Duration: 3h

What is the common difference of an A.P. in which a_{21} – a_{7} = 84?

Chapter: [2.02] Arithmetic Progressions

If the angle between two tangents drawn from an external point P to a circle of radius a and centre O, is 60°, then find the length of OP

Chapter: [3.01] Circles

If a tower 30 m high, casts a shadow `10sqrt3` m long on the ground, then what is the angle of elevation of the sun?

Chapter: [4.01] Heights and Distances

The probability of selecting a rotten apple randomly from a heap of 900 apples is 0.18. What is the number of rotten apples in the heap?

Chapter: [5.01] Probability

Find the value of *p*, for which one root of the quadratic equation px^{2} – 14x + 8 = 0 is 6 times the other.

Chapter: [2.03] Quadratic Equations

Which term of the progression 20, 19`1/4`,18`1/2`,17`3/4`, ... is the first negative term?

Chapter: [2.02] Arithmetic Progressions

Prove that the tangents drawn at the end points of a chord of a circle make equal angles with the chord.

Chapter: [3.01] Circles

A circle touches all the four sides of a quadrilateral ABCD. Prove that AB + CD = BC + DA

Chapter: [3.01] Circles

A line intersects the *y*-axis and *x*-axis at the points P and Q respectively. If (2, –5) is the mid-point of PQ, then find the coordinates of P and Q.

Chapter: [6.01] Lines (In Two-dimensions)

If the distances of P(*x, y*) from A(5, 1) and B(–1, 5) are equal, then prove that 3*x* = 2*y*

Chapter: [6.01] Lines (In Two-dimensions)

If ad ≠ bc, then prove that the equation (a^{2} + b^{2}) x^{2} + 2 (ac + bd) x + (c^{2} + d^{2}) = 0 has no real roots.

Chapter: [2.03] Quadratic Equations

The first term of an AP is 5, the last term is 45 and the sum is 400. Find the number of terms and the common difference.

Chapter: [2.02] Arithmetic Progressions

On a straight line passing through the foot of a tower, two points C and D are at distances of 4 m and 16 m from the foot respectively. If the angles of elevation from C and D of the top of the tower are complementary, then find the height of the tower.

Chapter: [4.01] Heights and Distances

A bag contains 15 white and some black balls. If the probability of drawing a black ball from the bag is thrice that of drawing a white ball, find the number of black balls in the bag.

Chapter: [5.01] Probability

In what ratio does the point `(24/11, y)` divide the line segment joining the points P(2, –2) and Q(3, 7)? Also find the value of y.

Chapter: [6.01] Lines (In Two-dimensions)

Three semicircles each of diameter 3 cm, a circle of diameter 4.5 cm and a semicircle of radius 4.5 cm are drawn in the given figure. Find the area of the shaded region

Chapter: [7.01] Areas Related to Circles

In Figure , two concentric circles with centre O, have radii 21cm and 42 cm. If ∠ AOB = 60°, find the area of the shaded region. [use π=22/7]

Chapter: [7.01] Areas Related to Circles

Water in a canal, 5·4 m wide and 1·8 m deep, is flowing with a speed of 25 km/hour. How much area can it irrigate in 40 minutes, if 10 cm of standing water is required for irrigation?

Chapter: [7.02] Surface Areas and Volumes

The slant height of a frustum of a cone is 4 cm and the perimeters (circumference) of its circular ends are 18 cm and 6 cm. find the curved surface area of the frustum.

Chapter: [7.02] Surface Areas and Volumes

The dimensions of a solid iron cuboid are 4·4 m × 2·6 m × 1·0 m. It is melted and recast into a hollow cylindrical pipe of 30 cm inner radius and thickness 5 cm. Find the length of the pipe

Chapter: [7.02] Surface Areas and Volumes

Solve for x :

`1/(x + 1) + 3/(5x + 1) = 5/(x + 4), x != -1, -1/5, -4`

Chapter: [2.03] Quadratic Equations

Two taps running together can fill a tank in `3 1/13` hours. If one tap takes 3 hours more than the other to fill the tank, then how much time will each tap take to fill the tank?

Chapter: [2.03] Quadratic Equations

If the ratio of the sum of the first n terms of two A.Ps is (7n + 1) : (4n + 27), then find the ratio of their 9^{th} terms.

Chapter: [2.02] Arithmetic Progressions

Prove that the lengths of the tangents drawn from an external point to a circle are equal.

Chapter: [3.01] Circles

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 ∠AOB=90°

Chapter: [3.01] Circles

Construct a triangle ABC with sides BC = 7 cm, ∠B = 45° and ∠A = 105°. Then construct a triangle whose sides are `3/4` times the corresponding sides of ∆ABC.

Chapter: [3.02] Triangles

An aeroplane is flying at a height of 300 m above the ground. Flying at this height, the angles of depression from the aeroplane of two points on both banks of a river in opposite directions are 45° and 60° respectively. Find the width of the river. [Use `sqrt3` = 1⋅732]

Chapter: [4.01] Heights and Distances

If the points A(k + 1, 2k), B(3k, 2k + 3) and C(5k − 1, 5k) are collinear, then find the value of k

Chapter: [6.01] Lines (In Two-dimensions)

Two different dice are thrown together. Find the probability that the numbers obtained have

1) even sum, and

2) even product.

Chapter: [5.01] Probability

In the given figure, ABCD is rectangle of dimensions 21 cm × 14 cm. A semicircle is drawn with BC as diameter. Find the area and the perimeter of the shaded region in the figure.

Chapter: [7.01] Areas Related to Circles

In a rain-water harvesting system, the rain-water from a roof of 22 m × 20 m drains into a cylindrical tank having diameter of base 2 m and height 3·5 m. If the tank is full, find the rainfall in cm. Write your views on water conservation.

Chapter: [7.02] Surface Areas and Volumes

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