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Q2 Conditional Probability Bayes' Theorem with Production/Source Identification View

A brand called Jogger's Pride produces pairs of shoes in three different units that are named $U _ { 1 } , U _ { 2 }$ and $U _ { 3 }$. These units produce $10 \% , 30 \% , 60 \%$ of the total output of the brand with the chance that a pair of shoes being defective is $20 \% , 40 \% , 10 \%$ respectively. If a randomly selected pair of shoes from the combined output is found to be defective, then what is the chance that the pair was manufactured in the unit $U _ { 3 }$?
(A) $30 \%$
(B) $15 \%$
(C) $\frac { 3 } { 5 } \times 100 \%$
(D) Cannot be determined from the given data.

Q5 Measures of Location and Spread Summation of Derived Sequence from AP View

If the $n$ terms $a _ { 1 } , a _ { 2 } , \ldots , a _ { n }$ are in arithmetic progression with increment $r$, then the difference between the mean of their squares and the square of their mean is
(A) $\frac { r ^ { 2 } \left( ( n - 1 ) ^ { 2 } - 1 \right) } { 12 }$
(B) $\frac { r ^ { 2 } } { 12 }$
(C) $\frac { r ^ { 2 } \left( n ^ { 2 } - 1 \right) } { 12 }$
(D) $\frac { n ^ { 2 } - 1 } { 12 }$

Q7 Inequalities Solving inequalities involving modulus View

Q10 Simultaneous equations View

For a real number $\theta$, consider the following simultaneous equations:
$$\begin{aligned} & \cos ( \theta ) x - \sin ( \theta ) y = 1 \\ & \sin ( \theta ) x + \cos ( \theta ) y = 2 \end{aligned}$$
The number of solutions of these equations in $x$ and $y$ is
(A) 0
(B) 1
(C) infinite for some values of $\theta$
(D) finite only when $\theta = \frac { m \pi } { n }$ for integers $m$, and $n \neq 0$.

Q11 Trig Graphs & Exact Values Solve trigonometric equation for solutions in an interval View

In the range $0 \leq x \leq 2 \pi$, the equation $\cos ( \sin ( x ) ) = \frac { 1 } { 2 }$ has
(A) 0 solutions.
(B) 2 solutions.
(C) 4 solutions.
(D) infinitely many solutions.

Q14 Polynomial Division & Manipulation Polynomial Construction from Root/Value Conditions View

Let $P ( X ) = X ^ { 4 } + a _ { 3 } X ^ { 3 } + a _ { 2 } X ^ { 2 } + a _ { 1 } X + a _ { 0 }$ be a polynomial in $X$ with real coefficients. Assume that
$$P ( 0 ) = 1 , P ( 1 ) = 2 , P ( 2 ) = 3 , \text { and } P ( 3 ) = 4 .$$
Then, the value of $P ( 4 )$ is
(A) 5
(B) 24
(C) 29
(D) not determinable from the given data.

Q15 Stationary points and optimisation Find critical points and classify extrema of a given function View

Let $f$ be a real-valued differentiable function defined on the real line $\mathbb { R }$ such that its derivative $f ^ { \prime }$ is zero at exactly two distinct real numbers $\alpha$ and $\beta$. Then,
(A) $\alpha$ and $\beta$ are points of local maxima of the function $f$.
(B) $\alpha$ and $\beta$ are points of local minima of the function $f$.
(C) one must be a point of local maximum and the other must be a point of local minimum of $f$.
(D) given data is insufficient to conclude about either of them being local extrema points.

Q16 Combinations & Selection Distribution of Objects into Bins/Groups View

A school allowed the students of a class to go to swim during the days March 11th to March 15, 2019. The minimum number of students the class should have had that ensures that at least two of them went to swim on the same set of dates is:
(A) 6
(B) 32
(C) 33
(D) 121 .

Q17 Probability Definitions Linear Diophantine Equations View

Let $a _ { 1 } < a _ { 2 } < a _ { 3 } < a _ { 4 }$ be positive integers such that
$$\sum _ { i = 1 } ^ { 4 } \frac { 1 } { a _ { i } } = \frac { 11 } { 6 }$$
Then, $a _ { 4 } - a _ { 2 }$ equals
(A) 11
(B) 10
(C) 9
(D) 8 .

Q19 Matrices Determinant and Rank Computation View

Let $M$ be a $3 \times 3$ matrix with all entries being 0 or 1 . Then, all possible values for $\operatorname { det } ( M )$ are
(A) $0 , \pm 1$
(B) $0 , \pm 1 , \pm 2$
(C) $0 , \pm 1 , \pm 3$
(D) $0 , \pm 1 , \pm 2 , \pm 3$.

Q20 Circles Inscribed/Circumscribed Circle Computations View

In the following picture, $A B C$ is an isosceles triangle with an inscribed circle with center $O$. Let $P$ be the mid-point of $B C$. If $A B = A C = 15$ and $B C = 10$, then $O P$ equals:
(A) $\frac { \sqrt { 5 } } { \sqrt { 2 } }$
(B) $\frac { 5 } { \sqrt { 2 } }$
(C) $2 \sqrt { 5 }$
(D) $5 \sqrt { 2 }$.

Q21 Function Transformations Evaluate Composition from Algebraic Definitions View

For every real number $x \neq - 1$, let $f ( x ) = \frac { x } { x + 1 }$. Write $f _ { 1 } ( x ) = f ( x )$ and for $n \geq 2 , f _ { n } ( x ) = f \left( f _ { n - 1 } ( x ) \right)$. Then,
$$f _ { 1 } ( - 2 ) \cdot f _ { 2 } ( - 2 ) \cdots \cdots f _ { n } ( - 2 )$$
must equal
(A) $\frac { 2 ^ { n } } { 1 \cdot 3 \cdot 5 \cdots \cdot ( 2 n - 1 ) }$
(B) 1
(C) $\frac { 1 } { 2 } \binom { 2 n } { n }$
(D) $\binom { 2 n } { n }$.

Q22 Binomial Theorem (positive integer n) Extract Coefficients Using Roots of Unity or Substitution Filter View

Let the integers $a _ { i }$ for $0 \leq i \leq 54$ be defined by the equation
$$\left( 1 + X + X ^ { 2 } \right) ^ { 27 } = a _ { 0 } + a _ { 1 } X + a _ { 2 } X ^ { 2 } + \cdots + a _ { 54 } X ^ { 54 }$$
Then, $a _ { 0 } + a _ { 3 } + a _ { 6 } + a _ { 9 } + \cdots + a _ { 54 }$ equals
(A) $3 ^ { 26 }$
(B) $3 ^ { 27 }$
(C) $3 ^ { 28 }$
(D) $3 ^ { 29 }$.

Q23 Probability Definitions Combinatorial Number Theory and Counting View

An examination has 20 questions. For each question the marks that can be obtained are either $-1$ or $0$ or $4$. Let $S$ be the set of possible total marks that a student can score in the examination. Then, the number of elements in $S$ is
(A) 93
(B) 94
(C) 95
(D) 96 .

Q24 Circles Inscribed/Circumscribed Circle Computations View

Chords $A B$ and $C D$ of a circle intersect at right angle at the point $P$. If the lengths of $A P , P B , C P , P D$ are $2,6,3,4$ units respectively, then the radius of the circle is:
(A) 4
(B) $\frac { \sqrt { 65 } } { 2 }$
(C) $\frac { \sqrt { 66 } } { 2 }$
(D) $\frac { \sqrt { 67 } } { 2 }$

Q25 Curve Sketching Locus or solution set characterization of a trigonometric relation View

The locus of points ( $x , y$ ) in the plane satisfying $\sin ^ { 2 } ( x ) + \sin ^ { 2 } ( y ) = 1$ consists of
(A) A circle that is centered at the origin.
(B) infinitely many circles that are all centered at the origin.
(C) infinitely many lines with slope $\pm 1$.
(D) finitely many lines with slope $\pm 1$.

Q26 Binomial Theorem (positive integer n) Quadratic Diophantine Equations and Perfect Squares View

The number of integers $n \geq 10$ such that the product $\binom { n } { 10 } \cdot \binom { n + 1 } { 10 }$ is a perfect square is
(A) 0
(B) 1
(C) 2
(D) 3

Q27 Indices and Surds Modular Arithmetic Computation View

Let $a \geq b \geq c \geq 0$ be integers such that $2 ^ { a } + 2 ^ { b } - 2 ^ { c } = 144$. Then, $a + b - c$ equals:
(A) 7
(B) 8
(C) 9
(D) 10 .

Q28 Solving quadratics and applications Existence or counting of roots with specified properties View

The number of integers $n$ for which the cubic equation $X ^ { 3 } - X + n = 0$ has 3 distinct integer solutions is:
(A) 0
(B) 1
(C) 2
(D) infinite.

Q29 Curve Sketching Number of Solutions / Roots via Curve Analysis View

The number of real solutions of the equation $x ^ { 2 } = e ^ { x }$ is:
(A) 0
(B) 1
(C) 2
(D) 3 .

Q30 Curve Sketching Number of Solutions / Roots via Curve Analysis View

The number of distinct real roots of the equation $x \sin ( x ) + \cos ( x ) = x ^ { 2 }$ is
(A) 0
(B) 2
(C) 24
(D) none of the above.

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