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18 maths questions

Q37 Vectors 3D & Lines Normal Vector and Plane Equation View
The equation of the plane passing through the point $( 1,1,1 )$ and perpendicular to the planes $2 x + y - 2 z = 5$ and $3 x - 6 y - 2 z = 7$, is
[A] $14 x + 2 y - 15 z = 1$
[B] $14 x - 2 y + 15 z = 27$
[C] $14 x + 2 y + 15 z = 31$
[D] $- 14 x + 2 y + 15 z = 3$
Q38 Vectors 3D & Lines Section Division and Coordinate Computation View
Let $O$ be the origin and let $P Q R$ be an arbitrary triangle. The point $S$ is such that
$$\overrightarrow { O P } \cdot \overrightarrow { O Q } + \overrightarrow { O R } \cdot \overrightarrow { O S } = \overrightarrow { O R } \cdot \overrightarrow { O P } + \overrightarrow { O Q } \cdot \overrightarrow { O S } = \overrightarrow { O Q } \cdot \overrightarrow { O R } + \overrightarrow { O P } \cdot \overrightarrow { O S }$$
Then the triangle $P Q R$ has $S$ as its
[A] centroid
[B] circumcentre
[C] incentre
[D] orthocenter
Q39 Differential equations Solving Separable DEs with Initial Conditions View
If $y = y ( x )$ satisfies the differential equation
$$8 \sqrt { x } ( \sqrt { 9 + \sqrt { x } } ) d y = ( \sqrt { 4 + \sqrt { 9 + \sqrt { x } } } ) ^ { - 1 } d x , \quad x > 0$$
and $y ( 0 ) = \sqrt { 7 }$, then $y ( 256 ) =$
[A] 3
[B] 9
[C] 16
[D] 80
Q40 Stationary points and optimisation Find concavity, inflection points, or second derivative properties View
If $f : \mathbb { R } \rightarrow \mathbb { R }$ is a twice differentiable function such that $f ^ { \prime \prime } ( x ) > 0$ for all $x \in \mathbb { R }$, and $f \left( \frac { 1 } { 2 } \right) = \frac { 1 } { 2 } , f ( 1 ) = 1$, then
[A] $f ^ { \prime } ( 1 ) \leq 0$
[B] $0 < f ^ { \prime } ( 1 ) \leq \frac { 1 } { 2 }$
[C] $\frac { 1 } { 2 } < f ^ { \prime } ( 1 ) \leq 1$
[D] $f ^ { \prime } ( 1 ) > 1$
Q41 Matrices Determinant and Rank Computation View
How many $3 \times 3$ matrices $M$ with entries from $\{ 0,1,2 \}$ are there, for which the sum of the diagonal entries of $M ^ { T } M$ is 5 ?
[A] 126
[B] 198
[C] 162
[D] 135
Q42 Combinations & Selection Selection with Group/Category Constraints View
Let $S = \{ 1,2,3 , \ldots , 9 \}$. For $k = 1,2 , \ldots , 5$, let $N _ { k }$ be the number of subsets of $S$, each containing five elements out of which exactly $k$ are odd. Then $N _ { 1 } + N _ { 2 } + N _ { 3 } + N _ { 4 } + N _ { 5 } =$
[A] 210
[B] 252
[C] 125
[D] 126
Q43 Probability Definitions Probability Involving Algebraic or Number-Theoretic Conditions View
Three randomly chosen nonnegative integers $x , y$ and $z$ are found to satisfy the equation $x + y + z = 10$. Then the probability that $z$ is even, is
[A] $\frac { 36 } { 55 }$
[B] $\frac { 6 } { 11 }$
[C] $\frac { 1 } { 2 }$
[D] $\frac { 5 } { 11 }$
Q44 Connected Rates of Change Parametric or Curve-Based Particle Motion Rates View
If $g ( x ) = \int _ { \sin x } ^ { \sin ( 2 x ) } \sin ^ { - 1 } ( t ) d t$, then
[A] $g ^ { \prime } \left( \frac { \pi } { 2 } \right) = - 2 \pi$
[B] $g ^ { \prime } \left( - \frac { \pi } { 2 } \right) = 2 \pi$
[C] $g ^ { \prime } \left( \frac { \pi } { 2 } \right) = 2 \pi$
[D] $g ^ { \prime } \left( - \frac { \pi } { 2 } \right) = - 2 \pi$
Q45 Addition & Double Angle Formulae Half-Angle Formula Evaluation View
Let $\alpha$ and $\beta$ be nonzero real numbers such that $2 ( \cos \beta - \cos \alpha ) + \cos \alpha \cos \beta = 1$. Then which of the following is/are true?
[A] $\tan \left( \frac { \alpha } { 2 } \right) + \sqrt { 3 } \tan \left( \frac { \beta } { 2 } \right) = 0$
[B] $\sqrt { 3 } \tan \left( \frac { \alpha } { 2 } \right) + \tan \left( \frac { \beta } { 2 } \right) = 0$
[C] $\tan \left( \frac { \alpha } { 2 } \right) - \sqrt { 3 } \tan \left( \frac { \beta } { 2 } \right) = 0$
[D] $\sqrt { 3 } \tan \left( \frac { \alpha } { 2 } \right) - \tan \left( \frac { \beta } { 2 } \right) = 0$
Q46 Exponential Functions True/False or Multiple-Statement Verification View
If $f : \mathbb { R } \rightarrow \mathbb { R }$ is a differentiable function such that $f ^ { \prime } ( x ) > 2 f ( x )$ for all $x \in \mathbb { R }$, and $f ( 0 ) = 1$, then
[A] $f ( x )$ is increasing in $( 0 , \infty )$
[B] $f ( x )$ is decreasing in $( 0 , \infty )$
[C] $f ( x ) > e ^ { 2 x }$ in $( 0 , \infty )$
[D] $f ^ { \prime } ( x ) < e ^ { 2 x }$ in $( 0 , \infty )$
Q47 Curve Sketching Continuity and Differentiability of Special Functions View
Let $f ( x ) = \frac { 1 - x ( 1 + | 1 - x | ) } { | 1 - x | } \cos \left( \frac { 1 } { 1 - x } \right)$ for $x \neq 1$. Then
[A] $\lim _ { x \rightarrow 1 ^ { - } } f ( x ) = 0$
[B] $\lim _ { x \rightarrow 1 ^ { - } } f ( x )$ does not exist
[C] $\lim _ { x \rightarrow 1 ^ { + } } f ( x ) = 0$
[D] $\lim _ { x \rightarrow 1 ^ { + } } f ( x )$ does not exist
Q48 Stationary points and optimisation Find absolute extrema on a closed interval or domain View
If $f ( x ) = \left| \begin{array} { c c c } \cos ( 2 x ) & \cos ( 2 x ) & \sin ( 2 x ) \\ - \cos x & \cos x & - \sin x \\ \sin x & \sin x & \cos x \end{array} \right|$, then
[A] $f ^ { \prime } ( x ) = 0$ at exactly three points in $( - \pi , \pi )$
[B] $f ^ { \prime } ( x ) = 0$ at more than three points in $( - \pi , \pi )$
[C] $f ( x )$ attains its maximum at $x = 0$
[D] $f ( x )$ attains its minimum at $x = 0$
Q49 Areas by integration View
If the line $x = \alpha$ divides the area of region $R = \left\{ ( x , y ) \in \mathbb { R } ^ { 2 } : x ^ { 3 } \leq y \leq x , 0 \leq x \leq 1 \right\}$ into two equal parts, then
[A] $0 < \alpha \leq \frac { 1 } { 2 }$
[B] $\frac { 1 } { 2 } < \alpha < 1$
[C] $2 \alpha ^ { 4 } - 4 \alpha ^ { 2 } + 1 = 0$
[D] $\alpha ^ { 4 } + 4 \alpha ^ { 2 } - 1 = 0$
Q50 Integration by Substitution Determine J−I or Compare Related Integrals via Substitution View
If $I = \sum _ { k = 1 } ^ { 98 } \int _ { k } ^ { k + 1 } \frac { k + 1 } { x ( x + 1 ) } d x$, then
[A] $I > \log _ { e } 99$
[B] $I < \log _ { e } 99$
[C] $I < \frac { 49 } { 50 }$
[D] $I > \frac { 49 } { 50 }$
Q51 Vectors 3D & Lines Vector Algebra and Triple Product Computation View
Let $O$ be the origin, and $\overrightarrow { O X } , \overrightarrow { O Y } , \overrightarrow { O Z }$ be three unit vectors in the directions of the sides $\overrightarrow { Q R } , \overrightarrow { R P }$, $\overrightarrow { P Q }$, respectively, of a triangle $P Q R$.
$| \overrightarrow { O X } \times \overrightarrow { O Y } | =$
[A] $\sin ( P + Q )$
[B] $\sin 2 R$
[C] $\sin ( P + R )$
[D] $\sin ( Q + R )$
Q52 Addition & Double Angle Formulae Function Analysis via Identity Transformation View
If the triangle $P Q R$ varies, then the minimum value of
$$\cos ( P + Q ) + \cos ( Q + R ) + \cos ( R + P )$$
is
[A] $- \frac { 5 } { 3 }$
[B] $- \frac { 3 } { 2 }$
[C] $\frac { 3 } { 2 }$
[D] $\frac { 5 } { 3 }$
Q53 Sequences and series, recurrence and convergence Multiple-choice on sequence properties View
Let $p , q$ be integers and let $\alpha , \beta$ be the roots of the equation, $x ^ { 2 } - x - 1 = 0$, where $\alpha \neq \beta$. For $n = 0,1,2 , \ldots$, let $a _ { n } = p \alpha ^ { n } + q \beta ^ { n }$.
FACT: If $a$ and $b$ are rational numbers and $a + b \sqrt { 5 } = 0$, then $a = 0 = b$.
$a _ { 12 } =$
[A] $a _ { 11 } - a _ { 10 }$
[B] $a _ { 11 } + a _ { 10 }$
[C] $2 a _ { 11 } + a _ { 10 }$
[D] $a _ { 11 } + 2 a _ { 10 }$
Q54 Sequences and series, recurrence and convergence Direct term computation from recurrence View
Let $p , q$ be integers and let $\alpha , \beta$ be the roots of the equation, $x ^ { 2 } - x - 1 = 0$, where $\alpha \neq \beta$. For $n = 0,1,2 , \ldots$, let $a _ { n } = p \alpha ^ { n } + q \beta ^ { n }$.
FACT: If $a$ and $b$ are rational numbers and $a + b \sqrt { 5 } = 0$, then $a = 0 = b$.
If $a _ { 4 } = 28$, then $p + 2 q =$
[A] 21
[B] 14
[C] 7
[D] 12