
(B) $f \left( \frac { 1 } { 2 } \right) > \frac { 1 } { 2 }$ and $f \left( \frac { 1 } { 3 } \right) > \frac { 1 } { 3 }$
(C) $f \left( \frac { 1 } { 2 } \right) < \frac { 1 } { 2 }$ and $f \left( \frac { 1 } { 3 } \right) < \frac { 1 } { 3 }$
(D) $f \left( \frac { 1 } { 2 } \right) > \frac { 1 } { 2 }$ and $f \left( \frac { 1 } { 3 } \right) < \frac { 1 } { 3 }$
Q24 Complex Numbers Arithmetic Geometric Interpretation and Triangle/Shape Properties View
Let $z = x + i y$ be a complex number where $x$ and $y$ are integers. Then the area of the rectangle whose vertices are the roots of the equation
$$z \bar { z } ^ { 3 } + \bar { z } z ^ { 3 } = 350$$
is
(A) 48
(B) 32
(C) 40
(D) 80
Q25 Conic sections Triangle or Quadrilateral Area and Perimeter with Foci View
The line passing through the extremity $A$ of the major axis and extremity $B$ of the minor axis of the ellipse
$$x ^ { 2 } + 9 y ^ { 2 } = 9$$
meets its auxiliary circle at the point $M$. Then the area of the triangle with vertices at $A , M$ and the origin $O$ is
(A) $\frac { 31 } { 10 }$
(B) $\frac { 29 } { 10 }$
(C) $\frac { 21 } { 10 }$
(D) $\frac { 27 } { 10 }$
Q26 Vector Product and Surfaces View
If $\vec { a } , \vec { b } , \vec { c }$ and $\vec { d }$ are unit vectors such that
$$( \vec { a } \times \vec { b } ) \cdot ( \vec { c } \times \vec { d } ) = 1$$
and $\quad \vec { a } \cdot \vec { c } = \frac { 1 } { 2 }$,
then
Q28 Permutations & Arrangements Forming Numbers with Digit Constraints View
The number of seven digit integers, with sum of the digits equal to 10 and formed by using the digits 1, 2 and 3 only, is
(A) 55
(B) 66
(C) 77
(D) 88
Q29 Areas Between Curves Select Correct Integral Expression View
Area of the region bounded by the curve $y = e ^ { x }$ and lines $x = 0$ and $y = e$ is
(A) $e - 1$
(B) $\int _ { 1 } ^ { e } \ln ( e + 1 - y ) d y$
(C) $e - \int _ { 0 } ^ { 1 } e ^ { x } d x$
(D) $\int _ { 1 } ^ { e } \ln y \, d y$
Q30 Taylor series Limit evaluation using series expansion or exponential asymptotics View
Let
$$L = \lim _ { x \rightarrow 0 } \frac { a - \sqrt { a ^ { 2 } - x ^ { 2 } } - \frac { x ^ { 2 } } { 4 } } { x ^ { 4 } } , \quad a > 0 .$$
If $L$ is finite, then
(A) $\quad a = 2$
(B) $\quad a = 1$
(C) $\quad L = \frac { 1 } { 64 }$
(D) $\quad L = \frac { 1 } { 32 }$
Q31 Sine and Cosine Rules Determine an angle or side from a trigonometric identity/equation View
In a triangle $A B C$ with fixed base $B C$, the vertex $A$ moves such that
$$\cos B + \cos C = 4 \sin ^ { 2 } \frac { A } { 2 }$$
If $a , b$ and $c$ denote the lengths of the sides of the triangle opposite to the angles $A , B$ and $C$, respectively, then
(A) $b + c = 4 a$
(B) $b + c = 2 a$
(C) locus of point $A$ is an ellipse
(D) locus of point $A$ is a pair of straight lines
Q32 Trig Proofs Trigonometric Equation Constraint Deduction View
If
$$\frac { \sin ^ { 4 } x } { 2 } + \frac { \cos ^ { 4 } x } { 3 } = \frac { 1 } { 5 } ,$$
then
(A) $\quad \tan ^ { 2 } x = \frac { 2 } { 3 }$
(B) $\quad \frac { \sin ^ { 8 } x } { 8 } + \frac { \cos ^ { 8 } x } { 27 } = \frac { 1 } { 125 }$
(C) $\quad \tan ^ { 2 } x = \frac { 1 } { 3 }$
(D) $\frac { \sin ^ { 8 } x } { 8 } + \frac { \cos ^ { 8 } x } { 27 } = \frac { 2 } { 125 }$
Q33 Matrices Structured Matrix Characterization View
Let $\mathscr { A }$ be the set of all $3 \times 3$ symmetric matrices all of whose entries are either 0 or 1. Five of these entries are 1 and four of them are 0.
The number of matrices in $\mathscr { A }$ is
(A) 12
(B) 6
(C) 9
(D) 3
Q34 Matrices Linear System and Inverse Existence View
Let $\mathscr { A }$ be the set of all $3 \times 3$ symmetric matrices all of whose entries are either 0 or 1. Five of these entries are 1 and four of them are 0.
The number of matrices $A$ in $\mathscr { A }$ for which the system of linear equations
$$A \left[ \begin{array} { l } x \\ y \\ z \end{array} \right] = \left[ \begin{array} { l } 1 \\ 0 \\ 0 \end{array} \right]$$
has a unique solution, is
(A) less than 4
(B) at least 4 but less than 7
(C) at least 7 but less than 10
(D) at least 10
Q35 Matrices Linear System and Inverse Existence View
Let $\mathscr { A }$ be the set of all $3 \times 3$ symmetric matrices all of whose entries are either 0 or 1. Five of these entries are 1 and four of them are 0.
The number of matrices $A$ in $\mathscr { A }$ for which the system of linear equations
$$A \left[ \begin{array} { l } x \\ y \\ z \end{array} \right] = \left[ \begin{array} { l } 1 \\ 0 \\ 0 \end{array} \right]$$
is inconsistent, is
(A) 0
(B) more than 2
(C) 2
(D) 1
Q36 Geometric Distribution View
A fair die is tossed repeatedly until a six is obtained. Let $X$ denote the number of tosses required.
The probability that $X = 3$ equals
(A) $\frac { 25 } { 216 }$
(B) $\frac { 25 } { 36 }$
(C) $\frac { 5 } { 36 }$
(D) $\frac { 125 } { 216 }$
Q37 Geometric Distribution View
A fair die is tossed repeatedly until a six is obtained. Let $X$ denote the number of tosses required.
The probability that $X \geq 3$ equals
(A) $\frac { 125 } { 216 }$
(B) $\frac { 25 } { 36 }$
(C) $\frac { 5 } { 36 }$
(D) $\frac { 25 } { 216 }$
Q38 Conditional Probability Conditional Probability with Discrete Random Variable View
A fair die is tossed repeatedly until a six is obtained. Let $X$ denote the number of tosses required.
The conditional probability that $X \geq 6$ given $X > 3$ equals
(A) $\frac { 125 } { 216 }$
(B) $\frac { 25 } { 216 }$
(C) $\frac { 5 } { 36 }$
(D) $\frac { 25 } { 36 }$
Q39 Differential equations Qualitative Analysis of DE Solutions View
Match the statements/expressions in Column I with the open intervals in Column II.
Column I
(A) Interval contained in the domain of definition of non-zero solutions of the differential equation $( x - 3 ) ^ { 2 } y ^ { \prime } + y = 0$
(B) Interval containing the value of the integral $$\int _ { 1 } ^ { 5 } ( x - 1 ) ( x - 2 ) ( x - 3 ) ( x - 4 ) ( x - 5 ) d x$$ (C) Interval in which at least one of the points of local maximum of $\cos ^ { 2 } x + \sin x$ lies
(D) Interval in which $\tan ^ { - 1 } ( \sin x + \cos x )$ is increasing
Column II
(p) $\left( - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right)$
(q) $\left( 0 , \frac { \pi } { 2 } \right)$
(r) $\left( \frac { \pi } { 8 } , \frac { 5 \pi } { 4 } \right)$
(s) $\left( 0 , \frac { \pi } { 8 } \right)$
(t) $( - \pi , \pi )$
Q40 Conic sections Conic Identification and Conceptual Properties View
Match the conics in Column I with the statements/expressions in Column II.
Column I
(A) Circle
(B) Parabola
(C) Ellipse
(D) Hyperbola
Column II
(p) The locus of the point $( h , k )$ for which the line $h x + k y = 1$ touches the circle $x ^ { 2 } + y ^ { 2 } = 4$
(q) Points $z$ in the complex plane satisfying $| z + 2 | - | z - 2 | = \pm 3$
(r) Points of the conic have parametric representation $x = \sqrt { 3 } \left( \frac { 1 - t ^ { 2 } } { 1 + t ^ { 2 } } \right) , y = \frac { 2 t } { 1 + t ^ { 2 } }$
(s) The eccentricity of the conic lies in the interval $1 \leq x < \infty$
(t) Points $z$ in the complex plane satisfying $\operatorname { Re } ( z + 1 ) ^ { 2 } = | z | ^ { 2 } + 1$