Pythagoras Theorem

Q.

The angle of elevation of the top of a tower from a certain point is $30°$. If the observer moves $20m$ towards the tower, the angle of elevation of the top increases by $15°$. Find the height of the tower.

Q. A ladder, 5 metre long, standing on a horizontal floor, leans against a vertical wall. If the top of the ladder slides down wards at the rate of 10 cm/sec, then find the rate at which the angle between the floor and ladder is decreasing when lower end of ladder is 2 metres from the wall.

Q. If a rectangle is inscribed in an equilateral triangle of side length $2\sqrt{2}$ as shown in the figure, then the square of the largest area of such a rectangle is

Q.

A point moves such that its distance from the point $(4,0)$ is half that of its distance from the line $x=16$.

The locus of this point is

1. $3x^2+4y^2=192$

2. $4x^2+3y^2=192$

3. $x^2+y^2=192$

4. None of these

Q. A circle of radius $2$ has centre at $(2,0)$ and another circle of radius $1$ has centre at $(5,0).$ A line is tangent to the two circles at points in first quadrant. The equation of tangent is

1. $\frac{x}{8}+\frac{y}{2\sqrt{2}}=1$
2. $\frac{x}{8}+\frac{y}{3}=1$
3. $\frac{x}{9}+\frac{y}{3}=1$
4. $\frac{x}{9}+\frac{y}{8}=1$

Q.

Two ships are anchored on opposite sides of a lighthouse. Their angles of depression as observed from the top of the lighthouse are found to be $30°$ and $45°$. The line joining the ships passes through the foot of the lighthouse. If the height of the lighthouse is $100\mathrm{m}$, find the distance between the ships.

Q.

Find the locus of a point such that the line segments having end points(2, 0) and (-2, 0) subtend a right angle at that point.

Q. Three circles of radii $a,4,16(a<4)$ touch each other externally. If they have $x-axis$ as a common tangent, then the value of $9a$ is

Q.

From the top of a tower $96m$ high, the angles of depression of two cars on a road at the same level as the base of the tower and on same side of it are $\theta$ and $\varphi ,$ where $\tan \left(\theta \right)=\frac{3}{4}$ and $\tan \left(\varphi \right)=\frac{1}{3}.$ Find the distance between the two cars.

Q. Let $AD$ and $BC$ be two vertical poles at $A$ and $B$ respectively on a horizontal ground. If $AD=8m,BC=11m$ and $AB=10m;$ then the distance (in meters) of a point $M$ on $AB$ from the point $A$ such that $M{D}^2+M{C}^2$ is minimum is

Q. Equations of four circles are $(x\pm 5{)}^2+(y\pm 5{)}^2=25.$ Then which of the following is (are) CORRECT?

1. The minimum radius of the circle touching all the four circles is $5(\sqrt{2}-1)$
   
2. The maximum radius of circle touching all the four circles is $5(\sqrt{2}+1)$
   
3. Area of region enclosed by all the four circles is $100-25\pi \text{sq. unit}$
   
4. The centres of four circles are the vertices of a rectangle

Q. If $x=e^{\frac{x}{y}}$ , prove that
$\frac{dy}{dx}=\frac{x-y}{xlogx}$

Q. How do you convert $-50$ degrees to radians?

Q. Ragnar leaves his office for his home. He walks towards east. After moving a distance of $15$ m, he turns south and walks $20$ m. Then he walks $40$ m towards the west and further $10$ m towards the north. He then turns towards east and walks $30$ m. What is the distance between his initial and final position?

1. $\sqrt{5}$ m
2. $5$ m
3. $25$ m
4. $5\sqrt{5}$ m

Q. A ladder is placed against a wall such that its foot is at a distance of 2.5 m from the wall and its top reaches a window 6 m above the ground. Find the length of the ladder.

Q. The diagonals $PR$ and $QS$ of a Rhombus $PQRS$ are $20cm$ and $48cm$ respectively. Find the length of side $PQ$.

Q. The area bounded by the circles $x^2+y^2=4,$ $x^2+y^2=16$ and the two rays given by $2x^2-3xy-2y^2=0(y\ge 0)$ is

1. $3\pi$ sq. unit
2. $\frac{3\pi }{2}$ sq. unit
3. $6\pi$ sq. unit
4. $\frac{3\pi }{4}$ sq. unit

Q. The equations $(q-r)x^2+(r-p)x+p-q=0$ and $(r-p)x^2+(p-q)x+q-r=0$ have a common root $x=a$. Find $a$.

Q. \begin{vmatrix}(y+z)^2&xy &zx xy&(x+z)^2&yz zx&yz&(x+y)^2\end{vmatrix} evaluate

Q. If f(x)=\begin{vmatrix}x-2&(x-1)^2&x^3 x-1&x^2&(x+1)^3 x&(x+1)^2&(x+2)^3\end{vmatrix} , then coefficient of x in f(x) is

Q. A ladder 13 m long leans against a wall. The foot of the ladder is pulled along the ground away from the wall, at the rate of 1.5 m/sec. How fast is the angle θ between the ladder and the ground is changing when the foot of the ladder is 12 m away from the wall.

Q. Assertion :If $x+ky=1$ and $x=a$ are the equations of the hypotenuse and a side of a right angled isosceles triangle then $k=\pm a$. Reason: Each side of a right angled isosceles triangle makes an angle $\pm \frac{\pi }{4}$ with the hypotenuse.

1. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
2. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
3. Assertion is incorrect but Reason is correct
4. Both Assertion and Reason are incorrect

Q. A circle of radius $2$ has centre at $(2,0)$ and another circle of radius $1$ has centre at $(5,0).$ A line is tangent to the two circles at points in first quadrant. The equation of tangent is

1. $\frac{x}{8}+\frac{y}{2\sqrt{2}}=1$
2. $\frac{x}{9}+\frac{y}{8}=1$
3. $\frac{x}{8}+\frac{y}{3}=1$
4. $\frac{x}{9}+\frac{y}{3}=1$

Q. A triangle ABC in which DE||BC, and intersects AB in D and AC in E
Hence,
$\frac{AB}{AD}=\frac{AC}{AE}$

1. True
2. False

Q. If $(p\wedge \sim q)\wedge (p\wedge r)\to \sim p\vee q$ is false, then the truth values of $p,q$ and $r$ are, respectively.

1. $T,F,T$
2. $F,F,F$
3. $T,T,T$
4. $F,T,F$

Q. If sec A = $\frac{17}{8}$, verify that $\frac{3-4{\sin }^{2}A}{4{\cos }^{2}A-3}=\frac{3-{\tan }^{2}A}{1-3{\tan }^{2}A}$

Q. Find the difference between CI and SI on Rs. $32,000$ at $12\%$ p.a. for $3$ years.

Q. A man is walking at at the rate of $4.5xkm/hr$ towards the foot of the tower $120xm$ high. At what rate is he approaching the top of the tower when he is $50xm$ away from the tower?

Q. If $f^{\prime \prime }\left(0\right)=k,k\ne 0$, then the value of $\underset{x\to 0}{lim}\frac{2f\left(x\right)-3f\left(2x\right)+f\left(4x\right)}{x^2}$ is

1. $k$
2. $3k$
3. $4k$
4. $2k$

Q. A line $L$ passing through the focus of parabola $y^2=4(x-1),$ intersects the parabola at two distinct points. If $m$ be the slope of line $L,$ then:

1. $m\in R-\left\{0\right\}$
2. $-1<m<1$
3. $m<-1\text{or}m>1$
4. None of these