If z z̅ = |z + z̅ |, where z = x + iy, i = $\sqrt{-1}$, then the locus of z is a pair of:

If 1! + 3! + 5! + 7! + ... + 199! is divided by 24, what is the remainder?

What is the value of $\sqrt{12+5 i}+\sqrt{12-5 i}$ where $i=\sqrt{-1}$ ?

If $A=\left[\begin{array}{l}1 \\ 2 \\ 3\end{array}\right]$, then what is the value of det(I + AA'), where I is the 3 × 3 identity matrix?

If A, B and C are square matrices of order 3 and det(BC) = 2 det(A), then what is the value of det(2A^-1BC)?

If the nth term of a sequence is $\frac{2 n+5}{7}$, then what is the sum of its first 140 terms? 

Let A be a skew-symmetric matrix of order 3.

What is the value of det(4A4) - det(3A3) + det(2A2) - det(A) + det(-I)  where I is the identity matrix of order 3?

If $A=\left[\begin{array}{rrr} 0 & 3 & 4 \\ -3 & 0 & 5 \\ -4 & -5 & 0 \end{array}\right]$, then which one of the following statements is correct?

If $A=\left[\begin{array}{lll} 2 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 4 \end{array}\right]$, then which of the following statements are correct?

1. An will always be singular for any positive integer n.

2. An will always be a diagonal matrix for any positive integer n.

3. An will always be a symmetric matrix for any positive integer n.

Select the correct answer using the code given below:

If (a + b), 2b, (b + c) are in HP, then which one of the following is correct?

Let t1, t2, t3 ... be in GP. What is $\rm \left(t_1 t_3 \ldots t_{21}\right)^{\frac{1}{11}}$ equal to ?

Which one of the following is a square root of $-\sqrt{-1} $?

What is the maximum number of points of intersection of 10 circles?

A set S contains (2n + 1) elements. There are 4096 subsets of S which contain at most n elements. What is n equal to?

If $\left|\begin{array}{ccc} x^2+3 x & x-1 & x+3 \\ x+1 & -2 x & x-4 \\ x-3 & x+4 & 3 x \end{array}\right|$ = ax4 + bx3 + cx2 + dx + e, then what is the value of e?"

If all elements of a third order determinant are equal to 1 or -1, then the value of the determinant is:

If $A=\left[\begin{array}{rrr} 2 & -1 & 0 \\ -1 & 3 & 0 \\ 1 & 0 & 1 \end{array}\right]$, then what is the value of det[adj(adjA)] ?

If $A=\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right]$, then what is 23A3 - 19A2 - 4A equal to ?

The value of the determinant of a matrix A of order 3 is 3. If C is the matrix of cofactors of the matrix A, then what is the value of determinant of C2?

If $A_k=\left[\begin{array}{cc} k-1 & k \\ k-2 & k+1 \end{array}\right] $, then what is det(A1) + det(A2) + det(A3) + ... + det(A100) equal to ?

The Cartesian product A × A has 16 elements among which are (0, 2) and (1, 3). Which of the following statements is/are correct?

1. It is possible to determine set A.

2. A × A contains the element (3, 2).

Select the correct answer using the code given below:

Let A = {1, 2, 3, ..., 20}. Define a relation R from A to A by R = {(x, y) : 4x - 3y = 1}, where x, y ∈ A. Which of the following statements is/are correct?

1. The domain of R is {1, 4, 7, 10, 13, 16).

2. The range of R is {1, 5, 9, 13, 17).

3. The range of R is equal to codomain of R.

Select the correct answer using the code given below:

Consider the following statements: 

1. The relation f defined by $f(x)= \begin{cases}x^3, & 0 \leq x \leq 2 \\ 4 x, & 2 \leq x \leq 8\end{cases}$ is a function.

2. The relation g defined by $g(x)= \begin{cases}x^2, & 0 \leq x \leq 4 \\ 3 x, & 4 \leq x \leq 8\end{cases}$ is a function.

Which of the statements given above is/are correct?

Consider the following statements 

1. A = (A ∪ B) ∪ (A - B),

2. A ∪ (B - A) = (A ∪ B)

3. B = (A ∪ B) - (A - B)

Which of the statements given above are correct?

A function satisfies $f(x-y)=\frac{f(x)}{f(y)}$, where f(y) ≠ 0. If f(1) = 0.5, then what is f(2) + f(3) + f(4) + f(5) + f(6) equal to ?

What is 2 cot $\left(\frac{1}{2} \cos ^{-1} \frac{\sqrt{5}}{3}\right)$ equal to ?

If sec-1 p - cosec-1q = 0, where p > 0, q > 0; then what is the value of p-2 + q-2 ?

What is $1+\sin ^2\left(\cos ^{-1}\left(\frac{3}{\sqrt{17}}\right)\right)$ equal to ?

If tan (π cos θ) = cot (π sin θ), $0<\theta<\frac{\pi}{2}$; then what is the value of $8 \sin ^2\left(\theta+\frac{\pi}{4}\right)$ ?

If $\tan \alpha=\frac{1}{7}$, $\sin \beta=\frac{1}{\sqrt{10}}$; $0<\alpha, \beta<\frac{\pi}{2}$, then what is the value of cos (α + 2β) ?

What is the number of real roots of the equation?

What is the sum of all the roots of the equation?

What are the roots of equation-I ?

Which one of the following is a root of equation-II?

What is the number of common roots of equation-I and equation-II?

If $\rm {k}=\frac{{c}}{2},({c} \neq 0)$, then the roots of the equation are :

If k = c, then the roots of the equation are:

What is T1 + 2T2 + 3T3 + ... + nTn equal to ?

What is 1 - T1 + 2T2 - 3T3 + ... + (-1)nnTn equal to ? 

What is T1 + T2 + T3 + ... + Tn equal to ?

Which one of the following is a possible expression for g(x)?

What is g(15) equal to ?

What is f(0.5) equal to ?

If f is differentiable, then what is f'(0.5) equal to?

The function is decreasing on :

The function attains local minimum value at :

What is the maximum value of y?

What is the maximum value of xy ?

What is the range of the function?

What is the period of the function?

What is the directrix of the parabola ?

What is the length of latus rectum of the parabola?

What is $\displaystyle \lim _{x \rightarrow 1} \frac{f(x)-1}{g(x)}$ equal to?

What is $\displaystyle \lim _{x \rightarrow 1} f(x)^{\frac{1}{g(x)}}$ equal to?

What is the domain of the function?

What is the greatest value of the function?

What is $\displaystyle \lim _{x \rightarrow 0+}$ h(x) equal to ?

What is $\displaystyle \lim _{x \rightarrow 0-}$ h(x) equal to ?

What is the value of a ?

What is the value of b?

What is $\displaystyle \int_0^\pi\left(\sin ^4 x+\cos ^4 x\right) d x$ equal to?

What is I equal to?

If the function f(x) is differentiable at x = 1, then what is the value of (a + b)?

What is $\displaystyle \lim _{x \rightarrow 0} $ f(x) equal to ?

If f(x) = |ln|x|| where 0 < x < 1, then what is f'(0.5) equal to ?

If f'(x) = cos (In x) and $y=f\left(\frac{2 x-3}{x}\right)$, then what is $\frac{d y}{d x}$ equal to ?

What is $\displaystyle \rm \int_0^{8 \pi}|\sin x| d x$ equal to?

What is the area between the curve f(x) = x |x| and x-axis for x = [-1, 1]?

What are the order and the degree respectively of the differential equation $\rm x^2\left(\frac{d^3 y}{d x^3}\right)^2+\left(\frac{d y}{d x}\right)^4+\sin x=0$ ? 

What is the differential equation of all parabolas of the type y2 = 4a (x - b)?

What is a1 + a5 - a10 - a15 - a20 - a25 + a30 + a34 equal to ?

What is $\displaystyle \sum_{n=1}^{34} a_n$ equal to ?

What is the value of p + q?

What is the value of pq?

What is pq equal to ?

For how many values of x does $\frac{1}{p}$ become zero?

What is a value of sin 3x + sin 3y?

What is a value of cos³ x + cos³ y?

What is the value of a + b + √2 c equal to ?

What is the ratio of a2 ∶ b2 ∶ c2 ?

What is the equation of directrix of parabola y2 = 4bx, where b < 0 and b2 + b - 2 = 0?

The points (-a, -b), (0, 0), (a, b) and (a2, ab) are:

Given that 16p2 + 49q2 - 4r2 - 56pq = 0. Which one of the following is a point on a pair of straight lines (px + qy + r) (px + qy - r) = 0?

If 3x + y - 5 = 0 is the equation of a chord of the circle x2 + y2 - 25 = 0, then what are the coordinates of the mid-point of the chord ?

Consider the following in respect of the equation $\frac{x^2}{24-k}+\frac{y^2}{k-16}=2$.

1. The equation represents an ellipse if k = 19.

2. The equation represents a hyperbola if k = 12.

3. The equation represents a circle if k = 20.

How many of the statements given above are correct?

Consider the following statements in respect of hyperbola $\frac{x^2}{\cos ^2 \theta}-\frac{y^2}{\sin ^2 \theta}=1$ :

1. The two foci are independent of θ.

2. The eccentricity is sec θ.

3. The distance between the two foci is 2 units.

How many of the statements given above are correct?

Consider the following in respect of the circle 4x2 + 4y2 - 4ax - 4ay + a2 = 0:

1. The circle touches both the axes. 

2. The diameter of the circle is 2a.

3. The centre of the circle lies on the line x + y = a.

How many of the statements given above are correct?

For what values of k is the line (k - 3)x - (5 - k2) y + k2 - 7k + 6 = 0 parallel to the line x + y = 1?

The line x + y = 4 cuts the line joining P(-1, 1) and Q(5, 7) at R. What is PR ∶ RQ equal to ?

What is the sum of the intercepts of the line whose perpendicular distance from origin is 4 units and the angle which the normal makes with positive direction of x-axis is 15°?

What is the length of projection of the vector $\rm \hat{i}+2 \hat{j}+3 \hat{k}$ on the vector $\rm2 \hat{i}+3 \hat{j}-2 \hat{k}$ ?

If $\rm (\vec{a} \times \vec{b})^2+(\vec{a} \cdot \vec{b})^2=144$ and $\rm|\vec{b}|=4 $, then what is the value of $\rm|\vec{a}|$ ?

If θ is the angle between vectors $\vec{a}$ and $ \vec{b}$ such that $\vec{a} \cdot \vec{b} \geq 0$, then which one of the following is correct?

The vectors $\rm 60 \hat{i}+3 \hat{j}, 40 \hat{i}-8 \hat{j}$ and $\rm \beta \hat{i}-52 \hat{j}$ are collinear if :

Consider the following in respect of the vectors $\rm \vec{a}=(0,1,1)$ and $\rm \vec{b}=(1,0,1) $ :

1. The number of unit vectors perpendicular to both $\rm \vec{a}$ and $\rm \vec{b}$ is only one.

2. The angle between the vectors is $\frac{\pi}{3}$.

Which of the statements given above is/are correct?

If L is the line with direction ratios < 3, -2, 6 > and passing through (1, -1, 1), then what are the coordinates of the points on L whose distance from (1, -1, 1) is 2 units?

Which one of the planes is parallel to the line $\rm \frac{x-2}{3}=\frac{y-3}{4}=\frac{z-4}{5}$ ?

What is the angle between the lines 2x = 3y = -z and 6x = -y = -4z?

What is the equation of the sphere concentric with the sphere x2 + y2 + z2 - 2x - 6y -8z - 5 = 0 and which passes through the origin?

A point P lies on the line joining A(1, 2, 3) and B(2, 10, 1). If z-coordinate of P is 7, what is the sum of other two coordinates? 

The sum of deviations of n numbers from 10 and 20 are p and q respectively. If (p - q)2 = 10000, then what is the value of n?

If X̅ = 20 is the mean of 10 observations x1, x2, ... x10; then what is the value of $\displaystyle \sum_{i=1}^{10}\left(\frac{3 x_i-4}{5}\right) ?$ ?

If the mean and the sum of squares of 10 observations are 40 and 16160 respectively, then what is the standard deviation?

Three dice are thrown. What is the probability of getting a sum which is a perfect square?

A, B, C and D are mutually exclusive and exhaustive events.

If 2P(A) = 3P(B) = 4P(C) = 5P(D), then what is 77P(A) equal to ?

Two distinct natural numbers from 1 to 9 are picked at random. What is the probability that their product has 1 in its unit place?

Two dice are thrown. What is the probability that difference of numbers on them is 2 or 3 ?

What is the mean of the numbers 1, 2, 3, ... 10 with frequencies ⁹C₀, ⁹C₁, ⁹C₂ ..., ⁹C₉, respectively?

The probability that a person recovers from a disease is 0.8. What is the probability that exactly 2 persons out of 5 will recover from the disease?

Suppose that there is a chance for a newly constructed building to collapse, whether the design is faulty or not. The chance that the design is faulty is 10%. The chance that the building collapses is 95% if the design is faulty, otherwise it is 45%. If it is seen that the building has collapsed, then what is the probability that it is due to faulty design?

If r is the coefficient of correlation between x and y, then what is the correlation coefficient between (3x + 4) and (-3y + 3)?

A fair coin is tossed 6 times. What is the probability of getting a result in the 6^th toss which is different from those obtained in the first five tosses ?

If H is the Harmonic Mean of three numbers 10C4, 10C5, and 10C6, then what is the value of $\frac{270}{H}$ ?

In a class, there are n students including the students P and Q. What is the probability that P and Q sit together if seats are assigned randomly?

In a Binomial distribution B(n, p), n = 6 and 9P(X = 4) = P(X = 2). What is p equal to ?

What is the probability that all three boys sit together?

What is the probability that boys and girls sit alternatively?

What is the probability that no two girls sit together?

What is the probability that P and Q take the two end positions?

What is the probability that Q and U sit together?