Relation between Variables and Equations

Q.

$3{(\sin \left(x\right)–\cos \left(x\right))}^4+6{(\sin \left(x\right)+\cos \left(x\right))}^2+4\left({\sin }^6\left(x\right)+{\cos }^6\left(x\right)\right)=?$

1. $11$

2. $12$

3. $13$

4. $14$

Q.

$\int \frac{sin\left(2x\right)}{si{n}^2\left(x\right)+2co{s}^2\left(x\right)}dx=?$

1. $–\log (1+{\sin }^{2}x)+c$

2. $\log |1+{\cos }^{2}x|+c$

3. $–\log |1+{\cos }^{2}x|+c$

4. $\log (1+{\tan }^{2}x)+c$

Q. You have been asked to plot graphs of the following equations:
Equation 1: $3x+2y=1$
Equation 2: $y-2x=2$
Fill the following tables (find p, q, r, s) which you will use to plot the graphs:
Equation 1:
$\begin{array}{ccc}x& 0& 5\\ y& p& q\end{array}$
Equation 2:
$\begin{array}{ccc}x& r& s\\ y& 0& 5\end{array}$

1. $p=\frac{1}{2},q=-7,r=-1,s=\frac{3}{2}$
2. $p=2,q=7,r=2,s=3$
3. $p=3,q=\frac{1}{2},r=\frac{2}{3},s=0$
4. $p=3,q=5,r=2,s=8$

Q.

The point of intersection of axes is __.

Q.

Find the coordinates of the point where the line 2x-y=3 meets both the Axes

Q.

The total height of the Statue of Liberty and its pedestal is $153$ feet more than the height of the statue. What is the height of the statue? Justify your answer.

Q.

Let $0<x<\frac{\pi }{4}$ then $\sec \left(2x\right)-\tan \left(2x\right)$ equals

1. $\tan \left(x-\frac{\pi }{4}\right)$

2. $\tan \left(\frac{\pi }{4}-x\right)$

3. $\tan \left(x+\frac{\pi }{4}\right)$

4. ${\tan }^2\left(x+\frac{\pi }{4}\right)$

Q. If abscissa of a point is zero, on which axis do the point lies

Q.

The point of intersection of the lines x = 4 and y = -3 is ________.

1. (4, -3)

2. (4, 3)

3. (-4, 3)

4. (-4, -3)

Q. If 3x+b1y+5=0 and 4x+b2y+10=0 cut the x-axis and y-axis in four concyclic points , then the value of b1b2 i

Q. Question 4
The taxi fare in a city is as follows: For the first kilometre, the fare is Rs. 8 and for the subsequent distance it is Rs. 5 per km. Taking the distance covered as x km and total fare as Rs. y, write a linear equation for this information, and draw its graph.

Q.

The parameter, on which the value of the determinant $\left|\begin{array}{ccc}1& a& a^2\\ \cos \left(\left(p-d\right)x\right)& \cos \left(px\right)& \cos \left(\left(p+d\right)x\right)\\ \sin \left(\left(p-d\right)x\right)& \sin \left(px\right)& \sin \left(\left(p+d\right)x\right)\end{array}\right|$ does not depend upon is

1. $a$

2. $p$

3. $d$

4. $x$

Q.

${}^{15}C_3+{}^{15}C_5+...+{}^{15}C_{15}$ is equal to

1. $2^{14}$

2. $2^{14}-15$

3. $2^{14}+15$

4. $2^{14}-1$

Q.

Represent the following situations mathematically:

(i) John and Jivanti together have $45$marbles. Both of them lost $5$marbles each, and the product of the number of marbles they now have is$124$. We would like to find out how many marbles they had to start with.

(ii) A cottage industry produces a certain number of toys in a day. The cost of production of each toy (in rupees) was found to be$55$ minus the number of toys produced in a day. On a particular day, the total cost of production was `$750$. We would like to find out the number of toys produced on that day.

Q.

If point P lies on the line y = 2, then what is its y-coordinate?

1. 2

2. -2

3. 0

4. Cannot be determined

Q.

Question 47
In the following question, state whether the given statement is true (T) or false (F).
The number LIV is greater than LVI.

Q.

You have a total of $18$ math and science exercises for homework.

You have six more math exercises than science exercises.

How many exercises do you have in each subject?

Q. Question 1
Draw the graph of each of the following linear equations in two variables:
$\left(i\right)x+y=4\left(ii\right)x-y=2\left(iii\right)y=3x\left(iv\right)3=2x+y$

Q. Question 5
From the choices given below, choose the equation whose graphs are given in Fig. $4.6$ and Fig. $4.7$.

$ForFig.4.6$ $ForFig.4.7$
$\left(i\right)y=x$ $\left(i\right)y=x+2$
$\left(ii\right)x+y=0$ $\left(ii\right)y=x-2$
$\left(iii\right)y=2x$ $\left(iii\right)y=-x+2$
$\left(iv\right)2+3y=7x$ $\left(iv\right)x+2y=6$

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Q. A line passes through A(1. 1) and B(100, 1000). The number of points with integral co-ordinates on the linestrictly between A and B is

Q.

A man is $7$ times as old as his son. If he is $56$ years, find the age of his son.

Q.

The function $\frac{a\sin \left(x\right)+b\cos \left(x\right)}{c\sin \left(x\right)+d\cos \left(x\right)}$ is decreasing, if

1. $ad-bc>0$

2. $ad-bc<0$

3. $ab-ad>0$

4. $ab-cd<0$

Q.

Gina has $15$ dollars more than twice as much money as her sister Maria. If Gina gives Maria $30$ dollars, then Gina will have half as much money as her sister. How many dollars does Gina have?

Q. Question 7
$\frac{2x}{3}=18$

Q.

To plot the graphs of the following equations:
Equation 1: $3x+2y=1$
Equation 2: $y-2x=2$
Fill in the values of p, q, r, and s in the given tables.
Equation 1:
$\begin{array}{ccc}x& 0& 5\\ y& p& q\end{array}$
Equation 2:
$\begin{array}{ccc}x& r& s\\ y& 0& 5\end{array}$

1. $p=\frac{1}{2},q=-7,r=-1,s=\frac{3}{2}$
2. $p=2,q=7,r=2,s=3$
3. $p=3,q=\frac{1}{2},r=\frac{2}{3},s=0$
4. $p=3,q=5,r=2,s=8$

Q.

Suppose you start at the origin, move along the $x$-axis a distance of $4$ units in the positive direction, and then move downward along the $z$-axis a distance of $5$ units.

What are the coordinates of your position? $(x,y,z)$

Q. What is co-ordinate geometry?

Q.

Apples are to be transferred from larger baskets to smaller baskets. When a larger box is emptied, the apples from it fill three smaller baskets and still, $16apples$ remain outside. If the number of apples in a small basket is taken to be $x$, what is the number of apples in the larger basket?

Q. Prove that composition of two continuous function is continuous.

Q.

If $x^2\ne n\pi +1$, $n\in N$, then $\int x\sqrt{\frac{2\sin \left(x^2-1\right)-\sin 2\left(x^2-1\right)}{2\sin \left(x^2-1\right)+\sin 2\left(x^2-1\right)}}dx$ is equal to:

1. $\ln \cos \left(\frac{x^2-1}{2}\right)+c$

2. $\frac{1}{2}\ln \cos \left(\frac{x^2-1}{2}\right)+c$

3. $\ln \sec \left(\frac{x^2-1}{2}\right)+c$

4. $\frac{1}{2}\ln \sec \left(\frac{x^2-1}{2}\right)+c$