Class 10 Maths - Hard Question

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### Real Numbers 1. Prove that $\sqrt{5}$ is irrational. 2. Prove that for any positive integer $n$, $n^2 - n$ is always even. 3. Show that $5 - 2\sqrt{3}$ is an irrational number. 4. If $p$ and $q$ are positive integers such that $p = ab^2$ and $q = a^3b$, where $a$ and $b$ are prime numbers, then find LCM $(p, q)$ and HCF $(p, q)$. 5. There is a circular path around a sports field. Sonia takes 18 minutes to drive one round of the field, while Ravi takes 12 minutes for the same. Suppose they both start at the same point and at the same time, and go in the same direction. After how many minutes will they meet again at the starting point? ### Polynomials 1. If $\alpha$ and $\beta$ are the zeroes of the quadratic polynomial $f(x) = x^2 - p(x+1) - c$, show that $(\alpha+1)(\beta+1) = 1-c$. 2. If the zeroes of the polynomial $x^2 + px + q$ are double in value to the zeroes of $2x^2 - 5x - 3$, find the values of $p$ and $q$. 3. If one zero of the polynomial $(a^2 + 9)x^2 + 13x + 6a$ is the reciprocal of the other, find the value of $a$. 4. If $\alpha, \beta, \gamma$ are the zeroes of the polynomial $f(x) = ax^3 + bx^2 + cx + d$, then find the values of $\alpha^2 + \beta^2 + \gamma^2$ and $\frac{1}{\alpha} + \frac{1}{\beta} + \frac{1}{\gamma}$. 5. If two zeroes of the polynomial $x^4 - 6x^3 - 26x^2 + 138x - 35$ are $2 \pm \sqrt{3}$, find the other zeroes. ### Pair of Linear Equations in Two Variables 1. Solve the following system of equations: $\frac{2}{x} + \frac{3}{y} = 13$ $\frac{5}{x} - \frac{4}{y} = -2$ 2. A boat goes 30 km upstream and 44 km downstream in 10 hours. In 13 hours, it can go 40 km upstream and 55 km downstream. Determine the speed of the stream and that of the boat in still water. 3. A man travels 600 km partly by train and partly by car. If he covers 400 km by train and the rest by car, it takes him 6 hours 30 minutes. But if he travels 200 km by train and the rest by car, he takes 7 hours. Find the speed of the train and the car. 4. The area of a rectangle gets reduced by 9 square units if its length is reduced by 5 units and breadth is increased by 3 units. If we increase the length by 3 units and the breadth by 2 units, the area increases by 67 square units. Find the dimensions of the rectangle. 5. For what values of $a$ and $b$ does the following pair of linear equations have an infinite number of solutions? $2x + 3y = 7$ $(a-b)x + (a+b)y = 3a+b-2$ ### Quadratic Equations 1. A takes 6 days less than B to finish a piece of work. If both A and B together can finish the work in 4 days, find the time taken by B to finish the work. 2. If the roots of the equation $(a^2+b^2)x^2 - 2(ac+bd)x + (c^2+d^2) = 0$ are equal, prove that $\frac{a}{b} = \frac{c}{d}$. 3. A train travels at a certain average speed for a distance of 63 km and then travels a distance of 72 km at an average speed of 6 km/h more than its original speed. If it takes 3 hours to complete the total journey, what is the original average speed? 4. Solve for $x$: $\frac{1}{x+1} + \frac{2}{x+2} = \frac{4}{x+4}$, where $x \neq -1, -2, -4$. 5. If $p, q$ are the roots of $x^2 - (k+1)x + b = 0$, then find the quadratic equation whose roots are $p^2+1$ and $q^2+1$. ### Arithmetic Progressions 1. If the $m^{th}$ term of an AP is $1/n$ and the $n^{th}$ term is $1/m$, show that its $(mn)^{th}$ term is 1. 2. If the sum of $n$ terms of an AP is $3n^2 + 5n$, find the AP. Also find its $16^{th}$ term. 3. The ratio of the sums of $m$ and $n$ terms of an AP is $m^2:n^2$. Show that the ratio of the $m^{th}$ and $n^{th}$ terms is $(2m-1):(2n-1)$. 4. If the sum of first $p$ terms of an AP is equal to the sum of first $q$ terms, then find the sum of its first $(p+q)$ terms. 5. The $n^{th}$ term of an AP is $p$. The sum of the first $n$ terms is $q$. Prove that the common difference is $\frac{2(q-np)}{n(n-1)}$. ### Triangles 1. In an equilateral triangle $ABC$, $D$ is a point on side $BC$ such that $BD = \frac{1}{3}BC$. Prove that $9AD^2 = 7AB^2$. 2. Prove that the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding medians. 3. State and prove Basic Proportionality Theorem (Thales Theorem). 4. In $\triangle ABC$, $AD$ is a median and $E$ is the midpoint of $AD$. $BE$ produced meets $AC$ at $F$. Prove that $AF = \frac{1}{3}AC$. 5. If a line intersects sides $AB$ and $AC$ of a $\triangle ABC$ at $D$ and $E$ respectively and is parallel to $BC$, prove that $\frac{AD}{AB} = \frac{AE}{AC}$. ### Coordinate Geometry 1. Find the ratio in which the line segment joining the points $(-3, 10)$ and $(6, -8)$ is divided by $(-1, 6)$. 2. If the points $A(1, -2)$, $B(2, 3)$, $C(a, 2)$ and $D(-4, -3)$ are the vertices of a parallelogram, find the value of $a$. 3. Find the coordinates of the vertices of a triangle whose sides are given by the equations $y = x$, $3y = x$, and $x + y = 8$. 4. If $P(x, y)$ is any point on the line joining the points $A(a, 0)$ and $B(0, b)$, then show that $\frac{x}{a} + \frac{y}{b} = 1$. 5. The area of a triangle is 5 sq units. Two of its vertices are $(2, 1)$ and $(3, -2)$. If the third vertex is $(x, y)$ where $y = x+3$, find the coordinates of the third vertex. ### Introduction to Trigonometry 1. Prove that: $\frac{\sin\theta - \cos\theta + 1}{\sin\theta + \cos\theta - 1} = \frac{1}{\sec\theta - \tan\theta}$ 2. If $\sin\theta + \cos\theta = \sqrt{3}$, then prove that $\tan\theta + \cot\theta = 1$. 3. If $\sec\theta = x + \frac{1}{4x}$, prove that $\sec\theta + \tan\theta = 2x$ or $\frac{1}{2x}$. 4. Prove that: $(1 + \cot A + \tan A)(\sin A - \cos A) = \frac{\sec^3 A - \csc^3 A}{\sec^2 A + \csc^2 A}$. 5. If $\angle A$, $\angle B$ and $\angle C$ are interior angles of a triangle $ABC$, then show that $\sin\left(\frac{B+C}{2}\right) = \cos\left(\frac{A}{2}\right)$. ### Some Applications of Trigonometry 1. A straight highway leads to the foot of a tower. A man standing at the top of the tower observes a car at an angle of depression of $30^\circ$, which is approaching the foot of the tower with a uniform speed. Six seconds later, the angle of depression of the car is found to be $60^\circ$. Find the time taken by the car to reach the foot of the tower from this point. 2. The angles of elevation of the top of a tower from two points at a distance of $4 \text{ m}$ and $9 \text{ m}$ from the base of the tower and in the same straight line with it are complementary. Prove that the height of the tower is $6 \text{ m}$. 3. From the top of a $7 \text{ m}$ high building, the angle of elevation of the top of a cable tower is $60^\circ$ and the angle of depression of its foot is $45^\circ$. Determine the height of the tower. 4. A $1.2 \text{ m}$ tall girl spots a balloon moving with the wind in a horizontal line at a height of $88.2 \text{ m}$ from the ground. The angle of elevation of the balloon from the eyes of the girl at any instant is $60^\circ$. After some time, the angle of elevation reduces to $30^\circ$. Find the distance travelled by the balloon during this interval. 5. A person standing on the bank of a river observes that the angle of elevation of the top of a tree on the opposite bank is $60^\circ$. When he retreats $20 \text{ m}$ from the bank, he finds the angle of elevation to be $30^\circ$. Find the height of the tree and the width of the river. ### Circles 1. Prove that the lengths of tangents drawn from an external point to a circle are equal. 2. Prove that the tangent at any point of a circle is perpendicular to the radius through the point of contact. 3. Two concentric circles are of radii $5 \text{ cm}$ and $3 \text{ cm}$. Find the length of the chord of the larger circle which touches the smaller circle. 4. In the given figure, $XY$ and $X'Y'$ are two parallel tangents to a circle with centre $O$ and another tangent $AB$ with point of contact $C$ intersecting $XY$ at $A$ and $X'Y'$ at $B$. Prove that $\angle AOB = 90^\circ$. (Assume a diagram where $O$ is the center, $XY$ and $X'Y'$ are parallel tangents. $AB$ is another tangent touching the circle at $C$. $A$ is on $XY$ and $B$ is on $X'Y'$) 5. A triangle $ABC$ is drawn to circumscribe a circle of radius $4 \text{ cm}$ such that the segments $BD$ and $DC$ into which $BC$ is divided by the point of contact $D$ are of lengths $8 \text{ cm}$ and $6 \text{ cm}$ respectively. Find the sides $AB$ and $AC$. ### Constructions 1. Draw a triangle $ABC$ with side $BC = 6 \text{ cm}$, $AB = 5 \text{ cm}$ and $\angle ABC = 60^\circ$. Then construct a triangle whose sides are $\frac{3}{4}$ of the corresponding sides of the triangle $ABC$. 2. Draw a circle of radius $3 \text{ cm}$. Take two points $P$ and $Q$ on one of its extended diameter each at a distance of $7 \text{ cm}$ from its centre. Draw tangents to the circle from these two points $P$ and $Q$. 3. Construct a triangle similar to a given triangle $ABC$ with its sides equal to $\frac{5}{3}$ of the corresponding sides of the triangle $ABC$. Give the steps of construction. 4. Draw a pair of tangents to a circle of radius $5 \text{ cm}$ which are inclined to each other at an angle of $60^\circ$. 5. Construct tangent to a circle from a point outside it, using alternate segment theorem. ### Areas Related to Circles 1. A car has two wipers which do not overlap. Each wiper has a blade of length $25 \text{ cm}$ sweeping through an angle of $115^\circ$. Find the total area cleaned at each sweep of the blades. 2. In a circular table cover of radius $32 \text{ cm}$, a design is formed leaving an equilateral triangle $ABC$ in the middle as shown in the figure. Find the area of the design. (Assume a diagram of a circle with an inscribed equilateral triangle $ABC$) 3. Find the area of the shaded region in the figure, where $ABCD$ is a square of side $14 \text{ cm}$ and $APDQ$ and $BPRC$ are semicircles. (Assume a diagram of a square $ABCD$ with two semicircles drawn on sides $AD$ and $BC$ as diameters, facing outwards, with points $P, Q, R$ on the arcs) 4. In the given figure, $ABCD$ is a square of side $10 \text{ cm}$ and semicircles are drawn with each side as diameter. Find the area of the shaded region. (Assume a diagram of a square $ABCD$ with four semicircles drawn on each side as diameter, facing inwards, forming a petal-like shaded region in the center) 5. A brooch is made with silver wire in the form of a circle with diameter $35 \text{ mm}$. The wire is also used in making 5 diameters which divide the circle into 10 equal sectors. Find: (i) the total length of the silver wire required. (ii) the area of each sector of the brooch. ### Surface Areas and Volumes 1. A solid is in the shape of a cone standing on a hemisphere with both their radii being equal to $1 \text{ cm}$ and the height of the cone is equal to its radius. Find the volume of the solid in terms of $\pi$. 2. A metallic sphere of radius $4.2 \text{ cm}$ is melted and recast into the shape of a cylinder of radius $6 \text{ cm}$. Find the height of the cylinder. 3. A well of diameter $3 \text{ m}$ is dug $14 \text{ m}$ deep. The earth taken out of it has been spread evenly all around it in the shape of a circular ring of width $4 \text{ m}$ to form an embankment. Find the height of the embankment. 4. A container shaped like a right circular cylinder having diameter $12 \text{ cm}$ and height $15 \text{ cm}$ is full of ice cream. The ice cream is to be filled into cones of height $12 \text{ cm}$ and diameter $6 \text{ cm}$, having a hemispherical shape on the top. Find the number of such cones which can be filled with ice cream. 5. A frustum of a right circular cone has a height of $14 \text{ cm}$. The radii of its two circular ends are $4 \text{ cm}$ and $2 \text{ cm}$. Find the volume of the frustum. ### Statistics 1. The mean of the following frequency distribution is 50. But the frequencies $f_1$ and $f_2$ in class intervals 20-40 and 60-80 are missing. Find the missing frequencies. | Class Interval | Frequency | |----------------|-----------| | 0-20 | 17 | | 20-40 | $f_1$ | | 40-60 | 32 | | 60-80 | $f_2$ | | 80-100 | 19 | | **Total** | **120** | 2. The median of the following data is 525. Find the values of $x$ and $y$, if the total frequency is 100. | Class Interval | Frequency | |----------------|-----------| | 0-100 | 2 | | 100-200 | 5 | | 200-300 | $x$ | | 300-400 | 12 | | 400-500 | 17 | | 500-600 | 20 | | 600-700 | $y$ | | 700-800 | 9 | | 800-900 | 7 | | 900-1000 | 4 | 3. A life insurance agent found the following data for distribution of ages of 100 policy holders. Calculate the median age. | Age (in years) | Number of Policy Holders | |----------------|--------------------------| | Below 20 | 2 | | Below 25 | 6 | | Below 30 | 24 | | Below 35 | 45 | | Below 40 | 78 | | Below 45 | 89 | | Below 50 | 92 | | Below 55 | 98 | | Below 60 | 100 | 4. The following distribution gives the daily income of 50 workers of a factory. Convert the distribution above into a less than type cumulative frequency distribution, and draw its ogive. | Daily Income (in Rs) | Number of Workers | |----------------------|-------------------| | 100-120 | 12 | | 120-140 | 14 | | 140-160 | 8 | | 160-180 | 6 | | 180-200 | 10 | 5. If the median of the distribution given below is 28.5, find the values of $y$ and $z$. | Class Interval | Frequency | |----------------|-----------| | 0-10 | 5 | | 10-20 | $y$ | | 20-30 | 20 | | 30-40 | 15 | | 40-50 | $z$ | | 50-60 | 5 | | **Total** | **60** | ### Probability 1. A bag contains 5 red balls and some blue balls. If the probability of drawing a blue ball is double that of a red ball, find the number of blue balls in the bag. 2. A box contains 12 balls out of which $x$ are black. If 6 more black balls are put in the box, the probability of drawing a black ball is now double of what it was before. Find $x$. 3. A card is drawn at random from a well-shuffled deck of 52 cards. Find the probability that the card drawn is: (i) a king or a queen (ii) a red card and a queen (iii) neither a king nor a queen (iv) a face card (v) a jack of hearts 4. Two dice are thrown simultaneously. What is the probability that: (i) the sum of the numbers appearing on the dice is 7? (ii) the product of the numbers on the dice is less than 9? (iii) the sum of the numbers is a multiple of 3? 5. A game consists of tossing a one rupee coin 3 times and noting its outcome each time. Hanif wins if all the tosses give the same result i.e., three heads or three tails, and loses otherwise. Calculate the probability that Hanif will lose the game.