List of practice Questions

If \( (2,3) \) is the focus and \( x - y + 3 = 0 \) is the directrix of a parabola, then the equation of the tangent drawn at the vertex of the parabola is:
The equation of the common tangent to the parabola \(y^2 = 8x\) and the circle \(x^2 + y^2 = 2\) is \(ax + by + 2 = 0\). If \(-\frac{a}{b}>0\), then \(3a^2 + 2b + 1 =\)

Consider the parabola \(25[(x-2)^2 + (y+5)^2] = (3x+4y-1)^2\), match the characteristic of this parabola given in List-I with its corresponding item in List-II. 

If \(6x-5y-20=0\) is a normal to the ellipse \(x^2 + 3y^2 = k\), then \(k =\)
If the common chord of the circles \( x^2 + y^2 - 2x + 2y + 1 = 0 \) and \( x^2 + y^2 - 2x - 2y - 2 = 0 \) is the diameter of a circle \( S \), then the centre of the circle \( S \) is:
The axis of a parabola is parallel to Y-axis. If this parabola passes through the points \( (1,0), (0,2), (-1,-1) \) and its equation is \( ax^2 + bx + cy + d = 0 \), then \( \frac{ad}{bc} \) is:
If the focus of an ellipse is \((-1,-1)\), equation of its directrix corresponding to this focus is \(x + y + 1 = 0\) and its eccentricity is \(\frac{1}{\sqrt{2}}\), then the length of its major axis is:
P(\( \theta \)) is a point on the hyperbola \( \frac{x^2}{a^2} - \frac{y^2}{9} = 1 \), S is its focus lying on the positive X-axis and Q = (0,1). If SQ = \( \sqrt{26} \) and SP = 6, then \( \theta \) is:
A right circular cone is inscribed in a sphere of radius 3 units. If the volume of the cone is maximum, then the semi-vertical angle of the cone is:
If \( f(x) = kx^3 - 3x^2 - 12x + 8 \) is strictly decreasing for all \( x \in \mathbb{R} \), then:
If \( \int x^3 \sin 3x \,dx = f(x) \cos 3x + g(x) \sin 3x + c \), then evaluate \( 27(f(x) + xg(x)) \):
Evaluate the integral: \[ \int \frac{2\cos 3x - 3\sin 3x}{\cos 3x + 2\sin 3x} dx. \]
Evaluate the integral: \[ I = \int_{\frac{-3\pi}{4}}^{\frac{\pi}{6}} \log(\sin(4x+3)) \,dx \]
Evaluate the integral: \[ \int_0^{16} \frac{\sqrt{x}}{1 + \sqrt{x}} dx. \]
Evaluate the integral: \[ \int \frac{4\tan^4 x + 3 \tan^2 x - 1}{\tan^2 x + 4} \, dx. \]
Evaluate the integral: \[ \int \frac{(\sin^4x + 2\cos^2x - 1) \cos x}{(1 + \sin x)^6} \, dx \]
Evaluate the integral: \[ \int (\log x)^3 \, dx \]
Evaluate the integral: \[ I = \int_{-\frac{\pi}{8}}^{\frac{\pi}{8}} \frac{\sin^4(4x)}{1 + e^{4x}} \, dx \]
If all the possible 3-digit numbers are formed using the digits 1, 3, 5, 7, 9 without repeating any digit, then the number of such 3-digit numbers which are divisible by 3 is:
A question paper has 3 parts A, B, C. Part A contains 7 questions, part B contains 5 questions and Part C contains 3 questions. If a candidate is allowed to answer not more than 4 questions from part A; not more than 3 questions from part B and not more than 2 questions from part C, then the number of ways in which a candidate can answer exactly 7 questions is:
The sum of all the 4-digit numbers formed by taking all the digits from \(2, 3, 5, 7\) without repetition is
The number of ways in which 15 identical gold coins can be distributed among 3 persons such that each one gets at least 3 gold coins is

The number of all possible combinations of 4 letters which are taken from the letters of the word ACCOMMODATION is

The sum of all the 4-digit numbers formed by taking all the digits from 0, 3, 6, 9 without repetition is:
The number of ways in which the number 831600 can be split into two factors which are relatively prime is: