Mathematics

CBSE Class 10

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In Fig. , PQ is tangent at point C to a circle with centre O. If AB is a diameter and ∠CAB = 30°, find ∠PCA.

In the given figure,

In Δ ACO,

OA=OC (Radii of the same circle)

Therefore,

ΔACO is an isosceles triangle.

∠CAB = 30° (Given)

∠CAO = ∠ACO = 30° (angles opposite to equal sides of an isosceles triangle are equal)

∠PCO = 90° …(radius is drawn at the point of

contact is perpendicular to the tangent)

Now ∠PCA = ∠PCO – ∠CAO

Therefore,

∠PCA = 90° – 30° = 60°

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3.

A ladder leaning against a wall makes an angle of 60° with the horizontal. If the foot of the ladder is 2.5 m away from the wall, find the length of the ladder.

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4.

A card is drawn at random from a well -shuffled fled pack of 52 playing cards. Find the probability of getting neither a red card nor a queen.

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5.

If -5 is a root of the quadratic equation 2x^{2} + px – 15 = 0 and the quadratic equation p(x^{2} + x)k = 0 has equal roots, find the value of k.

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6.

Let P and Q be the points of trisection of the line segment joining the points A(2, -2) and B(-7, 4) such that P is nearer to A. Find the coordinates of P and Q.

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7.

In Fig.2, a quadrilateral ABCD is drawn to circumscribe a circle, with centre O, in such a way that the sides AB, BC, CD and DA touch the circle at the points P, Q, R and S respectively. Prove that AB + CD = BC + DA.

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8.

Prove that the points (3, 0), (6, 4) and (-1, 3) are the vertices of a right-angled isosceles triangle.

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9.

The 4th term of an A.P. is zero. Prove that the 25th term of the A.P. is three times its 11th term.

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10.

In Fig, from an external point P, two tangents PT and PS are drawn to a circle with centre O and radius r. If OP=2r, show that ∠ OTS = ∠ OST = 30°.

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