Equations

Q. Which one of the following conditions must p, q and r satisfy so that the following system of linear simultaneous equation has at least one solution, such that p + q + r $\ne$ 0?
x + 2y - 3z = p
2x + 5y - 11z = q
x - 2y + 7z = r

1. 5q - 2q + r = 0
2. 5p - 2q - r = 0
3. 5p + 2q + r = 0
4. 5q + 2q - r = 0

Q. A polynomial $a{x}^3+b{x}^2$ + cx + d intersects the x - axis at 1 and -1, and y - axis at 2. The value of b is:

1. -2
2. 0
3. 2
4. 1

Q. If f(x, y) $=4^{x^y}+x^{4^y}$, then what is the total number of solutions for the equation f(x, y) = 5?

1. 0
2. 1
3. 2
4. More than 2

Q. 30 trees are planted in a straight line at intervals of 5m. To water them, the gardener needs to bring water for each tree, separately from a well, which is 10m from the first tree in line with the trees. How far will he have to walk in order to water all the trees beginning with the first tree? Assume that he starts from the well.

1. 4795 m
2. None of these
3. 4800 m
4. 4785 m

Q. Find the roots of the equation $32x^3-48x^2$ + 22x - 3 = 0, if it's given that the roots are in arithmetic progression (A.P).

1. $\frac{1}{4},\frac{1}{2},\frac{3}{4}$
2. 3, 5, 7
3. 1, 2, 3
4. $\frac{1}{2},1,\frac{3}{2}$

Q. If $x^2+y^2+\frac{1}{x^2}+\frac{1}{y^2}=4,$ then the value of $x^2+y^2$ is

1. 8
2. 16
3. 2
4. 4

Q. A boy throws a ball to the ground with force from a height of 10 m. After hitting the ground for the first time, it rises to a height of 20 m. Thereafter it rises only upto half the prior height on hitting the ground. What is the total distance travelled by the ball till it comes to rest?

1. 80 m
2. 40 m
3. 50 m
4. 90 m

Q. If $\alpha ,\beta ,\gamma$ are the roots of the equation $x^3-3x+11=0$, then the equation whose roots are $(\alpha +\beta ),(\beta +\gamma )$ and $(\gamma +\alpha )$ is :

1. $x^3+3x+11+0$
2. $x^3-3x+11=0$
3. $x^3+3x-11=0$
4. $x^3-3x-11=0$

Q. If the polynomial $x^3$ + px + q has three distinct roots, then which of the following is a possible value of p?

1. -1
2. 0
3. 1
4. 2

Q. In an examination, there are 30 questions, 1 mark is given for each correct answer and 0.25 is deducted for every incorrect answer. Ankur attempted all the questions and scored 13.75 How many incorrect answers did he have?

1. 10
2. 11
3. 12
4. None of these