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(1–
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The value of $\theta$ for which the system of equations $\left(\sin 3\theta \right)x-2y+3z=0,\left(\cos 2\theta \right)x+8y-7z=0$ and $2x+14y-11z=0$ has a non-trivial solution, is

($n\in Z)$

If the planes $x-3y+4z-1=0$ and $kx-4y+3z-5=0$ are perpendicular, then the value of $k$ is:

If $S$ is the set of all real values of $x$ such that $\frac{2x-1}{2x^3+3x^2+x}$ is positive, then $S$ contains

If $\sin \theta =-\frac{4}{5}$ and $\pi <\theta <\frac{3\pi }{2}$, then $\tan \theta +\cos \theta =$

How many elements will be there in the universal relation defined on the set A = {1,4, 9}

The equation of the hyperbola whose asymptotes are the lines 3x – 4y + 7 = 0 and 4x + 3y + 1 = 0 and which passes through origin is

Total number of values in $(-2\pi ,2\pi )$ and satisfying
$lo{g}_{|cosx|}|sinx|+lo{g}_{|sinx|}|cosx|=2$ is

If $f\left(x\right)=\underset{0}{\overset{x}{\int }}\frac{3^t}{1+t^2}dt,x>0$, then which of the following is/are correct?

Area of triangle formed by the points A(2, 0), B(6, 0) and C(4,6) is:
[4 MARKS]

The coefficient of $x^{50}$ in the expansion of $(1+x{)}^{1000}+2x(1+x{)}^{999}+3x^2(1+x{)}^{998}+......+1001x^{1000}$

The principal solution(s) for $\cos x=-\frac{1}{\sqrt{2}}$ is/are

Area bounded by the curve

f (x) = x² - 1; x-axis; lines x = 0 and x = 2 is

The value of is equal to ${\int }_3^6(\sqrt{x+\sqrt{12x-36}}+\sqrt{x-\sqrt{12x-36}})dx$is equal to

In an experiment with 15 observations on $x$, the following results were available: $\sum x^2=2830,\sum x=170$ One observation that was $20$ was found to be wrong and was replaced by the correct value $30$. The corrected variance is:

If $|z-1|\le 2$ and $|\omega z-1-{\omega }^2|=a$ (where $\omega$ is a cube root of unity) then complete set of values of $a$ is

If A = $\left[\begin{array}{cc}0& 2\\ 3& 5\end{array}\right]$, B = $\left[\begin{array}{ccc}3& 4& 5\\ 2& 3& 1\\ 1& 3& 7\end{array}\right]$ then A + B =

Let origin $O$ be the orthocentre of an equilateral triangle $ABC$. If $\overrightarrow{OA}=\overrightarrow{a},\overrightarrow{OB}=\overrightarrow{b},\overrightarrow{OC}=\overrightarrow{c},$ then $\overrightarrow{AB}+2\overrightarrow{BC}+3\overrightarrow{CA}=$

Differential equation of the family of parabolas whose vertex lie on the $x-$ axis and focus as origin is

• Write any nine words from the given list in the boxes. Put only one word in one box.
• The teacher will call out any six words. If the word she calls out is in the box put a cross on it. The one who crosses out all the words first shouts "Bingo" and is the winner.

Prove that
the following functions do not have maxima or minima:

(i) f(x)
= e^x (ii) g(x) = logx

(iii) h(x)
= x³ + x² + x + 1

If $x=\sin t,y=t\cos t.$ Then $\frac{dy}{dx}$ is equal to

The sum of the mean and variance of a binomial distribution is $15$ and the sum of their squares is $117$. Probability of atleast one success in case of above trials will be:

Evaluate $\underset{x\to 1}{lim}\frac{x^{15}-1}{x^{10}-1}$

If $\left|{\cos }^{-1}\left(\frac{1-x^2}{1+x^2}\right)\right|<\frac{\pi }{3}$ and $x\in (-\frac{1}{\sqrt{k}},\frac{1}{\sqrt{k}}),$ then the value of $k^2+1$ is equal to

If $a,b,c$ are the sides opposite to angles $A,B,C$ of a triangle $ABC,$ respectively and $\angle A=\frac{\pi }{3},b:c=\sqrt{3}+1:2,$ then the value of $\angle B-\angle C$ is

,
x
in quadrant II

If the lines (y-b)=$m_1$ (x+a) and (y-b)= $m_2$ (x+a) are the tangents of $y^2=4ax$then